SOME EXPERIMENTAL STUDIES ON VORTEX RING FORMATION AND INTERACTION DEEPAK ADHIKARI (B. Eng.(Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2009  Acknowledgements  ACKNOWLEDGEMENTS I would like to convey my gratitude to my project supervisor, Professor Lim Tee Tai for his supervision during my time at the Fluid Mechanics Laboratory. It is through him that I have learnt the art of scientific experimental research and gained much perspective to approach fundamental problems in fluid mechanics. I am thankful and deeply indebted to him for his patience and support which he has given me. I would like to thank Mr. Ramchandra Paudel, who has been like an elder brother to me. His words of encouragement have always been one of the greatest sources of inspiration and motivation behind the completion of my work. I am grateful and thankful to Assistant Professor Teo Chiang Juay, Dr. Lua Kim Boon, Dr. Cui Yongdong and Mr. Ng Yow Thye for the many constructive and enriching discussions. I have acquired knowledge in understanding the research world in fluid mechanics through casual conversations with them. I would also like to thank the laboratory technologists, Mr. Yap Chin Seng, Mr. Tan Kim Wah, Mr. Looi Siew Wah and Mr. James Ng Chun Phew for their valuable help in logistics and manufacturing some of the equipment which I used for my experiments. Last, but not least, I would like to thank all others who have helped me in one way or another during my time at the NUS Fluid Mechanics Laboratory.   i    Table of Contents   TABLE OF CONTENTS Acknowledgements (i) Table of Contents (ii) Summary (iv) List of Appendices (vi) List of Figures (vii) List of Tables (xix) Nomenclature (xx) Chapter 1 Introduction 1.1 Background 1.2 Motivation 1.3 Objectives 1.4 Organization of Thesis Chapter 2 Literature Review 1 1 5 7 8 9 2.1 Circular Vortex Rings 2.1.1 Formation of a Circular Vortex Ring 2.1.2 Structure of a Circular Vortex Ring 2.1.3 Stability of a Circular Vortex Ring 2.1.4 Turbulent Vortex Ring 10 10 17 22 23 2.2 Elliptic Vortex Rings 2.2.1 Characteristics of an Elliptic Vortex Ring 2.2.2 Elliptic Jets 2.2.3 Other Non-Circular Vortex Rings/Jets 25 25 33 37 2.3 Interaction of a Vortex Ring with a Circular Cylinder 2.3.1 Interaction of Vortex Ring with a Small Cylinder 2.3.2 Cut-and-Reconnection Phenomena 40 40 41 Chapter 3 Experimental Setup & Methodology 47 3.1 Vortex Ring Generator 3.1.1 Tank 3.1.2 Piston-Cylinder Arrangement 3.1.3 Motion Control Mechanism 48 48 49 49 3.2 Test Models 3.2.1 Circular Nozzle 3.2.2 Elliptic Nozzles 3.2.3 Cylinders 3.2.4 Cylinder/Nozzle Fixture 50 50 51 52 53 ii    Table of Contents   3.3 Dye Flow Visualization 55 3.4 Digital Particle Image Velocimetry (DPIV) 3.4.1 Illumination Source 3.4.2 Specifications of Image Acquisition 3.4.3 Synchronization of DPIV System and Motion Control 3.4.4 Seeding Specifications 3.4.5 Image Processing & Data Validation 57 57 57 58 60 60 3.5 Data Post-Processing 3.5.1 Vorticity Field 3.5.2 Dimensions of a Vortex Ring 3.5.3 Trajectory of a Vortex Ring 3.5.4 Circulation of a Vortex Ring 63 65 65 67 67 3.6 Experimental Conditions 69 Chapter 4 Results & Discussion 4.1 Circular Vortex Rings 4.1.1 Formation of the a Circular Vortex Ring 4.1.2 Variables Characterizing Circular Vortex Ring 4.1.3 Core Characteristics of Circular Vortex Ring 4.1.4 Section Summary 71 72 77 80 86 90 4.2 Elliptic Vortex Rings 4.2.1 Trajectory of Elliptic Vortex Rings 4.2.2 Stretching of the Vortex Core 4.2.3 Flow Field of Elliptic Vortex Rings 4.2.4 Effects of Higher Stroke Ratios (LN/DN ≥ 2) 4.2.5 Section Summary 91 101 108 111 115 133 4.3 Interaction of a Vortex Ring with a Circular Cylinder 4.3.1 Cut-and-Reconnection Process 4.3.2 Effects of Varying Stroke Ratio on the Interaction 4.3.3 Section Summary 134 135 151 155 Chapter 5 Conclusion 5.1 Circular Vortex Ring 5.2 Elliptic Vortex Rings 5.3 Interaction of a Vortex Ring with a Circular Cylinder Chapter 6 References Recommendations 156 156 157 159 160 161  iii    Summary   SUMMARY While the study of vortex rings has come a long way in research with myriad of interesting findings, there are still several areas in the study which have not been fully understood or explained. This thesis aims to open up some of the issues and provide experimental investigations in these areas. The experimental techniques used in the investigations are dye visualization and Digital Particle Image Velocimetry (DPIV). The current study is divided into three parts: 1. the study of circular vortex rings, 2. the study of elliptic vortex rings, and 3. the study of interaction of a vortex ring with a circular cylinder. In the study of circular vortex rings, relationship between the characteristics of the vortex rings and the fluid slug is investigated. This includes relating Reynolds number (based on circulation, ReΓ), the diameter of the ring (D), and the vortex core diameter (c), with the Reynolds number (based on nozzle diameter, ReN) and the stroke ratio (LN/DN). Next, the vorticity profile within the vortex core is investigated and found to be a close fit to a Gaussian function. The variation of this function with Reynolds numbers (nozzle) and stroke ratios is also presented. Elliptic vortex rings are studied with nozzles of aspect ratio (AR) 2 and 3. In the study, the spatial and temporal trajectories of the elliptic vortex rings are investigated for LN/DN = 1. A deviation of trajectory under some conditions is observed, and this is attributed to the effect of cross-linking of vortices. Next, the study on the existence of iv    Summary   core stretching of the elliptic vortex rings is carried out and findings reveal that stretching does not occur for both aspect ratios (AR = 2 and 3). The flow fields of the elliptic vortex rings, which reveals some interesting critical nodes upstream of the vortex ring, are also investigated. Lastly, the effects of high stroke ratios (LN/DN ≥ 2) on the formation of elliptic vortex rings are examined. Under this condition, visualization and DPIV results reveal the existence of vortex pair downstream and streamwise vortices upstream of the ring. These vortical structures are found to hasten the azimuthal instability and break down the vortex ring through complex interactions with the vortex core. Finally, in the study on the interaction of a circular vortex ring with a circular cylinder, it is found that the reconnection occurs only at cylinder diameters, dc ≤ 1.32 mm for LN/DN = 1. The interpretation of the reconnection mechanism is illustrated and explained. In the experiment, the behaviour of the vortex ring during the interaction with cylinders is observed to be insensitive to the Reynolds number (nozzle). However, for LN/DN ≥ 2, reconnection is observed for cylinder diameter as high as dc = 1.86 mm. Furthermore, vortex ring of large stroke ratios (LN/DN ≥ 2) reveal a “von Karman-like” vortex street which eventually interacts with the core of the vortex ring.   v    List of Appendices  LIST OF APPENDICES Appendix A: Program to track the coordinates of maximum and minimum vorticity values from Tecplot data file and determination of other quantities. 168 Appendix B: Program to calculate the circulation of a region defined by the relevant coordinates from the Tecplot data file. 172 Appendix C: Visualization and vorticity plots of circular vortex rings for varying LN/DN. 176 Appendix D: Vorticity plots of elliptic vortex rings for varying ReN, with LN/DN = 1. 181 Appendix E: Visualization of the interaction of a vortex ring with a circular cylinder for varying cylinder diameters, dc, and ReN with LN/DN = 1. 184   vi    List of Figures   LIST OF FIGURES Figure 2-1: Formation of a vortex ring (Adhikari, 2007). 10 Figure 2-2: Graph of Γ/Γslug against LN/DN (adapted from Lim and Nickels (1995)). 12 Velocity profile of fluid slug from the nozzle exit (a) at low stroke ratios and (b) at high stroke ratios. 13 Figure 2-4: Characteristics variables of the a) vortex ring and the b) fluid slug from ejecting from the nozzle. 18 Figure 2-5: Core composition of vortex ring. 20 Figure 2-6: (a) Diagrammatic view showing the entrainment by a vortex ring of the fluid from upstream. (b) Diagrammatic view of instantaneous streamlines relative to a moving vortex ring. (Maxworthy, 1972). 21 Figure 2-7: Dye visualization of a turbulent vortex ring (Glezer, 1988). 23 Figure 2-8: Schematic drawing of the trajectory of elliptic vortex ring in side, plan and perspective view. 26 Figure 2-9: Time evolution of elliptic vortex ring of AR = 3 (Oshima et. al., 1988). The third vortex ring from the nozzle shows the occurrence of partial bifurcation during deformation of the ring. 31 Perspective view of an elliptic vortex ring with crosslinked vortices (Zhao and Shi, 1997). 32   Figure 2-3:     Figure 2-10: vii    List of Figures   Figure 2-11: Pairing mechanism of two elliptic vortex rings (Husain & Hussain, 1991). 36 Formation of (a) vortex filaments in a jet, which gathers to form (b) rolls and braids, and subsequently through selfinduced velocity, results in the formation of (c) streamwise vortices (or ribs) (Husain & Hussain, 1993). The direction of self-induced (SI) and mutually induced (MI) velocities are labelled. 37 Figure 2-13: Schematic representation of vortex ring after interaction with a cylinder (Naitoh et. al., 1995). 41 Figure 2-14: Cut-and-reconnection mechanism of two anti-parallel vortex tubes (Melander & Hussain, 1989). 43 Mechanism of vortex reconnection. (a) A simple viscous cancellation of vorticity, and (b) Detailed mechanism of the reconnection process with the development of vortex bridges. Blank arrows indicate the flow direction that pushes the vortex tubes. Single and double arrows indicate the rotation of vorticity lines and direction of vorticity, respectively. Hatched areas show the region of vorticity calcellation through viscous diffusion. 44 Figure 2-16: Cut-and-reconnection mechanism when vortex core interacts with a series of wires (Adhikari and Lim, 2009). 45 Figure 3-1: Schematic diagram of the vortex ring generator (Plan view). The CCD camera refers to all cameras that were used in his experiment. 48 Figure 3-2: Sectional view of the circular nozzle. 50 Figure 3-3: Sectional view of the elliptic nozzles of aspect ratios (a) AR = 2 and (b) AR = 3. 51   Figure 2-12:   its   Figure 2-15:     viii    List of Figures   Diagram of the aluminium cylinder connected at its ends with thin wires to enable it to be mounted on the Perspex wall. 52 Figure 3-5: First angle orthographic and isometric view of the cylinder/nozzle fixture used for interaction of vortex ring with cylinders. 54 Figure 3-6: Data acquisition control network (visualization). 55 Figure 3-7: Sequence of images showing propagation of vortex ring from the right side of the image window. 56 Figure 3-8: Network of motion control system and DPIV system, together with the experimental setup. 59 An instant of a developed vortex ring (at ReN = 1000 & LN/DN = 1) visualized using dye. 63 Figure 3-9(b): An instant of a developed vortex ring (at ReN = 1000 & LN/DN = 1) visualized using DPIV. 64 Figure 3-10: Vorticity distribution of the vortex ring for ReN = 1420, LN/DN = 1 through the core centre. The diameter, D, is difined as the distance between the two peak vorticity values. Insert diagram shows the vortex ring and the dotted lines represent the location where the values were acquired. 66 Horizontal velocity of the vortex ring for ReN = 1420, LN/DN = 1 through the core centre. The core diameter, c, is defined as the distance between the two peak velocity values. Insert diagram shows the vortex ring and the dotted lines represent the location where the values were acquired. 67 Figure 3-4:       Figure 3-9(a):     Figure 3-11:   ix    List of Figures   Figure 3-12: The dotted points represent the data points taken around the vortex core in order to calculate the circulation of the vortex ring. 69 Dye visualization of the formation of a circular vortex ring at ReN = 1000 and LN/DN = 1. 73 Vorticity field of the formation of a circular vortex ring at ReN = 1000 and LN/DN = 1. The scales on axes are in mm and the vorticity contours are in s-1. 74 Instantaneous streamlines of the formation of a circular vortex ring at ReN = 1000 and LN/DN = 1. The scales on the axes are in mm. 75 Graph of circulation against time for vortex ring with ReN = 1000 and LN/DN = 1. 77 Reproduction of graph from figure 2-2 with results of current experiment. + ReN = 1000, - ReN = 1420, x ReN = 1740. 78 Vorticity field during the formation of a circular vortex ring at ReN = 1000 and LN/DN = 4. The scales on axes are in mm and the vorticity contours are in s-1. 79 Relationship between Reynolds number (circulation) and Reynolds number (nozzle). ○ LN/DN = 1, □ LN/DN = 2, ∆ LN/DN = 3. 82 Relationship between f1 and the stroke ratio, LN/DN. f1 represents the gradient of the lines in figure 4-7. 82   Figure 4-1:   Figure 4-2:   Figure 4-3:   Figure 4-4:   Figure 4-5:   Figure 4-6:   Figure 4-7:   Figure 4-8:       x    List of Figures   Relationship between dimensionless vortex ring diameter, D/DN, and the stroke ratio, LN/DN. Result of Maxworthy (1977) is depicted by broken line (----) while the solid line (──) represents the current experiment. ◊ ReN = 1000, □ ReN = 1230, ∆ ReN = 1420,x ReN = 1580, + ReN = 1740. 84 Relationship between core diameter (normalized by nozzle diameter), c/DN, and Reynolds number (nozzle), ReN for different stroke ratios, LN/DN. (○, ─) LN/DN = 1, 84 Figure 4-11: Relationship between dimensionless core diameter (normalized by ring diameter, D), c/D, and Reynolds number (nozzle), ReN for different stroke ratios, LN/DN. (○, ─) LN/DN = 1, (□, −·) LN/DN = 2, (∆, −−) LN/DN = 3. 86 Figure 4-12: Vorticity distribution within the core of the vortex ring of ReN = 1420, LN/DN = 1 with a Gaussian curve-fit. The figure is taken on the vortex core with positive vorticity value. 88 Relationship between A (see equation 4-6) and ReN. A corresponds to the peak vorticity value. (○, ─) LN/DN = 1, (□, −·) LN/DN = 2, (∆, −−) LN/DN = 3. 89 Relationship between w (see equation 4-6) and ReN. w is related to the width of the Gaussian vorticity profile. (○, ─) LN/DN = 1, (□, −·) LN/DN = 2, (∆, −−) LN/DN = 3. 90 Figure 4-15: Dye visualization of the formation of an elliptic vortex ring of AR = 2 at ReN = 1000 and LN/DN = 1. 92 Figure 4-16: Vorticity plot of the formation of a elliptic vortex ring of AR = 2 at ReN = 1000 and LN/DN = 1 in the x-y plane. The scales on axes are in mm and the vorticity contour values are in s-1. 93 Figure 4-9:   Figure 4-10: (□, −·) LN/DN = 2, (∆, −−) LN/DN = 3.     Figure 4-13:   Figure 4-14:     xi    List of Figures   Figure 4-17: Vorticity plot of the formation of a elliptic vortex ring of AR = 2 at ReN = 1000 and LN/DN = 1 in the x-z plane. The scales on axes are in mm and the vorticity contour values are in s-1. 94 Schematic view of elliptic vortex ring of AR = 2 in the y-z plane. The arrows represent the direction of the vortex lines. 95 Figure 4-19: Dye visualization of the formation of an elliptic vortex ring of AR = 3 at ReN = 1000 and LN/DN = 1. 97 Figure 4-20: Vorticity plot of the formation of an elliptic vortex ring of AR = 3 at ReN = 1000 and LN/DN = 1. The scales on axes are in mm and vorticity contour values are in s-1. 98 Vorticity plot of the formation of an elliptic vortex ring of AR = 3 at ReN = 1000 and LN/DN = 1. The scales on axes are in mm and vorticity contour values are in s-1. 99 Schematic view of an elliptic vortex ring of (a) AR = 2 and (b) AR = 3 at half-cycle of the oscillation in the y-z plane. The arrows represent the vortex lines. Shaded areas represent the cross-linking regions which appear as vortex pair in vorticity plot at x-z plane. 100 Figure 4-23(a): The spatial trajectory of an elliptic vortex ring with AR = 2 for all Reynolds number (nozzle). ◊ ReN = 1000, □ ReN = 1230, ∆ ReN = 1420, x ReN = 1580, + ReN = 1740. 104   Figure 4-18:     Figure 4-21:   Figure 4-22:     Figure 4-23(b): The temporal trajectory of an elliptic vortex ring with AR = 2 for all Reynolds number (nozzle). ◊ ReN = 1000, □ ReN = 1230, ∆ ReN = 1420, x ReN = 1580, + ReN = 1740. 104       xii    List of Figures   Figure 4-24(a): The spatial trajectory of an elliptic vortex ring with AR = 3 for all Reynolds number (nozzle). ◊ ReN = 1000, □ ReN = 1230, ∆ ReN = 1420, x ReN = 1580, + ReN = 1740. 105   Figure 4-24(b): The temporal trajectory of an elliptic vortex ring with AR = 3 for all Reynolds number (nozzle). ◊ ReN = 1000, □ ReN = 1230, ∆ ReN = 1420, x ReN = 1580, + ReN = 1740. 105   Vorticity plot of elliptic vortex ring with AR = 3 for ReN = 1230 in (a) x-y and (b) x-z plane. The scales on axes are in mm and vorticity contour values are in s-1. 106 Vorticity plot of an elliptic vortex ring with AR = 3 for ReN = 1420 in (a) x-y and (b) x-z plane. The scales on axes are in mm and vorticity contour values are in s-1. 107 Schematic view of an elliptic vortex ring of (a) ReN = 1230 and (b) ReN = 1420 in y-z plane. The arrows represent the direction of vortex lines. 108 Figure 4-28(a): Temporal variation of peak vorticity value of a vortex ring for ReN = 1000. ◊ Circular ring, □ AR = 2 (x-y plane), ∆ AR = 2 (x-z plane), x AR = 3 (x-y plane), + AR = 3 (x-z plane). 110 Figure 4-25:   Figure 4-26:   Figure 4-27:     Figure 4-28(b): Temporal variation of peak vorticity value of a vortex ring for ReN = 1740. ◊ Circular ring, □ AR = 2 (x-y plane), ∆ AR = 2 (x-z plane), x AR = 3 (x-y plane), + AR = 3 (x-z plane). 110   Figure 4-29: Instantaneous streamlines during the formation of a circular vortex ring at ReN = 1000 & LN/DN = 1. The scales on the axes are in mm. This figure is reproduced from figure 4-3. 112   xiii    List of Figures   Figure 4-30: The instantaneous streamlines of an elliptic vortex ring of AR = 2. (a) x-y and (b) x-z plane. The scales on axes are in mm. 113 Figure 4-31: The instantaneous streamlines of an elliptic vortex ring of AR = 3. (a) x-y and (b) x-z plane. The scales on axes are in mm. 114 Figure 4-32: Evolution of an elliptic vortex ring at ReN = 1740, LN/DN = 2 (a) AR = 2 and (b) AR = 3. 117 Figure 4-33: Vorticity field of an elliptic vortex ring of AR = 2, ReN = 1740 and LN/DN = 2. (a) x-y and (b) x-z plane. The scales on axes are in mm and the vorticity contours are in s-1. 118 Vorticity plot of an elliptic vortex ring AR = 3, ReN = 1740, LN/DN = 2. (a) x-y plane and (b) x-z plane. The scales on axes are in mm and vorticity contour values are in s-1. 119 Instantaneous streamlines of an elliptic vortex ring of AR = 2, ReN = 1740 and LN/DN = 2. (a) x-y and (b) x-z plane. The scales on axes are in mm. 121 Instantaneous streamlines of an elliptic vortex ring AR = 3, ReN = 1740, LN/DN = 2. (a) x-y plane and (b) x-z plane. The scales on axes are in mm. 122 Evolution of an elliptic vortex ring at ReN = 1000, LN/DN = 3. (a) AR = 2 and (b) AR = 3. 124 Evolution of an elliptic vortex ring at ReN = 1000, LN/DN = 4. (a) AR = 2 and (b) AR = 3. 125 Evolution of an elliptic vortex ring at ReN = 1000, LN/DN = 6. (a) AR = 2 and (b) AR = 3. 126     Figure 4-34:   Figure 4-35:   Figure 4-36:   Figure 4-37:   Figure 4-38:   Figure 4-39: xiv    List of Figures   Figure 4-40: Interpretation on the development of streamwise vortices for an elliptic vortex ring of AR = 2, ReN = 1000, LN/DN = 6. 129 Evolution of an elliptic vortex ring at ReN = 1740, LN/DN = 6. (a) AR = 2 and (b) AR = 3. 130 Figure 4-42: Vorticity plot of an elliptic vortex ring of AR = 2, ReN = 1740 and LN/DN = 6. (a) x-y and (b) x-z plane. The scales on axes are in mm and the vorticity contour values are in s-1. 131 Figure 4-43: Vorticity plot of an elliptic vortex ring of AR = 3, ReN = 1740 and LN/DN = 6. (a) x-y and (b) x-z plane. The scales on axes are in mm and the vorticity contour values are in s-1. 132 Figure 4-44: Interaction of a vortex ring with a cylinder of dc = 0.18mm at ReN = 1000 and LN/DN = 1. 137 Figure 4-45: Interaction of a vortex ring with a cylinder of dc = 0.18mm at ReN = 1420 and LN/DN = 1. 138 Interaction of a vortex ring with a cylinder of dc = 0.18mm at ReN = 1740 and LN/DN = 1. 139 Interaction of a vortex ring with a cylinder of dc = 0.39mm at ReN = 1420 and LN/DN = 1. 140 Author’s interpretation of the cut-and-reconnection mechanism for a vortex ring interacting with a cylinder. This mechanism is similar to that reported by Adhikari and Lim (2009). The arrow heads depict the direction of 141   Figure 4-41:     Figure 4-46:   Figure 4-47:   Figure 4-48: vorticity on half of the ring that is in the foreground.     xv    List of Figures   Figure 4-49: Vorticity plot of the interaction of a vortex ring with a cylinder of dc = 0.18mm at ReN = 1420 and LN/DN = 1. The scales on axes are in mm and the vorticity contour values are in s-1. 142 Interaction of a vortex ring with a cylinder of dc = 0.92 mm at ReN = 1420 and LN/DN = 1. 144 Interaction of a vortex ring with a cylinder of dc = 1.32 mm at ReN = 1420 and LN/DN = 1. 146 Vorticity plot of the interaction of a vortex ring with a cylinder of dc = 1.32 mm at ReN = 1420 and LN/DN = 1. The scales on axes are in mm and the vorticity contour values are in s-1. 147 Author’s interpretation of the interaction of a vortex ring with a cylinder of dc = 1.32 mm at ReN = 1420 and LN/DN = 1. The induced velocity field, represented by dotted lines, maintains the structure of vortex ring even after the interaction with the cylinder. The shaded area represents the core of the vortex ring. 148 Interaction of a vortex ring with a cylinder of dc = 1.86 mm at ReN = 1420 and LN/DN = 1. 150 Interaction of a vortex ring with a cylinder of dc = 1.86 mm at ReN = 1420 and LN/DN = 2. 152 Figure 4-56: Interaction of a vortex ring with a cylinder of dc = 1.86 mm at ReN = 1420 and LN/DN = 3. 153 Figure 4-57: Vorticity plot of the interaction of a vortex ring with a cylinder of dc = 1.89 mm at ReN = 1420 and LN/DN = 3. The scales on axes are in mm and the vorticity contour values are in s-1. 154   Figure 4-50:   Figure 4-51:   Figure 4-52:   Figure 4-53:   Figure 4-54:   Figure 4-55:   xvi    List of Figures   Figure C-1: The formation of a circular vortex ring at ReN = 1000 and LN/DN = 2. (a) Dye visualization and (b) vorticity field. The scales on axes in (b) are in mm and the vorticity contours are in s-1. 177 The formation of a circular vortex ring at ReN = 1000 and LN/DN = 3. (a) Dye visualization and (b) vorticity field. The scales on axes in (b) are in mm and the vorticity contours are in s-1. 178 The formation of a circular vortex ring at ReN = 1000 and LN/DN = 4. (a) Dye visualization and (b) vorticity field. The scales on axes in (b) are in mm and the vorticity contours are in s-1. 179 The formation of a circular vortex ring at ReN = 1000 and LN/DN = 5. (a) Dye visualization and (b) vorticity field. The scales on axes in (b) are in mm and the vorticity contours are in s-1. 180 Vorticity field of the formation of an elliptic vortex ring of AR = 3 at ReN = 1580 and LN/DN = 1. (a) x-y plane and (b) x-z plane. The scales on axes are in mm and the vorticity contours are in s-1. 182 Vorticity field of the formation of an elliptic vortex ring of AR = 3 at ReN = 1740 and LN/DN = 1. (a) x-y plane and (b) x-z plane. The scales on axes are in mm and the vorticity contours are in s-1. 183 Interaction of a vortex ring with a cylinder of dc = 0.39 mm at (a) ReN = 1420 and (b) ReN = 1740, LN/DN = 1. 185 Interaction of a vortex ring with a cylinder of dc = 0.92 mm at (a) ReN = 1420 and (b) ReN = 1740, LN/DN = 1. 186 Interaction of a vortex ring with a cylinder of dc = 1.32 mm at (a) ReN = 1420 and (b) ReN = 1740, LN/DN = 1. 187   Figure C-2:   Figure C-3:   Figure C-4:   Figure D-1:   Figure D-2:   Figure E-1:   Figure E-2:   Figure E-3: xvii    List of Figures   Figure E-4: Interaction of a vortex ring with a cylinder of dc = 1.86 mm at (a) ReN = 1000 and (b) ReN = 1420, LN/DN = 1. 188             xviii    List of Tables  LIST OF TABLES Table 3-1: Summary of DPIV image acquisition specification in all the experiments. 58 Table 3-2: Summary of the input parameters used for circular and elliptic vortex rings experiment. 70 Table 3-3: Summary of the input parameters used for vortex ring interaction with a cylinder experiment. 70   xix    Nomenclature  NOMENCLATURE Symbols c Core diameter of the vortex ring D Diameter of the vortex ring DN Inner diameter of the circular nozzle Do Outer diameter of the circular nozzle DP Diameter of the piston in the piston-cylinder arrangement dc Diameter of the cylinder LN Length of fluid slug ejected from the nozzle U Propagation of the vortex ring UN Velocity of fluid slug at the nozzle exit t Time of evolution of the vortex ring Greek Symbols Γ Circulation of the vortex ring λ Number of azimuthal waves cycles on the vortex ring ω Vorticity ν Kinematic viscosity δs Diameter of the vortex spiral Terminologies AR Aspect ratio LN/DN Stroke ratio ReΓ (=Γ/ν) Reynolds number (circulation) ReN (=UNDN/ν) Reynolds number (nozzle) t* (=UNt/DN) Non-dimensional time   xx    Chapter 1 Introduction   Chapter 1  Introduction    1.1 Background One can easily identify a vortex ring upon observing a smoke ring whether it is ejected by a smoker, a volcano or from the explosion of an atomic bomb. The vortex ring is observed to appear as a distinct smoke-filled torus propagating with a certain velocity, and it undoubtedly possesses some level of aesthetic pleasure. It is also interesting to note that this visual fascination of vortex rings is not only enjoyed by humans; Dolphins sometimes generate bubble vortex rings1 so that they can entertain themselves by playing with these bubble rings. Although the sight of vortex rings is amusing, it is their complete scientific understanding and application in technology that serves to be even more fascinating. The first recorded observation of vortex ring is debatable since it is a common occurrence in nature. However, its first proper scientific exploration was initiated by Hermann von Helmholtz through his “laws of vortex motion” during the mid 19th century. This incipient study of vortex ring was then developed further by William Thomson (later known as Lord Kelvin), who reasoned the existence of vortex rings using these laws. This reasoning was later complemented by P.G. Tait, a friend of Lord Kelvin, by constructing a simple technique to carry out an experiment to visualize the vortex ring (Eckert, 2006). This was the beginning of the scientific study                                                              1  Unlike the conventional vortex ring, the core of the bubble vortex ring produced by dolphins is mainly air blown from their blowhole.     1    Chapter 1 Introduction   of vortex rings, and since then, many researchers have worked on its theoretical, experimental and computational aspects. Towards the end of the 19th century, the first edition of a book on Hydrodynamics was published by Sir Horace Lamb (Lamb 6th ed., 1932) where he devoted a section of a chapter on vortex rings. In that section, he analytically derived the velocity of an inviscid thin-cored vortex ring and discussed the mutual influences of two or more rings travelling co-axially. A few years later, one of the most notable and significant analytical model for vortex ring was formulated. Hill (1894) formulated an analytical solution representing what is known today as the Hill’s spherical vortex. One of the characteristics of the Hill’s spherical vortex is that its vorticity is contained within a sphere, while the vortex still possesses a toroidal shape. This analytical model by Hill is known to be the most simplistic model of vortex ring to have been formulated and it is still used today. In the mid 20th century, a more wholesome contribution on the study of the vortex ring was published by Batchelor (1967). In his book, Batchelor described the different experimental techniques for generating vortex rings and reproduced some experimental work in the literature. He then acknowledged that the deficiency in studying vortex ring analytically is the ignorance of the core structure of the ring. He concluded by reasoning that a family of vortex rings can exist theoretically with different core structure assumptions, and that Hill’s spherical vortex is one such entity. 2    Chapter 1 Introduction   While early and mid 20th century saw mainly analytical models of simplified vortex rings with inviscid assumptions, it was during the 1970s that the fundamental study of vortex rings progressed rapidly. During this time, more theoretical and experimental works on its formation, structure and stability were published (Saffman, 1970, 1978; Kambe and Takao, 1971; Norbury, 1973; Maxworthy, 1972, 1977; Widnall and Sullivan, 1973; Moore and Saffman, 1972; Kambe and Oshima, 1975; Didden, 1979; Pullin, 1979). Furthermore, work on turbulent vortex rings was also initiated during this period (Maxworthy, 1974). In the late 20th century, the study on the interactions of vortex rings, the study of noncircular vortex ring and the computational study of vortex ring were initiated (Dhanak and De Bernardinis, 1981; Nitche and Krasny, 1994; Verzicco and Orlandi, 1994; Orlandi and Verzicco, 1993; Kiya et. al., 1992; Lim, 1989; Lim and Nickels, 1992). It was also during this period that significant findings of researchers were compiled and recorded to give a comprehensive insight on the understanding of vortex rings (Lim and Nickels, 1995; Saffman, 1992; Shariff and Leonard, 1992). Since the late 1990s, the study of vortex rings has been carried out to an even greater depth due to the rapid advancements in technology. With these improvements, some of the work carried out over the last two decades include: resolving controversial findings recorded in the past (Lim, 1997(a); Lim, 1997(b)), the generation of vortex ring with varying boundary conditions (Lim, 1998; Dabiri 2005) and the study on the optimal formation of vortex ring (Gharib et. al., 1998). A recent review has been 3    Chapter 1 Introduction   published which provides an overview on the optimal formation of vortex ring and its relation to biological propulsion system (Dabiri, 2009). Alongside the fundamental study of vortex rings that has progressed this far, its potential technological application has also been looked into with much interest. One of its major technological applications is in starting jet flow. Since vortex ring is regarded as the building block behind the mixing, entrainment, noise generation and heat transfer of the starting jet flow, understanding vortex rings will eventually enable suppression or enhancement of these properties (Lim and Nickels, 1995). Other potential applications of vortex rings include projecting smoke and other effluents to high altitudes in atmosphere without the use of tall chimneys (Turner, 1960; Fohl, 1967), underwater drilling (Chahine and Genoux, 1983) and combating fire in oil wells (Akhmetov et. al., 1980). Recently, vortex ring has been studied for its potential use in propulsion by drawing inspiration from the bio-locomotion of jellyfish and insect flights (Dabiri et. al., 2005; Dudley, 1999). 4    Chapter 1 Introduction   1.2 Motivation As discussed in the preceding section, the study of vortex ring is a very mature topic spanning more than a century. While many advances have been made to date, there are still several areas in this field that have not been fully investigated or understood. This thesis focuses on shedding light on some of these unanswered questions by means of experimentations. As mentioned earlier, Batchelor (1967) noted that theoretical investigation of vortex rings generally assumes the core structure and ignores the actual core composition of the ring. Several theoretical works on vortex ring assume thin, circular or even elliptic core structure (Batchelor, 1967; Saffman, 1970). One of the reasons behind such assumptions is due to the unavailability of proper record of the core structure in literature. Even Didden (1979), who has presented comprehensive work on the formation of vortex ring, does not have sufficient information on the evolution of the vortex core. Therefore, it is the author’s aim to carry out a systematic study on circular vortex ring with a particular focus on the core structure. This thesis will also attempt to answer the questions surrounding non-circular vortex rings through a detailed study of elliptic vortex rings. Several computational and theoretical studies have been carried out on elliptic vortex rings (Arms and Hama, 1965; Dhanak and De Bernardinis, 1981). However, the understanding of their evolution and characteristics is lacking mainly due to limited experimental investigation. The trajectory of the elliptic vortex rings and the effects on their characteristics by varying the piston profiles (i.e. velocity and length of fluid slug 5    Chapter 1 Introduction   ejecting from the nozzle) are some aspects of study which have not been recorded and fully understood. The motivation for the final study carried out in this thesis comes about from the author’s previous work on the Impact of vortex ring on porous screen which provides insights to the dynamics of the vortex ring when it interacts with porous screen (Adhikari and Lim, 2009). In the paper, the authors presented an interpretation of the mechanism by which vortex ring passes through the porous screens. However, the question on the effects of varying wire mesh thickness and the cut-and-reconnection mechanism of the ring was not elaborated thoroughly. In order to fully address this issue, vortex ring will be made to interact with cylinders of varying diameters, to investigate how the size of the cylinders affects the vortex reconnection. It should be noted that while the study of vortex ring interaction with thin cylinders has been carried out by Naitoh et. al. (1995), the issue of cut-and-reconnection was not addressed. 6    Chapter 1 Introduction   1.3 Objectives With the motivation discussed in the previous section, it is apparent that the main focus of this thesis can be divided in three parts: 1) study of circular vortex rings, 2) study of elliptic vortex rings and 3) study of vortex ring interaction with cylinders. These objectives encompass several specific goals which are as outlined below: 1. Study of circular vortex rings. a. To provide relationship between the piston profile and vortex ring characteristics; b. To examine the structure of the vortex core. 2. Study of elliptic vortex rings. a. To examine the trajectory and flow field of elliptic vortex rings; b. To examine the formation of an elliptic vortex ring at different piston profiles; c. To analyse the flow structures of elliptic vortex rings. 3. Study of vortex ring interaction with circular cylinders. a. To examine the reconnection of vortex ring after its interaction with a circular cylinder; b. To examine the effects of cylinders of different diameters; c. To examine the parameters affecting the vortex ring interaction with cylinders. 7    Chapter 1 Introduction   1.4 Organization of Thesis The thesis is written to provide a full overview of the author’s work in understanding the three topics of interest: the circular vortex ring, elliptic vortex ring and the vortex ring interaction with cylinder. Therefore, in the next chapter (Literature Review), the past works on these three topics are discussed separately to give a detailed insight into each of the topics. In Chapter 3 (Experimental Setup & Methodology), the experimental facility, the test models for the three different experiments, and the techniques used to acquire and process the data are discussed. Chapter 4 (Results & Discussion) presents the experimental results, some of which are new phenomena that have yet to be reported thoroughly in the literatures. Analysis and discussion of these results are also presented. Chapter 5 presents the conclusions of the work. Finally, Chapter 6 (Recommendations) proposes other experimental techniques which may be carried out to give further understanding to the topics of interest. 8    Chapter 2 Literature Review  Chapter 2  Literature Review  In this chapter, a review of topics related to the objectives of this thesis is presented. These topics are divided into three sections. The first section provides an overview of the current understanding on circular vortex rings. This is followed by a discussion on elliptic vortex rings which are known to exhibit interesting and complicated dynamics. The third section reviews some past studies on the interaction of a vortex ring with a circular cylinder, and discussion on the cut-and-reconnection phenomena of the vortex core. 9    Chapter 2 Literature Review  2.1 Circular Vortex Rings The evolution of a circular vortex ring has been studied extensively in the literature and it can be divided into several stages. The first stage in the evolution is the formation, where the fluid ejected out of an opening (either a nozzle or an orifice) rolls up to form a vortex ring. The second stage is when the vortex ring is fully developed and propagates with a certain velocity. The ring can then undergo azimuthal instability (third stage), depending on the initial flow condition, and eventually breakdown into a turbulent vortex ring (fourth stage). Each of these stages will be discussed in detail. 2.1.1 Formation of a Circular Vortex Ring During the formation of a circular vortex ring, a jet of fluid is ejected from a circular opening (either a nozzle or an orifice) and, due to Kelvin-Helmholtz instability, the vortex sheet rolls up into a toroidal spiral as shown in figure 2-1. Earlier works on the formation of the vortex ring aimed to analytically model the vortex sheet spiral and the circulation accurately. The two main models are the slug flow model and the selfsimilar rollup model (see Shariff and Leonard, 1992; Lim and Nickels, 1995). Rollup of vortex sheet   Figure 2-1: Formation of a vortex ring (Adhikari, 2007). 10    Chapter 2 Literature Review  The slug flow model is useful for predicting circulation, Γ, of a vortex ring. During the formation, the circulation flux “fed” by the fluid slug from the nozzle is accumulated within the spiral of the vortex ring. This circulation flux is given as an integral of the azimuthal vorticity, ωφ, of the ring and the horizontal velocity component of the fluid at nozzle exit, ux, and can be represented in equation form as: dΓ dt ω u dr ∂u u dr ∂r 1 2 ∂ u dr ∂x 1 U t                    (2-1) 2 N The first approximation sign in equation 2-1 is based on the assumption that there is no orthogonal velocity component to ux, and the second approximation sign assumes that the velocity of the fluid within the boundary layer near the nozzle exit has a uniform velocity profile of UN (i.e. a free slip boundary condition). After a certain formation time, tf, the vortex ring is fully developed and the circulation (from equation 2-1) can be approximated by: Γ 1 U t  x t 2 N 1   UN t LN                                                (2-2) 2 where LN is the length of the fluid slug ejected. Following the development of the slug flow model (equation 2-2), several researchers (Maxworthy, 1978; Lim and Nickels, 1995; Didden, 1979) conducted experiments on vortex rings and compared their measured circulation with the model (see figure 2-2). From the figure, it is clear that the slug flow model underestimates the experimental results of Didden (1979) and Lim et. al. (1992). This discrepancy has been explained by Didden (1979) to be the result of the peak velocity near the edge of the nozzle exit 11    Chapter 2 Literature Review  at lower stroke ratios, LN/DN (see figure 2-3(a)). This peak velocity results in a higher circulation flux as compared to the slug model. However, at higher stroke ratios, the slug model is observed to give a relatively better approximation since the high circulation flux generated initially, tends to be offset by lower circulation flux when the boundary layer at the nozzle thickens. Furthermore, the generation and entrainment of opposite vorticity outside the nozzle wall further reduces the overall circulation of the vortex ring (see figure 2-3(b)). 2 1.8 1.6 Γ/Γslug 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 LN/DN Slug Model Didden (1979) Didden's curve fit Pullin (1979) Lim et. al. (1992) Maxworthy (1977)   Figure 2-2: Graph of Γ/Γslug against LN/DN (adapted from Lim and Nickels (1995)). 12    Chapter 2 Literature Review  (a) (b) Vortex Sheet Vortex Sheet Vorticity generation and entrainment DN  DN  Peak velocity near the edge of nozzle exit Nozzle Velocity profile due to thickening of boundary layer Nozzle Figure 2-3: Velocity profile of fluid slug from the nozzle exit (a) at lower stroke ratios and (b) at higher stroke ratios. The second analytical model used to describe the formation of a vortex ring is the selfsimilar rollup model. Like the slug flow model, the formulation of self-similar model is based on inviscid assumptions, with a further assumption that the vortex sheet is two dimensional. This model is known to provide information about the vortex sheet evolution and gives an insight into the core structure (Shariff and Leonard, 1992). Pullin (1978) initiated the derivation of the self-similar rollup model by representing the velocity of the fluid slug ejecting from the nozzle as:                                                             2‐3 where UN(t) is the velocity of the fluid slug, t is time duration of piston motion, A and m are known constants (Note that for impulsive flow, m = 0). After some formulation, Pullin (1978) then derived the expression for the circulation, Γ, and the dimension of spiral, δs, of the vortex ring as follows: 13    Chapter 2 Literature Review  0.75 1 Γ 0.75 1                                            (2-4)                                             (2-5) where K1 and K2 are constants obtained from similarity theory calculations and a is given by the expression                                                              2‐6   where K3 is a constant that is a function of the geometry of the nozzle. Pullin (1978) further explained that in the case of a nozzle, K3 ≈ (2π)-1/2 is an extremely close approximate for the above equation. Equation 2-4 was subsequently used to derive another equation (2-7) which, when plotted together with the experimental curves in figure 2-2, seems to overestimate the experimental result of Didden (1979) at low stroke ratios (LN/DN < 0.5), and gives good agreement with the result of Maxworthy (1977), at higher stroke ratios (LN/DN ≥ 1.5). Note that equation 2-7 is formulated specifically for a nozzle with impulsive flow. Γ Γ 1.41                                                    (2-7) Didden (1979) reported that the self-similar model did not agree well with his experimental result as seen in figure 2-2. While the self-similar model showed that 14    Chapter 2 Literature Review  circulation varied with power -2/3 of stroke ratio (see equation 2-3), Didden’s experimental work suggested that the circulation varied with power -1 of the stroke ratio. This empirical equation of Didden is given by: Γ Γ 1.14 0.32                                                          2‐8 A wide range of postulations were proposed in several works to explain the differences between the self-similar model and the experimental results (Didden, 1979, 1982; Pullin and Perry, 1980; Blondeaux and De Bernardinis, 1983). However, it was Auerbach (1987) who later attributed the discrepancies to the variation of vortex sheet thickness and the generation of secondary vortices at the edge of the nozzle. To elaborate further on the disagreement between the self-similar model and experiment, Nitche and Krasny (1994) did a computational study of the vortex ring formation which showed remarkable visual resemblance with Didden’s experimental results. Like Didden, they also noted that the ring trajectory and circulation do not behave as predicted by the model. They found that the discrepancy between the selfsimilar model and the experiment by Didden are due to the following reasons: (1) omission of self-induced velocity of vortex ring in the similarity theory; (2) the absence of the downstream velocity in the similarity theory’s starting flow and; (3) the improbability of good impulsive flow in most experiments and simulations. 15    Chapter 2 Literature Review  In another investigation, a different aspect on the formation of vortex ring was studied by Gharib et. al. (1998) using Digital Particle Image Velocimetry (DPIV) technique. In the experiment, they varied the stroke ratio (LN/DN) of fluid ejecting from the nozzle and observed the formation of a vortex ring. It was noted that, at stroke ratio of approximately 4, the vortex ring attained the maximum circulation and pinched off from the emanating jet at the nozzle. They defined this stroke ratio as the “formation number” of the vortex ring. They also noted that, the existence of the formation number implies that the impulse and the kinetic energy of the vortex ring can be maximized. They further noted that the formation number varied between 3.5 and 4.2 depending on the initial and boundary conditions of the piston profile and the nozzle. The “universality”2 of the formation number has several implications. For instance, the maximum energy generated for a vortex ring produced at this number relates to the maximum thrust that can be generated in a pulsating flow. Further works based on this finding have led to its comparison with the in situ propulsion of jellyfish and its application in efficient propulsion system (Dabiri, 2005). In addition, studies have been conducted to assess the effect of changing the boundary conditions on the formation number (i.e. enhancing or suppressing it). For instance, Dabiri and Gharib (2005) highlighted that the formation number can be varied by temporally varying the diameter of the nozzle or the momentum flux of the fluid during the formation stage.                                                              2  While the formation number has been termed as “universal” by Gharib et. al. (1998), its tendency to vary from 3.5 to 4.2 suggests that it may not be deemed strictly as “universal”.  16    Chapter 2 Literature Review  Changing the boundary conditions affect not only the formation number, but also the formation characteristics and the structure of the vortex ring. One such condition, which affects the formation characteristics, is the fluid slug velocity profile. For example, Pullin (1979) showed that, varying the value of m in the equation UN = Atm (see equation 2-3), affects the spiral size and the circulation of the vortex ring (see equation 2-4 & 2-5). An example of a boundary condition, which affects the structure of the vortex ring, is the geometry of the nozzle. For instance, when a vortex ring is generated from an inclined nozzle, breakdown occurs quickly due to axial flow in the core of the vortex ring (Lim, 1998; Webster and Longmire, 1997). Non-circular nozzles have also been used to change the vortical dynamics of the vortex ring during and after formation (Grinstein et. al, 1995). Another aspect of the formation of vortex rings was studied by Fabris and Liepmann (1997). In their experiments, they focussed on the collection of vortical fluid at the front stagnation point of a vortex ring during the late stages of its formation. Their high-resolution DPIV results showed that when the piston stops ejecting the fluid, some vorticity is seen to accumulate at the stagnation point in front of the vortex ring. They concluded that, the collection of vortical fluid may give rise to the variation in stability, vorticity shedding and growth of a vortex ring as it propagates downstream. 2.1.2 Structure of a Circular Vortex Ring Didden (1979) noted that a fully developed vortex ring can be characterized by its diameter (D), circulation (Γ) and propagation velocity (U) (see figure 2-4(a)). These characteristic parameters are known to be dependent on the nozzle diameter (DN), the 17    Chapter 2 Literature Review  length (LN) and the velocity (UN) of the fluid slug (see figure 2-4(b)). The relationship between the characteristic parameters of the vortex ring and fluid slug will be discussed in the next few paragraphs. (a)  U LN (b)  Γ UN D DN Fluid Slug Vortex Ring Nozzle Figure 2-4: Characteristics variables of the (a) vortex ring and the (b) fluid slug from ejecting from the nozzle. In the paper by Maxworthy (1977), the relationship between dimensionless diameter (D/DN) of vortex ring and the stroke ratio (LN/DN) was presented, and he showed a positive non-linear correlation between them. Maxworthy also noted that D/DN has very weak dependence on ReN (=UNDN/ν), and that the diameter of the vortex ring depends on the stroke ratio only. Accordingly, he proposed the following empirical formula: . 1.18                                                          2‐9 Another relationship which may be inferred from work of Maxworthy (1977) is the linear correlation between ReΓ and ReN, for LN/DN = 1. Adhikari and Lim (2009) graphically represented this linear relationship for lower Reynolds numbers 18    Chapter 2 Literature Review  (500 < ReN < 3500) and found that the line passes through the origin and has a gradient of about 0.8. Another characteristic parameter of a vortex ring is the propagation velocity (U), which has been formulated analytically by various researchers. Lamb (1937) provided an expression for the velocity of an inviscid vortex ring with a thin core. His formula was later expanded by Saffman (1970) for a viscous vortex ring, which is given as: Γ 2 4 √4 0.558 4 4                       (2-10) where U is the propagation velocity, Γ is the circulation, D is the diameter, ν is the kinematic viscosity and t is the time. In deriving the above equation, Saffman assumed Gaussian vorticity profile in the vortex core to account for the growth of the core by viscous diffusion. This assumption shows that the knowledge of the core structure is important in the overall understanding of the vortex ring. A current knowledge on the structure of the vortex core is recorded by Lim and Nickels (1995) where they illustrated that the vortex core comprises of three regions (see figure 2-5). Region 1 consists of the vortex sheet; Region 2 is where the thickness of the vortex sheet is in the same order as the spacing amongst them; Region 3 consists of the viscous core in which the vorticity is distributed almost evenly, thus making it 19    Chapter 2 Literature Review  essentially a solid-body rotation. Further details on the core structure are unavailable in the literature, and this provides the motivation to study the structure of the vortex core in this thesis. 1 2 3 “Compressed” vortex sheet, where thickness ≈ spacing Viscous core, almost a solid-body rotation Vortex Sheet Figure 2-5: Core composition of vortex ring. It is important to note that the characteristic parameters of the vortex ring do not remain constant throughout the life of the ring due to viscous effects. The propagation velocity (U) and the circulation (Γ) of the vortex ring decreases, while the volume of the ring increases. The reason for these changes will be discussed in the next paragraph. During the propagation of a laminar vortex ring, a decrease in the propagation velocity will occur after some time (Maxworthy, 1972; Glezer and Coles, 1990). Maxworthy (1972) credited Reynolds on this phenomenon as it was first noted by Reynolds in late 19th century. Reynolds observed that the volume of the vortex ring increases with time due to entrainment of ambient fluid, and he concluded that, since the momentum of the vortex ring must be conserved, its propagation velocity should decrease as the volume increases. Maxworthy further elaborated on Reynolds’ findings and showed how the 20    Chapter 2 Literature Review  volume of the vortex ring increases through the entrainment of the ambient fluid (see figure 2-6). He explained that, due to viscous diffusion, the vorticity from the vortex core diffuses out in a radial manner. After which, vorticity dissipation takes place and causes the ambient, irrotational fluid outside the core to attain a reduced total pressure. With the reduction in total pressure, some of this ambient fluid, now with the acquired vorticity, will be entrained by the circulatory flow of the vortex ring, instead of passing over the ring. The rest of this ambient region which is “contaminated” by vorticity will be shed as the wake of the vortex ring. This also decreases the circulation of the ring. Maxworthy then formulated the equation for the rate of change of volume of the vortex ring, which was later validated by Dabiri and Gharib (2004) through DPIV measurements. Figure 2-6: (a) Diagrammatic view showing the entrainment by a vortex ring of the fluid from upstream. (b) Diagrammatic view of instantaneous streamlines relative to a moving vortex ring. (Maxworthy, 1972). 21    Chapter 2 Literature Review  2.1.3 Stability of a Circular Vortex Ring After a vortex ring propagates for some time, it will, under certain conditions, undergo azimuthal instability. Maxworthy (1972) noted that such instability is apparent when ReN > 600. He also showed that the azimuthal waviness grows and eventually causes the vortex ring to breakdown to turbulence. Furthermore, he conjectured that opposite vorticity which is produced on the outside wall of the nozzle is responsible for the azimuthal instability. However, a more credible explanation for this instability was provided by Widnall and Sullivan (1973) who pointed out that it is the inherent structure of the vortex ring which is responsible for causing such wavy formation. They further substantiated their explanation by providing similarities of this waviness to the one observed by Crow (1970) for wing tip vortices. Although Maxworthy (1972) lacked detailed explanation on the mechanism of instability, his experimental results (Maxworthy, 1977) showed many elaborate findings which are worth discussing. In his paper, Maxworthy (1977) observed azimuthal instability of vortex ring and noted that this waviness grows in amplitude as it forms. He also noted that the wave develops at 45° to the direction of propagation and these waves neither travel around the vortex core nor rotate in an axial flow when the amplitude of the wave is small. However, when the amplitude of wave becomes appreciable, axial flow (i.e. flow around the vortex core) was observed. Maxworthy (1977) noted that this axial flow is meant to suppress further instability from occurring. He empirically noted that the ratio of the wavelength of the azimuthal waves, λ, and the core diameter (i.e. λ/c ≈ 2.24). of the vortex ring, c, is approximately constant With this finding, he claimed that the core diameter can be approximated solely by obtaining the diameter of the vortex ring and counting the 22    Chapter 2 Literature Review  number of waves during the onset of instability. Maxworthy found that the number of waves monotonically increases with ReN, but has a weak relationship with the stroke ratio. Furthermore, the duration of instability is generally shorter for higher ReN. 2.1.4 Turbulent Vortex Rings Although the topic on turbulent vortex rings is beyond the scope of this thesis, a brief overview of it is provided here for completeness. A turbulent vortex ring is noticeable due to the existence of fine-scale structures surrounding the core of the ring and in the wake region (see figure 2-7). The formation of turbulent vortex ring is usually initiated after its breakdown from a laminar state via azimuthal instability. Shariff and Leonard (1992) classified another turbulent vortex ring (known as turbulent puff) which can be produced by placing a wire mesh at the nozzle exit during the formation of the vortex ring. Figure 2-7: Dye visualization of a turbulent vortex ring (Glezer, 1988). It is noted that at high Reynolds numbers, turbulent vortex rings form almost immediately, apparently bypassing the laminar stage (Glezer, 1988). According to 23    Chapter 2 Literature Review  Glezer (1988), the immediate onset of turbulence in the vortex ring results from the Kelvin-Helmholtz-like instability during the formation of the vortex ring. However, he did not provide sufficient evidence and information to substantiate how the instability leads to the breakdown of the laminar vortex ring. Lim (1997) reopened this issue and further explained that this Kelvin-Helmholtz-like instability leads to a generation of secondary vortex ring originating behind the primary ring. This is followed by a leap-frogging effect3 of the two rings, and thereafter, a slight misalignment in the rings would lead to complex interactions and the development of a turbulent vortex ring. Other works on turbulent vortex rings involve the formulation of mathematical models. These include the similarity model (Glezer and Coles, 1990), semi-empirical model (Maxworthy, 1974) and empirical model (Johnson, 1970). Interested readers are referred to Lim and Nickels (1995) for a review of these models.                                                              3   The leap frogging effect occurs when the secondary vortex ring is propelled forward through the primary ring by induced velocity. This will then lead to the secondary ring taking the place of the primary ring and vice versa. The process occurs until the vortex rings merge or set into turbulence in viscous fluid.  24    Chapter 2 Literature Review  2.2 Elliptic Vortex Rings Elliptic vortex rings have been studied in the past and are known to posses some unique characteristics such as vortex deformation (Dhanak and De Bernardinis, 1981; Hussain and Husain, 1989), and as a model for a two wave-number perturbed circular vortex ring (Zhao and Shi, 1997; Kambe and Takao, 1971). Some of the previous works on elliptic vortex rings are discussed in this section. First, the current understanding on the characteristics and structure of an elliptic vortex ring are discussed. Next, works on elliptic jets and their relevance to elliptic vortex rings are presented. Finally, previous works on vortex rings of other geometries are explained. 2.2.1 Characteristics of an Elliptic Vortex Ring An elliptic vortex ring is known to exhibit an oscillatory deformation while propagating. This oscillatory deformation causes the vortex ring to undergo axisswitching; a process when the major and the minor axes of the ring switch at half-cycle of the oscillation (Lim and Nickels, 1995; Dhanak and De Bernardinis, 1981). The orthographic and the perspective view of the deformation are illustrated in figure 2-8. 25    Chapter 2 Literaturee Review  Side View V Directio on of Motio on Direcction of Mo otion Plan View V At halff-cycle Direction of D Motion P Perspective View V Figure 2-8: Schematic drawing of the trajecto F ory of elliptiic vortex rinng in side, plan and p perspective v view. T This deform mation of thhe elliptic vortex v ring is i caused by b the self-iinduced vellocity on t vortex core, based on the o the Biot--Savart law,, which is given g by: 26    Chapter 2 Literature Review  Γ 4 ′ ′ | ′ |                                       (2-11) where u is the velocity at point x on the vortex core and x’(s) is the parametric variable representing the location of the points around the vortex ring for which s is the parameter. Equation 2-11 can be solved analytically for circular vortex ring to obtain the propagation velocity, which is constant in the binormal direction throughout the core (see equation 2-10). However, for elliptic vortex ring, the propagation velocity varies along the core and, thus, causes the oscillatory deformation which is shown in figure 2-8. One of the earlier theoretical works on elliptic vortex rings was carried out by Arms and Hama (1965). In their work, they derived a formula which governs the motion of a curved vortex filament4 using the Biot-Savart law, and arrived at a new equation which they referred to as the Localized-Induction Equation (LIE):                                                           (2-12) The LIE is a simplified equation, which assumes that the long distance effects and the filament size are negligible. Arms and Hama (1965) showed that the induced velocity of the vortex filament is inversely proportional to the local curvature at that point.                                                              4  Vortex filament here refers to vortex core with an infinitesimal core size, but with a finite circulation value.   27    Chapter 2 Literature Review  Using equation 2-12, they also deduced that stretching or shrinking of vortex filament does not occur, and this suggests that the length of the vortex filament of the elliptic vortex ring remains constant. Subsequent numerical works by Arms and Hama (1965) showed that the elliptic vortex ring oscillates periodically in time. In a related study, Kambe and Takao (1971) showed that the perturbation of circular vortex ring at the lowest mode gives rise to an elliptic vortex ring. Their analysis showed that the perturbed circular ring (i.e. elliptic vortex ring) does not stretch or shrink, which is consistent with the finding by Arms and Hama (1965). Kambe and Takao (1971) also carried out experimental study on elliptic vortex ring and found that under certain conditions, vortex ring was able to split into two or more smaller rings. However, they did not elaborate on how the splitting occurs. Another work on elliptic vortex ring was carried out by Dhanak and De Bernardinis (1981) where they conducted comprehensive simulations and experiments. In their numerical work, the Biot-Savart law was used to simulate the deformation of the elliptic vortex ring which, however, leads to a singularity problem when x = x’ (see equation 2-11). This makes the computational problem ill-conditioned. To handle this singularity problem, cut-off approximation derived by Moore and Saffman (1972) was used, which modifies equation 2-11 to give: Γ 4 | |                            2‐13 28    Chapter 2 Literature Review  where µ0 is proportional to the local core radius (see Dhanak and De Bernardinis, 1981). Dhanak and De Bernardinis (1981) compared their cut-off approximation with LIE, and noted that the drawbacks of LIE, used by Arms and Hama (1965), are in the assumption that the core size and long distance effects are negligible. They further noted that neglecting the contribution of long distance effect means that the LIE loses its behaviour of the Crow instability (Crow, 1970) which occurs when anti-parallel vortex tubes are in close proximity. These drawbacks of LIE are also emphasized by Saffman (1992). From their simulation results, Dhanak and De Bernardinis (1981) claimed that for elliptic vortex ring of aspect ratio (AR) 5/3 and 5/2, the vortex ring does not remain flat after it undergoes the first axis-switching processes. They also noted that the vortex ring oscillated with time for AR > 1, and these oscillations were periodic only for aspect ratios close to 1. At higher aspect ratios (AR ≥ 5), it was found that the core of the vortex ring deformed intensively, and touched each other, and they speculated that the vortex ring would split into two smaller vortex rings. However, due to numerical instability, the splitting of vortex ring could not be observed. Dhanak and De Bernardinis (1981) also conducted experiments on elliptic vortex rings, and found that the trajectory of the rings show qualitative resemblance to that depicted by their computational results. They also noted that elliptic vortex ring 29    Chapter 2 Literature Review  undergoes azimuthal instability and the wave number of this instability agreed with the findings by Saffman (1978). At high aspect ratios, the splitting of the vortex ring into two smaller rings was observed experimentally, and they referred to this process as bifurcation. A thorough analysis of the bifurcation of an elliptic vortex ring into two smaller rings was carried out experimentally by Oshima et. al. (1988). They noted that complete bifurcation occurred only at AR ≥ 5. They also introduced a new term, “partial bifurcation” of an elliptic vortex ring, which was observed for AR = 3. Partial bifurcation starts when the cores at the two extremities ends of the initially major axis, touch each other at half-cycle of the oscillation. Vorticity cancellation soon follows at the outer part of the adjacent vortex cores that are closest to each other (Oshima et. al., 1988), and this leads to the cross-linking of the vortex core. However, the ring does not split into two smaller rings, but remains in this partially bifurcated form. An example of this is illustrated in figure 2-9. Oshima et. al. (1988) further showed that, although the impulse and the enstrophy of the elliptic vortex ring remained almost constant, the circulation of the vortex core was affected by the partial bifurcation. 30    Chapter 2 Literature Review  Partial bifurcation Figure 2-9: Time evolution of elliptic vortex ring of AR = 3 (Oshima et. al., 1988). The third vortex ring from the nozzle shows the occurrence of partial bifurcation during deformation of the ring. A computational study to investigate elliptic vortex ring was carried out by Zhao and Shi (1997) using the pseudo-spectral method on the Navier-Stokes equation. Their results mainly confirmed many of the previous findings, such as the axis switching phenomenon at AR = 2, and an increase in the vortex core size due to vorticity diffusion. They also noted the cross-linking of the vortex core at AR = 4. They elucidated that during the deformation, the two extremities of the initially major axis would approach each other so closely that they would squeeze and deform. When the core eventually separates, patches of vorticity of opposite signs were observed between the cores which represent the cross-linking vorticity. Figure 2-10 shows the perspective view of the vortex ring during cross-linking. Note the vorticity patches formed by the cross-linked vortices in x = π plane. At AR = 6, Zhao and Shi (1997) noted a complete bifurcation of the vortex ring. In addition to the above, they also studied the variation of circulation with time and found that while the vortex ring with AR = 2 maintained its circulation, the circulation of the vortex ring with AR = 4 decreased after half-cycle. It was reasoned that, for AR = 4, the cross-linking vortices 31    Chapter 2 Literature Review  distort the vortex core and dissipate the vorticity, thus, leading to the decrease in circulation. For AR = 6, the circulation decreased immediately, but they did not elaborate the reason for the observed behaviour. Cross-linked vortices Figure 2-10: Perspective view of an elliptic vortex ring with cross-linked vortices (Zhao and Shi, 1997). In another related investigation, Ryu and Lee (1997) studied the sound generation by elliptic vortex ring. Through their computation, they found that a single elliptic vortex ring generated acoustic pressure fluctuations that were not found in circular vortex rings. These fluctuations were attributed to the axis-switching of the elliptic vortex ring. For two consecutively formed elliptic vortex ring, they observed even greater acoustic pressures radiated due to the vortex interaction during an attempted leapfrogging between the two rings. They concluded that the motions and acoustic signals for the elliptic vortex rings are strongly dependent on the aspect ratios and the initial separation of two or more rings. 32    Chapter 2 Literature Review  2.2.2 Elliptic Jets Since vortex rings are known to be the building blocks behind the formation of jets (Lim and Nickels, 1995), the dynamics of the rings are sometimes used to explain the phenomena in jets. For this reason, some of the current understanding of elliptic vortex rings has been reported in works related to elliptic jets. Gutmark and Ho (1986) studied forced (or excited) elliptic jets and found that at low excitation frequency, vortex merging is suppressed before the end of the potential core5, but at higher forcing frequency, merging is enhanced. They observed that when the vortices merged, the stretching and contraction due to mutually induced velocities enhanced mixing. A more comprehensive understanding of elliptic jets and the mechanism of the vortex merging has been carried out by Hussain and Husain, and reported in three separate papers (1989, 1991 and 1993) which are discussed below. In all three papers, quantitative measurements were carried out on an air jet using the Hot-Wire Anemometry (HWA), while water jet was used for flow visualization using fluorescent dye. They noted that the boundary layer thickness of the jet at the nozzle exit is different at the major and minor axes due to the construction mechanism of their nozzle. Therefore, to prevent uncertainty in results caused by the effect of boundary layer, they used a boundary layer suction technique to ensure that the thickness is                                                              5  Potential core of a jet is the region within the jet where the velocity profile is nearly uniform.  33    Chapter 2 Literature Review  consistent on both axes of the nozzle. For elliptic vortex rings, non-uniform boundary layer thickness is not an issue because the length of fluid slug ejected from the nozzle is not long enough to cause significant differences to the boundary layer thickness. In part 1 of the study, Hussain and Husain (1989) discussed the properties of excited and unexcited laminar jet emanating from an elliptic exit. For an initially unexcited laminar jet, they observed a change in the jet width, which they attributed to the axisswitching process of an elliptic vortex ring since the jet width resembled closely with that of the elliptic vortex ring trajectory. They also noted that the change in width occurred for a substantial length of the jet, and concluded that elliptic vortex ring plays a dominant role in the dynamics of elliptic jets. In addition to the unexcited jet, Hussain and Husain (1989) also conducted studies on excited elliptic jets to observe the response of these jets due to the excitation and how such excitation can enhance the entrainment and mixing processes in the fluid. Maximum turbulence intensity was observed in the jet centreline for nozzle with AR = 2 at the Strouhal number (St = fDN/UN) of 0.85, where f is the frequency of excitation. For the elliptic jet of AR = 4, high level of turbulence was observed for all Strouhal numbers. They highlighted that the high turbulence intensity at St = 0.85 for AR = 2 is due to preferred mode of pairing. However, for AR = 4, they noted bifurcation process for all Strouhal numbers. 34    Chapter 2 Literature Review  In part 2 of their investigation, Husain and Hussain (1991) studied the dynamics of the coherent structures that exist in an elliptic jet of AR = 2. Here, they found that exciting the jet at a particular mode, known as the stable pairing mode, causes successive and periodic pairing of elliptic vortex rings at the same location. They noted that the vortex interactions occur differently for major and minor axes due to non-planar and non-uniform self-induction of the elliptic coherent structures. In their diagrammatic representation (see figure 2-11), vortex pairing and merging occurs in the initially major axis through an entanglement process, while at the initially minor axis, the trailing vortex rushes through the leading vortex without pairing and subsequently breaks down. By comparing their results with that from a circular nozzle, they concluded that the pairing in elliptic jets is a major cause of the enhancement in mixing and entrainment of fluid. Husain and Hussain (1991) further elaborated that in an elliptic jet, a series of vortices would form, and the dynamical properties over many such vortices would bring about the development of streamwise vortices (or ribs), which are elaborated in part 3 of their study. 35    Chapter 2 Literature Review  Entanglement Figure 2-11: Pairing mechanism of two elliptic vortex rings (Husain & Hussain, 1991). In part 3 of their study, Husain and Hussain (1993) examined the formation of the streamwise vortices (or ribs) in elliptic jets. They observed that ribs were formed even without any forcing applied in the azimuthal direction of the jet and highlighted that the ribs formation can best be understood by identifying elliptic jet as a series of vortex filament ejecting from the nozzle (see figure 2-12). Once ejected, these vortex filaments will group together to form finite core vortices (or rolls) with some fine braid6 vortex lines (or filaments) between them. In the process, these vortex rolls and filaments will undergo self-induced deformation, and concurrently be influenced by the induced velocity from the preceding and succeeding vortices. As a result of these induced velocities, the vortex filament braid deforms and is directed streamwise into the vortex ring. These streamwise vortices constitute the ribs that interact with the                                                              6  Braids are coherent structures observed in jets which usually develop into streamwise vortices.  36    Chapter 2 Literature Review  vortex rolls, resulting in the generation of turbulence. They further noted that the onset of turbulence is one of the causes of higher fluid entrainment in the elliptic jet. Direction of jet (a)  (b)  Streamwise vortices (or ribs) (c)  Figure 2-12: Formation of (a) vortex filaments in a jet, which gathers to form (b) rolls and braids and subsequently results in the formation of (c) streamwise vortices (or ribs) through self-induced velocity (Husain & Hussain, 1993). The direction of self-induced (SI) and mutually induced (MI) velocities are labelled. 2.2.3 Other Non-Circular Vortex Rings/Jets The preceding sub-section described how the deformation and entanglement of the vortex structures enhanced entrainment and mixing in elliptic jets. Similar behaviours are also observed in other non-circular jets and vortex rings. 37    Chapter 2 Literature Review  Grinstein (1995), who studied the dynamics of a vortex ring evolving from a rectangular nozzle, explained that rectangular vortex ring, generally, undergoes nonplanar deformation due to the variation in the self-induced velocity. He further noted that at high aspect ratios (AR > 4), the ring splits into smaller rings due to intense deformation and cross-linking of vortex cores. In a related study, Auerbach and Grimm (1994) systematically studied a range of rectangular nozzles with aspect ratio from 2 to 20, and with different piston profiles, including forestroke and backstroke. Their results reveal the deformation of the vortex ring when it was ejected with forestroke at low aspect ratios (AR ≤ 3). At higher aspect ratios, the vortex ring deformed and split into smaller rings. However, when they introduced backstroke following the forestroke, they observed splitting of the vortex ring into 2, 3 or 4 smaller rings, even at the lowest aspect ratio of 2. While the numerical study of Kiya et. al. (1992) show that splitting of vortex ring occur at aspect ratio of 3, the experimental results of Auerbach and Grimm (1994) indicated otherwise unless a backstroke was applied to promote the splitting of the vortex ring. Auerbach and Grimm (1994) also observed that for aspect ratio of 3, the splitting angle was very sensitive to the stroke ratio, and for aspect ratio of 5 to 9, the angle appeared to be independent of the stroke ratio. They also found that the splitting angle of the vortex rings generated from the orifice is insensitive to the stroke ratio, unlike those ejected from a nozzle. Furthermore, they observed that the splitting angle seemed to be independent of the Reynolds number in all cases. Finally, they noted that vortex rings generated from the nozzle are more stable than those generated from an orifice. 38    Chapter 2 Literature Review  Grinstein et. al. (1995) presented a review on the dynamics of vorticity in jets where they discussed non-circular jets and how they enhance mixing and entrainment. They highlighted that vortex stretching, self-induced velocities and braid production in noncircular jets are the main causes of entrainment of ambient fluid. They emphasized that the vorticity at sharp corners will form hairpin vortices during the deformation and eventually take the role of the braids. 39    Chapter 2 Literature Review  2.3 Interaction of a Vortex Ring with a Circular Cylinder The interaction of vortex ring with a circular cylinder is discussed in this section with reference to the work of Naitoh et. al. (1995). This is followed by a brief review of the cut-and-reconnection phenomena. 2.3.1 Interaction of a Vortex Ring with a Small Cylinder In the paper by Naitoh et. al. (1995), circular vortex rings were made to interact with a small circular cylinder. They found that the interaction is governed by 3 dimensionless parameters, namely (1) the Reynolds number (Re = UD/ν), (2) the ratio of cylinder diameter to core diameter (dc/c) and (3) the ratio of core diameter to ring diameter (c/D). In their work, only dc/c was varied. Naitoh et. al. (1995) noted that when dc/c ≤ 0.025, there was no notable difference in the vortex ring, but when dc/c ≥ 0.025, the travelling distance of the vortex ring was reduced. At even higher ratio of dc/c ≥ 0.063, the travelling distance diminished by approximately 15%, thus, making the effects of the cylinder more significant. When the vortex ring interacted with the cylinders, Naitoh et. al. (1995) highlighted the formation of two counter rotating vortices (secondary vortex) on the lee-side of the cylinders. These vortices were noted to form as a result of the shear layer generated at the cylinders during interaction. Figure 2-13 shows their interpretation of the formation of the secondary vortices. Naitoh et. al. (1995) further observed the 40    Chapter 2 Literature Review  transformation of the vortex ring from a circular to an elliptic-shaped ring; based on the axis-switching-like trajectory of the ring after it has passed through the cylinder. Figure 2-13: Schematic representation of vortex ring after its interaction with a cylinder (Naitoh et. al., 1995). Although Naitoh et. al. (1995) examined the behaviour of a vortex ring after it has interacted with a cylinder in detail, they did not elaborate on the cut-and-reconnection of the vortex core as it passes through the cylinder. In the next sub-section, a brief review on the current understanding of cut-and-reconnection is presented. 2.3.2 Cut-and-Reconnection Phenomena One of the earliest recorded cut-and-reconnection phenomena was reported by Crow (1970). He provided a stability analysis for the two wingtip vortices from an airplane, and showed that they undergo a cut-and-reconnection process resulting in the formation of a series of vortex rings. Since then, several researchers have studied this phenomenon and their interpretations of the mechanism is presented in the following paragraphs. 41    Chapter 2 Literature Review  Ashurst and Meiron (1987) were one of the earliest to illustrate the cause of reconnection numerically. In their simulation, two vortex rings were made to propagate in the same direction near to each other, and the cut-and-reconnection of the vortex core at the closest distance was analysed. Their findings revealed that the low pressure that develops between the interacting vortex tubes causes the distortion of the initially circular vortex cross-section, and this forces the vorticity to rearrange itself, thus, causing the cut-and-reconnection of the cores. In another computational study by Melander and Hussain (1989), the mechanism of cut-and-reconnection of two anti-parallel vortex cores near each other was elaborated. They explained that there are three characteristic phases to this mechanism. The first phase is the core flattening and stretching when the vortex cores are very near to each other (see figure 2-14(d)). This is followed by the second phase, which is the bridging of the two vortices by the accumulation of the annihilated, and cross-linked vortex lines (see figure 2-14(e)). The last phase is the threading of the remnants of the initial vortex pair (i.e. vortex bridge) in between the two reconnected cores as they pull apart (see figure 2-14(f) and (g)). These phases can also be visually observed in Lim (1989) where two vortex rings were made to collide obliquely, causing the cut-andreconnection of the rings with a visible bridge. 42    Chapter 2 Literature Review  (a)  (e)  (b)  (f)  (c)  (d)  Vortex bridge (g)  Figure 2-14: Cut-and-reconnection mechanism of two anti-parallel vortex tubes (Melander & Hussain, 1989). The concept of bridging during the reconnection process of two anti-parallel vortex tubes has been explained by Kida and Takaoka (1994) (see figure 2-15). Figure 2-15(a) shows the simple identification of reconnection process, while figure 2-15(b) illustrates how the vorticity lines (indicated by double arrows in the figure) of the vortex tube undergo deformation and entanglement during reconnection process. In figure 2-15(b), the vortex tubes first approach each other (direction of approach is indicated by the blank arrows) and cancellation of the vorticity takes place at the point of contact. Subsequently, the vorticity lines from the tubes join with the adjacent tube, forming the vortex bridge, which link the reconnected tubes. 43    Chapter 2 Literature Review  Vorticity cancellation Vorticity cancellation Figure 2-15: Mechanism of vortex reconnection. (a) A simple viscous cancellation of vorticity, and (b) detailed mechanism of the reconnection process with the development of vortex bridges. Blank arrows indicate the flow direction that pushes the vortex tubes. Single and double arrows indicate the rotation of vorticity lines and the direction of vorticity, respectively. Hatched areas show the region of vorticity cancellation through viscous diffusion (Kida and Takaoka, 1994). Based on the mechanism proposed by Kida and Takaoka (1994) in figure 2-15, Adhikari and Lim (2009) elucidated the cut-and-reconnection of a vortex ring when it passes through a porous screen (see figure 2-16). They highlighted that when the core of the vortex ring impinges on a series of wires of the porous screen, parts of the core which are obstructed by the wires is stretched locally (see figure 2-16(b)). During stretching, the vorticity lines of opposite signs get in contact with each other (see 44    Chapter 2 Literature Review  figure 2-16(c)), leading to mutual cancellation of the vorticity through viscous diffusion. As they are subsequently connected to each other on the lee-side of the wires, vortex bridges are seen to form (figure 2-16(d)) while the rest of the reconnected vortex core continues to move away. The vortex bridges continue to stretch and are eventually dissipated by viscous diffusion. Figure2-16: Cut-and-reconnection mechanism when vortex core interacts with a series of wires (Adhikari and Lim, 2009). 45    Chapter 2 Literature Review  A different perspective for cut-and-reconnection mechanism was proposed by Saffman (1990). In his analytical model, he argued that the viscosity cancels the vorticity of the anti-parallel vortex core when they first touch each other. This weakens the centrifugal force in the vortex core touching each other, which will lead to a local increase in the pressure. As a result, the fluid in the core of the vortex will accelerate axially from the point of contact and thus, essentially “cutting” the vortex tube. Saffman then attributed the mechanism of the “reconnect” process to the Helmholtz law, stating that vortex line cannot terminate within the fluid domain and, thus, has to rejoin. During the cut-and-reconnection phenomena, dissipation of vorticity from the vortex tubes is known to occur. Chatelain et. al. (2003) explained such dissipative effects of vortex reconnection by numerical means. By simulation of two colliding vortex rings at low Reynolds number in different configurations, they found that reconnections are dissipative in nature due to the smoothing of vorticity gradients at reconnection kinks. Furthermore, they also explained that the stretched anti-parallel vortices (i.e. vortex bridge), which form as a by-product of reconnection, transfer kinetic energy to the small-scale structures, leading to an enhancement in dissipative effects. 46    Chapter 3 Experimental Setup & Methodology  Chapter 3  Experimental Setup & Methodology    The experiments were carried out in the Fluid Mechanics Laboratory at the National University of Singapore. To ensure acquisition of accurate results, several careful measures were taken in the fabrication and assembling of the experimental apparatus. This chapter describes, in detail, the apparatus and techniques used to acquire the data. 47    Chapter 3 Experimental Setup & Methodology  3.1 Vortex Ring Generator The in-house built vortex ring generator comprises of a tank, which is connected via a nozzle to the piston-cylinder arrangement, and the motion control mechanism (see figure 3-1). The motion control mechanism consists of a stepper motor, transmission gears, motor controller and a stand-alone workstation (PC 1). z  x  PC 1 Tank Motor controller Stepper motor Nozzle Vortex Piston Bevel gear Lead screw CCD camera Figure 3-1: Schematic diagram of the vortex ring generator (Plan view). The CCD camera refers to all the cameras that were used in this experiment. 3.1.1 Tank The tank, measuring 900 mm (L) x 500 mm (W) x 520 mm (H) in dimension, is made of Perspex in order to facilitate clear visualization of vortex motion. The tank was filled with water to a height of 500 mm from the bottom of the tank and the water was left for at least 24 hours to ensure that it attains a room temperature of about 23°C. 48    Chapter 3 Experimental Setup & Methodology  3.1.2 Piston-Cylinder Arrangement A piston-cylinder arrangement is connected to the tank via the nozzle. The piston has a diameter (DP) of 112 mm and the circular nozzle from which fluid is ejected has a diameter (DN) of 26 mm. Generally, there are two common techniques of generating vortex ring through a nozzle. One is the piston-cylinder arrangement (used in this experiment) and the other is the constant-pressure arrangement (Gharib et. al., 1998; Glezer, 1988). One of the advantages of the piston-cylinder arrangement, over the constant-pressure arrangement, is that the ejected fluid can be controlled with a predetermined motion. However, the disadvantage of this arrangement is the vibration caused by the motion of the stepper motor and the piston. The vibration effect of the piston-cylinder arrangement was minimised in the current experiment by regularly applying silicone grease on the piston, and by placing the tank on a separate platform. 3.1.3 Motion Control Mechanism The motion of the piston is actuated by a stepper motor, which runs on 400 pulses per revolution. The motor is connected to a bevel gear which converts the rotational motion into a linear motion of the piston through a lead screw with pitch of 2 mm. The stepper motor is controlled via a motor controller from the workstation, PC 1, equipped with a National Instruments (NI) Data Acquisition (DAQ) card and LabVIEW® software. The software is programmed to generate the required control signal to drive the motor. 49    Chapter 3 Experimental Setup & Methodology  3.2 Test Models The test models used in this experiment consist of (a) a circular nozzle, (b) two elliptic nozzles of aspect ratio (AR) 2 and 3, and (c) cylinders. To support the cylinders, a cylinder/nozzle fixture for the vortex ring/cylinder interaction was used. Each of these test models will now be discussed. 3.2.1 Circular Nozzle The circular nozzle used in this experiment is made of brass, and is painted black to minimize background light reflection, which may interfere with the visualization. The nozzle has an internal diameter (DN) of 26 mm and an external diameter (DO) of 46 mm (see figure 3-2). To carry out the visualization, the dye was directed from a dye reservoir into the cavity within the nozzle before being ejected from a small gap near the lip of the nozzle. Note that the vortex ring propagates in the positive x-direction. y z 46 mm y Dye Inlet x To Small gap 26 mm Piston Dye Inlet Cavity Figure 3-2: Sectional view of the circular nozzle. 50    Chapter 3 Experimental Setup & Methodology  3.2.2 Elliptic Nozzles Two elliptic nozzles with aspect ratios of AR = 2 and AR = 3 were similarly fabricated from brass and painted black. Also, the dye injection technique is the same as that described for circular nozzle. Their essential dimensions are shown in figure 3-3 and note that the minor axis of the nozzle is parallel to z-axis, and the major axis of the nozzle is parallel to y-axis. y y x z Dye Inlet 36.7 mm 60.0 mm Piston Small gap 18.4 mm (a) To Dye Inlet Dye Inlet 68.0 mm 45.0 mm To Piston Small gap (b) 15.0 mm Dye Inlet Figure 3-3: Sectional view of the elliptic nozzles (a) AR = 2 and (b) AR = 3. 51    Chapter 3 Experimental Setup & Methodology  To ensure that same momentum flux of the fluid is ejected from all the nozzles, the elliptic nozzles were constructed with the same exit area as the circular nozzle. The characteristic diameter of the elliptic nozzles is taken to be DN = 26 mm, which is also known as their equivalent diameter (Hussain and Husain, 1989). 3.2.3 Cylinders The circular cylinders which are used for vortex ring/cylinder interaction consist of both the extruded aluminium cylinders with diameters of 1.32 mm and 1.86 mm, and fishing lines with diameters of 0.18 mm, 0.39 mm and 0.92 mm. The ends of the aluminium cylinders were manually attached with wires to enable them to be fastened (see figure 3-4). wires Aluminium cylinder Figure 3-4: Diagram of the aluminium cylinder connected at its ends with thin wires to enable it to be mounted on the Perspex wall. 52    Chapter 3 Experimental Setup & Methodology  3.2.4 Cylinder/Nozzle Fixture In order for the circular vortex ring to interact with the cylinders, a fixture fabricated from Perspex was fixed to a Perspex rig as shown in figure 3-5. The Perspex rig was subsequently attached to the circular nozzle, and each of the desired cylinders was then mounted on the Perspex wall. The cylinders were put in place by securing the end wires of the aluminium cylinders, or the finishing lines to the bottom of the Perspex rig. Care was taken to ensure that these end wires, or the fishing lines, were taut. The cylinders were placed about 4DN away from the nozzle. The two sides of the Perspex walls supporting the cylinders are 7DN apart to minimise the influence of the walls during the interaction. Note that the axis of the cylinder is along the z-axis, while the vortex ring propagates along the x-axis.   53    y 4xDN Nozzle x Perspex wall Perspex wall Cylinder z Perspex wall Cylinder Wire Perspex rig Wire To piston Secured Wire Ends Front View Side View Secured Wire Ends Wire Perspex wall z Wire Perspex wall To piston x Cylinder Nozzle Perspex rig 7xDN Perspex rig Cylinder To piston Wire Plan View Perspex wall Wire Isometric View 54 Figure 3-5: First angle orthographic and isometric view of the cylinder/nozzle fixture used for interaction of vortex ring with cylinders.       ___________________________________Chapter 3 Experimental Setup & Methodology y Chapter 3 Experimental Setup & Methodology  3.3 Dye Flow Visualization Rekitt’s Bluo laundry brightener (Lim, 2000) was chosen as the tracer dye for visualization because of its relatively low Schmidt number (Adhikari and Lim, 2009) and its specific gravity, S.G. ≈ 1. To visualize the vortex ring, the dye was released slowly through the small gap at the nozzle lip (see figure 3-2). Flow images were captured using a SONY 3CCD Color video camera (Model DXC930P) fitted with FUJINON Aspheric 16x TV-Zoom lens. The camera has a frame rate of 25 frames-per-second (fps) and was set at a shutter speed of 1/1000s. The camera was connected to SONY digital video cassette recorder (Model DSR-45P), as well as a SONY Trinitron colour video monitor (Model PVM-14N5E) for real time viewing. The data from the digital cassette was subsequently transferred into a PC, via IEEE 1394 interface (FireWire). Figure 3-6 shows the connection network for visual data acquisition. Monitor Digital Video Recorder Video Camera Digital cassette PC Legend Data Flow Figure 3-6: Data acquisition control network (visualization). 55    Chapter 3 Experimental Setup & Methodology  Before acquiring the data, care was taken to ensure that the camera was aligned upright, and sufficiently bright lighting was provided by means of a high-powered spot light. Prior to capturing the flow images, the image of ruler was captured in order to calibrate the visualized dimensions. Figure 3-7 shows sequence of processed images, where the vortex ring is seen to propagate from the right side of the image window. 1  2  3  Figure 3-7: Sequence of images showing propagation of vortex ring from the right side of the image window. 56    Chapter 3 Experimental Setup & Methodology  3.4 Digital Particle Image Velocimetry (DPIV) Major aspects of the Digital Particle Image Velocimetry (DPIV) system include the illumination source and the image acquisition, which are synchronised with the motion control system. The acquired image of the seeded particles within the fluid is then processed and validated in order to obtain the required flow field in two-dimensions (2-D). Specific details on each of the components of the system, the seeding particles and the image processing and validation techniques will be described in the following sub-sections. 3.4.1 Illumination Source A Nd:YAG laser system (New Wave Solo) is used in the present study. The laser is capable of producing double 120 mJ laser pulses of 10 ns duration, and operating at a frequency of 8 Hz. A separate laser reflecting arm was used to guide the laser to the required illuminating area and spread the beam into a two-dimensional plane. To control the laser plane thickness, the laser plane was guided through a 2 mm slit before illuminating the test area. Maintaining 2 mm thick laser sheet is regularly used in DPIV measurement since too thin a sheet will promote the occurrence of erroneous vectors (Keane and Adrian, 1990). 3.4.2 Specifications of Image Acquisition Charge-coupled device (CCD) Kodak Redlake MegaPlus (Model ES 4.0 Megapixel 12-bit) camera with an array size of 2048 x 2048 pixels was used to capture the images. The pixel pitch of the CCD sensor measures 7.6 μm both vertically and horizontally. A Nikkor lens of focal length 60 mm was set at an f-stop (ratio of focal length to aperture size) of 2.8. The spatial resolution of the camera, as well as the 57    Chapter 3 Experimental Setup & Methodology  other essential information is summarized in Table 3-1. The maximum frame rate for a double-frame mode is 7.6 Hz, which translates into the minimum achievable time between image pairs of 132 ms. However, a precise synchronisation with the rest of the system could only take place at 250 ms. Table 3-1: Summary of DPIV image acquisition specification in all the experiments. Scale Factor Spatial Resolution (mm) Image Map Dimension (mm) Mean Particle Diameter (µm) Particle Image Size (pixels) 2048 x 2048 5.666 1.011 85.9 x 85.9 20 3-6 2048 x 2048 6.029 6.036 1.075 – 1.076 91.4 x 91.4 91.5 x 91.5 20 2-5 2048 x 2048 6.661 1.187 100.9 x 100.9 20 2-5 2048 x 2048 6.732 6.780 1.200 – 1.209 102.0 x 102.0102.8 x 102.8 20 2-5 Experiment Resolution of Camera Frame (pixels) Circular Vortex Ring Elliptic Vortex Ring (AR = 2) Elliptic Vortex Ring (AR = 3) Vortex Ring on Cylinders 3.4.3 Synchronization of the DPIV System and Motion Control The experimental execution requires synchronization of three devices: the CCD camera for acquiring the DPIV images, the laser and the stepper motor used to displace the piston to generate vortex rings. The Dantec’s FlowManager Software, installed in PC 2, was used to interface and operate the laser and the CCD camera, while LabVIEW® software, installed in PC 1, was used to interface and operate the stepper motor (see figure 3-8). Before execution, the FlowManager is on standby mode and await external TTL signal to trigger specified DPIV operations. The user then executes the LabVIEW® program from PC 1 58    Chapter 3 Experimental Setup & Methodology  to activate the NI-DAQ card, which sends TTL signal to the FlowMap System Hub, and control signal to activate the stepper motor via motor controller. The TTL signal sent to the System Hub will execute FlowManager Program in PC 2, which generates control signal to activate the laser and CCD camera. The frames captured by the camera are transferred in real time to the temporary storage at the System Hub, and at the end of the execution, the images are then transferred from the System Hub to the FlowManager in PC 2. FlowMap System Hub Workstation equipped with FlowManager Software To CCD camera y  x  Laser reflecting arm PC 2 Nd: YAG Laser NI – DAQ PCI Card Motor Controller Front view Legend Workstation equipped with LabVIEW® PC 1 Software Control signal Stepper motor TTL signal Figure 3-8: Network of motion control system and DPIV system, together with the experimental setup. 59    Chapter 3 Experimental Setup & Methodology  3.4.4 Seeding Specifications Polyamide Seeding Particles (PSP) of density 1030 ± 20 kg m-3 and a mean diameter of 20 µm were used in the experiments. The particles were carefully prepared so that they maintained neutral buoyancy in water. As the water in the tank is stagnant most of the time, attaining neutral buoyancy of the particles is extremely important. The size of the particles was chosen such that the seeding particles have an image diameter of at least 3 pixels so as to ensure sub-pixel accuracy, thus effectively increasing the resolution of the velocity measurements (FlowMap User’s Guide, 2000). In all cases considered here, it was observed that the images of the 20 μm particles were between 2 to 6 pixels (see table 3-1). The seeding density was ensured to be at least 7 particles per Interrogation area (IA) in order to increase the signal-to-noise ratio (FlowMap User’s Guide, 2000). When mixing the particles, soap solution was used in water as this would help to break down clumps of particles with ease. Since a homogenous mixture of particles in the flow was necessary for good quality DPIV images, the mixture was stirred overnight using a magnetic stirrer. After stirring, it was allowed to settle for several minutes before the top layer of foamy mixture was removed, and only the central portion of the homogenous mixture was eventually used in the experiment. 3.4.5 Image Processing & Data Validation The PC 2 workstation running Dantec Dynamics FlowManager Version 3.70 software was used to process the captured image pairs. Adaptive cross-correlation DPIV algorithm provided by the Dantec System was used. In this algorithm, 32 by 32 pixels 60    Chapter 3 Experimental Setup & Methodology  IA was used with a central difference window offset function. In addition, the 25% overlapping of IAs was used to improve the signal-to-noise ratio, as well as to increase the total number of velocity vectors obtained from the image map. Sub-pixel accuracy in the velocity vector was achieved using a three-point symmetrical Gaussian curve-fit interpolation scheme. The vector field derived using adaptive cross-correlation algorithm was further refined to remove spurious vectors using user-defined conditions. This involved rejecting vectors whose length exceeded a user-defined threshold value as well as replacing the rejected vectors with those approximated from the neighbouring vectors. The four data validation techniques are as follows: Peak-height Validation In cross-correlation, the highest correlated peak is assumed to be the signal and all other peaks are the noise. In order to achieve a signal with good confidence, this technique assigns a certain limit ratio between the highest peak and the second highest peak. In this experiment, the ratio was set to 1.2 as it gives minimum number of rejected vectors. Velocity-range Validation This validation rejects all velocity vectors which are beyond a certain threshold unattainable by the fluid in the domain. Once this threshold velocity was set, any vectors with value greater than this velocity in the domain were considered outliers 61    Chapter 3 Experimental Setup & Methodology  and therefore rejected. In this experiment, the maximum threshold velocity was calculated using the velocity profile of the piston recorded by Didden (1979). Moving-average Validation This method rejects velocity vectors based on the comparison amongst the neighbouring vectors, and replaces the rejected vectors with an estimated mean from its surrounding. Based on the user-defined acceptance factor, it will reject all vectors deviating by more than the maximum allowable percentage which varied between 5% and 15%, depending on the nature of the vortex ring. This validation essentially assumes the existence of continuity in fluid flow. Moving-average Filter This filter effectively substitutes the rejected vectors with a uniformly weighted average of the vectors in a neighbourhood of a specified size. This method is needed to replace all the rejected vectors. 62    Chapter 3 Experimental Setup & Methodology  3.5 Data Post-Processing The visualization images were analysed using the media player in the author’s PC. Sequential images were extracted from the video clip, and to further improve the clarity, the images were converted to greyscale using Adobe® Photoshop CS3 before the shades were inverted such that the vortex ring would appear white in black background. Figure 3-9(a) shows a typical vortex ring image captured. The video images were used mainly to visualize the flow structures of the vortex ring. Quantitative measurements such as the trajectory, diameter and core size of the vortex ring were obtained using DPIV measurement. y x Direction of propagation Figure 3-9(a): An instant of a developed vortex ring (at ReN = 1000 & LN/DN = 1) visualized using dye. 63    Chapter 3 Experimental Setup & Methodology  The DPIV results (i.e. velocity fields), after they were processed and validated by the FlowManager system, was exported into a data file for subsequent analysis using TecPlot® 360. It should be noted that due to the unsteady nature of the vortex ring, time-averaging in DPIV measurement is not feasible as it would smear the velocity field of the vortex ring due to ensemble averaging. Therefore, extreme care was exercised when acquiring the DPIV data. Typical acquired velocity and vorticity fields are shown in figure 3-9(b). Once the velocity fields are determined, they were used to derive vorticity field, diameter, core size, trajectory and the circulation of the vortex ring. The method of acquiring each of these quantities will be discussed in the next few sub-sections. y  x  Direction of propagation Figure 3-9(b): An instant of a developed vortex ring (at ReN = 1000 & LN/DN = 1) visualized using DPIV. 64    Chapter 3 Experimental Setup & Methodology  3.5.1 Vorticity Field The vorticity field was calculated using the algorithm provided by the FlowManager software. The basic principle involves the vorticity equation in a 2-D velocity field:                                                             3‐1 where ωz is the vorticity out of the plane (i.e. z direction), u and v are the velocity components in the x and y direction, respectively. 3.5.2 Dimensions of a Vortex Ring The two essential dimensions of a vortex ring are the diameter of the ring and the vortex core. These dimensions can be determined from the vector or vorticity field of the vortex ring. The diameter of the vortex ring (D) is obtained by taking the distance between the peak vorticity values as shown in figure 3-10. The vorticity distribution shown in the figure is acquired in the y-direction through the centre of the vortex core (see insert diagram in figure 3-10). 65    Chapter 3 Experimental Setup & Methodology  80 70 60 y (mm) 50 D 40 30 20 10 0 ‐30 ‐20 ‐10 0 10 20 30 ω (s-1) Figure 3-10: Vorticity distribution of the vortex ring for ReN = 1420, LN/DN = 1 through the core centre. The diameter (D) is defined as the distance between the two peak vorticity values. Insert diagram shows the vortex ring and the dotted lines represent the location where the values were acquired. The core of the vortex ring in viscous fluid has been defined by Saffman (1992) as the distance between the maximum and minimum tangential velocity within the vortex tube. Since Saffman’s definition of the core diameter also holds for an ideal Rankine vortex, it is reasonable to use this definition of core diameter for the present experiment. Figure 3-11 shows the plot of the tangential velocity profile of the vortex ring acquired in the y-direction through the centre of the vortex core (as shown in figure 3-11 (insert)), and the core diameter is denoted by c. 66    Chapter 3 Experimental Setup & Methodology  80 70 60 y(mm) 50 40 c 30 20 10 0 ‐50 ‐40 ‐30 ‐20 ‐10 0 10 20 U(mm/s) Figure 3-11: Horizontal velocity of the vortex ring for ReN = 1420, LN/DN = 1 through the core centre. The core diameter (c) is defined as the distance between the two peak velocity values. Insert diagram shows the vortex ring and the dotted lines represent the location where the values were acquired. 3.5.3 Trajectory of a Vortex Ring In measuring the trajectory of the vortex ring, the peak vorticity within the vortex core was tracked. Since acquiring the location of the peak vorticity values from several vorticity fields can be a tedious process, a program (written by the author) was used to aid the acquisition of these values from the Tecplot®. The complete source code, written in C++ language, is provided in Appendix A. 3.5.4 Circulation of a Vortex Ring The circulation of a region in a 2-D velocity field is given by: Γ ·                                                               3‐2 67    Chapter 3 Experimental Setup & Methodology  where u is the velocity vector and s is the direction vector that traverses around a known closed boundary. To find the circulation of a particular vortex ring, the closed region (or loop) of interest within the vector plot is selected (see figure 3-12). Equation 3-2 is then discretized and the data acquired from the selected loop is applied iteratively to the discretized formula to obtain the circulation. To ensure accuracy of the calculation, the interval spacing of the discrete points taken around the loop should not be greater than the spatial resolution of the vector field. A program written in C++ language by the author to calculate the circulation is provided in Appendix B. This program has been validated and verified using point vortex in a potential flow field by Paudel (2008). 68    Chapter 3 Experimental Setup & Methodology  Figure 3-12: The dotted points represent the data points taken around the vortex core in order to calculate the circulation of the vortex ring. 3.6 Experimental Conditions For the study on the circular and elliptic vortex ring, input variables are the velocity of fluid slug at the nozzle exit (UN), the length of the fluid slug (LN), and the aspect ratio of the nozzles (AR). The diameter of the circular nozzle (DN) was kept constant. In this part of the study, Reynolds numbers (nozzle) (ReN = UNDN/ν) ranges from 1000 to 1740, and stroke ratios (LN/DN) from 1 to 6. The aspect ratios, AR, of the nozzles are AR = 1, 2 and 3, with AR = 1 obviously being a circular nozzle. The time, t, was nondimensionalized to t* = UNt/DN. Note that the range of ReN is limited in order to increase the dimensionless temporal resolution of the DPIV measurement. Table 3-2 summarizes all the variables used for the experiment on circular and elliptic vortex rings. 69    Chapter 3 Experimental Setup & Methodology  Table 3-2: Summary of the input variables used for circular and elliptic vortex rings experiment. Aspect Ratio, AR 1, 2, 3 Reynolds number (nozzle), ReN 1000, 1230, 1420, 1580, 1740 Stroke Ratio, LN/DN 1, 2, 3, 4, 5, 6 For the experiments on the interaction of vortex ring with a circular cylinder, three Reynolds number (nozzle) were used (ReN = 1000, 1420, 1740), with the stroke ratio (LN/DN) ranging from 1 to 3, since a single vortex ring is known to pinch off at stroke ratio of approximately 4 (see Gharib et. al., 1998). The cylinder diameters (dc) used to interact with the vortex ring range from dc = 0.18 mm to 1.86 mm. Table 3-3 shows the summary of the parameters used for this part of the experiment. Table 3-3: Summary of the input parameters used for vortex ring interaction with a cylinder experiment. Diameters of Cylinders, dc (mm) 0.18, 0.39, 0.92, 1.32, 1.86 Reynolds number (nozzle), ReN 1000, 1420, 1740 Stroke Ratio, LN/DN 1, 2, 3   70    Chapter 4 Results & Discussion  Chapter 4  Results & Discussion  In this chapter, the experimental results are presented and discussed. The first part of this chapter focuses on the formation and structure of circular vortex rings with emphasis on the vortex core structures. This is followed by the discussion on elliptic vortex rings, where some unexpected findings are reported. Finally, the interaction of a vortex ring with a circular cylinder is presented with emphasis mainly on visualization and interpretation of the cut-and-reconnection phenomena. 71    Chapter 4 Results & Discussion  4.1 Circular Vortex Rings A typical formation of a circular vortex ring captured in the present study is depicted by dye pattern in figure 4-1 for ReN = 1000 and LN/DN = 1. Figure 4-2 and 4-3 shows the corresponding vorticity field and streamline plot, respectively. In figure 4-1, as the slug of fluid is ejected through the nozzle, the vortex sheet is observed to emanate from the edge of the nozzle and roll up to form a spiral. The ejection of fluid from the nozzle stops at t* = 1.00, after which, the vortex ring is seen to continue its propagation while drawing more fluid from the surrounding as can be inferred from the dye pattern in figure 4-1(c) – (h). In figure 4-2, the corresponding vorticity field of the vortex ring is shown. The figure clearly shows the generation and influx of vorticity into the vortex core. In the formation process, vorticity with the opposite sense of rotation are generated at the nozzle wall and entrained by the vortex ring (see figure 4-2(d)). Due to the finite spatial resolution of the DPIV images, the vorticity plot does not appear to be very smooth. Nevertheless, it should be noted that the velocity vectors, from which the vorticity plot was derived, are accurate to the subpixel level (see Chapter 3.4.4). Figure 4-3 shows the instantaneous streamlines of the vortex ring and these streamline patterns appear to be consistent with that of Dabiri and Gharib (2004). It can be noted that the streamlines develop complicated patterns upstream of the vortex ring (i.e. near the right side of the image) after its formation (see figure 4-3(f) – (h)). This may be attributed to the existence of the vortices at the nozzle wall. 72    Chapter 4 Results & Discussion  y (a)  x x t* = 0.00  t* = 2.85 y (b)  y (f) x x t* = 0.71  t* = 3.57 y (c)  y (g)  x x t* = 1.43  t* = 4.28 y (d)  y (h)  x t* = 2.14  y (e) x t* = 5.71 Figure 4-1: Dye visualization of the formation of a circular vortex ring at ReN = 1000 and LN/DN = 1. 73    Chapter 4 Results & Discussion  y (a)  x x t* = 2.85 t* = 0.00  y (b)  y (f) x x t* = 3.57 t* = 0.71  y (c)  x t* = 1.43  (d)  Vorticity at nozzle x wall y (g)  x t* = 2.14  y (e) t* = 4.28 y y (h) x t* = 5.71 Figure 4-2: Vorticity field of the formation of a circular vortex ring at ReN = 1000 and LN/DN = 1. The scales on axes are in mm and the vorticity contours are in s-1. 74    Chapter 4 Results & Discussion  y (a)  y (e) x x t* = 0.00  t* = 2.85 y (b)  y (f) x x Complicated streamline patterns t* = 0.71  t* = 3.57 y (c)  x x t* = 1.43  t* = 4.28 y (d)  (h) x t* = 2.14  y (g)  y x t* = 5.71 Figure 4-3: Instantaneous streamlines of the formation of a circular vortex ring at ReN = 1000 and LN/DN = 1. The scales on the axes are in mm. 75    Chapter 4 Results & Discussion  After its formation, the circular vortex ring is known to gradually shed its vorticity into the wake (see Maxworthy, 1972). However, in figure 4-2, the small-scale vortical structures shed in the wake are not observed after the formation of the ring, mainly because these structures are smaller than the spatial resolution of the vector field. To determine if vorticity is shed in the wake as noted by Maxworthy (1972), circulation of the vortex ring was calculated at every timestep of the DPIV results until the vortex ring propagated out of the image window. The result is shown in figure 4-4 which depicts the circulation (Γ) against non-dimensional time (t*). The decrease in circulation with time after t* ≈ 8 suggests that some of the vorticity has been lost into the wake of the vortex ring. Also, prior to t* = 2, the circulation flux from the nozzle causes the circulation of the vortex ring to increase before reaching a “plateau” regime (i.e. almost constant circulation within 5% error) from t* = 2 to 8. This “plateau” regime is where the circulation, diameter and core size of the vortex ring are measured. These measured values will be discussed later in this chapter. In the next few subsections, the characteristics of the circular vortex ring will be compared and discussed with the literature. Finally, an analysis on the core of the vortex ring will be presented. 76    Chapter 4 Results & Discussion  9.0E‐04 8.0E‐04 7.0E‐04 Γ(m2/s) 6.0E‐04 5.0E‐04 4.0E‐04 3.0E‐04 2.0E‐04 1.0E‐04 0.0E+00 0 2 4 6 8 10 12 14 16 18 t* (=UNtt/DN)   Figure 4-4: Graph of circulation against time for vortex ring with ReN = 1000 and LN/DN = 1. 4.1.1 Formation of a Circular Vortex Ring The current experimental results on the formation of a circular vortex ring will be compared against the slug flow and self-similar models discussed in Chapter 2.1.1, together with experimental results from other researchers. The comparison is made in figure 4-5, which shows that the current experimental result is in good agreement with those of Didden (1979) and Lim et. al. (1992). The result also shows that the slugflow model consistently underestimates the actual circulation (i.e. Γ/Γslug > 1). As pointed out earlier in Chapter 2.1.1, the reason for this underestimate is due to the high circulation flux, generated by the peak velocity of the fluid slug near the edge of the nozzle exit at lower stroke ratios. Furthermore, since the current experimental data points in figure 4-5, regardless of ReN, are relatively close to each other for every stroke ratio (LN/DN), it suggests that Γ/Γslug is weakly dependent on ReN. 77    Chapter 4 Results & Discussion  2 1.8 1.6 1.4 Γ/Γslug 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 LN/DN Slug Model Didden (1979) Didden's curve fit Pullin (1979) Lim et. al. (1992) Maxworthy (1977) Figure 4-5: Reproduction of graph from figure 2-2 with results from current experiment. + ReN = 1000, - ReN = 1420 , x ReN = 1740. Although the circular vortex ring in the current experiment was generated with stroke ratios ranging from LN/DN = 1 to 6, it was found that a single vortex ring could only be formed at a stroke ratio of less than 4 (Details are presented in Appendix C for LN/DN = 2 to 5). When the vortex ring was generated at LN/DN = 4, it occasionally developed trailing vorticity behind the ring (see figure 4-6). This finding is consistent with that of Gharib et. al. (1998) where they highlighted that optimal vortex ring formation occurs at stroke ratio of approximately 4. In view of this, the subsequent sub-sections will focus on the characteristics of the vortex ring up to LN/DN = 3. 78    Chapter 4 Results & Discussion  y (a)  y (e) x x t* = 5.35 t* = 0.00  y (b)  y (f) x x t* = 6.07 t* = 1.43  y (c)  y (g)  x x t* = 2.50  t* = 7.85 (d)  y x t* = 4.64  Trailing vorticity (h) Trailing vorticity y x t* = 9.28 Figure 4-6: Vorticity field during the formation of a circular vortex ring at ReN = 1000 and LN/DN = 4. The scales on axes are in mm and the vorticity contours are in s-1. 79    Chapter 4 Results & Discussion  4.1.2 Variables Characterizing Circular Vortex Ring Didden (1979) pointed out that a vortex ring can be characterized by its propagation velocity (U), ring diameter (D), and circulation (Γ) (see Chapter 2.1.2). However, the circular vortex ring can also be characterized by its circulation (Γ), ring diameter (D) and core diameter (c), since U is dependent on Γ and D (see equation 2-10). Taking into account the kinematic viscosity of the fluid (ν), the characteristic variables of the vortex ring in dimensionless forms are represented as: Reynolds number (circulation): Γ ν  Dimensionless ring diameter: Dimensionless core diameter:  These characteristic variables are dependent on how the fluid slug is being ejected through the nozzle, namely, the velocity of the fluid slug (UN), and the length of the fluid slug (LN). They are represented in dimensionless form as: Reynolds number (nozzle):  Stroke ratio:  How the characteristic variables of the vortex ring and the fluid slug are related to each other will be discussed next. 80    Chapter 4 Results & Discussion  The relationship between ReΓ and ReN was carried out for different stroke ratios (LN/DN) and the result is shown in figure 4-7. From this graph, it is clear that there is a linear relationship between ReΓ and ReN, regardless of the stroke ratio. This linear relationship was reported by Adhikari and Lim (2009) for LN/DN = 1 only, and is consistent with the finding of Maxworthy (1977). From figure 4-7, the relationship between ReΓ and ReN can be expressed as:                                                     4‐1 where f1 is a function which represents the gradient of the lines, and is dependent on the stroke ratio (LN/DN). By plotting f1 against LN/DN as shown in figure 4-8, and linearly curve-fitting the data, equation 4-1 can be expressed as: 0.4 1                                            4‐2 As far as the author is aware, this relationship has not been reported in the literature. 81    Chapter 4 Results & Discussion  3000 2500 ReΓ 2000 1500 1000 500 0 0 500 1000 1500 2000 ReN Figure 4-7: Relationship between Reynolds number (circulation), ReΓ, and Reynolds number (nozzle), ReN for different stroke ratios, LN/DN. ○ LN/DN = 1, □ LN/DN = 2, ∆ LN/DN = 3. 1.8 1.6 1.4 1.2 f1 1 0.8 0.6 0.4 0.2 0 0.5 1 1.5 2 2.5 3 LN/DN Figure 4-8: Relationship between f1 and the stroke ratio, LN/DN. f1 represents the gradient of the lines in figure 4-7. 82    Chapter 4 Results & Discussion  The relationship between the dimensionless diameter of the vortex ring (D/DN) and the characteristics of the fluid slug is also investigated. The result is presented in figure 4-9, for different ReN, and it clearly shows that D/DN is relatively insensitive to ReN, but positively correlated to the stroke ratio, thus;                                                                   4‐3 By curve fitting the data with the best-fit power law, equation 4-3 gives: . 1.12                                                        4‐4 The above empirical equation agree well with the empirical equation derived by Maxworthy (1977) (see equation 2-9), despite the sharp nozzle edge used by Maxworthy, compare to a pipe-like nozzle used in this experiment. In addition to the ring diameter, the core diameter of the vortex ring (c), normalized by DN, was also obtained for different ReN and LN/DN. Figure 4-10 shows the relationship between the dimensionless core diameter (c/DN) and ReN, for different LN/DN. 83    Chapter 4 Results & Discussion  1.8 1.6 1.4 D/DN 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 LN/DN Figure 4-9: Relationship between dimensionless vortex ring diameter, D/DN, and the stroke ratio, LN/DN. Result of Maxworthy (1977) is depicted by broken line (----) while the solid line (──) represents the current experiment. ◊ ReN = 1000, □ ReN = 1230, ∆ ReN = 1420, x ReN = 1580, + ReN = 1740. 0.7 0.6 c/DN 0.5 0.4 0.3 0.2 0.1 0 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 ReN Figure 4-10: Relationship between core diameter (normalized by nozzle diameter), c/DN, and Reynolds number (nozzle), ReN for different stroke ratios, LN/DN. (○, ─) LN/DN = 1, (□, −·) LN/DN = 2, (∆, −−) LN/DN = 3. 84    Chapter 4 Results & Discussion  From the figure above, it can be seen that the dimensionless core diameter (c/DN), increases with the stroke ratio, due to higher vorticity flux from the nozzle. However, an interesting observation is that c/DN shows a decreasing trend as ReN increases, and this has never been reported before. With this finding, c/DN is obviously dependent on both ReN and LN/DN, which may be represented mathematically as: ,                                                  4‐5 However, the relationship of the characteristic variables shown in figure 4-10 is insufficient to form a generic empirical equation based on the function in equation 4-5. If the core diameter (c) is normalized with the ring diameter (D) instead of DN, the three curves, shown in figure 4-10, appear to converge to each other slightly (see figure 4-11). However, the convergence of the curves is again not enough to generalize the form into an empirical equation. 85    Chapter 4 Results & Discussion  0.5 0.45 0.4 0.35 c/D 0.3 0.25 0.2 0.15 0.1 0.05 0 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 ReN Figure 4-11: Relationship between dimensionless core diameter (normalized by ring diameter, D), c/D, and Reynolds number (nozzle), ReN for different stroke ratios, LN/DN. (○, ─) LN/DN = 1, (□, −·) LN/DN = 2, (∆, −−) LN/DN = 3. 4.1.3 Core Characteristics of Circular Vortex Ring In this sub-section, a further analysis of the core will be made by investigating the vorticity distribution within the core of the vortex ring. The vorticity distribution in vortex core has been modelled in many ways. In potential flow, vortex (also known as point vortex) is treated with a singular point of infinite vorticity while possessing a finite circulation. In basic fluid mechanics, a finite core size vortex is modelled as Rankine vortex where the core is treated as a solid-body rotation with constant vorticity. 86    Chapter 4 Results & Discussion  Batchelor (1965) described analytically that the core structure of a vortex ring can be assumed with several different vorticity profiles. He further elaborated that different assumed core profile can collectively become a family of vortex rings. However, it was Saffman (1970) who treated the core of a vortex ring in viscous fluid with a Gaussian profile that allowed him to formulate the velocity of the vortex ring in viscous fluid (see equation 2-10). Following Saffman’s model, Maxworthy (1972) carried out experimental measurements of vorticity on vortex core and roughly curvefitted the vorticity values, within the core, with a Gaussian profile. Since then, several computational works have assumed Gaussian vorticity profiles in the vortex core (see Orlandi and Vezzico, 1993; Chatelain et. al., 2003). In this experiment, the vorticity values measured within the vortex core also suggest a Gaussian distribution and figure 4-12 shows this very clearly. However, the relationship between the variables of the Gaussian function and the characteristic variables of the fluid slug has not been carried out experimentally before. Such a relationship will allow accurate initial conditions to be used in the simulation of vortex rings. In the next few paragraphs, the above mentioned relationship will be presented. 87    Chapter 4 Results & Discussion  Figure 4-12: Vorticity distribution within the core of the vortex ring of ReN = 1420, LN/DN = 1 with a Gaussian curve-fit. The figure is taken on the vortex core with positive vorticity value. A generic Gaussian profile with reference to the variables in figure 4-12 is represented by the following equation:                                                           4‐6 where ω is the vorticity value, y is the displacement from the core centre, A represents the peak vorticity and w is related to the width of the Gaussian curve. It should be noted that equation 4-6 will always result in peak vorticity occurring at y = 0 mm, instead of y = 37 mm as shown in figure 4-12. However, this will not affect the shape of the Gaussian curve. From equation 4-6, the parameters that essentially characterize the Gaussian curve are A and w, and these variables are plotted with the 88    Chapter 4 Results & Discussion  Reynolds number (nozzle), ReN and stroke ratio, LN/DN, as shown in figures 4-13 and 4-14. In figure 4-13, there is a clear indication that the peak vorticity value, A, increases with ReN. However, the relationship between A and the LN/DN is not distinct. Since the curves are relatively close to each other, this may suggest that the peak vorticity, A, is insensitive to the stroke ratios, LN/DN. 35 30 A (s-1) 25 20 15 10 5 0 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 ReN Figure 4-13: Relationship between A (see equation 4-6) and ReN, for different stroke ratios, LN/DN. A corresponds to the peak vorticity value. (○, ─) LN/DN = 1, (□, −·) LN/DN = 2, (∆, −−) LN/DN = 3. Figure 4-14 shows the relationship between w and ReN for different stroke ratios, and it is observed that there is a decay of w when ReN increases. It is also noted that w increases with the stroke ratio, LN/DN. This relationship is not surprising since w is closely related to the core diameter, c, of the vortex ring (refer to figure 4-10). 89    Chapter 4 Results & Discussion  4.5 4 3.5 w (mm) 3 2.5 2 1.5 1 0.5 0 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 ReN   Figure 4-14: Relationship between w (see equation 4-6) and ReN. w is related to the width of the Gaussian vorticity profile. (○, ─) LN/DN = 1, (□, −·) LN/DN = 2, (∆, −−) LN/DN = 3.   4.1.4 Section Summary In this section, the study of the characteristics of a circular vortex ring has been carried out and the results are consistent with those reported in the literature. The core diameter of the vortex ring was found to increase with the stroke ratio, LN/DN, but decrease with the increasing Reynolds number (nozzle), ReN. Detailed investigations revealed that the vortex core possess a Gaussian vorticity profile. This supports the postulation by Saffman (1970) that the vorticity distribution of a viscous vortex ring follows a Gaussian distribution. 90      Chapter 4 Results & Discussion 4.2 Elliptic Vortex Rings An elliptic vortex ring is known to possess a complex oscillatory deformation while propagating, which makes it fascinating to study as compared to a circular vortex ring (Arms and Hama, 1965; Dhanak and De Bernardinis, 1981; Hussain and Husain, 1989; Oshima et. al., 1988). In this section, the results of elliptic vortex rings with aspect ratio (AR) 2 and 3 are presented. Sequential visualization images and vorticity plots of a typical evolution of an elliptic vortex ring are given in figures 4-15 to 4-17 for AR = 2, and figures 4-19 to 4-21 for AR = 3. Each of the figures will be discussed in the subsequent paragraphs. In figure 4-15, the images of an elliptic vortex ring of AR = 2 are shown. From the figure, vortex sheet emanating from the nozzle is observed to roll up to form the ring, and this is similar to the observation for a circular vortex ring. However, unlike the circular ring, the elliptic ring deforms as it rolls up (see figure 4-15(b)), and subsequently it propagates in an oscillatory manner. In the process, the elliptic vortex ring undergoes axis-switching at half-cycle of the oscillation. The axis-switching occurs when the orientation of the major and minor axes of the elliptic vortex ring swap during its deformation. This oscillatory deformation and axis-switching can also be observed in the corresponding 2D vorticity plot which are obtained in the x-y plane (see figure 4-16) and x-z plane (see figure 4-17). A sequential schematic view of the ring in y-z plane is illustrated in figure 4-18 where the axis-switching can be seen at half-cycle of the oscillation. The oscillatory deformation and axis-switching observed in the current experiment is also consistent with the literature (Dhanak and De Bernardinis, 1981; Hussain and Husain, 1989). 91    Chapter 4 Results & Discussion y (a)  x x t* = 5.00 t * = 0.00  y (b)  y (f) x x t* = 6.42 t*  = 0.71  y (c)  y (g) x x t * = 2.14  t* = 7.85 y (d)  y (h) x t * = 3.57  y (e) x t* = 11.42 Figure 4-15: Dye visualization of the formation of an elliptic vortex ring of AR = 2 at ReN = 1000 and LN/DN = 1. 92    Chapter 4 Results & Discussion y (a)  x x t* = 5.00 t* = 0.00  y (b)  y (f) x x t* = 0.71  t* = 6.42 y (c)  y (g) x x t* = 2.14  t* = 7.85 y (d)  y (h) x t* = 3.57  y (e) x t* = 11.42 Figure 4-16: Vorticity plot of the formation of a elliptic vortex ring of AR = 2 at ReN = 1000 and LN/DN = 1 in the x-y plane. The scales on axes are in mm and the vorticity contour values are in s-1. 93    Chapter 4 Results & Discussion z (a)  x x t* = 0.00  t* = 3.57 z (b)  z (f) x x t* = 0.71  t* = 5.00 z (c)  z (g) x x t* = 6.42 t* = 1.43  z (d)  z (h) x t* = 2.14  z (e) x t* = 8.56 Figure 4-17: Vorticity plot of the formation of a elliptic vortex ring of AR = 2 at ReN = 1000 and LN/DN = 1 in the x-z plane. The scales on axes are in mm and the vorticity contour values are in s-1. 94    Chapter 4 Results & Discussion y (a)  z (b)  (c)  At half-cycle (d)  (e)  Figure 4-18: Schematic of elliptic vortex ring of AR = 2 in the y-z plane. The arrows represent the direction of the vortex lines. 95    Chapter 4 Results & Discussion Figure 4-19 shows the visual images of an elliptic vortex ring of AR = 3. Like AR = 2, the oscillatory deformation of the vortex ring is also observed here. However, in this case, the vortex ring is seen to deformed more severely as compared to that of AR = 2. Furthermore, as seen in figure 4-19(e), deformation of the vortex core also occurs. This core deformation can also be seen clearly in the corresponding vorticity field in the x-y plane (see figure 4-20(e)). In figure 4-21, the vorticity field of the vortex ring with AR = 3 in the x-z plane is shown. Here, a vortex pair appears between the vortex cores when the ring switches its axes (see figure 4-21(f)), and disappears before the vortex ring makes a complete oscillation (see figure 4-21(g)). As noted in Chapter 2.2.1, the occurrence of this vortex pair indicates that cross-linking of the cores has occurred in the elliptic vortex ring (Zhao and Shi, 1997). Furthermore, the disappearance of the vortex pair implies that the vortex ring did not bifurcate into two rings after cross-linking; instead, it only partially bifurcated. Partial bifurcation occurs when vortex cores of the initially major axis of the elliptic vortex ring touch each other and cross-link at the outer part of the adjacent ends of the cores, but does not split into two smaller rings (Oshima et. al., 1988). Therefore, from the results presented here, the elliptic vortex ring of AR = 3, ReN = 1000 and LN/DN = 1, is observed to undergo partial bifurcation. The schematic view of the vortex ring in the y-z plane is depicted in figure 4-22. 96    Ch hapter 4 Reesults & Disscussion y (a)  y (e) x x Deformaation of vortexx core t * = 0.00  t * = 6.78 y (b)  x x t * = 1.07  t * = 8.92 y (c)  y (g) x x 5 t * = 10.35 t * = 2.50  y (d)  y (h) x t * = 3.93  y (f) x t * = 11.78 8 Figure 4-19: Dye visuallization of the F t formatio on of an elliiptic vortex ring of AR R = 3 at R N = 1000 and Re a LN/DN = 1. 97    Chapter 4 Results & Discussion y (a)  y (e) x x Deformation of vortex core t * = 0.00  t * = 6.78 y (b)  x x t * = 8.92 t * = 1.07  y (c)  y (g) x x t * = 2.50  t * = 10.35 y (d)  y (h) x t * = 3.93  y (f) x t * = 12.49 Figure 4-20: Vorticity plot during the formation of an elliptic vortex ring of AR = 3 at ReN = 1000 and LN/DN = 1. The scales on axes are in mm and vorticity contour values are in s-1. 98    Chapter 4 Results & Discussion z (a)  z (e) x x t * = 0.00  t * = 6.07 z (b)  z (f) x x Vortex pair t * = 0.71  t * = 6.42 z (c)  x x t * = 2.50  t * = 7.85 z (d)  z (h) x t * = 5.00  z (g) x t * = 9.99 Figure 4-21: Vorticity plot of the formation of an elliptic vortex ring of AR = 3 at ReN = 1000 and LN/DN = 1. The scales on axes are in mm and vorticity contour values are in s-1. 99    Chapter 4 Results & Discussion y (a)  z (b)  (c)  At half-cycle Cross-linked vortices (d)  (e)  Figure 4-22: Schematic of an elliptic vortex ring of a) AR = 2 and b) AR = 3 at half-cycle of the oscillation in the y-z plane. The arrows represent the vortex lines. Shaded areas represent the cross-linking regions which appear as vortex pair in vorticity plot at x-z plane. 100    Chapter 4 Results & Discussion The elliptic vortex rings in the current experiments were generated at several Reynolds numbers (nozzle), ReN, and stroke ratios, LN/DN. However, at LN/DN = 2, the elliptic vortex rings for both aspects ratios showed distinct vortex shedding and instability. Further increase in the stroke ratio caused the vortex ring to breakdown into fine-scale structures. Therefore, in the subsequent sub-sections, the study on an isolated elliptic vortex ring will be carried out only for those generated with LN/DN = 1. The spatial and temporal trajectory of the elliptic vortex rings will be discussed initially. This will be followed by the discussion on the existence of stretching or compression of the vortex core when the ring propagates. The flow field of the elliptic vortex ring will then be discussed. Finally, in the last sub-section, the effects of higher stroke ratios (i.e. LN/DN ≥ 2) on the formation of the elliptic vortex ring will be analysed. 4.2.1 Trajectory of Elliptic Vortex Rings To date, the literature on the trajectory of an elliptic vortex ring is reported by Dhanak and De Bernardinis (1981), where they noted that elliptic vortex rings of small aspect ratios will undergo periodic oscillations. However, at larger aspect ratios, the oscillations are not periodic. In the current experiment, the trajectory of an elliptic vortex ring in space and time are extracted from the vorticity field. Figures 4-23 and 4-24 show the trajectory of vortex rings with AR = 2 and AR = 3, respectively. Each of the figures contains the (a) spatial and (b) temporal trajectory of the oscillation. The number of oscillation cycles obtained was limited by the largest possible image size employable using the DPIV system whilst retaining sufficiently high resolution. Note that the centre of the 101    Chapter 4 Results & Discussion nozzle exit is taken to be the origin and the dimensionless time is t* = 0.00 when the formation of the vortex ring begins. Figures 4-23(a) and (b) show the spatial and temporal trajectory of the elliptic vortex ring of AR = 2 for different ReN. From the figures, the periodic behaviour of the oscillation, as mentioned by Dhanak and De Bernardinis (1981), cannot be confirmed from this experiment as more than one oscillation is needed to conclusively verify such phenomena. However, another observation made from the figures is that, the trajectories remain the same for the different ReN used. This finding suggests that, the trajectory of elliptic vortex ring, with AR = 2, is insensitive to the change to ReN. For the case of AR = 3, the spatial and temporal trajectories (see figure 4-24(a) and (b)) show that at lower ReN (ReN ≤ 1230), the vortex ring switches axis at half-cycle and maintains in that manner for some time (i.e. a prolonged half-cycle), before deforming back to its initial form. However, for ReN ≥ 1420, the trajectory appears to remain the same regardless of the ReN used. Therefore, this raises the question on the cause of the prolonged half-cycle of the oscillation. To investigate such phenomenon, the vorticity plots of the elliptic vortex rings at ReN = 1230 and ReN = 1420 were analysed on the x-y and x-z plane (see figures 4-25 and 4-26). The Reynolds numbers (nozzle), ReN = 1230 and ReN = 1420 were chosen since they differ in their trajectory as shown in figure 4-24(a) and (b). 102    Chapter 4 Results & Discussion From figure 4-25, it can be seen that for ReN = 1230, the vortex ring, as to be expected, undergoes oscillatory deformation (see figure 4-25(a)). In addition, a vortex pair is also observed between the vortex core during the half-cycle, before disappearing, thus suggesting a partial bifurcation (see figure 4-25(b)). These observations are consistent with the case of ReN = 1000 described earlier (see figures 4-16 and 4-17). However, for the case of ReN = 1420, such appearance of the vortex pair cannot be seen (see figure 4-26). This suggests that the elliptic vortex ring of AR = 3 and ReN = 1420 does not undergo partial bifurcation. From these observations, the schematic interpretations of the elliptic vortex rings of AR = 3 for both ReN = 1230 and 1420, at half-cycle of the oscillation, in the y-z plane are shown in figure 4-27. This finding shows that the “prolonged half-cycle of the oscillation” in figures 4-24(a) and (b) can be attributed to the existence of partial bifurcation. It suggests that the cross-linked vortices which occur during the bifurcation may be the cause of the delay in axis-switching process as seen in figures 4-24(a) and (b) for ReN = 1230. Furthermore, such cross-linked vortices are not observed in figure 4-26(b) and correspondingly, delay in axis-switching does not occur for ReN = 1420. The reader may also refer to Appendix D for further results of ReN = 1580 and 1740. Another finding which can be made from this analysis is that, partial bifurcation for elliptic vortex ring of AR = 3 only occur for ReN ≤ 1230. This finding has not been previously reported. 103    Chapter 4 Results & Discussion 1 0.8 0.6 0.4 y/DN 0.2 0 ‐0.2 0 0.5 1 1.5 2 2.5 3 3.5 ‐0.4 ‐0.6 ‐0.8 ‐1 x/DN Figure 4-23(a): The spatial trajectory of an elliptic vortex ring with AR = 2 for all Reynolds number (nozzle). ◊ ReN = 1000, □ ReN = 1230, ∆ ReN = 1420, x ReN = 1580, + ReN = 1740. 1 0.8 0.6 0.4 y/DN 0.2 0 ‐0.2 0 2 4 6 8 10 12 ‐0.4 ‐0.6 ‐0.8 ‐1 t* (=UNt/DN) Figure 4-23(b): The temporal trajectory of an elliptic vortex ring with AR = 2 for all Reynolds number (nozzle). ◊ ReN = 1000, □ ReN = 1230, ∆ ReN = 1420, x ReN = 1580, + ReN = 1740. 104    Chapter 4 Results & Discussion 1 0.8 0.6 Prolonged half-cycle of the oscillation 0.4 y/DN 0.2 0 ‐0.2 0 0.5 1 1.5 2 2.5 3 3.5 ‐0.4 ‐0.6 ‐0.8 ‐1 x/DN Figure 4-24(a): The spatial trajectory of an elliptic vortex ring with AR = 3 for all Reynolds number (nozzle). ◊ ReN = 1000, □ ReN = 1230, ∆ ReN = 1420, x ReN = 1580, + ReN = 1740. 1 0.8 0.6 Prolonged half-cycle of the oscillation 0.4 y/DN 0.2 0 ‐0.2 0 2 4 6 8 10 12 ‐0.4 ‐0.6 ‐0.8 ‐1 t* (=UNt/DN) Figure 4-24(b): The temporal trajectory of an elliptic vortex ring with AR = 3 for all Reynolds number (nozzle). ◊ ReN = 1000, □ ReN = 1230, ∆ ReN = 1420, x ReN = 1580, + ReN = 1740. 105    Chapter 4 Results & Discussion (a) y (i)  (b) z (i)  x x t* = 1.31 t* = 1.31 y (ii)  z (ii)  x x t* = 5.68 t* = 5.68 y (iii)  z (iii)  x x Cross-linked vortex pair t* = 6.99 t* = 6.99 y (iv)  x t* = 10.92 z (iv)  x t* = 10.92 Figure 4-25: Vorticity plot of an elliptic vortex ring with AR = 3 for ReN = 1230 in a) x-y and b) x-z plane. The scales on axes are in mm and vorticity contour values are in s-1. 106    Chapter 4 Results & Discussion (a) y (i)  (b) x x t* = 1.51  t* = 1.51 y (ii)  z (ii)  x x t* = 4.04  t* = 4.04 y (iii)  z (iii)  x x t* = 7.06  t* = 7.06 y (iv)  z (iv)  x t* = 10.09 z (i)  x t* = 10.09 Figure 4-26: Vorticity plot of an elliptic vortex ring with AR = 3 for ReN = 1420 in a) x-y and b) x-z plane. The scales on axes are in mm and vorticity contour values are in s-1. 107    Chapter 4 Results & Discussion y z (a)  (b)  Figure 4-27: Schematic of the elliptic vortex ring of a) ReN = 1230 and b) ReN = 1420 in y-z plane. The arrows represent the direction of vortex lines. 4.2.2 Stretching of the Vortex Core7 Stretching on the core of an elliptic vortex ring has been investigated theoretically by Arms and Hama (1965). In their paper, they derived the LIE (see equation 2-12) which was further formulated to show that stretching does not occur at any point on the vortex ring filament. This implies that the filament length of the vortex ring remains constant. However, in the derivation of the LIE, the long distance effect of velocity on the filament was neglected and the core size was assumed to be small (see Chapter 2.2.1). To date, no experiment has been reported to verify that vortex stretching is, in fact, absent when an elliptic vortex ring undergoes oscillatory motion. The stretching of the vortex core in this experiment can be interpreted from the 2-D DPIV measurement. A method to interpret existence of vortex stretching is provided by Perry and Chong (2000) who noted that in order to assess whether vortex is undergoing stretching or compression orthogonally in a 2-dimensional streamline,                                                               The findings in this sub-section were presented by the author in the 7th EUROMECH Fluid Mechanics Conference 2008 at University of Manchester, U.K. 7 108    Chapter 4 Results & Discussion there is a need to determine whether the streamline is a stable or unstable focus. For the observer moving in the frame of the translating vortex core, if the core exhibits a stable focus, then the vortex is stretching. If it is an unstable focus, the vortex core is compressing. However, in the cases studied here, the DPIV camera did not move with the vortex ring, and thus the method outlined by Perry and Chong (2000) to interpret stretching or compression is not feasible. Another method often used to find the occurrence of stretching is by locating the peak vorticity value during the propagation of the vortex ring. An increase in the absolute peak value implies stretching of the vortex core, while a decrease in the absolute peak value implies compression. Since the peak value in the core of the vortex ring gradually decreases over time due to viscous effects, the existence of stretching or compression is identified only by a local rise or drop in the vorticity value, respectively. Figure 4-28(a) and (b) show the maximum and minimum peak vorticity against time for ReN = 1000 and 1740, respectively. From the figure 4-28(a) and (b), it can be seen that the absolute peak vorticity value of the circular vortex ring and elliptic vortex rings (of AR = 2 and AR = 3) decreases monotonically for the two extreme ReN values used in the experiment. Since the decrease in vorticity is monotonic, this observation is interpreted to be the result of only viscous decay. Therefore, the elliptic vortex rings of AR = 2 and 3 neither stretch nor compress during their deformation for the different Reynolds numbers (nozzle), ReN, and stroke ratios, LN/DN, used in this experiment. This deduction supports the theoretical findings of Arms and Hama (1965). 109    Chapter 4 Results & Discussion 30 20 ω (s-1) 10 0 0 1 2 3 4 5 6 7 8 9 10 ‐10 ‐20 ‐30 ‐40 UNt/DN Figure 4-28(a): Temporal variation of peak vorticity value of a vortex ring and the time for ReN = 1000. ◊ Circular ring, □ AR = 2 (x-y plane), ∆ AR = 2 (x-z plane), x AR = 3 (x-y plane), + AR = 3 (x-z plane). 60 40 ω (s-1) 20 0 0 1 2 3 4 5 6 7 8 9 10 ‐20 ‐40 ‐60 ‐80 UNt/DN Figure 4-28(b): Temporal variation of peak vorticity value of a vortex ring and the time for ReN = 1740. ◊ Circular ring, □ AR = 2 (x-y plane), ∆ AR = 2 (x-z plane), x AR = 3 (x-y plane), + AR = 3 (x-z plane). 110    Chapter 4 Results & Discussion 4.2.3 Flow Field of Elliptic Vortex Rings In Chapter 4.1, the streamline plot of the circular vortex ring was discussed and the plot showed complicated patterns upstream of the ring after its formation. This streamline plot is reproduced in figure 4-29. Unlike the circular vortex ring, the streamlines patterns of the elliptic vortex rings are observed to form critical points. The nature of these critical points will be discussed in the next few paragraphs. In figure 4-30, streamlines patterns of the vortex ring of AR = 2 are shown. In both cases, a distinct saddle node is seen to form upstream of the ring, and sustains itself even when the elliptic ring reaches the left end of the image window (see figures 430a(iv) and 4-30b(iv)). In figure 4-31, streamline patterns of the vortex ring of AR = 3 are shown in the a) x-y and b) x-z plane. In this case, a saddle node is seen to form upstream of the ring in the x-y plane, and an unstable node forms in the x-z plane. An unstable node is interpreted as orthogonal motion of fluid with respect to the 2-D plane (Perry and Chong, 2000). The formation of the critical points (or nodes), discussed above, is not observed for circular vortex rings (see figure 4-29). Furthermore, such observations for elliptic vortex rings have not been reported before. However, the causes of these critical points and their influences cannot be fully explained from the streamline plot, and thus, are not feasible to be analysed further. 111    Chapter 4 Results & Discussion y (a)  y (e) x x t* = 0.00  t* = 2.85 y (b)  y (f) x x t* = 0.71  t* = 3.57 y (c)  y (g)  x x t* = 1.43  t* = 4.28 y (d)  (h) x y x Complicated streamline patterns t* = 2.14  t* = 5.71 Figure 4-29: Instantaneous streamlines during the formation of a circular vortex ring at ReN = 1000 and LN/DN = 1. The scales on the axes are in mm. This figure is reproduced from figure 4-3. 112    Chapter 4 Results & Discussion (a)  y (b)  (i)  x t* = 0.72  y x t* = 2.85  t* = 8.56  z (ii)  x t* = 2.85  Saddle node y x (iii)  z Saddle node x t* = 5.71  (iv)  x t* = 0.72  (ii)  (iii)  z (i)  t* = 5.71  y z (iv)  x x t* = 8.56  Figure 4-30: The instantaneous streamlines of an elliptic vortex ring of AR = 2. (a) x-y and (b) x-z plane. The scales on axes are in mm. 113    Chapter 4 Results & Discussion (a)  y (b)  (i)  z (i)  x x t* = 0.72  t* = 0.72  y (ii)  z (ii)  x x t* = 2.85  (iii)  t* = 2.85  Saddle node y (iii)  x t* = 5.71  t* = 5.71  y (iv)  z (iv)  x t* = 9.99  z Unstable node x x t* = 9.99  Figure 4-31: The instantaneous streamlines of an elliptic vortex ring of AR = 3. (a) x-y and (b) x-z plane. The scales on axes are in mm. 114    Chapter 4 Results & Discussion 4.2.4 Effects of Higher Stroke Ratios (LN/DN ≥ 2) The elliptic vortex rings discussed thus far are generated with LN/DN = 1. As mentioned earlier, for elliptic vortex rings with LN/DN ≥ 2, the vortex rings undergo instability fairly quickly and eventually break down to fine-scale structures at a very early stage. In this sub-section, the effects of higher stroke ratios are discussed and explanation of such instabilities and breakdown are elaborated. Figure 4-32(a) and (b) show the evolution of elliptic vortex ring of AR = 2 and AR = 3, respectively, for ReN = 1740, LN/DN = 2. The elliptic vortex ring is observed to propagate with similar oscillatory deformation as that of LN/DN = 1. However, in this case, the vortex ring develops instability within the first cycle of the oscillation (see figure 4-32a(iv) and 4-32b(iv)). From the visualization, the occurrence of the instability may be attributed to the formation and entrainment of the “flow structure 1” and “flow structures 2” into the core of the vortex ring (see figure 4-32a(iii) and 4-32b(iii)). During the analysis of the results, these flow structures were observed for all ReN with LN/DN = 2, but the instability was only seen at higher ReN (ReN > 1000). At lower ReN, the elliptic vortex ring decayed and the dye appeared to be “fossilized” after the first cycle of the oscillation. From the visualization images in figure 4-32, the “flow structure 1” seems to have developed from the vortex sheet which is generated at the nozzle. This vortex sheet appears to pass through the ring and form “flow structure 1” downstream of the ring. “Flow structures 2”, on the other hand, appear to resemble streamwise vortices which are developed from the jet upstream of the ring. However, in order to confirm these 115    Chapter 4 Results & Discussion observations and understand the development of flow structures 1 and 2, the DPIV flow fields will be examined. Figure 4-33 and 4-34 show the vorticity plots of the elliptic vortex ring of AR = 2 and 3, respectively. When these plots are observed in the x-y plane (see figures 4-33(a) and 4-34(a)), vortices are not seen downstream of the vortex ring. However, in the x-z plane (see figures 4-33(b) and 4-34(b)), a vortex pair is observed downstream of the vortex ring which appears to have evolved from the vortex sheet emanating from the nozzle. With this observation, “flow structure 1” (observed in figures 4-32) is essentially a vortex pair, which appears to have developed from the “excess” vortex sheet moving downstream of the vortex ring in the x-z plane.   116    Chapter 4 Results & Discussion (a) y (i)  Vortex sheet (b) y (i)  x x Vortex sheet t* = 2.77  t* = 2.77 y (ii)  y (ii)  x x Flow structure 1 Flow structure 1 t* = 3.95  Flow structures 2 y (iii)  t* = 3.95 x Entrainment of the flow structures t* = 7.91 y (iv)  Entrainment of the flow structures y (iv)  x x Wavy Instability t* = 9.89  y (iii)  x t* = 6.53  Flow structures 2 Wavy Instability t* = 11.86   Figure 4-32: Evolution of an elliptic vortex ring at ReN = 1740, LN/DN = 2. (a) AR = 2 and   (b) AR = 3. 117    Chapter 4 Results & Discussion (a)  y (i)  (b)  z (i)  x x t* = 1.24 t* = 1.24 y (ii)  z (ii)  Vortex  sheet  x t* = 3.09 x t* = 3.09 y (iii)  z (iii)  x x Vortex  pair  t* = 5.56 t* = 5.56 y (iv)  x t* = 8.65 z (iv)  x t* = 8.65 Figure 4-33: Vorticity field of an elliptic vortex ring of AR = 2, ReN = 1740 and LN/DN = 2. (a) x-y and (b) x-z plane. The scales on axes are in mm and the vorticity contours are in s-1. 118    Chapter 4 Results & Discussion (a)  y (i)  (b)  z (i)  x x t* = 1.24 t* = 1.24 y (ii)  z (ii)  x Vortex  sheet  t* = 3.09 x t* = 3.09 y (iii)  z (iii)  x x Vortex  pair  t* = 5.56 t* = 5.56 y (iv)  t* = 8.65 z (iv)  x x t* = 8.65 Figure 4-34: Vorticity field of an elliptic vortex ring of of AR = 3, ReN = 1740 and LN/DN = 2. (a) x-y and (b) x-z plane. The scales on axes are in mm and the vorticity contours are in s-1.   119    Chapter 4 Results & Discussion In addition to the vorticity field shown in figure 4-33 and 4-34, the corresponding streamline plots are presented in figure 4-35 and 4-36 for elliptic vortex ring of AR = 2 and 3, respectively. In figure 4-35, the vortex ring of AR = 2 is seen to evolve and form bifurcation lines (see figures 4-35a(iii) and 4-35b(iii)), which are essentially asymptotic lines in a flow field where the streamlines either converge or diverge (Perry and Chong, 2000). In fact, figure 4-35(a) reveals the existence of three bifurcation lines; two of them near the core of the ring, where the streamlines converge and the third, between the cores, where the streamlines diverge. However, in figure 4-35(b), there is only one bifurcation line between the cores, where the streamlines converge. Likewise, the streamline patterns of the vortex ring of AR = 3 (see figure 4-36) also display bifurcation lines. Three of such lines are observed in figure 4-36(a) and one in figure 4-36(b). This existence of bifurcation lines upstream of the elliptic vortex rings has not been previously reported. Perry and Chong (2000) highlighted that when bifurcation lines are adjacent to each other in a streamline plot, it suggests the existence of streamwise vortices. In figures 4-35(a) and 4-36(a), the three adjacent bifurcation lines, therefore, imply the occurrence of the streamwise vortices. This is consistent with the earlier observations in figure 4-32 that, the “flow structures 2” are streamwise vortices. 120    Chapter 4 Results & Discussion (a)  y (i)  (b)  z (i)  x x t* = 0.72  t* = 0.72  y (ii)  x t* = 2.85  z (ii)  x t* = 2.85  y (iii)  Bifurcation lines x z (iii)  t* = 5.71  Bifurcation line x t* = 5.71  y (iv)  t* = 8.56  x z (iv)  x t* = 8.56  Figure 4-35: Instantaneous streamlies of an elliptic vortex ring of AR = 2, ReN = 1740 and LN/DN = 2. (a) x-y and (b) x-z plane. The scales on axes are in mm.     121  Chapter 4 Results & Discussion (a)  y (b)  (i)  z (i)  x x t* = 0.72  t* = 0.72  y (ii)  x x t* = 2.85  (iii)  z (ii)  t* = 2.85  y Bifurcation lines (iii)  x z Bifurcation line x t* = 5.71  t* = 5.71  y (iv)  t* = 8.56  x z (iv)  x t* = 8.56  Figure 4-36: Instantaneous streamlines of an elliptic vortex ring of AR = 3, ReN = 1740 and LN/DN = 2. (a) x-y and (b) x-z plane. The scales on axes are in mm.   122    Chapter 4 Results & Discussion It has now been established that the “flow structure 1” refers to the vortex pair which develops downstream of the ring, and “flow structures 2” are the streamwise vortices upstream of the vortex ring. The vortex pair and the streamwise vortices eventually get entrained into the core of the elliptic vortex ring of LN/DN = 2, causing flow instability (see figure 4-32a(iv) & 4-32b(iv)). Notice that the vorticity direction of the streamwise vortices (x-direction) and vortex pair (y-direction) are orthogonal to each other, resulting in a rather complex interaction when they are entrained by the vortex core. Details of this complex interaction are beyond the scope of this thesis. However, in an attempt to understand how these flow structures may develop further, elliptic vortex ring of LN/DN ≥ 3 will be discussed next. Figure 4-37 shows the evolution of elliptic vortex ring at ReN = 1000, LN/DN = 3. The development of the vortex pair and the streamwise vortices are also evident here. They are also observed to be entrained by the core of the vortex ring before undergoing instability and developing fine-scale structures in the process. Further observation of elliptic vortex rings at ReN = 1000, LN/DN = 4 and 6 (see figures 4-38 and 4-39) also show the existence of vortex pairs and the streamwise vortices as described earlier. 123    Chapter 4 Results & Discussion (a)  y (i)  (b)  x x t* = 2.74  t* = 3.08 y (ii)  y (ii)  x x Vortex pair Vortex pair t* = 4.11  y (i)  Streamwise vortices t* = 5.14 y (iii)  Streamwise vortices y (iii)  x x t* = 6.17  t* = 6.51 y (iv)  y (iv)  x x Entrainment of vortical fluid t* = 8.56  t* = 10.28   Figure 4-37: Evolution of an elliptic vortex ring at Re = 1000, L /D = 3. (a) AR = 2 and   N N N (b) AR = 3. 124    Chapter 4 Results & Discussion (a)  y (i)  (b)  y (i)  x x t* = 3.08  t* = 3.43 y (ii)  y (ii)  x x Vortex pair Vortex pair t* = 4.57  Streamwise vortices t* = 5.71 y (iii)  y (iii)  x x t* = 5.71  t* = 6.62 y (iv)  y (iv)  x t* = 9.14  Streamwise vortices x t* = 9.82   Figure 4-38: Evolution of an elliptic vortex ring at ReN = 1000, LN/DN = 4. (a) AR = 2 and (b) AR = 3. 125    Chapter 4 Results & Discussion y (i)  (a)  (b)  y (i)  x x Vortex pair Vortex pair Streamwise vortices t* = 3.88  t* = 3.77 y (ii)  x t* = 5.02  t* = 5.71 y (iii)  x Secondary vortex ring t* = 6.85 y (iv)  y (iii)  x y (iv)  x x t* = 8.22  y (ii)  x t* = 6.51  Streamwise vortices t* = 9.14   Figure 4-39: Evolution of an elliptic vortex ring at ReN = 1000, LN/DN = 6. (a) AR = 2 and (b) AR = 3. 126    Chapter 4 Results & Discussion The formation of elliptic vortex ring at high stroke ratios resembles that of a starting elliptic jet. In elliptic jets, the development of the streamwise vortices (or ‘ribs’) has been discussed by Husain and Hussain (1993). They noted that the formation of these ribs is due to the deformation of the elliptic vortex filament braids in a streamwise manner (see Chapter 2.2.2). These filament braids are formed in consecutive series within the jet. The development of the ribs proposed by Husain and Hussain (1993) can also be used to explain the existence of the streamwise vortices in elliptic vortex rings at high stroke ratios in the current experiment. Considering an instant in figure 4-39, it can be observed that the formation of the elliptic vortex ring is followed by a jet upstream, which is seen to roll up into streamwise vortices. Using the explanation proposed by Husain and Hussain (1993), if the stream of jet upstream is treated as a series of vortex filament braids (see figure 4-40), the induced velocities amongst the vortex ring and braids cause the latter to deform into the streamwise vortices. The streamwise vortices then get entrained into the core, causing complex interactions (see figure 4-40(d)). The deformation of the vortex braid can be observed more clearly in the visualization at higher Reynolds number (nozzle) of ReN = 1740 (see figure 4-41). Here, secondary vortex rings are observed to form in the trailing jet, upstream to the primary ring, due to the Kelvin-Helmholtz-like instability (Lim, 1997(b)). These secondary vortex rings behave in the same way as the braids; they deform into the streamwise vortices in the manner illustrated in the previous paragraph (see figure 4-41a(iii) and 4-41b(iii)). 127    Chapter 4 Results & Discussion Another separate observation made at ReN = 1740 is the development of “arc-like” structure downstream of the vortex ring (see figure 4-41a(iv) and 4-41b(iv)). In order to investigate this structure, the vorticity plots of vortex ring of AR = 2 and 3 (see figures 4-42 and 4-43) were examined. In the figures, the vorticity plot in the x-y plane (see figures 4-42(a) and 4-43(a)) does not show any structures forming downstream of the vortex ring. However, in the x-z plane (see figures 4-42(b) and 443(b), a vortex pair is observed to form downstream of the ring through a “leapfrogging-like” mechanism. With this observation, it can be inferred that this “arc-like” structure is the vortex pair which does not get entrained by the vortex ring; instead, it travels downstream of the ring through a leap-frogging-like mechanism. 128    Chapter 4 Results & Discussion y (a)  x   nozzle t* = 0.91  y (b)  x t* = 1.94  y (c)  x Vortex filament braids t* = 3.08  y (d)  x t* = 3.88  Streamwise vortices Figure 4-40: Interpretation on the development of the streamwise vortices for an elliptic vortex ring of AR = 2, ReN = 1000, LN/DN = 6. 129    Chapter 4 Results & Discussion (a)  y (i)  (b)  x x t* = 2.97  t* = 3.36 y (ii)  x Secondary vortex ring Secondary vortex ring t* = 3.95  t* = 4.55 y (iii)  x Deformed into streamwise vortices y (iv)  y (iii)  x t* = 5.94 Deformed into streamwise vortices y (iv)  x x “arc-like” structure t* = 7.32  y (ii)  x t* = 4.94  y (i)  t* = 7.52 “arc-like” structure   Figure 4-41: Evolution of an elliptic vortex ring at ReN = 1740, LN/DN = 6. (a) AR = 2 and (b) AR = 3. 130    Chapter 4 Results & Discussion (a)  y (i)  (b)  x x t* = 0.62 t* = 0.62 y (ii)  x t* = 3.09 t* = 3.09 y (iii)  x t* = 4.32 y x   (iii) Leap-frogging-like x mechanism z t* = 4.32 (iv)    z (ii)  x t* = 5.56 z (i)  (iv)  “Arc-like” structure x z t* = 5.56 Figure 4-42: Vorticity plot of an elliptic vortex ring of AR = 2, ReN = 1740 and LN/DN = 6. (a) x-y and (b) x-z plane. The scales on axes are in mm and the vorticity contour values are in s-1. 131    Chapter 4 Results & Discussion (a)  y (i)  (b)  z (i)  x x t* = 0.62  t* = 0.62 y (ii)  z (ii)  x x t* = 3.09  t* = 3.09 y (iii)  (iii)  x t* = 4.94  “Arc-like” structure x t* = 4.94 y (iv)      z (iv)  x t* = 7.42  z x t* = 7.42 Figure 4-43: Vorticity plot of an elliptic vortex ring of AR = 3, ReN = 1740 and LN/DN = 6. (a) x-y and (b) x-z plane. The scales on axes are in mm and the vorticity contour values are in s-1. 132  Chapter 4 Results & Discussion 4.2.5 Section Summary In this section, the trajectory of the elliptic vortex rings of AR = 2 and 3 have been examined. For AR = 3, the trajectory showed deviation at lower ReN (ReN ≤ 1230) due to cross-linking of vortex cores during partial bifurcation. Also, the stretching and compression of vortex core of the elliptic vortex ring was not observed, which supports the earlier theoretical analysis by Arms and Hama (1965). Furthermore, critical nodes upstream of the ring were found to exist in streamline plot of the elliptic ring. These new findings have not been reported in the literature. The elliptic vortex ring at high stroke ratios (i.e. LN/DN ≥ 2) showed the development of the streamwise vortices and vortex pair. These structures may have a role in enhancing the instability and breakdown of the elliptic vortex ring. At even higher stroke ratios (LN/DN ≥ 4), the vortex pair formed an “arc-like” structure downstream of the vortex ring via a leap-frogging-like mechanism. While the occurrence of streamwise vortices has been noted by Husain and Hussain (1993) in their work on elliptic jets, the observation of the vortex pair downstream of the vortex ring has not been previously reported. 133    Chapter 4 Results & Discussion 4.3 Interaction of a Vortex Ring with a Circular Cylinder This part of the investigation arises from an earlier investigation that the author conducted on the impact of a vortex ring on a porous screen (Adhikari and Lim, 2009). In the study, a vortex ring at high Reynolds number impacting on the porous screen was found to pass through the screen via the mechanism of cut-and-reconnection. Here, the interaction of the vortex ring with a circular cylinder is studied in order to better understand this mechanism. Figures 44 to 46 show the interaction of a vortex ring with a circular cylinder of diameter, dc = 0.18 mm. The vortex rings are generated with ReN = 1000, 1420 and 1740, and stroke ratio of LN/DN = 1. In all the figures, the vortex rings are seen to pass through the cylinder and very light traces of dye, depicting vorticity, are observed in the wake of the ring. With these observations, it can be noted that varying ReN from 1000 to 1740 does not appear to significantly affect the behaviour of the vortex ring when it interacts with the cylinder (Interested readers are referred to Appendix E for further results showing similarities in interaction for other cylinder diameters). In view of this, the discussion on vortex ring/cylinder interaction is carried out with an arbitrarily chosen ReN (= 1420). As mentioned above, the limited range of ReN used in this experiment does not show significant difference in the vortex ring/cylinder interactions. Since the core diameter is dependent on ReN, this implies that there is a range of core sizes where similar phenomenon occurs during the interaction. Due to the existence of this range, the cylinder diameter (dc) is not normalized with the core diameter (c), and is taken in its 134    Chapter 4 Results & Discussion dimensional form in this study, although a related study by Naitoh et. al. (1995) used the core diameter as the normalizing parameter for dc (see Chapter 2.3.1). In the discussion to follow, the cut-and-reconnection process of a vortex ring of ReN = 1420, interacting with varying diameters of cylinders will be discussed first. This will be followed by the illustration on the effects of varying LN/DN on the interaction of the vortex ring with the cylinders. 4.3.1 Cut-and-Reconnection Process Figures 4-45 and 4-47 show the interaction of vortex ring with cylinder of diameter, dc = 0.18 mm and 0.39 mm, respectively. When a vortex ring interacts with these cylinders, the vortex core near the cylinder slows down while the core away from the cylinder continues to propagate forward. During this process, axial flow along the core is observed as a result of the pressure difference between the core near the cylinder and far away from the cylinder (sees figures 4-45(d) and 4-47(d)), thus, causing the core to stretch. The stretched vortex core then ‘cuts’ through the cylinder and ‘reconnects’ immediately after passing through the cylinder. The author’s interpretation of the cut-and-reconnection mechanism is illustrated in figure 4-48. The interpretation is similar to that proposed by Adhikari and Lim (2009), and is based on the mechanism recorded by Kida and Takaoka (1994). As depicted in figure 4-48, when the vortex ring interacts with the cylinder, the parts of the vortex core which are obstructed by the wires are stretched locally (see figure 4-48(b)). During the stretching, vorticity lines of opposite sign come into contact with each other, resulting in vorticity cancellation through viscous diffusion (see figure 4-48(c)). Consequently, 135    Chapter 4 Results & Discussion they are connected to their counterpart on the other side of the cylinder. Because of a lower angular velocity near the interaction area relative to the rest of the flow, the vorticity lines are entangled and subsequently forms cross-linking vortices (or bridge) as shown in figure 4-48(d) (see Melander and Hussain, 1989; Kida et al., 1991; Kida and Takaoka, 1994). As the rest of the vortex filament continues to convect away from the cylinder, the bridge is stretched until the vorticity is dissipated through viscous diffusion, and the flow relaxes to a reasonably stable, although slightly distorted vortex ring. From the visualization images (see figures 4-45 and 4-47), it can be observed that after the vortex ring interacts with the cylinder, there are secondary vortices generated at the lee-side of the cylinder. These secondary vortices may not be apparent from the visual images in figure 4-45, however the vorticity plot, under the same conditions, gives a clearer picture of these vortices present (see figure 4-49). The development of these secondary vortices is due to the shear generated when vortex ring passes through the cylinder, and is consistent with the observation by Naitoh et. al. (1995). The secondary vortices, after their formation, move around the core due to the induced velocity. Eventually, the secondary vortices induce vorticity of opposite sign from the core to form a vortex pair and convect away from the vortex ring (see figure 4-47(h)). After which, the ring is observed to propagate with a reduced diameter. This reduction in the diameter may be a manifestation of the transformation into an elliptic vortex ring observed by Naitoh et. al. (1995). 136    Chapter 4 Results & Discussion y (a)  y (e)  x x cylinder t* = 0.00  t* = 14.62  y (b)  x x t* = 5.02  t* = 17.81  y (c)  y (g)  x x t* = 6.85  t* = 21.13  y (d)  y (h)  x t* = 11.42  y (f)  x t* = 26.04  Figure 4-44: Interaction of a vortex ring with a cylinder of dc = 0.18mm at ReN = 1000 and LN/DN = 1. 137    Chapter 4 Results & Discussion y (a)  y (e)  x x cylinder t* = 0.00  t* = 12.92  y (b)  y (f)  x x t* = 4.84  t* = 16.15  y (c)  y (g)  x x t* = 5.98  t* = 17.76  y (d)  y (h)  x x Axial flow t* = 10.18  t* = 21.16  Figure 4-45: Interaction of a vortex ring with cylinder of dc = 0.18mm at ReN = 1420 and LN/DN = 1. 138    Chapter 4 Results & Discussion y (a)  y (e)  x x cylinder t* = 0.00  t* = 10.48  y (b)  x x t* = 3.96  t* = 12.86  y (c)  y (g)  x x t* = 5.14  t* = 14.44  y (d)  y (h)  x t* = 7.91  y (f)  x t* = 17.80  Figure 4-46: Interaction of a vortex ring with a cylinder of dc = 0.18mm at ReN = 1740 and LN/DN = 1. 139    Chapter 4 Results & Discussion y (a)  cylinder y (e)  x x t* = 0.00  t* = 9.69  y (b)  y (f)  x x t* = 3.23  t* = 12.11  y (c)  y (g)  x x Vortex bridge Induced vortices from vortex core t* = 4.84  t* = 13.72  y (d)  y (h)  x x Vortex pairs Axial flow t* = 7.27  t* = 16.93  Figure 4-47: Interaction of a vortex ring with a cylinder of dc = 0.39mm at ReN = 1420 and LN/DN = 1. 140    Chapter 4 Results & Discussion (a)  cylinder Direction of propagation (b)  (c)  Vorticity cancellation Vortex bridge (d)  Figure 4-48: Author’s interpretation of the cut-and-reconnection mechanism for a vortex ring interacting with a cylinder. This mechanism is similar to that reported by Adhikari and Lim (2009). The arrow heads depict the direction of vorticity on half of the ring that is in the 141  foreground.   Chapter 4 Results & Discussion y (a)  cylinder x x t* = 0.00  t* = 10.60  y (b)  Secondary vortices y (f)  x x t* = 12.61  t* = 4.54  y (c)  y (g)  x x t* = 6.56  t* = 14.64  y (d)  y (h)  x t* = 8.58  y (e)  x t* = 16.65  Figure 4-49: Vorticity plot of the interaction of a vortex ring with a cylinder of dc = 0.18mm at ReN = 1420 and LN/DN = 1. The scales on axes are in mm and the vorticity contour values are in s-1. 142    Chapter 4 Results & Discussion When the diameter of the cylinder was increased to dc = 0.92 mm, the interaction also caused an axial flow along the vortex core (see figure 4-50). However, unlike the previous case of dc = 0.18 mm and 0.39 mm, the axial flow in this case appears more intense. The dye, representing the core, moves to the far ends of the core (see figure 449(d)) and subsequently returns axially and reconnect in a “spiral-like” manner (see figure 4-50(e)). It should be noted that in this case, the axial flow observed causes the dye marker to move axially such that very limited dye (if any) is left to visualize the cut-andreconnection (see figure 4-50(d)). However, the dye is seen to reconnect in the “spirallike” manner and this suggests that reconnection mechanism illustrated in figure 4-48 applies for this case as well. The formation of the vortex pairs and the reduction in the diameter of the vortex ring after the interaction with the cylinder are also observed in this case. These characteristics are similar to the previous case of dc = 0.18 mm and 0.39 mm. 143    Chapter 4 Results & Discussion y (a)  y (e)  x x “Spiral-like” formation cylinder t* = 0.00  t* = 9.69  y (b)  y (f)  x x t* = 3.71  t* = 12.92  y (c)  y (g)  x x t* = 5.49  t* = 17.76  y (d)  x (h)  Reduction in diameter of ring x Vortex pairs Axial flow t* = 6.46  y t* = 24.23  Figure 4-50: Interaction of a vortex ring with a cylinder of dc = 0.92 mm at ReN = 1420 and LN/DN = 1. 144    Chapter 4 Results & Discussion When the diameter of the cylinder was increased to dc = 1.32 mm, axial flow is again observed from dye visualization (see figure 4-51). The axial flow, in this case, causes the vortex core to undergo complex deformation (see figures 4-51(d) and 4-51(e)). However, the deformed vortex cores eventually reorganize themselves and continue to propagate as a vortex ring (see figure 4-51(h)). After the interaction, the formation of vortex pairs and reduction in diameter are also observed here. To better understand the deformation of the vortex core during the interaction, it is necessary to examine the corresponding vorticity plot as shown in figure 4-52. Here, is can be seen that during the interaction, the vortex core deforms into a crescent shape, and subsequently, the vorticity peak of the core is seen to decrease as indicated by the reduced colour intensity (see figure 4-52(d)). The colour intensity is later observed to increase (see figure 4-52(g)), while the core appears to regenerate into a modified vortex ring (see figure 4-52(h)). After analysing the visualization and vorticity plot for the case of dc = 1.32 mm, the author proposes an interpretation on how the vortex ring is able to maintain its stable form after interacting with the cylinder. This is illustrated in figure 4-53. Here, when the vortex ring interacts with the cylinder, the core at the cylinder is stretched locally and the far ends of the core develop into a crescent-shaped vortex. However, the induced velocity around the vortex ring maintains the structure of the crescent-shaped core, and eventually causes the ring to reorganize and propagate as a modified vortex ring. 145    Chapter 4 Results & Discussion y (a)  y (e)  x x cylinder t* = 0.00  t* = 8.88  y (b)  x y (f)  Secondary vortices t* = 3.87  x t* = 12.44  y (c)  y (g)  x x Axial flow t* = 5.17  t* = 20.19  y (d)  x (h)  Reduction in diameter of ring y x Vortex pairs t* = 7.27  t* = 25.03  Figure 4-51: Interaction of a vortex ring with a cylinder of dc = 1.32 mm at ReN = 1420 and LN/DN = 1. 146    Chapter 4 Results & Discussion y (a)  y (e)  x x cylinder t* = 0.00  t* = 9.59  (b)  Development of secondary vorticesx y y (f)  x t* = 4.04  t* = 10.60  y (c)  y (g)  x x Crescent-shaped vortices t* = 7.06  t* = 12.11  y (d)  x t* = 8.08  y (h)  x t* = 14.64  Figure 4-52: Vorticity plot of the interaction of a vortex ring with a cylinder of dc = 1.32 mm at ReN = 1420 and LN/DN = 1. The scales are in mm and the vorticity plot values are in s-1. 147    Chapter 4 Results & Discussion Induced velocity field cylinder (a)  Direction of propagation (b)  (c)  Secondary vortices Secondary vortices Induced velocity field maintains the vortex ring structure Stretched vortex lines which gets dissipated eventually (d)  Flow induced from the core by the secondary vortex Secondary vortex Vortex pair Figure 4-53: Author’s interpretation of the interaction of vortex ring with cylinder of dc = 1.32 mm at ReN = 1420 & LN/DN = 1. The induced velocity field, represented by dotted lines, maintains the structure of vortex ring even after the interaction with the cylinder. The shaded area represents the core of the vortex ring. 148    Chapter 4 Results & Discussion For the cylinder diameter of dc = 1.86 mm, the reconnection process does not appear to take place after the vortex ring interacts with the cylinder (see figure 4-54). Here, the core of the vortex ring is seen to undergo deformation, and there are also evidences of streamwise vortices (see figure 4-54(f)). However, it is not clear how these streamwise vortices may affect the reconnection process. Eventually, fine-scale structures are developed and reconnection is not apparent from the visualization. 149    Chapter 4 Results & Discussion y (a)  y (e)  x x cylinder t* = 0.00  t* = 12.11  y (b)  x t* = 3.55  (f)  streamwise vortices x t* = 14.53  y (c)  y (g)  x x t* = 6.46  t* = 17.76  y (d)  y (h)  x x t* = 8.88  y t* = 22.61  Figure 4-54: Interaction of a vortex ring with a cylinder of dc = 1.86 mm at ReN = 1420 and LN/DN = 1. 150    Chapter 4 Results & Discussion 4.3.2 Effects of Varying Stroke Ratio on the Interaction Unlike the case for LN/DN = 1, vortex rings with higher stroke ratios (i.e. LN/DN = 2 and 3) are observed to reconnect even after interacting with the largest cylinder diameter, dc = 1.86 mm, used in this experiment (see figure 4-55 and 4-56). It is likely that larger core size may have helped to facilitate the reconnection process. Another observation made for vortex ring/cylinder interaction with high stroke ratios (i.e. LN/DN ≥ 2), is the formation of “von Karman-like” vortices which can be seen to form in the wake of the cylinder. Such observation is shown clearly in the vorticity plot for the case of LN/DN = 3 (see figure 4-57). The vortex street appears to cause a complex interaction with the core of the vortex ring and contribute to the formation of the fine-scale structures observed in the visualization (see figure 4-55(g) and 4-56(f)). Other observations, such as the deformation of the core into a crescent-shaped vortex and the regeneration into a modified vortex ring is made for dc = 1.86 mm and LN/DN = 2 and 3 (see figures 4-55, 4-56 and 4-57). These observations are consistent with that in for dc = 1.32 mm and LN/DN = 1 (see figure 4-51). However, further detailed investigation on the reconnection process is needed to confirm such similarity, and this is not in the scope of this experiment. 151    Chapter 4 Results & Discussion y (a)  y (e)  x x cylinder t* = 6.14  t* = 0.00  y (b)  x x t* = 2.75  t* = 8.07  y (c)  x y (g)  Fine-scale structures t* = 4.04  x t* = 10.01  y (d)  y (h)  x t* = 5.17  y (f)  x t* = 12.92  Figure 4-55: Interaction of a vortex ring with a cylinder of dc = 1.86 mm at ReN = 1420 and LN/DN = 2. 152    Chapter 4 Results & Discussion y (a)  y (e)  x x cylinder t* = 0.00  t* = 7.59  y (b)  y (f)  x x Fine –scale structures t* = 9.37  t* = 2.10  y (c)  y (g)  x x t* = 3.23  t* = 11.62  y (d)  y (h)  x x Vortex pairs t* = 5.49  t* = 15.50  Figure 4-56: Interaction of a vortex ring with a cylinder of dc = 1.86 mm at ReN = 1420 and LN/DN = 3. 153    Chapter 4 Results & Discussion y (a)  y (e)  x x cylinder t* = 0.00  t* = 5.55  y (b)  x x t* = 2.02  t* = 7.06  (c)  “von Karman-like” vortex street y t* = 4.04  y (g)  x x t* = 3.03  (d)  y (f)  t* = 8.08  crescent-shaped vortices x y y (h)  x t* = 9.09  Figure 4-57: Interaction of a vortex ring with a cylinder of dc = 1.89 mm at ReN = 1420 154  -1  and LN/DN = 3. The scales are in mm and the vorticity plot values are in s . Chapter 4 Results & Discussion 4.3.3 Section Summary In this section, the interaction of the vortex ring with circular cylinders has been studied. The results suggest that the behaviour of the vortex ring/cylinder interaction is not sensitive to the ReN, at least for the condition investigated here. When a vortex ring generated with stroke ratio of LN/DN = 1 was made to interact with a cylinder diameter of dc = 0.18 mm, 0.39 mm and 0.92 mm, the cut-and-reconnection of the vortex core was observed to occur. The mechanism for the reconnection process has been proposed. When the cylinder diameter was increased further to dc = 1.32 mm, the vortex ring was able to reorganize and reconnect into a modified vortex ring after the interaction. The reason for such phenomenon was interpreted to be the effect of the induced velocity. Further increase in the cylinder diameter to dc = 1.86 mm showed no reconnection. When the vortex ring of higher stroke ratios (i.e. LN/DN ≥ 2) were interacted with the cylinders, reconnection appeared to take place even at the largest cylinder diameter used in this experiment (i.e. dc = 1.86 mm). It is likely that the larger core size could have facilitated the reconnection process. Other observation made include, the “vonKarman-like” vortex street appearing on the lee-side of the cylinder. 155    Chapter 5 Conclusion   Chapter 5  Conclusion    The discussions carried out in the previous chapter, lead to the following conclusions. 5.1 Circular Vortex Ring a) The formation characteristics of a vortex ring have been studied and the results are consistent with those published in the literature. More importantly, an empirical equation of ReΓ as a function of ReN and LN/DN was proposed. As far as the author is aware, this relationship has not been reported in the literature before. The equation provides a valuable mean of converting from ReN to ReΓ, or vice versa. b) An empirical formula of D/DN as a function of LN/DN was also proposed and the results are in good agreement with that of Maxworthy (1977). c) The dimensionless core diameter of the vortex ring (c/DN) was found to be positively related to LN/DN, but inversely related to ReN. This finding has not been reported before, partly due to the lack of DPIV measurements in the earlier studies. d) The vorticity profiles within the core of the vortex rings were also investigated and found to be a close-fit to the Gaussian curve. This result provides experimental support of the assumption by Saffman (1970) in his analysis of viscous vortex ring. 156    Chapter 5 Conclusion   5.2 Elliptic Vortex Rings a) The spatial and temporal trajectories of the elliptic vortex rings were found to be independent of ReN for AR = 2, but dependent on ReN for AR = 3. This difference can be attributed to the formation of cross-linking vortices during partial bifurcation. b) Partial bifurcation of the elliptic vortex ring of AR = 3 was observed only for ReN ≤ 1230 at LN/DN = 1, and not when ReN >> 1230. c) No stretching or compression of the vortex core was observed during the deformation of the elliptic vortex ring for both AR = 2 and 3. This experimental finding provides experimental support to the theoretical analysis by Arms and Hama (1965). d) The streamline patterns of elliptic vortex rings showed saddle node (critical point) upstream of the ring vortex in both x-y plane and x-z plane for AR = 2. This is in contrast to elliptic vortex ring of AR = 3, which shows the saddle node in the x-y plane, and an unstable node in the x-z plane. No such critical points were observed for circular vortex ring. e) Larger stroke ratios (LN/DN ≥ 2) to generate elliptic vortex rings showed a faster onset of instability and breakdown, caused by the development of a vortex pair and streamwise vortices during the formation process. The entrainment of the vortex pair and the streamwise vortices by the elliptic vortex ring leads to complex interactions with the vortex core resulting in the 157    Chapter 5 Conclusion   generation of fine-scale structures. At even higher stroke ratios (LN/DN ≥ 4), the vortex pair propagates downstream of the vortex ring and form an “arclike” flow structure. However, it does not appear to have significant effect on the vortex ring. f) The formation and evolution of streamwise vortices in an elliptic vortex ring is similar to that explained by Husain and Hussain (1993) for an elliptic jet. 158    Chapter 5 Conclusion   5.3 Interaction of a Vortex Ring with a Circular Cylinder a) The cut-and-reconnection mechanism which occurs during vortex ring/cylinder interaction appears to be insensitive to ReN. In all ReN, and with LN/DN = 1, the cut-and-reconnection was observed for cylinders of diameters dc ≤ 1.32 mm, and no reconnection was observed for dc >> 1.32 mm. b) The cut-and-reconnection mechanism has been identified to be similar to that proposed by Adhikari and Lim (2009), although minor variations in the flow structures were observed for different cylinder sizes. For example, the reconnection occurs with the “spiral-like” formation at dc = 0.92 mm, and a crescent-shaped vortex core at dc = 1.32 mm. The latter stretched and eventually dissipated through viscous diffusion, while other part of the vortex core regenerated into a modified ring. No reconnection was observed for dc = 1.86 mm. c) At higher stroke ratios (LN/DN ≥ 2), vortex reconnection was observed at dc = 1.86 mm. This can be attributed to large core size relative to the cylinder diameter. Also, vortex ring generated using larger stroke ratio, after interacting with the cylinder, produces “von Karman-like” vortex street on the lee-side of the cylinder. This vortex street plays a role in the generation of the fine-scale structures observed during the interaction. 159    Chapter 6 Recommendations   Chapter 6  Recommendations    Since the development and propagation of an elliptic vortex ring is three-dimensional in nature, a 3-D stereo DPIV method would be better to analyse the flow field during the formation and subsequent motion. Similarly, for the case of circular vortex ring/cylinder interaction, the cut-andreconnection is a 3-D process and measurement of the 3-D vorticity field by the stereo DPIV method will provide a deeper insight into the mechanism of cut-andreconnection.   160    List of References  LIST OF REFERENCES Adhikari, D. (2007) “An Experimental Study on the Interaction of Vortex Ring on Porous Screens”. Final Year Dissertation. National University of Singapore. Adhikari, D. & Lim, T. T., (2009) “The impact of vortex ring on a porous screen”. Fluid Dynamics Research (Invited Paper). 41: 051404. Akhmetov, D. G., Lugovtsov, B. A. & Tarasov, V. F., (1980) “Extinguishing gas and oil well fires by means of vortex rings”. Combustion, Explosion & Shock Waves. 16: 490-494 Arms, R. J. & Hama, F. R. (1965) “Localized-Induction Concept on a Curved Vortex and Motion of an Elliptic Vortex Ring”. The Physics of Fluids. 8(4): 553-559. Ashurst, Wm. T. & Meiron, D. 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Zhao, Y. & Shi, X., (1997) “Evolution of single elliptic vortex rings”. Acta Mechanica Sinica (English Series). 13(1): 17-25. 167    Appendix A   Appendix A  Program to track the coordinates of maximum and minimum vorticity values from Tecplot data file and determination of other quantities.   ////////////////////////////////////////////////////// //Written By: Deepak Adhikari // // // //Description: This program tracks the coordinate of// // the maximum and minimum vorticity // // value from tecplot data file and gets// // all the other values at these points.// ////////////////////////////////////////////////////// #include #include #include #include #include void main() { ofstream outfile; ifstream infile; // List of Variables--------------------------------int i, j, numstart, numend, num; double x, y, u, v, c, t=0.0, maxc, minc, maxx, maxy, maxu, maxv, minx, miny, minu, minv, vel, dia; char garbage[101], filenamein[19] = "VortexRing0000.dat", filenameout[50]; // -------------------------------------------------- 168    Appendix A   // Output Query-------------------------------------cout numstart; cout numend; cout > filenameout; // -------------------------------------------------- // Output file header ------------------------------outfile.open(filenameout, ios::out); outfile u >> v >> c; if (c < minc) { minc = c; minx = x; miny = y; minu = u; minv = v; } 170    Appendix A   if (c > maxc) { maxc = c; maxx = x; maxy = y; maxu = u; maxv = v; } } vel = 0.5 * (sqrt(maxu*maxu + maxv*maxv) + sqrt(minu*minu + minv*minv)); dia = sqrt(pow(maxy-miny, 2) + pow(maxx-minx, 2)); outfile [...]... understanding on circular vortex rings This is followed by a discussion on elliptic vortex rings which are known to exhibit interesting and complicated dynamics The third section reviews some past studies on the interaction of a vortex ring with a circular cylinder, and discussion on the cut -and- reconnection phenomena of the vortex core 9    Chapter 2 Literature Review  2.1 Circular Vortex Rings The... generation of vortex ring with varying boundary conditions (Lim, 1998; Dabiri 2005) and the study on the optimal formation of vortex ring (Gharib et al., 1998) A recent review has been 3    Chapter 1 Introduction   published which provides an overview on the optimal formation of vortex ring and its relation to biological propulsion system (Dabiri, 2009) Alongside the fundamental study of vortex rings... C-1: The formation of a circular vortex ring at ReN = 1000 and LN/DN = 2 (a) Dye visualization and (b) vorticity field The scales on axes in (b) are in mm and the vorticity contours are in s-1 177 The formation of a circular vortex ring at ReN = 1000 and LN/DN = 3 (a) Dye visualization and (b) vorticity field The scales on axes in (b) are in mm and the vorticity contours are in s-1 178 The formation of... around the vortex core in order to calculate the circulation of the vortex ring 69 Dye visualization of the formation of a circular vortex ring at ReN = 1000 and LN/DN = 1 73 Vorticity field of the formation of a circular vortex ring at ReN = 1000 and LN/DN = 1 The scales on axes are in mm and the vorticity contours are in s-1 74 Instantaneous streamlines of the formation of a circular vortex ring at... ReN = 1000 and LN/DN = 1 137 Figure 4-45: Interaction of a vortex ring with a cylinder of dc = 0.18mm at ReN = 1420 and LN/DN = 1 138 Interaction of a vortex ring with a cylinder of dc = 0.18mm at ReN = 1740 and LN/DN = 1 139 Interaction of a vortex ring with a cylinder of dc = 0.39mm at ReN = 1420 and LN/DN = 1 140 Author’s interpretation of the cut -and- reconnection mechanism for a vortex ring interacting... the study on the interactions of vortex rings, the study of noncircular vortex ring and the computational study of vortex ring were initiated (Dhanak and De Bernardinis, 1981; Nitche and Krasny, 1994; Verzicco and Orlandi, 1994; Orlandi and Verzicco, 1993; Kiya et al., 1992; Lim, 1989; Lim and Nickels, 1992) It was also during this period that significant findings of researchers were compiled and recorded... an elliptic vortex ring of AR = 3 at ReN = 1580 and LN/DN = 1 (a) x-y plane and (b) x-z plane The scales on axes are in mm and the vorticity contours are in s-1 182 Vorticity field of the formation of an elliptic vortex ring of AR = 3 at ReN = 1740 and LN/DN = 1 (a) x-y plane and (b) x-z plane The scales on axes are in mm and the vorticity contours are in s-1 183 Interaction of a vortex ring with a... comprehensive work on the formation of vortex ring, does not have sufficient information on the evolution of the vortex core Therefore, it is the author’s aim to carry out a systematic study on circular vortex ring with a particular focus on the core structure This thesis will also attempt to answer the questions surrounding non-circular vortex rings through a detailed study of elliptic vortex rings Several... elliptic vortex rings a To examine the trajectory and flow field of elliptic vortex rings; b To examine the formation of an elliptic vortex ring at different piston profiles; c To analyse the flow structures of elliptic vortex rings 3 Study of vortex ring interaction with circular cylinders a To examine the reconnection of vortex ring after its interaction with a circular cylinder; b To examine the effects... circular vortex ring at ReN = 1000 and LN/DN = 4 (a) Dye visualization and (b) vorticity field The scales on axes in (b) are in mm and the vorticity contours are in s-1 179 The formation of a circular vortex ring at ReN = 1000 and LN/DN = 5 (a) Dye visualization and (b) vorticity field The scales on axes in (b) are in mm and the vorticity contours are in s-1 180 Vorticity field of the formation of an ... a Vortex Ring 3.5.4 Circulation of a Vortex Ring 63 65 65 67 67 3.6 Experimental Conditions 69 Chapter Results & Discussion 4.1 Circular Vortex Rings 4.1.1 Formation of the a Circular Vortex Ring. .. interesting and complicated dynamics The third section reviews some past studies on the interaction of a vortex ring with a circular cylinder, and discussion on the cut -and- reconnection phenomena... the boundary conditions affect not only the formation number, but also the formation characteristics and the structure of the vortex ring One such condition, which affects the formation characteristics,