BUOYANT JETS WITH TWO AND THREE-DIMENSIONAL TRAJECTORIES A thesis submitted in partial fulfilment of the requirements for the Degree of Doctor of Philosophy in Civil Engineering at the University of Canterbury by Gustaaf Adriaan Kikkert University of Canterbury Christchurch, New Zealand 2006 I Abstract Extensive experimental data is available from previous research into the behaviour of buoyant jets released into an unstratified ambient. The experimental data has been the basis for theoretical and numerical modelling work, and currently several numerical models exist that are employed in the design of engineering structures built for the disposal of wastewater in the ocean. However there are still flow configurations with limited or no available experimental data, and hence confidence in the use of the models under some circumstances is limited. These circumstances include two-dimensional trajectory flows that are discharged at oblique angles to the ambient and buoyant jet flows with three-dimensional trajectories. As part of the current project an experimental investigation is conducted into the behaviour of discharges that have either two-dimensional or three-dimensional trajectories, focussing particularly on those configurations with currently limited available experimental data. A light attenuation technique is developed for the investigation of such flows, largely because it enables the behaviour of discharges with three-dimensional trajectories to be recorded with relative ease. However, this technique provides integrated views of the flow and hence the interpretation of the integrated concentration data is aided by assumed mean cross-sectional concentration profiles. In the strongly advected region (with the exception of the weak-jet) a double-Gaussian approximation is shown to provide a reasonable representation of mean concentration profiles. In the weakly advected regions and the weak-jet region, it is well- known that a single Gaussian adequately represents the mean flow structure. A new numerical model, the Momentum Model, is developed to assist in the design and to monitor the performance of the experimental investigation. Unlike other models, the behaviour of the flow is determined by the relative magnitudes of the initial excess momentum flux, the buoyancy-generated momentum flux and the entrained ambient momentum flux. It is shown that ratios of these momentum fluxes are equivalent to the length-scales traditionally employed for this task. Predictions from the Momentum Model are compared with data from the current and previous experimental investigations and, in addition, predictions from two representative numerical models, VisJet and CorJet. Predictions from the Momentum Model are shown to be consistent with data for a wide variety of discharge configurations. These predictions are also generally consistent with those of VisJet and CorJet. However, the experimental results from the II buoyant jet discharged in a moving ambient show that the spreading rates of the strongly advected flows (puffs and thermals) differ, and while this difference is incorporated into the Momentum Model, it is not evident in the VisJet and CorJet predictions. Numerical model predictions of negatively buoyant discharges are shown to be inadequate. This discharge configuration is investigated in some detail experimentally and additional analytical solutions of the flow behaviour are developed to aid in the interpretation of the flow behaviour. The experimental results show that buoyancy-induced instabilities on the inner side of the jets, which generate additional vertical mixing, significantly alter the form of the mean concentration profiles in this region. This results in considerably higher integrated dilutions along the flow centreline. Another significant difference between the newly developed Momentum Model and the existing numerical models (VisJet and CorJet), is the approach taken to dealing with oblique discharges in a cross-flow. Experimental results in combination with additional analytical solutions show that for initial discharge angles of 20? and less, an oblique discharge in a cross-flow becomes a weak-jet in the strongly advected region, and for angles of 40? and above, the flow becomes a puff. The strongly advected behaviour predicted by the Momentum Model changes abruptly at the transition angle, and is reasonably consistent with the data. The gradual change in strongly advected behaviour employed by VisJet and CorJet does not appear to be appropriate in the puff region. Finally a preliminary experimental investigation of discharges with three-dimensional trajectories shows that there are significant discrepancies between the predicted behaviour and the experimental data. This is surprising given the numerical models are, for the most part, able to predict the behaviour of flows with two-dimensional paths with reasonable accuracy. It is evident that flows with three-dimensional paths are modified more severely by the different directions of the initial, buoyancy-generated, and entrained ambient momentum fluxes than the current models suggest. III Acknowledgements From day 1, the intension of the research was to investigate buoyant jets with a three- dimensional path. With this aim in mind, time was spent developing a flow visualization technique and an alternative numerical model to aid in the investigation. Both the flow visualization technique and the model were verified with experimental data obtained from buoyant jets with a two-dimensional path. This process included some major distractions in the form of the investigations of the negatively buoyant jet and obliquely discharged non- buoyant jets in a moving ambient. However, when it was time for the investigation of the buoyant jets with a three-dimensional path, all the knowledge that was gained from the flows with a two-dimensional path was applied in the design and analysis of that investigation. I would like to sincerely thank my supervisor, Dr. Mark Davidson, not just for suggesting the topic by presenting it as a challenge, but also for his continuous commitment, support and invaluable advice during the length of the investigation. I especially appreciate his honest insights into the wider research community, giving me a right perspective from which to grow as a researcher. I would like to thank my co-supervisor, Dr. Roger Nokes. His enthusiasm developed my interest in fluid mechanics as an under-graduate student. His comments as an expert in fluid dynamics slightly outside the current area of study were especially helpful in increasing the standard of the project. I am also grateful for his willingness to upgrade computer software at a moment?s notice. The laboratory staff, under the leadership of Mr. Ian Sheppard and including Mr. Colin Bliss, Mr. Ray Allan and Mr. Kevin Wines, have given me much support with the experimental investigation. Without their problem solving solutions and extra pair of hands, it would not have been possible to conclude the experimental investigation in time. Beside their help with research related issues I also appreciate the advice on and help with fixing anything from bicycles to vacuum cleaners. I would like to thank my fellow post-graduate students, Caroline, Dave, Langford, Bill and James, for being the first to complain to when computers, software or anything else that could go wrong, went wrong. I also appreciate the many small, but useful, time-wasters, which helped keep me sane during the many hours spent in the laboratory. IV Finally I would like to express my gratitude to my family, friends, and especially my wife, Heather. Their support from the outside made it possible for me to spend three years working on a single project. However they also helped me remember what is really important in life. V Table of Contents ABSTRACT------------------------------------------------------------------------------------------------I ACKNOWLEDGEMENTS-------------------------------------------------------------------------- III TABLE OF CONTENTS-------------------------------------------------------------------------------V LIST OF FIGURES--------------------------------------------------------------------------------------X LIST OF TABLES---------------------------------------------------------------------------------XVIII LIST OF NOTATIONS----------------------------------------------------------------------------- XIX CHAPTER 1 ? INTRODUCTION ------------------------------------------------------------------- 1 1.1 ? GENERAL INTRODUCTION ----------------------------------------------------------------------- 1 1.2 ? PROBLEM OVERVIEW ---------------------------------------------------------------------------- 2 1.3 ? SCOPE OF RESEARCH----------------------------------------------------------------------------- 3 CHAPTER 2 ? REVIEW OF PREVIOUS RESEARCH---------------------------------------- 5 2.1 ? INTRODUCTION ----------------------------------------------------------------------------------- 5 2.2 ? PROBLEM FORMULATION OF THE BUOYANT JET --------------------------------------------- 5 2.3 - RESEARCH HISTORY ------------------------------------------------------------------------------ 6 2.4 ? PREVIOUS EXPERIMENTAL INVESTIGATIONS-------------------------------------------------- 7 2.4.1 ? Flow measurement techniques------------------------------------------------------------ 7 2.4.2 ? Flow configurations------------------------------------------------------------------------ 8 2.4.2.1 - Jets --------------------------------------------------------------------------------------- 9 2.4.2.2 ? Pure Plumes ---------------------------------------------------------------------------10 2.4.2.3 ? Buoyant Jets---------------------------------------------------------------------------11 2.4.2.4 ? Advected Jets -------------------------------------------------------------------------13 2.4.2.5 ? Buoyant Discharges in an Ambient Flow-----------------------------------------14 2.4.3 ? Missing Experimental Data--------------------------------------------------------------17 2.5 - EXISTING MODELS-------------------------------------------------------------------------------19 2.5.1 ? Length-Scale Models----------------------------------------------------------------------19 2.5.2 ? Integral Models----------------------------------------------------------------------------21 2.5.3 ? Hybrid Models-----------------------------------------------------------------------------24 2.6 ? SUMMARY ----------------------------------------------------------------------------------------24 VI CHAPTER 3 ? FLOW VISUALIZATION TECHNIQUES-----------------------------------27 3.1 - INTRODUCTION -----------------------------------------------------------------------------------27 3.2 - LA -------------------------------------------------------------------------------------------------28 3.2.1 ? Light Attenuation System-----------------------------------------------------------------28 3.2.1.1 ? LA Experimental Configuration ---------------------------------------------------29 3.2.1.2 ? Theoretical Background-------------------------------------------------------------32 3.2.2 - Calibration experiments ------------------------------------------------------------------35 3.2.2.1 ? Experimental Set Up-----------------------------------------------------------------35 3.2.2.2 ? Experimental Method----------------------------------------------------------------40 3.2.2.3 ? Calibration Results-------------------------------------------------------------------42 3.2.2.4 ? Response of the Red Dye -----------------------------------------------------------48 3.2.3 ? Interpretation of the integrated information-------------------------------------------50 3.2.3.1 ? Weakly Advected-Flow, a Simple Jet Experiment ------------------------------51 3.2.3.2 ? Strongly Advected Flow, a Momentum Puff Experiment----------------------63 3.2.3.3 ? Angled Jet -----------------------------------------------------------------------------79 3.2.3.4 ? Parallax Issues ------------------------------------------------------------------------83 3.2.4 ? 3D LA ---------------------------------------------------------------------------------------84 3.2.4.1 ? 3D LA Equipment and Experimental Set Up ------------------------------------85 3.2.4.2 ? Calibration Length Scales-----------------------------------------------------------87 3.2.4.3 ? Verification of 3D LA system------------------------------------------------------89 3.2.5 - Summary-------------------------------------------------------------------------------------91 3.3 - LIF-------------------------------------------------------------------------------------------------93 3.3.1 ? Laser Induced Fluorescent System------------------------------------------------------93 3.3.2 ? LIF Calibration Methods-----------------------------------------------------------------95 3.3.3 ? Summary------------------------------------------------------------------------------------97 CHAPTER 4 ? MOMENTUM MODEL-----------------------------------------------------------99 4.1 - INTRODUCTION -----------------------------------------------------------------------------------99 4.2 ? MODEL CONFIGURATION AND INITIAL CONDITIONS----------------------------------------99 4.3 ? ORDINARY DIFFERENTIAL EQUATIONS ----------------------------------------------------- 102 4.3.1 ? Deriving Equations----------------------------------------------------------------------102 4.3.2 ? Spread Relationships--------------------------------------------------------------------105 4.3.3 ? Dimensionless ODEs--------------------------------------------------------------------111 4.3.4 ? Solving the Equations in MatLab -----------------------------------------------------112 VII 4.4 ? TOP-HAT CONVERSIONS---------------------------------------------------------------------- 114 4.5 ? DOUBLE-GAUSSIAN DILUTION RATIOS----------------------------------------------------- 116 4.6 ? SUMMARY -------------------------------------------------------------------------------------- 117 CHAPTER 5 ? TWO-DIMENSIONAL TRAJECTORY FLOWS------------------------- 119 5.1 ? INTRODUCTION -------------------------------------------------------------------------------- 119 5.2 ? STILL AMBIENT FLOWS----------------------------------------------------------------------- 120 5.2.1 ? Simple Jet---------------------------------------------------------------------------------121 5.2.1.1 ? Experimental Design -------------------------------------------------------------- 121 5.2.1.2 ? Experimental Results and Model Predictions ---------------------------------- 122 5.2.2 ? Plume--------------------------------------------------------------------------------------126 5.2.2.1 ? Experimental Design -------------------------------------------------------------- 126 5.2.2.2 ? Experimental Results and Model Predictions ---------------------------------- 127 5.2.3 ? Horizontal Buoyant Jet-----------------------------------------------------------------130 5.2.3.1 ? Experimental Design -------------------------------------------------------------- 130 5.2.3.2 ? Experimental Results and Model Predictions ---------------------------------- 131 5.2.4 ? Positively Buoyant Jet ------------------------------------------------------------------135 5.2.4.1 ? Experimental Design -------------------------------------------------------------- 135 5.2.4.2 ? Experimental Results and Model Predictions ---------------------------------- 136 5.3 ? MOVING AMBIENT FLOWS ------------------------------------------------------------------- 139 5.3.1 ? Vertically Discharged Non-Buoyant Jet in an Ambient Flow---------------------140 5.3.1.1 ? Experimental Design -------------------------------------------------------------- 140 5.3.1.2 ? Experimental Results and Model Predictions ---------------------------------- 142 5.3.2 ? Buoyant Jet in Moving Ambient Flow------------------------------------------------148 5.3.2.1 ? Experimental Design -------------------------------------------------------------- 148 5.3.2.2 ? Experimental Results and Model Predictions ---------------------------------- 149 5.3.3 ? Weak Jet ----------------------------------------------------------------------------------159 5.4 ? SUMMARY -------------------------------------------------------------------------------------- 161 CHAPTER 6 ? NEGATIVELY BUOYANT JETS-------------------------------------------- 163 6.1 - INTRODUCTION --------------------------------------------------------------------------------- 163 6.2 - ANALYTICAL SOLUTIONS --------------------------------------------------------------------- 164 6.2.1 - Discharge Configuration and Initial Conditions------------------------------------164 6.2.2 - Derivation of Equations-----------------------------------------------------------------165 VIII 6.3 - EXPERIMENTAL SET UP ----------------------------------------------------------------------- 170 6.3.1 - LA Experiments---------------------------------------------------------------------------170 6.3.2 - LIF Experiments--------------------------------------------------------------------------171 6.4 - EXPERIMENTAL RESULTS AND MODEL PREDICTIONS ------------------------------------- 172 6.4.1 - Gaussian Assumption--------------------------------------------------------------------172 6.4.2 - Conditions at Maximum Height--------------------------------------------------------176 6.4.3 - Conditions at impact point--------------------------------------------------------------183 6.4.4 ? More on the mean properties of negatively buoyant jets --------------------------188 6.4.5 ? Preliminary Cross-Sectional Results-------------------------------------------------197 6.5 ? SUMMARY -------------------------------------------------------------------------------------- 199 CHAPTER 7 ? OBLIQUE NON-BUOYANT DISCHARGES IN A MOVING AMBIENT---------------------------------------------------------------------------------------------- 201 7.1 - INTRODUCTION --------------------------------------------------------------------------------- 201 7.2 ? ANALYTICAL SOLUTIONS--------------------------------------------------------------------- 202 7.2.1 ? Weak Jet ----------------------------------------------------------------------------------203 7.2.2 ? Advected Line Momentum Puff Region-----------------------------------------------205 7.2.3 ? Theoretical Transition Angle----------------------------------------------------------207 7.2.4 ? Weakly Advected to Strongly Advected Transitions and Virtual Sources-------209 7.3 ? EXPERIMENTAL DESIGN ---------------------------------------------------------------------- 211 7.4 ? EXPERIMENTAL RESULTS AND MODEL PREDICTIONS------------------------------------- 212 7.4.1 - Comparison with Previous Experimental Results-----------------------------------212 7.4.2 - Comparison with Analytical Solutions------------------------------------------------215 7.4.2.1 ? Advected Line Momentum Puff Region---------------------------------------- 215 7.4.2.2 ? Weak-Jet Region ------------------------------------------------------------------- 218 7.4.2.3 ? Transition Region between Weak-Jet and Puff Regions --------------------- 221 7.4.3 ? Dilution Results--------------------------------------------------------------------------224 7.5 - SUMMARY--------------------------------------------------------------------------------------- 228 CHAPTER 8 ? BUOYANT JETS WITH THREE-DIMENSIONAL TRAJECTORIES ------------------------------------------------------------------------------------------------------------- 231 8.1 - INTRODUCTION --------------------------------------------------------------------------------- 231 8.2 ? FLOW CONFIGURATIONS---------------------------------------------------------------------- 232 8.3 - EXPERIMENTAL RESULTS AND MODEL PREDICTIONS ------------------------------------- 234 IX 8.3.1 ? Cross-Sectional Behaviour-------------------------------------------------------------234 8.3.2 ? Bulk Properties --------------------------------------------------------------------------246 8.3.2.1 ? Double-Integrated Dilution Results --------------------------------------------- 246 8.3.2.2 ? Trajectory Results------------------------------------------------------------------ 250 8.3.2.3 ? Integrated Dilution Results ------------------------------------------------------- 262 8.4 ? SUMMARY -------------------------------------------------------------------------------------- 266 CHAPTER 9 ? CONCLUSIONS ------------------------------------------------------------------ 269 REFERENCES ---------------------------------------------------------------------------------------- 277 APPENDIX A ? CODING AND REPRODUCING DIGITAL IMAGES----------------- 287 APPENDIX B ? ANALYSIS OF AVERAGE INTEGRATED CONCENTRATION IMAGE-------------------------------------------------------------------------------------------------- 291 APPENDIX C ? COMPUTER CODE OF MOMENTUM MODEL----------------------- 303 APPENDIX D - INITIAL CONDITIONS FOR EXPERIMENTS WITH 2D AND 3D TRAJECTORIES------------------------------------------------------------------------------------- 307 APPENDIX E ? ADDITIONAL FIGURES ----------------------------------------------------- 319 APPENDIX F ? TRAJECTORY SOLUTIONS FOR WEAK-JET AND PUFF REGIONS----------------------------------------------------------------------------------------------- 325 X List of Figures FIGURE 2.1 - INITIAL CONDITIONS AND CROSS-SECTIONAL VELOCITY PROFILE FOR JET EXPERIMENT...................................................................................................................... 10 FIGURE 2.2 - INITIAL CONDITIONS AND CROSS-SECTIONAL VELOCITY PROFILE FOR PURE PLUME .......................................................................................................................................... 11 FIGURE 2.3 - FLOW REGIONS OF A BUOYANT JET ....................................................................... 12 FIGURE 2.4 - DIFFERENT FLOW REGIONS OF ADVECTED JET....................................................... 14 FIGURE 2.5 ? FLOW REGIONS OF BUOYANT JET IN FLOWING AMBIENT WITH A 2D-TRAJECTORY 15 FIGURE 2.6 ? FLOW CONFIGURATION HORIZONTALLY DISCHARGED BUOYANT JET IN A CROSS- FLOW................................................................................................................................. 17 FIGURE 3.1 - SCHEMATIC DIAGRAM OF MAIN TANK .................................................................. 29 FIGURE 3.2 - MAGNETIC FLOW METER CALIBRATION TEST RESULTS ......................................... 30 FIGURE 3.3 - SCHEMATIC DIAGRAM OF EXPERIMENTAL EQUIPMENT ......................................... 31 FIGURE 3.4 - TYPICAL LA EXPERIMENTAL SET UP..................................................................... 32 FIGURE 3.5 - PATH OF A RAY OF LIGHT...................................................................................... 33 FIGURE 3.6 - PATH OF A RAY OF LIGHT INCLUDING DYED SOLUTION ......................................... 35 FIGURE 3.7 - ABSORPTION SPECTRUM FOR FOUR DIFFERENT TRACERS, (A) RED DYE (? (4000- 7000 ANGSTROMS) VS I/I0 (0-1)) WITH CONCENTRATIONS OF (FROM TOP) 0.05ML/L, 0.1ML/L, 0.2ML/L AND 0.4 ML/L, (B) GREEN DYE (? (3000-7000 ANGSTROMS) VS I/I0 (0-1)) WITH CONCENTRATIONS OF (FROM TOP) 0.05ML/L, 0.1ML/L, 0.2ML/L AND 0.4 ML/L, (C) BLUE DYE (? (3000-7000 ANGSTROMS) VS I/I0 (0-1)) WITH CONCENTRATIONS OF (FROM TOP) 0.05ML/L, 0.1ML/L, 0.2ML/L AND 0.4 ML/L, (D) POTASSIUM PERMANGANATE (? (4000-7000 ANGSTROMS) VS I/I0 (0-1)) WITH CONCENTRATIONS OF (FROM TOP) 0.6ML/L, 1.2ML/L, 2.4ML/L AND 4.8ML/L .......................................................................... 38 FIGURE 3.8 - RED DYE ABSORPTION SPECTRUM WITH AND WITHOUT GREEN FILTER ................ 39 FIGURE 3.9 - TEMPERATURE INFLUENCE ON GREEN INTENSITY RESPONSE ................................ 41 FIGURE 3.10 - THE RECORDED IMAGE (A) AND PROCESSED IMAGE (B) ...................................... 42 FIGURE 3.11 - COMPARISON OF GREEN INTENSITY RESPONSE OF TWO SEPARATE EXPERIMENTS 43 FIGURE 3.12 - RESPONSE OF THE GREEN GUN FOR TWO DIFFERENT BACKGROUND INTENSITIES 44 FIGURE 3.13 - RESPONSE OF CDYE FOR TWO PIXEL WITH DIFFERENT BACKGROUND INTENSITIES 45 FIGURE 3.14 - RESPONSE USING GREEN ABSORPTION FILTER FOR PIXEL WITH IREF = 240 ....... 46 FIGURE 3.15 ? RED DYE EXPERIMENT RESULTS ....................................................................... 49 XI FIGURE 3.16 - SINGLE FRAME OF JET VIDEO .............................................................................. 52 FIGURE 3.17 - AVERAGE IMAGE OF JET ..................................................................................... 53 FIGURE 3.18 ? INTEGRATED CONCENTRATION IMAGE OF JET .................................................... 54 FIGURE 3.19 - GREEN ABSORPTION IMAGE OF JET, BOUNDARIES ?1% TO 1% ........................... 55 FIGURE 3.20 - PLAN AND INTEGRATED VIEW OF CONCENTRATION PROFILES FOR A SIMPLE JET. 55 FIGURE 3.21 - SELF-SIMILARITY OF INTEGRATED CROSS-SECTIONAL PROFILES OF JET .............. 56 FIGURE 3.22 ? CONCENTRATION SPREAD OF JET ....................................................................... 58 FIGURE 3.23 ? INTEGRATED CENTRELINE AND POINT CENTRELINE DILUTION............................ 59 FIGURE 3.24 ? DOUBLE-INTEGRATED JET DILUTION.................................................................. 61 FIGURE 3.25 ? INTEGRATED CONCENTRATION FLUCTUATION ALONG THE JET CENTRELINE ...... 62 FIGURE 3.26 ? INTEGRATED CROSS-SECTIONAL PROFILES OF TURBULENT CONCENTRATION FLUCTUATIONS .................................................................................................................. 63 FIGURE 3.27 - Y-INTEGRATED (A) AND Z-INTEGRATED (B) MOMENTUM PUFF VIEWS................. 64 FIGURE 3.28 ? INTEGRATED CONCENTRATION IMAGE OF A Y-INTEGRATED MOMENTUM PUFF ... 65 FIGURE 3.29 ? INTEGRATED CONCENTRATION IMAGE OF A Z-INTEGRATED MOMENTUM PUFF ... 65 FIGURE 3.30 ? VORTEX PAIR AND DOUBLE-GAUSSIAN APPROXIMATION................................... 66 FIGURE 3.31 - CROSS-SECTIONAL INTEGRATED CONCENTRATION PROFILES INTEGRATED IN THE Y-DIRECTION ..................................................................................................................... 71 FIGURE 3.32 - CROSS-SECTIONAL INTEGRATED CONCENTRATION PROFILES INTEGRATED IN THE Z-DIRECTION...................................................................................................................... 71 FIGURE 3.33 - VALUE OF F AS A FUNCTION OF VERTICAL DISTANCE AWAY FROM SOURCE ........ 72 FIGURE 3.34 ? DOUBLE INTEGRATED DILUTION RESULTS FOR MOMENTUM PUFF EXPERIMENTS 73 FIGURE 3.35 ? DOUBLE-GAUSSIAN PARAMETER H AS A FUNCTION OF VERTICAL DISTANCE FROM THE SOURCE ...................................................................................................................... 74 FIGURE 3.36 - SPREAD COMPARISON OF TOP AND SIDE VIEW MOMENTUM PUFF EXPERIMENTS.. 75 FIGURE 3.37 - INTEGRATED DILUTION RESULTS........................................................................ 76 FIGURE 3.38 - CONVERTED PEAK DILUTION DATA USING FAR-FIELD AND NEAR-FIELD VALUES FOR H................................................................................................................................. 78 FIGURE 3.39 - MOMENTUM PUFF TRAJECTORY, COMPARING EXPERIMENTAL DATA WITH MODEL PREDICTIONS ..................................................................................................................... 79 FIGURE 3.40 - VERTICAL CROSS-SECTIONAL AND FRONTAL VIEW OF ANGLED JET EXPERIMENT 80 FIGURE 3.41 ? HORIZONTAL CROSS-SECTION............................................................................ 81 FIGURE 3.42 ? EXPERIMENTAL VERSUS THEORETICAL DILUTION OF ANGLED JET...................... 82 FIGURE 3.43 ? CROSS SECTIONAL VIEW OF SIMPLE JET EXPERIMENT INCLUDING THE EFFECTS OF PARALLAX ......................................................................................................................... 83 XII FIGURE 3.44 - 3D LA SYSTEM ? SET UP OF EXPERIMENTAL EQUIPMENT ................................... 86 FIGURE 3.45 ? UPGRADED CALIBRATION CELLS........................................................................ 87 FIGURE 3.46 - Y AND Z INTEGRATED VIEW LENGTH SCALES VERSUS DISTANCE AWAY FROM CAMERA ............................................................................................................................ 88 FIGURE 3.47 ? CONCENTRATION SPREAD RESULTS FOR Y AND Z-INTEGRATED SIMPLE JET EXPERIMENT...................................................................................................................... 89 FIGURE 3.48 ? INTEGRATED DILUTION RESULTS FOR Y AND Z-INTEGRATED SIMPLE JET EXPERIMENT...................................................................................................................... 90 FIGURE 3.49 ? DOUBLE INTEGRATED DILUTION RESULTS FOR Y AND Z-INTEGRATED MOMENTUM PUFF EXPERIMENT ............................................................................................................. 91 FIGURE 3.50 - LIF EXPERIMENTAL SET UP................................................................................. 95 FIGURE 3.51 - POLYNOMIAL FIELD CALIBRATION AT PIXEL (178,196) ..................................... 96 FIGURE 4.1 ? SCHEMATIC DIAGRAM OF COORDINATE SYSTEM OF MOMENTUM MODEL.......... 100 FIGURE 4.2 - SCHEMATIC DIAGRAM OF EXCESS MOMENTUM FLUX .......................................... 101 FIGURE 4.3 - SCHEMATIC DIAGRAM OF THE INITIAL EXCESS MOMENTUM FLUX DISCHARGE CONFIGURATION.............................................................................................................. 101 FIGURE 4.4 ? MOMENTUM MODEL SPREAD FUNCTION FLOW CHART....................................... 110 FIGURE 5.1 ? INTEGRATED CROSS-SECTIONAL CONCENTRATION PROFILES FROM JET RUNS 12, 14 & 15 AT VARIOUS LOCATION DOWNSTREAM FROM THE SOURCE ................................ 122 FIGURE 5.2 - CONCENTRATION SPREAD RESULTS VERSUS DISTANCE DOWNSTREAM, COMPARING THE EXPERIMENTAL JET RESULTS WITH THE MODEL PREDICTIONS................................... 124 FIGURE 5.3 - INTEGRATED CENTRELINE DILUTION RESULTS VERSUS DISTANCE DOWNSTREAM, COMPARING EXPERIMENTAL RESULTS WITH INTEGRATED JET THEORY............................ 125 FIGURE 5.4 - POINT CENTRELINE DILUTION RESULTS VERSUS DISTANCE DOWNSTREAM, COMPARING THE EXPERIMENTAL JET RESULTS WITH MODEL PREDICTIONS AND PREVIOUS EXPERIMENTAL RESULTS ................................................................................................. 126 FIGURE 5.5 ? INTEGRATED CROSS-SECTIONAL CONCENTRATION PROFILES FROM PLUME RUNS 1, 3 & 4 AT VARIOUS LOCATION DOWNSTREAM FROM THE SOURCE .................................... 128 FIGURE 5.6 - CONCENTRATION SPREAD RESULTS VERSUS DISTANCE DOWNSTREAM, COMPARING THE EXPERIMENTAL PLUME RESULTS WITH THE MODEL PREDICTIONS AND PREVIOUS EXPERIMENTAL RESULTS ................................................................................................. 129 FIGURE 5.7 - POINT CENTRELINE DILUTION RESULTS VERSUS DISTANCE DOWNSTREAM, COMPARING THE EXPERIMENTAL PLUME RESULTS WITH MODEL PREDICTIONS AND PREVIOUS EXPERIMENTAL RESULTS................................................................................. 130 XIII FIGURE 5.8 ? TRAJECTORY RESULTS HORIZONTAL BUOYANT JET IN STILL AMBIENT, COMPARING THE EXPERIMENTAL RESULTS WITH MODEL PREDICTIONS AND PREVIOUS EXPERIMENTAL RESULTS .......................................................................................................................... 132 FIGURE 5.9 - CROSS-SECTIONAL PROFILES FROM NEGATIVELY DISCHARGED BUOYANT JET RUN 38, INITIAL CONDITIONS: ?0 = 0O, FR0 = 56.51 AND RE0 = 4704...................................... 133 FIGURE 5.10 ? CONCENTRATION SPREAD RESULTS HORIZONTAL BUOYANT JET IN STILL AMBIENT, COMPARING THE EXPERIMENTAL RESULTS WITH MODEL PREDICTIONS AND PREVIOUS EXPERIMENTAL RESULTS................................................................................. 134 FIGURE 5.11 ? POINT CENTRELINE DILUTION RESULTS HORIZONTAL BUOYANT JET IN STILL AMBIENT, COMPARING THE EXPERIMENTAL RESULTS WITH MODEL PREDICTIONS AND PREVIOUS EXPERIMENTAL RESULTS................................................................................. 135 FIGURE 5.12 - TRAJECTORY RESULTS POSITIVELY BUOYANT JET EXPERIMENTS, COMPARING THE EXPERIMENTAL RESULTS WITH MODEL PREDICTIONS....................................................... 137 FIGURE 5.13 ? CONCENTRATION SPREAD RESULTS POSITIVELY BUOYANT JET EXPERIMENTS, COMPARING THE EXPERIMENTAL RESULTS WITH MODEL PREDICTIONS............................ 138 FIGURE 5.14 ? POINT CENTRELINE DILUTION RESULTS POSITIVELY DISCHARGED BUOYANT JET EXPERIMENTS, COMPARING THE EXPERIMENTAL RESULTS WITH MODEL PREDICTIONS .... 139 FIGURE 5.15 ? TRAJECTORY RESULTS FROM ADVECTED LINE MOMENTUM PUFF AND ADVECTED JET EXPERIMENTS ARE COMPARED WITH PREVIOUS RESULTS AND MODEL PREDICTIONS.. 143 FIGURE 5.16 ? CONCENTRATION SPREAD RESULTS FROM THE VERTICALLY DISCHARGED MOMENTUM PUFF AND ADVECTED JET EXPERIMENTS ...................................................... 145 FIGURE 5.17 - Y-INTEGRATED DILUTION RESULTS FROM ADVECTED LINE MOMENTUM PUFF AND ADVECTED JET EXPERIMENTS .......................................................................................... 146 FIGURE 5.18 ? DOUBLE INTEGRATED DILUTION RESULTS FROM ADVECTED JET AND JET EXPERIMENTS .................................................................................................................. 147 FIGURE 5.19 ? CROSS-SECTIONAL MINIMUM DILUTION RESULTS FROM VERTICALLY DISCHARGED MOMENTUM PUFF EXPERIMENTS...................................................................................... 148 FIGURE 5.20 - TRAJECTORY RESULTS FROM HORIZONTALLY DISCHARGED BUOYANT JET IN A MOVING AMBIENT............................................................................................................ 150 FIGURE 5.21 - LIMITING TRAJECTORY RESULTS FOR OBLIQUE DISCHARGED BUOYANT JETS IN A MOVING AMBIENT............................................................................................................ 152 FIGURE 5.22 ?TRAJECTORY RESULTS FROM BUOYANT JET IN MOVING AMBIENT EXPERIMENTS ........................................................................................................................................ 155 FIGURE 5.23 - CONCENTRATION SPREAD RESULTS FROM BUOYANT JET IN MOVING AMBIENT EXPERIMENTS WITH INITIAL DISCHARGE ANGLES OF 90? & 120? .................................... 157 XIV FIGURE 5.24 - CONCENTRATION SPREAD RESULTS FROM BUOYANT JET IN MOVING AMBIENT EXPERIMENTS WITH INITIAL DISCHARGE ANGLES OF -20? AND 0?.................................... 157 FIGURE 5.25 ?MINIMUM DILUTION RESULTS FROM BUOYANT JET IN MOVING AMBIENT EXPERIMENTS .................................................................................................................. 159 FIGURE 5.26 ? MOMENTUM MODEL PREDICTION FOR CONCENTRATION SPREAD IN WEAK JET REGION............................................................................................................................ 160 FIGURE 5.27 ? MOMENTUM MODEL PREDICTION FOR CENTRELINE DILUTION IN WEAK JET REGION............................................................................................................................ 160 FIGURE 6.1 - SCHEMATIC DIAGRAM OF THE GENERIC DISCHARGE CONFIGURATION FOR NEGATIVELY BUOYANT JET ............................................................................................. 165 FIGURE 6.2 - AVERAGED LIF IMAGE SHOWING THE PATH AND ADDITIONAL MIXING ASSOCIATED WITH A NEGATIVELY BUOYANT DISCHARGE. THE DASHED LINES REPRESENT ANALYTICAL MODEL PREDICTIONS OF THE FLOW?S PATH AND SPREAD. THE INITIAL CONDITIONS FOR THIS FLOW ARE ?0 = 45?, FR0 = 48.66 AND RE0 = 2945........................................................... 172 FIGURE 6.3 - LIF AND LA CONCENTRATIONS PROFILES FROM INCLINED JETS......................... 176 FIGURE 6.4 ? HORIZONTAL LOCATION OF MAXIMUM CENTRELINE HEIGHT FOR FLOWS WITH AN INITIAL ANGLE OF 45? AND REYNOLDS NUMBERS RANGING FROM 2144 TO 4639............ 177 FIGURE 6.5 - VERTICAL LOCATION OF MAXIMUM CENTRELINE HEIGHT FOR DISCHARGES WITH AN INITIAL ANGLE OF 45? AND REYNOLDS NUMBERS RANGING FROM 2144 TO 4639............ 178 FIGURE 6.6 - MAXIMUM HEIGHT OF EDGE OF JET FOR DISCHARGES WITH AN INITIAL ANGLE OF 45? AND REYNOLDS NUMBERS RANGING FROM 2144 TO 4639 ........................................ 178 FIGURE 6.7 - CENTRELINE INTEGRATED DILUTION AT MAXIMUM HEIGHT FOR DISCHARGES WITH REYNOLDS NUMBERS RANGING FROM 2144 TO 5207 ...................................................... 180 FIGURE 6.8 - THE COEFFICIENT FOR THE HORIZONTAL CENTRELINE LOCATION AT MAXIMUM HEIGHT AS A FUNCTION OF THE INITIAL DISCHARGE ANGLE............................................. 181 FIGURE 6.9 - THE COEFFICIENT FOR MAXIMUM CENTRELINE ELEVATION AS A FUNCTION OF INITIAL DISCHARGE ANGLE.............................................................................................. 182 FIGURE 6.10 - THE COEFFICIENT FOR MAXIMUM ELEVATION OF THE FLOW EDGE AS A FUNCTION OF INITIAL DISCHARGE ANGLE ......................................................................................... 182 FIGURE 6.11 ? HORIZONTAL LOCATION OF IMPACT POINT FOR FLOWS WITH INITIAL ANGLES OF 15? AND 45? .................................................................................................................... 184 FIGURE 6.12 - CENTRELINE INTEGRATED DILUTION AT IMPACT POINT FOR DISCHARGES WITH REYNOLDS NUMBERS RANGING FROM 2144 TO 5207 ...................................................... 185 FIGURE 6.13 - THE COEFFICIENT FOR THE HORIZONTAL CENTRELINE LOCATION AT THE IMPACT POINT AS A FUNCTION OF THE INITIAL DISCHARGE ANGLE .............................................. 187 XV FIGURE 6.14 - CENTRELINE INTEGRATED DILUTION DATA AT THE IMPACT POINT AS A FUNCTION OF THE INITIAL DISCHARGE ANGLE .................................................................................. 187 FIGURE 6.15 ? TRAJECTORY RESULTS FOR NEGATIVELY BUOYANT JETS WITH DISCHARGE ANGLES 15?, 30?, 45? AND 60?........................................................................................ 190 FIGURE 6.16 ? SPREAD COMPARISON BETWEEN EXPERIMENTAL DATA AND MOMENTUM MODEL FOR DISCHARGE ANGLES 15?, 30?, 45? AND 60?.............................................................. 193 FIGURE 6.17 - CONCENTRATION SPREAD RESULTS AS A FUNCTION OF DISTANCE FROM SOURCE ........................................................................................................................................ 194 FIGURE 6.18 - CENTRELINE INTEGRATED DILUTION DATA AS A FUNCTION OF DISTANCE DOWNSTREAM FOR DISCHARGE ANGLES 15?, 30?, 45? AND 60? ..................................... 196 FIGURE 6.19 - CROSS-SECTIONAL PROFILE AT THE MAXIMUM HEIGHT FOR A 45? BUOYANT JET ........................................................................................................................................ 198 FIGURE 6.20 - HORIZONTAL CROSS-SECTIONAL PROFILE AT SOURCE AND IMPACT HEIGHT FOR A 45? BUOYANT JET............................................................................................................ 198 FIGURE 6.21 - CROSS-SECTIONAL PROFILE AT THE IMPACT POINT FOR A 45? BUOYANT JET .... 199 FIGURE 7.1 - SCHEMATIC DIAGRAM OF THE GENERIC DISCHARGE CONFIGURATION FOR OBLIQUE MOMENTUM PUFF ............................................................................................................ 203 FIGURE 7.2 ? TRAJECTORY RESULTS FOR OBLIQUE DISCHARGES IN AMBIENT FLOW................ 214 FIGURE 7.3 - ANALYTICAL SOLUTIONS VERSUS EXPERIMENTAL TRAJECTORY DATA FOR PUFF REGION............................................................................................................................ 216 FIGURE 7.4 - ANALYTICAL SOLUTIONS VERSUS EXPERIMENTAL SPREAD DATA FOR PUFF REGION ........................................................................................................................................ 216 FIGURE 7.5 ? ANALYTICAL SOLUTIONS VERSUS Y-INTEGRATED CENTRELINE DILUTION RESULTS FOR PUFF REGION............................................................................................................. 217 FIGURE 7.6 - ANALYTICAL SOLUTIONS VERSUS EXPERIMENTAL TRAJECTORY DATA FOR WEAK- JET REGION ...................................................................................................................... 219 FIGURE 7.7 - ANALYTICAL SOLUTIONS VERSUS EXPERIMENTAL CONCENTRATION SPREAD DATA FOR WEAK-JET REGION .................................................................................................... 220 FIGURE 7.8 ? ANALYTICAL SOLUTIONS VERSUS Y-INTEGRATED CENTRELINE DILUTION RESULTS FOR WEAK-JET REGION .................................................................................................... 220 FIGURE 7.9 - ANALYTICAL SOLUTIONS VERSUS EXPERIMENTAL TRAJECTORY DATA FOR TRANSITION REGION ........................................................................................................ 222 FIGURE 7.10 - ANALYTICAL SOLUTIONS VERSUS EXPERIMENTAL CONCENTRATION SPREAD DATA FOR TRANSITION REGION ................................................................................................. 223 XVI FIGURE 7.11 - ANALYTICAL SOLUTIONS VERSUS Y-INTEGRATED CENTRELINE DILUTION RESULTS FOR TRANSITION REGION ................................................................................................. 223 FIGURE 7.12 ? PEAK DILUTION RESULTS FOR OBLIQUE DISCHARGES IN MOVING AMBIENT...... 227 FIGURE 8.1 ? INTEGRATED CROSS-SECTIONAL CONCENTRATION PROFILES RUN 1................... 237 FIGURE 8.2 - VALUES FOR F AS A FUNCTION OF THE DISTANCE IN THE Y AND Z DIRECTIONS.... 238 FIGURE 8.3 ? INTEGRATED CROSS-SECTIONAL CONCENTRATION PROFILES RUN 2................... 242 FIGURE 8.4 - INTEGRATED CROSS-SECTIONAL CONCENTRATION PROFILES RUN 3.................... 243 FIGURE 8.5 - INTEGRATED CROSS-SECTIONAL CONCENTRATION PROFILES RUN 4.................... 245 FIGURE 8.6 - INTEGRATED CROSS-SECTIONAL CONCENTRATION PROFILES RUN 5.................... 246 FIGURE 8.7 - DOUBLE INTEGRATED DILUTION RESULTS FROM RUNS 1 AND 2.......................... 248 FIGURE 8.8 - RUN 1 TRAJECTORY RESULTS IN X-Y PLANE COMPARED WITH TRAJECTORY RESULTS FROM NON-BUOYANT DISCHARGE IN MOVING AMBIENT (REYNOLDS NUMBER = 2133, VELOCITY RATIO = 0.033) ..................................................................................... 250 FIGURE 8.9 - TRAJECTORY RESULTS BUOYANT JET WITH THREE-DIMENSIONAL TRAJECTORIES RUN 1 .............................................................................................................................. 252 FIGURE 8.10 - RUN 2 TRAJECTORY RESULTS IN X-Y PLANE COMPARED WITH TRAJECTORY RESULTS FROM NON-BUOYANT DISCHARGE IN MOVING AMBIENT (REYNOLDS NUMBER = 4379, VELOCITY RATIO = 0.029) ..................................................................................... 254 FIGURE 8.11 - TRAJECTORY RESULTS BUOYANT JET WITH THREE-DIMENSIONAL TRAJECTORIES RUN 2 .............................................................................................................................. 256 FIGURE 8.12 - TRAJECTORY RESULTS BUOYANT JET WITH THREE-DIMENSIONAL TRAJECTORIES RUN 3 .............................................................................................................................. 258 FIGURE 8.13 - TRAJECTORY RESULTS BUOYANT JET WITH THREE-DIMENSIONAL TRAJECTORIES RUN 4 .............................................................................................................................. 260 FIGURE 8.14 - TRAJECTORY RESULTS BUOYANT JET WITH THREE-DIMENSIONAL TRAJECTORIES RUN 5 .............................................................................................................................. 261 FIGURE 8.15 - Y-INTEGRATED DILUTION RESULTS ................................................................... 264 FIGURE 8.16 - Z-INTEGRATED DILUTION RESULTS ................................................................... 265 FIGURE A.1 - THE CIE LUMINOUS EFFICIENCY FUNCTION....................................................... 287 FIGURE A.2 - EXAMPLES OF SPECTRAL WEIGHING FUNCTIONS (POYNTON, 1996) ................. 288 FIGURE A.3 - CIE COLOUR MATCHING FUNCTION (POYNTON, 1996) ...................................... 289 FIGURE E.1 ? HORIZONTAL LOCATION OF MAXIMUM CENTRELINE HEIGHT FOR FLOWS WITH INITIAL ANGLES OF 30? AND 60?...................................................................................... 319 FIGURE E.2 - VERTICAL LOCATION OF MAXIMUM CENTRELINE HEIGHT FOR DISCHARGES WITH INITIAL ANGLES OF 30? AND 60?...................................................................................... 320 XVII FIGURE E.3 - MAXIMUM HEIGHT OF EDGE OF JET FOR DISCHARGES WITH INITIAL ANGLES OF 30? AND 60?........................................................................................................................... 321 FIGURE E.4 - HORIZONTAL LOCATION OF IMPACT POINT FOR FLOWS WITH INITIAL ANGLES OF 30? AND 60? .................................................................................................................... 322 FIGURE E.5 - THREE-DIMENSIONAL VIEW OF TRAJECTORY RESULTS BUOYANT JET WITH THREE- DIMENSIONAL TRAJECTORIES RUN 2 ................................................................................ 323 FIGURE E.6 - THREE-DIMENSIONAL VIEW OF TRAJECTORY RESULTS BUOYANT JET WITH THREE- DIMENSIONAL TRAJECTORIES RUN 3 ................................................................................ 323 FIGURE E.7 - THREE-DIMENSIONAL VIEW OF TRAJECTORY RESULTS BUOYANT JET WITH THREE- DIMENSIONAL TRAJECTORIES RUN 4 ................................................................................ 324 FIGURE E.8 - THREE-DIMENSIONAL VIEW OF TRAJECTORY RESULTS BUOYANT JET WITH THREE- DIMENSIONAL TRAJECTORIES RUN 5 ................................................................................ 324 XVIII List of Tables TABLE 2.1 - DOMINANT PARAMETERS IN FLOW REGIONS .......................................................... 20 TABLE 2.2 ? TRANSITION LENGTH-SCALES FOR TRANSITION BETWEEN FLOW REGIONS ............ 20 TABLE 2.3 ? CHARACTERISTIC RELATIONS OF FLOW PARAMETERS WITH DISTANCE WITHIN FLOW REGIONS ............................................................................................................................ 21 TABLE 3.1 - COMPARISON OF VELOCITY AND CONCENTRATION SPREAD VALUES...................... 57 TABLE 4.1 - DOMINANT MOMENTUM FLUX RATIOS PER FLOW REGION.................................... 109 TABLE 4.2 - TRANSITION LENGTH-SCALES FOR RELEVANT FLOW TRANSITIONS ...................... 109 TABLE 4.3 ? MOMENTUM FLUX RATIO AT FLOW TRANSITION FOR RELEVANT FLOW TRANSITION ........................................................................................................................................ 110 TABLE 8.1 ? LENGTH-SCALE ANALYSIS AND OBSERVATION FROM BUOYANT JET IN CROSS-FLOW EXPERIMENTS, CHEUNG (1991)....................................................................................... 240 TABLE D.1 - INITIAL CONDITIONS FOR SIMPLE JET RUNS ........................................................ 307 TABLE D.2 - INITIAL CONDITIONS FOR PURE PLUME RUNS ...................................................... 308 TABLE D.3 - INITIAL CONDITIONS FOR HORIZONTAL BUOYANT JET RUNS ............................... 308 TABLE D.4 - INITIAL CONDITIONS FOR POSITIVELY BUOYANT JET RUNS ................................. 308 TABLE D.5 - INITIAL CONDITION FOR ADVECTED LINE MOMENTUM PUFF RUNS ..................... 309 TABLE D.6 - INITIAL CONDITION FOR ADVECTED JET RUNS .................................................... 310 TABLE D.7 - INITIAL CONDITIONS FOR BUOYANT JET IN MOVING AMBIENT RUNS.................... 311 TABLE D.8 - INITIAL CONDITIONS FOR LA NEGATIVELY BUOYANT JET RUNS ......................... 312 TABLE D.9 - INITIAL CONDITIONS FOR LIF NEGATIVELY BUOYANT JETS ................................ 314 TABLE D.10 - INITIAL CONDITION FOR NON-BUOYANT OBLIQUE DISCHARGES IN MOVING AMBIENT ......................................................................................................................... 315 TABLE D.11 - INITIAL CONDITION FOR BUOYANT JETS WITH 3D TRAJECTORIES ..................... 317 XIX List of Notations a Calibration constant b, b1, b2, b3, b4, bjp, bmt, bwj Velocity spread bc Concentration spread bth Average Gaussian velocity spread bth-mt Average double-Gaussian velocity spread C Local tracer concentration C0 Initial tracer concentration C0rp Trajectory coefficient c1 Constant in empirical relationship for h c2-c7 Transition constants C1-C6 Reflection loss constants Cair, Cglass, Cperspex, Cwater, Cdye Absorption loss constants Cc Tracer concentration at centre of double-Gaussian approximation Ci Local integrated tracer concentration Ci0 Integrated initial tracer concentration Cii Double-integrated tracer concentration Cii0 Double-integrated initial tracer concentration Cil, Cil-assumed, Cil-actual Integrated centreline tracer concentration Cily Integrated single-Gaussian centreline tracer concentration in the y-direction Cilz Integrated single-Gaussian centreline tracer concentration in the z-direction Ciy Local integrated tracer concentration in the y-direction Ciz Local integrated tracer concentration in the z-direction Ciz-peak Integrated peak tracer concentration in the z-direction Cjk Constant in weak-jet spread assumption Cl, Cl-1 Centreline tracer concentration Cl-vortex Tracer concentration at centre of double-Gaussian approximation Cm Experimental maximum tracer concentration within cross-section Cpeak Tracer concentration at peak of double-Gaussian approximation Cth-m Average tracer concentration in the advected line momentum puff region Cth-t Average tracer concentration in advected thermal region XX cwj1, cwj2 Constants in derivation of weak-jet trajectory equation d Diameter of the source f Double-Gaussian approximation constant Fr, Fr0 Initial Froude number g Gravitational constant h Double-Gaussian approximation constant h* Asymptotic value of h Icam Intensity of ray of light at camera Igreen Intensity of green gun Im, Iq, Ic, Iqc, Icdg, Iqdg, Shape constants Iref Intensity of reference with no dye present Iref_green Intensity of reference green gun Isource Intensity of ray of light at source k Gaussian spread constant kcm Dilution relationship coefficient kme, kxm, kx0R, kzm Trajectory relationship coefficients ksg Concentration spread-rate for the single-Gaussian of the double-Gaussian approximation divided by ? kth Average Gaussian spread constant L, L1 Distance from source along trajectory in angled jet experiment ljp Length-scale for the transition between the strong-jet and advected plume regions ljm Length-scale for the transition between the strong-jet and advected line momentum puff regions lmt Length-scale for the transition between the advected line momentum puff region and the advected thermal region Lmz* Length-scale for the transition between weakly advected and strongly advected regions for the trajectory data of the obliquely discharged non-buoyant jet in a moving ambient M0 Initial momentum flux M0' Reflection of M0 in the x-y plane Ma Entrained ambient momentum flux Ma0 Initial ambient momentum flux Mb Buoyancy-generated momentum flux Me Excess momentum flux Me0 Initial excess momentum flux Me0x, Me0y, Me0z Components of the initial excess momentum flux in the x, y and z-directions XXI Ms Total momentum flux Ms0 Initial total momentum flux mth Average double-Gaussian spread constant mth-m Average double-Gaussian advected line momentum puff spread constant mth-t Average double-Gaussian advected thermal spread constant Mx Momentum flux in the x-direction My Momentum flux in the y-direction Mz Momentum flux in the z-direction n Coordinate representing distances in the direction perpendicular to the initial velocity p Instantaneous modified pressure P Mean Modified Pressure p' Fluctuating modified pressure Q0 Initial flow rate Q?0 Initial density deficit flux r Radial coordinate Re, Re0 Discharge Reynolds number s Distance from source along trajectory si Coordinate representing distances in the direction of the initial velocity Sjp Length-scale for the transition between jet and plume regions Sm Distance from source to the maximum centreline height in the s-direction syz Distance from the source, travelled along the projection of the direction of the initial discharge in the y-z plane t Time u Local velocity U0 Initial velocity Ua Ambient velocity ue Entrainment velocity Ue Excess velocity Ue0 Initial excess velocity Ueth Average excess velocity ui Instantaneous velocity Ui Mean velocity ui' Fluctuating velocity Ul Single-Gaussian cross-sectional centreline velocity Ur Ratio of ambient velocity over the initial velocity w Substitute variable in derivation of weak-jet trajectory equation XXII x Cartesian coordinate in the same direction as the ambient velocity; (in the direction of the horizontal component of the initial velocity for still ambient flows) xamp Horizontal distance from advected line momentum puff virtual source x0r Horizontal distance from source to impact point x0rj Horizontal distance from source to impact point in the jet region x0rp Horizontal distance from source to impact point in the plume region xjp Horizontal distance from source to jet to plume transition point xm Horizontal distance from source to maximum centreline height xtrp Horizontal distance from transition point to impact point in the plume region xwj Horizontal distance from weak-jet virtual source y Cartesian coordinate perpendicular to the x-coordinate in the horizontal plane y' Distance in y-direction from centre of double-Gaussian cross-sectional profile yl Distance in y-direction from source to centre of double-Gaussian cross-sectional profile z Cartesian coordinate in the same direction as the vertical component of the initial velocity z' Distance in z-direction from centre of double-Gaussian cross-sectional profile zamp Vertical distance from advected line momentum puff virtual source zjp Vertical distance from source to jet to plume transition point zl Distance in z-direction from source to centre of double-Gaussian cross-sectional profile zm Vertical distance from source to maximum centreline height zme Vertical distance from source to edge of flow at maximum centreline height zt Vertical distance from source to the point of transition zv Vertical distance from the point of transition to the virtual source zwj Vertical distance from weak-jet virtual source ? Angle between the total momentum flux and its reflection in the x-y plane ?0 Angle between the initial excess momentum flux and the horizontal plane (3D trajectory flow) ? Angle between the reflection of the total momentum flux in the x-y plane and the ambient momentum flux ?amp Ratio of the puff edge radius to the nominal radius b ?0 Angle between the projection of the initial excess momentum flux in the x-y plane and x-axis (3D trajectory flow) ?wj Ratio of the weak-jet edge radius to the nominal radius b ? Local density deficit ? Entrainment coefficient ?0 Angle between the excess momentum flux and the ambient momentum flux XXIII ? Angle between the assumed camera position and the actual camera position in relation to parallax issues ? Kinematic viscosity of water ? Angle between the trajectory and the velocity spread ? Density of jet fluid ?a Density of ambient fluid ? Ratio of the spread of the mean concentration distribution to the spread of the mean velocity distribution ?r Ratio of ?-values of the weak-jet region to the puff region XXIV Chapter 1 ? Introduction 1 Chapter 1 ? Introduction 1.1 ? General Introduction In the modern society urban areas are expanding rapidly and at the same time the environmental awareness of its citizens has increased. Both these processes have lead to a more critical point of view with respect to the effluent disposal problem. Of all the major cities in the world, about 80% of them are near the coast. This makes the disposal of wastewater into the ocean an attractive option. It is close-by, it is well buffered for both pH and temperature changes, and vast quantities of dissolved oxygen are available to biodegrade the organics. Also the concentration of the contaminants can be reduced with the help of initial dilution and the dilution capabilities of the ocean. Engineering structures have been built for years to dispose of wastewater into the ocean. The wastewater is normally disposed of through an outfall. The main purpose of the outfall is to enhance the dilution as it is released into the receiving water and thus reduce the impact on the local environment. Models of the outfall are needed to determine whether or not the outfall meets the environmental requirements set by the local agencies. Communities are now generally favouring higher degrees of land-based treatment and in many cases the environmental requirements can be met at the end of the initial dilution zone. This is the region where the essentially fresh water effluent rises to the surface of the higher density oceanic waters. Extensive mixing takes place in this region. If it can be shown that the environmental requirements are satisfied within this region then the need to model the behaviour of the effluent in the ocean mixing region is largely eliminated. The development of ocean mixing models is an expensive option, because these models require extensive quantities of field data for calibration and validation procedures, if reasonable predictions are to be obtained. A decision as to whether or not such models are needed is based on the degree of confidence with which predictions of dilution in the initial dilution zone can be made. The accuracy of the initial dilution zone models is therefore of particular importance. The release of fast flowing wastewater into a large body of stagnant or slow moving unstratified ocean water, creates a jet flow. The body of water is normally large enough so Chapter 1 ? Introduction 2 that the boundaries do not interfere with the flow of the jet. The fluid of the jet is fully turbulent when the Reynolds number, based on the initial conditions of the jet (source diameter, initial velocity), is larger than approximately two thousand. The relative densities of the two fluids determine whether the flow is buoyant or non-buoyant. During the initial dilution phase the wastewater will generally go through three distinct regions. The first one is the jet-region where the initial momentum flux of the wastewater dominates its behaviour. The second region is the plume region, where the buoyancy forces dominate the behaviour of the flow. The third phase is the advected thermal region, where the ambient current dominates the behaviour. Many experiments have been conducted to understand the behaviour of the flow released into a stagnant unstratified ambient, and a large amount of knowledge is now available. However a high percentage of this work has focused on the jet fluid being released into the ambient horizontally or vertically. Fewer experimental investigations have been carried out for discharges that are released into a moving unstratified ambient current. In most cases these experiments were carried out with the source flow either released vertically or in the same direction as the ambient current. The flow trajectories in these stagnant and moving discharge configurations are all two-dimensional and, as a result of previous studies, reasonably well understood. However, in general effluent is released at some angle to the ambient current and the trajectory of the discharge is three-dimensional. Few studies have focussed on this type of discharge. 1.2 ? Problem Overview To be able to estimate the dilution and trajectories of the wastewater plume released into an unstratified ambient, relatively simple integral methods are routinely employed to predict the behaviour of the plume with two- and three-dimensional trajectories. Experimental investigations provide data to which the model predictions can be compared. Positive matches between experimental data and model prediction build confidence in the use of the model. However due to gaps in experimental evidence the confidence in the use of the models under some circumstances is limited. These circumstances include two-dimensional trajectory flows in a still ambient that are discharged at oblique angles and buoyant jet flows with three- dimensional trajectories. As part of this project an experimental investigation will be conducted into the behaviour of discharges with two-dimensional trajectories (including jets and plumes) and three-dimensional trajectories. Chapter 1 ? Introduction 3 1.3 ? Scope of Research For the past century researchers have actively investigated the phenomenon of the buoyant jet. In the last 50 years several mathematical models have been presented to predict the trajectory and dilution of buoyant jets with different discharge configurations. An overview of the different models as well as their mathematical background is given in Chapter 2. All models have been verified by laboratory and field data. Experimental studies into the behaviour of buoyant jets started at approximately the same time the first model was presented, and the database has been expanding ever since. Considerable amounts of laboratory and field data are now available for particular flow regions and these are summarized in Chapter 2, where discharge configurations for which there is very limited experimental data are also discussed. To create a high quality data set two quantitative flow visualizations techniques are used. These are presented in Chapter 3. The first is LIF or Laser Induced Fluorescence. LIF has been used for nearly two decades and has been used to visualize buoyant jets for the past decade. LIF performs well for investigations into the behaviour of flows with two- dimensional trajectories and is used in the present study during the investigation of discharges in a still ambient fluid (in particular the negatively buoyant jet). An alternative flow visualization technique is developed to help with the investigation into the behaviour of three- dimensional trajectory flows. It is referred to as LA or Light Attenuation and is also employed for the two-dimensional trajectory flow experiments. The technique is based on a linear relationship between the increase in concentration of dye in the water and decrease in the intensity of the light passing through it. A new model has been set up to aid in the design and to monitor the performance of the experiments. The model is based on a relatively simple framework when compared to existing models. The behaviour of the discharge at any particular location is determined by the relative magnitudes of three distinct forms of momentum flux (the initial momentum flux, buoyancy- generated momentum flux, and the entrained ambient momentum flux). Hence the new model is called the Momentum Model. The development of the model is outlined in Chapter 4. Its performance is verified by comparing it with existing experimental data for different flow configurations. Chapter 1 ? Introduction 4 A wide range of LA experiments have been conducted. This is firstly done to verify the visualization technique, secondly to provide additional insight into the behaviour of discharges with two-dimensional trajectories, and finally to investigate the behaviour of discharges with three-dimensional trajectories. In Chapter 5 results for a range of initial discharge configurations are presented and comparisons are made with existing data and model predictions, as well as predictions from the Momentum Model. A more detailed investigation is carried out into the behaviour of negatively buoyant jets and this is described in Chapter 6. There is a growing interest in the process of desalination to produce drinking water, because of increased uncertainties in water supply associated with unstable weather patterns. Discharges from desalination plants are in the form of negatively buoyant jets (wastewater with a higher density than the receiving ambient). Earlier investigations have primarily focused on a single angle of discharge (60?). The present investigation expands that to a range of angles and includes a comparison with both analytical and numerical model predictions. Another flow with a two-dimensional trajectory that is investigated in more detail is the non- buoyant oblique discharge in a moving ambient. Both horizontally and vertically released non-buoyant discharges in a moving ambient have been studied in the past and the influence of the ambient on the flow in the strongly advected region was found to be significantly different for the two cases. The attention of the current investigation is focused on the transition angle that defines the distinctly different final forms of flow behaviour. The theoretical and experimental investigations can be found in Chapter 7. Finally, the results of the experimental investigation into the behaviour of three-dimensional trajectory flows are presented in Chapter 8. As the knowledge of these flows is limited, the main purpose of the study is to observe these flows in general, comment on their behaviour and compare the results with predictions from current models. Conclusions from the current research and possible suggestions for future work are put forward in Chapter 9. Chapter 2 ? Review of Previous Research 5 Chapter 2 ? Review of Previous Research 2.1 ? Introduction In this chapter an overview is given of relevant research undertaken into the behaviour of buoyant jets released into an unstratified ambient. For over a century research into this topic has been carried out and this has resulted in extensive knowledge about the theory of buoyant jets. The theories developed have formed the basis for several mathematical models and a summary of the models as well as their theoretical background is presented. Experimental investigations have led to considerable quantities of relevant experimental data on buoyant jets. Previous experimental investigations are discussed in this chapter and flow configurations with very limited experimental data are highlighted. 2.2? Problem Formulation of the Buoyant Jet A (buoyant) jet is generated when relatively fast flowing fluid, from a continuous source, is discharged in a reservoir of relatively slow flowing fluid and the density difference between the two fluids is small. The high velocity gradients at the interface between jet and the ambient fluid make it highly unstable, and cause the jet fluid to rotate. These turbulent vortices entrain the ambient fluid into the jet, causing the mixing processes and the dissipation of the energy from the discharge. Previous studies suggest that a buoyant jet flow can be divided into distinct flow regions (for example Pun (1998) and Jirka (2004)). These flow regions are the initial region, strong jet region, weak jet region, advected line momentum puff region, advected plume region, and the advected thermal region, however the influence of the initial region on the overall behaviour of the flow is minimal. In each of these regions the flow is dominated by a group of independent flow parameters, and the overall flow behaviour can therefore be described by a sequence of these distinct flow regions. Research, both experimental and theoretical modelling, has therefore focused on increasing the understanding of the behaviour of the flow within the distinct flow regions, and to a lesser extend on the transition regions that connect the flow regions. Chapter 2 ? Review of Previous Research 6 As the distinct flow regions are the same for buoyant jet flows with two-dimensional and three-dimensional paths, it is assumed that the understanding of the behaviour of the flow, gained from experiments with two-dimensional trajectories, can also be applied to buoyant discharges with three-dimensional trajectories. Models based on this approach are assumed to predict with reasonable accuracy the behaviour of discharges with two-dimensional and three- dimensional paths (Cheung et al. 2000; Jirka 2004). 2.3 - Research History Buoyant jets have been observed and commented on since the beginning of modern science, for example, they were observed coming out of smokestacks and volcanoes. Jirka (2004) gives an extended overview of the history of research on the buoyant jet. Some key features are given here. It was not until the beginning of the twentieth century that the first detailed experimental measurements and an analytical explanation were completed on the subject. The investigation was lead by L. Prandtl in the 1920?s; he applied boundary layer theory to the jet flow. Soon it was followed by measurements on the round non-buoyant jet (Zimm 1921) and the plane non- buoyant jet (F?rthmann 1934). These measurements were the basis for the development of the similarity solutions for the spread and the velocity decay of the jet (G?rtler 1942; Reichardt 1942; Tollmien 1926). Prandtl?s turbulent mixing length hypothesis was used to relate the shear stresses to the mean flow of the jet and this method was taken a step further to include a pure vertically rising plume by Schmidt (1941). Reichardt (1943) was the first to determine that the cross-sectional properties of the jet could be approximated by Gaussian profiles, forming the foundation of the jet-integral method. The method was further developed into a jet integral model with the results of more detailed experiments carried out on the simple jet (Albertson et al. 1949) and the pure plume (Rouse et al. 1952). Morton et al. (1956) introduced the idea that fluid momentum, vorticity and scalars in a jet are spread by turbulent entrainment rather than turbulent diffusion. They hypothesised that non- turbulent, irrotational fluid from outside the jet was entrained into the turbulent jet. This viewpoint was quickly incorporated into the already existing jet-integral methods. Jordinson (1956) and Keffer and Baines (1963) included a cross flowing ambient current in their experimental studies of the pure jet. Scorer (1959) introduced dimensional analysis and the use of length scales to separate regions of strong and weak deflections. With a similar analysis Chapter 2 ? Review of Previous Research 7 Turner (1960) and Richards (1963) showed that an internal double vortex pair significantly affected the velocity and scalar distributions when the jet or plume flow was strongly deflected by the ambient current. 2.4 ? Previous Experimental Investigations Over the past six decades many experimental studies have been carried out in the field of buoyant jets. Most of these studies have been carried out in the laboratory. Data from field studies is available, but it was not possible to measure all the major factors influencing the flow behaviour and therefore the data is difficult to interpret. Because of the less-controlled environment outside the laboratory the results were less accurate. Inside the laboratory it is possible to separate the important independent parameters, the initial momentum flux, the buoyancy-generated momentum flux and the ambient momentum flux, from outside influences and from each other. This enables the researcher to carefully determine the influence of each of the factors on the flow. The laboratory studies differed in the use of measurement techniques, aims and types of flow. Because of the increase in the technology available to researchers over time, the studies have become more detailed; the flow measurement techniques more accurate and more complex flow configurations have been monitored. Data from experimental investigations has been used for verification of models as well as determination of the empirical parameters in the length-scale and integral models (see section 2.5). 2.4.1 ? Flow measurement techniques An overview of the main techniques used to measure velocity and concentrations in buoyant jets is given below. Mean velocity measurements in an air jet where made by Albertson et al (1949) using pitot tubes. Rouse et al. (1952) measured velocities above a heated air plume using a vane anemometer (a thermocouple was used to measure temperatures). Pitot tubes were subsequently replaced by Hot-Wire Anemometry (Lassiter 1957) and Laser-Doppler Anemometry (LDA) (Abbis et al. 1975; Capp 1983). These changes expanded the research into the turbulent properties of the flow, as the newer techniques were capable of measuring the fluctuating components. The Hot-Wire Anemometry (HWA) was later upgraded to reduce Chapter 2 ? Review of Previous Research 8 the high local turbulence intensity, caused by the flow around the stationary wires, with the Flying Hot-Wire anemometry (FHW) (Hussein et al. 1994). With the introduction of Particle Image Velocimetry (PIV) (Simoens and Ayrault 1994), velocity measurements were no longer confined to a point, but a planar velocity field could be observed and measured Concentration measurements were made by Ayoub (1971) using conductivity probes, which were used to determine mean cross-sectional concentration profiles, and a black and white still camera in combination with potassium permanganate dye was used to determine the flow trajectory. For the buoyant jet in a cross-flow experiments a second black and white camera was added to record the trajectory in both the x-y and x-z planes. Papanicolau (1984) investigated the concentration profiles of a buoyant jet in a still ambient using laser-induced fluorescence (LIF), a non-intrusive optical technique, giving both trajectory and instantaneous concentration measurements. Knudsen (1988) added red dye to her experiments and recorded the trajectory using either a photographic or video camera. The trajectory of the centreline was determined by the averaging the two points defining the visible edges of the flow. With the known trajectory, a set of suction probes was inserted into the flow to measure the concentration at a pre-determined location. The upgrade of LIF to Planar Laser-Induced Fluorescence (PLIF) (van Cruyningen et al. 1990) did for the concentration measurements what PIV has done for the velocity measurements. Hereafter Laser Induced Fluorescence and Planar Laser-Induced Fluorescence are both referred to as LIF. However, simpler techniques continue to provide valuable information. For example, Cheung (1991) used hot water, rather than salt water, to create a difference in density between ambient and jet fluid. Rows of thermistor probes or a single thermilinear probe were then used to find the cross-sectional concentration field. The centre of mass of the concentration field defined the trajectory. Experimental studies have also employed more than one measurement technique to obtain both velocity and concentration data, or to compare results from more than one technique as an internal verification. Examples are Papanicolau (1984) and Chu (1996) using both LDA and LIF, and Wang (2000a) using both PIV and LIF. 2.4.2 ? Flow configurations In this section several different flow configurations are discussed and sources of experimental data listed. These sources are not described in detail here, but data from these studies is Chapter 2 ? Review of Previous Research 9 incorporated into subsequent chapters where appropriate. Note that x is the Cartesian coordinate in the same direction as the ambient velocity; or in the direction of the horizontal component of the initial velocity for still ambient flow, z is the Cartesian coordinate in the same direction as the vertical component of the initial velocity, and y is the Cartesian coordinate perpendicular to the x-coordinate in the horizontal plane. The angle ?0uniF029 is the angle between the excess momentum flux and the ambient momentum flux; or between the excess momentum flux and the x-axis for still ambient flows, and s is the distance from the source along the trajectory of the flow. 2.4.2.1 - Jets The first experimental investigations were on the behaviour of the simple jet. A simple jet flow has no buoyancy flux, as the density of the fluid in the jet (?) and the density of the ambient fluid (?a) are the same, and the ambient environment is stationary (the ambient velocity (Ua) is zero). The behaviour of the flow is therefore dominated by the initial momentum flux (M0 = Q0 U0, where U0 is the initial (uniform) velocity of the flow, and Q0 is the flow rate of the discharge) and is independent of the initial angle of discharge (?0uniF029). Investigations by Corrsin (1943), Hinze and van der Hegge Zijnen (1949), Albertson et al (1949), Corrsin and Uberoi (1950), Forstall and GayLord (1954), Sunavala et al. (1957), Ricou and Spalding (1961), Kiser (1963), Rosler and Bankoff (1963), Becker et al. (1967), Wygnanski and Fiedler (1969), Crow and Champagne (1971), Labus and Symons (1972), Birch et al. (1978), Capp (1983), Hussein et al. (1994), Pun (1998), Law and Wang (2000) and others have led to a firm understanding of the spread, velocity and dilution profiles as well as the rate of entrainment of the simple jet. The mean cross-sectional velocity and concentration profiles were both shown to fit the Gaussian shape well. Later experiments involving a simple jet have been carried out as a first step towards more complicated flow configurations or to investigate the instantaneous behaviour, including the turbulent properties of the jet. A schematic representation of a jet flow can be seen in Figure 2.1, including a mean cross-sectional velocity profile at some distance away from the source. The cross-sectional centreline velocity is represented by Ul, the local velocity in the cross-section by u, and the cross-sectional velocity spread by b. By defining b as the distance from the centre of the cross-section to the e-1Ul contour, the total width of the jet is approximately 4b. C0 is the Chapter 2 ? Review of Previous Research 10 initial concentration of an inert pollutant added to the jet fluid, and d is the diameter of the source. s u d x 2b ?0 ? =?a Ua=0 u(b)=e-1 U l b Ul U0 u(b) Source of jet Cross-sectional velocity profile of jet of jet C0 z Figure 2.1 - Initial conditions and cross-sectional velocity profile for jet experiment 2.4.2.2 ? Pure Plumes Like the simple jet, the pure plume flows into a still ambient. But unlike the simple jet there is a buoyancy flux due to the difference in density between the ambient fluid and the plume fluid, and the initial velocity of the flow is almost negligible (Figure 2.2). Due to the low initial velocity of the flow, the source direction does not influence the behaviour of the flow. In the environment the ambient fluid is generally more dense than the pure plume fluid and therefore the buoyancy flux acts vertically upwards. Examples are heated air released from a vertical smokestack or wastewater (fresh water) released into an ocean (salt water). Adding salt to the pure plume fluid most often generates the density difference in the laboratory and therefore the buoyancy force is acting vertically downwards. Due to the relatively small density differences used (up to approximately three percent) it is appropriate to use Boussinesq?s assumption and the results of the laboratory experiments can be used to describe the phenomena in the outside world. The mean cross-sectional concentration and velocity profiles of pure-plumes have also been shown to fit the Gaussian assumption. It is worth noting that generally pure plume experiments are not conducted in the laboratory, because laboratory flows normally have a notable initial momentum flux. Chapter 2 ? Review of Previous Research 11 d z x ?0 ? 0 ? = ? a U0 Strong Jet Region Advected Line Momentum Puff Plume Region Weak Jet Region u Strong Jet Region Ua Ua a) Advected Jet in Cross Flow b) Advected Jet in Co-Flow Ue < Ua C0 z y Ul Jet-to-Puff transition Region Ua > 0 ? = ? a Ul Ul Strong Jet-to-Weak Jet transition Region y z x x Figure 2.4 - Different flow regions of advected jet 2.4.2.5 ? Buoyant Discharges in an Ambient Flow All previously discussed flows can be described as two-dimensional trajectory flows (including jets and plumes). In all cases it was possible to define a plane that encompasses the complete trajectory of the flow. This may no longer be possible for some configurations of the buoyant jet in an ambient flow, because the densities of the ambient and the source fluid are no longer the same and hence three different forms of momentum flux are generated in the flow. As the buoyancy-generated momentum flux and the entrained ambient momentum flux act in perpendicular directions, the initial momentum flux determines whether a two- Chapter 2 ? Review of Previous Research 15 dimensional or a three-dimensional trajectory flow is created. If the initial momentum flux acts in the same plane as the buoyancy-generated and ambient entrained momentum flux, the resulting flow has a two-dimensional trajectory. The focus of past experimental investigations for buoyant discharges in an ambient flow has largely been on 2D trajectory flows, and in particular on either the vertically discharged buoyant jet (Figure 2.5a) or the co-flowing case where the flow is discharged horizontally (Figure 2.5b). u d d U0 Ua > 0 ? < ? a U0 (Advected Line Momentum Puff /Plume Region) Strong Jet Region (+ Advected Plume Region) Ua Ua a) Vertically Discharged Buoyant Jet in an Ambient Flow b) Horizontally Discharged Buoyant Jet in an Co-Flow C0 z y Ul Ua > 0 ? < ? a Ul y z x Advected Thermal Region u U0 Advected Plume Region Strong Jet Region Ul Advected Thermal Region x Figure 2.5 ? Flow regions of buoyant jet in flowing ambient with a 2D-trajectory The vertically discharged buoyant jet in an ambient flow moves through three regions. At first the flow has jet-like behaviour (the strong jet). In the strong jet region, both the buoyancy- generated momentum flux and the entrained ambient momentum flux increase in size. If the buoyancy-generated momentum flux dominates after the strong jet region, the flow behaves like a plume. However, if the entrained ambient momentum flux dominates the flow after the Chapter 2 ? Review of Previous Research 16 strong jet region, the flow is an advected line momentum puff. After either the plume or puff regions a second flow transition takes place and the flow is then in the advected thermal region where both the buoyancy-generated and the entrained ambient momentum flux dominate the flow. In the weakly advected flow regions (strong jet and plume regions) the deflections due to the ambient current are small. In the strongly advected region (advected line momentum puff and advected thermal regions) the flow is noticeably bent over due to the ambient current and the velocity and concentration profiles resemble a counter-rotating vortex pair. Experimental investigations into this flow configuration have been carried out by Fan (1967), Chu and Goldberg (1974), Wright (1977) and Cheung (1991). Hewett (1971) investigated the vertical heated jet in a cross-flow. By changing the angle of release of the discharge so that it is in line with the ambient current a buoyant jet in a co-flow is produced (Figure 2.5b). In this situation it is not possible for an advected line momentum puff to form and the presence of buoyancy effectively eliminates the possibility of the formation of a weak jet. Thus only three different flow regions are possible (strong-jet, plume, advected thermal), making their identification relatively simple. Ayoub (1971), Knudsen (1988), Davidson et al (1991) , Wong and Lee (1991) and Gaskin and Wood (1993) have studied the horizontal buoyant jet in a coflow. Experimental studies into the behaviour of buoyant jets with two-dimensional trajectories in an ambient flow with discharge configurations that differ from the two mentioned above are less common. Knudsen (1988) studied the horizontal buoyant jet in a counter-flow. Chu (1975b) and Anderson et al. (1973) have investigated the behaviour of negatively buoyant jets in a cross-flow. Chu released the initial momentum flux perpendicular to the ambient flow; Anderson et al. released the initial momentum flux at 60? to the cross-flow. If the source outlet is not lying in the plane of the ambient motion and buoyancy force, the initial momentum flux is not in the direction of the buoyancy-generated and entrained ambient momentum flux plane. The flow trajectory is therefore three-dimensional and the flow is called a buoyant jet in a cross-flow (Figure 2.6). The flow can pass through the same regions as the equivalent two-dimensional trajectory flow, but the transformations take place along a three-dimensional path. Experimental investigations into these flows are inherently difficult because the flow measurement techniques employed to investigate buoyant jet flows are not easily adapted for measuring concentration and velocity profiles along a three- dimensional path. The early techniques (Ayoub 1971; Chu 1975a) measured the concentration Chapter 2 ? Review of Previous Research 17 or velocity profiles of the flow at one or several points, setting up the measuring equipment at the point of interest (perpendicular to the direction of flow). However with none of the trajectory co-ordinates fixed, locating the trajectory and the direction of flow required a separate investigation. The introduction of LIF made it possible to obtain more detailed information at a particular cross-section, but locating the cross-section so that it was perpendicular to the flow direction remained problematic. Determining the trajectory co- ordinates from photographs or video images is difficult because of the changing calibration length-scales, due to the flow not travelling perpendicular to the plane of view. However, the above-mentioned constraints are largely eliminated for areas of the flow that travel predominantly in a single direction, reducing the flow to one with an essentially two- dimensional trajectory (Cheung 1991). Ayoub (1971), Chu (1975a) and Cheung (1991) studied the horizontally discharged buoyant jet in a cross-flow and Wallingford Hydraulic Research Station (1977) the horizontally discharged heated jet in a cross-flow. Ul U0 d Advected Thermal Region Advected Plume Region Strong Jet Region Ua > 0 ? < ? a Ua z x y Figure 2.6 ? Flow configuration horizontally discharged buoyant jet in a cross-flow 2.4.3 ? Missing Experimental Data Even though there is extensive experimental data available for the two-dimensional trajectory flows as a whole, there still remain flow configurations with limited or no available data. The horizontal and vertical flow configurations have dominated past experimental investigations. Relatively few studies have focussed on discharges with oblique angles. Those studies that Chapter 2 ? Review of Previous Research 18 have investigated these flows have largely been limited to trajectory results. These flows are of particular interest because they provide an opportunity to study more closely the nature of transitions between the different strongly advected regions. In the non-buoyant case these regions are the weak-jet and the advected line momentum puff. More generally, currently available models predict oblique discharge behaviour and validation of these predictions with experimental data is desirable. With the increase in the demand for clean water as well as decreasing costs for the desalination process, desalination plants are becoming an increasingly viable option as a supplementary reliable main water supply for many communities. The effluent from desalination plants has relatively high salinity concentrations. Discharging the effluent into less dense surrounding fluid makes the effluent fall rather than rise. If the ambient motion is relatively small or non-existent the discharge essentially becomes a negatively buoyant jet. Except for the vertical discharge configuration, the negatively buoyant jet has not received a great deal of attention. Past experimental investigators have primarily studied the behaviour of the discharge with a 60? angle, and to a lesser extent the 30? and 45?. The experimental results have focused on the rise height of the flow, the distance from the point of release to the impact point (the point at which the flow returns to the source height), and the dilution at the impact point. Widening the scope of the investigation into the behaviour of the negatively buoyant jets (including a range of discharge angles and determining spread and dilution data along the trajectory of the flow), will increase the knowledge of the mixing characteristics of these flows. This understanding can eventually lead to more effective discharge techniques. As indicated, relatively few studies have focused on the three-dimensional flow trajectory, in part because of the difficulties described previously. While the more recent study by Cheung (1991) provides valuable information about these flows, the study was necessarily limited in its coverage of flow regions. In addition, the experiments were carried out in a flume where the influence of mean shear and ambient turbulence was difficult to assess. Two earlier studies (Ayoub 1971; Chu 1975a) were completed over thirty years ago and hence limited information could be obtained from the flows. Coverage of flow regions was again also limited. With current technology more detailed information can be obtained and a broader range of discharge configurations more easily explored. The past studies have been limited to a single discharge configuration. The source outlet was horizontal and perpendicular to the ambient current. Chapter 2 ? Review of Previous Research 19 2.5 - Existing Models The considerable research activity in this area over the past 50 years has resulted in a number of different models to mathematically describe the trajectory and dilution of buoyant discharges. Over time these models have expanded to incorporate more complex flow configurations. Most models are now able to predict the behaviour of a buoyant jet with a three-dimensional flow trajectory. The different models can generally be split into three different categories, the length-scale models, the integral models, and the models that use a combination of both length-scales and integral techniques. 2.5.1 ? Length-Scale Models The first group of models is based on the length-scale approach (for a more detailed explanation, see Pun (1998)). The first step in this approach is to determine the different flow regions. For buoyant jets in a cross-flow, the different flow regions or limiting cases are the initial, strong jet, weak jet, line momentum puff, advected plume and advected thermal. These flow regions are determined by the relative magnitude of the independent parameters of the flow. The independent parameters are the initial flow rate (Q0), initial excess momentum flux (Me0), the initial density deficit flux (Q?0) and the ambient velocity (Ua). Table 2.1 shows the dominant parameters for each of the different flow regions. U0 represents the initial (uniform) velocity of the flow at the end of the round source, d the diameter of the source, g the gravitational constant, ?a the density of the ambient fluid, ? the density of the jet fluid, and ?0 the initial discharge angle. By combining the dominant parameters of two neighbouring flow regions different length- scales can be formed. The length-scales are along the predominant direction of movement of the flow before the transition (with the exception of the weak-jet to advected thermal length- scale which is along the direction of the secondary movement), and determine approximate locations where the flow transforms from one flow region into another (Table 2.2). Length- scales are compared to determine the form of the flow at a particular location. If the independent parameters are all non-zero then buoyant jet discharges start in the initial flow region and end up as an advected thermal. The relative magnitudes of the independent parameters determine the sequence of flow regions that form as the flow develops. Transition points are located more accurately through the introduction of transition constants into the Chapter 2 ? Review of Previous Research 20 transition length-scale relationships. These constant are obtained from comparisons with experimental data. Table 2.1 - Dominant parameters in flow regions Flow Region Dominant Parameters Initial ( )20 04Q d Upi= Strong Jet [ ]( )aee UUQUQM ?== 00000 Weak Jet Me0 and Ua Line Momentum Puff Me0 and Ua Advected Plume ( )( )0 0 aQ Q g ? ? ?? = ? Advected Thermal 0Q? and Ua Table 2.2 ? Transition length-scales for transition between flow regions Flow Region Transition Transition Length-Scale Initial ? Strong Jet 01 2 0e Q M? Initial ? Advected Plume 3 5 0 1 5 0 Q Q?? Initial ? Advected Thermal 2 0 0 aQU Q?? Strong Jet ? Weak Jet 1 2 0 0cos e a M U ?? Strong Jet ? Advected Line Momentum Puff 1 2 0 0sin e a M U ?? Strong Jet ? Advected Plume 3 4 0 0 eM Q?? Weak Jet ? Advected Thermal 0 0 0 cose aM UQ ? ? ? Advected Plume ? Advected Thermal 03 a Q U ?? Advected Line Momentum Puff ? Advected Thermal 0 0 0 sine aM UQ ? ? ? Chapter 2 ? Review of Previous Research 21 The second step involves working out relationships that describe flow behaviour within a flow region using dimensional analysis. With the help of the Buckingham ? theorem the trajectory, spread, velocity and dilution can be related to the distance from the source or the virtual source (see Table 2.3). The virtual source of a flow region is the location of the source if the flow was released in that particular flow region. Appropriate constants are introduced that can again be obtained from experimental data. Table 2.3 ? Characteristic relations of flow parameters with distance within flow regions Flow Region Trajectory Spread Velocity Dilution Strong Jet (vertical discharge) 1 2z x? 1z? 1z?? 1z? Weak Jet (horizontal discharge) - 1 3x? 2 3x?? 2 3x? Line Momentum Puff (vertical discharge) 1 3z x? 1z? 2z?? 2z? Advected Plume 3 4z x? 1z? 1 3z?? 5 3z? Advected Thermal 2 3z x? 1z? 1 2z?? 2z? The length-scale approach has been used by Scorer (1959), Hoult et al. (1969), Chu and Goldberg (1974), Wright (1977), Knudsen (1988) and Doneker and Jirka (1990). The commercially available outfall model CorMix (Doneker and Jirka 1990) is also based on the length-scale approach. The main advantage of this approach is its simplicity and it enhances the physical insight into the flow. But it relies heavily on empirical data and is unable to predict flow behaviour in the transition zones. Nevertheless the models based on the length- scale approach are considered accurate enough to be used for engineering applications. 2.5.2 ? Integral Models All incompressible fluid dynamical phenomena are governed by the Navier Stokes equations. Equation 2.1 is the conservation of mass equation using index notation. Equation 2.2 is the conservation of momentum equation using index notation. The gravity term is absorbed into the pressure term, thus p is the modified pressure. 0i i u x ? = ? (2.1) Chapter 2 ? Review of Previous Research 22 jj i ij i j i xx u x p x uu t u ?? ?+ ? ??= ? ?+ ? ? 21 ? ? (2.2) At this moment in time it is difficult to solve the Navier Stokes equation directly for a turbulent flow, as the equations are four non-linear partial differential equations with four dependent and four independent parameters. Osborne Reynolds suggested solving the Navier Stokes equations for average values of the velocity and pressure rather than the instantaneous values. For turbulent flows that are steady in the mean, the parameters ui and p randomly change in time around some mean value. These parameters can be decomposed into a mean (Ui and P) and a fluctuating part (ui` and p`). This is known as Reynolds decomposition. ' iii uUu += (2.3) 'pPp += (2.4) Inserting equation 2.3 and 2.4 into the Navier Stokes equations and averaging gives the average continuity equation (2.5) and the Reynolds equation (2.6). The Reynolds equation governs the mean velocity and pressure fields in a turbulent flow. 0i i U x ? = ? (2.5) ??? ? ??? ? ? ? ? ? ?+ ? ??= ? ?+ ? ? ''1 ji j i jij i j i uu x U xx P x UU t U ? ? (2.6) When comparing equations (2.1) with (2.5) and equation (2.2) with (2.6), it can be noted that the terms in the average equation are almost the same as in the instantaneous equation. The only extra term in the Reynolds equation is '' jiuu? , the turbulent stress per unit density. When multiplied by ? it is also known as the Reynolds stress. This term came into the equation because of the averaging of the non-linear advection term in the original equation. The term involves the product of the fluctuating components. The Reynolds stress is a symmetric second order tensor and non-zero. Thus the averaging process produces six new unknowns. Together with four mean flow parameters this gives a total of ten unknown parameters and only four equations. This problem is known as the closure problem of turbulence: how to relate the turbulent stress terms to the mean velocity and pressure terms, or provide additional equations. As a solution to the buoyant jet problem the assumption of self-similarity for all cross- sectional properties, velocities or concentration, was proposed by Morton et al. (1956). The self-similarity of the cross-sectional properties was used to integrate across the cross-section Chapter 2 ? Review of Previous Research 23 and then solve for the velocity and concentration distributions along the flow. The unknown parameters velocity, spread, density of jet fluid and concentration of the tracer were all related to the entrainment of ambient fluid into the jet. This simplified the problem to finding just one equation, the relationship between the entrainment and the mean flow parameters. Morton et al. arrived at the entrainment assumption. It stated that the entrainment velocity is proportional to the centreline velocity of the buoyant jet. e lu U?= (2.7) where ue is the entrainment velocity and Ul the centreline velocity. The entrainment coefficient (?) has been shown experimentally to be approximately 0.055 for a strong jet based on Gaussian cross-sectional profiles; but this changes for different flow configurations, for example, if the flow is dominated by the buoyancy-generated momentum flux (a plume forms) the value for the entrainment coefficient is 0.083. Morton et al. developed the integral model of the buoyant jet by using the entrainment assumption and integrating the equations of motion over a control volume. Abraham (1963) arrived at the spread assumption using a second approach to solve the set of equations. It stated that the rate of spread of a jet is a constant. b kx= (2.8) where b is velocity spread and x the distance from the source, the spread constant (k) is approximately 0.11 for the strong jet. The advantage of the spread assumption over the entrainment assumption is that it can be generalized more easily. The rate of spread of a plume is also a constant and not dissimilar to that of the jet. Abraham was the first to present a more general jet-integral model that included different source and ambient conditions. Abraham?s model was however based on the jet-diffusion approximation (fluid momentum and vorticity spread by turbulent diffusion) rather than turbulent entrainment. The first to incorporate the entrainment approach into a general jet model was Fan (1967). He used an Eulerian integral method, in which the flow passes through a fixed control volume, and integrated the equations of motion over that control volume. Others who have used this approach are Fan and Brooks (1969), Muellenhoff et al. (1985), Wood (1993) and Chu and Lee (1996) and Jirka (2004) [CorJet]. Cheung et al (2000) [VisJet] and Baumgartner et al (1993) used the Lagrangian integral method for the development of their models. The Lagrangian integral method integrates the equations of motion over a control volume that moves with the flow. Both CorJet and VisJet are commercially available models. Chapter 2 ? Review of Previous Research 24 2.5.3 ? Hybrid Models A significant benefit of the Length Scale approach is that the problem is simplified, while the results predicted by the Length Scale models are still acceptable for engineering applications. However that is at the same time their weakness: Length Scale models are unable to accurately predict the behaviour of flow, especially in the zones of transition between different regions, because of their simplicity and their dependence on experimental data. The integral models are more sophisticated, and at the same time this can be a drawback with the need for numerical solutions. They are, however, capable of producing relatively accurate results and do not have a high dependence on experimental data. Attempts have been made to combine the two approaches. An example is the hybrid model of Davidson & Pun (1998) and Davidson & Pun (2000). The Length Scale approach was used to define the different flow regions and analytical integral solutions were used to define the flow behaviour within the flow regions. Thus combining the simplicity of a length-scale model with the reduced empirical dependence of the integral solutions. 2.6 ? Summary For over a century researchers have investigated the behaviour of the buoyant jet. The observations and quantitative results from early experimental research led to the formulation of empirical and analytical solutions. These solutions formed the basis of the mathematical models that are used today to predict the behaviour of these flows. Later experimental studies have benefited from newer flow measurement techniques that have increased the accuracy and detail of the data available. The new results have been used to verify the output of the models, as well as increasing the general knowledge of flows, particularly with respect to instantaneous behaviour. However, as pointed out above, when considering mean flow behaviour several important experimental data sets are missing, and these are the focus of the current study. In the current study, separate investigations have been carried out into the behaviour of negatively buoyant jets, the non-buoyant oblique discharge in a moving ambient flow, and the buoyant jet with three-dimensional trajectories. The results of these investigations can be found in Chapters 6, 7 and 8. In Chapter 5 experimental results from other two-dimensional trajectory flows that Chapter 2 ? Review of Previous Research 25 were carried out, with a view to creating a more complete experimental data set for buoyant jet discharges, are presented. These included simple jets, plumes, positively buoyant jets, non- buoyant discharges released perpendicular to the ambient flow and oblique buoyant discharges in a moving ambient. Chapter 2 ? Review of Previous Research 26 Chapter 3 ? Flow Visualization Techniques 27 Chapter 3 ? Flow Visualization Techniques 3.1 - Introduction As indicated in the previous chapter, buoyant jet flows can be classified based on their trajectory. For those flows with two-dimensional trajectories (including jets and plumes) it is possible to observe the flow through a camera located such that it is perpendicular to the flow path. Flows in the second group have three-dimensional trajectories. For these flows it is no longer possible to set the camera up in such a way that it is perpendicular to the central flow path at all points. An example of a flow from this group is a buoyant jet in a cross-flow, discharged horizontally and perpendicular to the ambient flow. LIF (Laser Induced Fluorescence) has been used for the last two decades to investigate two- dimensional trajectory flows in a qualitative as well as a quantitative way (Papanicolaou 1984; Papantoniou and List 1989). LIF is a flow visualization technique that is well suited to these types of flows, because the laser sheet used for LIF can be set up in such a way that it coincides with the central trajectory plane. As this is no longer possible for the three- dimensional trajectory buoyant jet flows, LIF is not suitable unless the laser sheet is rapidly scanned in the third dimension and image recording is coordinated with that scan (Tian and Robert 2003). Although traditional planar LIF can be employed to obtain cross-sections from these flows, large numbers of these would be required to adequately characterize each flow. A relatively simple alternative approach, employed here, is to use the light attenuation (LA) technique. LA is based upon the relationship between the increase in dye concentration in the fluid and the decrease of intensity of the light passing through the dyed fluid. Cenedese and Dalziel (1998) showed that attenuation of light that passes through a dyed fluid could be used to measure the concentration of dye in the fluid or the thickness of a fluid layer. The technique has been successfully applied to 2 dimensional fluid flows (Zhang and Chu 2003; Gaskin et al. 2004). In the present study a LA system has been developed to study 3 dimensional fluid flows and the technique is then applied to problems of interest with both two-dimensional and three-dimensional trajectories. Chapter 3 ? Flow Visualization Techniques 28 LA and LIF are not strictly alternatives as LA provides information that has been integrated over a flow depth, whereas LIF provides information that has been integrated over the width of the laser sheet (typically of the same scale as the source). Due to the integrated nature of the LA concentration data, the signal does not diminish as quickly with increasing distance from the source, when compared to the point concentration data from LIF. Thus the LA system can provide high quality quantitative information for the flow as a whole. However this same integrated nature makes the interpretation of the data obtained more challenging. This chapter firstly presents the details of the LA technique developed for the buoyant jet flows with three-dimensional trajectories, including the limitations and interpretation of the results from LA experiments. A LIF system employed for parts of the present study is also briefly outlined at the end of the chapter. 3.2 - LA An overview of the light attenuation flow visualization technique and its application to the problem at hand is given below. Initially the experimental and theoretical aspects of the LA system are discussed. This is followed by the set up, method and results of the calibration experiments; which were carried out to explore the experimental configuration and the limitations of the method for the current application. The data acquired from a LA experiment is integrated concentration data. The interpretation of the integrated concentration data is discussed for weakly advected (strong-jet and plume) and strongly advected flows (puff, thermal and weak-jet). In addition a flow angled towards the camera is also dealt with and this in turn is used to explore parallax issues. Finally the upgrade from the standard LA system (used for two-dimensional trajectory flows), to the 3D LA system (used for three-dimensional trajectory flows), is presented. This includes verification of the results using a weakly advected and a strongly advected flow. 3.2.1 ? Light Attenuation System As indicated, LA is based upon the relationship between the increase in dye concentration in the fluid and the decrease of intensity of the light passing through the dyed fluid. This is explained in some detail when discussing the theoretical background of the light attenuation Chapter 3 ? Flow Visualization Techniques 29 technique. The experimental configuration of a typical LA experiment is used as an aid in that explanation and is therefore discussed first. 3.2.1.1 ? LA Experimental Configuration The light attenuation technique requires controlled lighting conditions and hence the LA experiments were carried out in a darkroom. The tank used for the experiments had inside dimensions of (length x width x height) 6220mm x 1540mm x 1080mm. Both the ends and the sides of the tank were made up of glass windows with a viewable area of (width x height) 700mm x 980mm. The ends had two such windows, the sides eight. A trolley-system with a variable speed control was mounted on top of the tank, driving along the length of the tank. The source was attached to the trolley and a magnetic flow meter and timer were situated on top of the trolley. The trolley-system included a cable-tray that was used to feed through the electrical cables and hoses to the source. A schematic diagram of the tank can be seen in Figure 3.1. The results of the magnetic flow meter calibration test can be seen in Figure 3.2. The error was within 2%. 6220mm 1540mm 1080mm Trolley-System, including cable-tray, flowmeter and source Trolley Rails Figure 3.1 - Schematic Diagram of main tank Chapter 3 ? Flow Visualization Techniques 30 0 0.01 0.02 0.03 0.04 0.05 0.060 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Actual Flow (m3/hr) F lo wm ete r R ea di ng (m 3 /h r) Figure 3.2 - Magnetic flow meter calibration test results A second, smaller tank with a volume of 320 litres was situated next to the main tank. This tank was filled with the source solution. The temperature of the solution was controlled with cooling cells, so that the source fluid had the same temperature as the water in the main tank. Before and during experiments, frequent temperature measurements of the source fluid, at the source, and the water in the main tank were carried out to make sure the temperature variations were within 0.1o C. Covering the tank when not in use minimized evaporation of the source solution. The source solution was pumped up to a constant head tank, 3870mm above the floor of the main tank. From the constant head tank, the solution ran back down through the flow meter and out the source. A second outlet was inserted in the system to enable draining of the hoses before each experiment to make sure the temperature of the source solution matched the temperature of the water in the main tank. The overflow of the constant head tank ran back into the source tank. After each experiment the source fluid left in the constant head tank was pumped back into the source tank. The light source for the light attenuation experiments was a white-light bank consisting of twelve 100Hz fluorescent tubes. This light bank was positioned behind the third window of the main tank (see Figure 3.3). The light bank was supported by a dexion frame, and placed at a height so that the light bank covered the complete viewable area of the window. An opaque Chapter 3 ? Flow Visualization Techniques 31 Perspex diffusion sheet with a thickness of 2mm was placed in between the lights and the window. The Perspex sheet acted as a diffuser creating a more uniform light sheet. The light bank was covered with black cloth to minimize the chance of light reaching the camera other than through the selected window. On the opposite side of the tank a digital video camera was positioned up to approximately 5 metres from the middle of the tank. The camera was either a Canon MV4i DV (or MV30 DV) or Jai CV M7+(or Jai CV M7+ CL). The video camera was mounted on a tri-pod and positioned in such a way that the lens of the camera was in the middle of, and perpendicular to the third window. The camera was connected to the computer via a data-cable and either a IEEE Firewire (for the Canon cameras) or Bitflow video board (for the Jai cameras). The ImageStream Software (Nokes 2005) was used to analyse the data. Two calibration cells were placed inside the tank. These calibration cells consisted of a glass picture frame and its supports, including wheels. The visible area of the glass had dimensions of 770mm x 550mm. The width between the glass plates was 32mm (?0.3%). The wheels of the calibration cells were set inside two L-shaped rails that were positioned lengthwise on the bottom of the tank. The rails were fixed in place using magnets. The first calibration cell was left empty, with its valves open. Therefore the main tank and this cell were filled simultaneously. The second calibration cell was filled with diluted source solution and then closed. Figure 3.3 shows the positions of the light source, video camera and the calibration cells relative to the main tank. Fluorescent Lights Perspex Diffusion Sheet Digital Video Camera Calibration Cell Rails Towing Tank (lxbxh = 6m x 1.5m x 1m) Figure 3.3 - Schematic diagram of experimental equipment Chapter 3 ? Flow Visualization Techniques 32 The source solution was made up from water, red food dye and salt. The relative quantity of each depended on the type of discharge. After mixing the source solution was left alone for at least one day so that the salt was properly dissolved, and the air bubbles were dissolved. A cross-sectional cut at the third window creates Figure 3.4, the set up for a typical LA experiment. Camera Trolley Light Source Perspex Sheet Computer 00:00:25 0.0025 m3/hr Source Tank Figure 3.4 - Typical LA experimental set up 3.2.1.2 ? Theoretical Background The attenuation of light intensity and the operation of a digital video camera and computer are combined to form the basis for the flow visualization technique. Additional details of the attenuation of light technique can be found in a paper by Cenedese and Dalziel (1998) and the operation of a digital video camera is described in depth by Poynton (1996) In Figure 3.4, light travels from the light source through the Perspex diffuser sheet and the tank and is caught by the digital video camera located at the other side of the tank. The camera sends the data to the computer. The computer stores the data and is the tool used to analyse the recorded data. Chapter 3 ? Flow Visualization Techniques 33 Looking in more detail at the individual components of the system, one can consider what happens to a ray of light from when it departs the light source until it is analysed using the computer (see Figure 3.5). After leaving the light source the ray of light will immediately start to decrease in intensity. The air absorbs the light and thus the further away from the light source the less bright it appears. Absorption of light also occurs when the ray is travelling through the Perspex diffuser sheet, the glass walls of the tank and the water in the tank. Upon hitting the Perspex sheet some of the incident ray of light will be reflected back due to the air/Perspex interface, and the ray loses some of its intensity. Similar reflections occurs when the ray hits the Perspex/air, air/glass interface, glass/water interface, water/glass interface and glass/air interface. If these losses were calculated as the intensity of the light after the loss divided by the intensity before the loss occurred, ten constants with a value between zero and one would be determined. These constants can be referred to as Cair, CPerspex, Cglass and Cwater for the four different absorption losses and C1, C2, C3, C4, C5 and C6 for the reflection losses as mentioned above. Figure 3.5 shows the path of the ray of light for a typical system including the losses of intensity along the way. Glass Walls of Tank Ray of Light C6 C4 C3 C2 C1 CPerspex Cglass Cglass Cair Cair Cwater Camera Tank Light Source Perspex Sheet C5 Figure 3.5 - Path of a ray of light The intensity of the ray of light at the camera (Icam) is then related to the intensity of the ray at the light source (Isource) by taking into account all the loss terms along its path. This gives equation (3.1). In LA experiments the intensity of the light at the camera will be used as a reference intensity, Iref. Substituting Iref for Icam in equation (3.1) gives equation (3.2). Chapter 3 ? Flow Visualization Techniques 34 sourcewaterglassPerspexaircam ICCCCCCCCCCI ??????????= 654321 (3.1) sourcewaterglassPerspexairref ICCCCCCCCCCI ??????????= 654321 (3.2) Upon receiving the light, the camera codes the colours by assigning three values to represent each colour. The three values correspond to the red, blue and green intensities of the colour. The camera sends the codes to the computer via a data-cable. The computer receives the data and converts it back to colours to reproduce the image. An introduction to the processes involved can be found in Appendix A Figure 3.6 is as Figure 3.5 but with the addition of some dyed fluid in the path of the ray of light. This dyed fluid can be thought of as a jet-flow in the middle of the tank or a stationary dyed fluid covering the whole tank. The image recorded in this situation is similar to an image recorded during a typical LA calibration experiment. However, due to the added dye in the path of the light, there is an extra loss term, Cdye, as the dye will absorb some of the intensity of the ray while it is passing through the dye. If the colour of the dye is red and the concentration of the dye is low enough, the red signal transmitted through the fluid is unaltered, but part of the green and blue signals will be absorbed. If the dye is green, the green signal is unaltered but the red and blue signals attenuate etc. This extra loss term changes the relationship between the intensity at the camera and the intensity at the light source. The relationship is shown in equation (3.3). 1 2 3 4 5 6cam air Perspex glass water dye sourceI C C C C C C C C C C C I= ? ? ? ? ? ? ? ? ? ? ? (3.3) Dividing equation (3.3) by equation (3.2) gives 1 2 3 4 5 6 1 2 3 4 5 6 air Perspex glass water dye sourcecam dye ref air Perspex glass water source C C C C C C C C C C C II C I C C C C C C C C C C I ? ? ? ? ? ? ? ? ? ? ?= = ? ? ? ? ? ? ? ? ? ? (3.4) Equation (3.4) shows the relationship between the intensity at the camera of the image during an experiment and the intensity at the camera of a reference image, recorded when the dye is not present. The dye loss term can therefore be used as a quantitative measure of dye concentration, but if and only if there is a known relationship between the increase in dye concentration in the flow, and decrease of intensity of the light. Chapter 3 ? Flow Visualization Techniques 35 Red Dye Solution Cdye Cwater Glass Walls of Tank Ray of Light C6 C4 C3 C2 C1 CPerspex Cglass Cglass Cair Cair Camera Tank Light Source Perspex Sheet C5 Figure 3.6 - Path of a ray of light including dyed solution 3.2.2 - Calibration experiments To determine the relationship between the increase in dye concentration and the loss term Cdye, calibration experiments were carried out. These experiments were designed to explore the best possible set up and to provide insights into the limitations of the method. The specific experimental set up (including the video cameras, light source and dye), employed to carry out the calibration experiments is discussed below, and this is followed by the method for these experiments. The results from the calibration experiments demonstrate the repeatability of the LA technique. They also confirm the exponential relationship between the increase in dye concentration and the decrease of light passing through it (Cenedese and Dalziel 1998). The constant in this exponential relationship is determined from the calibration data. Finally the response of the red dye under different circumstances is determined with a separate set of experiments. 3.2.2.1 ? Experimental Set Up The experimental set up was similar to that as shown in Figure 3.4. To record the experiments three different cameras were used. Two of them were Canon Digital Video Cameras, the Canon MV30i DV (720x576 pixels, progressive scan, firewire enabled) and the Canon MV4i Chapter 3 ? Flow Visualization Techniques 36 DV (720x576 pixels, progressive scan, firewire enabled). Both cameras are so-called handycams and designed for home-videoing. They set most of their parameters automatically to make it easier for inexperienced people to get reasonable images. This affected the experiments in a negative way, as the camera altered some of the parameters during an experiment making the results inconclusive. This issue was resolved by manually setting some of the camera?s parameters. During an experiment it was important that the only change in intensity recorded by the camera was due to the increase in dye concentration in the tank. Therefore the room was blackened so that only the light from the light source was able to reach the camera. The camera automatically changed the shutter-speed and the exposure if less light reached the camera. In order to avoid this the shutter-speed was set to 1/120, and the exposure was set so that the intensity signal was close to 255 (normally between 230-240). If the exposure was set such that the intensity signal was at 255 the image was over-exposed, and the colour was saturated. The closer the maximum intensity was set to 255, the bigger the range of intensity recorded. A bigger range of intensity increased the accuracy of the measurement technique. The camera also changed the recorded intensity by altering the white-balance during an experiment. The camera itself set the white-balance so that pure-white consisted of blue, green and red guns with the same intensity. As only red dye was added to the water the camera attempted to compensate and create a more uniform picture by adjusting the relative magnitude, the gain, of the other two guns (colours). The white-balance could be set manually. The camera had to be focussed on a pure-white background and the camera would adjust the relative magnitudes, of the red, green and blue guns at each pixel so that they all had the same value at that pixel. As the pure-white background strongly depended on the lighting conditions and the relative position of the camera, it was difficult to recreate and thus the experiments using the manually set white-balance were not repeatable. An alternative approach involved setting the white-balance manually using one of the camera?s two pre-sets white-balances; one was for recording inside and one for recording outside. These set the white-balances independently of outside light sources and thus the experiments were repeatable. Because the lighting conditions during the experiments were substantially different from normal lighting conditions, neither pre-set white-balance gave an image that represented the original flow well in terms of colour. But the green signal was adequate for analysis purposes. The last parameter to set manually was the focus. Chapter 3 ? Flow Visualization Techniques 37 The third camera used was the Jai CV M7+ in combination with a Cosmicar/Pentax Precision 12.5 ? 75mm zoom lens. The Jai is a more expensive scientific camera and gives the user full control over exposure, shutterspeed and white-balance (by adjusting the black-level and the gain). The shutter-speed was set to 1/100s and the maximum intensity of 255 was accomplished by setting the black level to 250, the gain to 67 and the lens aperture to 16mm. The camera records 1292x1028 pixels at a frame rate of 24Hz and has the ability to record at higher frame rates by sampling a portion of the image. Just as with the Canon handycams, the Jai camera is a bayer filter camera, meaning that at each pixel one colour is recorded. In every 2x2 matrix of pixels, the pixels on one diagonal both record a value of the green gun and the pixels on the other diagonal record one value of the blue and one value of the red gun. The whole image consists of a multiple of these 2x2 matrices. The actual image can be recreated by calculating the values of the missing guns at each pixel by linearly interpolating the values of the guns surrounding the pixel. The Canon cameras have a built in feature that does the conversion from recorded image to actual image automatically. The data recorded with the Jai were converted using ImageStream (Nokes 2005). During the calibration experiments several light sources were tested. The chosen light source was to give the most constant and uniform light intensity and had to be independent of time. The combination of the camera with the fluorescent lights at 100Hz had the least variation and was the most uniform. Eight tubes were used to make up a light bank of 1.5 meters by wide 0.5 high meters. It had less variation then the fluorescent lights at 50Hz, because the 100Hz lights had a frequency that was further removed from the approximately 25Hz at which the camera recorded images. Both OHP-lamps and Halogen lights had more variation, because it was not possible to configure these lamps to create a relatively uniform light sheet. It was important that the air temperature surrounding the lights was constant as the light intensity depends on this temperature. The heat from the lights was the main factor influencing the temperature of the surrounding air. The lights were turned on at least half an hour before an experiment for the temperature of the surrounding air to reach a steady state. This steady state was checked by comparing the intensity results of a blank image recorded before and after the experiment. In order for the relationship between the increase in dye concentration and Cdye to be useful, the dye employed as the tracer should have a constant absorption response with different wavelengths, so that the absorption of dye is independent of wavelength. Figure 3.7 (Figure 7, Cenedese and Dalziel (1998)) shows the absorption spectrum for red dye (a), blue dye (b), Chapter 3 ? Flow Visualization Techniques 38 green dye (c) and potassium permanganate solution (d). The concentrations of red, green and blue dye are from the top 0.05ml/l, 0.1ml/l, 0.2ml/l and 0.4 ml/l respectively. The concentration of potassium permanganate is from the top 0.6ml/l, 1.2ml/l, 2.4 ml/l and 4.8ml/l respectively. From Figure 3.7 it can be seen that, for low concentrations, only the red-dye and the potassium permanganate have a region where the absorption is more or less constant and thus independent of the wavelength (between 4500-5600 and 5200-5800 Angstroms respectively). The absorption responses of the blue and green dyes show large gradients, and therefore strongly depend on the wavelength. For the experiments the red dye was chosen over potassium permanganate because of the toxic nature of the latter. (a) (b) (c) (d) Figure 3.7 - Absorption spectrum for four different tracers, (a) red dye (? (4000-7000 Angstroms) vs I/I0 (0-1)) with concentrations of (from top) 0.05ml/l, 0.1ml/l, 0.2ml/l and 0.4 ml/l, (b) green dye (? (3000-7000 Angstroms) vs I/I0 (0-1)) with concentrations of (from top) 0.05ml/l, 0.1ml/l, 0.2ml/l and 0.4 ml/l, (c) blue dye (? (3000-7000 Angstroms) vs I/I0 (0-1)) with concentrations of (from top) 0.05ml/l, 0.1ml/l, 0.2ml/l and 0.4 ml/l, (d) potassium permanganate (? (4000-7000 Angstroms) vs I/I0 (0-1)) with concentrations of (from top) 0.6ml/l, 1.2ml/l, 2.4ml/l and 4.8ml/l Chapter 3 ? Flow Visualization Techniques 39 With red-dye as the tracer, red light was transmitted through the discharge and thus the intensity of the red signal is unaltered by the presence of the red-dye (for low concentrations of red-dye). Therefore, only the blue and green light was attenuated, and given the relatively constant response of the absorption spectrum of the green signal it was used to determine the attenuation of the light during an experiment. Evidence that the absorption spectrum of the green light was relatively constant for the red dye is evident in Figure 3.8 (Figure 10, (Cenedese and Dalziel 1998)). Cenedese and Dalziel (1998) set up a green filter between the video camera and the flow. This filter only let light through with a wavelength between 450- 610 nm. The continuous line is the absorption spectrum without filter; the dotted line is the absorption spectrum with filter. The filtered response was significantly improved in this context when compared to the unfiltered response. For the calibration experiments described here no physical filter was used, but instead the camera and the computer-software were able to effectively isolate the green signal as indicated earlier. Therefore Cdye from equation (3.4) is calculated by dividing the green component during an experiment ( )greenI by the green component of the reference image ( )greenrefI _ at the same pixel. Figure 3.8 - Red Dye Absorption Spectrum with and without green filter Chapter 3 ? Flow Visualization Techniques 40 The investigation could also have been carried out with the results from the blue signal, but the blue signal is not as strong as the green one. Seventy two percent of the energy of light is in the green colour (Poynton 1996). Use of the weaker blue signal therefore amplifies errors. 3.2.2.2 ? Experimental Method The calibration experiments were conducted by mixing a volume of red dye solution (with a known concentration) in a tank of water, recording a video of the tank using Adobe Premier 6.0 or Labview ImageGrabber, and analysing it using ImageStream, The red food dye used for the calibration experiments was Ariavit Carmoisine manufactured by Quest International, Auckland, New Zealand. It is also known as Food Additive Code No 122. Two litres of red dye solution were made up with a nominal concentration of 0.05 g/l. The weight of dye in the dyed solution was determined to a precision of 0.0001g (0.1%) and the volume of the solution to a precision of 0.7ml (0.035%). The tank used during the experiments was a rectangular based Perspex tank with inside dimensions (length x base x height) of 751mm x 100mm x 382mm. A Perspex diffusion sheet with a thickness of approximately 2mm was placed behind the Perspex tank, and in front of the bank of lights. The tank was filled with approximately 24 litres of filtered water. The weight of the water in the tank was determined using a set of scales to a precision of 0.5 gram (0.002%). The weight of water was converted to a volume based on the density at the measured temperature. The temperature was measured using a thermometer. For the duration of the experiment the temperature of the water in the tank was kept constant (within 0.1 of a degree). The attenuation of green light due to a fixed amount of red dye depended on temperature. This is demonstrated in Figure 3.9. It shows the response of the green signal for four different experiments each with an increased water temperature and a corresponding increase in green intensity. Chapter 3 ? Flow Visualization Techniques 41 0 100 200 300 400 500 600 700 800120 130 140 150 160 170 180 190 200 210 220 Location (pixels) R ela tiv e G re en In ten sit y 12.2o C 13.0o C 13.5o C 14.0o C Figure 3.9 - Temperature influence on green intensity response The camera was set at a distance that provided a view of the complete tank. During an experiment the distance between the light source and the tank and the distance between the tank and the camera were not altered. At least two background videos were recorded before the start of each experiment. The background videos showed the tank filled with water, without red dye solution. Two videos were necessary to confirm that the fluorescent lights had properly warmed up and the emitted light intensity was constant. From the two litres of red dye solution 50ml was taken with a pipette to a precision of 0.053ml (0.1%) and added to the water in the tank. After it had been mixed in properly another video sequence was recorded. The process of adding 50ml of the red dye solution and recording videos was repeated until 1.5 litres was added to the water in total. The videos consisted of 100 frames with a frame rate of approximately 24 frames per second. These frames were exported as .tif files and used to create time-averaged images in ImageStream. At each step the integrated concentration was calculated to define the amount of red dye present between the light source and the camera. For the calibration experiments the integrated concentration was calculated by dividing the mass of the added red dye by the total volume of water in the tank and multiplying this concentration by the width of the tank (100mm). Chapter 3 ? Flow Visualization Techniques 42 3.2.2.3 ? Calibration Results For the planned buoyant jet experiments the Jai camera had significant advantages over the Canon handycams. The total manual control of the camera gave better repeatability and its higher resolution would give more detailed results. Therefore all results of calibration experiments presented here come from runs using the Jai camera. Figure 3.10a is an image from one of the calibration experiments. Here 12 litres of water was mixed with 12 litres of red dye solution of a certain concentration and repeatability issues were explored. This image clearly shows the tank, the red coloured water, the white background and the non-uniformity of the intensity due to the non-uniformity of the background lighting. One hundred of these images were recorded and averaged to create one average colour image. This average image was used to create an intensity field based on the green intensity. The processed image is shown in Figure 3.10b. Again it clearly shows the non-uniformity of the background lighting. The green intensity values at the pixels along the black line in Figure 3.10b have been plotted in Figure 3.11. The results of a second experiment have been plotted as well, the experiment used the same set up as the one mentioned above including the same temperature of both the water and the red dye solution. Comparing the two data sets shows that the repeatability of the experiments is reasonable when the temperature of the water as well as the temperature of the lights do not change between experiments. (a) (b) Figure 3.10 - The recorded image (a) and processed image (b) Chapter 3 ? Flow Visualization Techniques 43 0 50 100 150 200140 150 160 170 180 190 200 210 220 Location (pixels) R ela tiv e G re en In ten sit y Figure 3.11 - Comparison of green intensity response of two separate experiments Figure 3.12 shows the response of the green gun for a complete calibration experiment at two pixels with different background intensities( )greenrefI _ . As expected, the intensity of the green gun decreases with an increase of dye in the tank, and the decrease reduces the lower the background intensity. Thus the initial slope for the green intensity response is steeper for pixels with higher background intensities. Chapter 3 ? Flow Visualization Techniques 44 0 1 2 3 x 10-4 0 50 100 150 200 250 Integrated Concentration (g/l*m) R ela tiv e G re en In ten sit y Iref=250 Iref=190 Figure 3.12 - Response of the green gun for two different background intensities Figure 3.13 shows Cdye versus the volume of dyed solution added, Cdye being the ratio of the green intensity with the red dye present divided by the green intensity of the background. As expected there is no loss at 0ml, thus Cdye was 1. Cdye decreases towards zero for increasing intensity. It also shows this ratio depends weakly on the background intensity as the slopes for the two experiments in the initial region of the plot are not the same, resulting in diverging curves for higher concentrations. Thus Cdye at a particular pixel is a function of the background intensity. This can be written as ??? ? ??? ?= ref greenref green i II IfC , _ (3.5) where iC is the integrated concentration. Chapter 3 ? Flow Visualization Techniques 45 0 1 2 3 x 10-4 0 0.2 0.4 0.6 0.8 1 Integrated Concentration (g/l*m) C dy e = I g re en /I re f-g re en Iref=250 Iref=190 Figure 3.13 - Response of Cdye for two pixel with different background intensities The decay of light intensity of the green gun is exponential (Cenedese and Dalziel 1998), therefore an extra transformation is applied to the data. This is called the green absorption transformation. In Figure 3.14 the natural logarithm of the inverse of Cdye is plotted versus the integrated concentration, this value is called the green absorption value (GA). The background green intensity at this particular pixel was 240. This graph confirms that there is a linear relationship between increase of dye and decrease of intensity, but only for low integrated concentrations. With red dye as the tracer, red light is transmitted through the buoyant jet, however only for low concentrations of red dye. At some point there is enough red dye in the water to decrease the intensity of the red gun as well. Prior to this equation (3.5) can be re- written as ??? ? ??? ?= green greenref i I IaC _ln (3.6) where a is a constant determining the slope of the linear relationship. The value of the constant depends on the background intensity. The upper limit was found by fitting a straight line through the data using the first and the last point of the dataset. This was repeated for multiple datasets; including pixels with the maximum and minimum expected background green intensity during a LA experiment. The error of this approximation was calculated for all points in between and if the error at any of the points from any of the datasets was more than Chapter 3 ? Flow Visualization Techniques 46 five percent the last point in the datasets was removed and a straight line was fitted through the remaining data. This process was necessary because the actual calibration before the LA experiments would be a two-point calibration. The process was repeated until the error at all points was less than five percent. An example of the result of this process can be seen in Figure 3.14, the upper integrated concentration limit was 0.00016 g/l*m. 0 1 2 3 x 10-4 0 0.2 0.4 0.6 0.8 1 Integrated Concentration (g/l*m) G A (=L N (1/ C d ye ) = LN (I r ef -g re en /I g re en )) Iref=240 Figure 3.14 - Response using Green Absorption Filter for pixel with Iref = 240 During the experimental investigation into the behaviour of simple jets, the integrated initial concentration ( )0iC was set lower than the upper integrated concentration limit. Due to the internal structure of the simple jet and the turbulent fluctuations during the flow, the maximum instantaneous integrated concentration is approximately 48% higher than 0iC ( (1 0.1*2) 0.812 1.48+ = , see equation (3.16) and Figure 3.25 and including 2 standard deviations (95%)). The integrated initial concentration was set at 0.00014g/l*m and therefore the maximum integrated concentration during a jet experiment was 0.00021 g/l*m. Using the same approximation as before, the maximum instantaneous error at the maximum integrated concentration was between 5% and 9% depending on the pixel. However for values up to 0.00018 g/l*m (29% above 0iC ) the instantaneous error remained within 5%. Due to the small time-scales, the instantaneous error at the maximum integrated concentration is believed not Chapter 3 ? Flow Visualization Techniques 47 to have a significant effect on the mean integrated concentration values and it is reasonable to assume that the error remained less than five percent at all points. For the investigation into all other buoyant jet flows the same integrated initial concentration was used as for the simple jet flows. The turbulent fluctuations were no longer as much of a concern as the centreline-integrated concentration decreased with distance downstream and was therefore no longer in the vicinity of the upper limit. To increase the accuracy of the data at a location far downstream, the initial integrated concentration can be significantly increased. During the analysis any points with an integrated concentration of more than 0.00018g/l*m would then be discarded. As mentioned previously the slope of the straight line depends on the background green intensity, but repeating the straight line fit procedure for a pixel with a background green intensity of 190 gave the same upper-limit for the integrated concentration. As the slope of the line represents the actual relationship between the increase in dye concentration in the flow and decrease of intensity of the light, the value of the slope must be known at each pixel. Before a LA experiment, the value of the slope of the line at each pixel (a) was found using two calibration cells (see section 3.2.1.1). The first calibration cell was filled with water from the main tank; the second calibration cell was filled with diluted source solution. The source solution was diluted down to an integrated concentration of 0.00014g/l*m. The width between the glass panels was 32mm; therefore the concentration in the calibration cell was 0.0044g/l. The green intensity had a temporal variability of ?2, therefore four hundred frames were recorded of both cells and then averaged. A third series of four hundred images was recorded of the background lighting (no cells or dye present) and this series too was averaged. This averaged image was called the reference image. The average calibration cell images were converted into GA images using the reference image. The slope of the calibration line at each pixel is solely related to the attenuation due to the red dye. However, the glass plates of the calibration cell also attenuated some of the light. To compensate for this the GA value from the red dye solution calibration cell was reduced by the value obtained from the calibration cell filled with clear water. The resulting image gave the initial GA value at each pixel. To determine the integrated concentration values at each pixel for a particular LA image, firstly the GA values for the image are calculated using the background green intensity. With the GA values known at all pixels for the integrated initial concentration and for the zero Chapter 3 ? Flow Visualization Techniques 48 integrated concentration, the GA value at a particular pixel can be converted into the integrated concentration value using linear interpolation. 3.2.2.4 ? Response of the Red Dye To be able to carry out a set of experiments in succession it was important that the concentration of red food dye, in both the secondary (storage) tank and the calibration cells, did not change over time. Therefore the red dye solution was tested for variations in light attenuation with time. The two calibration cells were filled with the red dye solution and the tank was filled with water. At each time interval the digital camera recorded 200 images of both calibration cells and 200 reference images. These were averaged over time and GA images were created. From the GA images a square area of approximately 209.0 m was taken and the GA values averaged. The values were plotted versus the elapsed time since filling the calibration cell, and the results of these tests can be seen in Figure 3.15. The tests are divided by the method of composing the red dye solution in the calibration cell. It was either done by diluting the red dye solution in the secondary tank by the appropriate amount (the dilution tests), or by mixing dry red dye matter into water to create the same concentration as above (the solution tests). The red dye solution for dilution test 1 was made at the same time as the red dye solution in the secondary tank was made. The results of the test do not show a decrease in light attenuation. The red dye solution for dilution test 2 was made 5 days later from the same batch of red dye solution in the secondary tank as dilution test 1. As the result from test 2 matches the one from test 1, it can also be concluded that there had been no decrease of concentration of red dye in the secondary tank during that period. During the dilution process for dilution test 3, Sodium Chloride ( )NaCl was added to the solution to create a density difference of approximately 2.5% with the water in the tank. The results show that the light attenuation for dilution test 3 is slightly lower than for tests 1 and 2, however the light attenuation does not decrease with time. The lower initial light attenuation is taken into account by the field calibration and that makes NaCl suitable to be used in conjunction with the red dye. Chapter 3 ? Flow Visualization Techniques 49 0 1000 2000 3000 4000 5000 600070 75 80 85 90 95 100 105 110 L ig ht A tte nu ati on (% ) Time (minutes) Dilution Test 1 Dilution Test 2 Dilution Test 3 Solution Test 1 Solution Test 2 Solution Test 3 Figure 3.15 ? Red Dye Experiment Results For solution test 1 the red dye solution was made up from water and red dye. The results from solution test 1 match the results from dilution test 1 and 2. For the second solution test, Sodium Sulphite ( )32SONa was added to the red dye solution. The Sodium Sulphite reacts with the dissolved oxygen in the water used to create the red dye solution, stopping bubbles forming on the inside of the calibration cells. However Figure 3.15 shows that the light attenuation results from solution test 2 are lower than expected initially and that they decrease during the remainder of the test, making it difficult to use the Sodium Sulphite for the LA experiments. Solution test 3 used a similar red dye solution as solution test 2, but at a time of zero minutes it had already been in the calibration cell for approximately 50 days. It confirms the results of solution test 2 and shows that the maximum decrease of the light attenuation due to the Sodium Sulphite is about 9 percent. The red dye in combination with Sodium Chloride showed very little variation in light attenuation over time. The maximum error was 2.8%, but most of the results fell within 1% of the average recorded integrated concentration. However the error in the test results was difficult to assess. Both the changing water temperature and the purity of the ambient water had an influence on the results. By comparing the results from the Sodium Chloride test with Solution test 1 (which was recorded at the same time as the Sodium Chloride test) part of Chapter 3 ? Flow Visualization Techniques 50 these uncertainties could be assessed. This comparison determined that the variation was consistent, between 1.4% and 1.7%, throughout the recording period. The changes in ambient temperature and purity do not have an influence on the results of a jet experiment as the field calibration takes them into account. But to reduce the possible error due to variations in light attenuation further, the red dye solution in the calibration tanks was replaced every week. 3.2.3 ? Interpretation of the integrated information When using the LIF flow visualization technique for a buoyant jet experiment, the thickness of the plane illuminated by the laser is in the order of millimetres. Because of the finite thickness of the plane, the data received from a LIF experiment is in principle integrated concentration data. The error involved in assuming the data to be point-values is small, given that the data is typically integrated over a source diameter. This assumption does not hold for data received from a LA experiment as the data is integrated over the depth of the flow. To be able to use the integrated concentration data, a method for relating the integrated to the non- integrated data is required. This can either be done by integrating the governing equations and comparing the data with the predictions of the new equations, or by transforming the integrated data to non-integrated data and comparing the converted data with the existing equations and data. As the data is integrated over the flow, the interpretation of the integrated data depends on the internal concentration structure of the flow. The study of axi-symmetric buoyant discharges has identified two distinct types of internal structures; for weakly advected flows the mean cross-sectional concentration distributions are found to be Gaussian, and for strongly advected flows the mean concentration distributions resemble those of a vortex pair. Here two example experiments, a simple jet flow as an example of a weakly advected flow and a momentum puff flow as an example of a strongly advected flow, are used to investigate the interpretation of the integrated information. These experiments will also be used to examine how well the quantitative flow visualization technique performs. In addition a third flow is investigated, that of a jet where the source discharges at an angle to the camera. This flow is an example of a flow that is no longer flowing perpendicular to the camera view. Although it is possible to carefully choose the position of the source for flows with two-dimensional trajectories so that they flow perpendicular to the camera axis, this is Chapter 3 ? Flow Visualization Techniques 51 not possible for flows with three-dimensional trajectories. Therefore it is important to investigate the LA technique under those circumstances. Parallax issues are assumed to have a negligible effect on the results of the experiments that are used to investigate the interpretation of the integrated information. In section 3.2.3.4 the angled jet flow analysis will be used to verify this assumption. For the interpretation of the integrated information the following subscripts are used: i, integrated; iy, integrated in the y-direction; iz, integrated in the z-direction; l, centreline of single-Gaussian; c, centreline of vortex pair; peak, maximum in vortex pair cross-section; 0, initial; dg, double-Gaussian; sg, single Gaussian. 3.2.3.1 ? Weakly Advected-Flow, a Simple Jet Experiment A simple jet experiment was set up using the experimental system as shown in Figure 3.4. For a more precise description of the experimental set up see section 3.2.1.1. The dimensions of the tank were (length x base x height) 6220mm x 1540mm x 1080mm. The source of the jet was mounted on a frame situated above the tank. It pointed vertically down into the tank and was positioned approximately in the middle of the tank, so that the boundaries were not having a significant effect on the flow. The port diameter used was 2.45mm and the initial velocity 1.47 m/s. This gave a Reynolds number for the flow of approximately 3600. A red dye solution was made up to a concentration ( )0C of 0.0571g/l. This takes into account both the 0.00014g/l*m upper concentration limit ( )0iC as well as the 0.00245m diameter of the source ( )d . More information about this relationship can be found in section 3.2.2.3. As the camera was set further back than during the calibration experiments to reduce parallax issues, the gain was increased to 200 so that initial background green values were between 190 and 250. The other parameters were the same as those employed for the calibration experiments. The first video sequence recorded was of a cross-shaped ruler inserted vertically down into the tank at the position where the jet would be. During the analysis this image was used to deduce the calibration length-scales in the horizontal and the vertical direction of the images. The second video sequence recorded was the reference video. The cross-shaped ruler had Chapter 3 ? Flow Visualization Techniques 52 been removed from the tank, but the jet source had not been opened at this stage, and this created a background image. To eliminate the influence of small light fluctuations with time, 400 frames were recorded and later averaged. With about 24 frames being recorded per second, the length of the reference video was approximately 16 seconds. The third video sequence was of the calibration cell filled with the red dye solution with a uniform integrated concentration of 0.00014g/l*m. Video sequence four was of a second calibration cell, but this time filled with just water. The last video sequence was of the jet itself after the source had been opened. About a minute of footage was recorded, approximately 1300 frames. A single frame can be seen in Figure 3.16. The 1300 frames were used to create the averaged image, independent of time, shown in Figure 3.17. Figure 3.16 - Single frame of jet video Chapter 3 ? Flow Visualization Techniques 53 Figure 3.17 - Average image of jet In ImageStream the average image of the jet, as well as the average background image, and the average images of both calibration cells were used to calculate the integrated concentration values at each pixel. Figure 3.18 shows the integrated concentration image of the jet. The upper integrated concentration limit of the image is 0.18mg/l*m and the lower limit is ?0.02mg/l*m. Values outside this range show up as white on the image. The image appears to match expectations as the integrated concentration values are the highest in the middle of the jet and fall away towards zero on the edges. Also the integrated concentration values along the centre line of the jet are constant. This can be explained by the fact that the integrated concentration is made up of two parts, the concentration of the dye ( lC ) and the distance of the light path through the dye (?2*2bc, where bc is the concentration spread, and 2bc is generally associated with the distance from the centreline to the edge of the jet). For a jet, the spread increases linearly with distance and the concentration decreases linearly with distance. Multiplying these together (? lC *4 bc) gives a constant value (see equation (3.16)). The trajectory of the flow was defined as the point with the maximum integrated concentration value in a cross-section, and a cross-section of the flow was taken perpendicular to the trajectory. The integrated concentration image was saved as a txt-file and opened in Chapter 3 ? Flow Visualization Techniques 54 MatLab. A MatLab algorithm was written to analyse the file and find the position of the trajectory and the corresponding maximum values, and also the cross-sectional position and values. An example of the algorithm can be found in Appendix B. Figure 3.18 ? Integrated concentration image of jet In theory the integrated concentration values outside the jet should be zero as nothing should change from the reference image to the jet image. The blue colour outside the jet seen in Figure 3.18 is an indication that this was indeed what was found. To provide a more thorough review of the values outside the jet, the upper and lower bounds of the colours were narrowed to ?1% of the maximum integrated concentration (blue regions) for the lower limit of the colour map and 1% for the upper limit (red regions). The result can be seen in Figure 3.19. The colours cover almost the entire image, except where the jet is located. The yellow colour shows regions where the background noise is less than ?0.1% of the maximum integrated concentration. Chapter 3 ? Flow Visualization Techniques 55 Figure 3.19 - Green absorption image of jet, boundaries ?1% to 1% The jet problem has been studied experimentally and analytically over many years and hence its expected behaviour is well known. Work by Morton et al.(1956), Abraham (1963), Fan and Brook (1969) and others led to the integral methods commonly used now to predict the behaviour of buoyant jet discharges using the assumption that the mean cross-sectional concentration distributions are Gaussian. A coordinate system can be defined as shown in Figure 3.20 where the jet is discharged along the x- axis. Source of Jet z y d x, C Integrated View Plan View Cil Ci Cl C0 Figure 3.20 - Plan and integrated view of concentration profiles for a simple jet In this coordinate system the Gaussian assumption relates the local concentration in a cross- section ( )C to the centreline concentration ( )lC . This relationship is: 22 ??? ? ??? ?? ??? ? ??? ?? = cc b z b y l eeCC (3.7) Chapter 3 ? Flow Visualization Techniques 56 where bc is the concentration spread of the flow. Integrating this equation in the y-direction gives the relationship between the local integrated concentration ( )iC and the centreline concentration. 222 ??? ? ??? ??? ?? ?????????? ? ??? ?? == ? ccc b z cl b z b y li ebCdyeeCC pi (3.8) The integrated centreline concentration ( )ilC is then: piclil bCC = (3.9) or alternatively pic il l b CC = (3.10) Substituting equation (3.10) back into equation (3.8) to find a relationship between local integrated concentration and the centreline integrated concentration gives: 2 ??? ? ??? ?? = cb z ili eCC (3.11) Figure 3.21 shows concentration profiles for the jet. Seven different cross-sections were used from 14 port-diameters downstream to 132 port-diameters downstream. As expected, with appropriate scaling, the profiles are self-similar and there is a good match to the Gaussian curve of equation (3.11). -2 -1.5 -1 -0.5 0 0.5 1 1.5 20 0.2 0.4 0.6 0.8 1 z/bc C i/C il x/d=13.61 x/d=32.69 x/d=52.57 x/d=72.45 x/d=92.37 x/d=112.20 x/d=132.08 Gaussian Profile Figure 3.21 - Self-similarity of integrated cross-sectional profiles of jet Chapter 3 ? Flow Visualization Techniques 57 From the Eulerian Integral method comes the relationship between the centreline concentration and the initial concentration ( )0C . x d kI ICC qc m l 40 pi= (3.12) where d is the diameter of the source of the flow, k is the velocity spread value, and mI and qcI are both shape factors (Pun 1998). Table 3.1 shows the experimental values found by various researchers for both the velocity spread and the concentration spread. Note that the values by Fischer et al. (1979) were an average of all experimental spread data that was available at the time of publishing. In the process of averaging the spread values, the results from the most recent studies were given the most significance, as the newer measuring techniques used in those studies increased the accuracy of the results. With an average value for k of 0.106 and a value for k? of 0.129, the value for ? is 1.22. Both Papanicolau (1984) and Wang (2000a) found that 7% of the total mass flux was carried by turbulence, therefore qcI can be calculated to have a value of 2.03. The jet experiments by Hussein et al. (1994) gave a value of approximately 10% for that part of the total momentum flux that is carried by turbulence. Similar experiments carried out by Wang (2000a) gave a value of approximately 9%. mI can therefore calculated to be approximately 1.7. Table 3.1 - Comparison of velocity and concentration spread values k (velocity spread) k? (concentration spread) (Fischer et al. 1979) 0.107 0.127 (Chen and Rodi 1980) 0.103 0.136 (Papanicolaou 1984) 0.112 0.139 (Papanicolaou and List 1988) 0.104 0.126 (Panchapakesan and Lumley 1993) 0.115 (Hussein et al. 1994) 0.106 (Wang 2000a) 0.105 0.127 Weighted-Average spread values 0.106 0.129 The above determined concentration spread value is verified using experimental concentration spread results, non-dimensionalised by the port diameter, shown in Figure 3.22 The experimental data was found by fitting Gaussian curves through the cross-sectional integrated concentration values. The cross-sectional profile was perpendicular to the direction of the Chapter 3 ? Flow Visualization Techniques 58 trajectory at that point. The theoretical line was calculated as k times the distance from the source times ? (the spread assumption). The figure shows that theory is consistent with the experimental data. 0 20 40 60 80 100 120 140 160 180 2000 5 10 15 20 25 30 x/d b c /d Figure 3.22 ? Concentration spread of jet As indicated in second paragraph of section 3.2.3.1., the before the relationship between the maximum integrated initial concentration ( )0iC and the initial concentration is dCCi *00 = (3.13) where d is the diameter of the source. Substituting equation (3.13) into equation (3.12) gives xkI IC x d kI I d CC qc m i qc mi l 1 44 0 0 pipi == (3.14) To find the relationship between the integrated centreline concentration and the integrated initial concentration, equation (3.14) is substituted into (3.10) ?pipipi 214 00 qc m ic qc m iil I ICb xkI ICC == (3.15) or 0 2qci il m IC C I pi?= (3.16) The experimentally determined ratio of the initial integrated concentration and the integrated centreline concentration has been plotted versus non-dimensional distance downstream in Chapter 3 ? Flow Visualization Techniques 59 Figure 3.23. The figure confirms that the integrated centreline dilution is indeed constant with distance downstream. A least squares fit line was plotted through the integrated centreline dilution data from seven jet experiments (see Chapter 5) to determine an average dilution value of 0.829 (see Figure 3.23). Equation (3.16) and the above-determined values for ? , mI and qcI combine to give a theoretical value for the integrated dilution of 0.812. The difference between the experimental and the theoretical value is associated with a decrease of approximately 2.0% in the value of the ratio qc mI I (from 1.59 to 1.56). This suggests a lower value for the total mass flux carried by turbulence and/or a higher value for the part of the total momentum flux that is carried by turbulence. 0 20 40 60 80 100 120 140 160 180 2000 0.2 0.4 0.6 0.8 1 1.2 x/d C i0/ C i l Ci0/Cil C0/Cl 0 20 40 60 10 30 50 C 0/ C l Figure 3.23 ? Integrated centreline and point centreline dilution Close to the source the values for the ratio of the integrated dilution increase. This is due to the small jet scale relative to the pixel resolution. In this situation the pixels smooth the peak concentration significantly, and significantly affected data was discarded. Equation (3.10) can also be used to convert the integrated concentration data back to point concentration data, by combining it with equation (3.13) to give: picl i il b C C C Cd 00 = (3.17) Chapter 3 ? Flow Visualization Techniques 60 or alternatively to calculate the dilution d b C C C C c il i l pi00 = (3.18) The data from the simple jet experiment was converted using the above equation and plotted versus non-dimensional distance downstream in Figure 3.23. The experimental data depended on a second experimental value ( )cb (as seen in Figure 3.22). Theoretical predictions are shown for comparison (equation (3.12)). There is a good agreement between the theoretical predictions and experimental results. The opposite of converting the integrated dilution data into point dilution data is to convert the data into double integrated dilution data. This can again be done using the spread at the cross-section or by calculating the area under the cross-sectional profile. The theoretical relationship between the double integrated concentration ( )iiC and the centreline concentration is found by integrating equation (3.7) in both the y-direction and z- direction. pi2 22 cl b z b y lii bCdydzeeCC cc == ? ? ? ?? ? ?? ??? ? ??? ?? ??? ? ??? ?? (3.19) The double-integrated initial condition ( )0iiC is defined as dCdCC iii 44 0200 pipi == (3.20) Substituting equations (3.19) and (3.20) into equation (3.12) gives the relationship for the double-integrated dilution. 0 1 2 2 1 1 2 qcii ii m IC d C I k x?pi= (3.21) The experimental results are compared with equation (3.21) in Figure 3.24. Here again the experimental results match the predictions. Chapter 3 ? Flow Visualization Techniques 61 0 20 40 60 80 100 120 140 160 180 2000 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 x/d C ii0 /C ii Figure 3.24 ? Double-integrated jet dilution By creating integrated concentration images from the instantaneous jet images before the images are averaged, the integrated turbulent statistics can be investigated. Previous investigators have shown that the dimensionless turbulent intensity of the concentration fluctuations along the centreline of the jet is approximately 0.225 (Papanicolaou 1984; Wang 2000a). Figure 3.25 shows that the intensity of the dimensionless integrated concentration fluctuations along the flow centreline was approximately 0.12 for these experiments. The instantaneous jet consists of multiple rotating eddies of different sizes, entraining the surrounding ambient fluid. Large eddies dominate the flow and therefore have a major influence on the turbulent fluctuations. The relatively good correlation between the centreline and integrated turbulent intensity confirms the presence of the large eddies. If the instantaneous flow had existed out of numerous small eddies, the reduction in turbulent intensity, due to the integrated view, would have been significantly greater. Chapter 3 ? Flow Visualization Techniques 62 0 10 20 30 40 50 60 70 80 90 1000 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 x/d [(C il') 2 ] 0.5 /C il Figure 3.25 ? Integrated concentration fluctuation along the jet centreline The cross-sectional variation of the integrated turbulent fluctuations can be seen in Figure 3.26. The double peak shaped profile matches the cross sectional variations found by previous researchers for the point turbulent concentration fluctuations. However the peaks are more pronounced for the integrated case. Moving away from the centreline of the jet, the distance over which the integration takes place becomes smaller and therefore the dampening effect of the integrated view decreases. Hence, the maximum difference in the turbulent intensity between the point and integrated values, happens at the centreline. Moving away from the centreline the integrated values increase more than the point intensity values, so that these values are more closely matched. The ratio of the integrated turbulent fluctuation to the non- integrated turbulent fluctuation (Papanicolaou 1984; Wang 2000a) increases from 0.53 at the centreline to 0.61 and 0.67 at z/x = 0.1 and 0.2 respectively. Chapter 3 ? Flow Visualization Techniques 63 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.250 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 z/x [(C iL* )2 ] 0.5 /C iL x/d = 40 x/d = 71 x/d = 117 Figure 3.26 ? Integrated cross-sectional profiles of turbulent concentration fluctuations 3.2.3.2 ? Strongly Advected Flow, a Momentum Puff Experiment For the momentum puff experiments, two separate series of runs were required to characterize the flow, providing perpendicular views of the discharge. One series provided a side view (see Figure 3.27a), the camera view is along the y-axis of the flow and therefore records the y- integrated view of the flow. The experimental set up for the side view experiment was similar to that for the simple jet experiment described above. However, in this case the source was attached to a trolley that moved through the camera view during the experiment. These experiments will be referred to as y-integrated momentum puff experiments. More details about the experimental set up of the camera recording the y-integrated view can be found in section 3.2.1.1. For the second series the view was along the z-axis of the flow (see Figure 3.27b) such that the initial discharge was facing towards the camera. These experiments will be referred to as z-integrated momentum puff experiments, and more details about the recording of the z-integrated view can be found in section 3.2.4.1. The momentum puff flow is independent of the discharge angle as long as the flow is discharged perpendicular to the ambient. Therefore by rotating the source and the coordinate system by 90o, the z-integrated momentum puff experiments could be recorded without altering the position of the camera. Chapter 3 ? Flow Visualization Techniques 64 The calibration length-scales change with the changing distance between camera and flow trajectory during the z-integrated momentum puff experiments. Details of the method used to determine the calibration length-scales for the z-integrated momentum puff experiments can be found in section 3.2.4.2. a) b) y x z x z y Figure 3.27 - y-integrated (a) and z-integrated (b) momentum puff views The source diameter for the momentum puff experiment was either 2.45mm or 3.00mm. The Reynolds number ranged between 2600 and 5100, and the ambient velocity (the velocity of the trolley) ranged between 33mm/s and 98mm/s. Again videos were recorded of the background and both calibration cells before the experiments. The videos of the momentum puff were approximately 1500 frames long. In ImageStream the images were converted into integrated concentration images and then transformed into a reference frame moving with the source. Figure 3.28 and Figure 3.29 show the integrated concentration plots for the y-integrated view and z-integrated view respectively. The white colour indicates an integrated concentration of 0 and the black approximately 0.00014 g/l*m. Chapter 3 ? Flow Visualization Techniques 65 Figure 3.28 ? Integrated concentration image of a y-integrated momentum puff Figure 3.29 ? Integrated concentration image of a z-integrated momentum puff In the strongly advected phase of the line momentum puff the internal concentration structure no longer has a Gaussian shape, but the structure resembles that of a vortex pair. This can be seen in Figure 3.29, where the integrated concentration values are clearly lower in the centre of the flow. It has been shown (Richards 1963; Knudsen 1988; Wong and Lee 1991; Gaskin 1995; Chu 1996) that the fully developed vortex pair is self-similar in the mean. To date the vortex pair distribution has not been approximated by a relatively simple function. This is necessary to be able to relate the integrated concentration to the point concentration values of the vortex pair and is achieved with a pair of stretched, merging Gaussians. This approximation is not particularly good near the edges of the flow, but it allows for the peaks to be resolved with reasonable accuracy. A schematic of the approximation can be seen in Figure 3.30. Figure 3.30a shows the approximation of the vortex-pair by the double-Gaussian and its integrated counterpart in the z-direction. The integration in the y-direction can be seen in Figure 3.30b. The x-coordinate is in the direction of the ambient flow and perpendicular to the cross-sectional plane. Note that the spread value bc is the spread of the single Gaussian and f and h are constants if the distribution is self-similar. Plotting least-squares fit double- Gaussian curves through the y-integrated and z-integrated cross-sectional concentration data at certain locations can determine values for f and h. The value for h is the ratio of the y- integrated spread in the z-direction to the z-integrated spread in the y-direction, and f is the ratio of the distance between the centreline and peak of a single Gaussian to the concentration spread of the single Gaussian. Chapter 3 ? Flow Visualization Techniques 66 Cross - Sectional view of vortex pair (Based on Gaskin, 1995) z - integrated view of Double - Gaussian approximation y z y z Ciz b c b c f b c f b c y Cross - Sec tional view of Double - Gaussian approximation Cross - Sectional view of vortex pair (Based on Gaskin, 1995) y-integrated view of Double - Gaussian approximation z y z y b c b c h b c h b c z Cross - Secti onal view of Double - Gaussian approximation a) b) Ciy Figure 3.30 ? Vortex pair and double-Gaussian approximation Using this approximation, the equation relating the local concentration to the centreline concentration in a cross-section (equation (3.7)) can be rewritten to incorporate the double Gaussian. 2 2 2 c c c c c y fb y fb z b b hb lC C e e e ? ? ? ? ? ?? +? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?= + ? ?? ? (3.22) Chapter 3 ? Flow Visualization Techniques 67 For a given experiment the assumed location of the centreline of the vortex pair and the actual location may differ. To allow for this the following adjustment is made: ' ly y y= ? (3.23) and ' lz z z= ? (3.24) where ly and lz are the coordinates of the actual flow centreline. Replacing y and z in equation (3.22) with y? and z? gives: 222 ''' ??? ? ??? ?? ??? ? ??? ? +? ??? ? ??? ? ?? ? ? ? ? ? ? ? ? += cc c c c hb z b fby b fby l eeeCC (3.25) Again this equation can be integrated, but the distribution is no longer axi-symmetric and therefore integrations in the y and z-directions must be dealt with separately. Integrating in the y-direction gives the relationship between the local integrated concentration in the y-direction ( )iyC and the centreline concentration. The result is equation (3.26). ( ) 22 222222 '' '''''' 2 ? ? ? ? ??? ?? ??? ? ??? ?? ??? ? ??? ??? ?? ? ?? ??? ? ??? ? +? ??? ? ??? ? ?? ??? ? ??? ??? ?? ??? ? ??? ? +? ??? ? ??? ? ?? =+= ? ? ? ? ? ? ? ? += ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ?? ? ? ? += ? ?? cc cc c c c cc c c c hb z cl hb z ccl hb z b fby b fby l hb z b fby b fby liy ebCebbC edyedyeCedyeeCC pipipi (3.26) or alternatively 2' ??? ? ??? ?? = chb z ilyiy eCC (3.27) Substituting z? = 0 into equation (3.26) gives the relationship between ilyC (y-integrated centreline concentration) and lC piclily bCC 2= (3.28) Note that the centreline concentration ( )lC in equation (3.28) is based on the single Gaussian distribution and does not represent either the centreline value or the peak value of the vortex pair. To find the relationship between ilyC and the concentration at the centreline of the vortex pair ( )cC , first the relationship between lC and cC is found by substituting ' 0y = and ' 0z = into equation (3.25) to create equation (3.29). Then equation (3.29) is rearranged for lC and substituted into equation (3.28) to form equation (3.30) ( )2 2f fc lC C e e? ?= + (3.29) Chapter 3 ? Flow Visualization Techniques 68 2f ily c cC C b epi= (3.30) The same approach leads to the relationship between ilyC and the peak concentration of the vortex pair ( )peakC . This time 'y f b?= and ' 0z = are substituted into equation (3.25) ( )( )221 f peak lC C e ?= + (3.31) ( )22 2 1 c ily peak f bC C e pi ? = + (3.32) Integrating equation (3.25) in the z-direction gives the relationship between the local integrated concentration in the z-direction( )izC and the centreline concentration of the single Gaussian. ?? ? ? ? ?? ? ? ? += ?? ? ? ? ?? ? ? ? += ? ? ? ? ??? ? +? ??? ? ??? ? ??? ?? ??? ? ??? ?? ??? ? ??? ? +? ??? ? ??? ? ?? ? 22222 ''''' c c c c cc c c c b fby b fby cl hb z b fby b fby liz eehbCdzeeeCC pi (3.33) or ?? ? ? ? ?? ? ? ? += ? ? ? ? ??? ? +? ??? ? ??? ? ?? 22 '' c c c c b fby b fby ilz iz ee C C (3.34) Pun (1998) gives the relationship between the initial concentration and the centreline concentration. Modified to include the double-Gaussian assumption, this can be written as: 2 5.0 0 20 0 ??? ? ??? ?= ae sgcdg l a UM zkI C C U U (3.35) where aU is the ambient velocity, 0U the initial velocity, sgk the concentration spread-rate for the single Gaussian divided by ? , 0eM the initial excess momentum flux and cdgI a shape constant defined by the integral b zd b ydeeeI fbyfbyhbz cdg ccc? ? ? ?? ? ?? ??? ? ??? ? +? ??? ? ??? ? ?? ??? ? ??? ?? ?? ? ? ? ?? ? ? ? += 222 ''' (3.36) cdgI has a value of pi? 22h . Note that by defining cdgI by equation (3.36), lC in equation (3.35) is again based on the single Gaussian distribution and cdgI is a constant value if the Chapter 3 ? Flow Visualization Techniques 69 cross-sections are self-similar. Rearranging equation (3.25) for lC and inserting it into equation (3.35) gives 2 2 22 ' ' ' 2 00 0.5 0 c c c c c y fb y fb z b b hbcdg sg a e a U I kC z e e e U M UC ? ? ? ? ? ?? +? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? ?? ? ? ?? ?? ?= +? ? ? ?? ?? ? ? ?? ?? ? ? ?? ? (3.37) Integrating in the y-direction gives the relationship between initial concentration and iyC . ?? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ??? ? ??? ?= ????????? 2'2 5.0 0 2 00 2 chb z c aea sgcdg iy ebUM zU kIUCC pi (3.38) Inserting 0'=z , and combining equations (3.13) and (3.38) gives the relationship between the integrated initial condition and the integrated centreline concentration in the y-direction ae sg aeac sgcdg ily i UM zkh UM z Ub kIdU C C 5.0 0 2 5.0 0 2 00 2 2 ?pi =? ? ? ? ??? ?= (3.39) Integrating equation (3.37) in the z-direction gives the relationship between initial condition and izC 2 22 ' ' 2 00 0.5 0 c c c c y fb y fb b bcdg sg c a e aiz U I kC z hb e e U M UC pi ? ? ? ?? +? ?? ? ? ? ? ? ? ? ? ?? ?? ?? ? ? ?? ?? ?= +? ? ? ?? ?? ? ? ?? ?? ? ? ?? ? (3.40) Inserting 0'=y , and combining equations (3.13) and (3.40) gives the relationship between the integrated initial condition and the integrated centreline concentration in the z-direction 2 2 22 00 0.5 0.5 0 0 22 f cdg sg fi sg ilz e a e ac a dU I kC z e zk e C M U M Uhb U ?pi ? ?= =? ? ? ? (3.41) Combining equations (3.39) and (3.41) gives a second experimental means (the first being the ratio of the spread at a cross-section) to find an estimate for h ilzi ilyif CC CCeh 0 02= (3.42) An assessment of the value of the double-Gaussian approximation is determined through comparisons with experimental data from the momentum puff using the LA technique. At 20 different points along the trajectory of the flow, cross-sections were taken from a y-integrated momentum puff experiments (Re = 4363, Ur = 0.0289). The cross-sectional values were non- dimensionalised, using the strong jet to advected line momentum puff transition length-scale (Me00.5/Ua), and plotted versus the Gaussian theory of equation (3.27). The results are shown Chapter 3 ? Flow Visualization Techniques 70 in Figure 3.31. The double-Gaussian assumption matches the cross-sectional experimental data well. It also shows that the cross-sections are self-similar which is consistent with the observations of Knudsen (1988), Wong and Lee (1991), Gaskin (1995) and Chu (1996) For a comparison of the z-integrated double-Gaussian assumption, ten non-dimensional integrated cross-sectional concentration profiles from a z-integrated momentum puff experiment (Re = 4042, Ur = 0.0317) were compared with the theory of equation (3.34) in Figure 3.32. There is good agreement between the assumed and experimental profiles and this agreement suggests that the double-Gaussian approximation for tracer profiles is reasonable in the strongly advected region of the flow. The assumed profile in Figure 3.32 used a value for f of 0.88. Values of the parameter f determined from least-squares fits to the profiles in the strongly advected region, such as those shown in Figure 3.32, are presented in Figure 3.33. The value for f is reasonably constant in the strongly advected region, but did have a slight downward trend in some of the datasets. The average value for f is 0.88 with an error of ? 0.05. The error reduced with increasing distance from the source. Much of the error is believed to be caused by the sensitivity of the flow to the generated turbulence, especially in the region close to the transition. Richards (1963) carried out line thermal experiments in a still ambient. A horizontal cylinder of fluid with a higher density than the surrounding fluid was released and carried away by buoyancy forces. His work clearly showed the presence of a vortex pair. He also showed that the flow was approximately self-similar in the mean and the width of the line-thermal was proportional to the (vertical) distance travelled. However the proportionality constant he found was heavily dependent on the method of release. In the present experimental set up, it is possible that the random nature of the turbulence in the transition zone to the puff region generates different circumstances under which the vortex pair is formed, influencing both the position and the behaviour of the vortex pair. Using equations (3.29) and (3.31) the theoretical ratio of maximum concentration to centreline concentration is determined to be 1.13. This is lower than the 1.18 that Chu (1996) found experimentally. However as Figure 3.32 shows the double-Gaussian curve has a tendency to smooth the peaks and troughs of the experimental cross-sectional data, decreasing the expected ratio of maximum to centreline concentration. Chapter 3 ? Flow Visualization Techniques 71 -2 -1.5 -1 -0.5 0 0.5 1 1.5 20 0.2 0.4 0.6 0.8 1 z'/bc C iy/ C i ly z=2.40 Me00.5/Ua z=2.72 Me00.5/Ua z=2.93 Me00.5/Ua z=3.09 Me00.5/Ua z=3.22 Me00.5/Ua z=3.33 Me00.5/Ua z=3.44 Me00.5/Ua z=3.53 Me00.5/Ua z=3.63 Me00.5/Ua z=3.72 Me00.5/Ua z=3.80 Me00.5/Ua z=3.89 Me00.5/Ua z=3.98 Me00.5/Ua z=4.05 Me00.5/Ua z=4.12 Me00.5/Ua z=4.19 Me00.5/Ua z=4.26 Me00.5/Ua z=4.28 Me00.5/Ua Equation (3.27) Figure 3.31 - Cross-Sectional Integrated Concentration profiles integrated in the y-direction -3 -2 -1 0 1 2 30 0.2 0.4 0.6 0.8 1 y'/bc C iz/ C i lz z=2.01 Me00.5/Ua z=2.49 Me00.5/Ua z=2.98 Me00.5/Ua z=3.24 Me00.5/Ua z=3.46 Me00.5/Ua z=3.64 Me00.5/Ua z=3.79 Me00.5/Ua z=3.93 Me00.5/Ua z=4.06 Me00.5/Ua z=4.18 Me00.5/Ua Equation (3.34) Figure 3.32 - Cross-Sectional Integrated Concentration profiles integrated in the z-direction Chapter 3 ? Flow Visualization Techniques 72 3 3.2 3.4 3.6 3.8 4 4.2 4.40 0.2 0.4 0.6 0.8 1 1.2 1.4 z/(Me00.5/Ua) f Figure 3.33 - Value of f as a function of vertical distance away from source The double integrated concentration results from both the y and z-directions are presented in Figure 3.34. For most runs the results were within 5% of the expected results. Part of that error came from the determined calibration length scale for the z-integrated runs (see section 3.2.4.2). However for some runs the error was much larger. This was related to the use of Sodium Sulphite during the experimental investigation (see section 3.2.2.4). To take the influence of the Sodium Sulphite into account, a correction based on the double integrated results was applied to the experimental integrated concentration data. The double integrated dilution equation can be calculated from the conservation of mass equation for the momentum puff ( )aiiUCQC =00 or by double integrating equation (3.37), 100 = ii ii a C C U U (3.43) where Cii0 is the double integrated initial concentration and Cii is the double integrated concentration. The theoretical solution in the jet region was determined by modifying equation (3.21) to include the velocity ratio. Of interest is the location of the transition from weakly to strongly advected behaviour, which appears to occur gradually and not be complete until at least 3.5 length-scales from the source in the z-direction. Comparing Figure 3.24 with Figure 3.34 it can be noted that, in the jet region, the current results are over predicted by the Chapter 3 ? Flow Visualization Techniques 73 double-integrated theory while the theory under predicted the results shown in Figure 3.24. In section 5.3.1.2 this is investigated further. 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 z/(Me00.5/Ua) U 0/ U a *C ii0 /C ii Momentum Puff Results equation (3.43) equation (3.21) Figure 3.34 ? Double integrated dilution results for momentum puff experiments The distortion of the individual Gaussians is taken into account by the value h, where h is the ratio of the y-integrated spread in the z-direction to the z-integrated spread in the y-direction of the single Gaussian. As mentioned before the ratio of the integrated centreline dilutions provides information with respect to this distortion (equation (3.42)). Ratios of the centreline z-integrated dilution to the y-integrated centreline dilution are shown in Figure 3.35. A direct measure of the distortion is given by the ratio of the spread in the z-direction to that in the y- direction as determined from least-squares fits of the double-Gaussian approximation to the integrated profiles. This data is also shown in Figure 3.35. The two sets of data are consistent and show the profiles are more distorted near the transition region and that this distortion eases in the strongly advected region where the value of h approaches 1.46 (Figure 3.35). These variations in h can be approximated empirically with the relationship 0.5 01* e c a Mch h b U= + (3.44) Chapter 3 ? Flow Visualization Techniques 74 where h* is equal to 1.46 and c1 is equal to 0.04. It is worth noting that for distances less than three length-scales from the source the angle of the cross-sectional profiles is more than 10? from the vertical. With the vertical cross-sectional assumption that is built into the experimental technique this leads to significantly increasing errors closer to the source 3 3.5 4 4.5 5 5.5 60 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 z/(Me00.5/Ua) h h (from centreline integrated dilution data h (from spread data) Figure 3.35 ? Double-Gaussian parameter h as a function of vertical distance from the source Figure 3.36 compares the y-integrated concentration spread in the z-directions and the z- integrated concentration spread in the y-direction. Fitting linear least-square fit lines to both data sets gives concentration spread constants of 0.36 and 0.24 for the concentration spread in the y and z-directions respectively. Relationships predicting the dilution are based on the concentration spread divided by ? of one of the single Gaussians ( sgk , equation (3.35)), and this is given as 0.24 0.20sgk ?= = . Chapter 3 ? Flow Visualization Techniques 75 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 z/(Me00.5/Ua) b c /(M e00.5 /U a) bc h bc least-square fit lines Figure 3.36 - Spread comparison of top and side view momentum puff experiments As the vortex pair distribution has not been approximated by a relatively simple function definitions of the characteristic width of the momentum puff have differed in the past (Gaskin 1995). The y-integrated spread constant can be compared with previous data from point cross- sectional concentration profiles measured in the z-direction over the whole cross-section. The value for the y-integrated spread constant compares well with the value found by Knudsen (1988) of 0.36 and to a lesser degree with 0.41 found by Gaskin (1995). Wong and Lee (1991) calculated a ratio of horizontal to vertical width for a strongly advected cross-section. The boundary of the cross-section was defined by a mCeC 1?= contour, where mC is the maximum concentration in the cross-section. The experiments they carried out showed that the ratio increased from 1 (in the jet region of the flow) to about 1.1-1.3. The average value they found was 1.2. Chu (1996) determined an average value for the puff aspect ratio of 1.23. Using the definition of the aspect ratio as explained above, the double-Gaussian cross-sectional assumption gives a value for the width ratio of 1.26, comparing well with the experimental values found by Wong and Lee, and Chu. Chapter 3 ? Flow Visualization Techniques 76 Figure 3.37 shows the results of a comparison between the experimental results and the integrated dilution equations. The Reynolds numbers ranged between 2631 and 5100 and the ambient velocity between 33mm/s and 98mm/s. Integrating in the y-direction gives a more comprehensive view as the complete flow has been recorded by the camera and thus the two different regions, the jet and momentum puff regions were visible. On Figure 3.37 only the strongly advected results have been plotted. The flow recorded from the z-integrated direction has the limitation that it cannot be interpreted clearly until it is sufficiently bent over so that the vertical cross-section assumption is reasonable. Both data sets collapse well onto a single line, although there is more scatter in the z-integrated data. The theory of equation (3.39) matches the experimental data from 3.5 length-scales onwards. This is consistent with the end of the transition zone determined with the double-integrated data. Up to the transition the theoretical integrated dilution value is 0.812 (see section 3.2.3.1). The y-integrated dilution results therefore suggest a transition point at a distance of 1.9 length-scales in the z-direction, the z-integrated results suggest a transition point at 1.7 length-scales. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 z/(Me00.5/Ua) C i0/ C i l Ci0/Cily Ci0/Cilz equation (3.39) equation (3.41) Figure 3.37 - Integrated Dilution results Within the puff region the integrated dilution in the y-direction can be converted into point dilution. The minimum point dilution occurs at the peaks. The theoretical equation for the minimum dilution can be found by inserting equation (3.31) into equation (3.35). Rearranging gives Chapter 3 ? Flow Visualization Techniques 77 ( )( ) 2 5.0 0 2 2 0 0 21 ?? ? ? ??? ? + = ? ae f sgcdg peak a UM z e kI C C U U (3.45) The experimental y-integrated dilution data can be converted into minimum point dilution data by combining equation (3.28), equation (3.13) and equation (3.31). Solving the equation for the actual dilution gives ( )( ) ( ) ( )( )2 2 0 0 0 0.52 2 0 0 0 2 1 1 1 a i c a i c f fpeak ily ily e a U C C b U C b U C C d U C M Ue e pipi ? ? = = + + (3.46) Similarly combining equation (3.33) ( )cfby =' , equation (3.13) and equation (3.31) gives the relationship between the minimum point dilution and the integrated dilution when integrating in the z-direction ( )0 0 0 0.50 0 02a i c a i cpeak iz peak iz peak e a U C C b h U C h b U C C d U C M U pi pi ? ? = = (3.47) Figure 3.38 shows the minimum point dilution of the momentum puff. The figure presents the experimental results from the y-integrated momentum puff experiments converted into point peak dilution data using equation (3.46) and the experimental results from the z-integrated momentum puff experiments converted into point peak dilution data using equation (3.47). The data is compared with predictions based on equation (3.45), with appropriate allowance for a virtual source, and those of VisJet (Cheung et al. 2000) and CorJet (Jirka 2004). The model prediction from VisJet overestimates the dilution in the strongly advected region and the prediction from CorJet underestimates the dilution. All experimental data collapses onto a single line that matches the theory well. The agreement between the data obtained from two different integrated perspectives further indicates the value of the double-Gaussian approximation in interpreting the strongly advected behaviour of these flows. Chapter 3 ? Flow Visualization Techniques 78 100 101 10-1 100 101 z/(Me00.5/Ua) C 0/ C p ea k* U r y-integrated experiments z-integrated experiments equation(3.45) CorJet VisJet Figure 3.38 - Converted peak dilution data using far-field and near-field values for h For the present experimental configurations the trajectory of the momentum puff can only be seen using the y-integrated view. It is defined as the location with the maximum integrated concentration value in a cross-section, taken perpendicular to the flow trajectory. The result of a y-integrated experiment is plotted in Figure 3.39 and it is compared with trajectory predictions from two different mathematical models. The experimental data agrees well with the experimental data of Hung (1998) and Pratte and Baines (1967), however there is less agreement with the experimental data of Jordinson (1956), and Keffer and Baines (1963). Model prediction from CorJet and VisJet appear to match the experimental data reasonably well, but the CorJet predictions have a tendency to under predict the data further downstream, whereas VisJet tends to under predict initially, but is more consistent with the data further downstream. Chapter 3 ? Flow Visualization Techniques 79 10-1 100 101 102 10-1 100 101 x/(Me00.5/Ua) z/ (M e00.5 /U a) Present experimental study Jordinson (1956) Keffer and Baines (1963) Pratte and Baines (1967) Hung (1998) VisJet CorJet Figure 3.39 - Momentum puff trajectory, comparing experimental data with model predictions 3.2.3.3 ? Angled Jet As mentioned previously the angled jet flow is an example of a flow that is no longer flowing perpendicular to the camera view. The experimental set up for the angled jet experiment is the same as for the simple jet with the exception of the source position. It was no longer positioned vertically downwards, but makes a certain angle, ?0, with the horizontal. This can be seen in Figure 3.40. The angle ?? was 65o for these experiments, determined to a precision of 0.25 of a degree. The diameter of the source was 3mm and 0Q was 0.010l/s. To be able to convert the integrated concentration values, obtained from the angled jet experiment, into centreline concentration values, the internal concentration structure along the ray of light that reached the camera has to be known. The cross-sectional view of the angled jet in Figure 3.41 has been cut through the centre of the jet. Therefore the integrated centreline concentration has been integrated over the local concentration along dimensionless distance 21 bb + . At a certain distance dkbdsL ?== / downstream, where ds/ is the non- dimensional distance downstream from the source along the trajectory of the jet, 1b and 2b Chapter 3 ? Flow Visualization Techniques 80 can be written as function of angles 0? , ? and L. The value of ? is approximately equal to ?k . ( ) 1 0 sin cos 2 b L ?pi ? ? = ? ? ? ?? ?? ? (3.48) ( ) 2 0 sin cos 2 b L ?pi ? ? = ? ? ? +? ?? ? (3.49) ?0 ?0 k?L k?L b1 b2 Side view Camera Camera view k?L k?L L/sin(?0) ? L = s/d Figure 3.40 - Vertical cross-sectional and frontal view of angled jet experiment Using the above derived non-dimensional distances for 1b and 2b , a horizontal cross-section of the flow in the plane of the camera can be drawn (Figure 3.41). A coordinate system with the origin at the maximum concentration value has been added. The cross-section consists out of two half ovals meeting at the y-axis and can be mathematically described by equations (3.50) and (3.51). Chapter 3 ? Flow Visualization Techniques 81 for 0?x : ( ) ( ) 2 2 2 2 0 1 sin cos 2 x y k L L ? ? pi ? ? + = ? ? ? ? ? ?? ? ? ?? ?? ?? ? ? ?? ? (3.50) for 0?x : ( ) ( ) 2 2 2 2 0 1 sin cos 2 x y k L L ? ? pi ? ? + = ? ? ? ? ? ?? ? ? ?? +? ?? ? ? ?? ? (3.51) k?L k?L Lsin(?) cos(pi/2??0??) Lsin(?) cos(pi/2??0+?) y x Figure 3.41 ? Horizontal cross-section The cross-sectional centreline concentration profiles of a simple jet are known to have a Gaussian shape, therefore the cross-sectional concentration profiles of the angled jet can be described by two half Gaussians, again meeting at the y-axis. Equation (3.7) is split up into two integrals, from minus infinity to zero and zero to infinity, and used to calculate the theoretical integrated centreline concentration. ( ) 2 2 1 2 0 0 0 0 1 1 1sin 2 cos cos 2 2 x x b b il l lC C e dx e dx C Lpi ? pi pi ? ? ? ? ? ? ? ??? ? ? ? ? ? ? ? ?? ? ?? ? ? ?? ? ? ?= + = +? ? ? ? ? ? ? ?? ? ? +? ? ? ? ? ? ? ?? ? ? ?? ? ? ? (3.52) Chapter 3 ? Flow Visualization Techniques 82 or alternatively to convert the integrated value into the actual centreline concentration ( ) 0 0 2 1 sin 1 1 cos cos2 2 l ilC C Lpi ? pi pi? ? ? ? = ? ? ? ? ? ?+? ? ? ? ? ?? ? ? +? ? ? ?? ? ? ? ? ?? ? (3.53) Combining equation (3.53) with equation (3.13) gives the following relationship for dilution. ( )0 0 0 0 sin 1 1 2 cos cos 2 2 i l il LC C C C d pi ? pi pi? ? ? ? ? ? ? ? ? ?= +? ? ? ? ? ?? ? ? +? ? ? ? ? ?? ? ? ?? ? (3.54) Figure 3.42 compares the dilution results (converted using equation (3.54)) from the angled jet experiment with the theoretical equation (3.12). Just as in the simple jet case the accuracy of the dilution results is affected by the need to use the concentration spread values to convert the integrated values to point values and in this case a value for ? must also be calculated at each point. However the agreement with theory is reasonable and this shows that LA can be used under these circumstances as long as the internal concentration structure is known. 0 20 40 60 80 100 120 1400 5 10 15 20 25 L=s/d C 0/ C l Figure 3.42 ? Experimental versus theoretical dilution of angled jet Chapter 3 ? Flow Visualization Techniques 83 3.2.3.4 ? Parallax Issues The above derived equations can be used to calculate the errors that are involved in LA due to parallax. The dimensions of the recorded camera view are approximately 570mm x 700mm (horizontal x vertical). The highest error due to parallax will occur when a flow is recorded in the corner of the screen. For these experiments, the maximum vertical distance from the image centre is 350mm and the maximum horizontal distance is 285mm. Therefore the distance to the corner is approximately 450mm. The averaged distance of the camera from the flow in the tank was 4000mm. The two camera positions, where the camera is assumed to be and where it actually is can be seen on the left hand side of Figure 3.43, creating the angle ? . Returning to the angled jet, rotating the source of the flow over an angle ? generates the same parallax issue. The result can be seen on the right hand side of Figure 3.43. Note that 0 2 pi? ?= ? (3.55) k?L k?L ? ?0 L L 4000mm 450mm L1 Assumed Camera Position Actual Camera Position k?L k?L b3 b4 b4 b3 Actual Camera View Rotated Camera View (Angled Jet Experiment) Figure 3.43 ? Cross sectional view of simple jet experiment including the effects of parallax To measure the error involved in neglecting the effects of parallax, the ratio of the assumed integrated centreline concentration over the actual integrated centreline concentration is Chapter 3 ? Flow Visualization Techniques 84 calculated. The assumed integrated centreline concentration assumedilC ? can be calculated using equation (3.9) rewritten as pi?pi? kLCbCC llassumedil ==? (3.56) The actual integrated centreline concentration can be found using equation (3.52), but as the centreline distance from the source has increased form L to L1, the centreline concentration has decreased from lC to 1?lC . Inserting into equation (3.52) gives ( ) ( ) ( )1 11 1 1sin2 cos cosil actual lC C Lpi ? ? ? ? ?? ? ? ?= +? ?? + ? ? (3.57) and therefore the error involved is ( ) ( ) ( )1 1 21 1 1 1sin cos cos il assumed l il actual l C C L kerror C C L ? pi pi ? ? ? ? ? ? ? ? = ? = ? ? ? +? ?? + ? ? (3.58) As mentioned before the spread of a jet increases linearly with distance and the concentration decreases linearly with distance. As the spread is linearly related to the distance downstream from source, LCl is a constant and therefore the ratio of LCl over 11LCl? is 1 and equation (3.58) can be simplified to ( ) ( ) ( ) 21 1 1sin cos cos kerror ? ? ? ? ? ? = ? ? ? +? ?? + ? ? (3.59) This gives a maximum error in the measured integrated centreline concentration value due to the effects of parallax of 0.0099 or 0.99%. Using a different experimental set up will slightly alter the error involved, but will still be well within the maximum 5% error of the linear intensity versus concentration assumption used for calibration and therefore parallax issues are not considered significant. 3.2.4 ? 3D LA The established LA flow visualization technique was upgraded and used to visualize buoyant jet flows with three-dimensional trajectories. The upgraded experimental set up is discussed including a section on the issues surrounding calibration length-scales of the flows with three- dimensional trajectories. The system is verified by presenting experimental data from two 3D LA experiments, a weakly advected simple jet experiment and a strongly advected momentum puff experiment. Chapter 3 ? Flow Visualization Techniques 85 3.2.4.1 ? 3D LA Equipment and Experimental Set Up As the trajectory of a three-dimensional trajectory flow is no longer located in a plane perpendicular to a single camera, a second camera is needed to establish where the trajectory is at any given point. The second camera records a perpendicular view relative to the first camera, and both cameras are positioned perpendicular to the direction of the ambient flow. The camera locations during the experimental investigation can be seen in Figure 3.44. The digital video camera on the side of the tank, recording the y-integrated view of the experiment, was the Jai CV M7+, recording a maximum of 1400 images per experiment. The digital video camera situated above the tank, recording the z-integrated view of the flow, was the Jai CV M7+ CL, recording a maximum of 1046 images per experiment. When the tank was filled with water both cameras were approximately 3.7 metres away from the centre of the body of water. The light source for the z-integrated view, a set of twelve 100Hz fluorescent light bulbs were mounted on an aluminium frame and placed on the floor of the main tank (see Figure 3.44). The light bulbs were spaced 55mm apart, the total length and width of the light sheet were 1.60 and 0.84 metres respectively. Extension cords were attached to either end of the bulbs. They connected the submerged light bulb with the light fitting situated outside the tank. To generate enough heat for the submerged light bulbs to reach standard operating temperatures the bulbs were inserted into clear plastic tubes. The air in the tube worked as a layer of insulation protecting the submerged light bulbs from the cold water. A second aluminium frame was built to fit over top of the lights and support the Perspex diffusion sheet. The distance between the frame supporting the light bulbs and the frame supporting the diffusion sheet was approximately 110mm. A significant gap was needed to stop the lights from overheating. The sides along the length of the light bulbs were also covered with diffusion sheets to reduce reflections of the light in the glass panels of the tank. Chapter 3 ? Flow Visualization Techniques 86 Fluorescent Lights Perspex Diffusion Sheet Digital Video Camera (y-integrated view) Towing Tank (lxbxh = 6m x 1.5m x 1m) Submerged Fluorescent Lights and Perspex Diffusion Sheet Digital Video Camera (z-integrated view) Figure 3.44 - 3D LA system ? set up of experimental equipment For the 3D LA system, the calibration cells had to be upgraded. Due to the size of the submerged light sheet the distance between the rails had to be increased and therefore the trolleys needed an increase in track width. The trolleys also had to be able to be driven over the top of the light sheet and therefore needed increased ground clearance, however the glass picture frame had to stay at the same height above the floor of the tank. With two directions of view a vertical and horizontal field calibration needed to be carried out before each experiment. Therefore one of the calibration trolleys was rebuilt and used for the field calibration in the z-integrated direction (see Figure 3.45). As there was only enough room in the tank for two trolleys, the upgraded trolleys were filled with clear water before a set of experiment to record the attenuation due to the glass, and during the experiments the trolleys were filled with the diluted red dye mixture, to be used for the field calibration. A comparison of several GA images, recorded at different times, displaying the attenuation due to the glass plates, showed that the error involved by not having a recording of the attenuation due to the glass before each single experiment was minimal (<<1%). Chapter 3 ? Flow Visualization Techniques 87 Figure 3.45 ? Upgraded calibration cells During the recording of the field calibration images in the z-integrated direction, the frame of the calibration cell created a shadow. This could be remedied during the recording of the background images for the attenuation of the glass by placing a piece of aluminium framing on the trolley to recreate the shadow. However, this was not possible during the recording of the background images when determining the attenuation of the red dye during an experiment. The result was a slightly overestimated attenuation due to the red dye in the ten percent of the field calibration closest to the vertical light sheet. During the experimental investigation, the flow configurations were chosen so as to minimize its impact. 3.2.4.2 ? Calibration Length Scales For the standard LA system the calibration length-scales were found by inserting a cross- shaped ruler in the trajectory plane of the flow. This approach is no longer sufficient for the 3D LA system as the flow is travelling either towards or away from the camera during the length of the experiment, and the calibration length-scales change with the changing distance between camera and flow trajectory. To be able to define the length-scales in both the y and z- integrated directions, the length-scales were found as a function of distance from the camera. The results can be seen in Figure 3.46. Chapter 3 ? Flow Visualization Techniques 88 2800 3000 3200 3400 3600 3800 4000 4200 44000.4 0.45 0.5 0.55 0.6 0.65 Distance from Camera (mm) y- int eg ra ted vi ew le ng th sc ale (m m/ pix el) 2800 3000 3200 3400 3600 3800 4000 42000.4 0.45 0.5 0.55 0.6 0.65 Distance from Camera (mm) z- int eg ra ted vi ew len gt h s ca le (m m/ pix el) In air In water In air In water Figure 3.46 - y and z integrated view length scales versus distance away from camera As it is not possible to change the calibration length-scales for individual images during the creation of the transformed average image, two calibration length-scales were used to analyse each experiment, creating two different transformed average integrated concentration images. The first one was the initial length scale at the source. The distance between the camera and the source was found and with the help of the results in Figure 3.46 transformed into a calibration length-scale. The second calibration length-scale was the final length scale at the end of the experiment. Because the camera situated above the tank recorded fewer images than the one on the side, the end of the experiment was defined by image 1040 for both recordings. From image 1040 the distance in pixels that the flow had travelled from the source was determined. Using these distances, the length-scales at the source, and the equations from the least-squares fit lines from the data in Figure 3.46, four equations with four unknown were determined. These equations were solved simultaneously and the results were the distances from the cameras to the end points of the flow and the corresponding calibration length- scales. The two transformed average integrated concentration images were both analysed in MatLab. The results from the MatLab analysis for both the y-direction and the z-direction Chapter 3 ? Flow Visualization Techniques 89 were then combined using an iterative process to determine the results at intermediate locations. The iterative process reduced the error due to the inaccuracies in the length-scales at any point of the flow to less than 1%. 3.2.4.3 ? Verification of 3D LA system The first experiment testing the 3D LA system was a simple jet experiment. The Reynolds number of the flow was 3479, the source diameter 2.45 millimetres and the initial velocity 1.42m/s. As the flow is axi-symmetric in the mean the results were expected to be the same from both cameras. Figure 3.47 shows the concentration spread results for both the y- integrated and z-integrated results. The two data series match each other and the theoretical prediction very well, indicating that the correct length-scales were used in both directions. Figure 3.48 shows the integrated dilution results of the same experiment. Again the two data series collapse onto the same line, the theoretical prediction is slightly lower as explained in 3.2.3.1. The collapse of the integrated dilution results indicates that the field calibration for both the horizontal and vertical direction was successful. 0 20 40 60 80 100 120 140 160 180 2000 5 10 15 20 25 30 x/d b c /d y-integrated jet results z-integrated jet results Integral Theory Figure 3.47 ? Concentration spread results for y and z-integrated simple jet experiment Chapter 3 ? Flow Visualization Techniques 90 0 20 40 60 80 100 120 140 160 180 2000 0.2 0.4 0.6 0.8 1 x/d C i0/ C i l y-integrated jet results z-integrated jet results Integral Theory Figure 3.48 ? Integrated dilution results for y and z-integrated simple jet experiment Here again the vertically discharged momentum puff is used as an example of a strongly advected flow. These flows are no longer axi-symmetric, however the double integrated dilution results are still expected to be the same for both in the y and the z-directions when the ambient momentum flux is dominating the flow. Figure 3.49 shows the double integrated dilution results for a vertically discharged momentum puff recorded in both the y and the z- direction. The Reynolds number of the flow was 4340, the velocity ratio 0.029 and the source diameter 2.45 millimetre. Up to a non-dimensional vertical distance of two, the results are very different. This can be explained by the source discharging the red-dye straight towards the z-integrated camera. The flow is still in the jet-like region and can therefore only be properly analysed with the y-integrated data. However with the influence of the ambient momentum flux growing, the flow becomes more bent over and thus flows more parallel to the camera recording the z-integrated view. The error due to the assumption of horizontal cross-sectional profiles is negligible by a non-dimensional vertical distance of 2.5, as at that point the double integrated dilutions of the two different viewpoints match. This happens in the middle of the transition zone from jet to puff. The transition zone is not complete until a non-dimensional vertical distance of approximately 3.5. The three-dimensional trajectory flows are constantly moving towards or away from the camera, and both cameras will be recording views similar to the z-integrated momentum puff view. The results from this z- Chapter 3 ? Flow Visualization Techniques 91 integrated momentum puff experiment show that the three-dimensional trajectory flows can also be analysed quantitatively with reasonable accuracy as long as the flow direction is predominantly perpendicular to the line of sight of the camera. 1 1.5 2 2.5 3 3.5 4 4.5 50.5 1 1.5 2 2.5 3 z/(Me00.5/Ua) U 0/ U a *C ii0 /C ii y-integrated momentum puff results z-integrated momentum puff results equation (3.43) Figure 3.49 ? Double integrated dilution results for y and z-integrated momentum puff experiment 3.2.5 - Summary The Light Attenuation flow visualization technique is an alternative technique based on the relationship between the increase in dye concentration and the corresponding decrease of light intensity that reaches the camera. The technique can be applied to visualize buoyant jet flows with two- and three-dimensional trajectories and provides integrated concentration data for the flow as a whole. However, it cannot be used to directly acquire cross-sectional data. The calibration experiments and the jet and puff experiments show that LA is a quantitative flow visualization technique that provides very good results with very little background noise and errors due to parallax. The main limitation is the relatively low upper limit of concentration of the red dye. In flow situations where high dilutions occur this upper limit could be a constraint on the accuracy of the data. Chapter 3 ? Flow Visualization Techniques 92 The results for the simple jet experiment show that LA can be used for weakly advected flows. For other weakly advected flows such as the plume and the buoyant jet the equations derived above have to be altered to incorporate the specific dilution rates of the flows. The same is true for the strongly advected flows such as the advected thermal. As both the advected flows as well as the angled jet show good agreement with the theory, LA can be confidently used for more complex flows with three-dimensional trajectories. The challenge with LA is related to the fact that the internal concentration distribution has to be known to be able to calculate point dilutions. For weakly advected flows this was shown to be straightforward as the distribution is well documented, however this was not the case for strongly advected flows. The double-Gaussian approximation has been shown to be an effective representation of the distribution for strongly advected flows. Chapter 3 ? Flow Visualization Techniques 93 3.3 - LIF The laser induced fluorescent flow visualization technique is an established technique to investigate buoyant jet flows with two-dimensional trajectories. In the present study the LIF system was employed for the investigation of negatively buoyant jets (see Chapter 6). The LIF system was used to provide comparisons with LA results and to also provide additional information on the flow structure. The LIF system that was used during these experiments is described below in two parts. In the first part the equipment used and the experimental set up of the system are explained and in the second part the calibration method used is explained. Because it was only used for the investigation of negatively buoyant jets, the discussion of the system relates directly to those experiments. 3.3.1 ? Laser Induced Fluorescent System A schematic overview of the LIF system can be seen in Figure 3.50. The LIF experiments were carried out under controlled lighting conditions in a darkroom. The tank for these experiments had inside dimensions of (length x width x height) 1070mm x 1070mm x 610mm. The bottom of the tank was approximately 810mm above the floor. The source with a diameter of 2.45mm was set to the desired angle using a setsquare with a precision of 0.25?. The source fluid was stored in a 300 litre tank positioned approximately 2300mm above the end of the source. The source tank was connected to the source via a magnetic flow meter. The Laser system used during the LIF experiments was a Millennia IIs, diode-pumped, cw Visible Laser, which has a maximum output of 2Watts at 532nm. The Laser Head was connected to a laser sheet generator via a fibre-optic cable. The maximum output from the cable was approximately 1 Watt. The laser sheet generator consisted of a cylindrical lens producing a non-uniform fan-like laser sheet. The laser sheet generator was set up in such a way that it was vertical, and in line with the centre of the source, therefore displaying the centreline of the flow during an experiment. Chapter 3 ? Flow Visualization Techniques 94 Before an experiment the storage tank was filled with 150 litres of water up to a precision of 0.05 litres (0.033%). Salt was added to the water to create a density difference of approximately 2.5%. The actual density difference was checked before and after a series of experiments with an Anton Paar density meter. The tracer was initially made up as two litres of Rhodamine 6G dye solution with a concentration of 50mg/l using distilled water. The weight of dye in the dyed solution was determined to a precision of 0.0001g (0.1%) and the volume of the solution to a precision of 0.7ml (0.035%). From the two litres of solution 400ml (precision 1ml or 0.25%) was added to the salt-water solution. When illuminated by the laser sheet the Rhodamine 6G fluorescent molecules emitted light with a wavelength of 590nm. A 10-bit Pulnix 1010 DV digital video camera was used for recording the flows. For most of the experiments it was set up approximately 2.5 metres away from the side of the tank and perpendicular to the laser sheet. A Cosmicar/Pentax 50mm F/1.4 TV lens was attached to the camera while in that position. An alternative set up was used for recording non-vertical cross- sections of the flow. For those experiments the camera was positioned approximately one meter above the tank and set to an angle so that the camera was perpendicular to the recorded cross-section. The lens used was a Cosmicar/Pentax 25mm F/1.4 CCTV camera lens. At all times a Heliopan ES 46 filter was placed in front of the lens. This (532nm) filter reduced the amount of laser light reflected from particulate matter reaching the camera. The camera recorded 1008x1008 pixels at a rate of 15Hz. The height of the camera was adjusted to be approximately in between the maximum and the minimum height of the flow. The camera was connected to the computer via a data-cable and a Bitflow video board. Due to the size of the 10-bit images, 888 images could be recorded per experiment. To increase the recording time and therefore the time over which an average was taken only every second or third image displayed was recorded. A software based control programme, Pulnix TM-1010 Controller v1.02, was used to set the camera?s manual parameters. The intensity signal of the 10-bit camera had a range between zero and 1023, where zero is black and 1023 saturation. The manual parameters were set so that the intensity of the camera was not saturated at any pixel, but still gave the largest practical range of intensity. For the LIF experiments the camera had an open shutter, the Gain was set to 50, the white-level (VRef) was set to 170, and the black-level (URef) to 70. The recorded images were analysed in ImageStream (Nokes 2005). Chapter 3 ? Flow Visualization Techniques 95 Source Laser-Sheet Generator Fibre-Optic Cable 10-bit digital video camera Laser-Sheet Magnetic Flow-meter Tank (1.07m x 1.07m x 0.61m) Figure 3.50 - LIF experimental set up 3.3.2 ? LIF Calibration Methods The LIF system was calibrated before each series of LIF experiments so the results from the experiments could give quantitative as well as qualitative information. The relationship between light intensity and Rhodamine dye concentration is linear (Pun 1998). However the response of the camera was non-linear and to correct for the camera response a multipoint calibration was carried out. Due to the fan-like appearance of the laser sheet, the light intensity of the sheet decreased with distance from the laser sheet generator. Therefore the system had to be calibrated for each pixel separately. Before a series of experiments, the tank was filled with approximately 660l of water up to a precision of 0.05l. Then 175ml, ?1ml, of the Rhodamine 6G solution was stirred into the tank. When the dyed solution was uniformly mixed through the water the laser was turned on and set to the maximum setting of 2 Watts. A hundred images were recorded of the tank at this setting, then the output of the laser was reduced to 1.8 Watts and another hundred images Chapter 3 ? Flow Visualization Techniques 96 were recorded. This process was repeated until the output of the laser was 0 Watts. This gave eleven series of one hundred images that were transformed into eleven averaged images. These images were plotted versus their respective laser output and provided a field calibration for each pixel. The result of one pixel can be seen in Figure 3.51, note that the laser output has been non-dimensionalised by the maximum output. A second order polynomial was fitted through the data as it was found to better match the data than a linear fit, especially through the lower quarter of the data. The calibration error was within 4%. The second order polynomial at each pixel was used to transform the 10-bit intensity data into relative concentration data. 0 0.2 0.4 0.6 0.8 1 1.2 0 50 100 150 200 250 300 350 400 450 500 10-bit Intensity Re lat ive L as er O ut pu t Figure 3.51 - Polynomial Field Calibration at pixel (178,196) The relative concentration found after applying the polynomial field calibration to the raw data was converted into the actual concentration by multiplying the relative concentration by the ratio in concentration between the source fluid and the concentration of the fluid during the recording of the calibration images. This was found in two different ways. The first was to calculate the concentration of Rhodamine 6G dye in the source fluid and divide it by the calculated value for the concentration of dye in the tank during calibration. The second method was using the TD-700 Laboratory Fluorometer. The Fluorometer was firstly calibrated so that no sample exceeded the value 999 and water with no fluorescent dye gave a value of zero. Then a sample from both the source fluid as well as the tank (during calibration) was taken and inserted into the fluorometer. The respective values were noted and Chapter 3 ? Flow Visualization Techniques 97 the first was divided by the second. The fluorometer experienced drift and hence each sample had to be referenced to a zero value. The error involved in comparing the values from the two methods was approximately 6%. As the concentration of fluorescent dye is known to decrease over time when mixed in tap-water (Pun 1998), a sample was taken from the source fluid every day. 3.3.3 ? Summary The LIF system as used during the negative buoyant jet experiments is presented above. The error due to the second order polynomial assumption is of the same order as the error in the LA system when determining relative concentrations. However this is increased when the actual concentration values are calculated. The results of the negative buoyant jet experiments can be found in chapter 6. Chapter 3 ? Flow Visualization Techniques 98 Chapter 4 ? Momentum Model 99 Chapter 4 ? Momentum Model 4.1 - Introduction To assist in the design and to monitor the performance of the experiments a new numerical model was set up. The new model also provided the opportunity to independently confirm parameters currently used to model buoyant discharges, as well as the ability to isolate issues that required more detailed investigations. The model is referred to as the Momentum Model and is based on the integral approach, similar to VisJet (Cheung et al. 2000) and CorJet (Jirka 2004). However unlike the other models, the behaviour of the discharge at any particular location is determined by the relative magnitudes of three distinct forms of momentum flux. The three distinct forms of momentum flux are the entrained ambient momentum flux per unit density, Ma, the buoyancy-generated momentum flux per unit density, Mb, and the initial momentum flux per unit density, M0. Hereafter the momentum flux per unit density is referred to as the momentum flux. Employing momentum fluxes to determine the behaviour of the flow is both simplistic and physically consistent with the real flow. It therefore potentially gives a greater insight into the behaviour of the flow. Nine differential equations are set up in MatLab, and the set of equations is closed by the spread assumption. An ODE-solver is employed to solve the ten differential equations simultaneously. A comparison between the Momentum Model and previous experimental studies is carried out to confirm the accuracy of the new model. The results of this comparison can be found in Chapter 5. 4.2 ? Model Configuration and Initial Conditions A coordinate system is defined so that the ambient fluid ( )aU flows in the same direction as the x-axis (Figure 4.1). The z-axis is defined to be in line with the buoyancy force and thus vertical. As the ambient flow is assumed to flow horizontally or perpendicular to the buoyancy force, the y-axis can be defined as perpendicular to both the ambient flow and the buoyancy force. The direction of the initial momentum flux is related to the coordinate system by angles 0? and 0? . The angle 0? is defined in the z- 0M plane, that is, between the initial Chapter 4 ? Momentum Model 100 momentum flux and its projection in the x-y plane ( )'0M . The second angle, 0? , is the angle in the x-y plane between '0M and the x-axis. Further downstream, away from the source, Ms is the total momentum flux at a distance s from the source along the trajectory of the flow. This total momentum flux is calculated by taking into account the magnitudes of the different forms of momentum flux. The direction of the total momentum flux is related to the coordinate system by angles ? and ?. Angle ? is the angle between the total momentum flux and its projection in the x-y plane. Angle ? is the angle between this projection of the total momentum flux and the ambient momentum flux. x z y M0 M0? ?0 ?0 Ua M0 Mb Ma Ms ? ? Figure 4.1 ? Schematic diagram of coordinate system of Momentum Model The entrained ambient momentum flux, Ma, and the initial momentum flux, M0 are defined as follows: a aM U Q?= where 0QQQ ?=? (4.1) pi4 2 2 0000 dUQUM == (4.2) where Q is the flow rate. The initial zone of establishment has not been modelled due to its negligible effect on the flow away from the source. An excess momentum flux, Me, is calculated as well. This excess momentum flux represents the initial plus buoyancy-generated momentum fluxes in excess of the entrained ambient Chapter 4 ? Momentum Model 101 momentum flux (Figure 4.2). Angle ? is the angle between the excess momentum flux and the ambient momentum flux. The initial excess momentum flux (Figure 4.3) includes the assumption that coflowing discharges are immediately aware of the ambient flow (a jet only forms because of the initial excess velocity), but cross-flowing discharges are not aware of the ambient flow until ambient fluid is entrained. The initial excess momentum flux can therefore be written as: ( )0 0 0 0 0cose aM U Q U Q ?= ? (4.3) and ( ) ( )10 0 0cos cos cos? ? ??= ? ?? ? (4.4) ?0 ?0 ?0 Me0 Me0x Me0y Me0z Me y x z Ms Mb Ma Figure 4.2 - Schematic diagram of excess momentum flux Ua Q0 cos (?0) U a Q0 sin (?0) Ua Q0 ?0 Me0 Ms0 M0 = U0 Q0 Figure 4.3 - Schematic diagram of the initial excess momentum flux discharge configuration The initial excess momentum flux can be split up into three different vectors in the x, y and z direction respectively using angles 0? and 0? . This is shown in Figure 4.2. ( ) ( )0 0 0 0 0 0 0cos cos cos( )e x aM U Q U Q ? ? ?= ? ?? ?? ? (4.5) ( ) ( )0 0 0 0 0 0 0cos cos sin( )e y aM U Q U Q ? ? ?= ? ?? ?? ? (4.6) Chapter 4 ? Momentum Model 102 ( ) ( )0 0 0 0 0 0cos sine z aM U Q U Q ? ?= ? ?? ?? ? (4.7) Applying the same assumption as for the initial excess momentum flux, the initial ambient momentum flux is given by ( )0 0 0cosa aM U Q ?= (4.8) Assuming there is no significant drag on the buoyant fluid, the momentum flux of the flow increases with distance downstream because of increases in the entrained ambient momentum flux and the buoyancy-generated momentum flux. 4.3 ? Ordinary Differential Equations Six parameters are needed to describe the behaviour of the flow at any particular location. The x, y and z coordinates give the location of the flow centreline, b is spread of the flow, Q the volume flux of the flow and ? the buoyancy at the particular location. In addition four different forms of momentum fluxes, that indicate the flow condition, are determined, that is, the entrained ambient momentum flux, the buoyancy-generated momentum flux, the excess momentum flux and the total momentum flux, giving ten unknowns. To solve the problem in this form, ten equations are needed. These ten equations are set up as ordinary differential equations with respect to the step-size, s, along the flow trajectory and rewritten in dimensionless form. The velocity and the concentration in a cross-section of the flow are assumed to have top-hat distributions. 4.3.1 ? Deriving Equations The equations for the location of the flow centreline can be derived by comparison of the different forms of momentum flux in a particular direction to the total momentum flux at a specific location. The relative change in the x-direction along the trajectory is equal to the momentum flux in the x-direction over the total momentum flux. There are two possible sources of momentum flux in the x-direction, the entrained ambient momentum flux and the initial excess momentum flux in the x-direction. The change in the y-direction with respect to the step-size is equal to the momentum flux in the y-direction over the total momentum flux. There is only one possible source of momentum flux in the y-direction, the initial excess momentum flux in that direction. In the z-direction there are again two possible sources of Chapter 4 ? Momentum Model 103 momentum flux, the buoyancy-generated momentum flux and the initial excess momentum flux in the z-direction. This gives the following equations for the change in location due to a step along the trajectory. 0x e x a s s M M Mx s M M +? ? = ? (4.9) 0y e y s s M My s M M ? ? = ? (4.10) 0e z bz s s M MMz s M M +? ? = ? (4.11) Letting the step-size approach zero gives the ordinary differential equations for the location of the flow. 0e x a s M Mdx ds M += (4.12) 0e y s Mdy ds M= (4.13) 0e z b s M Mdz ds M += (4.14) The buoyancy-generated momentum flux can be calculated by determining the buoyancy force. The buoyancy force is generated by the density deficit ? (also known as reduced gravity), and can be calculated as ( )ga ? ?? ?=? (4.15) The change in buoyancy-generated momentum flux is then given by: 2 bM b spi? ? ?? ?? (4.16) Dividing both sides by ?s and letting the step-size go to zero gives the ODE for the buoyancy- generated momentum flux. 2b ds dMb ?=pi (4.17) The conservation of density deficit, the mass flux relationship, shows that there cannot be a change in the total amount of buoyancy flux in the flow. Therefore ( ) 0=? ds Qd (4.18) Chapter 4 ? Momentum Model 104 where Q is the total volume flux of the flow. The product in equation (4.18) can be split up into two parts 0=?+? dsdQdsdQ (4.19) Rearranging the above equation gives the fifth ODE, the change in buoyancy with distance downstream ds dQ Qds d ??=? (4.20) The change in volume flux along the trajectory of the flow can be found by relating the total volume flux to the total momentum flux. 2 2 22 b QbUM s pipi == (4.21) or pi22 bMQ s= (4.22) Differentiating both sides of the equation with respect to the distance along the trajectory, s, and rearranging gives the sixth ODE, which describes the change in volume flux: ds db Q bM ds dM Q b ds dQ ss pipi += 2 2 (4.23) The total momentum flux is equal to the square root of the momentum fluxes in the x, y and z- direction squared. The ODE for the second of the four different kinds of momentum flux, sM , can be found by a differentiation with respect to s of that relationship ( ) ( ) 222 222 1 2 1 zyx zyx s MMM ds d MMMds dM ++ ++ = (4.24) or ds dM M M ds dM M M ds dM M M ds dM z s zy s yx s xs ++= (4.25) The change in momentum flux in the x-direction will solely be caused by the change in entrained ambient momentum flux, as the initial excess momentum flux in the x-direction is constant. As the initial excess momentum flux is the only momentum in the y-direction, there will be no change in momentum flux in the y-direction and the change in the z-direction will be caused by the change in buoyancy-generated momentum flux. Therefore the ODE for the total momentum flux becomes ( ) ( )0 0e x a e z bs a b s s M M M MdM dM dM ds M ds M ds + += + (4.26) Chapter 4 ? Momentum Model 105 Figure 4.2 can be used to find the ODE for the excess momentum flux. It shows that the total excess momentum flux is equal to the square root of the excess momentum fluxes in the x, y and z-directions. In both the x and y-directions the only excess momentum flux is the initial excess momentum flux, in the z-direction the excess momentum flux is the initial excess momentum flux plus the buoyancy-generated momentum flux . Differentiating both sides with respect to s gives ( )( ) ( )( )22 20 0 0 22 2 0 0 0 1 1 2 e e x e y e z b e x e y e z b dM d M M M M ds dsM M M M= + + ++ + + (4.27) or ( )0b e ze b e M MdM dM ds M ds += (4.28) The ODE for the entrained ambient momentum flux can be found by differentiating equation (4.1) with respect to the distance along the trajectory ds dQU ds dM a a = (4.29) 4.3.2 ? Spread Relationships The final differential equation cannot be derived. The spread equation is based on the spread assumption and it is used to solve the closure problem of turbulence (see section 2.5.2). It uses experimentally determined spread ratios to predict the flow spread at a given location. There are essentially three forms of the spread relationship. One is for Gaussian flows and has the following form (Wang 2000b): 1.55 e a e Udb k ds U U= + (4.30) where eU is the centreline excess velocity. The relationship adequately predicts the spreading rates of the weakly advected flows (jet and plumes) where it becomes: jpdb k ds = (4.31) The relevant strongly advected Gaussian flow is the weak-jet and equation (4.30) reduces to the following in this region 1.55 wj e a db Uk ds U= (4.32) Chapter 4 ? Momentum Model 106 In employing equation (4.30) the assumption is made that in the weak-jet region the initial excess momentum flux rather than the horizontal component of the initial excess momentum flux is the dominant parameter. This assumption has not been verified by experimental evidence. However, the maximum difference in spreading rates is 6% for a discharge angle of 25?, and the difference reduces with the decrease of angle ? further downstream. For larger discharge angles a weak-jet is not expected to form (see Chapter 7). The second spread relationship has been developed (from the standard spread equation used by models, db/dz = m) for the strongly advected line momentum puff region and is based on the double-Gaussian assumption. It has the form: 2 2 2 2 yzm sg m sg m dsdb y dy z dzk k ds ds ds dsy z y z? ? ? ? ? ?= = +? ? + +? ? (4.33) where syz is the distance travelled along the projection of the direction of the initial discharge in the y-z plane. The third spread relationship has been developed for the strongly advected thermal region and has the form: t sg t db dzk ds ds?= (4.34) Note that the experimental results in Chapter 5 indicate that the spreading rate in the advected line momentum puff region of the flow is not equal to the spreading rate in the advected thermal region. Hence the need for separate values for the spreading rate in the momentum puff region (ksg-m) and thermal region (ksg-t). However these spread relationships were determined using the Gaussian (see 3.2.3.1) or double-Gaussian assumption (see 3.2.3.2). As the velocity and concentration distributions for the Momentum Model were assumed to be top-hat and hence based on the top-hat spread, the values for k and ksg (as defined in Chapter 3), have to be converted to be used with the top-hat assumption ( thk and thm ). Details of these conversions are given in section 4.4. Furthermore, the centreline excess velocity in equation (4.30) has to be converted into the average excess velocity. The conversion equations for the weak-jet region are equations (4.67) and (4.68). Inserting equation (4.68) into (4.67) and solving for the centreline excess velocity gives: 2 2 wj wj e e eth q q MU U I I Q pi? pi?= = (4.35) where Ueth is the average excess velocity, and ?wj represents the ratio of the jet edge radius to the nominal radius and has a value of 2 (see Chapter 7 for more details). The spread Chapter 4 ? Momentum Model 107 relationships, using the parameters as defined above, including the assumptions that a jet in a cross-flow is not immediately aware of the ambient flow, and the top-hat conversions, become 2 2 020 1 1.55 cos1.55 cos wj eth e th th ewj eq aa wjq Mdb M Qk k MMds I Q UU QI Q pi? pi? ?? ? = = ++ (4.36) 2 2 2 2 th m th m db y dy z dzm ds ds dsy z y z ? ? ? ? ? ?= +? ? + +? ? (4.37) th t th t db dzm ds ds ? ?= (4.38) In the Momentum Model the application of the relevant spread relationships per region is determined by the relative magnitudes of the momentum fluxes, that control the flow behaviour, and in one instance the initial discharge angle. Table 4.1 shows the momentum flux ratios and corresponding relationships for the regions that are of interest in terms of the spread equation. Therefore only regions before a transition zone are listed. The momentum flux ratios defined in Table 4.1 are functions of the distance from the source. The momentum flux ratios increase with increasing distance, and at some point the ratio exceeds the transition constant and the flow enters a different region. These transition constants are represented by the symbols c2 to c6. Note that the physical transitions are not abrupt, but a transition between region exists. The current approach is an approximation for modelling purposes only. Setting the momentum flux ratios equal to the transition ratios and solving for the distance from the source gives the length-scales in the centre column of Table 4.2. These derived length-scales are compared with the length-scales determined by Pun (1998) using dimensional analysis (column 3, Table 4.2). The exception is the transition from the advected line momentum puff to the advected thermal, where Pun?s length-scale was not obtained directly from length-scale analysis, but was obtained from the constraints of a constant momentum flux, dilution, velocity and spread at the transition point. By combining the two results, the constants c2 to c6 can be determined. The results can be seen in Table 4.3. The constants smC , spC , ptC wtC and mtC come from experimental work and are set to 1, 2.3, 0.5 1.1 and 1.35 respectively (Pun 1998). The transition point between the strong jet and weak jet region has not been calculated because the spread function, as defined by equation (4.35), Chapter 4 ? Momentum Model 108 automatically changes the spread depending on the relative sizes of both the excess centreline, and the ambient velocity. Each scenario starts in the strong-jet region. With the first region known, the relevant momentum flux ratios in that region are calculated at each step. If any of the ratios exceeds the value of the transition constants, the flow region changes, and the spread function changes accordingly. If no transition ratios are exceeded the current spread function is retained. The initial discharge angle (?0) adds an additional complexity to the strong-jet region. The strongly advected behaviour of a non-buoyant jet discharge in a moving ambient is dependent on the initial discharge angle. For angles close to 90? the strong-jet turns into an advected line momentum puff, for angles close to 0?, the strong-jet turns into a weak-jet. As the discharge angle at which the strongly advected form changes is not known (Pun 1998), an experimental investigation was carried out to determine the angle. Details of the investigation are presented in Chapter 7. Based on this study, the Momentum Model uses a transition discharge angle of 25?. By comparing the results from the momentum flux analysis with the results from the dimensional analysis for the strong-jet to advected line momentum puff transition it can be noted that c2 appears to depend on the initial discharge angle. Pun assumed that the dominant parameter in the puff region was the component of the initial excess momentum flux perpendicular to the ambient flow. The results presented in Chapter 7 indicate that this assumption is correct and that the initial discharge angle cannot be ignored. In Table 4.3 the relevant momentum flux ratio has been modified accordingly. Interestingly, Table 4.3 shows that a transition involving the entrained ambient momentum flux is triggered when the momentum flux ratio is approximately one-third, whereas a transition involving the buoyancy-generated momentum flux is triggered when the momentum flux ratio is closer to one. Using the momentum flux ratios, a spread function flow chart can be set up to show all possible paths through the different flow regions (Figure 4.4). Note that there is no distinct separation between the strong-jet and weak-jet regions, and there is no direct path from the weak-jet region to the advected thermal region. The momentum flux ratio determining the transition from strong-jet to advected plume region is the same as the momentum flux ratio determining the transition from the weak-jet to advected thermal region. The value of the Chapter 4 ? Momentum Model 109 momentum flux ratio at the transition of the former is lower than that for the latter, and hence a flow in the weak-jet region automatically enters the advected plume region after such a transition. However in moving from the weak-jet region, the flow has already entrained a significant amount of ambient fluid. Therefore the flow immediately exceeds the transition ratio in the advected plume region and enters the advected thermal region. Table 4.1 - Dominant momentum flux ratios per flow region Current Flow Region Dominant Momentum Flux Ratio Flow Region After Transition Advected Line Momentum Puff Strong Jet 0.5 20.5 0 0 a a th e e M U d sk c M M dpi? = 23 0 3 0 04 b th e e M d xk c M M dpi ? ? ?= = ? ?? ? Advected Plume Advected Plume 1 3 1 33 2 2 0 43 0 16 3 a a th b M U zk Fr c M U d ? ?? = ? ?? ? Advected Thermal Weak Jet 0 5 0 0 b e a e M d s c M U U d ?= = Advected Thermal Advected Line Momentum Puff 0 6 0 0 b e a e M d s c M U U d ?= = Advected Thermal Table 4.2 - Transition length-scales for relevant flow transitions Flow region transition Momentum flux analysis Dimensional analysis Strong Jet ? Advected Line Momentum Puff 0.5 0 2 0.5 1 e th a Ms c d k U dpi= 0.5 0 0sin e sm a Ms c d U d ?= Strong Jet ? Advected Plume ( ) 0.75 0 3 0.5 0 0 2 e th Ms c d k Q dpi= ? ( ) 0.75 0 0.5 0 0 e sp Ms c d Q d= ? Advected Plume ? Advected Thermal 3 0 0 4 3 3 4 th a Qz c d k U dpi ?= 0 0 3pt a Qz c d U d ?= Weak Jet ? Advected Thermal 0 5 0 0 e aM Us c d Q d= ? 0 0 0 e a wt M Us c d Q d= ? Advected Line Momentum Puff ? Advected Thermal 0 6 0 0 e aM Us c d Q d= ? 1 32 0 0 0 0 sine mt a Ms c d Q U ?? ?? ?? ?? ?= ?? ?? ? Chapter 4 ? Momentum Model 110 Table 4.3 ? Momentum flux ratio at flow transition for relevant flow transition Flow region transition Momentum flux ratio Strong Jet ? Advected Line Momentum Puff ( ) 0.5 0 0 0.255sina th sm e M k c M pi? = = Strong Jet ? Advected Plume 0.5 2 0 0.6734b th sp e M k C M pi? ?= =? ? ? ? Advected Plume ? Advected Thermal 351.034 31 2 =? ? ?? ? ?= ptth b a Ck M M pi Weak Jet ? Advected Thermal 0 1.1b wt e M C M = = Advected Line Momentum Puff ? Advected Thermal 1 3 4 0 2 0 0 0 0 1.35 sin b e a mt e M M U C M Q ? ? ? ? ? = = ? ?? ? ?? ?? ? or 1 32 0 0 0 7 74 0 0 sin 1.35b mt mt e e a QM C c C c M M U ?? ?? ? ?? ?? ?= = = ? ?? ? k Start of Scenario Advected Thermal Region Advected Plume Region Strong Jet Region Weak Jet Region Advected Line Momentum Puff Region 0 0.351a e M M > End of Scenario 7 0 1.35b e M c M > 0 0.673b e M M > 0 0 0 25 & 0.255sino a e M M? ?> > Figure 4.4 ? Momentum Model spread function flow chart Chapter 4 ? Momentum Model 111 4.3.3 ? Dimensionless ODEs Before the equations are entered into MatLab they are first non-dimensionalised. This eases the interpretation of the results when comparing them with either results from other models or experimental data. To make the equations dimensionless the following dimensional scales are used. The length scale is the diameter of the source, d, the velocity scale is the initial velocity of the flow, 0U , the discharge scale is the initial volume flux of the flow, 0Q , the momentum flux scale is the initial momentum flux, 0M , and the buoyancy scale is the initial density deficit, 0? . The subscript ? indicates the non-dimensional variables The geometric relationships become: 0e x a s M Mdx ds M ? ?? ? ? += (4.39) 0e y s Mdy ds M ?? ? ? = (4.40) 0e z b s M Mdz ds M ? ?? ? ? += (4.41) The buoyancy-generated momentum flux: 2 2 0 4bdM b ds Fr ? ? ? ? = ? (4.42) where Fr0 is the initial Froude number of the flow defined by 0 0 0 UFr d= ? (4.43) The density-deficit flux: ? ? ? ? ? ? ??=? ds dQ Qds d (4.44) Chapter 4 ? Momentum Model 112 The spread: 02 2 2 2 2 1.55 cos th e th e a wj th m th m th t th t db M Qk Mds U Q db y dy z dzm ds ds dsy z y z db dzm ds ds ?? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? = + ? ? ? ?= +? ? + +? ? = (4.45) The entrained ambient momentum flux: ? ? ? ? ? = ds dQU ds dM a a (4.46) The excess momentum flux: ( )0b e ze b e M MdM dM ds M ds ? ?? ? ? ? ? += (4.47) The total momentum flux: ( ) ( )0 0e x a e z bs a b s s M M M MdM dM dM ds M ds M ds ? ? ? ?? ? ? ? ? ? ? ? + += + (4.48) And the volume flux of the flow: ? ? ? ?? ? ? ? ? ? ? += ds db Q bM ds dM Q b ds dQ ss 42 2 (4.49) 4.3.4 ? Solving the Equations in MatLab In MatLab the ten equations are solved simultaneously at a particular location along the trajectory. Simultaneously means that MatLab integrates the system of differential equations and finds answers for all ten variables at a specific location and then checks the error involved in the integration. MatLab only advances a step size if the error is smaller than a maximum tolerance that is specified beforehand. If the error is outside the tolerance level, MatLab shortens the step size and goes through the set of equations again. The order in which the derivatives are calculated is of importance. The differential equations for x, y, z, bM and b are calculated first, as these do not depend on the results of any of the other equations. With the buoyancy-generated momentum flux equation now known, the excess momentum flux equation can be calculated and is therefore next. For MatLab to Chapter 4 ? Momentum Model 113 calculate values for the other equations, the total momentum flux equation (4.48) has to be rewritten so that the result is no longer dependent on the change in ambient momentum flux. The reason for this is that the answer from the entrained ambient momentum flux equation (4.46) depends on the change in volume flux and the answer from the volume flux equation (4.49) depends on the change in total momentum flux, therefore creating a loop that cannot be solved. By inserting equation (4.49) into (4.46) and inserting the result into equation (4.48), the total momentum flux equation can be modified so that it only depends on the results of the spread equation and the buoyancy-generated momentum flux equation. As these results are already known the total momentum flux equation can be determined. ( )2 00 42 e z bs e x a s s b a s s M MdM M M dM M b dMb dbU ds M Q ds Q ds M ds ? ?? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? +? ? ? ?+= + + ? ? ? ?? ?? ? (4.50) Solving equation (4.50) for the change in total momentum flux gives the final non- dimensional equation ( ) ( ) ( ) 0 0 2 0 4 21 e x a a e z b b s s e x a a s M M U b M M dMdb dM Q ds M ds M M U bds Q M ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ?? ? ? + ++ = + ? (4.51) The volume flux equation will be the eighth equation solved, and this is followed by the last two, ? and aM , both depending on the change in volume flux. A MatLab m-file, MomentumModel7, is used to enter the initial conditions of a flow scenario, the source diameter, the initial velocity of the flow, the ambient velocity, the density difference used to calculate the initial buoyancy flux, and both the initial angles of the discharge, 0? and 0? . These initial conditions are used to calculate the scenario dependent constants, 0? , 0eM , 0e xM , 0e yM , 0e zM , aU , 0Fr and c7 as well as the first values for the ten dependent variables. The scenario dependent constants and the initial values for the ten dependent variables are then non-dimensionalised. A row-vector is created with the initial values of all the ten non-dimensional ODE?s. The MomentumModel7 m-file calls a second m- file, dy7, in which the ten equations are defined in the above-mentioned order. The universal constants for the velocity spread of a weakly advected flow, kth, a strongly advected puff flow , mth-m, and a strongly advected thermal flow, mth-t, are calculated in dy7 as these do not change from one scenario to another. The MatLab function ODE45 solves the ten differential equations. This function is specifically programmed to evaluate the right-hand side of differential equations by using an explicit Runge-Kutta (4,5) formula. It is a one-step solver Chapter 4 ? Momentum Model 114 needing only the solution at the immediately preceding point. The output of the model are the values for the ten parameters at each step size. It stores these values in a matrix for later access and uses them as the initial conditions for the next step. A printed version of both algorithms can be found in Appendix C. 4.4 ? Top-Hat Conversions As indicated in section 4.3.2 it is necessary to relate the Gaussian or double-Gaussian parameters to the top-hat parameters employed by the model. From integral methods the momentum, volume and tracer fluxes using the Gaussian assumption can be determined. These have to be equal to their top-hat counterparts. This is shown below. Momentum flux: 2 2 2 2m e th ethI U b b Upi= (4.52) Volume flux: 2 2q e th ethI U b b Upi= (4.53) Tracer flux: 2 2qc e l eth th thI U Cb U C bpi= (4.54) mI , qI and qcI are shape factors with values of 1.7, pi and 2.03 respectively (see section 3.2.3.1). The momentum flux equation can be divided by the volume flux equation to show the conversion equation for the velocity: 1.7m eth e e q IU U U I pi= = (4.55) Equation (4.52) is inserted into the volume flux equation to find the conversion equation for the spread: 1.7 q th m Ib b b I pi pi= = ` (4.56) and 0.1361.7thk k kpi= = (4.57) Both equation (4.55) and equation (4.56) are inserted into the tracer flux equation to find the conversion equation for the concentration: 0.646qcth l q IC C C I= = (4.58) For the determination of the conversion equations from the double-Gaussian assumption to the top-hat assumption for the momentum puff region, equations for the volume and tracer flux are changed due to the influence of the ambient velocity. Chapter 4 ? Momentum Model 115 Volume flux: 22 thaaqdg mUbUI pi= (4.59) Tracer flux: 22 ththalacdg mCUbCUI pi= (4.60) The expressions for cdgI can be found in Chapter 3.2.3.2. 2qdgI b is the area under the double- Gaussian. qdgI is determined below, using the double-Gaussian assumption as suggested in Chapter 3. In the calculation of qdgI , the integration limits were ?2bc, because 2bc is generally associated with the distance from the centreline to the edge of the single-Gaussian jet. However the spread relationship employs a velocity spreading rate, where the velocity spreading rate is the concentration spreading rate divided by ?. This is taken into account by multiplying a factor 21 ? in the calculation of qdgI . ( ) ( ) 02 2 2 2 1 2 2 2 2 2 81 1 4 2 4 4 2sin 2 2 c c c qdg c c b fb b h f fI b h hx hfb dx h f b b pi pi ? ? ? ? + ? ?? ?= ? ? ? = + ? + ? ? ? ?? ?? ?? (4.61) Using the values for f and h, as found in 3.2.3.2, the spread and concentration conversion equations for the momentum puff become 3.00qdgth m Ib b bpi? = = (4.62) and 3.00 0.60qdgth m sg sgIm k kpi? = = = (4.63) ( )( )220.482 0.482 0.4611 cdg peak th m l l peakf qdg I CC C C C I e? ?= = = =+ (4.64) The experimental results from the buoyant jet in an ambient flow experiments (see section 5.3.2.2) are used to determine the spread in the advected thermal region. It gave the following relationship for the spread equation: 0.73th tm ? = (4.65) Therefore the concentration conversion equation becomes: ( )( )220.324 1 peak th t f CC e? ? = + (4.66) Note that due to the influence of the ambient flow, the standard conversion equations based on the Gaussian assumption cannot be applied in the weak jet region. Altering equations (4.52) to (4.54) to take the ambient into account, gives: Chapter 4 ? Momentum Model 116 Momentum flux: 2 2q a e th eth aI U U b b U Upi= (4.67) Volume flux: ( )2 2a wj th aU b b Upi ? pi= (4.68) Tracer flux: 2 2c a a th thI U Cb U C bpi= (4.69) The concentration conversion equation becomes: 2 2 2 2 c th l l th wj I bC C C b ? pi ?= = (4.70) 4.5 ? Double-Gaussian Dilution Ratios The use of the double-Gaussian profile as the approximation of the cross-sectional concentration profile in the strongly advected regions makes it possible to determine minimum cross-sectional and minimum centreline dilutions, and convert between the two. It also gives a tool to convert dilution predictions from other models, to make a direct comparison possible. In the strongly advected regions VisJet gives average dilution results that can be converted to minimum point centreline dilution (C0/Cc) by dividing the average dilution by 2.3 (Cheung 1991). For the data to be compared with the cross-sectional minimum point dilution experimental data (Cpeak), the predicted values by VisJet are converted using the following equation: ( )( ) 2 2 0 0 0 2 2 2.3 1 f ave c fpeak ave c peak ave C C C C C e C C C C C e ? ? = = + (4.71) The dilution results given by CorJet are centreline minimum dilution results and cross- sectional average dilution results, however no definition is given as to how this relates to the double-vortex concentration distribution (Jirka 2004). For a comparison with the experimental data, the average dilution values are multiplied by 1.4. Assuming the centreline maximum concentration is the maximum expected concentration in the cross-section the conversion equation becomes: 0 0 0 1 1.4 ave peak ave peak ave C C C C C C C C= = (4.72) In the strongly advected regions of the flow previous experimental studies have both determined the minimum dilution of the cross-section and the minimum centreline dilution of the cross-section. Using equations (4.64) and (4.66) enables a comparison between the Chapter 4 ? Momentum Model 117 experimental data and the predictions from the Momentum Model. To find the centreline minimum dilution of the cross-section, ' 0y = and ' 0z = are entered into the equation for the double-Gaussian approximation to give the equation relating the average dilution with the minimum centreline dilution cC . For the momentum puff region this becomes: 2 0 00.482 2 f c ave C C e C C= (4.73) and for the advected thermal region: 2 0 00.310 2 f c ave C C e C C= (4.74) It should be noted that each model has its own definition for Cave. This definition is based on the spread definition used by the model. A more complete discussion of the comparison between models is given in Kikkert et al. (2006a). 4.6 ? Summary Ten ordinary differential equations that describe the dependent parameters of the flow along its trajectory are the basis of the Momentum Model. The behaviour of the discharge at any particular location is determined by the relative magnitudes of three distinct forms of momentum flux, the initial excess, buoyancy-generated and entrained ambient momentum flux. This lead to the derivation of nine ordinary differential equations. The tenth equation, the spread equation, came from empirical relationships and depended on the flow type. Momentum flux ratios within flow regions determined the transition points between flow regions. The equations were solved simultaneously at a particular location along the trajectory. Top-hat profiles were assumed in both the strongly and weakly advected regions of the flow and conversions equations were included to properly interpret the data and make it possible for the results to be compared with previous experimental work. To assist in the design and to monitor the performance of the experiments the accuracy of the Momentum Model predictions has to be verified. This verification is presented in the following chapter. Chapter 4 ? Momentum Model 118 Chapter 5 ? Two-Dimensional Trajectory Flows 119 Chapter 5 ? Two-Dimensional Trajectory Flows 5.1 ? Introduction This chapter gives an overview of the all the experiments with two-dimensional trajectories (including jets and plumes) that were carried out as part of the current investigation. First the flows in a still ambient are discussed followed by the two-dimensional trajectory flows in a moving ambient. As most of these flows have been investigated in the past, in the present study these flows were not studied in detail. However, the present investigation did give special attention to oblique discharge configurations for flows in a stagnant and moving ambient, as data from these flows is limited. The specifics of the experimental set up for each of the flows investigated is discussed, a more thorough discussion of the experimental technique can be found in Chapter 3. Before the Momentum Model (Chapter 4) can be used as a guide in the design and monitoring of the performance of experimental investigations, the output of the model has to be verified against experimental data. The experimental results obtained include trajectory, concentration spread and dilution data and these are used as part of the verification of the model. One flow region not covered by the current investigation is the weak-jet, because of the large horizontal distances needed to record weak-jet behaviour properly. Momentum Model predictions in the weak-jet region are compared with the most recent weak-jet data by Wang (2000b). Both the new data and the Momentum Model predictions are compared with predictions from VisJet (Cheung et al. 2000) and CorJet (Jirka 2004) and experimental data from previous researchers whenever possible. The results from the two-dimensional trajectory flow experiments were also used as a verification of the LA flow visualization technique (see Chapter 3) and to provide the foundation for the three-dimensional trajectory experiments (see Chapter 8). A more thorough investigation was carried out into the behaviour of negatively buoyant jets in a still ambient and non-buoyant jets discharged at oblique angles to a moving ambient. The theoretical and experimental findings of these investigations can be found in Chapters 6 and 7 respectively. Chapter 5 ? Two-Dimensional Trajectory Flows 120 5.2 ? Still Ambient Flows The behaviour of a buoyant jet flow in the environment can be separated into weakly advected (close to the source) and strongly advected behaviour. In the strongly advected region the flow is dominated by the influence of the ambient motion. In the weakly advected region the flow is dominated by the initial momentum flux, and the buoyancy-generated momentum flux. In the laboratory a special case can be created by reducing the ambient flow velocity to zero. The entire flow then exhibits the behaviour of the weakly advected region and it can be studied in detail. Three distinctly different flows can be observed depending on the initial conditions of the flow. The first is the simple jet. Besides having an ambient velocity of zero there is also no density difference between the fluid from the jet and the surrounding fluid. Therefore the behaviour of the simple jet depends only on the initial momentum flux. The second distinct flow in a still ambient is the plume. Compared with the simple jet there is a density difference between the buoyant jet fluid and the ambient fluid, but the initial velocity is relatively small. The behaviour of the flow is therefore dominated by the buoyancy-generated momentum flux rather than the initial momentum flux. The third distinct flow is the buoyant jet in a still ambient. It combines the characteristics of the simple jet and the plume. The buoyant jet flow goes through a region where its behaviour is jet-like and dominated by the initial momentum flux, and a region where its behaviour is plume-like and dominated by the buoyancy-generated momentum flux. The two flow regions are connected by a relatively short transition region. Depending on the angle of discharge the buoyant jet flow is either a positively buoyant jet (the vertical component of the initial velocity of the jet acts in the same direction as the buoyancy-generated momentum flux), a horizontal buoyant jet or a negatively buoyant jet. The horizontal buoyant jet has in the past received by far the most attention from researchers. Simple jet, plume and buoyant jet experiments have been carried out as part of the present study. An overview of the experimental set up and experimental results for the three flows are presented below. Chapter 5 ? Two-Dimensional Trajectory Flows 121 5.2.1 ? Simple Jet Twenty-seven simple jet experiments were carried out as part of the present investigation. First the experimental set up will be discussed, including the initial conditions of each run, and then the results will be presented. 5.2.1.1 ? Experimental Design The light attenuation flow visualization system was used for all jet runs. For more details of the LA technique see section 3.2. The simple jet runs were used as an integral part of the investigation into the LA technique. They enabled the exploration of various parameters that influence the accuracy of the LA technique, and in addition an exploration of how the LA system handled a jet run flowing towards the camera under a pre-determined angle. They were also used to verify the integrated dilution theory, and to investigate and verify the 3D LA technique. The specifics of each run are listed in Table D.1 in Appendix D The orientation of the source does not affect the behaviour of the flow, because there is no buoyancy-generated momentum flux or ambient flow. Therefore the source could be positioned anywhere in the tank as long as the trajectory of the flow was perpendicular to the camera (with the exception of the Angled Jet experiments ? runs 16.2 and 17.2). The source was placed so that it discharged either vertically downwards or horizontally. The advantage of the vertical position was the longer path of the flow that could be recorded, because of the orientation of the camera and calibration cells. But the disadvantage was the potential influence of the bottom boundary on the flow near the tank floor. The initial conditions of all jet runs can be seen in Table D.1. It shows that the Reynolds numbers ranged from 2988 to 9762, thus ensuring the discharges were turbulent at all times. The source diameter was reduced during the investigation from 10.15mm to 2.45mm. This was partly due to the low concentration of red dye needed for flows with bigger diameter sources and partly due to the fact that with a smaller diameter source the flow could be investigated further downstream in terms of non-dimensional distances. The initial volume flux of the flow, recorded with the magnetic flow meter, varied between 0.0630l/s for the Chapter 5 ? Two-Dimensional Trajectory Flows 122 largest diameter source and 0.0058l/s for the smallest diameter source. Due to the small temperature differences between the fluid of the jet and the ambient fluid, the density differences were minute and the initial Froude number was effectively infinite for all flows. 5.2.1.2 ? Experimental Results and Model Predictions The jet experiments provided concentration spread and centreline dilution results (both point and integrated). The concentration spread and point dilution data from the experiments, and the model predictions of the same quantities depend on the assumption that the cross-sectional profile of the jet has a Gaussian shape. From the Gaussian assumption, the integrated cross- sectional profile is shown to also have a Gaussian shape (equation 3.11). Figure 5.1 shows integrated cross-sectional concentration profiles from jet runs 12, 14 and 15. The location of the cross-sections increases from under 14 port diameters (run 15) to over 175 port diameters (run 14). The results, as expected, show that the Gaussian assumption is a very good representation of the cross-sectional integrated concentration values. -2 -1.5 -1 -0.5 0 0.5 1 1.5 20 0.2 0.4 0.6 0.8 1 y/bc C i/C il Run 12 - x/d=17.22 Run 12 - x/d=25.99 Run 12 - x/d=52.13 Run 12 - x/d=78.35 Run 12 - x/d=104.46 Run 12 - x/d=121.94 Run 14 - x/d=56.56 Run 14 - x/d=80.31 Run 14 - x/d=104.06 Run 14 - x/d=127.81 Run 14 - x/d=151.56 Run 14 - x/d=175.31 Run 15 - x/d=13.61 Run 15 - x/d=32.69 Run 15 - x/d=52.57 Run 15 - x/d=72.45 Run 15 - x/d=92.37 Run 15 - x/d=112.20 Run 15 - x/d=132.08 Best-fit Gaussian profile Figure 5.1 ? Integrated cross-sectional concentration profiles from Jet Runs 12, 14 & 15 at various location downstream from the source Chapter 5 ? Two-Dimensional Trajectory Flows 123 The concentration spread results (bc) of the jet experiments, non-dimensionalised by the port diameter (d), can be seen in Figure 5.2. The experimental data collapses very well for the first 100 port diameters downstream, but for increasing distances the data shows more scatter. The increasing scatter can be attributed to the increasing time-scale of the turbulence, and therefore increasing number of frames needed to create a suitable average image. Experimental data from Pun (1998) reached approximately 80 port diameters downstream. This data is consistent with the current results. The model predictions from the Momentum Model and CorJet for the concentration spread are nearly identical, VisJet predicts a higher concentration spread. The difference in predictions can be traced back to the constants employed in the relationship that closes the set of equations in the various models, entrainment constants in CorJet and VisJet, and a spread constant in the Momentum Model The most accurate experimental data, data up to 100 port diameters downstream, is most consistent with the predictions from the Momentum Model and CorJet. Further downstream the average concentration spread value of all the experimental data falls approximately in line with the predicted value by the two models. The predicted values from VisJet appear to be too high when compared to the experimentally determined values. CorJet models the zone of flow establishment and a difference in predictions can be seen in the first 5 port diameters after the fluid has left the source. However this has no discernable effect on the predicted results further downstream when compared to the Momentum Model predictions, validating the assumption in the Momentum Model that the zone of flow establishment has a negligible effect on the flow at distances greater than 10 port-diameters from the source. Chapter 5 ? Two-Dimensional Trajectory Flows 124 0 20 40 60 80 100 120 140 160 180 2000 5 10 15 20 25 x/d b c /d Present Study Pun (1998) Momentum Model VisJet CorJet Figure 5.2 - Concentration spread results versus distance downstream, comparing the experimental jet results with the model predictions The integrated centreline dilution theory for the jet is discussed in section 3.2.3.1. Equation (3.16) shows that the integrated centreline dilution (Ci0/Cil) for a jet is independent of distance downstream and has a theoretical value of 0.812. Figure 5.3 shows the theoretical line and the comparison with the experimental values found. The experimental results collapse well to form a single horizontal line with an average value of 0.829, and approximately 4% scatter. The numerical models are not designed to give integrated dilution results and therefore have been omitted from Figure 5.3. Chapter 5 ? Two-Dimensional Trajectory Flows 125 0 20 40 60 80 100 120 140 160 180 200 2200 0.2 0.4 0.6 0.8 1 x/d C i0/ C i l Run 20 Run 22 Run 23 Run 24 Run 25 Run 26 Run 27 Theory Figure 5.3 - Integrated centreline dilution results versus distance downstream, comparing experimental results with integrated jet theory The integrated centreline dilution results can be combined with the concentration spread results to find the point centreline dilution (C0/Cl) in a cross-section. Equation (3.18) describes the relationship between the integrated centreline and the point centreline dilution. The equation was used to convert the integrated dilution data from the jet experiments into point dilution data and the results can be seen in Figure 5.4. The results collapse reasonably well, however the scatter is higher than for the concentration spread results. This is expected as the calculation of the point dilution involves two experimentally determined quantities. As with the concentration spread results, Figure 5.4 shows that the predictions by CorJet and the Momentum Model are very similar, with no noticeable difference due to the omission of the zone of flow establishment in the Momentum Model. They are more consistent with the experimental data than the predictions of VisJet. The experimental values of Pun (1998) fall within the scatter of the present study and also indicate that VisJet is overestimating the dilution. Chapter 5 ? Two-Dimensional Trajectory Flows 126 0 20 40 60 80 100 120 140 160 180 2000 5 10 15 20 25 30 35 40 x/d C 0/ C l Present Study Pun (1998) Momentum Model VisJet CorJet Figure 5.4 - Point centreline dilution results versus distance downstream, comparing the experimental jet results with model predictions and previous experimental results 5.2.2 ? Plume Four plume (or vertically discharged buoyant jet) runs were carried out as part of the present investigation. The experimental set up and results are discussed below. 5.2.2.1 ? Experimental Design The plume experiments were carried out directly after the first series of jet experiments that were used to guide the set up of the LA system. The plume experiments were therefore the first test for the LA system with a different flow configuration. Salt was added to the red-dye solution to create the density difference with the ambient fluid and therefore the source was discharged vertically downwards to create a plume flow. The density difference between source fluid and ambient fluid was checked using an Anton Paar density meter. Chapter 5 ? Two-Dimensional Trajectory Flows 127 Details and initial conditions of each plume run can be seen in Appendix D, Table D.2. To be able to record the plume flow the Froude number had to be low enough so that the jet-to- plume transition length-scale was small. The jet region was then nearly non-existent and the majority of the recorded flow was in the plume-region. Due to the reasonably small port- diameter (3.2mm) and the low Froude number needed to create a plume flow, the Reynolds numbers were not high enough to expect fully turbulent flow at the end of the source. This was confirmed by the experimental results. A virtual source was found for each run by projecting the spread results back to a virtual source. The initial Froude number ranged from 5.66 to 20.63, and the Reynolds number from 557 to 2029. The density difference between the red-dye solution and the ambient fluid was kept constant for the duration of the experiments at just over three percent. The value for ?, used to convert the velocity spread data predicted by the models into concentration spread data, was set to 1.05 (Papanicolaou and List 1988; Wang 2000a) for the plume flows. 5.2.2.2 ? Experimental Results and Model Predictions As for the simple jet, the cross-sectional velocity and concentration distributions are assumed to be Gaussian and self-similar for the plume, and therefore the integrated cross-sectional profiles are also assumed to be Gaussian. This assumption was verified by collapsing the integrated experimental cross-sectional data and comparing the outcome with the theoretical Gaussian curve. The results can be seen in Figure 5.5. It shows cross-sectional profiles from runs 1, 3 & 4 at various locations downstream from the source, varying between 16 port- diameters (run 3) and 138 port-diameters (run 4). The graph confirms that the Gaussian assumption is a very reasonable approximation of the cross-sectional distribution of the plume. It can therefore be confidently used in the calculation of the point dilution. Chapter 5 ? Two-Dimensional Trajectory Flows 128 -2 -1.5 -1 -0.5 0 0.5 1 1.5 20 0.2 0.4 0.6 0.8 1 y/bc C i/C il Run 1 - x/d=16.85 Run 1 - x/d=33.05 Run 1 - x/d=49.26 Run 1 - x/d=65.47 Run 1 - x/d=81.68 Run 1 - x/d=97.89 Run 1 - x/d=105.99 Run 3 - x/d=15.81 Run 3 - x/d=27.92 Run 3 - x/d=48.10 Run 3 - x/d=68.28 Run 3 - x/d=88.46 Run 3 - x/d=108.64 Run 4 - x/d=28.69 Run 4 - x/d=47.63 Run 4 - x/d=66.58 Run 4 - x/d=85.53 Run 4 - x/d=104.48 Run 4 - x/d=132.43 Run 4 - x/d=137.64 Gaussian Profile Figure 5.5 ? Integrated cross-sectional concentration profiles from plume runs 1, 3 & 4 at various location downstream from the source The concentration spread results from the plume runs, non-dimensionalised by the port diameter and the Froude number, can be seen in Figure 5.6. The experimental results from the present study collapse well onto a single line, and are consistent with the results of Papanicolaou and List (1988) and Fisher (1995). The results of Pun (1998) are consistently lower than the results found in the present study. The experimental results are most consistent with the predictions of the Momentum Model and VisJet. The predicted spread result of CorJet is lower than the predictions of the other models, but still lies above the data of Pun. Chapter 5 ? Two-Dimensional Trajectory Flows 129 0 5 10 15 20 250 0.5 1 1.5 2 2.5 3 z/d/Fr b c /d/ Fr Present Study Momentum Model VisJet CorJet Papanicolaou (1988) Fisher (1995) Pun (1998) Figure 5.6 - Concentration spread results versus distance downstream, comparing the experimental plume results with the model predictions and previous experimental results The point centreline dilution data, divided by the Froude number is plotted versus the non- dimensional distance downstream, also divided by the Froude number in Figure 5.7. The experimental values collapse well and are consistent with all the model predictions and the data from Fisher (1995) and Wang (2000a). The experimental values found by Papanicolaou and List and Pun do not match the experimental plume data, and therefore do not agree with the model predictions either. Chapter 5 ? Two-Dimensional Trajectory Flows 130 0 2 4 6 8 10 12 14 16 18 200 5 10 15 z/d/Fr C 0/ C l /Fr Present study Momentum Model VisJet CorJet Papanicolaou (1988) Fisher (1995) Pun (1998) Wang (2000a) Figure 5.7 - Point centreline dilution results versus distance downstream, comparing the experimental plume results with model predictions and previous experimental results 5.2.3 ? Horizontal Buoyant Jet Three horizontal buoyant jet runs were carried out. The experimental set up is described below, followed by the results of the experiments. The experimental results are compared with the model predictions and data from previous experimental investigations. 5.2.3.1 ? Experimental Design To create the density difference between the source solution and the ambient fluid, salt was added to the red-dye solution. The heavier source solution would therefore fall towards the bottom of the tank and hence the source was positioned horizontally in the top quarter of the image that was recorded by the camera. Table D.3 in appendix D shows the initial conditions of all horizontal buoyant jet flows. The Froude number ranged from 14 to 50 and the Reynolds number ranged from 1300 to 4900. Chapter 5 ? Two-Dimensional Trajectory Flows 131 The source was not changed during the experiments and had a diameter of 3.2 millimetres. The density difference was just over three percent. The higher Froude number flows (runs 1 and 3) disappeared from the camera view before having reached the bottom of the recorded image. After the recording of the first part of Runs 1 and 3 the trolley (with the source attached to it) was moved a sufficient amount so that during a second recording of the same flow the lower part of the flow was recorded. The new source location was measured and used during the analysis of the second part of runs 1 and 3. The Reynolds number of run 2 was not sufficiently large for the flow to be fully turbulent by the end of the source. The concentration-spread results from run 2 were used to find a virtual source. The centreline dilution results were used to confirm the location of the virtual source. The point at which the buoyant jet flow changed its behaviour from that resembling a jet flow into that resembling a plume is called the jet-plume transition point. Using the length-scale method, Pun (1998) defined the distance from the source to the transition point as ( ) dFrQ Ml e jp 0 41 21 00 43 0 43.23.2 ?? ?? ? ?= ?= pi (5.1) The values for jpl can be found in the 11th column of Table D.3. The relatively low values for jpl indicate that for each experiment the flow passed from the jet-like region into the plume- like region. 5.2.3.2 ? Experimental Results and Model Predictions Unlike the trajectory of a simple jet, or a pure plume, the trajectory of the horizontal buoyant jet is no longer a straight line. The initial momentum flux drives the flow in the horizontal direction and the buoyancy-generated momentum flux drives the flow in the vertical direction. The experimental trajectory results, non-dimensionalised by the port diameter and the Froude number, can be seen in Figure 5.8. The Figure shows that the experimental data is consistent with results found by previous researchers (Anwar 1969; Davidson 1989; Hansen and Schroder 1968). The prediction of the Momentum Model also matches the experimental data well. The predictions of CorJet and VisJet for the trajectory of the horizontal buoyant jet indicate a higher degree of curvature due to the buoyancy force than the Momentum Model. The prediction by VisJet is still a reasonable prediction of the experimental data, CorJet overestimates the distance travelled in the z-direction. Chapter 5 ? Two-Dimensional Trajectory Flows 132 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50 1 2 3 4 5 6 7 8 9 10 x/d/Fr z/ d/F r Present Study Momentum Model VisJet CorJet Hanson & Schroder (1968) Anwar (1969) Davidson (1989) Figure 5.8 ? Trajectory results horizontal buoyant jet in still ambient, comparing the experimental results with model predictions and previous experimental results To predict the trajectory data shown in Figure 5.8, all models use the Gaussian assumption that was used to predict the simple jet and pure plume results. However due to the buoyancy- generated momentum flux and the initial momentum flux no longer acting in the same direction, the inner edge of the jet is inherently unstable, and the cross-sectional profiles are no longer axi-symmetric. Figure 5.9 shows integrated cross-sectional concentration profiles from a horizontal buoyant jet run (this run was part of the investigation into the behaviour of the negatively discharged buoyant jets, for more details see Chapter 6). The cross-sections were located between 0.22 and 1.88 transition length-scales downstream from the source. It confirms that the cross-sectional profiles are clearly no longer axi-symmetric when the distance downstream is greater than approximately 1 transition length-scale. However, the fact that the models all assume axi-symmetrical cross-sectional profiles and still predict the trajectory of the horizontal buoyant jet with reasonable accuracy, suggests that the assumption is also still a reasonable one. It is worth noting that the left-hand side of the cross-sectional profiles, which corresponds with the outside spread of the flow, still matches the Gaussian profile and can be used to calculate the spread and dilution data from the experimental runs. Chapter 5 ? Two-Dimensional Trajectory Flows 133 -2 -1.5 -1 -0.5 0 0.5 1 1.5 20 0.2 0.4 0.6 0.8 1 y/bc C i/C il s/ljp = 0.22 s/ljp = 0.48 s/ljp = 1.09 s/ljp = 1.35 s/ljp = 1.88 Gaussian Profile Figure 5.9 - Cross-sectional profiles from negatively discharged buoyant jet run 38, initial conditions: ?0 = 0o, Fr0 = 56.51 and Re0 = 4704 The concentration spread results were non-dimensionalised by the jet-plume transition point length scale, jpl , and therefore the jet-like region is the region with 13500; CrossSectionLength=CrossSectionLength-5; CrossSectionPoint1(1,2)=YPositionOfTrajectory(i-3,:)- CrossSectionLength*sin(CrossSectionAngleNew(i-3,:)); CrossSectionPoint1(1,1)=XPositionOfTrajectory(i-3,:)- CrossSectionLength*cos(CrossSectionAngleNew(i-3,:)); end q=q+1; end CrossSectionLength=400; for q=1:100; if CrossSectionPoint1(1,1)>600; CrossSectionLength=CrossSectionLength-5; CrossSectionPoint1(1,1)=XPositionOfTrajectory(i-3,:)- CrossSectionLength*cos(CrossSectionAngleNew(i-3,:)); CrossSectionPoint1(1,2)=YPositionOfTrajectory(i-3,:)- CrossSectionLength*sin(CrossSectionAngleNew(i-3,:)); end q=q+1; end CrossSectionLength=400; for q=1:100; if CrossSectionPoint1(1,2)<626; CrossSectionLength=CrossSectionLength-5; CrossSectionPoint1(1,2)=YPositionOfTrajectory(i-3,:)- CrossSectionLength*sin(CrossSectionAngleNew(i-3,:)); CrossSectionPoint1(1,1)=XPositionOfTrajectory(i-3,:)- CrossSectionLength*cos(CrossSectionAngleNew(i-3,:)); Appendix B ? Analysis of Average Integrated Concentration Image 297 end q=q+1; end CrossSectionLength=400; for q=1:100; if CrossSectionPoint2(1,1)<0; CrossSectionLength=CrossSectionLength-5; CrossSectionPoint2(1,1)=XPositionOfTrajectory(i- 3,:)+CrossSectionLength*cos(CrossSectionAngleNew(i-3,:)); CrossSectionPoint2(1,2)=YPositionOfTrajectory(i- 3,:)+CrossSectionLength*sin(CrossSectionAngleNew(i-3,:)); end q=q+1; end CrossSectionLength=400; for q=1:100; if CrossSectionPoint2(1,2)>3500; CrossSectionLength=CrossSectionLength-5; CrossSectionPoint2(1,2)=YPositionOfTrajectory(i- 3,:)+CrossSectionLength*sin(CrossSectionAngleNew(i-3,:)); CrossSectionPoint2(1,1)=XPositionOfTrajectory(i- 3,:)+CrossSectionLength*cos(CrossSectionAngleNew(i-3,:)); end q=q+1; end CrossSectionLength=400; for q=1:100; if CrossSectionPoint2(1,1)>600; CrossSectionLength=CrossSectionLength-5; CrossSectionPoint2(1,1)=XPositionOfTrajectory(i- 3,:)+CrossSectionLength*cos(CrossSectionAngleNew(i-3,:)); CrossSectionPoint2(1,2)=YPositionOfTrajectory(i- 3,:)+CrossSectionLength*sin(CrossSectionAngleNew(i-3,:)); end q=q+1; end CrossSectionLength=400; for q=1:100; Appendix B ? Analysis of Average Integrated Concentration Image 298 if CrossSectionPoint2(1,2)<626; CrossSectionLength=CrossSectionLength-5; CrossSectionPoint2(1,2)=YPositionOfTrajectory(i- 3,:)+CrossSectionLength*sin(CrossSectionAngleNew(i-3,:)); CrossSectionPoint2(1,1)=XPositionOfTrajectory(i- 3,:)+CrossSectionLength*cos(CrossSectionAngleNew(i-3,:)); end q=q+1; end CrossSectionLength=400; % Using the coordinates of the begin and end point of the cross-section and the amount of point within the % cross-section, the cross-section position vector is determined for k=0:CrossSectionPoints; CrossSectionCoordinatesVector(k+1,1)=((CrossSectionPoint2(1,1)- CrossSectionPoint1(1,1))/CrossSectionPoints*k+CrossSectionPoint1(1,1)); CrossSectionCoordinatesVector(k+1,2)=((CrossSectionPoint2(1,2)- CrossSectionPoint1(1,2))/CrossSectionPoints*k+CrossSectionPoint1(1,2)); end % Use cross-section position vector to calculate integrated concentration values along cross-section for k=0:CrossSectionPoints; CrossSectionIntensityVector(k+1,i- 3)=(interp2(X,Y,Z,CrossSectionCoordinatesVector(k+1,1),CrossSectionCoordinatesVector(k+1,2))); CrossSectionPositionVector(k+1,i-3)=(k)*sqrt((CrossSectionPoint2(1,2)- CrossSectionPoint1(1,2))^2+(CrossSectionPoint2(1,1)-CrossSectionPoint1(1,1))^2)/CrossSectionPoints; end % Determine the maximum integrated concentration in the cross-section and its position Cl(i-3,:)=max(CrossSectionIntensityVector(:,i-3),[],1); % Find maximum values of cross section MaximumIsOneInCrossSection=(CrossSectionIntensityVector(:,i-3)==Cl(i-3,:)); % Make all non-maxima values zero and maxima-values one in cross-section MaximaAdded=MaximumIsOneInCrossSection'*CrossSectionPositionVector(:,i-3); % Multiply maxima-cross-section times position vector clear Ones Ones(1:CrossSectionPoints+1,1)=1; % Create ones vector MaximaMultiplier=MaximumIsOneInCrossSection'*Ones; Appendix B ? Analysis of Average Integrated Concentration Image 299 % Multiply maxima-cross-section times ones vector CrossSectionPositionOfTrajectory=MaximaAdded./MaximaMultiplier; % Find maximum along cross section % Determine integrated concentration (intensity) of cross section at concentration spread 'bc' SpreadIntensity=exp(-1)*Cl(i-3,:); % Determine distance from point of maximum integrated concentration to ?b? on both sides of the maximum ?. CrossSectionIntensityVectorDataLimited1=CrossSectionIntensityVector(:,i-3); CrossSectionIntensityVectorDataLimited2=CrossSectionIntensityVector(:,i-3); CrossSectionPositionVectorDataLimited1=CrossSectionPositionVector(:,i-3); CrossSectionPositionVectorDataLimited2=CrossSectionPositionVector(:,i-3); j=1; k=1; while CrossSectionPositionVectorDataLimited2(k,:)CrossSectionPositionOfTrajectory; CrossSectionIntensityVectorDataLimited1(j-k+1,:)=[]; CrossSectionPositionVectorDataLimited1(j-k+1,:)=[]; k=k+1; end end [m,n]=size(CrossSectionIntensityVectorDataLimited2); for k=1:m CrossSectionIntensityVectorDataLimited3(k,1)=CrossSectionIntensityVectorDataLimited2(m-k+1,1); CrossSectionPositionVectorDataLimited3(k,1)=CrossSectionPositionVectorDataLimited2(m-k+1,1); end SpreadPositionOfCrossSection(1,1)=interp1q(CrossSectionIntensityVectorDataLimited1,CrossSectionPositionV ectorDataLimited1,SpreadIntensity); Appendix B ? Analysis of Average Integrated Concentration Image 300 SpreadPositionOfCrossSection(1,2)=interp1q(CrossSectionIntensityVectorDataLimited3,CrossSectionPositionV ectorDataLimited3,SpreadIntensity); Spread(i-3,1)=CrossSectionPositionOfTrajectory-SpreadPositionOfCrossSection(1,1); Spread(i-3,2)=CrossSectionPositionOfTrajectory-SpreadPositionOfCrossSection(1,2); % Remove all data in cross-section integrated concentration (intensity) and corresponding position vector that % has an integrated concentration less than 37% of the maximum CrossSectionIntensityVectorDataLimited = CrossSectionIntensityVector(:,i-3); CrossSectionPositionVectorDataLimited = CrossSectionPositionVector(:,i-3); j=0; k=0; while j25/180*pi; if (Y(8)*Ua)/(Me0*sin(Phi0))<(0.255) & Y(4)/Me0<=0.673345; dy7(6)=k*abs(Y(5)/Y(8)/(Y(5)/Y(8)+1.55*Ua/2*abs(cos(Phi0)))); elseif Y(4)/Me0>0.673345; FlowRegion = 2; dy7(6)=k*abs(Y(5)/Y(8)/(Y(5)/Y(8)+1.55*Ua/2*abs(cos(Phi0)))) else; dy7(6)= mp*abs((Y(2)/sqrt(Y(2)^2+Y(3)^2)*dy7(2)+Y(3)/sqrt(Y(2)^2+Y(3)^2)*dy7(3))); FlowRegion = 3; end; elseif Y(4)/Me0>0.673345; FlowRegion = 2; dy7(6)=k*abs(Y(5)/Y(8)/(Y(5)/Y(8)+1.55*Ua/2*abs(cos(Phi0))));; else dy7(6)=k*abs(Y(5)/Y(8)/(Y(5)/Y(8)+1.55*Ua/2*abs(cos(Phi0)))); end; end; if FlowRegion == 2; if Y(10)/Y(4)<=0.3509; dy7(6)=k*abs(Y(5)/Y(8)/(Y(5)/Y(8)+1.55*Ua/2*abs(cos(Phi0)))); Appendix C ? Computer Code Of Momentum Model 306 else dy7(6)=mt*abs(dy7(3)); end; end; if FlowRegion == 3; if Y(4)/Me0>c7*1.35 dy7(6)= mt*abs(dy7(3)); else dy7(6)= mp*abs((Y(2)/sqrt(Y(2)^2+Y(3)^2)*dy7(2)+Y(3)/sqrt(Y(2)^2+Y(3)^2)*dy7(3))); end; end; % dy(7)=(4*(Me0x+Ma)*Ua*b/Q*dy(6)+(Me0z+Mb)/Ms*dy(4))/(1-2*(Me0x+Ma)*Ua*b^2/Ms/Q); dy7(7)=(4*(Me0x+Y(10))*Ua*Y(6)/Y(8)*dy7(6)+(Me0z+Y(4))/Y(7)*dy7(4))/(1- 2*(Me0x+Y(10))*Ua*Y(6)^2/Y(7)/Y(8)); % Y(7)=dMsds % dy(8)=2*b^2/Q*dy(7)+4*Ms*b/Q*dy(6) dy7(8)=2*Y(6)^2/Y(8)*dy7(7)+4*Y(7)*Y(6)/Y(8)*dy7(6); % Y(8)=dQds % dy(9)=-delta/Q*dy(8); dy7(9)=-Y(9)/Y(8)*dy7(8); %Y(9)=ddeltads % dy(10)=Ua*dy(8) dy7(10)=(Ua)*dy7(8); %Y(10)=dMads Appendix D ? Initial Conditions for Experiments with 2D and 3D trajectories 307 Appendix D - Initial Conditions for Experiments with 2D and 3D Trajectories Table D.1 - Initial Conditions for simple jet runs Jet Run Date Q0 d U0 Fr0 Re0 ?a ? Camera Angle Comment No. (l/s) (mm) (m/s) (kg/m3) (kg/m3) (?f030uniF0290) 1 21/05/2003 0.0630 10.15 0.78 273 7903 998.85 998.77 Canon 90? No analysis 2 22/05/2003 0.0425 10.15 0.53 5331 1000.10 1000.10 Canon 90? No analysis 3 9/06/2003 0.0397 10.15 0.49 100 4980 999.22 998.98 Canon 90? Investigating LA 4 11/06/2003 0.0460 6.00 1.63 1831 9762 999.22 999.23 Canon 90? No analysis 5 24/06/2003 0.0460 6.00 1.63 407 9762 999.42 999.15 Canon 90? Investigating LA 6 25/06/2003 0.0460 6.00 1.63 502 9762 999.50 999.32 Canon 90? Investigating LA 7 25/06/2003 0.0224 3.20 2.79 1997 8913 999.39 999.32 Canon 90? Investigating LA 8 26/06/2003 0.0220 3.20 2.74 893 8754 999.35 999.05 Canon 90? Investigating LA 9 29/06/2003 0.0220 3.20 2.74 1533 8754 999.54 999.43 Canon 90? No analysis 10 29/06/2003 0.0220 3.20 2.74 1533 8754 999.54 999.43 Canon 90? Investigating LA 11 30/06/2003 0.0222 3.20 2.76 1296 8833 999.47 999.32 Canon 90? Investigating LA 12 1/07/2003 0.0220 3.20 2.74 ?! 8754 999.42 999.42 Canon 90? Investigating LA 13 14/07/2003 0.0206 3.20 2.56 1292 8196 999.59 999.71 Canon 90? No analysis 14 14/07/2003 0.0207 3.20 2.57 2071 8236 999.61 999.66 Canon 90? Investigating LA 15 16/07/2003 0.0217 3.20 2.70 1579 8634 999.49 999.40 Canon 90? Investigating LA 16.1 21/11/2003 0.0100 3.00 1.41 1471 4244 998.91 998.88 Canon 90? No analysis 16.2 21/11/2003 0.0100 3.00 1.41 1471 4244 998.91 998.88 Canon 80? No analysis 17.1 1/12/2003 0.0100 3.00 1.41 1037 4244 998.91 998.85 Canon 90? Investigating Jet on angle 17.2 1/12/2003 0.0100 3.00 1.41 1471 4244 998.91 998.88 Canon 65? Investigating Jet on angle 18 5/12/2003 0.0100 3.20 1.24 1020 3979 998.91 998.87 Canon 0? Investigating 3D LA 19 8/12/2003 0.0100 3.00 1.41 1165 4244 998.77 998.72 Canon 90? Investigating Integrated Theory 20 11/10/2004 0.0094 2.45 1.99 ? 4879 998.87 998.87 Jai CV M7+ 90? Investigating Integrated Theory 21 8/11/2004 0.0072 2.45 1.53 ? 3753 998.82 998.82 Jai CV M7+ 90? Investigating Integrated Theory 22 12/11/2004 0.0069 2.45 1.47 ? 3609 998.91 998.91 Jai CV M7+ 90? Investigating Integrated Theory 23 12/11/2004 0.0069 2.45 1.46 ? 3566 998.90 998.90 Jai CV M7+ 90? Investigating Integrated Theory 24 21/04/2005 0.0058 2.45 1.22 ? 2988 998.70 998.70 Jai CV M7+ 0? Verifying 3D LA 25 21/04/2005 0.0058 2.45 1.22 ? 2988 998.70 998.70 Jai CV M7+ CL 0? Verifying 3D LA 26 28/04/2005 0.0067 2.45 1.42 ? 3479 999.14 999.14 Jai CV M7+ 0? Verifying 3D LA 27 28/04/2005 0.0067 2.45 1.42 ? 3479 999.14 999.14 Jai CV M7+ CL 0? Verifying 3D LA Appendix D ? Initial Conditions for Experiments with 2D and 3D trajectories 308 Table D.2 - Initial Conditions for pure plume runs Plume Date Q0 d U0 Fr0 Re0 ?? ?a ? CameraAngle Comment Run No. (l/s) (mm) (m/s) (kg/m3) (kg/m3) (?f030uniF0290) 1 23/07/2003 0.0025 3.20 0.31 10.11 995 0.0301 999.46 1029.50 Canon 90? 2 25/07/2003 0.0051 3.20 0.63 20.63 2029 0.0301 999.42 1029.50 Canon 90? 3 14/08/2003 0.0014 3.20 0.17 5.66 557 0.0301 999.27 1029.30 Canon 90? 4 21/08/2003 0.0020 3.20 0.25 8.09 796 0.0301 999.40 1029.50 Canon 90? Table D.3 - Initial Conditions for horizontal buoyant jet runs Buoyant Jet Date Q0 d U0 Fr0 Re0 ?? ?a ? ljp Camera Angle Comment Run No. (l/s) (mm) (m/s) (kg/m3) (kg/m3) (m) (?f030uniF0290) 1.1 30/07/2003 0.0124 3.20 1.54 50.16 4934 0.0301 999.47 1029.55 0.35 Canon 0? 1.2 30/07/2003 0.0124 3.20 1.54 50.16 4934 0.0301 999.47 1029.55 0.35 Canon 0? 2 3/08/2003 0.0034 3.20 0.42 13.75 1353 0.0301 999.19 1029.27 0.10 Canon 0? 3.1 9/09/2003 0.0061 3.20 0.76 24.68 2427 0.0301 999.05 1029.12 0.17 Canon 0? 3.2 9/09/2003 0.0061 3.20 0.76 24.68 2427 0.0301 999.05 1029.12 0.17 Canon 0? Table D.4 - Initial Conditions for positively buoyant jet runs Positively Buoyant Jet Date Q0 d U0 Fr0 Re0 ?? ?a ? ljp Camera Angle Comment Run No. (l/s) (mm) (m/s) (kg/m3) (kg/m3) (m) (?f030uniF0290) 1 18/12/2003 0.0060 2.91 0.90 27.15 2625 0.0301 998.77 1028.8 0.1939 Canon 50? 2 18/12/2003 0.0060 2.91 0.90 27.15 2625 0.0301 998.77 1028.8 0.1939 Canon 60? 3 18/12/2003 0.0060 2.91 0.90 27.15 2625 0.0301 998.63 1028.7 0.1939 Canon 70? 4 18/12/2003 0.0060 2.91 0.90 27.15 2625 0.0301 998.63 1028.7 0.1939 Canon 80? 5 18/12/2003 0.0060 2.91 0.90 27.15 2625 0.0301 998.63 1028.8 0.1939 Canon 90? 6 18/12/2003 0.0060 2.91 0.90 27.15 2625 0.0301 998.67 1028.8 0.1939 Canon 40? 7 18/12/2003 0.0060 2.91 0.90 27.15 2625 0.0301 998.67 1028.7 0.1939 Canon 30? 8 19/12/2003 0.0060 2.91 0.90 27.15 2625 0.0301 998.67 1028.7 0.1939 Canon 20? 9 19/12/2003 0.0060 2.91 0.90 27.15 2625 0.0301 998.67 1028.8 0.1939 Canon 10? 10 19/12/2003 0.0060 2.91 0.90 27.15 2625 0.0301 998.67 1028.8 0.1939 Canon 0? Appendix D ? Initial Conditions for Experiments with 2D and 3D trajectories 309 Table D.5 - Initial Condition for advected line momentum puff runs Advected Line Momentum Puff Date Q0 d U0 Fr0 Re0 Ua Ur ?a ? Camera Angle Comment Run No. (l/s) (mm) (m/s) (m/s) (kg/m3) (kg/m3) (?f030uniF0290) 1 20/10/2003 0.0181 3.00 2.56 910 7682 0.1041 0.0406 Canon 90? No analysis 2.1 22/10/2003 0.0062 3.00 0.88 849 2631 0.0336 0.0383 998.96 998.65 Canon 90? No analysis 2.2 22/10/2003 0.0062 3.00 0.88 1012 2631 0.0338 0.0385 998.74 998.38 Canon 90? No analysis 2.3 22/10/2003 0.0062 3.00 0.88 1012 2631 0.0334 0.0380 998.67 998.42 Canon 90? No analysis 2.4 22/10/2003 0.0062 3.00 0.88 2006 2631 0.0339 0.0387 998.67 998.42 Canon 90? No analysis 3.1 5/11/2003 0.0062 3.00 0.88 1623 2631 0.0333 0.0380 998.90 998.84 Canon 90? No analysis 3.2 5/11/2003 0.0062 3.00 0.88 1241 2631 0.0338 0.0385 998.87 998.77 Canon 90? No analysis 3.3 5/11/2003 0.0062 3.00 0.88 2875 2631 0.0337 0.0384 998.84 998.67 Canon 90? No analysis 4.1 6/11/2003 0.0062 3.00 0.88 2847 2631 0.0333 0.0380 998.91 998.95 Canon 90? No analysis 4.2 6/11/2003 0.0062 3.00 0.88 1414 2631 0.0338 0.0386 998.90 998.87 Canon 90? No analysis 5.1 10/11/2003 0.0062 3.00 0.88 2256 2631 0.0327 0.0373 998.79 998.91 Canon 90? No analysis 5.2 10/11/2003 0.0062 3.00 0.88 ? 2631 0.0331 0.0377 998.75 998.70 Canon 90? No analysis 6.1 13/11/2003 0.0062 3.00 0.88 3783 2631 0.0337 0.0385 998.53 998.53 Canon 90? Investigating LA 6.2 13/11/2003 0.0062 3.00 0.88 3772 2631 0.0334 0.0380 998.53 998.55 Canon 90? Investigating LA 6.3 13/11/2003 0.0062 3.00 0.88 2663 2631 0.0334 0.0381 998.53 998.51 Canon 90? Investigating LA 6.4 13/11/2003 0.0062 3.00 0.88 2046 2631 0.0334 0.0381 998.53 998.49 Canon 90? Investigating LA 7.1 19/11/2003 0.0062 3.00 0.88 2923 2631 0.0332 0.0379 998.99 998.93 Canon 90? Investigating LA 7.2 19/11/2003 0.0062 3.00 0.88 ? 2631 0.0334 0.0380 998.99 999.02 Canon 90? Investigating LA 8.1 19/11/2003 0.0100 3.00 1.41 3301 4244 0.0334 0.0236 998.99 998.99 Canon 90? Investigating LA 8.2 19/11/2003 0.0100 3.00 1.41 2899 4244 0.0340 0.0240 998.99 998.93 Canon 90? Investigating LA 9 21/11/2003 0.0100 3.00 1.41 5029 4244 0.0334 0.0236 998.91 998.84 Canon 90? Investigating LA Test 1 4/05/2004 0.0094 2.91 1.41 5083 4113 0.0695 0.0492 999.22 999.19 Jai CV M7+ 90? z-integrated view Test 2 4/05/2004 0.0095 2.91 1.43 9842 4157 0.0297 0.0208 999.22 999.19 Jai CV M7+ 90? z-integrated view 10.1 13/10/2004 0.0092 2.45 1.95 9842 4770 0.0976 0.0501 998.87 998.85 Jai CV M7+ 90? y-integrated view 10.2 13/10/2004 0.0092 2.45 1.95 ? 4770 0.0977 0.0502 998.87 998.85 Jai CV M7+ 90? y-integrated view 11.1 20/10/2004 0.0098 2.45 2.07 ? 5067 0.0966 0.0467 998.82 998.82 Jai CV M7+ 90? z-integrated view 11.2 20/10/2004 0.0098 2.45 2.07 ? 5081 0.0979 0.0472 998.82 998.82 Jai CV M7+ 90? z-integrated view 12.1 12/11/2004 0.0071 2.45 1.51 5529 3710 0.0546 0.0360 998.99 998.99 Jai CV M7+ 90? z-integrated view 12.2 12/11/2004 0.0071 2.45 1.51 ? 3710 0.0547 0.0361 998.98 998.95 Jai CV M7+ 90? z-integrated view 13.1 23/02/2005 0.0071 2.45 1.50 ? 3667 0.0371 0.0248 998.63 998.63 Jai CV M7+ 90? y-integrated view 13.2 23/02/2005 0.0071 2.45 1.50 ? 3667 0.0372 0.0249 998.56 998.56 Jai CV M7+ 90? y-integrated view 14.1 25/02/2005 0.0071 2.45 1.50 ? 3667 0.0375 0.0251 998.72 998.72 Jai CV M7+ 90? z-integrated view 14.2 25/02/2005 0.0070 2.45 1.48 ? 3623 0.0377 0.0255 998.67 998.67 Jai CV M7+ 90? z-integrated view 14.3 25/02/2005 0.0071 2.45 1.50 ? 3667 0.0377 0.0252 998.67 998.67 Jai CV M7+ 90? z-integrated view 14.4 25/02/2005 0.0070 2.45 1.49 ? 3652 0.0377 0.0253 998.67 998.67 Jai CV M7+ 90? z-integrated view 15.1 14/04/2005 0.0084 2.45 1.79 ? 4374 0.0518 0.0290 999.07 999.07 Jai CV M7+ 90? y-integrated view 15.2 14/04/2005 0.0084 2.45 1.79 ? 4374 0.0518 0.0290 999.07 999.07 Jai CV M7+ 90? y-integrated view 15.3 14/04/2005 0.0083 2.45 1.77 ? 4331 0.0518 0.0293 999.07 999.07 Jai CV M7+ 90? y-integrated view 16.1 14/04/2005 0.0084 2.45 1.79 ? 4374 0.0518 0.0290 999.07 999.07 Jai CV M7+ CL 90? z-integrated view 16.2 14/04/2005 0.0084 2.45 1.79 ? 4374 0.0518 0.0290 999.07 999.07 Jai CV M7+ CL 90? z-integrated view 16.3 14/04/2005 0.0083 2.45 1.77 ? 4331 0.0518 0.0293 999.07 999.07 Jai CV M7+ CL 90? z-integrated view Appendix D ? Initial Conditions for Experiments with 2D and 3D trajectories 310 17.1 18/05/2005 0.0083 2.45 1.77 ? 4331 0.0512 0.0290 999.08 999.08 Jai CV M7+ 90? y-integrated view 17.2 18/05/2005 0.0086 2.45 1.81 ? 4446 0.0513 0.0283 999.05 999.05 Jai CV M7+ 90? y-integrated view 17.3 18/05/2005 0.0084 2.45 1.77 ? 4345 0.0513 0.0289 999.05 999.05 Jai CV M7+ 90? y-integrated view 17.4 18/05/2005 0.0083 2.45 1.77 ? 4331 0.0513 0.0290 999.05 999.05 Jai CV M7+ 90? y-integrated view 18.1 18/05/2005 0.0083 2.45 1.77 ? 4331 0.0512 0.0290 999.08 999.08 Jai CV M7+ CL 90? z-integrated view 18.2 18/05/2005 0.0086 2.45 1.81 ? 4446 0.0513 0.0283 999.05 999.05 Jai CV M7+ CL 90? z-integrated view 18.3 18/05/2005 0.0084 2.45 1.77 ? 4345 0.0513 0.0289 999.05 999.05 Jai CV M7+ CL 90? z-integrated view 18.4 18/05/2005 0.0083 2.45 1.77 ? 4331 0.0513 0.0290 999.05 999.05 Jai CV M7+ CL 90? z-integrated view 19.1 15/07/2005 0.0066 2.45 1.41 ? 3450 0.0621 0.0441 999.48 999.48 Jai CV M7+ 90? y-integrated view 19.2 15/07/2005 0.0066 2.45 1.41 ? 3450 0.0621 0.0441 999.41 999.41 Jai CV M7+ 90? y-integrated view 19.3 15/07/2005 0.0066 2.45 1.41 ? 3450 0.0621 0.0441 999.41 999.41 Jai CV M7+ 90? y-integrated view 20.1 25/10/2005 0.0078 2.45 1.65 ? 4042 0.0522 0.0317 999.14 999.14 Jai CV M7+ 90? y-integrated view 20.2 25/10/2005 0.0078 2.45 1.65 ? 4042 0.0522 0.0317 999.11 999.11 Jai CV M7+ 90? y-integrated view 20.3 25/10/2005 0.0078 2.45 1.65 ? 4042 0.0522 0.0317 999.07 999.07 Jai CV M7+ 90? y-integrated view 21.1 25/10/2005 0.0078 2.45 1.65 ? 4042 0.0522 0.0317 999.14 999.14 Jai CV M7+ CL 90? z-integrated view 21.2 25/10/2005 0.0078 2.45 1.65 ? 4042 0.0522 0.0317 999.11 999.11 Jai CV M7+ CL 90? z-integrated view 21.3 25/10/2005 0.0078 2.45 1.65 ? 4042 0.0522 0.0317 999.07 999.07 Jai CV M7+ CL 90? z-integrated view 22.1 26/10/2005 0.0098 2.45 2.07 ? 5081 0.0522 0.0252 999.07 999.07 Jai CV M7+ 90? y-integrated view 22.2 26/10/2005 0.0098 2.45 2.07 ? 5081 0.0522 0.0252 999.05 999.05 Jai CV M7+ 90? y-integrated view 22.3 26/10/2005 0.0098 2.45 2.07 ? 5081 0.0522 0.0252 999.04 999.04 Jai CV M7+ 90? y-integrated view 23.1 26/10/2005 0.0098 2.45 2.07 ? 5081 0.0522 0.0252 999.07 999.07 Jai CV M7+ CL 90? z-integrated view 23.2 26/10/2005 0.0098 2.45 2.07 ? 5081 0.0522 0.0252 999.05 999.05 Jai CV M7+ CL 90? z-integrated view 23.3 26/10/2005 0.0098 2.45 2.07 ? 5081 0.0522 0.0252 999.04 999.04 Jai CV M7+ CL 90? z-integrated view Table D.6 - Initial Condition for advected jet runs Advected Jet Date Q0 d U0 Fr0 Re0 Ua Ur ?a ? Camera Angle Comment Run No. (l/s) (mm) (m/s) (m/s) (kg/m3) (kg/m3) (?f030uniF0290) 1 1/07/2005 0.0068 2.45 1.43 ? 3508 0.0046 0.00319 999.51 999.51 Jai CV M7+ 90? y-integrated view 2 7/07/2005 0.0066 2.45 1.40 ? 3436 0.0141 0.01005 999.32 999.32 Jai CV M7+ 90? y-integrated view 3 11/07/2005 0.0065 2.45 1.38 ? 3392 0.0104 0.00751 999.49 999.49 Jai CV M7+ 90? y-integrated view 4.1 12/07/2005 0.0066 2.45 1.39 ? 3407 0.0257 0.01845 999.40 999.40 Jai CV M7+ 90? y-integrated view 4.2 12/07/2005 0.0066 2.45 1.39 ? 3407 0.0257 0.01845 999.35 999.35 Jai CV M7+ 90? y-integrated view 5 22/07/2005 0.0076 2.45 1.62 ? 3970 0.0371 0.02292 999.48 999.48 Jai CV M7+ 90? y-integrated view 6 25/07/2005 0.0072 2.45 1.53 ? 3753 0.0125 0.00816 999.48 999.48 Jai CV M7+ 90? y-integrated view Appendix D ? Initial Conditions for Experiments with 2D and 3D trajectories 311 Table D.7 - Initial conditions for buoyant jet in moving ambient runs Buoyant Jet in Ambient Date Q0 d U0 Fr0 Re0 Ua Ur ?a ? Camera Angle Comment Flow Run No. (l/s) (mm) (m/s) (m/s) (kg/m 3) (kg/m3) (? f030uniF0290) 1.1 22/12/2003 0.0060 2.91 0.90 27.15 2625 0.0336 0.0372 998.84 1037.16 Canon -40? 1.2 22/12/2003 0.0060 2.91 0.90 27.15 2625 0.0336 0.0372 998.84 1037.16 Canon -40? 2.1 23/12/2003 0.0080 2.91 1.20 36.20 3500 0.0339 0.0282 998.84 1037.16 Canon -30? 2.2 23/12/2003 0.0080 2.91 1.20 36.20 3500 0.0339 0.0282 998.84 1037.16 Canon -30? 3.1 23/12/2003 0.0100 2.91 1.50 45.25 4375 0.0447 0.0297 998.77 1037.42 Canon -20? 3.2 23/12/2003 0.0100 2.91 1.50 45.25 4375 0.0451 0.0300 998.77 1037.42 Canon -20? 4.1 23/12/2003 0.0100 2.91 1.50 45.25 4375 0.0448 0.0298 998.80 1037.34 Canon -10? 4.2 23/12/2003 0.0100 2.91 1.50 45.25 4375 0.0448 0.0298 998.80 1037.34 Canon -10? 5.1 23/12/2003 0.0100 2.91 1.50 45.25 4375 0.0448 0.0298 998.80 1037.34 Canon 0? 5.2 23/12/2003 0.0100 2.91 1.50 45.25 4375 0.0448 0.0298 998.80 1037.34 Canon 0? 6.1 24/12/2003 0.0100 2.91 1.50 45.25 4375 0.0449 0.0298 998.87 1037.27 Canon 10? 6.2 24/12/2003 0.0100 2.91 1.50 45.25 4375 0.0453 0.0301 998.87 1037.27 Canon 10? 7.1 24/12/2003 0.0100 2.91 1.50 45.25 4375 0.0454 0.0302 998.87 1037.27 Canon 20? 7.2 24/12/2003 0.0100 2.91 1.50 45.25 4375 0.0455 0.0302 998.87 1037.27 Canon 20? 8.1 29/12/2003 0.0100 2.91 1.50 45.25 4375 0.0450 0.0299 998.84 1037.16 Canon 30? 8.2 29/12/2003 0.0100 2.91 1.50 45.25 4375 0.0455 0.0302 998.84 1037.16 Canon 30? 9.1 29/12/2003 0.0100 2.91 1.50 45.25 4375 0.0449 0.0299 998.84 1037.16 Canon 40? 9.2 29/12/2003 0.0100 2.91 1.50 45.25 4375 0.0450 0.0299 998.84 1037.16 Canon 40? 10.1 30/12/2003 0.0100 2.91 1.50 45.25 4375 0.0447 0.0297 998.79 1037.34 Canon 50? 10.2 30/12/2003 0.0100 2.91 1.50 45.25 4375 0.0443 0.0295 998.79 1037.34 Canon 50? 11.1 30/12/2003 0.0100 2.91 1.50 45.25 4375 0.0444 0.0295 998.79 1037.34 Canon 60? 11.2 30/12/2003 0.0100 2.91 1.50 45.25 4375 0.0444 0.0295 998.79 1037.34 Canon 60? 12.1 11/03/2004 0.0086 2.91 1.29 46.44 3741 0.0598 0.0465 998.75 1025.39 Canon 70? 12.2 11/03/2004 0.0086 2.91 1.29 46.44 3741 0.0598 0.0465 998.75 1025.39 Canon 70? 13.1 11/03/2004 0.0086 2.91 1.29 46.44 3741 0.0598 0.0465 998.75 1025.39 Canon 80? 13.2 11/03/2004 0.0086 2.91 1.29 46.44 3741 0.0598 0.0465 998.75 1025.39 Canon 80? 14.1 12/03/2004 0.0095 2.91 1.43 51.60 4157 0.0598 0.0419 998.67 1025.48 Canon 90? 14.2 12/03/2004 0.0095 2.91 1.43 51.60 4157 0.0601 0.0421 998.67 1025.48 Canon 90? 14.3 12/03/2004 0.0095 2.91 1.43 51.60 4157 0.0601 0.0421 998.67 1025.48 Canon 90? 15.1 16/03/2004 0.0103 2.91 1.55 55.35 4507 0.0595 0.0384 998.84 1026.26 Canon 100? 15.2 16/03/2004 0.0103 2.91 1.55 55.35 4507 0.0601 0.0388 998.84 1026.34 Canon 100? 16.1 16/03/2004 0.0096 2.91 1.44 51.59 4200 0.0602 0.0417 998.84 1026.23 Canon 110? 16.2 16/03/2004 0.0096 2.91 1.44 51.59 4200 0.0603 0.0418 998.84 1026.09 Canon 110? 17.1 17/03/2004 0.0096 2.91 1.44 51.59 4200 0.0599 0.0415 998.87 1026.13 Canon 120? 17.2 17/03/2004 0.0096 2.91 1.44 51.59 4200 0.0603 0.0418 998.84 1026.09 Canon 120? 18.1 17/03/2004 0.0098 2.91 1.47 52.66 4288 0.0594 0.0403 998.87 1026.19 Canon 130? 18.2 17/03/2004 0.0098 2.91 1.47 52.66 4288 0.0600 0.0407 998.87 1026.18 Canon 130? 19.1 16/12/2005 0.0079 2.43 1.70 78.16 4119 0.1245 0.0735 998.88 1018.59 Jai CV M7+ 0? Trajectory only 19.2 16/12/2005 0.0079 2.43 1.70 78.16 4119 0.1246 0.0735 998.88 1018.59 Jai CV M7+ 0? Trajectory only 19.3 16/12/2005 0.0079 2.43 1.70 78.16 4119 0.1246 0.0735 998.88 1018.59 Jai CV M7+ 0? Trajectory only 20.1 21/12/2005 0.0060 2.43 1.30 59.93 3158 0.2420 0.1862 998.53 1018.23 Jai CV M7+ 0? Trajectory only 20.2 21/12/2005 0.0060 2.43 1.30 59.93 3158 0.2423 0.1865 998.53 1018.23 Jai CV M7+ 0? Trajectory only 20.3 21/12/2005 0.0060 2.43 1.30 59.93 3158 0.2423 0.1865 998.53 1018.23 Jai CV M7+ 0? Trajectory only Appendix D ? Initial Conditions for Experiments with 2D and 3D trajectories 312 21.1 17/01/2006 0.0033 2.43 0.72 23.62 1747 0.2420 0.3367 998.75 1037.54 Jai CV M7+ 0? Trajectory only 21.2 17/01/2006 0.0033 2.43 0.72 23.62 1747 0.2423 0.3372 998.72 1037.50 Jai CV M7+ 0? Trajectory only 21.3 17/01/2006 0.0033 2.43 0.72 23.62 1747 0.2427 0.3377 998.72 1037.50 Jai CV M7+ 0? Trajectory only 22.1 17/01/2006 0.0061 2.43 1.32 43.31 3202 0.1253 0.0951 998.72 1037.50 Jai CV M7+ 0? Trajectory only 22.2 17/01/2006 0.0061 2.43 1.32 43.31 3202 0.1253 0.0951 998.70 1037.49 Jai CV M7+ 0? Trajectory only 22.3 17/01/2006 0.0061 2.43 1.32 43.31 3202 0.1254 0.0952 998.69 1037.47 Jai CV M7+ 0? Trajectory only 23.1 24/01/2006 0.0024 2.43 0.53 17.32 1281 0.1250 0.2372 998.67 1037.45 Jai CV M7+ 0? Trajectory only 23.2 24/01/2006 0.0024 2.43 0.53 17.32 1281 0.1252 0.2376 998.62 1037.40 Jai CV M7+ 0? Trajectory only 23.3 24/01/2006 0.0024 2.43 0.53 17.32 1281 0.1253 0.2378 998.62 1037.40 Jai CV M7+ 0? Trajectory only Table D.8 - Initial conditions for LA negatively buoyant jet runs Buoyant Jet in Ambient Date Q0 d U0 Fr0 Re0 ?? ?a ? ljp Camera Angle Comment Flow Run No. (l/s) (mm) (m/s) (kg/m 3) (kg/m3) (m) (?f030uniF0290) 1 19/12/2003 0.0060 2.91 0.90 27.15 2625 0.0301 998.67 1028.73 0.17 Canon 0? 2 19/12/2003 0.0060 2.91 0.90 27.15 2625 0.0301 998.67 1028.73 0.17 Canon 10? 3 19/12/2003 0.0060 2.91 0.90 27.15 2625 0.0301 998.67 1028.73 0.17 Canon 20? 4 19/12/2003 0.0060 2.91 0.90 27.15 2625 0.0301 998.67 1028.73 0.17 Canon 30? 5 19/12/2003 0.0060 2.91 0.90 27.15 2625 0.0301 998.67 1028.73 0.17 Canon 40? 6 23/01/2004 0.0105 2.91 1.58 59.47 4594 0.0247 998.84 1023.49 0.37 Canon 47? 7 23/01/2004 0.0050 2.91 0.74 28.04 2166 0.0247 998.84 1023.49 0.18 Canon 47? 8 23/01/2004 0.0075 2.91 1.12 42.20 3260 0.0247 998.84 1023.49 0.27 Canon 47? 9 23/01/2004 0.0025 2.91 0.38 14.16 1094 0.0247 998.84 1023.49 0.09 Canon 47? 10 23/01/2004 0.0037 2.91 0.56 20.96 1619 0.0247 998.84 1023.49 0.13 Canon 47? 11 11/02/2004 0.0049 2.91 0.74 27.32 2144 0.0255 998.67 1024.10 0.17 Canon 45? 12 11/02/2004 0.0098 2.91 1.47 54.65 4288 0.0255 998.67 1024.10 0.34 Canon 45? 13 11/02/2004 0.0074 2.91 1.11 41.27 3238 0.0255 998.67 1024.10 0.26 Canon 45? 14 11/02/2004 0.0034 2.91 0.51 18.96 1488 0.0255 998.67 1024.10 0.12 Canon 45? 15 11/02/2004 0.0010 2.91 0.15 5.58 438 0.0255 998.67 1024.10 0.04 Canon 45? 16 12/02/2004 0.0065 2.91 0.98 36.25 2844 0.0255 998.80 1024.24 0.23 Canon 75? 17 12/02/2004 0.0052 2.91 0.78 29.00 2275 0.0255 998.80 1024.24 0.18 Canon 75? 18 19/02/2004 0.0062 2.91 0.93 33.72 2713 0.0268 998.53 1025.26 0.21 Canon 75? 19 23/02/2004 0.0055 2.91 0.83 29.84 2406 0.0269 998.72 1025.59 0.19 Canon 30? 20 23/02/2004 0.0104 2.91 1.56 56.42 4550 0.0269 998.72 1025.59 0.36 Canon 30? 21 23/02/2004 0.0066 2.91 0.99 35.81 2888 0.0269 998.72 1025.59 0.23 Canon 30? 22 23/02/2004 0.0034 2.91 0.51 18.45 1488 0.0269 998.72 1025.59 0.12 Canon 30? 23 23/02/2004 0.0010 2.91 0.15 5.43 438 0.0269 998.72 1025.59 0.03 Canon 30? 24 3/02/2004 0.0119 2.91 1.79 65.97 5207 0.0258 998.84 1024.57 0.42 Canon 30? 25 3/02/2004 0.0091 2.91 1.36 50.17 3960 0.0258 998.84 1024.57 0.32 Canon 30? 26 3/02/2004 0.0065 2.91 0.97 35.76 2822 0.0258 998.84 1024.57 0.23 Canon 30? 27 3/02/2004 0.0078 2.91 1.17 43.24 3413 0.0258 998.84 1024.57 0.27 Canon 30? 28 3/03/2004 0.0118 2.91 1.77 65.42 5163 0.0258 998.98 1024.72 0.41 Canon 15? 29 3/03/2004 0.0091 2.91 1.36 50.17 3960 0.0258 998.98 1024.72 0.32 Canon 15? Appendix D ? Initial Conditions for Experiments with 2D and 3D trajectories 313 30 3/03/2004 0.0065 2.91 0.97 35.76 2822 0.0258 998.98 1024.72 0.23 Canon 15? 31 4/03/2004 0.0081 2.91 1.21 44.63 3522 0.0258 998.96 1024.70 0.28 Canon 45? 32 4/03/2004 0.0070 2.91 1.04 38.53 3041 0.0258 998.96 1024.70 0.24 Canon 45? 33 8/03/2004 0.0083 2.91 1.24 45.74 3610 0.0258 998.79 1024.52 0.29 Canon 60? 34 8/03/2004 0.0069 2.91 1.03 37.98 2997 0.0258 998.79 1024.52 0.24 Canon 60? 35 8/03/2004 0.0115 2.91 1.73 63.75 5032 0.0258 998.79 1024.52 0.40 Canon 60? 36 25/03/2004 0.0089 2.91 1.34 46.79 3894 0.0287 999.02 1027.65 0.29 Canon 0? 37 25/03/2004 0.0071 2.91 1.07 37.33 3107 0.0287 999.02 1027.65 0.24 Canon 0? 38 25/03/2004 0.0108 2.91 1.62 56.51 4704 0.0287 999.02 1027.65 0.36 Canon 0? 39 12/04/2004 0.0066 2.91 0.99 38.29 2888 0.0235 999.22 1022.73 0.24 Canon 0? 40.1 13/04/2004 0.0077 2.91 1.16 44.40 3369 0.0238 998.98 1022.77 0.28 Canon 15? Investigate time-averaging 40.2 13/04/2004 0.0077 2.91 1.16 44.40 3369 0.0238 998.98 1022.77 0.28 Canon 15? Investigate time-averaging 41 14/04/2004 0.0073 2.91 1.09 40.86 3172 0.0249 999.22 1024.14 0.26 Canon 15? 42 29/10/2004 0.0085 2.94 1.26 48.64 3699 0.0233 999.02 1022.32 0.31 Jai CV M7+ 45? 43 29/10/2004 0.0096 2.94 1.42 54.97 4181 0.0233 999.02 1022.32 0.35 Jai CV M7+ 45? 44 29/10/2004 0.0106 2.94 1.57 60.68 4615 0.0233 999.02 1022.32 0.39 Jai CV M7+ 45? 45 29/10/2004 0.0106 2.94 1.57 60.68 4615 0.0233 999.02 1022.32 0.39 Jai CV M7+ 45? 46 29/10/2004 0.0096 2.94 1.42 54.97 4181 0.0233 999.02 1022.32 0.35 Jai CV M7+ 45? 47 26/01/2005 0.0064 2.94 0.95 35.28 2796 0.0253 998.49 1023.77 0.22 Jai CV M7+ 45? 48 26/01/2005 0.0107 2.94 1.58 58.54 4639 0.0253 998.49 1023.77 0.37 Jai CV M7+ 45? 49 26/01/2005 0.0107 2.94 1.58 58.54 4639 0.0253 998.49 1023.77 0.37 Jai CV M7+ 45 50 26/01/2005 0.0083 2.94 1.23 45.62 3615 0.0253 998.49 1023.77 0.29 Jai CV M7+ 45 51 26/01/2005 0.0083 2.94 1.23 45.62 3615 0.0253 998.49 1023.77 0.29 Jai CV M7+ 45 52 27/01/2005 0.0068 2.94 1.00 37.10 2940 0.0253 998.58 1023.87 0.24 Jai CV M7+ 60? 53 27/01/2005 0.0077 2.94 1.13 41.97 3326 0.0253 998.58 1023.87 0.27 Jai CV M7+ 60? 54 27/01/2005 0.0083 2.94 1.22 45.16 3579 0.0253 998.58 1023.87 0.29 Jai CV M7+ 60? 55 27/01/2005 0.0093 2.94 1.38 50.94 4037 0.0253 998.58 1023.87 0.32 Jai CV M7+ 60? 56 27/01/2005 0.0106 2.94 1.57 58.24 4615 0.0253 998.49 1023.77 0.37 Jai CV M7+ 60? 57 27/01/2005 0.0106 2.94 1.57 58.24 4615 0.0253 998.49 1023.77 0.37 Jai CV M7+ 60? 58 28/01/2005 0.0062 2.94 0.92 33.91 2687 0.0253 998.56 1023.85 0.22 Jai CV M7+ 30? 59 28/01/2005 0.0081 2.94 1.20 44.55 3531 0.0253 998.56 1023.85 0.28 Jai CV M7+ 30? 60 28/01/2005 0.0070 2.94 1.03 38.32 3037 0.0253 998.56 1023.85 0.24 Jai CV M7+ 30? 61 28/01/2005 0.0106 2.94 1.57 58.24 4615 0.0253 998.56 1023.85 0.37 Jai CV M7+ 30? 62 28/01/2005 0.0106 2.94 1.57 58.24 4615 0.0253 998.56 1023.85 0.37 Jai CV M7+ 30? 63 31/01/2005 0.0062 2.94 0.92 34.06 2699 0.0253 998.49 1023.77 0.22 Jai CV M7+ 15? 64 31/01/2005 0.0069 2.94 1.02 37.71 2988 0.0253 998.49 1023.77 0.24 Jai CV M7+ 15? 65 31/01/2005 0.0085 2.94 1.26 46.53 3687 0.0253 998.49 1023.77 0.30 Jai CV M7+ 15? 66 31/01/2005 0.0085 2.94 1.26 46.53 3687 0.0253 998.49 1023.77 0.30 Jai CV M7+ 15? 67 31/01/2005 0.0106 2.94 1.57 58.24 4615 0.0253 998.46 1023.74 0.37 Jai CV M7+ 15? 68 31/01/2005 0.0106 2.94 1.57 58.24 4615 0.0253 998.46 1023.74 0.37 Jai CV M7+ 15? 69 7/02/2005 0.0068 2.94 1.00 36.95 2928 0.0253 998.60 1023.88 0.23 Jai CV M7+ 45? 70 7/02/2005 0.0074 2.94 1.09 40.45 3205 0.0253 998.67 1023.96 0.26 Jai CV M7+ 45? 71 7/02/2005 0.0079 2.94 1.16 43.03 3410 0.0253 998.67 1023.96 0.27 Jai CV M7+ 45? Appendix D ? Initial Conditions for Experiments with 2D and 3D trajectories 314 Table D.9 - Initial conditions for LIF negatively buoyant jets Buoyant Jet in Ambient Date Q0 d U0 Fr0 Re0 ?? ?a ? ljp Camera Angle Comment Flow Run No. (l/s) (mm) (m/s) (kg/m3) (kg/m3) (m) (?f030uniF0290) 1 14/09/2004 0.0069 2.45 1.47 54.69 3595 0.0299 998.99 1028.91 0.29 Pulnix TM 1010 5? 2 14/09/2004 0.0092 2.45 1.94 72.48 4764 0.0299 998.99 1028.91 0.38 Pulnix TM 1010 5? 3 14/09/2004 0.0060 2.45 1.28 47.66 3133 0.0299 998.99 1028.91 0.25 Pulnix TM 1010 5? 4 14/09/2004 0.0081 2.45 1.71 63.91 4201 0.0299 998.99 1028.91 0.34 Pulnix TM 1010 5? 5 16/09/2004 0.0061 2.45 1.28 47.88 3147 0.0299 998.99 1028.91 0.25 Pulnix TM 1010 10? 6 16/09/2004 0.0090 2.45 1.91 71.16 4677 0.0299 998.99 1028.91 0.38 Pulnix TM 1010 10? 7 16/09/2004 0.0065 2.45 1.38 51.61 3392 0.0299 998.99 1028.91 0.27 Pulnix TM 1010 10? 8 16/09/2004 0.0080 2.45 1.70 63.47 4172 0.0299 998.99 1028.91 0.34 Pulnix TM 1010 10? 9 20/09/2004 0.0061 2.45 1.30 48.32 3176 0.0299 998.96 1028.87 0.26 Pulnix TM 1010 15? 10 20/09/2004 0.0090 2.45 1.91 71.16 4677 0.0299 998.96 1028.87 0.38 Pulnix TM 1010 15? 11 20/09/2004 0.0068 2.45 1.44 53.59 3522 0.0299 998.96 1028.87 0.28 Pulnix TM 1010 15? 12 20/09/2004 0.0080 2.45 1.70 63.47 4172 0.0299 998.96 1028.87 0.34 Pulnix TM 1010 15? 13 21/09/2004 0.0062 2.45 1.31 48.76 3205 0.0299 998.98 1028.89 0.26 Pulnix TM 1010 30? 14 21/09/2004 0.0090 2.45 1.91 71.16 4677 0.0299 998.98 1028.89 0.38 Pulnix TM 1010 30? 15 22/09/2004 0.0060 2.45 1.27 47.44 3118 0.0299 998.99 1028.91 0.25 Pulnix TM 1010 41.4? 16 22/09/2004 0.0067 2.45 1.42 52.93 3479 0.0299 998.99 1028.91 0.28 Pulnix TM 1010 41.4? 17 23/09/2004 0.0061 2.45 1.28 47.88 3147 0.0299 998.98 1028.89 0.25 Pulnix TM 1010 37.5? 18 23/09/2004 0.0093 2.45 1.97 73.58 4836 0.0299 998.98 1028.89 0.39 Pulnix TM 1010 37.5? 19 23/09/2004 0.0071 2.45 1.51 56.23 3696 0.0299 999.12 1029.04 0.30 Pulnix TM 1010 37.5? 20 23/09/2004 0.0082 2.45 1.74 65.01 4273 0.0299 999.12 1029.04 0.34 Pulnix TM 1010 37.5? 21 23/09/2004 0.0059 2.45 1.26 47.00 3089 0.0299 998.98 1028.89 0.25 Pulnix TM 1010 52.5? 22 23/09/2004 0.0093 2.45 1.97 73.58 4836 0.0299 998.98 1028.89 0.39 Pulnix TM 1010 52.5? 23 23/09/2004 0.0069 2.45 1.47 54.69 3595 0.0299 998.98 1028.89 0.29 Pulnix TM 1010 52.5? 24 23/09/2004 0.0083 2.45 1.75 65.23 4287 0.0299 998.98 1028.89 0.35 Pulnix TM 1010 52.5? 25 23/09/2004 0.0062 2.45 1.32 49.20 3234 0.0299 998.98 1028.89 0.26 Pulnix TM 1010 60? 26 23/09/2004 0.0082 2.45 1.73 64.57 4244 0.0299 998.98 1028.89 0.34 Pulnix TM 1010 60? 27 24/09/2004 0.0059 2.45 1.26 47.00 3089 0.0299 998.98 1028.89 0.25 Pulnix TM 1010 65.5? 28 24/09/2004 0.0071 2.45 1.50 56.01 3681 0.0299 998.98 1028.89 0.30 Pulnix TM 1010 65.5? 29 24/09/2004 0.0085 2.45 1.81 67.43 4432 0.0299 998.98 1028.89 0.36 Pulnix TM 1010 65.5? 30 24/09/2004 0.0091 2.45 1.93 71.82 4721 0.0299 998.98 1028.89 0.38 Pulnix TM 1010 65.5? 31 6/01/2005 0.0046 2.45 0.97 39.46 2388 0.0254 998.72 1024.08 0.21 Pulnix TM 1010 45? 32 6/01/2005 0.0057 2.45 1.20 48.66 2945 0.0254 998.72 1024.08 0.26 Pulnix TM 1010 45? 33 6/01/2005 0.0068 2.45 1.44 58.44 3537 0.0254 998.72 1024.08 0.31 Pulnix TM 1010 45? 34 6/01/2005 0.0083 2.45 1.77 71.56 4331 0.0254 998.72 1024.08 0.38 Pulnix TM 1010 45? 35 6/01/2005 0.0104 2.45 2.20 88.97 5385 0.0254 998.72 1024.08 0.47 Pulnix TM 1010 45? 36 6/01/2005 0.0092 2.45 1.94 78.71 4764 0.0254 998.72 1024.08 0.42 Pulnix TM 1010 45? 37 6/01/2005 0.0108 2.45 2.30 93.02 5630 0.0254 998.72 1024.08 0.49 Pulnix TM 1010 45? 38 12/01/2005 0.0072 2.45 1.53 62.02 3753 0.0254 998.72 1024.08 0.33 Pulnix TM 1010 45? 39 12/01/2005 0.0083 2.45 1.77 71.56 4331 0.0254 998.72 1024.08 0.38 Pulnix TM 1010 45? 40 17/01/2005 0.0058 2.45 1.23 49.61 3003 0.0254 998.70 1024.06 0.26 Pulnix TM 1010 45? Appendix D ? Initial Conditions for Experiments with 2D and 3D trajectories 315 Table D.10 - Initial condition for non-buoyant oblique discharges in moving ambient Oblique Discharge Date Q0 d U0 Fr0 Re0 Ua Ur ?a ? Camera Angle Comment Run No. (l/s) (mm) (m/s) (m/s) (kg/m3) (kg/m3) (?f030uniF0290) 1.1 9/12/2003 0.0100 3.00 1.41 1471 4244 0.0330 0.0233 998.91 998.88 Canon 70? 1.2 9/12/2003 0.0100 3.00 1.41 1471 4244 0.0330 0.0233 998.91 998.88 Canon 70? 2.1 10/12/2003 0.0100 3.00 1.41 941 4244 0.0334 0.0236 998.91 998.99 Canon 100? 2.2 10/12/2003 0.0100 3.00 1.41 2090 4244 0.0335 0.0237 998.91 998.93 Canon 100? 3.1 11/12/2003 0.0100 2.91 1.50 850 4375 0.0335 0.0223 998.96 998.85 Canon 60? 3.2 11/12/2003 0.0100 2.91 1.50 2256 4375 0.0335 0.0223 998.91 998.93 Canon 60? 4.1 11/12/2003 0.0100 2.91 1.50 1307 4375 0.0336 0.0223 998.96 998.91 Canon 110? 4.2 11/12/2003 0.0100 2.91 1.50 1009 4375 0.0337 0.0224 998.96 998.88 Canon 110? 5.1 12/12/2003 0.0100 2.91 1.50 1587 4375 0.0337 0.0224 998.91 998.88 Canon 120? 5.2 12/12/2003 0.0100 2.91 1.50 1015 4375 0.0333 0.0222 998.91 998.99 Canon 120? 6.1 12/12/2003 0.0100 2.91 1.50 1587 4375 0.0332 0.0221 998.91 998.88 Canon 130? 6.2 12/12/2003 0.0100 2.91 1.50 842 4375 0.0332 0.0221 998.91 998.80 Canon 130? 7.1 12/12/2003 0.0100 2.91 1.50 583 4375 0.0331 0.0220 998.63 998.87 Canon 30? 7.2 12/12/2003 0.0100 2.91 1.50 1515 4375 0.0332 0.0221 998.63 998.67 Canon 30? 8.1 15/12/2003 0.0100 2.91 1.50 ? 4375 0.0332 0.0221 998.63 998.63 Canon 20? 8.2 15/12/2003 0.0100 2.91 1.50 ? 4375 0.0334 0.0222 998.63 998.63 Canon 20? 9.1 16/12/2003 0.0100 2.91 1.50 983 4375 0.0330 0.0220 998.84 998.75 Canon 10? 9.2 16/12/2003 0.0100 2.91 1.50 ? 4375 0.0331 0.0220 998.84 998.84 Canon 10? 10.1 16/12/2003 0.0100 2.91 1.50 1562 4375 0.0331 0.0220 998.84 998.80 Canon 140? 10.2 16/12/2003 0.0100 2.91 1.50 983 4375 0.0331 0.0220 998.84 998.75 Canon 140? 11.1 17/12/2003 0.0100 2.91 1.50 1273 4375 0.0329 0.0219 998.84 998.79 Canon 50? 11.2 17/12/2003 0.0100 2.91 1.50 1273 4375 0.0331 0.0220 998.84 998.79 Canon 50? 12.1 17/12/2003 0.0100 2.91 1.50 1273 4375 0.0330 0.0219 998.84 998.79 Canon 0? 12.2 17/12/2003 0.0100 2.91 1.50 1273 4375 0.0331 0.0220 998.84 998.79 Canon 0? 13.1 17/12/2003 0.0080 2.91 1.20 712 3500 0.0330 0.0275 998.80 998.70 Canon 80? 13.2 17/12/2003 0.0080 2.91 1.20 548 3500 0.0331 0.0275 998.80 998.63 Canon 80? 14.1 18/12/2003 0.0100 2.91 1.50 828 4375 0.0337 0.0224 998.84 998.72 Canon 50? No analysis 14.2 18/12/2003 0.0100 2.91 1.50 983 4375 0.0338 0.0225 998.84 998.75 Canon 50? No analysis 15.1 21/10/2004 0.0099 2.45 2.10 ? 5154 0.0804 0.0382 999.02 999.02 Jai CV M7+ 10? No dilution results 15.2 21/10/2004 0.0099 2.45 2.10 ? 5154 0.0809 0.0385 999.02 999.02 Jai CV M7+ 10? No dilution results 16.1 22/10/2004 0.0098 2.45 2.07 ? 5081 0.0800 0.0386 999.12 999.12 Jai CV M7+ 12.5? No dilution results 16.2 22/10/2004 0.0098 2.45 2.07 3536 5067 0.0809 0.0391 999.12 999.11 Jai CV M7+ 12.5? No dilution results 17.1 26/10/2004 0.0097 2.45 2.06 3567 5038 0.0798 0.0388 999.18 999.16 Jai CV M7+ 15? No dilution results 17.2 26/10/2004 0.0097 2.45 2.06 ? 5038 0.0807 0.0392 999.18 999.18 Jai CV M7+ 15? No dilution results 18.1 27/10/2004 0.0097 2.45 2.06 ? 5053 0.0805 0.0390 999.11 999.11 Jai CV M7+ 17.5? No dilution results 18.2 27/10/2004 0.0098 2.45 2.07 ? 5067 0.0808 0.0391 999.11 999.11 Jai CV M7+ 17.5? No dilution results 19.1 28/10/2004 0.0099 2.45 2.10 ? 5154 0.0805 0.0383 999.02 999.02 Jai CV M7+ 20? No dilution results 19.2 28/10/2004 0.0099 2.45 2.10 ? 5154 0.0809 0.0384 999.02 999.02 Jai CV M7+ 20? No dilution results 20.1 22/03/2005 0.0076 2.45 1.62 ? 3970 0.0512 0.0316 998.87 998.87 Jai CV M7+ 13.5? 20.2 22/03/2005 0.0077 2.45 1.63 ? 3999 0.0517 0.0317 998.87 998.87 Jai CV M7+ 13.5? 20.3 22/03/2005 0.0077 2.45 1.64 ? 4013 0.0518 0.0316 998.84 998.84 Jai CV M7+ 13.5? 21.1 24/03/2005 0.0083 2.45 1.77 ? 4331 0.0516 0.0292 998.77 998.77 Jai CV M7+ 15? 21.2 24/03/2005 0.0083 2.45 1.76 ? 4302 0.0517 0.0294 998.70 998.70 Jai CV M7+ 15? Appendix D ? Initial Conditions for Experiments with 2D and 3D trajectories 316 21.3 24/03/2005 0.0084 2.45 1.79 ? 4374 0.0517 0.0289 998.70 998.70 Jai CV M7+ 15? 22.1 4/04/2005 0.0083 2.45 1.76 ? 4316 0.0519 0.0294 998.84 998.84 Jai CV M7+ 12.5? 22.2 4/04/2005 0.0083 2.45 1.77 ? 4331 0.0518 0.0293 998.84 998.84 Jai CV M7+ 12.5? 22.3 4/04/2005 0.0083 2.45 1.77 ? 4331 0.0518 0.0293 998.82 998.82 Jai CV M7+ 12.5? 23.1 29/04/2005 0.0084 2.45 1.79 ? 4374 0.0517 0.0290 999.30 999.30 Jai CV M7+ 0? 23.2 29/04/2005 0.0084 2.45 1.78 ? 4360 0.0517 0.0291 999.27 999.27 Jai CV M7+ 0? 23.3 29/04/2005 0.0084 2.45 1.79 ? 4374 0.0517 0.0289 999.26 999.26 Jai CV M7+ 0? 24.1 5/05/2005 0.0083 2.45 1.77 ? 4331 0.0516 0.0292 998.93 998.93 Jai CV M7+ 10? 24.2 5/05/2005 0.0084 2.45 1.78 ? 4360 0.0518 0.0291 998.93 998.93 Jai CV M7+ 10? 24.3 5/05/2005 0.0084 2.45 1.77 ? 4345 0.0518 0.0292 998.93 998.93 Jai CV M7+ 10? 25.1 6/05/2005 0.0083 2.45 1.77 ? 4331 0.0517 0.0293 999.19 999.19 Jai CV M7+ 20? 25.2 6/05/2005 0.0084 2.45 1.77 ? 4345 0.0517 0.0292 999.15 999.15 Jai CV M7+ 20? 25.3 6/05/2005 0.0084 2.45 1.79 ? 4374 0.0518 0.0290 999.14 999.14 Jai CV M7+ 20? 26.1 9/05/2005 0.0084 2.45 1.79 ? 4374 0.0517 0.0290 999.30 999.30 Jai CV M7+ 45? 26.2 9/05/2005 0.0085 2.45 1.81 ? 4432 0.0517 0.0286 999.27 999.27 Jai CV M7+ 45? 26.3 9/05/2005 0.0083 2.45 1.77 ? 4331 0.0517 0.0293 999.27 999.27 Jai CV M7+ 45? 27.1 10/05/2005 0.0083 2.45 1.77 ? 4331 0.0517 0.0293 999.34 999.34 Jai CV M7+ 15? 27.2 10/05/2005 0.0084 2.45 1.77 ? 4345 0.0517 0.0292 999.34 999.34 Jai CV M7+ 15? 27.3 10/05/2005 0.0084 2.45 1.79 ? 4374 0.0518 0.0290 999.34 999.34 Jai CV M7+ 15? 28.1 11/05/2005 0.0083 2.45 1.76 ? 4316 0.0517 0.0294 999.30 999.30 Jai CV M7+ 12? 28.2 11/05/2005 0.0084 2.45 1.77 ? 4345 0.0517 0.0292 999.30 999.30 Jai CV M7+ 12? 28.3 11/05/2005 0.0084 2.45 1.79 ? 4374 0.0518 0.0290 999.30 999.30 Jai CV M7+ 12? 29.1 30/08/2005 0.0084 2.45 1.78 ? 4360 0.0521 0.0293 999.39 999.39 Jai CV M7+ 40? 29.2 30/08/2005 0.0083 2.45 1.76 ? 4316 0.0522 0.0296 999.32 999.32 Jai CV M7+ 40? 29.3 30/08/2005 0.0083 2.45 1.76 ? 4316 0.0521 0.0296 999.32 999.32 Jai CV M7+ 40? 30.1 31/08/2005 0.0083 2.45 1.77 ? 4331 0.0521 0.0295 999.40 999.40 Jai CV M7+ 35? 30.2 31/08/2005 0.0083 2.45 1.76 ? 4316 0.0522 0.0296 999.34 999.34 Jai CV M7+ 35? 30.3 31/08/2005 0.0084 2.45 1.78 ? 4360 0.0522 0.0293 999.32 999.32 Jai CV M7+ 35? Appendix D ? Initial Conditions for Experiments with 2D and 3D trajectories 317 Table D.11 - Initial Condition for buoyant jets with 3D trajectories Buoyant Jet Date Q0 d U0 Fr0 Re0 Ua Ur ?a ? Camera Angle Angle View Run No. (l/s) (mm) (m/s) (m/s) (kg/m3) (kg/m3) (Jai CV) (?f030uniF0290) (?f030uniF0290) 1.1 19/09/2005 0.0065 2.43 1.40 64.79 3391 0.0052 0.0374 999.37 1018.83 M7+ 0? 90? y-integrated 1.2 19/09/2005 0.0065 2.43 1.40 64.79 3391 0.0052 0.0374 999.37 1018.83 M7+ CL 0? 90? z-integrated 1.3 19/09/2005 0.0064 2.43 1.38 63.95 3348 0.0052 0.0379 999.30 1018.75 M7+ 0? 90? y-integrated 1.4 19/09/2005 0.0064 2.43 1.38 63.95 3348 0.0052 0.0379 999.30 1018.75 M7+ CL 0? 90? z-integrated 1.5 19/09/2005 0.0063 2.43 1.37 63.40 3319 0.0052 0.0383 999.30 1018.75 M7+ 0? 90? y-integrated 1.6 19/09/2005 0.0063 2.43 1.37 63.40 3319 0.0052 0.0383 999.30 1018.75 M7+ CL 0? 90? z-integrated 2.1 06/10/2005 0.0078 2.43 1.69 77.51 4104 0.0052 0.0309 999.30 1019.20 M7+ 0? 45? y-integrated 2.2 06/10/2005 0.0078 2.43 1.69 77.51 4104 0.0052 0.0309 999.30 1019.20 M7+ CL 0? 45? z-integrated 2.3 06/10/2005 0.0078 2.43 1.68 77.24 4090 0.0052 0.0311 999.29 1019.19 M7+ 0? 45? y-integrated 2.4 06/10/2005 0.0078 2.43 1.68 77.24 4090 0.0052 0.0311 999.29 1019.19 M7+ CL 0? 45? z-integrated 2.5 06/10/2005 0.0078 2.43 1.68 77.24 4090 0.0052 0.0311 999.30 1019.16 M7+ 0? 45? y-integrated 2.6 06/10/2005 0.0078 2.43 1.68 77.24 4090 0.0052 0.0311 999.30 1019.16 M7+ CL 0? 45? z-integrated 3.1 10/10/2005 0.0078 2.43 1.68 76.96 4075 0.0052 0.0311 999.32 1019.23 M7+ 0? 135? y-integrated 3.2 10/10/2005 0.0078 2.43 1.68 76.96 4075 0.0052 0.0311 999.32 1019.23 M7+ CL 0? 135? z-integrated 3.3 10/10/2005 0.0078 2.43 1.68 76.96 4075 0.0052 0.0311 999.31 1019.21 M7+ 0? 135? y-integrated 3.4 10/10/2005 0.0078 2.43 1.68 76.96 4075 0.0052 0.0311 999.31 1019.21 M7+ CL 0? 135? z-integrated 3.5 10/10/2005 0.0078 2.43 1.68 76.96 4075 0.0052 0.0311 999.26 1019.16 M7+ 0? 135? y-integrated 3.6 10/10/2005 0.0078 2.43 1.68 76.96 4075 0.0052 0.0311 999.26 1019.16 M7+ CL 0? 135? z-integrated 4.1 17/10/2005 0.0078 2.43 1.68 76.96 4075 0.0052 0.0311 999.32 1019.23 M7+ 45? 90? y-integrated 4.2 17/10/2005 0.0078 2.43 1.68 76.96 4075 0.0052 0.0311 999.32 1019.23 M7+ CL 45? 90? z-integrated 4.3 17/10/2005 0.0077 2.43 1.67 76.41 4046 0.0052 0.0312 999.31 1019.21 M7+ 45? 90? y-integrated 4.4 17/10/2005 0.0077 2.43 1.67 76.41 4046 0.0052 0.0312 999.31 1019.21 M7+ CL 45? 90? z-integrated 4.5 17/10/2005 0.0078 2.43 1.68 76.96 4075 0.0052 0.0311 999.30 1019.20 M7+ 45? 90? y-integrated 4.6 17/10/2005 0.0078 2.43 1.68 76.96 4075 0.0052 0.0311 999.30 1019.20 M7+ CL 45? 90? z-integrated 5.1 18/10/2005 0.0078 2.43 1.68 76.96 4075 0.0052 0.0310 999.32 1019.23 M7+ 60? 90? y-integrated 5.2 18/10/2005 0.0078 2.43 1.68 76.96 4075 0.0052 0.0310 999.32 1019.23 M7+ CL 60? 90? z-integrated 5.3 18/10/2005 0.0078 2.43 1.69 77.51 4104 0.0052 0.0309 999.32 1019.23 M7+ 60? 90? y-integrated 5.4 18/10/2005 0.0078 2.43 1.69 77.51 4104 0.0052 0.0309 999.32 1019.23 M7+ CL 60? 90? z-integrated 5.5 18/10/2005 0.0078 2.43 1.68 76.96 4075 0.0052 0.0311 999.29 1019.19 M7+ 60? 90? y-integrated 5.6 18/10/2005 0.0078 2.43 1.68 76.96 4075 0.0052 0.0311 999.29 1019.19 M7+ CL 60? 90? z-integrated Appendix D ? Initial Conditions for Experiments with 2D and 3D trajectories 318 Appendix E ? Additional Figures 319 Appendix E ? Additional Figures 0 10 20 30 40 50 60 70 80 90 1000 20 40 60 80 100 120 140 160 x m /d Fr0 LA data LIF data Cipollina et al (2005) Analytical Solution VisJet CorJet E.1(a) - Initial angle of 30? and Reynolds numbers ranging from 2406 to 5207 0 10 20 30 40 50 60 70 80 90 1000 20 40 60 80 100 120 140 160 180 x m /d Fr0 LA data LIF data Cipollina et al (2005) Analytical Solution VisJet CorJet E.1(b) - Initial angle of 60? and Reynolds numbers ranging from 2997 to 5032 Figure E.1 ? Horizontal location of maximum centreline height for flows with initial angles of 30? and 60? Appendix E ? Additional Figures 320 0 10 20 30 40 50 60 70 80 90 1000 10 20 30 40 50 60 70 80 z m /d Fr0 LA data LIF data Cipollina et al (2005) Analytical Solution VisJet CorJet E.2(a) - Initial angle of 30? and Reynolds numbers ranging from 2406 to 5207 0 10 20 30 40 50 60 70 80 90 1000 20 40 60 80 100 120 140 160 180 z m /d Fr0 LA data LIF data Cipollina et al (2005) Analytical Solution VisJet CorJet E.2(b) - Initial angle of 60? and Reynolds numbers ranging from 2997 to 5032 Figure E.2 - Vertical location of maximum centreline height for discharges with initial angles of 30? and 60? Appendix E ? Additional Figures 321 0 10 20 30 40 50 60 70 80 90 1000 20 40 60 80 100 120 z m e/ d Fr0 LA data LIF data Zeitoun et al (1972) Cipollina et al (2005) Analytical Solution VisJet CorJet E.3(a) - Initial angle of 30? and Reynolds numbers ranging from 2406 to 5207 0 10 20 30 40 50 60 70 80 90 1000 50 100 150 200 z m e/ d Fr0 LA data LIF data Zeitoun et al (1972) Cipollina et al (2005) Analytical Solution VisJet CorJet E.3(b) - Initial angle of 60? and Reynolds numbers ranging from 2997 to 5032 Figure E.3 - Maximum height of edge of jet for discharges with initial angles of 30? and 60? Appendix E ? Additional Figures 322 0 10 20 30 40 50 60 70 80 90 1000 50 100 150 200 250 300 x 0 R/ d Fr0 LA data LIF data Zeitoun et al (1972) Cipollina et al (2005) Analytical Solution E.4(a) - Initial angle of 30? and Reynolds numbers ranging from 2406 to 5207 0 10 20 30 40 50 60 70 80 90 1000 50 100 150 200 250 300 x 0 R/ d Fr0 LA data Zeitoun et al (1972) Cipollina et al (2005) Analytical Solution E.4(b) - Initial angle of 60? and Reynolds numbers ranging from 2997 to 5032 Figure E.4 - Horizontal location of impact point for flows with initial angles of 30? and 60? Appendix E ? Additional Figures 323 0 20 40 60 80 100 120 140 160 010 20 0 5 10 15 x/(Q0?0/Ua3) y/(Q0?0/Ua3) z/ (Q 0???? 0/ U a3 ) Figure E.5 - Three-dimensional view of trajectory results buoyant jet with three-dimensional trajectories run 2 0 20 40 60 80 100 0 10 20 0 10 20 x/(Q0?0/Ua3) y/(Q0?0/Ua3) z/ (Q 0???? 0/ U a3 ) Figure E.6 - Three-dimensional view of trajectory results buoyant jet with three-dimensional trajectories run 3 Appendix E ? Additional Figures 324 0 20 40 60 80 100 120 140 160 180 200 0102030 -10 -5 0 5 10 15 20 y/(Q0?0/Ua3) x/(Q0?0/Ua3) z/ (Q 0???? 0/ U a3 ) Figure E.7 - Three-dimensional view of trajectory results buoyant jet with three-dimensional trajectories run 4 0 20 40 60 80 100 010 20 -10 -5 0 5 y/(Q0?0/Ua3) x/(Q0?0/Ua3) z/ (Q 0???? 0/ U a3 ) Figure E.8 - Three-dimensional view of trajectory results buoyant jet with three-dimensional trajectories run 5 Appendix F ? Trajectory Solutions for Weak-Jet and Puff Regions 325 Appendix F ? Trajectory Solutions for Weak-Jet and Puff Regions Weak-Jet Region In the weak-jet region, the flow is dominated by the component of the excess momentum flux that is parallel to the ambient current and the entrained ambient momentum flux. The component of momentum flux perpendicular to the ambient current will deflect the discharge and thereby alter its trajectory. The relationship for this is given by: 0 0 0 0 0 0 sin cos wj e wj a e dz U Q dx U Q U Q ? ?= + (F.1) where 0 0 0 2 2 2 e e a a wj U Q M U Q U b? pi= (F.2) Application of the equations for conservation of mass and momentum, combined with the spread assumption, yields the following relationship for the spread in the weak-jet region (Wang 2000b): 1 3 1/3 0 0.5 0.5 0 0 3 cos wj e a q jk e a xkb M U I C M U ?? ? ? ?= ? ? ? ? ? ? ? ?? ? (F.3) Inserting equation (F.3) into (F.2) gives: 0 0 2 3 2 3 2 0 1 2 0 1 3 cos e a wj wj q jk e a U Q U Q xk I C M U ?pi? = ? ? ? ? ? ? ? ?? ? ? ?? ? (F.4) Equation (F.1) can be re-written as: 0 0 0 0 0 0 sin cos1 e wj a ewj a U Q dz U Q U Qdx U Q ? ?= + (F.5) Inserting equation (F.4) into (F.5) gives: 0 2 3 2 3 2 20 1 2 2 3 0 0 1 2 3 2 3 2 3 2 0 1 2 0 sin 3 cos cos1 1 3 cos wj wj wj q jk e awj wj wjwj wjwj wj q jk e a x ck I C M Udz x cdx xxk I C M U ? ?pi? ? ?pi? ? ? ? ? ? ? ? ?? ? ? ?? ?= = + + ? ? ? ? ? ? ? ?? ? ? ?? ? (F.6) Therefore: Appendix F ? Trajectory Solutions for Weak-Jet and Puff Regions 326 2 2 3 2 2 3 1 1 2 31 wj wj wj wj wj wj wj wj wj wj c x cdz dx dx c x c x = = + + (F.7) Let 1 3wjw x= => 23dx w dw= . Thus: 2 2 12 2 22 2 2 1 1 1 13 3 3 1 1 1 wj wj wj wj wj wj wj wj c w cdz w dw c dw c w c w c w c ? ? ? ?= = = ? ? ? ? ?? ? ? ?+ + + ? ? ? ? (F.8) 2 1 2 1 13 1 wj wj wj wj dz c c dwc w? ?= ?? ?? ?+ ? ? (F.9) Integrating equation (F.9), and applying the virtual source assumption gives: 1 2 1 1 3 tanwj wj wj wj wz c w c c ? ? ?? ? ? ?? ?= ? ? ? ? ?? ?? ? (F.10) 1 3 1 3 1 2 1 1 3 tan wjwj wj wj wj wj xz c x c c ? ? ?? ? ? ?? ?= ? ? ? ? ?? ?? ? (F.11) Inserting cwj1 and cwj2 into equation (F.10) gives the analytical trajectory solution for the weak-jet region: 1 32 3 1 3 1 31 2 1 2 2 32 3 0 010 0 2 3 2 3 1 2 * 0 * 0 * tan sin3sin 3 costan 3 cos cos wj q jk wj q jk wj wj mz wj mz wj q jk mz z I C x I C xk l k l k I C l ? pi ? ?? ? pi? ? pi ? ? ? ? ?? ?? ? ? ? ? ? ? ?= ? ? ?? ? ? ? ? ?? ? ? ?? ? ? ? ? ?? ?? ? (F.12) where ( )0.50 0sine mz a ML U ? ? = (F.13) Puff Region In the puff region, the flow travels in the z-direction due to the initial excess momentum flux in that direction; mean motion in the x-direction is a combined effect of the entrained ambient momentum flux and the appropriate component of the initial excess momentum flux. The relationship for this is given by: 0 0 0 0 0 0 0 0 0 0 0 0 sin sin 1 cos 1 cos amp e e amp a e a e a dz U Q U Q dx U Q U Q U Q U Q U Q ? ? ? ? ? ?= = ? ?+ +? ? (F.14) where Appendix F ? Trajectory Solutions for Weak-Jet and Puff Regions 327 0 0 0 2 2 2 e e a a amp U Q M U Q U b? pi= (F.15) The spread assumption, based on the double-Gaussian approximation (see 3.2.3.2) yields the following relationship for the spread in the puff region: 0.5 0.5 0 0 0 0sin sin ampc sg e a e a zb hk M U M U?? ?= (F.16) Inserting equation (F.16) into (F.15) gives: 0 0 2 2 2 1 2 0 1e a amp sg e a U Q U Q z k M Upi? = ? ? ? ?? ? (F.17) Inserting equation (F.17) into (F.14) gives: 0 2 2 1 2 0 02 2 2 21 2 0 21 2 0 0 02 2 22 2 1 2 0 sin sin cos cos1 1 amp e a amp sg amp sg ampamp e a amp e a amp amp sg amp amp sg e a z M Uk k zdz M U dx M U z k zk M U ? ?pi? pi? ? ? pi?pi? ? ?? ? ? ?? ? ? ? ? ? ? ?= = ? ?+ + ? ?? ? ? ? ? ?? ? ? ? (F.18) Therefore: 21 2 0 0 22 2 2 2 2 1 21 2 0 0 00 0 2 2 cos1 1 tan sinsin e a amp sg amp amp sg amp amp amp amp e ae a amp sg amp M U k z k zdx dz dz M UM U k z ? pi? pi? ? ?? pi? ? ?+ ? ?? ? ? ? ? ?= = + ? ? ? ? ? ? ? ?? ? ? ? (F.19) Integrating equation (F.19), and applying the virtual source assumption gives: 22 2 3 1 2 0 0 0 1 1 tan 3sin amp sg amp amp amp e a kx z z M U pi? ? ? ? ?= + ? ?? ? (F.20) or 32 2 * 0 * * 1 tan 3 amp amp amp sg amp mz mz mz x z k z l l l pi? ? ? ?= + ? ?? ? (F.21)