Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

An experimental investigation for resistance reduction on displacement-type ships by parabolization of… Tan, Beng-Yeow 2004

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_2004-0714.pdf [ 25.02MB ]
Metadata
JSON: 831-1.0080733.json
JSON-LD: 831-1.0080733-ld.json
RDF/XML (Pretty): 831-1.0080733-rdf.xml
RDF/JSON: 831-1.0080733-rdf.json
Turtle: 831-1.0080733-turtle.txt
N-Triples: 831-1.0080733-rdf-ntriples.txt
Original Record: 831-1.0080733-source.json
Full Text
831-1.0080733-fulltext.txt
Citation
831-1.0080733.ris

Full Text

AN EXPERIMENTAL INVESTIGATION FOR RESISTANCE REDUCTION ON DISPLACEMENT-TYPE SHIPS BY PARABOLIZATION OF HULL FORM AT WATER LINE by Beng-Yeow Jeff TAN B.Eng., Memorial University of Newfoundland, 2002 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 2004 © Beng-Yeow Jeff TAN, 2004 Abstract Hull optimization for minimum total resistance exhibits great interests among ship hydrodynamic research community for the attractive benefits it brings along in terms of fuel consumption, power requirement, payload capacity, cruising speed, traveling range, endurance and operational cost effectiveness. Even minute improvement can translate into significant savings throughout the design life of the vessel. Ship total hydrodynamic resistance mainly consists of frictional resistance, form resistance and wave resistance. Considerable improvement in frictional resistance is less likely to be attained through hull form modification. With form resistance being a fraction of frictional resistance and that wave resistance dominates as speed is increased, it becomes clear to turn toward minimizing wave resistance for hull form optimization studies. Hull parabolization was originally tested successfully on a tanker. The present investigation was initiated as a feasibility study to expand the idea of hull form parabolization for wave resistance reduction on to a small craft. The objective is to attain beneficial wave resistance reduction over moderate to relatively higher operating speed range between 0.30 < FnL < 0 .40. The parabolization was done by continuously extending the hull form at waterline and replacing the conventional parallel middle body section with parabolic side bulbs that function as 3-D wave maker. The bulbs modify the pressure field in the vicinity and generate stronger shoulder wave system to interact with bow and stern wave systems. Beneficial reduction in wave resistance, thus total resistance or effective horsepower, is practically attainable through the wave interactions upon cautious hull modifications. The side bulbs also increase payload capacity and improve ship stability. ii A parent hull was selected and modified with increased beam gradually up to 2 0 % using various add-on side bulbs. The model was built at a scale of 13.75:1. Systematical experimental investigations were conducted in towing tank to understand the effects of beam increment, influences of longitudinal location of maximum beam and fairing extensions on hull form factor as well as resistance characteristics. Wave resistances were calculated upon experimental determination of form factors using Hughes-Prohaska's method. Additionally, direct wave resistance values were calculated by applying Sharma's longitudinal wave cut method on a rectangular patch of wave pattern. The wave elevations in the patch were acquired during experiment by a laser-camera wave pattern profiling system. Limitations of the system were addressed. The direct wave resistance values calculated from experiments were used to support the theoretical wave resistance predicted using Michell's integral from thin ship theory. Qualitative and quantitative comparisons between experimental and theoretical results were made. Upon studying using the add-on bulbs, an optimized hull form with built-in parabolization is conceived and built. The resistance-reduction capability of this recommended parabolized hull was substantiated by experiments. Over the targeted speed range, the model scale results have shown the consistent 2 5 % reduction in wave resistance leads to the encouraging 1 5 % reduction in total resistance (RTM) and 3 8 % improvement in stability [GMM). iii Table Of Contents Abstract ii Table Of Contents iv List of Tables viii List of Figures xii List of Figures xii List of Symbols, Nomenclatures and Abbreviations xix Acknowledgements xxiii C H A P T E R 1 Introduction 1 1.1 Previous Work 4 1.1.1 Wave Resistance Reduction by Hull Form Parabolization 4 1.1.2 Theoretical Wave Resistance by MichelPs Integral 6 1.1.3 Direct Experimental Wave Resistance by Wave Cuts 10 Transverse Wave Cut Analysis 11 Longitudinal Wave Cut Analysis 12 X-Y Wave Cut Analysis 14 1.2 Current Work 15 1.2.1 Motivation 15 1.2.2 Scope. . . . 16 1.2.3 Subject of Studies 17 C H A P T E R 2 Feasibility Studies by Theoretical Wave Resistance (2-D body)20 C H A P T E R 3 Experimental Work 23 3.1 Testing Facility 23 3.2 Experimental Approach 25 3.3 Test Programs 28 3.3.1 Calmwater Resistance Towing Tests 28 [PI] Effects of Beam Increment Using Single Bulb 30 [P2] Repeatability Tests on Selected Single Bulb 30 [P3] Effects of Maximum Beam Location Using Single Bulb 31 iv [P4] Effects of Fairing Extension at Rear of Single Bulb 31 3.3.2 Underwater Flow Visualization Tests 32 3.3.3 Wave Pattern Profiling Tests 33 C H A P T E R 4 Numerical Work 38 4.1 Theoretical Wave Resistance by Michell's Integral (3-D body) 38 4.2 Experimental Wave Resistance by Longitudinal Wave Cuts 39 4.2.1 Camera Calibration Target 39 4.2.2 Wave Cut Coordinate System 40 4.2.3 Wave Cut Reference & Truncation Points 41 C H A P T E R 5 Results & Discussions 44 5.1 Experimental Work 44 5.1.1 [P1] Effects of Beam Increment Using Single Bulb 45 [P1] Summary 61 5.1.2 [P2] Repeatability Analysis 61 [P2] Summary 64 5.1.3 [P3] Effects of Maximum Beam Location Using Single Bulb 65 [P3] Summary 74 5.1.4 [P4] Effects of Fairing Extension at Rear of Single Bulb 75 [P4] Summary 84 5.1.5 Summary of Influences on Hull Form Factors & Hydrostatics .. 84 5.1.6 Recommended New Hull Form 89 5.2 Numerical Work 99 5.2.1 Comparison of Theoretical & Experimental Wave Resistance . 99 5.2.2 Wave Pattern profiling System & Wave Cut 102 Mathematical Explanation of Wave Resistance-Reduction Capability of Hull Parabolization 117 Effect of Truncation of Wave Number on Wave Resistance... 118 Effect of Location of Wave Cut on Wave Resistance 122 Effect of Truncation Correction & Limitations of Wave Cut Method 127 C H A P T E R 6 Conclusions & Future Improvements 129 v 6.1 Calmwater Model Testing... , 132 6.2 Wave Pattern Profiling Testing 133 6.3 Theoretical Wave Resistance Using Michell's Integral 134 6.4 Recommendations & Future Work 135 References 138 APPENDIX A Ship Resistances & Extrapolation Methods 143 A.1 Breakdown of Total Resistance 143 A.2 Wave Interactions 144 APPENDIX B General Pictorial Nomenclature of Ship Form 147 APPENDIX C Analysis Procedures 148 C.1 ITTC 1957 Method 148 C. 2 Hughes' Method 151 APPENDIX D Model-Ship Conversion Tables 154 D. 1 Model-Ship Scaling Table 154 D.2 Physical Conditions Table 155 APPENDIX E Offset Tables 156 APPENDIX F Records of Testing Conditions 167 F.1 Model Testing Records 167 F.2 Wave Pattern Profiling System Records 178 APPENDIX G Hydrostatics at Design Waterline 180 APPENDIX H Hull Form Factors 183 H.1 Experimental Based - Hughes-Prohaska's Method 183 H. 2 Statistical Based - Empirical Formulae (PNA, Wright, Granville) 194 APPENDIX I Repeatability Analysis of Model Scale Total Resistance 196 I. 1 Parent Hull (B0p_single) 196 1.2 Parent Hull with 1 5 % Increased beam (B15p_single) 198 1.3 Recommended Hull with 1 1 % Increased beam (B11p_single) 200 APPENDIX J Matlab Script Files & Execution Manual 202 J.1 S T E P 1: Spatial Transformation Matrix (transform.M) 202 J.2 S T E P 2: Spatial Transformation & Noise Filtering 204 J.2.1 S T E P 2a: Spatial Transformation (spatial.M) 204 vi J.2.2 S T E P 2b: Data smoothing & fitting subroutine (splinefit.M).... 209 J.3 S T E P 3: Processed Wave Patch & Wave Cuts Analysis 214 J.3.1 S T E P 3a: Processed Wave Patch (Z_patch.M) 214 J.3.2 S T E P 3b: Multiple Cut Analysis (sharma_multiple_cut.M) 219 J.3.3 S T E P 3c: Fresnel's Integrals (fresnelCS.M) 231 J.3.4 S T E P 3d: Plottings (reproduce_plottings_T25.M) 233 vii List of Tables Table 1-1: General particulars of parent hull (UBC series-model #3) (scale=1:13.75) 17 Table 3-1: Multi-phase test programs for experimental work 29 Table 5-1: [P1] Preliminary tests using various single midship bulb configurations. 45 Table 5-2: [P1] Resistance summary of the effects of increased beam using single midship bulb (scale=1:13.75, averaged within design speed regime of FnL=0.35+/-0.02) 60 Table 5-3: [P1] Hydrostatics summary of the effects of increased beam using single midship bulb (scale=1:13.75) 60 Table 5-4: [P2] Repeatability tests using selected single-bulb configuration 62 Table 5-5: [P2] Data sets used for repeatability analysis (n=3sets) 63 Table 5-6: [P3] Tests on effects of maximum beam location (forebody & aftbody) 65 Table 5-7: [P3] Hydrostatics summary of the effects of increased beam at model scale (1:13.75) 74 Table 5-8: [P4] Tests on effects of fairing extension at rear of single-midship bulb configuration 75 Table 5-9: [P4] Hydrostatics summary of the effects of increased beam at model scale (1:13.75) 84 Table 5-10: Comparison of hydrostatics between the understudied parabolized hull and recommended hull at model scale (1:13.75) 90 Table D-1: Model-Ship Scaling Table 154 Table D-2: Table of Kinematic Viscosity (source: PNA vol. II [22]) 155 Table D-3: Table of Density (source: PNA vol. II [22]) 155 viii Table E-1: [P1&2] Offset tables of parent hull [BOp_single]. 156 Table E-2: [P1] Offset tables of parent hull using single bulb at midship of 5 % increased beam [B5p_single_midship] 157 Table E-3: [P1] Offset tables of parent hull using single bulb at midship of 10% increased beam [B10p_single_midship] 158 Table E-4: [P1 & 2] Offset tables of parent hull using single bulb at midship of 1 5 % increased beam [B15p_single_midship] 159 Table E-5: [P1] Offset tables of parent hull using single bulb at midship of 2 0 % increased beam [B20p_single_midship] 160 Table E-6: [P3] Offset tables of parent hull using single bulb at forebody (F) of 1 5 % increased beam [B15p_single_forebody] 161 Table E-7: [P3] Offset tables of parent hull using single bulb at aftbody (A) of 1 5 % increased beam [B15p_single_aftbody] 162 Table E-8: [P4] Offset tables of parent hull using single bulb at midship of 1 5 % increased beam with 2 5 % LWL fairing extension [B15p_single_midship_alpha1] 163 Table E-9: [P4] Offset tables of parent hull using single bulb at midship of 1 5 % increased beam with 3 0 % LWL fairing extension [B15p_single_midship_alpha2] 164 Table E-10: [P4] Offset tables of parent hull using single bulb at midship of 1 5 % increased beam with 3 5 % LWL fairing extension [B15p_single_midship_alpha3] 165 Table E-11: [Recommended] Offset tables of recommended UBC series model #3 [B11p_single_midship_recommended] 166 Table F-1: [P1 & 2] Model testing condition records for parent hull [B0p_single] 167 Table F-2: [P1] Model testing condition records for parent hull using single bulb at midship of 5 % increased beam [B5p_single_midship] 168 Table F-3: [P1] Model testing condition records for parent hull using single bulb at midship of 1 0 % increased beam [B10p_single_midship] 169 ix Table F-4: [P1 & 2] Model testing condition records for parent hull using single bulb at midship of 1 5 % increased beam [B15p_single_midship].. 170 Table F-5: [P1] Model testing condition records for parent hull using single bulb at midship of 2 0 % increased beam [B20p_single_midship] 171 Table F-6: [P3] Model testing condition records for parent hull using single bulb at forebody of. 1 5 % increased beam [B15p_single_forebody] 172 Table F-7: [P3] Model testing condition records for parent hull using single bulb at aftbody of 1 5 % increased beam [B15p_single_aftbody] 173 Table F-8: [P4] Model testing condition records for parent hull using single bulb at midship of 1 5 % increased beam and 2 5 % extended fairing length at rear of bulb [B15p_single_midship_alpha1] 174 Table F-9: [P4] Model testing condition records for parent hull using single bulb at midship of 1 5 % increased beam and 3 0 % extended fairing length at rear of bulb [B15p_single_midship_alpha2] . 175 Table F-10: [P4] Model testing condition records for parent hull using single bulb at midship of 1 5 % increased beam and 3 5 % extended fairing length at rear of bulb [B15p_single_midship_alpha3] 176 Table F-11:' [Recommended] Model testing condition records for the revised UBC series model #3 [B11p_single_midship_recommended] 177 Table G-1 : [P1 & 2] Hydrostatics at design waterline for parent hull using single bulb at midship of 0 - 2 0 % increased beam 180 Table G-2: [P3] Hydrostatics at design waterline for parent hull using single bulb at forebody (F) / aftbody (A) of 1 5 % increased beam 181 Table G-3: [P4] Hydrostatics at design waterline for parent hull using single bulb at midship of 1 5 % increased beam with 2 5 - 3 5 % extended fairing length'at rear of bulb 182 Table H-1: [P1 & 2] Hughes-Prohaska's form factor for parent hull [B0p_single] 183 x Table H-2: [P1] Hughes-Prohaska's form factor for parent hull using single bulb at midship of 5 % increased beam [B5p_single_midship] 184 Table H-3: [P1] Hughes-Prohaska's form factor for parent hull using single bulb at midship of 10% increased beam [B10p_single_midship] 185 Table H-4: [P1 & 2] Hughes-Prohaska's form factor for parent hull using single bulb at midship of 1 5 % increased beam [B15p_single_midship].. 186 Table H-5: [P1] Hughes-Prohaska's form factor for parent hull using single bulb at midship of 2 0 % increased beam [B20p_single_midship] 187 Table H-6: [P3] Hughes-Prohaska's form factor for parent hull using single bulb at forebody of 1 5 % increased beam B15p_single_forebody] 188 Table H-7: [P3] Hughes-Prohaska's form factor for parent hull using single bulb at aftbody of 15% increased beam [B15p_single_aftbody] 189 Table H-8: [P4] Hughes-Prohaska's form factor for parent hull using single bulb at midship of 1 5 % increased beam and 2 5 % extended fairing length at rear of bulb [B15p_single_midship_alpha1] 190 Table H-9: [P4] Hughes-Prohaska's form factor for parent hull using single bulb at midship of 1 5 % increased beam and 3 0 % extended fairing length at rear of bulb [B15p_single_midship_alpha2] 191 Table H-10: [P4] Hughes-Prohaska's form factor for parent hull using single bulb at midship of 15% increased beam and 3 5 % extended fairing length at rear of bulb [B15p_single_midship_alpha3] 192 Table H-11: [Recommended] Hughes-Prohaska's form factor for the revised UBC series model #3 [B11p_single_midship_recommended] 193 Table 1-1: [P2] Repeatability analysis of model scale total resistance on parent hull: RTM , % R T M 1 9 6 Table I-2: [P2] Repeatability analysis of model scale total resistance on parent hull with single bulb at midship of 1 5 % increased beam: R T M , % R T M 198 Table I-3: [Recommended] Repeatability analysis of model scale total resistance on recommended hull with 1 1 % increased beam: R T M , % R T M 200 xi List of Figures Figure 1-1: Multi-directional wave generator in wave basin (Tan [36]) 3 Figure 1-2: Oscillations on resistance curves due to wave interactions 7 Figure 1-3: Eggers' transverse wave cut and Sharma's longitudinal wave cut... 12 Figure 1-4: Longitudinal wave cut (Moran & Landweber 1972, Tsai & Landweber 1975) 13 Figure 1-5: X -Y method wave cut (Ward 1963, 1966) 14 Figure 1-6: General lines plan for parent hull control baseline - UBC series model #3 19 Figure 2 -1 : Feasibility studies of increased beam from theoretical wave resistance using MichelPs integral at model scale 2-D body (1:13.75): RWM, % R W M • 21 Figure 3 -1 : Towing tank at.Ocean Engineering Centre of Vizon SciTec 23 Figure 3-2: Plan view & specifications of testing facility at Ocean Engineering Centre - B R R Inc 24 Figure 3-3: Front view of the various add-on side bulbs 25 Figure 3-4: Various bulbous bow configurations for reduction of wave resistance. 26 Figure 3-5: Schematic illustration of experimental approach to study various effects of side bulb configurations 27 Figure 3-6: Typical underwater flow visualization test with attached tufts (parent hull at design speed, Fn=0.35) 32 Figure 3-7: Wave pattern profiling system calibration setup 33 Figure 3-8: Camera-captured reflected laser line on seeding... 36 Figure 3-9: Styrofoam seeded free surface patch 37 xii Figure 4-1: Percentage change in effects of increased beam on model scale resistance coefficients: % C W M 38 Figure 4-2: Camera calibration target 39 Figure 4-3: Coordinate system used for wave cut analysis 40 Figure 4-4: Longitudinal wave cut truncation point 42 Figure 5-1: [P1] Profiles of tested parabolized hull at design waterline level 46 Figure 5-2: [P1] Effects of increased beam on measured total resistance at model scale (1:13.75): R T M , % R T M 47 Figure 5-3: [P1] Effects of increased beam on resistance coefficients at model scale (1:13.75): C T M , CFOM, C r m •• 50 Figure 5-4: [P1] Percentage change in effects of increased beam on resistance coefficients at model scale (1:13.75): % C T M , % C F O M , % C R M 51 Figure 5-5: [P1] Effects of increased beam on hull form factors at model scale (1:13.75): (1+k), %(1+k) 54 Figure 5-6: [P1] Effects of increased beam on resistance coefficients at model scale (1:13.75): C V M , CWM 56 Figure 5-7: [P1] Percentage change in effects of increased beam on resistance coefficients at model scale (1:13.75): % C V M , % C W M — 57 Figure 5-8: [P1] Percentage change in effects of increased beam on effective horsepower at full scale (1:13.75): E H P S , 3 D , % E H P S , 3 D 59 Figure 5-9: [P3] Profiles of tested parabolized hull at design waterline level 66 Figure 5-10: [P3] Effects of maximum beam location on measured total resistance at model scale (1:13.75): RTM, % R T M 67 Figure 5-11: [P3] Effects of maximum beam location on resistance coefficients at model scale (1:13.75): CTM, CFOM, C r m 68 Figure 5-12: [P3] Percentage change in effects of maximum beam location on resistance coefficients at model scale (1:13.75): % C T M , % C F O M , % C r m 69 Figure 5-13: [P3] Effects of increased beam on hull form factors at model scale (1:13.75): (1+k), %(1+k) 70 xiii Figure 5-14: [P3] Effects of maximum beam location on resistance coefficients at model scale (1:13.75): CVMI CWM 71 Figure 5-15: [P3] Percentage change in effects of maximum beam location on resistance coefficients at model scale (1:13.75): % C V M , % C W M 72 Figure 5-16: [P3] Percentage change in effects of increased beam on effective horsepower at full scale (1:13.75): E H P S , 3 D , % E H P S , 3 D 73 Figure 5-17: [P4] Profiles of tested parabolized hull at design waterline level.... 76 Figure 5-18: [P4] Effects of increased beam on hull form factors at model scale (1:13.75): (1+k), %(1+k) 77 Figure 5-19: [P4] Effects of increased beam on measured total resistance at model scale (1:13.75): R T M , % R T M 78 Figure 5-20: [P4] Effects of increased beam on resistance coefficients at model scale (1:13.75): C T M , C F O M , C r m '. 79 Figure 5-21: [P4] Percentage change in effects of increased beam on resistance coefficients at model scale (1:13.75): % C T M , % C F O M , % C R M 80 Figure 5-22: [P4] Effects of increased beam on resistance coefficients at model scale (1:13.75): CVM, C W M • • 81 Figure 5-23: [P4] Percentage change in effects of increased beam on resistance coefficients at model scale (1:13.75): % C V M , % C W M 82 Figure 5-24: [P4] Percentage change in effects of increased beam on effective horsepower at full scale (1:13.75): E H P S , 3 D , % E H P S , 3 D 83 Figure 5-25: Effect of increased beam and maximum beam location on hull form factor by Hughes-Prohaska's method at model scale (1:13.75): (1+k), %(1+k) 87 Figure 5-26: Summary on effects of increased beam on hydrostatics performance at model scale (1:13.75) 88 Figure 5-27: Profiles of tested parabolized hull and recommended hull at design waterline level 91 Figure 5-28: 3-D' rendered perspective view of (a) the parent hull and (b) the recommended hull 91 xiv Figure 5-29: Comparison of lines plans for the original & revised optimized hull. 92 Figure 5-30: [Recommended] Effects of increased beam on measured total resistance at model scale (1:13.75): RTM, % R T M . 93 Figure 5-31: [Recommended] Effects of increased beam on resistance coefficients at model scale (1:13.75): CTM, CFOM, CRM ....... . . .94 Figure 5-32: [Recommended] Percentage change in effects of increased beam on resistance coefficients at model scale (1:13.75): % C T M , % C F O M , % C r m 95 Figure 5-33: [Recommended] Effects of increased beam on resistance coefficients at model scale (1:13.75): CVM, CWM 96 Figure 5-34: [Recommended] Percentage change in effects of increased beam on resistance coefficients at model scale (1:13.75): % C V M , % C W M - 97 Figure 5-35: [Recommended] Percentage change in effects of increased beam on effective horsepower at full scale (1:13.75): E H P S , 3 D , % E H P s , 3 D 9 8 Figure 5-36: Comparison of percentage change in model scale wave resistance between theoretical predictions and experimental results at model scale (1:13.75): % C W M • 100 Figure 5-37: Comparison of percentage change in model scale theoretical wave resistance between the understudied & recommended configuration at model scale (1:13.75): % C W M , theoretical • 101 Figure 5-38: Top view of the acquired wave patch (Fn=0.30, 512 cuts) 102 Figure 5-39: Top view of the acquired wave patch (Fn=0.31, 512 cuts) 103 Figure 5-40: Top view of the acquired wave patch (Fn=0.32, 512 cuts)... 103 Figure 5-41: Top view of the acquired wave patch (Fn=0.33, 512 cuts) 104 Figure 5-42: Top view of the acquired wave patch (Fn=0.34, 512 cuts) 104 Figure 5-43: Top view of the acquired wave patch (Fn=0.35, 512 cuts) 105 Figure 5-44: Wave elevation along a longitudinal wave cut at fixed lateral positions (Fn=0.30) 106 Figure 5-45: Wave elevation along a longitudinal wave cut at fixed lateral positions (Fn=0.31) 106 xv Figure 5-46: Wave elevation along a longitudinal wave cut at fixed lateral positions (Fn=0.32) 107 Figure 5-47: Wave elevation along a longitudinal wave cut at fixed lateral positions (Fn=o!33) 107 Figure 5-48: Wave elevation along a longitudinal wave cut at fixed lateral positions (Fn=0.34) 108 Figure 5-49: Wave elevation along a longitudinal wave cut at fixed lateral positions (Fn=0.35) 108 Figure 5-50: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.30) 114 Figure 5-51: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.31) 114 Figure 5-52: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.32) 115 Figure 5-53: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.33) 115 Figure 5-54: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.34) 116 Figure 5-55: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.35) 116 Figure 5-56: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.30) 119 Figure 5-57: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.31) 120 Figure 5-58: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.32) 120 Figure 5-59: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.33) 121 Figure 5-60: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.34) 121 xvi Figure 5-61: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.35, 512 cuts) 122 Figure 5-62: Illustration of variation in wave resistance from longitudinal wave cut at different lateral positions at Fn=0.2236.... . 123 Figure 5-63: Variation in wave resistance from longitudinal wave cut at different lateral positions (Fn=0.30, 512 cuts) 124 Figure 5-64: Variation in wave resistance from longitudinal wave cut at different lateral positions (Fn=0.31,512 cuts) 124 Figure 5-65: Variation in wave resistance from longitudinal wave cut at different lateral positions (Fn=0.32, 512 cuts) 125 Figure 5-66: Variation in wave resistance from longitudinal wave cut at different lateral positions (Fn=0.33, 512 cuts) 125 Figure 5-67: Variation in wave resistance from longitudinal wave cut at different lateral positions (Fn=0.34, 512 cuts) 126 Figure 5-68: Variation in wave resistance from longitudinal wave cut at different lateral positions (Fn=0.35, 512 cuts) 126 Figure 5-69: Comparisons of wave resistances of parent hull calculated from empirical formulae & longitudinal cut 128 Figure A - 1 : Breakdown of Total Resistance (source: Couser et. al [6]) 143 Figure A -2 : Performance of specific resistance components with speed (source: Harvald [13]) 144 Figure A -3 : Interfering Wave Systems (source: Harvald [13]) 145 Figure A-4: Typical resistance exhibiting "humps" and "hollows" (source: Harvald [13]) 146 Figure B-1: Schematic illustration of basic naval architectural terminology (source: Zubaly [48]) 147 Figure C -1 : Standard skin friction lines (source: PNA vol. II [22]) 149 xvii Figure C-2: Schematic representation of ITTC 1957 method (source: PNA vol. II [22]) 151 Figure C-3: Schematic representation of Hughes's method (source: PNA vol. II [22]) : : 153 Figure 1-1: [P2] Repeatability analysis of model scale total resistance on parent hull: R T M , % R T M 197 Figure I-2: [P2] Repeatability analysis of model scale total resistance on parent hull with single bulb at midship of 15% increased beam: R T M , % R T M 199 Figure I-3: [Recommended] Repeatability analysis of model scale total resistance on recommended hull with 1 1 % increased beam: R T M , % R T M 201 xviii List of Symbols, Nomenclatures and Abbreviations A (orW) V (or V) 9 A 7 P C (or TJ, z, Z ) ATTC b B BCRI (or BCR Inc.) C C D CA CB CF CP CR Displacement Displaced volume Wave propagating or radiating direction Wave length Integration variable used in MichelPs integral Water density Wave heights American Towing Tank Conference Overall width of towing tank Beam British Columbia Research Incorporation (currently operating as Vizon SciTec Inc.) Couple-Charged Devices (used in camera) Model-ship correlation allowance (applicable only to full scale, typically 0.0004) Block coefficient Frictional resistance coefficient (subscript'm' denotes model scale, and 's ' denotes full scale) Prismatic coefficient Residuary resistance coefficient (subscript'm' denotes model scale, and's ' denotes full scale) Total resistance coefficient (subscript'm' denotes model scale, and's ' denotes full scale) xix Cv Viscous resistance coefficient (subscript'm' denotes model scale, and's ' denotes full scale) Cw Wave resistance coefficient (subscript'm' denotes model scale, and's ' denotes full scale) EHP Effective horsepower (applicable only to full scale with subscript's') f(x,y) Half breadth or offsets of hull fx(x,y) Slope of hull Fnh (or Fnh) Froude number (depth based) FnL (or FnL) Froude number (length based) FW (or fw) Fresh water 9 Gravitational acceleration GM Metacentric height h Water depth / Second moment of area ITTC International Towing Tank Conference ITU Istanbul Technical University k0 Wave number LCB Longitudinal center of buoyancy LCF Longitudinal center of floatation LCG Longitudinal center of gravity L O A Length overall L p p Length between perpendiculars Length of load waterline NALAB Offshore Engineering & Naval Architecture Laboratory at University of British Columbia o Origin of coordinates xx OEC Ocean Engineering Centre at Vizon SciTec Inc. PNA Principles of Naval Architecture (see ref. [22]) RF Frictional resistance (subscript'm' denotes model scale, and's' denotes full scale) Rn {or Rn) Reynolds number (length based) (subscript'm' denotes model scale, and's' denotes full scale) RR Residuary resistance (subscript'm' denotes model scale, and's' denotes full scale) RT Total resistance (subscript'm' denotes model scale, and's' denotes full scale) Rv Viscous resistance (subscript'm' denotes model scale, and's' denotes full scale) Rw Wave resistance (subscript'm' denotes model scale, and's' denotes full scale) s Non-dimensionalized longitudinal wave number (kinematically linked to wave propagation direction 6 such that: s = secO) and u = sVs2 -1 S Wetted surface area of hull (subscript'm' denotes model scale, and's' denotes full scale) SNAME Society of Naval Architects & Marine Engineers SW {or sw) Sea water SWL (or swl) Still water level T Draft xxi u Non-dimensionalized transverse wave number also known as circular wave number (kinematically linked to wave propagation direction 6 such that: u = secOtanO) and u = sVs2 - 1 U Ship speed UBC University of British Columbia xxii Acknowledgements The author would like to thank Dr. Sander M. Calisal for his supervision during the course of studies and for providing the opportunity to work on this project. Throughout this period, he has provided the author with many useful advices, constructive guidance, unselfish patience and solid trust. Also sitting on the author's thesis committee are respected members, to whom the author would like to be thankful to for their time, effort and many useful suggestions in the thesis revision, they are Dr. Omer Goren (ITU, Dept. of Naval Architecture) & Dr. Steven Rogak (UBC, Dept. of Mechanical Engineering). Many people have helped out with conducting the experimental work in the towing tank, which being the provision of technical assistance, model fabrication, and model preparation for experimental work. The author would like to express sincere appreciation to those who lent their expertise in the above-mentioned tasks, particularly: - Mr. Gerald N. Stensgaard & Mr. Gary Novlesky Staff, Ocean Engineering Centre @ Vizon SciTec (formerly BCR Inc.) - Dr. Sander M. Calisal & Mr. Jon Mikkelsen Faculty staff, Department of Mechanical Engineering @ UBC - Mr. Eyup M. Sireli The author's good friend and former colleague, NALAB @ UBC. Thanks are extended to summer students who worked in the laboratory over different periods to assist in model testing work, they are Mr. George W. Rawlings, Mr. Wouter Schiferli, Mr. Christophe Thetiot, Mr. Vincent Riss, Mr. xxiii Adrian Black, Mr. Kelvin McNally and Ms. Monique Staals; and the fellow graduate students, Mr. Eyup M. Sireli, Mr. Voytek Klaptocz, Mr. Dan Vyselaar. Thanks are also extended to Mr. Brian Konesky for providing the normalized offset table of the UBC series-model#3 (parent hull). Without the dedication of time and effort from all of the above-mentioned personnel, the work would have taken much longer time or be incomplete. Last but not least, not to be forgotten are the many advice and friendships from the author's colleagues in the laboratory, they are Dr. Peter Ostafichuk, Dr. Jian Wang, Mr. Ye Li and Mr. Arlon Ratcliff. Thank you all, once again. xxiv CHAPTER 1 Introduction Ship designers and naval architects bear traditional mindset that displacement-type vessels operating at moderate to higher speeds are to be built of slender body with low block coefficient in order to minimize resistance or effective horsepower. This means smaller beam and frontal area projection for the hull below the waterline. A first attempt using systematical experimental studies to investigate the effects of beam and the extent of parallel middle body section including beam-to-draft ratios on ship resistance were done by Kent [20]. His study was limited to length based Froude numbers (FnL) below 0.21. He concluded that the addition of a parallel middle body section with a decreased beam would be good for ship resistance reduction. Further studies were also performed by Gertler [11], Wehausen et al. [46], Weinblum [47], and others. The general understanding is that any increase in beam will result in increasing resistance, especially at moderate to higher speeds by Harvald [13], Lewis [21], Schneekluth [33], and etc. However, these knowledge were built upon studies focused mainly in the low speed range for FnL < 0.25. At relatively higher speeds, there might be a different story. It is interesting to note is that in recent years, the studies from applied theoretical and experimental work on hull form optimization for minimum total resistance arrived at different conclusion, particularly for ships traveling at moderate to higher speeds. An experimental study performed on a coastal tanker by Calisal et al. [5] showed that by modifying the parallel middle body section on a conventional hull form into parabolic-shaped side bulb, a ship could in fact attain beneficial reduction in wave resistance while operating at higher I speed range between 0.25 < FnL < 0.45. On the other hand, Gotman [12] in her theoretical analysis using Michell's integral from thin-ship theory had showed that least wave resistance is possible with implementation of midship side bulb. The side bulbs were derived from parabolization of hull form that extends the waterlines continuously to eliminate the parallel middle body section. As a result, the beam is increased yet resistance is reduced. This leads to relatively lower total resistance or effective horsepower and renders the excess power for good use in traveling at higher speeds or for longer endurance. Remarkably, this is accompanied by increment in payload capacity and improvement in ship stability. Increased wave excitation on free surface trailing behind the ship is observed as speed is increased. Such deformation to the free surface is more readily known as waves. The waves around the hull are results of the variation in pressure field at the air-fluid interface arise from flow acceleration and deceleration as fluid particles pass by the moving hull. By replacing the wall-sided parallel middle body section on conventional hull form with side bulbs, the flow pattern in the vicinity is changed and further agitates the localized pressure field. Consequently, a stronger shoulder waves system is generated. Within the context of this thesis work, experimental investigations using various single-bulb configurations were studied. The bulbs function as 3-D wave makers by mimicking a segment of the zig-zagging wave petals in the multidirectional wave basin except that the zig-zagging wave petals are in fixed position in our case. In Figure 1-1 a & b, the close-up view of the wave petals on the wave generator and the top view of the wave basin are shown, respectively. Beneficial reduction in wave resistance is possible upon wave interactions if the newly generated shoulder wave system and the original bow-stern wave systems are out of phase. Consequently, a reduction in ship's total resistance and effective horsepower can be attained. From a designer's point of view, it 2 would be interesting to understanding such flow mechanism and the effects of increasing beam on form factor for hull optimization in achieving practical degree of wave resistance reduction and for future reference to integrate side bulbs into conventional hull forms. Figure 1-1: Multi-directional wave generator in wave basin (Tan [36]). 3 Explanations on the components of resistance can be found in typical naval architecture textbooks, such as Principles of Naval Architecture, vol. II [22], Zubaly [48], etc. A quick glance of this breakdown of resistance can be found in APPENDIX A . Explanations on wave resistance using Michell's integral can be found in Gotman [12], Michell [25], Tuck [38], Tuck et al. [39], [40] & [41], Wehausen [45], Wehausen & Laitone [46], etc. Further explanations and derivations of wave mechanics and wave theory can be found in some classical or newly published textbooks on marine hydrodynamics and fluid dynamics, e.g. Acheson [1], Lamb [21], Newman [29], etc. As such, their details are omitted within the context of this thesis. A conversion table including the depth based Froude number and various model testing conditions are tabulated in APPENDIX D . 1.1 Previous Work Prior to launching investigation into the present thesis work for wave resistance reduction and direct measurement, background studies being looked into included the findings of hull form parabolization at waterline, numerical prediction of wave resistance by Michell's integral and experimental work on wave pattern profiling for wave resistance computation by wave cuts. 1.1.1 Wave Resistance Reduction by Hull Form Parabolization While modifying the hull of an oceanographic vessel with the addition of sponsons to improve ship stability, Calisal et al. [5] found that the modified hull with the increased beam showed improved resistance characteristics. They then performed a study on a coaster tanker to investigate both experimentally and numerically the contradictory concept that increasing 4 beam could have reduced resistance. The beam increment at midship was derived at the waterline to replace the wall-sided parallel middle body on conventional hull with parabolic-shaped side bulbs leading to beam size that is significantly larger than before. Yet, the resulted total resistance exhibited reduction trend upon such bulb adaptation. Similar application has been successfully integrated on the bulbous bow of typical newly built vessels for minimizing bow wave system and reducing pressure variation prevailing along the hull length. Provided the dimensions of the parabolic shape and location are correctly optimized at certain speed, the wave interactions caused by these wave maker-like parabolic side bulbs would lead to a beneficial wave resistance reduction as the newly generated and original wave systems are out of phase by as much as possible. Accordingly, the total resistance or the effective horsepower is reduced. The improvement is due to a modified pressure field around the hull that alters the near and far-field flow pattern thereby generating new wave systems to interact with the original wave systems from the bow, stern and shoulders. More studies are needed to understand these resistance characteristics and the complex wave interactions to be able to successfully integrate side bulbs on performance-oriented vessels and for any innovative ship design purposes. From the economical standpoints of initial costs due to construction and materials, it may be less desirable to have parabolic-shaped side bulbs as oppose to wall-sided parallel middle body sections. But when overlooking the operation of such ship throughout its design life, it is still an attractive option considering the savings on fuel consumptions, which could compensate the high initial costs. Much investigation is needed to justify the economical feasibility on the concept of having side bulbs. 5 Nevertheless, the conclusion of these studies interestingly supports parabolic side bulb for wave resistance reduction and is worth further investigation. The reduction in resistance can easily lend to the increment of propagating speed or cruising range, whilst the increased beam size allows higher payload and ship stability. These are of particular importance for performance-oriented ships like naval vessels and for year round operating commercial ships measured based on payload capability like containers, bulk carriers, tankers, etc. 1.1.2 Theoretical Wave Resistance by Michell's Integral From dispersion theory of waves, we know that for a surface wave of wavelength X traveling in deepwater, a relationship between the phase speed (C, also known as wave speed or celerity) and wavelength can be expressed as: °2=T = 9 k 2K If we consider the transverse wave system of a ship, we can establish a relationship between the wavelength and the wave speed (or ship speed, since waves generated by a moving ship travel with the ship, the two must equal for a given wavelength) such that: f—) where N -1,2,3,4,. The troughs of the wave systems will coincide when N is of odd multiples and vice versa (see Figure 1-2a). Such interactions tend to produce "humps (correspond to resultant crests yielding higher resistances) or "hollows" 6 (correspond to resultant troughs yielding lower resistances) on the resistance characteristics curve. This "oscillating" trend suggests possibility for wave resistance reduction. As such, this strikes great interest in the ship research community for hull form optimizations (see Figure 1-2b). (b) Figure 1-2: Oscillations on resistance curves due to wave interactions. 7 In 1898, Australian mathematician J .H. Michell first illustrated the calculation of wave resistance for a thin ship moving steadily in inviscid calmwater of infinite water depth based on linear wave theory [25]. The complex nature of the formulation, which involves time-consuming triple-integral calculations at a time when computers were unavailable, and the lack of proper data acquisition system, had combined to limit its practicality for experimental validation. Thus, further theoretical development was inhibited. Michell's integral was first introduced to the naval architectural design community by Weinblum in 1930 for prediction of wave resistance. It has since then, stirred up many investigations. The formulation itself allows the calculation of wave pattern for entire wave field but such calculations were not completed until recent years. Many works had been performed for comparisons between experimental results and numerical calculations to examine the practicability of the theory Havelock [14], Tuck [38], Tuck et al. [39], [40] & [41], Gotman [12], Calisal et al.[5], etc. The knowledge had been significantly advanced over the last few decades. Michell's integral offers a potential theoretical approach to solving wave resistance. Within the limits of thin-ship theory and linear wave theory, it is good for qualitative comparisons with experimental results. Still, a century later in the present day, more efforts are needed. Many alternative forms of Michell's integral were used. One useful form of the theory reported by Tuck and used in the software called Michlet, is described below. The vertical deformation of free surface, namely the wave elevation z = Z (x , y ) , caused by a thin ship with hull offsets of y = ± V ( x , y ) passing by at steady speed V in negative x-direction in water of infinite depth with wave number being k0 .= g/V2 can be written as: 8 Z(x,y) = ~j\Yt&C)Gx(x-£,y,?,C)d£dC K0 R . where G is the velocity potential for a unit Havelock source that satisfies the linearized free surface boundary condition k0<pz + <pzz = 0 at z = 0 given by the combination of kinematic and dynamic boundary conditions, with r being the radial distance from source center, can be written as (Wehausen and Laitone [46]): G(x,y,z,C) = - — + - L R f Te i k { x c o s 9 + y s i n d ) k + k ° s e c 9 ek^^dkdO ^ , / ' Anr An2 Jf J0 k-kosec20 A slightly more numerically convenience alternative expression known as "Fourier-Kochin" representation is used, based on the interchange of order of (g,C) and (k,0), though still complicated by four integrations can be written as: Z(x,y) = - W f 7 e - ' ^ c o s ^ s i n e > k 2 ( P + i Q ) d k d e v } n2 J f J k-kosec20 where P + iQ = — ff K (£ C )eikcosei+kid£dC ikcosO^ 5 The total energy in maintaining the wave pattern at far field gives rise to the triple integral calculations in MichelPs integral for wave resistance (Rw) of a ship at s p e e d y , i.e.: R w = ^ t ( ( P 2 + Q2)sec5Od0 9 1.1.3 Direct Experimental Wave Resistance by Wave Cuts During model testing in the towing tank, the total resistance is being measured directly and the frictional resistance is being calculated using ITTC-57 (International Towing Tank Conference 1957) correlation line (see APPENDIX C ). Consequently, the residuary resistance is obtained indirectly by subtracting the approximated frictional resistance from the measured total resistance, such that: where the subscript'm' denotes model scale This approximation is based on the hypothesis by Froude and should be reinforced further by direct experimental approach. As such, the ability to accurately determine wave resistance and the method to directly measure it becomes the main goals for hull form design in resistance reduction. Much attention had been received since the idea of direct experimental determination of wave spectra and wave resistance from towing tests first being suggested on the H-5 Panel of S N A M E in 1960 by Korvin-Kroukovsky. Inui (1962) recommended a combined approach by using wave height measurements obtained from experiments to compliment analytical approach. This is because of the limitation of analytical approach, which does not provide useful insights into the working relationship between wave characteristics and hull form. He then worked towards the developmental work on waveless hull forms. Since then, there were many others who worked on experimental approach for direct measurement of wave resistance from the wave patterns. This was followed by various reported wave cut methodology, which utilized 10 the finite length of the "sliced" free surface profiles, either longitudinally or transversely relative to towing tank centerline. These ingenious methods were the transverse wave cut by Eggers [9], the longitudinal wave cut by Sharma [34], Moran & Landweber [28], Tsai & Landweber [37] and the "X-Y" method by Ward [43] & [44]. Each method has its own merits and involves different experimental setup in the towing tank. The studies by Kobus (1967), Landweber et al. (1968), Maruo et al. (1969) and Eggers et al. (1969) concluded that the contribution form the near-field velocity potential to the surface disturbance is negligible. Consequently, this renders the appropriateness of measuring wave resistance by the far-field wave pattern through the wave cut methods. But limitation of the problem lies with actual implementation of the methodology. The successful implementation of such methods could increase the effectiveness of conventional towing tests and improve hull form designs. Transverse Wave Cut Analysis Eggers' [9] transverse wave cut method requires two or more surface profile measurements in direction normal to the ship motion (see Figure 1-3). This can be done by using a movable wave probe fixed on the towing carriage. The corresponding wave amplitude functions F and G at each transverse wave number u are to be determined from the acquired wave profile. This method offers possibility to analyze wave motion in a rectangular channel of finite breadth and depth. The associated wave resistance is given as (see Eggers et al. [10], equation 32): Rw - du 11 Longitudinal Wave Cut Analysis Sharma's [34] longitudinal wave cut method requires one or more surface profile measurements in direction parallel to the ship motion (see Figure 1-3). In this case, a stationary wave probe can be used to measure the time dependent wave height records in order to determine the wave amplitude functions C and S at each corresponding longitudinal wave number s by applying Fourier transformation to the acquired single longitudinal cut. Thereby, yields the entire free-wave spectrum. The associated wave resistance is given as (see Eggers et al. [10], equation 44): R „ = i j r ( C - ) 2 + ( s - ) 2 i x - — ' d u 7t . J o oL v ' v J s^(2s - 1 ) where the transverse wave numbers (u) and longitudinal wave numbers (s) are related to other such that u = sVs2 - 1 , where 0 < u < + ° o and 1 < S < +oo . REGION OF SIGNIFICANT OUTER BOUNDARY REFLECTION FROM TANHWALL OF KELVIN PATTERN X X X X X V V N X \ V \ X \ \ \ \ \ \ \ S S A \ V N v N NNN N x N\N WUIDS-20 g 0 u r c e . TRANSVERSE CUTS Eggers et '•//•;/;ssr.ss/s .>sr?"/,'y ///;;/s;s/ /Tj ////////•?/ s al. [10] TANK WALL Figure 1-3: Eggers' transverse wave cut and Sharma's longitudinal wave cut. The above-mentioned methods have a common drawback as the wave profiles is to be truncated up to the point when the first reflected wave from 12 the sidewall bounces back to the towing line. To account for the loss in wave momentum as a result of the truncation, an asymptotic correction is applied to continue to wave records to infinity. This prompted the studies of Moran & Landweber [28], Tsai & Landweber [37] for wave cut methods that is possible to take into account of up to certain number of wave reflection from the wall (see Figure 1-4). The associated wave resistance given by Moran et al. [28] and Tsai et al. [37] are very similar. The non-dimensionalized wave resistance as given by Tsai & Landweber [37] is: 2/2 (c0)2+(s0)2+i; 7=1 k0 + kj x ( C y ) 2 + ( S y ) 2 where kj = J(k0f + v b j and Rw=^-pV2l2Cw source: Moran et al. [28] Figure 1-4: Longitudinal wave cut (Moran & Landweber 1972, Tsai & Landweber 1975). 13 X-Y Wave Cut Analysis Ward's [43] X -Y method, on the other hand, works like a combination of transverse and longitudinal cut method (see Figure 1-5). The method is complicated by the fact that the towing line is offset from the tank centerline that may require certain modifications to the towing carriage. On top of that, a vertical cylinder is to be fixed over the tank for force measurements in the directions parallel (X) to and normal (Y) to the ship motion. The associated wave resistance is given as see Eggers et al. [10], equation 47): ^ BC TANK SIDE , , , / , / / / f . , > / / . w / y ntttttitttitttst'ftttL\ VUNDER^y^-V/AVE t TANK •— i j j a J CEUJERUNEj REFLECTED WAVE,. I ) / T")7"7 / ' 1 * ' ' I t J i Figure 1-5: X-Y method wave cut (Ward 1963,1966) source: Eggers al. [10] et The wave resistances obtained by wave cuts are sensitive to wave prove location and may induce experimental error. Employing multiple cuts can minimize such error, which then requires multiple wave probes that may introduce interference in the flow field. Measurement of a free surface patch using multiple cuts without interference from the multiple wave probes in the tank is desired. A detailed list of literature review on development of wave resistance measurement and formulations was given by Eggers, Sharma & Ward [10]. The end result of wave pattern profiling is primarily for wave resistance, also 14 being acquired is the free wave spectrum, which is useful for design optimization of multi-hulls configuration. Many research works focused on result validations, ease of wave measurement techniques and adaptation into existing towing tanks. The potential of applying wave pattern measurement into practical work is encouraging. The thesis work employed a wave profiling system by using the laser line reflected by seedings on the free surface. This is going to be very useful with the number of cuts limited only by camera resolution. The conventional towing tank testing makes no use at all the wave pattern generated by the towed model, by maturing such implementation technique could increase the effectiveness of towing tests and improve hull form designs. 1.2 Current Work In this section, the readers are informed of the motivating reason behind our studies, the encompassed work scope and a brief introduction to the subject of the studies, i.e. the model in used. 1.2.1 Motivation The total resistance encountered by any moving ship forms are added burden that cannot be physically avoided. It is of great concern to be able to minimize resistance from an economical and operational point of view. A hull form that can overcome this challenge without compromising payload and safety is a significant breakthrough. Side bulbs designed using parabolization of the ship hull at the waterline are proving to be a viable option for total resistance reduction. The present study is initiated to understand resistance characteristics for conventional hull form integrated with add-on parabolic side bulbs to reduce 15 wave resistance through beneficial wave interactions. Wave resistance is the key contributor to total resistance. Reducing this component of resistance results in a significant reduction in total resistance and effective horsepower. The acquired knowledge is useful for ship designers and naval architects on preliminary design optimization for effects of increasing beam on hull form factors and resistance trend. It is with hope that the work will advance knowledge in ship hydrodynamics in this aspect of innovative ship design and applications. 1.2.2 Scope This thesis work attempts to investigate, by experimental means on a small craft, the resistance-reduction capability of side bulbs that is derived from parabolization of hull form at waterline. The effects that beam increment and its maximum location have on hydrodynamic performance are to be studied through model testing in towing tank. The study is conducted in an effort to minimize wave resistance, so as to obtain beneficial reduction in total resistance, over a targeted speed range between 0.30 < FnL < 0 .40. It should be noted that no effort is made herein for prediction of degree of flow separation and corresponding location through the experimental work. Wave pattern profiling system is used to map the wave heights of a finite length of rectangular free surface patch trailing behind the towed hull. The wave heights are used for the computation of wave resistance using Sharma's method on longitudinal wave cut. No detail study on the effect of numerical truncation is looked into. 16 The work encompasses experimental and numerical aspects for qualitative and quantitative comparisons through a systematic approach for design optimization and feasibility studies of hull form parabolization. Based on the experimental studies, recommendations are made to revise the parent hull into an optimized version with built-in parabolization of hull forms. 1.2.3 S u b j e c t o f S t u d i e s The subject of the studies is the UBC series - model #3 (see Calisal and McGreer [4]). It is a 1:13.75 scale model derived from a typical fishing vessel hull form suitable for use on the west coast of Canada. The design called for a larger aft deck area and volumetric coefficients for better functionality and space usage. It was a collaborated design between the University of British Columbia (UBC) and the Ocean Engineering Centre of Vizon SciTec (formerly BC Research Inc). This hull form was referred to as the parent hull. Results obtained from this configuration were established as the control baseline for comparisons of any modification. The general particulars of the parent hull, in both the full scale and model scale, are tabulated in Table 1-1 and the detailed hydrostatics are discussed later in section 5.1.5 and listed in APPENDIX G . Table 1-1: General particulars of parent hull (UBC series-model #3) (scale=1:13.75). Full scale Model scale LWL 27.734 m 2.017 m LPP 27.734 m 2.017 m B 6.968 m 0.507 m T 2.799 m 0.204 m A 3202999 N 1250 N C B 0.604 0.604 17 The general lines plan of UBC series-model #3 is illustrated in Figure 1-6. This model is to be referred to as the parent hull. The results obtained from this model will be used as control baseline for comparison to results obtained upon any modification to the parent hull. As such, one can have an idea if the modification has improved or degraded the resistance performance. Thereby, this allows the understanding of the effect that a certain modification has on the hull form factors and resistance characteristics. 18 CHAPTER 2 Feasibility Studies by Theoretical Wave Resistance (2-D body) A feasibility study on the wave resistance reduction capability of the modified hull with parabolization at waterline was looked into. The wave resistance values were calculated using Michell's integral based on the formulation given by Wehausen & Laitone [46] while simplifying it for a 2-D body of wall sided with unit depth and coded by the author in Matlab version 6.5 environment: The modified parent hull has its waterline extended continuously along the / x L wall-sided parallel middle body located between — -^ L < — < - ^ L . Consequently, the modified hull has its beam widened where the parallel middle body is transformed into bulb-like parabolic section. The preliminary studies used beam increment ranging from 5 % to 2 0 % with maximum beam location fixed at midship. The results are shown in Figure 2-1. W e see that there is a hump in the vicinity of FnL = 0 .25 , where the increased where the wave amplitude functions are given by: 20 beam showed no beneficial reduction at all. However, within 0.28 < FnL < 0 .34, all of the increased beam configurations exhibited encouraging resistance-reduction capability. 600 500 400 =- 300 200 -t 100 x%0f^^>a^x-sx-^ X ... r^rZ A •  i 0 0.20 (a) 0.25 0.30 0.35 0.40 0.45 Froude no. (b) -40 -I i 1 1 i 1 0.20 0.25 0.30 0.35 0.40 0.45 Froude no. Figure 2 - 1 : Feasibility studies of increased beam from theoretical wave resistance using Michell's integral at model scale 2 - D body ( 1 : 1 3 . 7 5 ) : RWM, %RWM Interesting to point out is the fact that the reductions were able to maintain over a short range of speeds. Such robustness is good for actual operating condition on the rough sea. Above FnL > 0 .25, the increased beam only made worse the resistance performance. In general, the trend effect indicated that 21 above FnL > 0.35, the resistance increased drastically. This corresponds to the model's hull speed of approximately FnL = 0 .40. At this speed, the generated transverse waves length almost equals the model length. At this point, the hull is practically "climbing" its own bow waves. To go beyond this point, the hull speed requires significant amount of power due to the drastic increment of resistance encountered. The percentage change in wave resistance for each increased beam configuration in the vicinity of the concerned speeds, between 0.30 < FnL < 0 .40, is shown. A closer look at a few selected speeds within this speed range indicated useful reduction by as much as 2 7 % at FnL = 0.30. The observation strongly implies that increased beam brings beneficial wave resistance reduction and that the test case of 1 5 % beam increment is the favoured one. These interesting results prompted the experimental work in order to gain insight to the actual resistance behaviour, and to allow for the validation work to be done. 22 CHAPTER 3 Experimental Work 3 . 1 Testing Facil ity The model tests were conducted in the commercial towing tank at the Ocean Engineering Centre of Vizon SciTec (formerly BC Research Inc). It is located within the University of British Columbia (UBC) campus. A front view spanning the entire length of towing tank including the cantilever type towing carriage and general experimental setup for our tests is shown in Figure 3-1. F i g u r e 3 - 1 : T o w i n g t a n k at O c e a n E n g i n e e r i n g C e n t r e of V i z o n S c i T e c . The towing tank is 67m long, 3.7m wide and 2.4 m deep. The towing carriage is capable of speeds up to 6m/s. More specifications and the plan view of the facilities are shown in Figure 3-2. 23 \A—s-s yifflTTT me CD 2 1 c & 2, Jj 5 l 5s I a s t s « IN <>l «q a; » -O lO I- O) p s -tr JT O) *^  < o LL o a < a o E a . a o I Vi in -«*-OL u. o Lb a; m UJ 1 z < h-Z. o UJ o 5 tr U) I 3 i n f o c a: cc CQ c o o o c o a> c "5> c ui c (0 a> o O +-* ro o ro OJ c (/) c o ro o o 0) Q. (0 <* > c ro csi CO a> >_ 3 3.2 Experimental Approach As systematic experimental approached was devised to investigate the performance of the modified parent hull. The modifications were done by using various configurations of add-on side bulbs to increase the beam (see Figure 3-3). This made the model testing more feasible. Firstly, the removable side bulbs greatly cut down model preparation time. Secondly, this approach rendered the project very cost-effective by eliminating the need to build various new hulls of increased beam. The side bulbs were derived from a concept called parabolization of waterline by Calisal et al.[5]. They extend continuously along the waterline and over the parallel middle body. The modified parent hull had its wall-sided parallel middle body section replaced by parabolic section of various configurations: - Beam increment (from 5 % to 20%) - Bulb arrangements (single/double bulb, fore/midship/aft bulb) F i g u r e 3 - 3 : F r o n t v i e w of the v a r i o u s a d d - o n s i d e b u l b s . 25 The side bulbs act in much same way as the bulbous bow of a vessel (see Figure 3-4) only that the bulbous bow protrudes from the bow to generate a localized low-pressure region that interacts with the high stagnating pressure at the bow. In turn, the bulbous bow is there to create a new wave system to countermeasure the bow wave system. A carefully designed bulbous bow could attain useful wave cancellations. Similarly, a carefully designed side bulb configuration could potentially do the same with added benefits such as improved payload capacity and ship stability. (a) Typical bulbous bow (b) Experimented bulbous bow (source: Keppel Shipyard). (source: US Navy). F i g u r e 3-4: V a r i o u s b u l b o u s b o w c o n f i g u r a t i o n s for r e d u c t i o n of w a v e r e s i s t a n c e . The side bulbs protrude from the side to generate new wave system to achieve the very same purpose. This resistance-reduction concept is the core reason that drives the investigation to understand such hull form characteristics and resistance performance in order for future consideration in integrating bulb designs into conventional hull form to achieve reduced ship resistance. 26 The present work outlines a step-by-step experimental investigation aimed at understanding three main effects of side bulb configurations on the resistance characteristics (see Figure 3-5): Effects of beam increment using single bulb Baseline (B0%) Modifications Parabolized @ midship (+B5% ~ +B20%) . . ' • • " i f " " - - . . . Effects of max. beam location of single bulb Baseline Parabolized @ midship (+B15%) Modification Parabolized @ forebody (+B15%, F) Modification Parabolized @ aftbody (+B15%, A) 1 ^ Effects of fairing extension to rear of single bulb Baseline Parabolized @ midship (+B15%, +L0%) Modifications Parabolized @ midship (+B15%, +L25% ~ +L35%) F i g u r e 3-5: S c h e m a t i c i l l u s t r a t i o n of e x p e r i m e n t a l a p p r o a c h to s t u d y v a r i o u s ef fects of s i d e b u l b c o n f i g u r a t i o n s . 27 3.3 Test Programs The measurements in the experimental studies consist of three parts: (1) Calmwater resistance tests \ Primary measurement - Resistances Secondary measurement- Heave - Trim Others - Still frame photos - Videos (2) Flow Visualization Tests ( Primary measurement - Still frame photos 1 - Videos (3) Wave Pattern Profiling Tests f Primary measurement - Wave heights 3.3.1 Calmwater Resistance Towing Tests The experiments investigating the three main effects of side bulb configurations on the resistance characteristics were conducted over four separated time periods. The reasons being to accommodate the availability of the commercial towing tank for student research work and lead-time of model fabrication. During each phase, calmwater resistance tests were conducted in the towing tank to measure total resistance, heave and trim responses. For each different bulb configuration, the hull was subjected to low-speed tests between 0.10 < FnL < 0.20 and high-speed tests between 0.20 < FnL < 0 .45. The low-speed tests were conducted to determine hull form factors (1+k) using Hughes-Prohaska's method (see APPENDIX H ). The high-speed tests were conducted to measure resistances, heaves and trims. These 28 model-scale measurements were then extrapolated into full-scale performance following procedures as outlined in ITTC methodology (see [16] &[18]). Each configuration bears distinctive designation for database identification purpose. The physical conditions during model testing, such as water temperature, density, viscosity, model ballast weights and etc., are tabulated in APPENDIX F . The hydrostatics calculations are listed in APPENDIX G . And a generalized overall test matrix is shown in Table 3-1. Table 3-1: Multi-phase test programs for experimental work Phase 1 - Investigation on effects of beam increment using single midship bulb test configurations test designations Parent hull BOp_single Parent hull + 5% parabolization @ midship B5p_single Parent hull + 10% parabolization @ midship B10p_single Parent hull + 15% parabolization @ midship B15p_single Parent hull + 20% parabolization @ midship B20p_single Phase 2 - Confirmation & repeatability tests on selected single bulb parabolization test configurations test designations Parent hull B0p_single Parent hull + selected 15% parabolization @ midship B15p_single Phase 3 - Investigation on effects of max. beam location using selected single bulb parabolization test configurations test designations Parent hull + selected 15% parabolization @ midship B15p_single Parent hull + selected 15% parabolization @ forebody B15p_single_F Parent hull + selected 15% parabolization @ aftbody B15p_single_A Phase 4 - Investigation on effects of fairing extension using selected single bulb parabolization test configurations test" designations Parent hull + selected 15% parabolization @ midship B15p_single Parent hull + selected 15% parabolization @ midship + fairing extension of 25%L W L B15p_single_alpha1 Parent hull + selected 15% parabolization @ midship + fairing extension of 30%L W L B15p_single_alpha2 Parent hull + selected 15% parabolization @ midship + fairing extension of 35%L W L B15p_single_alpha3 29 T P H E f f e c t s of B e a m I n c r e m e n t U s i n g S i n g l e B u l b Phase 1 is referred to as [P1]. This was the first decision-making criteria for the bulb optimization studies. The single-bulbs with increased beam of 5 % , 10%, 1 5 % and 2 0 % were used to modify the parent hull in order to quickly incorporate the concept of hull form parabolization without having to build new modified hulls. Calmwater resistance trends were studied to understand the effects of various beam increment configurations. Thus, allowing a specific single-bulb configuration to be chosen for further studies at later phases. These parabolic-shaped single-bulbs have their maximum beam fixed at midship (Station 5). The bulbs blend into the parent hull between station 2.5 and station 7.5 from lower chine to deck level. All opened edges during attaching were sealed with yellow plasticine and taping to maintain surface smoothness. Both the slow and high-speed tests were performed. Upon completion, the single bulb configurations of 1 5 % beam increment (B15p_single) was found to outperform the remaining bulb configurations in terms of hull form factor and resistance characteristics with respect to the control baseline, i.e. the parent hull without any bulb attached (B0p_single). Therefore, this configuration was chosen for further studies. Detail discussions of the results and analysis are presented in section 5.1.1.' \P2] R e p e a t a b i l i t y T e s t s o n S e l e c t e d S i n g l e B u l b Phase 2 is referred to as [P2]. This was conducted to confirm the calmwater resistance trends of the control baseline configuration and the single bulb configuration of selected beam increment from Phase 1. 30 All opened edges during attaching were sealed with yellow plasticine and taping to maintain surface smoothness. Both the slow and high-speed tests were performed. Detail discussions of the results and analysis will be reported later. Each of the two re-tested configuration consisted of three sets (set A, B and C) for a repeatability check. Besides the typical calmwater resistance tests to measure the heave, trim and resistances (under set A, B and C), also being measured were the wave heights using the wave pattern profiling system (under set A and B) and underwater flow visualizations (under set C). TP31 Effects of Maximum Beam Location Using Single Bulb Phase 3 is referred to as [P4]. This was conducted to understand the resistance trends of having the single bulb of selected beam increment from Phase 1 located at different longitudinal locations, namely at forebody (B15p_single_F), midship (B15p_single) and aftbody (B15p_single_A). All opened edges during attaching were sealed with yellow plasticine and taping to maintain surface smoothness. Both the slow and high-speed tests were performed. Detail discussions of the results and analysis will be reported later. 1P41 Effects of Fairing Extension at Rear of Single Bulb Phase 4 is referred to as [P4]. This was conducted to complete the understanding of the trend effect to hull form factor for having the rear or the trailing edge of the single bulb of selected beam increment from Phase 1 being gradually faired to modify the flow exit angles. This could have suppressed the flow separation. 31 The fairings extended in lengths of 490mm (B15p_double_F_alpha1), 610mm (B15p_double_F_alpha2) and 720mm (B15p_double_F_alpha3). The applied extensions blended in between midship and trailing edge. They corresponded to waterline length of approximately 2 5 % , 3 0 % and 3 5 % , respectively. All opened edges during attaching were sealed with yellow plasticine and taping to maintain surface smoothness. Only slow-speed tests were performed to determine hull form factor. Detail discussions of the results and analysis will be reported later. 3.3.2 Underwater Flow Visualization Tests The flow visualization tests were performed on the control baseline configuration and the single bulb configuration of selected beam increment from Phase 1A typical test run for flow visualization photo is shown in Figure 3-6. Figure 3-6: Typical underwater flow visualization test with attached tufts (parent hull at design speed, Fn=0.35). 32 As seen in Figure 3-6, the tests were performed using black colour yarn tufts of equal length and attached at equal spacing to the hull below design waterline at each station. These underwater photos were taken through a side window, which is located just halfway along the sidewall of the towing tank and below water level. 3.3.3 Wave Pattern Profi l ing Tests The wave profiling system consists of a low power (35mW) Helium-Neon laser gun and MAPP2500 Smart Vision System camera capable of acquiring images at 25fps. The laser gun emits a single beam of red colour light rays to a minuscule 360° rotating mirror, which then projects a sheet of light onto the free surface. The sheet of lights now becomes a visible red line shined across the width of the towing tank. The calibration setup of the system is showed in Figure 3-7. F i g u r e 3-7: W a v e pat tern p r o f i l i n g s y s t e m c a l i b r a t i o n s e t u p . 33 The camera was tilted at an angle of 45° with respect to water surface. It was mounted at about 1.35 m vertically above the free surface and 1.27 m horizontally away from the red line. These positions were fixed at all time during the wave pattern profiling testing. The calibration was done using a dummy dominos board. For stronger contrast, the dots are painted in black and leaving the background in white. The centroids of the dots are of known distances and captured by the camera. During the wave pattern profiling tests, the ambient lights were diminished for improved screen contrast. The camera acquired the vertical variation of the laser-projected line on the free surface in 2-D over a time period. A series of analysis were performed to transform the 2-D wave elevations into 3-D for a finite free surface rectangular patch. The data were then used to calculate wave resistance using longitudinal wave cut method of Sharma [34]. For practical limitation of the camera due to resolution, the camera was used to capture a length of approximately 0.60m along the laser-projected line on the free surface. As a result, the acquired rectangular free surface patch is of the same width and spanning longitudinally in time until data acquisition is stopped. The nearest edge of the rectangular patch to the towing tank centerline was approximately 0.55m. It must be emphasized that the towing tank centerline was essentially the carriage towing line, which also coincided with the longitudinal centerline of the model. Due to the poor reflection of the laser-projected red line on the free surface, the camera captured no visible line on the screen for analysis even with the sensitivity threshold adjusted to maximum level. To allow the camera to be able to capture any useful reflection from the free surface, seeding were 34 required to be spread over a patch of the free surface in the vicinity of the laser-projected red line. Various seeding agents were tried. These attempted agents being: (1) Fog - Dry ice (2) Floating particles - Dust - Punched hole paper - Styrofoam - Quartz microspheres - Z-light spheres (3) Dye - Laser-induced fluorescence - Light-emitted fluorescein - Rhodamine (red colour) (4) Thin film - Magnetized ferrofluid - Oil spray The constraints to employing a particular seeding agent in the towing tank are: - Impose no contamination to water in towing tank (non poisonous, non oil based, non colour based, non evaporative) - Minimal interference to towing measurements (small, light and neutrally buoyant, non sticky base) - Good reflection - Inexpensive and availability in large quantity - Ease of seeding collection upon tests - Salvageable and reusable - Required no necessary draining of water in towing tank as cleaning option 35 p 512 Cam 1, AlgSensorHorThrSub512 (a) Threshold mode Taj: 535 (b) Normal mode F i g u r e 3-8: C a m e r a - c a p t u r e d re f lec ted l a s e r l i n e o n s e e d i n g . Considering above-mentioned constraints and the highly maintained swimming pool water quality in the towing tank of O E C , we selected the 36 styrofoam particles as the candidate. They reflected the laser-projected line relatively better and performed quite well during the wave profiling tests. Views as captured by the camera in two different modes are shown in Figure 3-8a & b. The two shortcomings using the styrofoam particles were: - Re-spreading & re-seeding of the particles after each towing test as they tend to be pushed apart by the waves and the free surface boundary layer generated by the towed hull - Extremely tedious salvaging process at the end of the experiments as they tend to stick on the sidewall of tank and spread across the towing tank. A typical wave pattern profiling tests with the model towed across the seeded styrofoam patch are shown in Figure 3-9a & b. (a) Reflected laser light (b) Typical tests with the seeds F i g u r e 3-9: S t y r o f o a m s e e d e d free s u r f a c e p a t c h . 37 CHAPTER 4 Numerical Work 4.1 Theoretical Wave Resistance by Michel l 's Integral (3-D body) The Michell's integral studies were extended from the previous 2-D body during feasibility studies to the present 3-D body. A commercial suite called Michlet Pro ® version 8.03 was used for such calculations. It too, employs Michell's integral for the computations of wave resistance (see section 1.1.2 for mathematical formulations). Input to the program includes hull offsets and physical conditions (water depth, density & viscosity). The 3-D results will be more realistic when comparing to experimental values. Results from Michlet for effects of increased beam using single-bulb were calculated and compared to baseline (see Figure 4-1). 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Froude no (achieved) Figure 4-1: Percentage change in effects of increased beam on model scale resistance coefficients: % C W M 38 4.2 Experimental Wave Resistance by Longitudinal Wave Cuts When performing longitudinal wave cut analysis using the wave pattern acquired from the model testing, there are few important things to take note, i.e. the calibration setup, the employed coordinate system and the truncation point downstream. 4.2.1 Camera Calibration Target A calibration target was purposely made to calibrate the camera (see Figure 4-2). Several important camera threshold settings were adjusted in order to capture the laser light reflected by the styrofoam seeding on the free surface. These settings are mainly for fine-tuning the grayscale sensitivity of the C C D camera settings and to determine the spatial transformation matrix. The matrix was used to correlate screen positions to real life positions. a. Actual view b. Camera view (calibration mode) Figure 4-2: Camera calibration target. The acquisition and transformation matrix were performed by Ranger System ®, a software accompanied the camera system. The calibration target had high contrast black dots of known proximity on crystal white background. The central block of known height was used to calibrate the vertical reference points. 39 4.2.2 Wave Cut Coordinate System The coordinate system in used is a body-fixed right-handed Cartesian coordinate system as shown below (see Figure 4-3): +z Originally produced by the author in Solidworks Vship F i g u r e 4-3: C o o r d i n a t e s y s t e m u s e d f o r w a v e c u t a n a l y s i s Consequently, the acquired wave profiles downstream are in the negative x-direction. The x-coordinate (X,) at each acquired point of wave height (Z ( ) corresponds to the model speed and accumulated acquisition time of camera: X , = l / x ( / x A f ) where i = 0, 1, 2, N V = Model speed [metre/second] At = 40000, resolution due to camera acquisition time [micro second] 40 The y-coordinate (V^) corresponds to the lateral position of each longitudinal wave cut (V C ( J f , ) : ^ = n* > / = n * * + ( ' 'xAy) where i 0, 1 ,2 n 512, camera resolution [pixel] 0.675m, distance away from longitudinal centerline of model [m] 0.585m, effective length of laser line as captured within camera frame [m] Lljne/n, horizontal distance resolution [m], n offset L, mline Ay The z-coordinate (Z ( ) represents the acquired wave heights with positive value indicating wave elevation above the still water level (SWL) and vice-versa. 4.2.3 Wave Cut Reference & Truncation Points The acquired wave elevations within the region between 0.675 < Y; < 1 .260 were transformed from 2-D into 3-D by taking into account the progression of time line. The wave patch was then treated explicitly to filter out any random noise while preserving as detail as possible the corresponding wave pattern. The downstream truncation point differs from cut to cut as this very location is governed by Kelvin wave angle of 19.5° and the lateral position of the cut (see Figure 4-4). 41 REGION OF SIGNFICm OUTER BOUNDARY REFlEtnONFfKHTANIWALL OF KELVIN PATTERN l^»W.WNNy\^xy\X\N\\\)M\N\s\\\\\^.\J>'i.N— Ysldemi,= (b/2) = 1,830m TRUNCATION POINTS J**' \ REFERENCE POINTS % —-=£>•<-• | "^o^r ^ LONGtflMNAL CUTS «- L, •EFFECTIVE, 1 ^-EFFECTNE. 1 >! 9=19.5? "-TANK AND MODEL CENTER PLANE ^^-^ffiODEL Yatl 512 - Yatt> et+ f-Bie = ?-260m Yait 1 = Y0ffset= 0.675m b/2 b/2 TANK WALL F i g u r e 4-4: L o n g i t u d i n a l w a v e c u t t r u n c a t i o n p o i n t . The truncation points {^TRUNCATION,i) a t downstream can be worked out by first finding the reference points (XREFERENCEJ) at upstream, which correspond to the starting points of bow wave. Thereby, defines half of the signal length (LEFFECriVE,l) 9 ' V e , _(b/2)-YcuU '-EFFECTIVE, i tan (19.5°) thus, XTRUNCATION^ i = XREFERENCE^ j — (2 x LEFFECTIVE^ i) where b is the overall width of the towing tank. The closer the lateral position of the longitudinal cut to the towing tank centerline, the stronger the near-field effect of the wake in the vicinity of the turbulent boundary layer, thereby, the less accurate the calculated wave resistance. At the same time, the closer the lateral position of the longitudinal cut to the towing tank sidewall, the shorter the effective length of the wave records, thereby, the less accurate the calculated wave resistance. The 42 chosen region, i.e. the wave patch to be captured, has taken into account these effects. For the wave cut analysis, the near-field wave pattern within the wake zone, 0 < Yt < 0 .675m, was neglected to dismiss the waves of random short wavelength and breaking waves. With the fully stretched tank width at b=3.660m, that means half of the tank width is 1.830m. The region nearby the sidewall of the towing tank, 1.260m < Yj < 1.830m, was also neglected to account for wave reflection from the sidewall. As such, only the region between 0.675m < < 1 .260m was considered. This improved the resolution of horizontal distance between each successive longitudinal cut due to camera resolution, which was limited at 512 pixels (see Figure 4-4). 43 CHAPTER 5 Results & Discussions 5.1 Experimental Work The parabolic bulbs, extended laterally from the beam, alter the pressure field around the hull without causing significant hull form discontinuity that promotes flow separation and eddy formation. As a result of the alteration to the near-field pressure at the free surface in the vicinity of the add-on bulb, new shoulder wave system is generated. As the near-field wave systems propagate downstream, they modify the far-field wave-making pattern through series of interactions between these wave systems. If the crests from the new shoulder wave system are in phase and coincide with crests from bow and stern wave systems, the resultant interacting waves give rise to constructive increment in wave height and contain higher energy. Consequently, the wave resistance encountered by the ship is increased and a hump can be observed on the measured resistance curve. The higher the magnitude of the hump, the greater the energy shed away by the ship to overcome its forward motion. On the other hand, if the crests from new shoulder wave system are out of phase and coincide with troughs from bow and stern wave systems, the resultant interacting waves leads to constructive decrement in wave height and contain lower energy. Consequently, the wave resistance encountered by the ship is decreased and a hollow can be observed on the measured resistance curve. The lower the magnitude of the hollow, the lower the energy shed away by the ship to overcome its forward motion. 44 If sufficient understanding of such interactions can be achieved, then it is possible to engineer the way wave interact to achieve beneficial reduction in wave resistance through proper modification of hull shape. Such interactions are very non-linear in nature and presents great difficulties to simulate numerically for qualitative predictions using existing mathematical tools derived from linearized free surface theory and thin ship theory in ideal fluid bounded by the hull at air-fluid interface. Moreover, the fluid-structure interactions present yet another level of complexity. This thesis work is focused on studying the effects of side bulb configuration (single / double) and bulb location (fore body / midship / aft body) to the ship hydrodynamic resistance performance. 5.1.1 [P1] Effects of Beam Increment Using Single Bulb The purpose of this test series was to validate the theoretical prediction made in feasibility studies, in which a potential single-bulb configuration was identified (i.e. 1 5 % beam increment). Preliminary experimental studies on the effects of increased beam using single-bulb were conducted first in Phase 1 [P1] and then confirmed in Phase 2 [P2]. The test programs are briefed in Table 5 -1 . T a b l e 5-1: [P1] P r e l i m i n a r y t e s t s u s i n g v a r i o u s s i n g l e m i d s h i p b u l b c o n f i g u r a t i o n s . Phase 1 - Investigation on effects of beam increment using single midship bulb test configurations Parent hull test designations BOp_single B5p_single B10p_single B15p_single B20p_single Parent hull + 5% parabolization @ midship Parent hull + 10% parabolization @ midship Parent hull + 15% parabolization @ midship Parent hull + 20% parabolization @ midship 45 For this test series, the maximum beam location was fixed to midship (station 5). The hull parabolization was implemented along the hull between x - 0 . 2 5 / . ^ < — <0.25LWL, which corresponds respectively to between station 2.5 (aftbody) and station 7.5 (forebody). As such, the total bulb length is 5 0 % of waterline length. The profiles of the tested parabolized hulls at the waterline level are shown in Figure 5 -1 . •B 1-4 I 1.2 ca 0.8 ® 0.6 £ 0.4 0. 2 2 E m ; : ! I I ! 1 1 ! 1 1 i 1 | B0p_single_midship (baseline) — B15p_single_midship (selected) B5p_single_midship B20p_single_midship 1—1 1—1 1 — B10p_single_midship 1 — l — 1 — 1 — 1 — I — 1 — I — 1 — i — 1 — i — 1 — -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 x/L, normalized length along design waterline F i g u r e 5-1: [P1] P r o f i l e s of t e s t e d p a r a b o l i z e d h u l l at d e s i g n w a t e r l i n e l e v e l . In Figure 5-2a, one can observe the model scale total resistance characteristics of increased beam using various side bulbs at speeds between 0.20 < FnL < 0 .45. In the author's opinion, it is important to show firstly these actual total resistances as measured directly off the model, rather than in their non-dimensionalized forms. They should be sufficient in conveying to the reader the resistance-reduction capability of parabolized hull through increased beam. 46 16 14 12 10 8 6 4 2 —x-B0p_single [P1] (baseline) + B5p_single [P1] * B10p_single [P1] —>-B15p_single [P1] * B20p_single [P1] *r 3 0 H—1—1—1—1—I—1—1—1—1—I—1—1—1—1—I—1 0.20 0.25 0.30 0.35 Froude no (achieved) 0.40 0.45 — x — B 0 p _ s i n g l e [P1] (baseline) — < • — B5p_single [P1] B10p_single [P1] B15p_single [P1] B20p__single [P1] 0.20 0.25 0.30 0.35 Froude no (achieved) 0.40 0.45 (a) R. TM (b) TM 'TM,i TM.baseline TM,baseline x100 F i g u r e 5 - 2 : [ P 1 ] E f f e c t s of i n c r e a s e d b e a m o n m e a s u r e d tota l r e s i s t a n c e at m o d e l s c a l e ( 1 : 1 3 . 7 5 ) : R T M , % R T M This will remove the suspicion when presenting in non-dimensionalized form, which is dependent on wetted surface area ( S M ) in the denominator. Because when presenting in non-dimensionalized forms, as shown below, one might be misled to interpret the actual resistance-reduction to be due to the increased wetted surface area in the denominator and inadvertently denies the resistance-reduction capability of the parabolized hull: o r — ™ ™ ~ l n <5 V 2 2 HM^M vM We presented in Figure 5-2b the percentage variation of the effect on resistance with respect to parent hull. It clearly illustrates speed regions where hull parabolization is beneficial or make worse the resistance performance. For each increased beam configuration, we see a very intense peak in the moderately high-speed range between 0.25 < FnL < 0.30 that corresponds to intense resistance increment and multiple small peaks in the high-speed range between 0.38 < FnL < 0.45 that correspond to subtle resistance increment. Within these two above-mentioned speed ranges, the single-bulbs exhibited no practical resistance-reduction capability at all. However, we see encouraging improvements in the slow-speed range between 0.20 < FnL < 0.235 and in the targeted high-speed range between 0.30 < FnL < 0 .40. This trend is most noticeable for the single midship bulb with 1 5 % beam increment (B15p_single), which coincides with prediction made in feasibility studies. The trend of this percentage reduction holds rather consistent at around 6 % (see Figure 5-2b). It suggests that beneficial reduction in total resistance is practically possible simply by parabolizing the hull form at the waterline. Perhaps proper "tailoring" will yield better improvement. To further assess the effects of increased beam on individual resistance components for each modified parent hull, it is more convenient to look at their non-dimensionalized version and work from here. The corresponding total resistance coefficients, frictional resistance coefficients and residuary resistances are shown in Figure 5-3a, b & c, respectively. Similar to the previous finding, the non-dimensionalized resistances also point to the single midship bulb with 1 5 % beam increment (B15p_single) as a 48 favoured configuration for resistance reduction. Through the trend in percentage variation as shown in Figure 5-4a, b & c, one can more conveniently observe that within the slow-speed range between 0.20 < FnL < 0.235 and the targeted high-speed range between 0.30 < FnL < 0 .40, beneficial reductions in total resistance of - 8 % can be achieved (see Figure 5-4a). This is due to significant reduction in residuary resistance by - 1 0 % (see Figure 5-4c). Moreover, the reductions hold rather consistent over a small range of speeds. This is very good for practical ship operation since a fixed speed is hard to maintain in the rough sea. There were no considerable variations in frictional resistance since it depends on Reynolds number, which is mainly affected by waterline length that was fixed in all experiments. The minute difference within 0 . 3 % (see Figure 5-4b) in frictional resistance between models is mainly due to fluctuations in physical conditions in the towing tank, i.e. water temperature, density, viscosity and water level. Both the dimensionalized and the non-dimensionalized results lead to same conclusion that the resistance-reduction capability of parabolized hull is particularly true for the single midship bulb configuration of 1 5 % beam increment (B15p_single). 49 0.025 0.020 c 0.015 CD o 0.010 -x-B0p_single [P1] (baseline) + B5p_single [P1] * B10p_single [P1] ->-B15p_single [P1] x B20p_single [P1] 0.005 + 0.000 0.20 0.25 0.30 0.35 Froude no (achieved) 0.40 0.45 0.005 0.004 I 0.003 + o 0.002 0.001 0.000 —x-B0p_single [P1] (baseline) + B5p_single [P1] * B10p_single [P1] -»-B15p_single [P1] » B20p_single [P1] -+-0.20 0.25 0.30 0.35 Froude no (achieved) 0.40 0.45 0.025 0.020 to t= 0.015 'o o 0.010 0.005 —x-B0p_single [P1] (baseline) + B5p_single [P1] * B10p_single [P1] -o-B15p_single [P1] * B20p_single [P1] 44 0.000 0.20 0.25 0.30 0.35 0.40 0.45 Froude no (achieved) (a) T M (b) FOM 0.075 [ log(f ln M ) -2] (c) = CTM-CFOM F i g u r e 5 - 3 : [ P 1 ] E f fec ts of i n c r e a s e d b e a m o n r e s i s t a n c e c o e f f i c i e n t s at m o d e l s c a l e ( 1 : 1 3 . 7 5 ) : CTM, CFOM, C R M 50 o — x —BOp_s ing l e [P1] (baseline) — i - — B5p_single[P1] B10p_single [P1] B15p_single [P1] B20p_single [P1] 0.20 0.25 0.30 0.35 Froude no (achieved) 0.40 0.45 0.3 0.2 0.1 — x — B0p_s i ng l e [P1] (baseline) — • — B5p_single [P1] — a — B10p_single [P1] —°—B15p_s ing le [P1] -••«•••• B20p_single [P1] 0.20 0.25 0.30 0.35 Froude no (achieved) 0.40 0.45 100 — x —B0p_s ing l e [P1] (baseline) —•—B5p_s ing le [P1] B10p_single [P1] B15p_single [P1] B20p_single [P1] o.io (a) TM CJMJ " C T M t b a s e n n e y ^TM,baseline j x100 (b) %C, FOM r c - r A ^FOMJ ^ FOM,baseline y ^FOM,baseline j x100 (C) %C, RM CRMJ " C R M p a s e i j n e y ^RM,baseline j x100 Froude no (achieved) Figure 5-4: [P1] Percentage change in effects of increased beam on resistance coefficients at model scale (1:13.75): % C T M , %CFOM, %CRM 51 Hull parabolization was originally tested successfully on lengthier vessels like coastal tanker for resistance reduction (see Calisal et al. [5]). The present tests extended the positive results of hull parabolization to a smaller vessel, so far the results seemed to concur that beam increment is fit to be adapted for small crafts too. The small craft results imply possible practical applications for ships traveling at either slow speeds or moderately high speeds upon proper "tailoring" of the bulb configuration. Although parabolized hull was initially tested on a coastal tanker, our theoretical predictions and experimental results had compliment each other to show that the concept could be implemented on small crafts. However at this point, we shall reserve any decision-making until hull form factors are taken into account. By then, the performance in term of wave resistance, form resistance and viscous resistance can be understood upon resolving the residuary resistance by making use the hull form factor. This will allow a better judgment to be made as to how much the beam can be increased. A different model testing analytical approach other than the standard ITTC approach will be necessary for the determination of hull form factor (k), i.e. Hughes' method working in conjunction with Prohaska's method (see APPENDIX C ). Hughes stated that total resistance is comprised of viscous resistance and wave resistance. Its non-dimensionalized form is such that. CT (R„.Fn )=CF(R„)+ CF0RM (Rn) + Cw (Fn) = CF(Rn)+kCF(Rn)+Cw(Fn) = (-\ + k)CF(Rn)+Cw(Fn) = Cv{Rn)+Cw{Fn) 52 This constant k can be determined experimentally by towing the model in the towing tank at low-speed typically between 0.10 < FnL < 0 .20. The method defines a Reynolds number dependent form drag coefficient that is proportional to frictional resistance coefficient by some constant k, i.e.: CFORM{F(n)VkCF(Rn). The lumped hull form factors (1+k) is found by assuming the wave-making characteristic of the towed model is relatively minimized to such a degree that wave resistance becomes insignificant compare to frictional resistance, i.e.: Cw(Fn)0 0 Thus, CT(Rn,Fn)=(l + k)CF(Rn) -> =(1 + k) But for the determination of hull form factor, it is more common to employ a modified formulation known as Hughes-Prohaska's method: FnN + (\ + k) where 4</V<6 >L *X y-axis x-axis y-intercept For our study, a typical value of 4 is used for N. However, the application of Hughes-Prohaska's method is to be treated with caution for vessels of wet transom, as in this case. As such, the form factors were also calculated alternatively using previously published empirical formulae obtained by Granville (1956), Wright (1984) and Principles of Naval Architects (1988) using regression analysis from statistical database (results are shown in APPENDIX H ) . 53 It is to be noted that the alternative empirical formulae are, either not truthfully reflecting hull characteristics to account for increasing beam while fixing hull length or that they were obtained statistically from model testing not conforming to these effects. Thus, the hull form factors so calculated may only be served as qualitative reference. In Figure 5-5, a comparison was made for hull form factors obtained from experiment using Hughes-Prohaska's method and with those from the alternative empirical formulae. Baseline B5% B10% B15% B20% Figure 5-5: [P1] Effects of increased beam on hull form factors at model scale (1:13.75): (1+k), %(1+k) 54 The results of the form factors using all methods exhibited increasing trend in general, as is expected. However, it is interesting to see that experimental results using Hughes-Prohaska's method showed a reduction for the single-bulb configuration of 1 5 % beam increment (B15p_single). This is an indication that is in agreement with the directly measured total resistance trend (see Figure 5-2b), but a characteristic that is Unable to be revealed by conventional empirical formulae. At this time, the result of form factors by experiment is crucial from design point of view. It suggests that references to form factor using empirical formulae for future hull design or modification with increased beam while fixing hull length is to be proceed with extreme care, at least not until sufficient database is built up. The inferiority of empirical formulae is mainly shown in Figure 5-5b. We can then evaluate how the wave-making and viscous resistance characteristics are affected by the single-bulb configuration affects, such that: - CFOM + kCFOM = (1 + * ) C , FOM ^WM ~ Cm ~ CVM The so-calculated coefficients of viscous resistance and wave resistance and their corresponding percentage variation with respect to the baseline parent hull are shown in Figure 5-6 and Figure 5-7, respectively. 55 0.007 0.006 -*_XX 0.005 -^x->U 0.004 4-0.002 0.001 0.000 —x- B0p_single [P1] (baseline) + B5p_single [P1] A B10p_single [P1] - & - B 15p_ s i n g l e [P1] * B20p_single [P1] -4-0.20 0.25 0.30 0.35 Froude no (achieved) 0.40 0.025 0.020 £ 0.015 -x-B0p_single [P1] (baseline) + B5p_single [P1] & B10p_single [P1] -<>--B15p_single [P1] * B20p_single [P1] 0.000 0.20 0.25 0.30 0.35 0.40 (a) 'VM FOM 0.45 0.45 (b) Froude no (achieved) F i g u r e 5 -6: [P1] E f f e c t s of i n c r e a s e d b e a m o n r e s i s t a n c e c o e f f i c i e n t s at m o d e l s c a l e (1 :13.75) : C V M , C W M 56 35 30 f 25 CD g 20 to —x—BOp_s i ng l e [P1] (baseline) — ^ — B5p_single [P1] —<=•—B10p_single [P1] —°—B15p_s ing l e [P1] B20p_single [P1] •X— X- - X- • • ~ • x— x- - x- - -x- - x- - x- - -x- - -x- • -x- - -x- - x- - -x- - -x- - x- • x- - x- - -X- • -X- - -x- - x-15 10 5 f 0 -X—x—x-0.20 -><—X—X—X—X—><—X—X—X—<x—jx—JX—IX—IX—X—^<—X—X—X—X 0.25 0.30 0.35 Froude no (achieved) 0.40 0.45 (a) VM 'r -r N ^VM.i ^VM,baseline y ^VM,baseline j x100 Froude no (achieved) Figure 5-7: [P1] Percentage change in effects of increased beam on resistance coefficients at model scale (1:13.75): % C V M , % C W M Although quantitative comparison is inappropriate between the experimental wave resistance using Hughes-Prohaska's method and the theoretical wave resistance using Michell's integral, since the theoretical results were only 2-D calculations (see Figure 2-1 a & Figure 5-6b). But a qualitative comparison between the experimental and theoretical results is plausible during the feasibility studies (see Figure 2-1 b and Figure 5-7b). 57 Throughout the tested speeds, we observed similar trend of humps and hollows that only differ by their magnitude and location between the experimental and theoretical results. We see one particularly intense hump of around +50% at FnL • 0.25 in the theoretical results while the experimental results showed +45% at FnLD0.28. And there is a hollow of around - 2 7 % at FnL • 0.30 in the theoretical results while the experimental results gives - 1 8 % at FnL • 0.34. The primary difference between the theoretical and experimental wave resistances lie at the low-speed range 0.20 < FnL < 0.25 where the experimental results indicate significant reductions up to around - 6 0 % whereas the theoretical results indicate the other way around, that being a minute reduction to only - 5 % and then rises steeply to - 4 7 % (see Figure 2-1 b & Figure 5-7b). The qualitative comparisons allow the correlation between the trend effects of the experimental and theoretical results in order to support the experimental findings of the hull form factors that when the hull form factor is lowered, so did the resistance characteristics. Both the experimental and theoretical methods suggest that the test case of 1 5 % beam increment is likely to be a recommended hull-bulb combination, as they lead to better resistance reduction than others. The extrapolation to full-scale effective horsepower was calculated. The results are shown in Figure 5-8. In general when comparing to theoretical results, the humps and hollows in the experimental results showed a lag in speed by around 0.03-0.04 in term of Froude number, and a 5 - 9 % smaller magnitude in percentage change of wave resistance. 58 Q . . C i CD o Q . B5p_single [P1] A B10p_single [P1] ->-B15p_single [average of P1 & P2] * B20p_single [P1] 2000 1800 | | - x - B0p_s ing le [average of P1 & P2] (baseline) j 1600 1400 1200 1000 800 600 400 -E 200 0 7 0.20 + 0.25 0.30 0.35 Froude no (achieved) 0.40 0.45 80 -r O) C - B10p_single •B15p_single B20p_single —x—BOp_s ing le average of P1 & P2] (baseline) B5p_single [P1] P1] average of P1 & P2] ?1] •+-0.20 0.25 0.30 0.35 Froude no (achieved) 0.40 0.45 (a) (EHPs)3D = (RTS\DxVs (b) °HEHPS\D J(EHPs\o.r ( E H P s l x100 Figure 5-8: [P1] Percentage change in effects of increased beam on effective horsepower at full scale (1:13.75): EHPs,3D, % E H P s , 3 D In Table 5-2, the effects of increased beam on various resistances at the model- and full-scale, including the effective horsepower, are summarized. They are average at the design speed regime, i.e. FnL = 0.35 ±0.02. Despite a 14% worsened hull form factor and an increased in viscous resistance by the same order of 13.8%, still, the single-bulb of 1 5 % beam increment outperformed the remaining bulb configurations thanks to the well-reduced14.3% wave resistances. As a result, a rather attractive reduction in effective horsepower of 7.4% was achieved. 59 Table 5-2: [P1] Resistance summary of the effects of increased beam using single midship bulb (scale=1:13.75, averaged within design speed regime of FnL=0.35+/-0.02) B5p_single [P1] B10p_single [P1] B15p_single [P1] B20p_single [P1] [% change wrt baseline] [% change wrt baseline] [% change wrt baseline] [% change wrt baseline] (1+k) 6.0 18.0 14.0 19.0 -2.6 -1.6 -4.1 -0.1 C-TM -2.6 -1.7 -4.4 -0.7 CFOM 0.0 0.0 -0.2 -0.2 CRM -3.6 -2.4 -6.2 -0.9 CVM 6.2 17.8 13.8 18.6 CWM -7.3 -12.2 -14.3 -11.1 CFOS 0.0 0.0 0.0 0.0 CRS -3.6 -2.4 -6.2 -0.9 CTS (2D) -2.9 -1.9 -4.9 -0.7 Cvs 6.2 17.8 14.1 18.9 Cws -7.3 -12.2 -14.3 -11.1 CTS (3D) -4.1 -5.3 -7.6 -4.2 E H P S (3D) -4.3 -5.2 -7.4 -3.7 The hydrostatics performance as shown in Table 5-3 was calculated from keel up to design waterline level. Table 5-3: [P1] Hydrostatics summary of the effects of increased beam using single midship bulb (scale=1:13.75) SUMMARY @ DWL : B0p_single B5p_single B10p_single B15p_single B20p_single V, total [mm3l 1.26E+08 1.27E+08 1.28E+08 1.30E+08 1.31 E+08 W, total TNI 1232.11 1248.19 1260.54 1271.01 1280.93 LCB %LPP wrt midship (+fore / -aft) -3.84 -3.82 -3.79 -3.75 -3.72 LCF %LPP wrt midship (+fore / -aft) -7.60 -7.52 -7.40 -7.28 -7.17 KB [mm] 118.54 118.85 119.13 119.40 119.66 KM (=KB+BM) [mm] 241.44 246.38 252.64 259.51 266.89 KG (=draft, T) [mm] 203.53 203.53 203.53 203.53 203.53 GM (=KM-KG) [mm] 37.91 42.85 49.11 55.99 63.36 Cb - 0.60 0.58 0.56 0.54 0.52 Cp - 0.68 0.66 0.65 0.64 0.64 Cw - 0.82 0.79 0.77 0.74 0.72 Cm - 0.89 0.88 0.86 0.84 0.82 60 The results also indicated improved stability of the parabolized hull over the baseline parent hull. The improvement are exhibited in terms of reduced block coefficient from 0.60 to 0.54, reduced prismatic coefficient from 0.68 to 0.64, increased metacentric height from 37.91mm to 55.99mm. fP11 Summary The preliminary investigations on the effect of increased beam and influence on hull form factor are convincing enough at both the experimental and theoretical level to conclude that, a certain amount of beam increment and the elimination of parallel middle body is "good" for reduction in resistance over certain speed range. This is doable upon proper "tailoring". Finally, the resistance trend had shown that the single-bulb with 15% beam increment at midship (B15p_single) is in favoured of beneficial resistance-reduction. This was justified in experimental studies to compliment the theoretical findings. Questions remained to be asked now are probably: - What will the repeatability tests say about the selected configuration? 5.1.2 [P2] Repeatability Analys is The repeatability tests were done in Phase 2 [P2] to confirm the conclusion in Phase 1 [P1] that the 1 5 % beam increment single-bulb is the favoured hull-bulb combination. For this test series, only two configurations were tested, i.e. the parent hull (B0p_single) and the parent hull modified with single midship bulb of 15% beam increment (B15p_single). The test programs are briefed in Table 5-4. 61 Table 5-4: [P2] Repeatability tests using selected single-bulb configuration. Phase 2 - Confirmation & repeatability tests on selected single bulb parabolization test configurations test designations Parent hull BOp_single Parent hull + selected 15% parabolization @ midship B15p_single Besides the routine towing tests, included in this test series were the underwater flow visualization tests and the wave pattern profiling system tests. The wave pattern profiling system was put to test to acquire the wave pattern trailing behind the towed model in order to compute the wave resistance values. For the towing tests, similar resistance test program as in Phase 1 [P1] were conducted three times (i.e. test set A, B and C) in this phase for each configuration in order to allow for repeatability checking (see Table 5-5). But strictly speaking, only two out of the three data sets served this purpose (i.e. test set A and B) whereas test set C is irrelevant since yarn tufts were used on the underside of the hull for flow visualization tests. The yarn tufts incurred excess drag on the towed hull and increased the total resistance. Although results from test set C is inapplicable for repeatability checking, it is still useful in the sense that the results can serve as a comparable trend over the range of speeds tested. The repeatability analysis using n=3 data sets (see Table 5-5) was found that the maximum upper and lower limit of all the experimental results are bounded within ±5% with respect to the average values. Summary and details of the repeatability analysis is tabulated in APPENDIX I. 62 Table 5-5: [P2] Data sets used for repeatability analysis (n=3sets). n=3 sets Parent hull Selected modified hull Phase 1 [P1] Data set 1 BOp_single B15p_single Phase 2a [P2a] Data set 2 BOp_single_setA B15p_single_setA Data set 3 BOp_single_setB B15p_single_setB Considering the fact that the tests were not conducted consecutively one by one or even day-by-day, but rather over different time periods. Thereby, the results incurred variation in testing conditions in the towing tank, that being fluctuation in water temperature, density, viscosity and water level in towing tank. Although it is believed that their effects are relatively smaller than others, but since we are interested in small-scale changes in resistance characteristics, details such as this might be a concern too. Some other possible sources of error are listed below: - Attaching and re-attaching of the side bulbs - Application & manual shape molding of plasticine - Increased surface roughness due to scraping off of plasticine layer - Ballast weights & locations - Ballast weights securing - Model trimming condition - Towing carriage vibration & jerkiness - Coincidence of centerlines between ship model and towing carriage Although great efforts was put in to maintain consistency for all of the above-mentioned settings during each test, but unquestionably human factors are to be factored in as they could very possibly contribute in one way or another to experimental measurement deviations. But with the repeatability analysis established using low sampling of only n=3 data sets and taking into 63 account many of the above-mentioned sources of error, the maximum deviations of ±5% from average is deemed very good reliability in this case. This marks the end of test series conducted in Phase 2 [P2]. But for the purpose of comparison of this test series, it will make more sense to compare with results from Phase 3 [P3], which were also of single-bulb configuration only that the maximum beam location is shifted to investigate such effect on resistance characteristics. fP21 Summary In conclusion, the repeatability analysis showed that the maximum deviations from average are within ±5%. This is deemed acceptable within the limitation of using the add-on bulbs. The repeatability tests results support the conclusion from Phase 1 [P1] by seconding the elimination of parallel middle body by hull parabolization. There exists an optimum beam increment that enjoys beneficial resistance reduction over the targeted speed range while improving payload capacity. In this case, the "good" configuration was found to be of 1 5 % beam increment. It is to be emphasized that any comparisons made this point onward to control hull (BOP_single) and the selected 1 5 % beam increment single-bulb (B15p_single) are based on their corresponding average values from Phase 1 & 2 [P1 & P2], i.e.: For the control hull: use average of results from \ - Phase_1_B0P_single - Phase_2_B0P_single_setA - Phase_2_B0P_single_setB 64 For the selected single-bulb of 1 5 % beam increment: ' - Phase_1_B15P_single use average of results from I - Phase_2_B15P_single_setA - Phase_2_B15P_single_setB Questions remained to be asked now are probably: - What will happen if max. beam location is shifted forward or aftward? - What will happen if the rear of the bulb is being faired with extended length along the hull? 5.1.3 [P3] Effects of Maximum Beam Location Using Single Bulb The main purpose of this test series was to study effects of bulb location, i.e. how much influence the location of maximum beam has on the resistance if the bulb is shifted to forebody and aftbody as oppose to the previously tested midship location. The test programs are briefed in Table 5-6. Table 5-6: [P3] Tests on effects of maximum beam location (forebody & aftbody) Phase 3 - Investigation on effects of max. beam location using selected single bulb parabolization test configurations test designations Parent hull + selected 15% parabolization @ midship B15p_single Parent hull + selected 15% parabolization @ forebody B15p_single_F Parent hull + selected 15% parabolization @ aftbody B15p_single_A The forebody bulb has its maximum beam location fixed at station 6 (-0.1 O L ^ ) and blended into the hull between - 0 . 2 5 / ^ < < 0 ^ . Whereas the aftbody bulb has its maximum beam location fixed at station 4 (+0.10/.^) x and blended into the hull between OL^ < — <+0.25LWL. 65 The profiles of the tested parabolized hulls at the waterline level are shown in Figure 5-9. • BOp_single_midship (baseline) - • B15p_single_midship B15p_single_aft (selected) B15p_single_fore 0.3 0.4 0.5 x/L, normalized length along design waterline Figure 5-9: [P3] Profiles of tested parabolized hull at design waterline level. Together with selected single-bulb parabolization of 15% beam increment from Phase 2 [P2], we will have sufficient results after this test series to address the resistance characteristics with respect to bulb location at fore body, midship and aft body. The fore and aft bulbs tested were also of the same selected 15% beam increment. The measured model scale resistance and the corresponding percentage variation with respect to parent hull are shown Figure 5-10a & b, respectively: It is to be noted that both the fore bulb and the aft bulb span only half the length of the originally used single midship bulb. Nevertheless, it can be shown that having the bulb at fore body offers no benefit at all whereas with the bulb at aft body, minute beneficial reduction in resistance could be attained at higher-end of the speeds Fn L >0.35. It is likely that the flow separation in this case occurs much earlier near the midship and trails a long distance to the stern. As such, it leads to severe 66 growth of boundary layer that induced excess resistance. Same observation can be drawn even when the results are presented in non-dimensionalized forms as shown in Figure 5-11 & Figure 5-12. 20 15 .a CD is 10 CD DC co o 5 + — x — B 0 p _ s i n g l e [average of P1 & P2] (baseline) — 0 —B15p_s i ng l e_m id sh i p [average of P1 & P2] -•-•+•-•-B15p_single_forebody [P3] & B15p_single_aftbody [P3] H — ' 0.20 0.25 ' — I — 1 0.30 0.35 0.40 0.45 Froude no (achieved) 160 — x — B 0 p _ s i n g l e [average of P1 & P2] (baseline) —o—B15p_s ing le_midsh ip [average of P1 & P2] •••-+•-• B15p_single_forebody [P3] • A — B15p_single_aftbody [P3] 0.20 0.25 0.30 0.35 0.40 0.45 (a) TM (b) %RT ^TM.i " ^TM,baseline y TM.baseline J x100 Froude no (achieved) Figure 5-10: [P3] Effects of maximum beam location on measured total resistance at model scale (1:13.75): R TM, %RTM Interestingly, it appeared that we might be able to adapt the parabolized single-bulb of 1 5 % beam increment and shift the maximum beam location from the originally tested midship location to aft body in order to open up the window of resistance reduction to higher speed range between 0.35 < FnL < 0 .45. This may offer minute resistance reduction in the order of 1 - 3 % particularly (see Figure 5-10b).. Details of the resistance coefficients 67 and their corresponding percentage variation with respect to baseline parent hull are shown in Figure 5-11 and Figure 5-12, respectively. Q o O 0.035 0.030 0.025 0.020 0.015 0.010 0.005 0.000 : — X — •B0p_single [average of P1 & P2] (baseline) Q •B15p_single_midship [average of P1 & P2] .: - - + - B15p_single_forebody [P3] •,+-: & ••• B15p_single_aftbody [P3] : .. + ' 1 ..,,+..' ; „.ja*cift : : i : r : - A £ 2 ^ ^ ^ f ; \ > i : (a) 'TM n TM IPMSMK M 0.20 0.25 0.30 0.35 Froude no (achieved) 0.40 0.45 c 'o o O 0.005 0.004 0.003 0.002 0.001 i 0.000 0.20 — x — B 0 p _ s i n g l e [average of P1 & P2] (baseline) — o — B15p_single_midship [average of P1 & P2] - -+ - • B15p_singleJorebody [P3] A B15p_single_aftbody [P3] '—I— L 0.25 4-(b) •'FOM 0.075 [\og(RnM)-2] 0.30 0.35 Froude no (achieved) 0.40 0.45 o O 0.030 0.025 0.020 0.015 0.010 0.005 — x — BOp_single [average of P1 & P2] (baseline) -—-°—B15p_single_midship [average of P1 & P2] - -+ - B15p_singleJorebody [P3] & ••- B15p_single_aftbody [P3] 0.000 (c) 'RM - C -C ' ^TM FOM 0.20 0.25 0.40 0.45 0.30 0.35 Froude no (achieved) Figure 5 - 1 1 : [ P 3 ] Effects of maximum beam location on resistance coefficients at model scale ( 1 : 1 3 . 7 5 ) : C T M , C F O M , C R M 68 CO o> c CO 160 140 120 100 80 60 40 20 0 -20 — x—B 0 p _ s i n g l e [average of P1 & P2] (baseline) — ° — B15p_single_midship [average of P1 & P2] ••-•+••• B15p_single_forebody [P3] • - -A- - - - B15p_single_aftbody [P3] 2.0 1.5 1.0 0.5 0.0 -0.5 0.20 i i • • . i " A , . . i . . . . (a) TM 1r -r A ^TM,i ^TM,baseline y ^TM,baseline j x100 0.20 0.25 0.30 0.35 Froude no (achieved) 0.40 0.45 ••ft •& - x — B0p_single [average of P1 & P2] (baseline) -o—B15p_s ingle_midship [average of P1 & P2] •-+••-• B15p_single_forebody [P3] -A B15p_single_aftbody [P3] (b) 0.25 0.30 0.35 0.40 0.45 %c, FOM rr -r A FOM,i v-yFOM, baseline y ^FOM,baseline j x100 260 240 220 200 180 160 140 120 100 80 60 40 20 0 -20 Froude no (achieved) — x — B 0 p _ s i n g l e [average of P1 & P2] (baseline) — 0 —B15p_s ing le_midsh ip [average of P1 & P2] •-•+••-B15p_single_forebody [P3] • • - A — B15p_single_aftbody [P3] (C) RM 'r -r A ^RMJ ^RM,baseline RM,baseline x100 0.20 0.25 0.40 0.45 0.30 0.35 Froude no (achieved) Figure 5-12: [P3] Percentage change in effects of maximum beam location on resistance coefficients at model scale (1:13.75): % C T M , % C F O M I % C R M 69 By observing the trend of the residuary resistances as shown in Figure 5-12c, it seems that at speeds between 0.20 < FnL < 0.35 we observed that: R > R > R 7M, single fore bulb TM, single aft bulb TM, single midship bulb Such resistance trend when overlapping onto the trend of hull form factor, is only being more closely matched by those obtained experimentally using Hughes-Prohaska's method (see Figure 5-13b): e fore bulb ^  0 O^single aft bulb single midship bulb 22% 17% 14% 1.60 1.55 4 1.50 1.45 1.40 1.35 1.30 1.25 1.20 I (1 +k) experiment • (1 +k1) PNA (1988) I (1+k) Wright (1984) • (1 +k)Granville (1956) 1.49 1.23 Baseline B15% (single, B15% (single, fore B15% (single, aft midship) body) body) 30 25 —i—(1+k) experiment -x-(1+k1) PNA (1988) -o— (1+k) Wright (1984) (1+k)Granville (1956) Baseline B15% (single, midship) 3 4 B15% (single, fore B15% (single, aft body) body) (a) (1 + /C) (b) %(1 + /c) x100 Figure 5-13: [P3] Effects of increased beam on hull form factors at model scale (1:13.75): (1+k), %(1+k) 70 For all of the occasions, we found that the hull form factors calculated using empirical formulae were not able to match Hughes-Prohaska's trend at all. Moreover, the hull form factors calculated using empirical formulae showed no correlation with the measured total resistance. As previously established in Phase 1 [P1], it was deemed more appropriate to use Hughes-Prohaska's method to resolve the residuary resistance in Order to study the characteristics of wave resistance and viscous resistance (see Figure 5-14 & Figure 5-15). 0.008 0.006 x - x - x •M 0.004 f o O 0.002 0.000 X ~ X " X ~ X ~ X ~ X - X - x - x - x - > < - > < - > < - > < - x ^ < - ^ ^ - x—B0p_s i ng l e [average of P1 & P2] (baseline) -°—B15p_single_midship [average of P1 & P2] -+ - B15p_single_forebody [P3] -a B15p_single_aftbody [P3] 4-0.025 0.020 | 0.015 0.20 0.25 0.30 0.35 Froude no (achieved) 0.40 — x — B 0 p _ s i n g l e [average of P1 & P2] (baseline) — o — B15p_single_midship [average of P1 & P2] - -+ - • B15p_single_forebody [P3] A B15p_single_aftbody [P3] „ x_X-x^5S». A" 0.000 0.20 0.25 0.30 0.35 0.40 (a) 'VM = (l + k)C, FOM 0.45 (b) ^WM ^  = - &VM 0.45 Froude no (achieved) Figure 5 - 1 4 : [ P 3 ] Effects of maximum beam location on resistance coefficients at model scale ( 1 : 1 3 . 7 5 ) : C V M , C W M 71 40 35 30 25 20 15 10 5 0 - X — x - x -0.20 400 350 300 250 — x—B 0 p _ s i n g l e [average of P1 & P2] (baseline) —o—B15p_s ing le_midsh ip [average of P1 & P2] ----+•-.. B15p_singleJorebody [P3] — A — B15p_single_aftbody [P3] • •'•--+ •+ + + +•-+•-+•• (a) VM rr -r ^ ^VM,i ^VMfiaseline VM,baseline x100 < X—X—X—X—X—X—X—X—X-X—X— X—X—X—X—X—X—X—X— X—I-0.25 0.30 0.35 0.40 0.45 Froude no (achieved) c CO -j — x — B0p_single [average of P1 & P2] (baseline) —o—B15p_s ing le_midsh ip [average of P1 & P2] •---+••-• B15p_single_forebody [P3] • - - -A- • • • B15p single aftbody [P3] ..:- • •+• -& > $ . ' i : I i . . . i • . . . i . . . . i . . . . (b) %C, WM 'WM.i 'WM,baseline WM,baseline J x100 100 50 0 -50 -100 0.20 0.25 0.30 0.35 0.40 0.45 Froude no (achieved) Figure 5-15: [P3] Percentage change in effects of maximum beam location on resistance coefficients at model scale (1:13.75): % C V M , %CwM 72 I Q. 2500 2000 1500 1000 500 + - x — B0p_single [average of P1 & P2] (baseline) -o—B15p_single_midship [average of P1 & P2] -+ - B15p_single_forebody [P3] -A B15p_single_aftbody [P3] 0.25 0.30 0.35 Froude no (achieved) 0.40 (a) (EHPs\D = (RTS)3DxVs 0.45 0.20 — x—B 0 p _ s i n g l e [average of P1 & P2] (baseline) B15p_single_midship [average of P1 & P2] B15p_single_forebody [P3] B15p_single_aftbody [P3] 0.25 0.30 0.35 0.40 0.45 (b) \EHPs\o,r(EHPs\ (EHPs\ x100 Froude no (achieved) Figure 5-16: [P3] Percentage change in effects of increased beam on effective horsepower at full scale (1:13.75): E H P s > 3 D , % E H P s , 3 D At speeds FnL < 0.32, though the aft bulb performed relatively better than the fore bulb, still they give no beneficial resistance reduction. But it is between 0.35 < FnL < 0.45, that the aft bulb outperforms the midship bulb by averaging almost a 10% more reduction in wave resistance (see Figure 5-15b). This corresponds to an averaging 8 % more reduction in total resistance over the same range of speeds (see Figure 5-12a). 73 The hydrostatics performance as shown in Table 5-7 was calculated from keel up to design waterline level. The results also indicated improved stability of the parabolized hull over the baseline parent hull. The improvements are exhibited in terms of reduced block coefficient from 0.60 to 0.546, increased metacentric height from 37.91mm to 43.76mm. The prismatic coefficient maintained at approximately 0.689 and showed insignificant variation. Table 5-7: [P3] Hydrostatics summary of the effects of increased beam at model scale (1:13.75) SUMMARY @ DWL : B0p_single B15p_single B15p_single_ forebody B15p_single_ aftbody V, total [mm3] 1.26E+08 1.30E+08 130475842.21 130512529.42 W, total [Nl 1232.11 1271.01 1279.97 1280.33 LCB %LPP wrt midship (+fore / -aft) -3.84 -3.75 -3.21 -4.19 LCF %LPP wrt midship (+fore / -aft) -7.60 -7.28 -6.94 -7.77 KB [mml 118.54 119.40 118.86 118.32 KM (=KB+BM) [mm] 241.44 259.51 248.98 247.29 KG (=draft, T) [mm] 203.53 203.53 203.53 203.53 G M (=KM-KG) [mm] 37.91 55.99 45.45 43.76 Cb - 0.60 0.54 0.545 0.546 Cp - 0.68 0.64 0.689 0.689 Cw - 0.82 0.74 0.732 0.727 Cm - 0.89 0.84 0.792 0.792 Therefore, it can be concluded that having bulb at the aft body is likely to improve to certain extend the resistance characteristics in the relatively higher speed range without compromising on stability performance. TP31 Summary W e concluded that the trend of hull form factors using Hughes-Prohaska's method that is based on experimental results matches more closely with resistance trend. Thus, Hughes-Prohaska's method is indeed a better choice to be used for wave resistance and viscous resistance calculations. 74 In conclusion, shifting the 1 5 % beam increment bulb to fore body only make worse the resistance performance whereas shifting to aft body offers beneficial reduction in resistance at moderate to higher speeds. Question remained to be asked now is probably: - What will happen if the rear of the bulb is being faired with extended length along the hull? 5.1.4 [P4] Effects of Fairing Extension at Rear of Single Bulb This test series is concerned about the effects in hull form factors with the fairing extension being applied to the rear of the selected single midship bulb of 1 5 % beam increment (B15p_single) from Phase 1 & 2. The fairings were meant to smoothen the flow exit angle upon leaving the bulb. In this matter, the hull form factors were major concern. As such, only slow-speed tests were conducted to determine the hull form factors and calmwater resistance tests at design speed regime (FnL = 0.30 , 0.35) were tested. The fairings were applied from midship and extended aftward at lengths of 2 5 % , 3 0 % and 3 5 % of waterline length, which were equivalent to 490mm (alphal), 610mm (alpha2) and 720mm (alpha3), respectively. The test activity is outlined in Table 5-8. Table 5-8: [P4] Tests on effects of fairing extension at rear of single-midship bulb configuration. Phase 4 - Investigation on effects of fairing extension using selected single bulb parabolization test configurations test designations Parent hull + selected 15% parabolization @ midship B15p_single Parent hull + selected 15% parabolization @ midship + fairing extension of 25%LWL B15p_single_alpha1 Parent hull + selected 15% parabolization @ midship + fairing extension of 30%LWL B15p_single_alpha2 Parent hull + selected 15% parabolization @ midship + fairing extension of 35%L W L B15p_single_alpha3 75 The profiles of the tested parabolized hulls at the waterline level are shown in Figure 5-17. TJ CO CD -Q "CO -o CD _N "CO E i— o c s E m 1.4 1.2 1 0.8 0.6 0.4 0.2 0 i \. ' ; ; : : --^0 i i BOp_single_midship (baseline) 1 1 B15p_s ing le_midsh ip_L35p (selected) B15p_s ing le_midsh ipJ_25p B15p_s ing le_midsh ip_L30p -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 x/L, normalized length along design waterline Figure 5-17: [P4] Profiles of tested parabolized hull at design waterline level. In Figure 5-18, the trend effect of fairing extension on hull form factors are shown. With the exception of the 18% from the case of the second-longest fairing extension (B_15p_single_midship_alpha2), the remaining configurations of the fairing extension point out that hull form factors improved with increased fairing extension at rear of the bulb (see Figure 5-18b), such that: 14% (without fairing extension, B_15p_single_midship) JJ- improved using shortest fairing extension 1 3 % (B_15p_single_midship_alpha1) U- further improved using longest fairing extension 1 2 % (B_15p_single_midship_alpha3). 76 1.60 1.55 1.50 1.45 1.40 1.35 1.30 1.25 1.20 • (1+k) experiment B(1+k1) PNA (1988) B (1+k) Wright (1984) • (1+k)Granville (1956) Baseline B15% (single, B15% (single, B15% (single, B15% (single, midship) midship, midship, midship, alphal) alpha2) alpha3) 30 25 4 (1+k) experiment -(1+k) Wright (1984) * - ( l + k 1 ) PNA (1988) (1+k)Granville (1956) B15% (single, midship) B15% (single, B15% (single, B15% (single, midship, alphal) midship, alpha2) midship, alpha3) (b) %(1 + k) 'I v 'baseline > baseline x100 Figure 5-18: [P4] Effects of increased beam on hull form factors at model scale (1:13.75): (1+k), %(1+k) 77 Although the fairing extensions did not improve significantly the resistance performance between 0.20 < FnL < 0 .25, the test runs between 0.30 < FnL <0.35 did show reduction by around 8 - 1 0 % with respect to the selected single midship bulb (B15p_single) (see Figure 5-19b, Figure 5-20a & Figure 5-21 a). The improvement was more noticeable with increased extension lengths. Even though no beneficial total resistance could be obtained between 0.20 < FnL < 0.25 where the intense peaks were observed, the magnitude of the peaks did settle down to certain degree (see Figure 5-19b, Figure 5-21 a). 20 15 CD cr - x—B0p_ s i n g l e [average of P1 & P2] (baseline) -o—B15p_single_midship [average of P1 & P2] -+ - - B15p_single_aftbody [P3] A B15p_single_midship_alpha1 [P4] * B15p_single_midship_alpha2 [P4] o B15p_single_midship_alpha3 [P4] 0.20 0.25 0.30 0.35 0.40 (a) R. TM 0.45 CD D ) cz CB SZ o 160 140 120 100 80 60 40 20 0 -20 Froude no (achieved) — x — B 0 p _ s i n g l e [average of P1 & P2] (baseline) —o—B15p_s ing l e_m idsh ip [average of P1 & P2] B15p_single_aftbody [P3] B15p_single_midship_alpha1 B15p_single_midship_alpha2 B15p_single_midship_alpha3 (b) %R-0.20 0.25 0.30 0.35 0.40 0.45 TM fa a \ TMJ ^TMfiaseline y TM,baseline J x100 Froude no (achieved) Figure 5 - 1 9 : [ P 4 ] Effects of increased beam on measured total resistance at model scale ( 1 : 1 3 . 7 5 ) : R T M , % R T M 78 0.030 0.025 m 0.020 c 0) in 0.015 o ° 0.010 0.005 . 0 — x — B 0 p _ s i n g l e [average of P1 & P2] (baseline) — o — B15p_single_midship [average of P1 & P2] — - B 1 5 p _ s i n g l e _ a f t b o d y [P3] A B15p_single_midship_alpha1 [P4] * B15p_single_midship_alpha2 [P4] n B15p_single_midship_alpha3 [P4] \ b ^ p * ^ 4-(a) 'TM R-TM 0.20 0.25 0.30 0.35 Froude no (achieved) 0.40 0.45 0.005 0.004 0.003 o 0.002 0.001 0.000 — x—B 0 p _ s i n g l e [average of P1 & P2] (baseline) —o—B15p_single_midship [average of P1 & P2] — + - B15p_single_aftbody [P3] A B15p_single_midship_alpha1 [P4] x B15p_single_midship_alpha2 [P4] o B15p_single_midship_alpha3 [P4] (b) 'FOM 0.075 [log(/?nM) -2] 0.20 0.25 0.30 0.35 Froude no (achieved) 0.40 0.45 0.025 0.020 <n I 0.015 ° 0.010 0.005 0.000 : — x — -B0p_single [average of P1 & P2] (baseline) •B15p_single_midship [average of P1 & P2] — 1 ~ B15p_single_aftbody [P3] : - - - • ! A B15p_single_midship_alpha1 [P4] X B15p_single_midship_alpha2 [P4] • B15p_single_midship_alpha3 [P4] --- V — X m -. - •F— -_ _ +^rr ' ' I ' ' ' ' i ' ' ' ' i ' ' (c) 'RM ~ ^TM'^FOM 0.20 0.25 0.40 0.45 0.30 0.35 Froude no (achieved) F i g u r e 5 - 2 0 : [ P 4 ] E f f e c t s of i n c r e a s e d b e a m o n r e s i s t a n c e c o e f f i c i e n t s at m o d e l s c a l e ( 1 : 1 3 . 7 5 ) : C T M , C F O M , C R M 79 CD 160 140 120 100 80 60 — x — B 0 p _ s i n g l e [average of P1 & P2] (baseline) —o—B15p_s ing le_midsh ip [average of P1 & P2] --+--B15p_single_aftbody [P3] A B15p_single_midship_alpha1 [P4] x B15p_single_midship_alpha2 [P4] a B15p_single_midship_alpha3 [P4] 0.20 0.25 0.30 0.35 0.40 0.45 (a) TM TM.i TM,baseline y TM .baseline j x100 8 7 6 5 4 3 2 1 0 -6—o—o--1 0.20 0.25 Froude no (achieved) *—B0p_s ing le [average of P1 & P2] (baseline) °—B15p_single_midship [average of P1 & P2] + - - B15p_single_aftbody [P3] A B15p_single_midship_alpha1 [P4] x B15p_single_midship_alpha2 [P4] Q B15p_single_midship_alpha3 [P4] 0.30 0.35 0.40 0.45 (b) FOM ^FOM.i " ^FOM.baseline yFOM,basetine x100 CD D) C CO 350 300 250 f 200 150 100 -50 Froude no (achieved) - x — B0p_single [average of P1 & P2] (baseline) -o—B15p_single_midship [average of P1 & P2] - + - - B15p_single_aftbody [P3] B15p_single_midship_alpha1 B15p_single_midship_alpha2 B15p_single_midship_alpha3 P4] P4] P4] 0.20 0.25 0.30 0.35 0.40 (C) 0.45 %C, RM f e e A ^RMj ^RM,baseline y ^RM,baseline j x100 Froude no (achieved) F i g u r e 5 - 2 1 : [ P 4 ] P e r c e n t a g e c h a n g e in e f fects of i n c r e a s e d b e a m o n r e s i s t a n c e c o e f f i c i e n t s at m o d e l s c a l e ( 1 : 1 3 . 7 5 ) : % C T M , % C F O M , % C R M 80 CD O O 0.008 0.002 4-0.000 — x — B 0 p _ s i n g l e [average of P1 & P2] (baseline) —°—B15p_s i ng l e_m id sh i p [average of P1 & P2] —••- B15p_single_aftbody [P3] A B15p_single_midship_alpha1 [P4] x B15p_single_midship_alpha2 [P4] a B15p_single_midship_alpha3 [P4] (a) 0.20 0.25 0.30 0.35 0.40 0.45 Froude no (achieved) 0.025 0.020 0.015 o 0.010 + 0.005 0.000 •x—B0p_s ing le [average of P1 & P2] (baseline) o—B15p_s ing le_midsh ip [average of P1 & P2] +- B15p_single_aftbody [P3] A B15p_single_midship_alpha1 [P4] x B15p_single_midship_alpha2 [P4] a B15p_single_midship_alpha3 [P4] 'WW (i + k)C, FOM (b) ^WM ~ CJM ~ CVM 0.20 0.25 0.40 0.45 0.30 0.35 Froude no (achieved) Figure 5-22: [P4] Effects of increased beam on resistance coefficients at model scale (1:13.75): C V M , C W M 81 40 35 30 CD 25 I 20 15 10 5 0 400 350 -I 300 -!• 250 -\ CD O) 200 -\ c CO .c 150 --_ o : # 100 50 0 -X A X a -B0p_single [average of P1 & P2] (baseline) -B15p_single_midship [average of P1 & P2] -B15p_single_aftbody [P3] B15p_single_midship_alpha1 [P4] B15p_single_midship_alpha2 [P4] B15p_single_midship_alpha3 [P4] (a) %C, —x— BOps ing le [average of P1 & P2] (baseline) — ° — B 1 5 p single midship [average of P1 & P2] - - + - - B15pIsingleIaftbody [P3] A B15p_single_midship_alpha1 x B15p_single_midship_alpha2 n B15p_single_midship_alpha3 VM C -C ^VM,baseline y VM.baseline j x100 •><—X—X—I ' X— X — X — X — X — > < — X — X — X — X—fc— I X — X — X — X — > < — X — X — X — X -0.20 0.25 0.30 0.35 0.40 0.45 Froude no (achieved) (b) fc-r A ^WM.i ^WM, baseline WM,baseline x100 0.20 0.25 0.40 0.45 0.30 0.35 Froude no (achieved) Figure 5 - 2 3 : [ P 4 ] Percentage change in effects of increased beam on resistance coefficients at model scale ( 1 : 1 3 . 7 5 ) : % C V M , % C W M 82 I o CL 1800 1600 1400 1200 1000 800 600 400 200 0 - x—B 0 p _ s i n g l e [average of P1 & P2] (baseline) -°—B15p_s ing le_midship [average of P1 & P2] -+--B15p_single_aftbody [P3] A B15p_single_midship_alpha1 [P4] x B15p_single_midship_alpha2 [P4] o B15p_single_midship_alpha3 [P4] 0.20 180 160 140 120 100 80 60 40 20 0 -20 H — ' — ' — ' — 1 — I — 0.25 0.30 Froude no (achieved) — x — B O o s i n g l e [average of P1 & P2] (baseline) — ° — B15p single midship [average of P1 & P2] — ' B15pIsingleIaftbody [P31 A B15p_single_midship_alpna1 [P4 x B15p_single_midship_alpha2 P4 o B15p_single_midship_alpha3 [P4 (a) (EHPs\D = (RTS\DxVs 0.35 0.40 0.45 (b) °HEHPS\D J(EHPsXDJ-(EHPsX^ x100 0.20 0.25 0.40 0.45 0.30 0.35 Froude no (achieved) Figure 5-24: [P4] Percentage change in effects of increased beam on effective horsepower at full scale (1:13.75): E H P s > 3 D , % E H P s , 3 D 83 The hydrostatics performance as shown in Table 5-9 was calculated from keel up to design waterline level. The results also indicated improved stability of the parabolized hull over the baseline parent hull. The improvements are exhibited in terms of reduced block coefficient from 0.604 to 0.547, reduced prismatic coefficient from 0.680 to 0.651 and increased metacentric height from 37.91mm to 59.40mm. Assumption was made such that the center of gravity from keel (KG) is approximately equal to the draft. Table 5-9: [P4] Hydrostatics summary of the effects of increased beam at model scale (1:13.75) SUMMARY @ DWL : BOp single B15p_single B15p_single_ alphal B15p_single_ alpha2 B15p_single_ alpha3 V, total [mm3] 1.256E+08 1.296E+08 1.303E+08 1.306E+08 1.308E+08 W, total fNl 1232.11 1271.01 1278.04 1281.06 1283.11 LCB %LPP wrt midship (+fore / -aft) -3.84 -3.75 -3.84 -3.89 -3.92 LCF %LPP wrt midship (+fore / -aft) -7.60 -7.28 -7.36 -7.40 -7.43 KB [mml 118.54 119.40 119.52 119.59 119.64 KM (=KB+BM) [mml 241.44 259.51 261.69 262.08 262.93 KG (=draft, T) [mml 203.53 203.53 203.53 203.53 203.53 G M (=KM-KG) [mm] 37.91 55.99 58.17 58.55 59.40 Cb - 0.604 0.542 0.545 0.546 0.547 Cp - 0.680 0.645 0.648 0.650 0.651 C w - 0.817 0.743 0.748 0.749 0.750 Cm - 0.887 0.840 0.840 0.840 0.840 TP41 Summary In conclusion, it was found that hull form factor and resistance characteristics improved with increased extended fairing length at rear of the bulb. 5.1.5 Summary of Influences on Hull Form Factors & Hydrostatics Since it was already shown in previous sections that trend of hull form factors obtained using Hughes-Prohaska's method based on experimental 84 results are of closer match with trend of resistance, the hull form factors using empirical formulae will not be addressed herein. The effects of increased beam / maximum beam location / fairing extension on hull form factors for each tested single- & double-bulb configuration are plotted in Figure 5-25a with the percentage variation with respect to baseline plotted in Figure 5-25b. When the beam is widened step-wise from 0 % to 2 0 % using single-bulb at midship, we see that the hull form factors varied between 0 - 1 9 % . It was found that there is indeed a preferred beam increment when attempting to parabolize the hull. It was determined that this "sweet" setting is of 15% beam increment. Although the hull form factor is in fact increased by 14% with respect to baseline at this very setting, nevertheless, it exhibited better beneficial resistance-reduction capability over the tested speeds. Within the targeted speed range between 0.30 < FnL <0.40, it resulted in significant reduction in wave resistance by nearly 18% that leads to reduction in total resistance by 8 % . It is to be noted that this improvement is obtained simply by parabolizing the hull to eliminate the parallel middle body, and it was done without detail "tailoring". As such, a properly "tailored" bulb will definitely see more improvement. Tests for the effects of maximum beam location on resistance indicate that having maximum beam at fore body is a disaster (steep rose of hull form factor to 22%), while a location somewhere between the midship and aft body is preferred (subtle rose of hull form factor to 14-17%) . Tests for the effects of fairing extension at rear of the original single-bulb (the one with the parallel middle body being eliminated) showed that hull form factors improved with increased fairing length. This corresponds to the 1 3 % 85 from the shortest fairing length (approximately 2 5 % of waterline length, i.e. 490mm beginning from midship) and the 1 2 % from the longest fairing length (approximately 3 5 % of waterline length, i.e. 702mm beginning from midship). The test for the intermediate length is inconclusive as the form factor simply jumped to a high 18%. Nevertheless, it is still convincing enough based on three data points (without any extension, extensions of 2 5 % and 3 5 % waterline length) that hull form factors reduced with increased extension length at rear of the single-bulb. Based on the influences on the hull form factor and results from the resistance characteristics, it is recommended that the optimized hull should have such features: - Parabolize the hull with single-bulb (15% beam increment). - Adapt the maximum beam location of the bulb to aft body (Station 4). - Apply fairing extension to rear of bulb with the longest length (35% L W J . These features when applied, though will increase the hull form factor between 1 2 - 1 4 % (see Figure 5-25a & b), will still provide potential reduction of the total resistance around 10% and increase the payload capacity by approximately 3 . 2 - 4 . 1 % and ship stability by approximately 4 7 . 7 - 5 6 . 7 % (see Figure 5-26 and detailed in APPENDIX G ). 86 CO + o o ro (0 ro .c o i_ CL CO 4) vO - CD .9- CO r : •- - a a . co w F m -p CD .y- CN £ o>^  J T" -— "O Q. 5^ of .9- ^  & g-5 5 "T" • - T3 Q. ffl CD~ >* ^ .E n CO co « V CO CD g S - , _ .— "O m .n — # - 'S 8 g « CM .E -a CD _co g •— "D CD S, g O 0 5 ' CO ^ | CO m LO —1— o 1 LO i o LO i o LO O CO LO co CO CM CM H— D) o C i — 03 CD CO .o o & ion nq o CO CD c — a> CD ext = .9-c0 o o E 03 CD CO E o SB o •— "O CL C0 co v5 CD .y- CM & g ^ J CQ « i . _ "n Q. CD >< °^ ^3 LO O) o ID io c - — CO CD •— "a DO at, | ai o c -O X O ro " E c o c o + + ro m E -ro cu .a E 3 E "S ro E T3 C ro E ro o n T3 O <n ro a> k_ o c u a) «^ «^ ui LO CM LO 3 ro o </> aj •o o E ro "D o <D E r--co (>1+L) % CD O ) c ca sz o 'co o C5 3 o CD I o I CO o CO o CD O 0 E CD O 03 Q. CO b 03 > | CD -a | £ Z5 -t-> O ^ (IAJ0) 9BUBL|0 o/0 LO O LO I"1 O D ) 'CD JC o CD O 03 CD o o f i • -A -f+ d* HI S w TT-— "O Q. O-ir- -— T3 «) a o m 'to .1 £ •- "a S, 1>° •—• p CO -4—' c CD g^ "CD o O 5 O c « CL a) rt c g -*—' o CD CO Q. !c CO "O O E 8 O m Q. o g rt E CO -f! K O I I 4 — K • 'E <D .9- <N •g CD . 9 - " CO <r> •= m vP fli >• 8s- —-Lf> O) o .£ -O LO g> n -— "O m «. -| <D "o LO D) O 5 « £ CO rt u (0 (pa»8MV 'A 'M) sBueqo % sjuepioeoo 5.1.6 Recommended New Hull Form The optimized hull form will adapt to findings as discussed in section 5.1.5. It was decided within our group (Dr. Sander M. Calisal, E.M. Sireli & the author) that the revised optimized hull form is to be built such that the displacement is scaled down to match the parent hull while maintaining constant hull length and the parabolized hull feature. This allows the comparison of effects of a parabolized hull at fixed displacement. Upon such revision, the recommended UBC series model #3 hull form is to be modified with respect to baseline parent hull such that: - Displacement (weight & volume) - Bow entrance angle - Single parabolized bulb feature - Stern exit angle (transom width) scaled down to match baseline i.e. 0 % change Reduced from 29.1° to 26.0° i.e. - 9 . 4 % change maximum beam increment shift to aftbody (station 4) increased from 506.8mm to 561.4mm i.e. +10.8% change from 237.7mm to 227.3mm i.e. - 4 . 4 % change The hydrostatics performance as shown in Table 5-10 was calculated from keel up to design waterline level. As can be seen, there is insignificant percentage change (0%) in the submerged volume and displacement between the baseline parent hull and the recommended hull. Although there is also no change in block coefficient, a minute 2 .6% reduction in prismatic coefficient is observed. 89 Best of all, the ship stability is not compromised by such drastic change but in fact improved significantly, the waterplane coefficient (Cw), vertical center of buoyancy (KG), height of transverse metacenter above keel (KM) have all contributed to the significant 3 8 . 2 % improvement in the metacentric height (GM). Moreover, the longitudinal center of buoyancy (LCB) and floatation (LCF) presented insignificant variation (see summary in Table 5-10). Table 5-10: Comparison of hydrostatics between the understudied parabolized hull and recommended hull at model scale (1:13.75) SUMMARY @ DWL : BOps ing le baseline B15p_single midship B15p_single _aftbody B15p_single _midship_L35p B11p_single _recommended actual values actual values % change wrt baseline actual values % change wrt baseline actual values % change wrt baseline actual values % change wrt baseline V, total [mm31 1.256E+08 1.296E+08 (+3.2%) 1.305E+08 (+3.9%) 1.308E+08 (+4.1%) 1.256E+08 (+0%) W, total IN) 1232.11 1271.01 (+3.2%) 1280.33 (+3.9%) 1283.11 (+4.1%) 1232.27 (+0%) LCB % L P P wrt midship (+fore / -aft) -3.84 -3.75 (-2.3%) -4.19 (+9%) -3.92 (+2%) -3.93 (+2.3%) LCF % L P P wrt midship (+fore/-aft) -7.60 -7.28 (-4.2%) -7.77 (+2.2%) -7.43 (-2.3%) -7.59 (-0.1%) KB [mm] 118.54 119.40 (+0.7%) 118.32 (-0.2%) 119.64 (+0.9%) 119.50 (+0.8%) KM =KB+BM, [mml 241.44 259.51 (+7.5%) 247.29 (+2.4%) 262.93 (+8.9%) 255.91 (+6%) KG ~draft=T, [mml 203.53 203.53 (+0%) 203.53 (+0%) 203.53 (+0%) 203.53 (+0%) GM =KM-KG, [mml 37.91 55.99 (+47.7%) 43.76 (+15.4%) 59.40 (+56.7%) 52.39 (+38.2%) Cb - 0.60 0.54 (-10.3%) 0.546 (-9.6%) 0.547 (-9.4%) 0.604 (+0%) Cp - 0.68 0.64 (-5.2%) 0.689 (+1.2%) 0.651 (-4.3%) 0.663 (-2.6%) Cw - 0.82 0.74 (-9%) 0.727 (-11%) 0.750 (-8.1%) 0.837 (+2.4%) Cm - 0.89 0.84 (-5.4%) 0.792 (-10.8%) 0.840 (-5.4%) 0.911 (+2.7%) (1+k) Huqhes-Prohaska 1.23 1.40 (+14%) 1.437 (+17.2%) 1.367 (+11.5%) 1.303 (+6.3%) The recommended hull form is arrived through experimental and numerical means using the understudied add-on bulbs on the effects of beam increment, maximum beam longitudinal location and fairing extension. The worsening of hull form factor (1+k) accompanied by the beam increment have been reduced from 1 4 - 1 7 % down to 6 . 3 % with respect to baseline (see summary in Table 5-10). 90 In Figure 5-27, the selected hull profiles at the waterline level of the understudied hull-bulb configurations and the recommended parabolized hull are plotted on top of each other for a clear visualization. BOp_single_midship (baseline) B15p_single_midship [effect #1] B15p_single_aft [effect #2] B15p_single_midship_L35p [effect #3] B11p_single_midship [recommended] -0.4 -0.3 0.5 Figure x/L, normalized length along design waterline 5-27: Profiles of tested parabolized hull and recommended hull at design waterline level. (a) Parent hull (b) Recommended hull (11 % beam increment) Originally produced by E.M. Sireli in AutoCad (UBC, NALAB) Figure 5-28: 3-D rendered perspective view of (a) the parent hull and (b) the recommended hull. 91 In Figure 5-30, the model scaled total resistance as measured directly off the towing carriage between the understudied hull-bulb configurations and the recommended new hull form is plotted for comparison. In general, the intense peak normally appeared in Figure 5-30b between 0.15 < FnL <0.30 are greatly minimized, accompanied by such modifications are the intriguingly improved and consistent resistance reduction in the order of 1 5 % . This improvement is seen in the higher speeds between 0.30 < FnL < 0.45, which also coincides with our concerned speed range for the resistance reduction. 20 15 CP rr c — x — B 0 p _ s i n g l e [average of P1 & P2] (baseline) --o--B15p_single_midship [average of P1 & P2] _ - B15p_single_aftbody [P3] a B15p_single_midship_alpha3 [P4] —«—B11p_s ing le_midsh ip [recommended] 0.10 160 140 120 100 80 60 40 20 0 -20 0.15 0.20 0.25 0.30 0.35 0.40 0.43 Froude no (achieved) (a) — x — B O p s i n g l e [average of P1 & P2] (baseline) . - B15p_single_midsnip [average of P1 & P2J - -+- - B15p_single_aftbody [P3] o B15p_single_midship alpha3 [P4] — a — B11 p_single_midship~[recommended] R. TM (b) % R TM TM,i nTM,baseline y TM,baseline j x100 0.45 Froude no (achieved) Figure 5-30: [Recommended] Effects of increased beam on measured total resistance at model scale (1:13.75): R T M , % R T M 93 0.030 0.025 „ 0.020 c g 0.015 6 0.010 0.005 0.000 — x — B 0 p _ s i n g l e [average of P1 & P2] (baseline) - - B15p_single_midship [average of P1 & P2] - -+--B15p_s ing le_af tbody [P3] • B15p_single_midship_alpha3 [P4] —t,— B11 p_single_midship [recommended] 0.10 0.15 0.20 0.25 0.30 0.35 Froude no (achieved) 0.40 0.45 0.005 0.004 + 0.003 0.002 0.001 0.000 * "ft "ft «fc >A » n rt; &L K—B 0 p _ s i n g l e [average of P1 & P2] (baseline) • - - B15p_single_midship [average of P1 & P2] -B15p_single_aftbody [P3] » B15p_single_midship_alpha3 [P4] » — B1 I p s i n g l e m i d s h i p [recommended] -+- -+-0.10 0.15 0.20 0.25 0.30 0.35 Froude no (achieved) 0.40 0.45 0.025 0.020 I 0.015 § 0.010 -f-O 0.005 — x — B 0 p _ s i n g l e [average of P1 & P2] (baseline) - - o - - B15p_single_midship [average of P1 & P2] - -+- - B15p_single_aftbody [P3] • B15p_single_midship_alpha3 [P4] —«—B11p_s ing le_midsh ip [recommended] 0.000 (a) 'TM R TM 2 PM$MVM 2 (b) 'FOM 0.075 [log(fl/)M)-2] (c) RM - C -C ' ^TM ^FOM 0.10 0.15 0.40 0.45 0.20 0.25 0.30 0.35 Froude no (achieved) Figure 5 - 3 1 : [Recommended] Effects of increased beam on resistance coefficients at model scale ( 1 : 1 3 . 7 5 ) : C T M , C F O M , C R M 94 160 140 120 100 80 60 40 20 0 -20 — x — B 0 p _ s i n g l e [average of P1 & P2] (baseline) - -> - - B15p_single_midship [average of P1 & P2] --f--B15p_single_aftbody [P3] o B15p_single_midship_alpha3 [P4] — » — B11 p_single_midship [recommended] 0.10 0.15 V i i V + -x -x -x ^« - K ^ j^^Sr* -x ^ *E*&7^££§£ 0.20 0.25 0.30 0.35 0.40 0.45 (a) TM 'r -r ^ TM,i ^TM, baseline y TM.baseline j x100 CO 8 7 6 5-fe 4 3 2 1 0 -fe. -1 0.10 Froude no (achieved) — x — B 0 p _ s i n g i e [average of P1 & P2] (baseline) - - ° - - B15p_single_midship [average of P1 & P2] --+--B15p_single_aftbody [P3] • B15p_single_midship_alpha3 [P4] B11 p_single_midship [recommended] 0.15 0.20 0.25 0.30 0.35 0.40 0.45 (b) %C, FOM ( CFQM,i ~ C'FOM,baseline y ^FOM,baseline j x100 co o 0.10 Froude no (achieved) — x — B 0 p _ s i n g l e [average of P1 & P2] (baseline) - -«- -B15p_s ing le_midsh ip [average of P1 & P2] --+--B15p_single_aftbody [P3] •> B15p_single_midship_alpha3 [P4] • B11p_single_midship [recommended] (C) %C, 0.15 0.20 0.25 0.30 0.35 0.40 0.45 RM yRM,i RM,baseline yRM,baseline x100 Froude no (achieved) F i g u r e 5 - 3 2 : [ R e c o m m e n d e d ] P e r c e n t a g e c h a n g e in ef fects of i n c r e a s e d b e a m o n r e s i s t a n c e c o e f f i c i e n t s at m o d e l s c a l e ( 1 : 1 3 . 7 5 ) : % C T M , % C F O M , % C R M 95 In Figure 5-32a, when the resistances are normalized, we also observed an improved total resistance in the order of 1 2 - 1 5 % reduction between 0.30 < FnL < 0 .45, which also coincides with our concerned speed range for the resistance reduction. No significant change in frictional resistance (see Figure 5-32B). This significant improvement between 0.30 < FnL < 0.45 is mainly contributed by the minimized residuary resistance in the order of 1 5 - 2 5 % reduction (see Figure 5-32c). 0.008 0.002 0.000 0.025 0.020 •£ 0.015 4-§ 0.010 4-o 0.005 4-— x — B 0 p _ s i n g l e [average of P1 & P2] (baseline) _ -<,_ - B15p_single_midship [average of P1 & P2] - -+- - B15p_single_aftbody [P3] ° B15p_single_midship_alpha3 [P4] —a— B11 p_single_midship [recommended] 4- -+- -+-0.10 0.15 0.20 0.25 0.30 0.35 Froude no (achieved) 0.40 — x — B 0 p _ s i n g l e [average of P1 & P2] (baseline) - - ° - - B15p_single_midship [average of P1 & P2] __ f _-B15p_s ing le_aftbody [P3] • B15p_single_midship_alpha3 [P4] —a—B11p_s ing le_midsh ip [recommended] 0.000 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.45. (a) 'VM = (i + k)C, FOM (b) ^WM - C -C ~ ^TM ^VM Froude no (achieved) Figure 5 - 3 3 : [Recommended] Effects of increased beam on resistance coefficients at model scale ( 1 : 1 3 . 7 5 ) : C V M , C W M 96 The hull form factor was determined to be 1.303, which corresponds to only 6 .3% increment with respect to baseline as oppose to the previously 1 4 - 1 7 % increment. One can expect that the viscous resistance should also reduced by almost the same order as is observed in Figure 5-34a. As a result, the wave resistance can be computed and was found to be improved in the order of 2 0 - 3 0 % between 0.30 < FnL < 0.45 (see Figure 5-34b). Such improvement, although not over the same speed range nor of the same magnitude, can still be observed in the calculated theoretical wave resistance using Michell's integral within the limitation of thin ship theory (see section 5.2.1). 40 35 30 <u 25 o> | 20 o # 15 — x — B 0 p _ s i n g l e [average of P1 & P2] (baseline) - - » - - B15p_single_midship [average of P1 & P2] - -+- - B15p_single_aftbody [P3] • B15p_single_midship_alpha3 [P4] —a—B11p_single_midship [recommended] (a) VM ^VM,i " ^VM,baseline y VM.baseline j x100 Froude no (achieved) 400 350 300 250 200 150 100 50 0 -! -50 -100 0. — x — B O p s i n g l e [average of P1 & P2] (baseline) - -o - - B15p_single_midsnip [average of P i & P2] - -+- - B15p_single_aftbody [P3] ° B15p_single_midship alpha3 [P4] —a— B11 p_single_midship~[recommended] (b) WM rr -r A ^WM.i ^WM,baseline y ^WM,baseline j x100 0.45 Froude no (achieved) F i g u r e 5 - 3 4 : [ R e c o m m e n d e d ] P e r c e n t a g e c h a n g e in ef fects of i n c r e a s e d b e a m o n r e s i s t a n c e c o e f f i c i e n t s at m o d e l s c a l e ( 1 : 1 3 . 7 5 ) : % C V M , % C W M 97 When extrapolating the model scale to full-scale performance, one can expect an improvement in effective horsepower in the order of 15-20%, which is a very significant amount of reduction. Considering the typical design life cycle of a vessel of 25 years, such performance can easily translate to enormous amount of fuel savings and environmental impact. 1800 1600 1400 1200 1000 800 600 400 200 0 x —B0p_s ing l e [average of P1 & P2] (baseline) o--B15p_single_midship [average of P1 & P2] +--B15p_single_aftbody [P3] • B15p_single_midship_alpha3 [P4] a—B11p_single_midship [recommended] (a) (^P s) 3 D = (flr e)3 Dxl/ s 0.20 0.25 0.30 0.35 0.40 0.45 Froude no (achieved) 180 160 140 120 100 80 60 40 20 0 -20 -40 _ x — B O o s i n g l e [average of P1 & P2] (baseline) --«-"B15p_single_midsnip [average of P1 & P2] ._+__B15p_single_aftbody [P3] ° B15p_single_midship alpha3 [P4] —«—B11p_single_midship~[recommended] — x — x — g — (b) 0.35 0.40 3DJ)aseline x100 0.45 Froude no (achieved) Figure 5-35: [Recommended] Percentage change in effects of increased beam on effective horsepower at full scale (1:13.75): EHP S,3D, % E H P s , 3 D 98 5.2 Numerical Work The pros and cons of Michell's integral are discussed herein in comparison to experimental results. Limitation of longitudinal wave cut and experimental implementations are reported. 5.2.1 Compar ison of Theoretical & Experimental Wave Resistance The wave resistance reduction capability of hull parabolization is predicted in theory by Michell's integral calculated in 3-D (see Figure 5-36a) as oppose to the 2-D calculations did in earlier feasibility studies (see Figure 2-1 b). This prediction has been validated in the experimentation (see Figure 5-36b). However, no single optimal configuration could not be identified through the theoretical results. As seen in Figure 5-36a, it can be shown that the higher the beam increment, the higher the resistance between 0.23 < FnL < 0.28 and the lower the resistance between 0.28 < FnL < 0 .37. Whereas in the experimental results, though similar trends of high and low are exhibited within close vicinity to the speed windows in the theoretical predictions, yet experimental results suggested an optimal configuration, i.e. 1 5 % beam increment (B15p_single). Another noticeable contradiction between the theoretical predictions and experimental results are in the lower bound and upper bound of the speeds tested, which corresponds to approximately 0.20 < FnL < 0.25 and 0.40 < FnL < 0.45 , respectively. In the lower bound region, experimental results of the optimal case of 15% beam increment showed reduction by as much as 6 0 % as opposed to roughly 5 % in theoretical predictions. In the upper bound region, experimental results 99 of the optimal case of 1 5 % beam increment showed reduction within 3 - 5 % as opposed to 3 0 % drastic surge in theoretical predictions. 60 50 40 -+—B0p_single (baseline) -o-- - B5p_single B10p_single - • B15p_single • B20p_single (a) %C, WM, Michell (3D) Froude no (achieved) (b) %C, WM, experiment Froude no (achieved) Figure 5-36: Comparison of percentage change in model scale wave resistance between theoretical predictions and experimental results at model scale (1:13.75): % C W M In general, an agreement is established between the two sets of results for the optimal case of 1 5 % beam increment such that reduction by as much as 2 0 % of resistance is indeed possible, particularly at speeds between 0.30 < FnL < 0.37 . 100 Based on the present studies, it is true that theoretical wave resistance calculated using Michell's integral within thin ship theory is able to predict the trend effect that is qualitatively comparable to experimental results. But it lacks of the ability to quantitatively identify a certain optimal configuration, not to mention the contradiction of results at the lower and upper bound regions. A comparison was made between the theoretical wave resistances of the parent hull, the recommended new hull and the selected understudied single-bulb configurations of 15% beam increment (midship bulb / aftbody bulb / midship bulb + longest rear fairing extension) (see Figure 5-37). Figure 5-37: Comparison of percentage change in model scale theoretical wave resistance between the understudied & recommended configuration at model scale (1:13.75): % C W M , theoretical As can be seen, upon adapting the features based on the understanding from the experiments, we see great improvement in wave resistance over a much broader speed window between 0.10 < FnL < 0 .39. An average 101 reduction by 1 0 % was achieved between 0.10<Fn L< 0.25 and peaks at 2 5 % reduction between 0.30 < FnL < 0 .35. 5.2.2 Wave Pattern profil ing System & Wave Cut Typical wave pattern in the acquired rectangular patch are processed. They are made up of 512 longitudinal cuts as limited by the resolution of the camera. Five samples of the analyzed wave patches within the concerned speed regime corresponding to Froude numbers at 0.30, 0.31, 0.32, 0.33, 0.34 and 0.35 are showed from Figure 5-38 to Figure 5-43, respectively. 102 2-D WAVE PATCH @ Fn=0.31 , Vm=1.379 [m/s] 200 400 | 600 o I 800 E E >- 1000 v 1 1200 1400 1600 IS. 41': K f o | £ j f ; -1 - £ I •" 1 1 I :•: j . . I |: 3 1 I i ; . '*! » 'I M r ! , | * „ * 1*1 ? i ', | 111 , ,; 1800 -3.5 3 -2.5 -2 -1.5 -1 -0.5 away from model <— X [mm] —> approaching model x 10 Figure 5-39: Top view of the acquired wave patch (Fn=0.31, 512 cuts). 2-D WAVE PATCH @ Fn=0.32 , Vm=1.4234 [m/s] 200 400 800 E E >- 1000 1 1200 1400 1600 1800 I! ! 120 10 -20 -30 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 away from model <— X[mm] —> approaching model x 1 0 4 Figure 5-40: Top view of the acquired wave patch (Fn=0.32, 512 cuts). 103 2-D WAVE PATCH @ Fn=0.33 , Vm=1.4679 [m/s] 200 400 £ 600 I 800 f E, >- 1000 1 1200 1400 1600 1800 10 -1 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 away from model <— X[mm) —> approaching model x 1 0 4 Figure 5-41: Top view of the acquired wave patch (Fn=0.33, 512 cuts). 200 400 § 6 0 0 o i 800 E E >- 1000 •5 1200 5 1400 1600 1800' 2-D WAVE PATCH @ Fn=0.34, Vm=1.5124 [m/s] . • •: i i t •: i ' : ' i l l j B | B H H K ' i . ' ! 11 " I i f I' I I ! 130 H20 10 1-20 -30 -40 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 away from model <— X[mm] —> approaching model x 1 0 4 Figure 5-42: Top view of the acquired wave patch (Fn=0.34, 512 cuts). 104 2-D WAVE PATCH @ Fn=0.35 , Vm=1.5569 [m/s] 200 | 400 "c V o I 600 o A I 800 I E > 1000 I v 1200 1400 1600 1800 i f [ H i :> H, t i i . i j I ! I 1 ' i jf 10 0 -10 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 away from modeK— X[mm] —> approaching model x 1 0 ' Figure 5-43: Top view of the acquired wave patch (Fn=0.35, 512 cuts). Note the very obvious zigzagging pattern, which is due to the wave reflections from the sidewall being bounced back and forth. The reflection is a main concern when attempt to apply the wave cut analysis, be it transversally or longitudinally. A few important things that affect the accuracy of wave resistance as calculated from the longitudinal wave cut analysis based on Sharma's [34] method are discussed herein. The main concerns are the effects of the lateral cut position where the longitudinal cut analysis is executed, the effects of truncation along the longitudinal cut to discard wave height records beyond the first reflection point from the sidewall, and the effects of truncation correction to account for this loss of wave momentum. 105 The longitudinal cuts corresponding to the previous five wave patches for which the maximum wave resistances are obtained are shown from Figure 5-44 to Figure 5-49, respectively. E i s O i o o o 0.1 0.05 0 •0.05 -0.1 •0.15 LONGITUDINAL CUT WAVE PROFILE: Fn=0.3, Vm=1.3345m/s @ y=678.369mm (wrt tank centerline) 20 r -2.5 -2 -1.5 -1 -0.5 downstream <- LONGITUDINAL DISTANCE, X [mm] -> upstream x 1 0 < -160 -140 -120 -100 -80 -60 -40 -20 C downstream <- LONGITUDINAL DISTANCE. (X/1000*ko) mon-diml -> upstream Figure 5-44: Wave elevation along a longitudinal wave cut at fixed lateral positions (Fn=0.30). LONGITUDINAL CUT WAVE PROFILE: Fn=0.31, Vm=1.379m/s @ y=599.502mm (wrt tank centerline) downstream <- LONGITUDINAL DISTANCE, X [mm] -> upstream x 1 0 < ; -180 -160 -140 -120 -100 -80 -60 -40 -20 0 downstream <- LONGITUDINAL DISTANCE. (X/1000*kot Tnon-diml -> UDstream Figure 5-45: Wave elevation along a longitudinal wave cut at fixed lateral positions (Fn=0.31) 1 0 6 E T> i c 0 * o o o N I I LU I LU I 0.15 0.1 0.05 0 •0.05 -0.1 0.15 LONGITUDINAL CUT WAVE PROFILE: Fn=0.32, Vm=1.4234m/s @ y=671.511mm (wrt tank centerline) 40 r 20 -20 -40 + filtered (untruncated) — filtered + fitted (untruncat - o starting point — filtered + fitted (truncated i ed) i i i i ) i i i i -3.5 -3 -2.5 -2 -1.5 -1 -0.5 downstreams LONGITUDINAL DISTANCE, X [mm] -> upstream Y10' -180 -160 -140 -120 -100 -80 -60 -40 -20 ( downstreams LONGITUDINAL DISTANCE. (X71000*ko)fnon-diml -> uostream Figure 5-46: Wave elevation along a longitudinal wave cut at fixed lateral positions (Fn=0.32). E -C o 0.15-0.1 -o o 0.05-o (Z/1 0 -h- -0.05 -I -0.1 -UJ LU -0.15 -> < -1 LONGITUDINAL CUT WAVE PROFILE: Fn=0.33, Vm=1.4679m/s @ y=766.38mm (wrt tank centerline) 40, 1 1 I i W I 111 I 20 •20 -40 f filtered (untruncated) — filtered + fitted (untruncated) O starting point —— filtered + fitted (truncated) -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 downstream <- LONGITUDINAL DISTANCE, X [mm] -> upstream x 10< -180 -160 -140 -120 -100 -80 -60 -40 -20 ( downstreams LONGITUDINAL DISTANCE. (X/1000*koWnon-diml - > U D s t r e a m Figure 5-47: Wave elevation along a longitudinal wave cut at fixed lateral positions (Fn=0.33). 107 E T3 i § o" * o o o T -N LU I LU I 0.2 0.1 •0.1 •0.2 LONGITUDINAL CUT WAVE PROFILE: Fn=0.34, Vm=1.5124mfc @ y=730.947mm (wrt tank centerline) 50 r -3.5 -3 -2.5 -2 -1.5 -1 -0.5 downstream <- LONGITUDINAL DISTANCE, X [mm] -> upstream „ 1 0 4 -180 -160 -140 -120 -100 -80 -60 -40 -20 C downstream <- LONGITUDINAL DISTANCE. (X71000*ko) Inon-diml ->uDstream Figure 5-48: Wave elevation along a longitudinal wave cut at fixed lateral positions (Fn=0.34). V 0.2 8 0.1 t o o o & 0 i -0.1 LU UJ I •0.2 L LONGITUDINAL CUT WAVE PROFILE: Fn=0.35, Vm=1.5569m/s @ y=805.242mm (wrt tank centerline) -3.5 -3 -2.5 -2 -1.5 -1 -0.5 downstream <- LONGITUDINAL DISTANCE, X [mm] -> upstream x10 160 -140 -120 -100 -80 -60 -40 -20 ( downstreams LONGITUDINAL DISTANCE. [X/1000*koHnon-diml -> UDstream Figure 5-49: Wave elevation along a longitudinal wave cut at fixed lateral positions (Fn=0.35). 108 The subjects of studies, that being the parent hull and any modifications to it using side bulbs, are all of transversally symmetric hull forms. Thus, the entire free wave spectrum can be obtained from the Fourier transforms using only a single longitudinal cut. Otherwise, one longitudinal cut on either side of the ship is required for the wave analysis. Ideally, the longitudinal wave cut requires that both the length of the cut ( - 0 0 < X < +00) and the wave numbers (0 < u < +<» or equivalently 1 < s < +°°) be infinite. Thus, the wave resistance is given by (Sharma, equation 25): +00 R X — rCJU n 0 S 2 ( 2 S 2 - 1 ) However, in practice, only a finite length of record (-X2 <X<-X,) and wave number (0 < u < utwncated or equivalently 1 < s< struncated) is available. In the acquired longitudinal wave records, X0 is the location where the wave pattern data acquisition is initiated, at which point the model is stationary and the free surface is rather calm. -X, is the location of first detection of bow wave in that particular longitudinal cut. -X2 is the location when the wave pattern acquisition is stopped, at which point the free surface is disturbed. As such, the wave resistance computation can be represented by: A u truncated r~ n , ^ 4 i k 0 X „ . \—rdu S 2 ( 2 S 2 - 1 ) where C* , S* are the weighted Fourier transformed amplitude coefficients from the acquired wave height records (x,y) . A mathematical form given 109 by Sharma [34] (equation 17) that eliminates the numerical ambiguity at s=1 is given by: C* (s,y) + IS' (s,y) = | V i ^ T • C(x,y)-eisxdx -x2 When applying the method to towing tank of restricted width, not all of the acquired wave records in the cut will be utilized in the wave analysis. The waves records beyond the first reflection from the sidewall of the towing tank are not taken into consideration due to the complex mathematical nature. In order to account for this loss of momentum, truncation correction to the records is necessary. This truncating location, -Xtruncate, is different from one lateral position to another due to the Kelvin wave angle (see Figure 4-4). The truncation point is executed first with user intervention on an arbitrary cut to establish reference positions and then performed automatically by the Matlab script files. By using a longitudinal cut at an arbitrary lateral position as reference Ycul r e f e r e n c e , the corresponding bow-wave peaking location -preference i s identified manually. Since the lateral position is known, the distance between this position and the sidewall of towing tank. The truncating location at downstream, -Xtruncate,reference' c a n b e f o u n d °y applying Kelvin wave angle and taking into account r e f e r e n c e • Details were briefed in section 4.2.3. With all the reference locations known for this arbitrary cut, the starting j) and truncating (-Xtruncatei) points for each remaining cut (Ycuti) can be calculated. This latter process is performed automatically by the script files. 110 The truncation resulted in loss ot wave momentum along the longitudinal cut, thus, underestimated the actual wave resistance. Therefore, a truncation correction is necessary. To account for the truncation corrections, Sharma [34] estimated the wave elevations beyond truncation by using the asymptotic decaying behaviour as postulated in analogy to results using usual stationary phase analysis of any theoretical wave system (Sharma's equation 18): where cv c2, c 3 are unknown constants to be determined by least square fitting to the tail end of the truncated finite cut between -Xtruncate <X<-X1. As such, the wave elevations can then be extrapolated to the far-field domain - c o < X < -Xtruncate. It is convenient to assume c3 ~ 0 for sake of simplicity in calculations without incurring significant error when the acquired longitudinal wave records are sufficiently long ([10]). Therefore, each of the resultant longitudinal wave record is to yield two parts of weighted Fourier transformed amplitude coefficients in order to simulate a close to complete dissipation of wave energy for proper wave analysis. The first part of the weighted Fourier transformed amplitude coefficients is from the original wave records but truncated between -Xtruncate <X<-X, (chopped-off at the first reflected wave front as bounced back by the sidewall): qcosx + CjSinx asymptotic -x, truncated 111 The second part of the weighted Fourier transformed amplitude coefficients is constructed using the postulated asymptotic behaviour of theoretical wave system calculated between -X2< X <-Xtruncate in order to account for truncation correction due to momentum loss: AC* (s,y)+ /AS* (s,y) = " J~~ V T ^ i • C a s y m p t o t i c (x,y)-e isxdx -x2 The numerical ambiguity at s=1 is eliminated in this integral by rewriting the above expression and solve using Fresnel integrals (C F, SF) a described by Sharma [34] (equation 20, 21, 22 and 23): AC* = ^ [d,CF (z+ ) + d2SF (z+ ) + d3CF (z- ) + d4SF (z- )] AS-=^[-dA{z +)+d2CF(z +)-d3SF(z-)+d4CF(z)] where the Fresnel integral terms are given by: CF(z)+iSF(z)= je^ 2 y and each Fresnel integral term is being evaluated at z that corresponds to wave number s in such way: ^ _ 2 ( c 3 - X f m n c a t e ) ( s ± 1 ) V 112 where d, = V s - 1 cos[c3 (s +1)] + c2 sin[c3 (s +1 )]^ d, = V s - 1 ^ sin[c3 (s +1)] - c2 cos[c3 (s +1)]^ c/3 = + C O S [ C 3 (s -1)] - c2 sin[c3 (s -1)]^ of4 = -VsTT sin [c3 (s -1 ) ] + c2 cos [c3 (s - 1 ]Jj Hence, the complete mathematical representation of the total wave resistance used for our computation is given by: •4 "truncated r~ o O ~l 1 « k - = 7 i ( C - + A C - ) 2 + (S' + A S - f 0 L uncated r~ 2 ' J (CL) + (sLJ ( 2 S 2 - 1 ) S 2 ( 2 S 2 - 1 ) du du where 1 < s < 2 0 , or equivalently 0 < L/< 400 , since u - sVs2 - 1 - X 2 = - 5 0 0 , already non-dimensionalized by wave number The reconstructed longitudinal wave records correspond to the previously shown five original longitudinal wave records are illustrated from Figure 5-50 to Figure 5-55, respectively: They showed the above-mentioned two parts of wave records, first part being the truncated original records and second part being the asymptotically derived free wave for truncation correction. They were used for the wave resistance computations. 113 LONGITUDINAL CUT WAVE5PRO'FIEE::Fn=0:3\:Vm='i-;3'3>'5mfe;C6^y^K8i369mm (wrt tank centerline) without truncation correction with truncation correction. ~y-500 -450 -400 -350: -300, -250; -200 -150 -100 -50, downstream <F, LONGITUDINAL DISTANeE,,(X/l600*ko);[non-dirn] -> upstream Figure 5-50: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.30). LONGITUDINAL CUT WAVE.PRdFlLE: ;Fn=0.3l, ym=1.37.9m/s@;y=599:502mm (wrt;tank centerline) 0.15. 8 5. •40.05. -0:15 -02 ; - without itruncation correction - withtruncatjon correction -450: -400 -350 -300 -250 -200 -150 -100 -50 0 downstream <- LONGITUDINALblSTANGE,,(X>;io66*ko) (non-dimj -*;upstream Figure 5-51: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.31). 114 L O N G I T U D I N A L C U T W A V E P R O F I L E . F n = 0 . 3 2 . V m = 1 . 4 2 3 4 m / s @ y * - - 6 7 1 . 5 1 1 m m (wrt tank centerl ine) 0 . 1 5 , 0.05 5= - 0 . 0 5 CD ; : w i thout t runcat ion .cor rect ion v/ith t runcat ion c o r r e c t i o n : " - 5 0 0 - 4 5 0 - 4 0 0 - 3 5 0 - 3 0 0 - 2 5 0 - 2 0 0 : - 1 5 0 ? - 1 0 0 - 5 0 0 d o w n s t r e a m <- L O N G I T U D I N A L D I S T A N C E . (X/1000*ko) [non-d im] -> u p s t r e a m Figure 5-52: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.32). L O N G I T U D I N A L C U T W A V E P R O F I L E : >n=^0.33, V m = 1 4 6 7 9 m / s @ y ^ 7 6 6 . 3 8 m m (wrt tank centerl ine) , 0 , 0 5 E ,=:0:05 CD 1X1 zn UJ 5: -0.1 without t runcat ion cor rec t ion with t runcat ion c o r r e c t i o n : -bOO - 4 5 0 : - 4 0 0 - 3 5 0 - 3 0 0 - 2 5 0 - 2 0 0 - 1 5 0 - 1 0 0 - 5 0 0 d o w n s t r e a m <- L O N G I T U D I N A L D I S T A N C E , (X/1000*ko) |non-dim] -> u p s t r e a m : Figure 5-53: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.33). 1 1 5 L O N G I T U D I N A L C U T W A V E P R O F I L E : F n = 0 . 3 4 , . V m = 1 5 T 2 4 m / s , @ y = 7 3 0 . 9 4 7 m m (wrt tank centerl ine) 0 .2 I ' ' .. i . ' . . . . i i i ; r ^— n ! ~!~ I — without t runcat ion co r rec t ion L; with truncation: cor rect ion ; _n 2 1 L J i i J J * 3 J - • • - 5 0 0 - 4 5 0 ^400 - 3 5 0 - 3 0 0 T 2 5 0 -200^ - 1 5 0 - 1 0 0 - 5 0 0 d o w n s t r e a m <- L O N G I T U D I N A L D I S T A N C E , (X/1000"ko) [non-dim] -> u p s t r e a m Figure 5-54: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.34). L O N G I T U D I N A L C U T W A V E P R O F I L E : Fn=0'.3'5, V m = 1 . 5 5 6 9 m / s @ y = 8 0 5 : 2 4 2 m m (wrt t a n k c e n t e r l i n e ) 0 . 2 , •e 0.1 S ~ 0 : 0 5 CD m - c o s I -0.1 -0 . .15 without t runcat ion :cor rect ion '•with: t runcation: c o r r e c t i o n : -0:2l =500 - 4 5 0 ; - 4 0 0 - 3 5 0 - 3 0 0 T 2 5 0 - 2 0 0 : - 1 5 0 -1 ,00 - 5 0 d o w n s t r e a m <- L O N G I T U D I N A L D I S T A N C E , : ( X / 1 0 0 0 * k o ) [non-d im] -> u p s t r e a m Figure 5-55: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.35). 116 M a t h e m a t i c a l E x p l a n a t i o n of W a v e R e s i s t a n c e - R e d u c t i o n C a p a b i l i t y of H u l l P a r a b o l i z a t i o n For the baseline parent hull: ^total, baseline ^baseline ^ ^baseline S t o f a / , b a se , ine=^ a se , ine + A ^ a S e , i n e Thus, ( r f \2 (cu—r=[(c£-.)+(ACL-.)J 2 \2 = (^baseline ) + ^ (^baseline ) ^baseline ) + (^Qoseline ) {^total, baseline ) ~ [ ( b a s e l i n e ) + ba se l ine ) ] 2 \2 = (^baseline ) + ^ ( b a s e l i n e )(^^aseline ) + (^^aseline ) For the parabolized hull: Ctotal, parabolized = (^baseline + ^ ^baseline ) + ( Q u l b + A Q>ulb ) ^total, parabolized = ( b a s e l i n e ^ b a s e l i n e ) + (*^ulb ^^Julb ) Thus, iptotal, parabolized ) "~ [(^baseline + ^ ^baseline ) + ( Q u l b + ^ Q i u l b )] 2 2 = (^baseline + A ^baseline ) + ( Q u l b + A Q ) u l b ) + ^ ( b a s e l i n e + A ^baseline )(Q)ulb + A Q u l b ) iptotal, parabolized ) = [ ( b a s e l i n e + ^ b a s e l i n e ) + (^julb + A ^ u l b )] 2 2 - ( b a s e l i n e + ^^aseline ) + ( §Ju lb + A ^ u l b ) + ^ ( b a s e l i n e + ^^aseline ) ( ^ u l b + A l ^ u l b ) 117 The squared terms are the one used in the wave resistance computations. Between the parent hull and the parabolized hull, the main difference in the squared terms are due to the cross-multiplication terms. Depending on the effective sign upon multiplications, the crossed terms could lead to either higher wave resistance (when the components are of the same sign, i.e. when both are '+ve' or when both are '-ve') or lower wave resistance (when the components are of different sign, either '+ve' or '-ve'). Similar explanation was given by Calisal et al. [5]. As such, the side bulbs that were used to widen the parent hull offer resistance-reduction capability. At this point, detail studies on the truncation error was only briefly looked into since the main purpose was to demonstrate the feasibility of the multiple longitudinal wave cut method. Effect of Truncation of Wave Number on Wave Resistance According to Sharma [34] and Eggers et al.[10], theoretical analysis suggested that constant amplitude oscillations at ever-increasing wave number, in terms of either u or s, would be seen as X - * - « > . ' Sharma [34] showed that for the practical aim to wave analysis, truncation to wave number spectrum could be applied to the higher-end without prejudice to the calculation of wave resistance. Such truncation is possible since this end of the spectrum contributes little or none to wave resistance. In one case as analyzed by Sharma [34], truncating the wave spectrum at s=3 resulted in no more than 0 .5% to wave resistance. Similar trend were observed in our analysis, as can be seen from the wave energy spectrum below that the oscillations tend to settle down anywhere above s=3. For the conservative considerations, the truncations in our case were executed at s=20 or equivalent to u=400. 118 The wave amplitude functions ff o t a l , G t o t a l (Sharma [34] equation 24) correlate to the previously derived weighted Fourier transformed amplitude coefficients C*otal , S t o t a l are given as: Ftota](u)=[F-(u)+AF'(uj_ An 2s2 ^ \{p]otat •s\n(uy)+S'total-cos(uy)) G t o t a l (^) = [ G - ( a ) + A G - ( a ) ] M Y 4 ' l ^ l 2 s 2 - 1 y {0'total-cos(uy)-S;otars\n(uy)) The wave spectrums correspond to the previously shown five longitudinal cuts are plotted from Figure 5-56 to Figure 5-61, respectively: WAVE ENERGY SPECTRUM: Fn=0.3. Vm-1.3345m/s @ y=678.3SGmm (wrt tank contoriine) 2:5 - - without truncation correction: (C+S ) 3.5 4!5 0:04 «.0:03 0.01 • with tnjncatipn:correctjon: (Ft;dF)2 - (G+dGr 1.5 2:5: 3.5 TRANSVERSE WAVE NUMBER,:u (computationstruncated at!u=99:,:i.e: s=10) Figure 5-56: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.30). 119 WAVE ENERGY SPECTRUM: Fn=0.31, Vm=i:379m/s @i y=599.502mm (wrt tank centerline), TRANSVERSA WAVE NUMBER, u (computationstruncated at:U=99 , i.e; s=,10) Figure 5-57: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.31). WAVE ENERGY SPECTRUM;;Fn=0:32, Vm=1:4234m/s @y=67l.511mm (wrtiank centerline) co ro 0,15 t Oil i .0:05. CO 01025 (K 0.02 i CO :0.5 1; ! 1: ' 'Kjif - ...... without truncation correction: (C2+S2) - with1 truncation,correction: (C+dC)2 t:.(S+dS)2 -% " f — 4 - ,..,.,,.f,:,, ; ;v^,4,,^,,,.::^:,L,, ,,-: : : \ f > * > J I ] I 1 | 1 . 5 2 2.5 3 3.5 4 4.5 £ I I 1 1- withouttruncation correction: (F2+G2) •—" with truncation correction: (F+dF)2 + (G+dG)2 • »:::::. r'"  ,v" VP" ' .--^ii--^ :.-:±-_ :> i -:-y... ,;• 1:5 2.5 3.5 4.5 TRANSVERSE WAVE NUMBER,ufcomputationstruncat'edat u=99:, i.e. s=10) Figure 5-58: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.32). 120 WAVE: ENERGY SPECTRUM: Fn=0.33,.Vm=T.4679rTVs:@,y=766.38mm (wrt tank centerline) TRANSVERSE WAVE NUMBER, u (computationsltruncated at u=99 , i.e. s=10j Figure 5-59: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.33). 0.2 O 0.15 £ 0:1. :WAVErENERGY:SPECTRUM:,Fn=0:34,'V^ (wrttank centerline) \. \ | : 1 1 1 L 1 tf\ \ |; : ^ i - ! : without iruncationxorTection: (C2+S2) / \ — • with truncation correction: (C+dC)2 - (S*dS)2 • • i' .•„...::•..:-i f. .i,^"^™ :o:o5 1.5 2 2.5 3:5 4.5 0.015 with truncation cprrection:;(F+dF)2 * (G+cG)2 0.5 1 1b 2 2.5 3 3:5 4 4.5 5 TRANSVERSE WAVE NUMBER.u (computations^ tmncated at u=99 , i.e: s=10) Figure 5-60: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.34). 121 WAVE ENERGY SPECTRUM: Fn-0.35, Vrrv-=1 55S9m/s @ y=805.242mm (wrt tank cen(erline) S?:v without truncation.correction::(G2+,S2) ••—- with truncation coTection: (C-dC)2 • (S<dS)2 2:5 3.5 4:5 • 0:02 •— with truncation correction:; (j-tdF)2 • (G'dGr 3.5 •4.5 TRANSVERSE WAVE NUMBER, u (computations truncated atu=99, i.e. s=10): Figure 5-61: Asymptotic correction to wave truncated beyond first reflection point of a longitudinal cut (Fn=0.35, 512 cuts). It was found that the wave resistance computation is more sensitive to step size changes in wave numbers rather than the truncations. The curves can be smoothened using finer step size, which obviously incurs longer computation time and computation costs when calculating over the 512 longitudinal cuts. But is it clearly illustrated that truncating beyond wave number at s=3 (or equivalent to u=8.5) seemed reasonable and would incur negligible error, as suggested by Sharma with error of less than 0 .5% in so doing. Effect of Location of Wave Cut on Wave Resistance Sharma [34] reported that the wave resistance at any given speed as calculated from the longitudinal wave cut method varies greatly with the lateral position where the cut was applied (see Figure 5-62). Same conclusion was arrived from our experiments (see sample results from Figure 122 5-63 and Figure 5-68, which correspond to the previously shown five wave patches). OS 5 : in : \ TRUN CATION CORRECTION TRUNCATION PQU ^^WITHC ITxg-SO XJT TRUNCATION CORRECTION source: Sharma [34] )0 20 30 O 5 Q5\ — 9UNCATION CORRECTION TRUNCATIONS WITHOUT TRUN 0INTxe^-IOO CATION CORRECTION 10 20 TRANSVERSE LOCATION Y • 30 Figure 5-62: Illustration of variation in wave resistance from longitudinal wave cut at different lateral positions at Fn=0.2236. Sharma [34] also reported that the actual wave resistance is to be taken at the peak of the curve, whereas the peak location varies from speed to speed. As such, any attempt to quantify the wave resistance by measuring wave heights in a longitudinal cut using conventional wave probe at a fixed lateral position would introduce significant error. To avoid such error would require multiple wave probes. But the problem with having more wave probes is the escalated interference to the flow field. Even with the implementation of multiple wave probes in the towing tank, the proximity between each wave probe still presents a great concern. A plausible solution would be to capture the entire patch of elevated wave surface exactly as it is. Thus, the wave pattern profiling system comprised of laser - camera system was employed, with styrofoam particles being used to 123 reflect the laser sheet projected onto the free surface. This allowed the longitudinal wave cut to be extended from single to multiple cuts as shown from Figure 5-63 to Figure 5-68. The actual wave resistance was taken at the peak at each speed (Eggers et al. [10] & Sharma [34]). max(Cw.uncorrected)=0.00C98652., ma«ewx6rrected)=b.0018736; v i n - MULTIPLE LONG-CUT @jFh=0:3;Vni=13345[rn7s]->ma^ Iff! L i L J L - L J 0 500 600^  700 800 900 1000 1100 1200 (y=0) to tank centerline <- TRANSVERSE LOCATION. Y [mm] -> to tank sidewall (y=1830) I :i 3 L i L !: i ':...:! 3 3.5 " 4 4.5 5 5.5 6 6.5 fv=01 to tank centerline <- TRANSVFRSF I OCATION (Y/1000*ko> Inon-riiml -> to tank sidewall (v=10 081) Figure 5-63: Variation in wave resistance from longitudinal wave cut at different lateral positions (Fn=0.30, 512 cuts). maxi'Cw.uncorrected)=0.001'8804. max(Cw.ccrrected)=0.0023945 0:2 -• uncorrected ' ^ ' ' \ . • . — corrected O maximum: :i |? j = [ 500 600 700 800 900 1000 1100 1200 (y=0) to tank centerline <- TRANSVERSE LOCATION, Y [mm] -> to tank sidewail (y=1830) I ' , 1 i' t i 1—— 1 j J 3 3.5 4 4.5 5 :5.5 6 (v=0) to tank centerline <- TRANSVERSE LOCATION. (Y/1000*ko) [non-diml-> to tank side-Afail (v=9.4411) Figure 5-64: Variation in wave resistance from longitudinal wave cut at different lateral positions (Fn=0.31, 512 cuts). 124 :max(Gw.uhcorrectea)=0.0022922:. max(Cw.corrected)=0.0029895 3 MULTIPLE LONG-CUT @ Fn=0.32,Vm=1.4234[m/s]-->max(Rw.unccrTected)=079471[lb].max(Rw.corrected)=1.0364[lb] - x 1 ° - 1.5 i : 1 r n 1 1 ? 3= •O 2 5 0.5 ;o — uncorrected — corrected 0 .maximum; 500 600: 700 800 900 1000 1100 . , 1 2 0 0 (y=0) to tank centerline <- TRANSVERSE LOCATION. Y [mm] ~> to tank sidewall (y=1830) LL J 2.5 3 3 . 5 4 4.5 5 5.5 fv=01 to tank centerline <- TRANSVERSE LOCATION. (Y/1000«kol Inon-diml > to tank sidewall (v=8.8602) Figure 5-65: Variation in wave resistance from longitudinal wave cut at different lateral positions (Fn=0.32, 512 cuts). x 10 max(Cw.uncorrected)=0.0028157.. max(Cw.corrected)=0.002975 MULTIPLE LONG-CUT @ Fn=0.33.Vm=1.467S[rn/s]->rnax(Rw,uncorrected)=1.0382[lb].max(Rw.corrected)=1.0969;ib] o 2 1.5. 0.5 0. uncorrected •—corrected O maximum 500 600 700 800 900 1000 1100 1200 (y=0) to tank centerline <- TRANSVERSE LOCATION, Y [mm] to tank sidewall (y=1830) 2.5 3 3.5 4 4.5 5 to tank centerline <- TRANSVERSE LOCATION. (Y/1000'ko) Inorvdiml -> to tank sidewall fv=8.3314) Figure 5-66: Variation in wave resistance from longitudinal wave cut at different lateral positions (Fn=0.33, 512 cuts). 125 x 10 rr,ax(Cw.uncorrected)-0.0040442. max(Cw.corrected)-0.0044809 MULTIPLE LONG-CUT @ Fn=0.34Vm=1.5124[m^ 1.5' or. 0:5 uncorrected — corrected, O maximum-500 600 700 800 900 1000 1100 1200 (y=C) to tank centerline <- TRANSVERSE LOCATION, Y [mm] :^ > to tank sidewall (y=1830) L • " 2 . 5 3 3 5 4 4.5 5 fv=0i to tank centerline <- TRANSVERSE LOCATION. (Y/1000'kol inon-diml -> to tank sidewall (v=7.84851 Figure 5-67: Variation in wave resistance from longitudinal wave cut at different lateral positions (Fn=0.34, 512 cuts). .,x 10' max(Cw.uncorrected)=0.C041485. :max(Cw,cdrrected)=0:0048779 3 MULTIPLE LONG-CUT @ Fn=0.35,Vm=1.5569[m/s]->max(Rw,uncorrected)=1.7206[lb],max(Rw,corrected)=2.0231[lb] ,o. J 2.5 2 S -1.5 a- f 0.5 uncorrected — corrected O maximum 500 600 700 800 900 1000 1100 1200 (y=0) to tank centerline <- TRANSVERSE LOCATION. Y [mm] :-> to tank sidewall:(y=1830) 2.5 3 3.5 4 45 (v=0)to tank centerline <- TRANSVERSE LOCATION. (Y/1000«ko) Inon-diml -> to tank sidewali i'v=7.4034) Figure 5-68: Variation in wave resistance from longitudinal wave cut at different lateral positions (Fn=0.35, 512 cuts). 126 Effect of Truncation Correction & Limitations of Wave Cut Method The areas enclosed under the curves of the corrected wave energy spectrum (C'+tiC'f + (S*+dS*)2 , as shown in the samples exhibited from Figure 5-56 to Figure 5-61, correspond to the wave resistance at that speed. The results from these wave cuts using the combined laser - camera system with styrofoam seeding were quite good at higher speeds but relatively poor at slow speeds. It was suspected that the reason for being so was due to the inferiority in sensitivity and resolution of the C C D camera for not able to interpret small change in wave amplitudes. Another suspected reason was the piling up of the styrofoam particles, which introduced "artificial surface elevations". Moreover, the small amplitudes at slow speeds together with this "artificial elevation" might have rendered the wave records to be inconclusive. Whereas at higher speeds, the relatively higher wave amplitudes might have agitated the "artificial elevated" styrofoam particles and spread them out to some extend, which could have helped to result in more conclusive wave records. In Figure 5-69, a comparison is made between the experimental wave resistances as calculated firstly from the empirical formulae and secondly from the longitudinal cuts (taken at the corresponding maximum point of each speed as shown in samples from Figure 5-63 to Figure 5-68). Obviously, without the truncation correction to account for the momentum loss in wave energy contained in a longitudinal cut, the calculated wave resistance by the cut will be underestimated. 127 CD O O 0.020 0.018 0.016 0.014 0.012 0.010 -t 0.008 0.006 0.004 0.002 •+• 0.000 -F— 0.20 - X — C R M @ BOp_single_midship [average of PI & P2] (baseline) -+• • - CWM@ B0p_single [average of PI & P2] (baseline) -o—— CWM@ B0p_single_midship [long cuts] 0.25 0.30 0.35 0.40 (C) Q ? M = CTM-CF0M 0.45 Froude no (achieved) Figure 5-69: Comparisons of wave resistances of parent hull calculated from empirical formulae & longitudinal cut. In anyways, the tests suggested that a better seeding agent is yet to be employed, not just because the inferiority exhibited during the tests at slow speeds but also for its tedious clean up at the end of the tests. A camera of higher screen resolution would be helpful too. In conclusion, the longitudinal wave cut formulation proved to be useful through the direct analytic of the acquired wave records, providing that wave amplitudes can be properly mapped particularly at slow speeds. 128 CHAPTER 6 Conclusions & Future Improvements The present study investigated the resistance characteristic and hull form factor of a displacement-type small craft when it is being introduced to hull form parabolization while traveling between 0.20 < FnL <0.45. They were meant for the study of resistance-reduction capability when the beam is being increased strategically while fixing the hull length and draft. Three main influences were being looked into in our study: - Effects of increased beam using single-bulb (0%, 5 % , 10%, 15%, 2 0 % of B) - Effects of maximum beam location (fore body / midship / aft body) - Effects of fairing extension at rear of single-bulb (20%, 2 5 % , 3 0 % of L^) The parent hull selected as the control baseline was a small craft, i.e. the UBC series-model #3. It is a model built to the scale of 13.75:1. Based on our investigations, it was shown that hull form parabolization possesses feasible resistance-reduction capability. Similar conclusions were arrived at previously on other displacement-type vessels by Calisal et al. [5] and Gotman [12]. Preliminary studies had indicated that optimal performance (with respect to control baseline at model scale), that being beneficial reduction in total resistance up to 1 0 % despite the worsened hull form factor to 14%, can be achieved by adopting single-bulb configuration at midship to 1 5 % beam increment (B15p_single) while traveling at intermediate to moderately high speeds between 0.30 < FnL < 0 .40. The key contributor to resistance-reduction is wave resistance. The findings were supported by theoretical prediction and justified by model testing. 129 As a benchmark for further studies, the beam increment was fixed at 1 5 % based on the preliminary finding. The further studies performed had looked into the shifting of the maximum beam location (fore body / midship / aft body) and the application of fairing extension to the rear of the single midship bulb (between 2 5 - 3 5 % of waterline length). Studies had shown that (when comparing to the control baseline at model scale) parabolizing the model with single bulb at fore body had worsened the hull form factor by 2 2 % , which incurred earlier flow separation and the growth of boundary layer. Consequently, no beneficial reduction in resistance were measured throughout the tested speed range. Whereas having the single bulb at aft body (when comparing to the control baseline at model scale), despite a worsened 1 7 % in hull form factor, could still attain beneficial reduction in resistance up to 10% while traveling at moderate to high speeds between 0.35 < FnL < 0 .45. Within this high speeds range, such resistance reduction was almost an 8 % improvement (with respect to the single bulb at midship at model scale). The tests on the application of fairing extension to the rear of the midship bulb indicated that the longer the extension length the better the resistance performance. With the longest tested fairing extension of 3 5 % waterline length, the hull form factor was improved from 1 4 % down to 12%. When tested at the design speed of FnL = 0.35 , the resistance was improved by another 2 - 3 % (with respect to the single bulb at midship at model scale). Although the parabolized UBC series-model #3 exhibited resistance-reduction capability upon the systematical studies, the reduction was confined to certain speed windows, in this case primarily between the targeted 0.30 < FnL < 0.40 and secondarily between 0.20 < FnL < 0.235. Fortunately the reductions was able to maintain a rather consistent trend over a short range of speeds. This 130 would be helpful for practical purpose when operating the vessel in robust conditions when a specific speed would be hard to maintained. The recommended hull form combines the characteristics from these selected understudied add-on side bulbs while at the same time with its displacement being scaled down to match the control baseline. As a result, the middle body section is parabolized by 1 1 % beam increment, has its maximum beam location fixed at aft body (station 4) and implemented with the longest fairing extension of 3 5 % waterline length to the rear of the bulb. The anticipated improvements were validated in the model testing. When comparing the recommended hull form to the control baseline at model scale, the hull form factor was downed to a mere 6 . 3 % while the total resistance was improved by as much as 1 5 % rather steadily between the targeted 0.30 < FnL < 0 .40. Impressively, the combined characteristics of having the maximum beam location at aftbody and the fairing extension kick in to push the resistance improvement further up to the higher speeds between 0.40 < FnL < 0.45 by as much as 1 1 % . Moreover, the stability (in term of metacentric height, GM) was improved by as much as 3 8 % . In short, the recommended hull form outperforms the control baseline significantly in which hull parabolization plays important role. The implementation of hull parabolization indicated that resistance reduction is practically possible without compromising stability, and in fact, increased the stability significantly. These two distinct advantages are what made the hull parabolization an attractive idea. It may also offer solution for vessels to grow laterally (parabolization) since vessels traveling in certain locations are subjected to length regulation. 131 For the present time, this thesis work would be a first approach in order to understand the feasibility and to answer certain aspects of design considerations for hull parabolization. The accumulated knowledge built on will benefit future ship design in integrating increased beam features for resistance reduction. It is foreseeable that ship designs in the future will be revolutionized pending on sufficient studies in this area. Such research of "tailoring" the hull form for optimal resistance performance would continue to stir the interests within the ship hydrodynamics research community. Irradical idea as it may sound at this stage, but for practical purpose and for future improvement on a performance-oriented vessel, it may be desirable to be able to achieve favorable resistance reduction at any given speed. An active-control side bulb that is interchangeable in beam size and location could be the solution. This will definitely require extensive studies in order to mature the technology and science of hull form parabolization, the method of implementation and the material of selection. 6.1 Calmwater Model Testing The main problem encountered during towing tests was at the slow speed range. The hull form factors were obtained by towing the models at slow speeds between 0.10 < FnL < 0 .20. Unfortunately, due to the drive train gain adjustment and jerking tendency of the towing carriage at lower bound of the slow speeds, it was found that most of the time the actual useful data points fell between 0.15 < FnL < 0 .20. Although the results were acceptable and the resistance trends were compatible with theoretical calculations, it would be desirable to have data points in the lower bound to compare the variations in hull form factors. No problems were faced in the moderate to higher speeds range. Hull form factors determined using Hughes-Prohaska's method seemed to describe the hull curvatures better than all other alternative empirical methods. 132 Another important thing to note during the model testing was the implementation of hull form parabolization using the add-on side bulbs. Great efforts were paid to maintain consistency when applying the plasticine fairing to these add-ons, particularly regions below the chine lines. Nevertheless, this automatically introduced human factor error. The repeatability tests of the parent hull (n=3 data sets) and the selected 1 5 % beam increment (n=3 data sets) showed that their corresponding maximum deviation from the average were acceptable based on the given testing conditions. To be more accurately speaking, that would be: +4.6%/-3.2% (n=3, max. root mean square=1.9%) for the control baseline. +6.0%/-5.1% (n=3, max. root mean square=2.2%) for 1 5 % beam increment. When evaluated in similar manner, the maximum deviation from the average of the recommended hull with 1 1 % beam increment (n=2 data sets) were also found to be acceptable based on the same given testing conditions. To be more accurately speaking, that would be: +1.5%/-1.5% (n=2, max. root mean square=0.8%) for the recommended hull. Although such error margin was deemed acceptable given the current situation, it would have been improved had a hull with built-in side bulb features were used. However, this would have introduced longer model fabrication time and most notably, higher model fabrication costs. 6.2 Wave Pattern Profi l ing Testing In general, the styrofoam seedings were performing rather well. They proved to possess good reflectivity and followed the deformation of the free surface prior to wave breaking. But within the close vicinity to the moving hull, they were repelled by the free surface boundary layer and rendered no reflection of the 133 laser light in this region. Thus, leaving voids in the captured wave profiles. These voids tend to incur computation errors in the calculated wave resistance. Unfortunately, the styrofoam seeding left a mess in the towing tank at the end of the tests. They tend to spread all over the tank and stick to the sidewall. When cleaning up using vacuum cleaner and sump pump, they tend to choke the cleaner and pump, even flew around spread over the floor since they are very light. As such, the clean up of the towing tank after the tests was very tedious and time consuming. Hence, it is recommended that a better seeding agent is to be allocated for this purpose. On the other hand, the camera screen resolution was limited to a maximum of 512x512 pixels. One problem observed was the inferiority in detecting small amplitude waves. This was noticeable particularly for speeds somewhere below FnL < 0.38 which were test runs that were conducted during the initial tests. But another suspicious point was that, upon calibration during the initial tests, the camera was not tightened enough or that the camera support frame got shifted due to vibration. That may have explained the discrepancy at the lower bound speeds. As such, experimental wave resistances calculated from longitudinal wave cut below these speeds were unreliable. Nevertheless, Sharma's longitudinal wave cut methodology works in general and that the laser-camera system together with the styrofoam seeding fulfilled the requirement for now. 6.3 Theoretical Wave Resistance Using Michel l 's Integral The accuracy of the theoretical wave resistance calculated using Michell's integral increases with the increase in station numbers and waterline numbers. This is effective when computing wave resistance at higher wave angles, mainly The theoretical wave resistances calculated from Michell's integral were higher in magnitude. As such, no quantitative comparisons were made. But qualitative comparisons based on the trend effects showed good consistency with experimental results. The trend effects were obtained from percentage variation with respect to parent hull. In general, there was good agreement between theoretical and experimental results such that beneficial wave resistance was indeed attainable. The main differences between the two results lie with the magnitude of predicted reductions in resistance and the speed range where the reductions were achieved. Hence, the qualitative comparisons render Michell's integral a useful tool for preliminary hull form design in evaluating optimal wave resistance performance. But it is to be proceed with caution and validated with experimental results. 6.4 Recommendations & Future Work The present work is a very interesting topic and strikes many other interesting areas. Some recommendations and foreseeable future improvements are listed below: (1) Current thesis work encompassed experimental validations and theoretical predictions of hull form parabolization focused only on mono-hull, it would be interesting to see also work on multi-hull in the future. (2) The model testing at this design stage is sufficient by only studied the calmwater resistance characteristics. It would be interesting to see future studies on the seakeeping characteristics. (3) There are still rooms for further possible improvement to the recommended new hull form, for instance, the transom region, how the 135 aft of the bulb blend into the transom, and the connecting region between chine and side wall at the transom where flow streams were sucked down as observed during flow visualization tests. These improvements will reduce flow separation, minimize eddies and vortex. (4) In search of seeding agents more suitable for model testing in towing tank that are inexpensive, neutrally-buoyant, good reflectivity, non-toxic, salvageable and easy to clean up. (5) With the introduction of larger newly-built vessels, the models used become comparatively smaller. On the other hand, the wave resistance obtained indirectly from current ITTC standard model testing procedures is shrouded by the accuracy of the frictional resistance, which is calculated using the interim solution from the ITTC 1957 formula. There is a need to improve wave resistance measurement. Maturing the non-intrusive multiple wave cut experimental methods may prove to be useful for this purpose. The technique may be applied to model testing in towing tank and to in-situ measurements during full-scale sea trials. This will allow examination of the model-ship correlation. In a reverse engineering sense, this offers insight to improve the existing resistance extrapolation method. This is very important for performance prediction. (6) If the results from model testing of the revised hull form of UBC series model #3 shown agreement with expectations, it is with hope that a full-scale trial version can be built for further studies. The work may lay the pathway for detail investigation on model-ship correlation and could be referred to in the future for improvement on resistance extrapolation method. 136 (7) Wave cut measurement also yields the free wave spectrum for particular mono-hull geometry. The free wave spectrum allows superposition. It would be interesting to see future work in this capacity dealing with multi-hull design optimization studies, both experimental and numerical aspects, by utilizing the characteristics of free wave spectrum. (8) Possible improvement was identified for the wave cut truncation correction in which the wave amplitude coefficients (C, S) can be calculated at different spacing in term of wave numbers. In short, the region in the vicinity of the numerical ambiguity (u=0 or s=1), or more effectively between 0<tv<1 should use finer spacing while for region of high frequency u > 1 should have no problem when subjected to coarse spacing. This is a conclusion drawn from the wave amplitude coefficients as plotted in Figure 5-56 to Figure 5-61 where spikes are observed on the curves. These spikes can be smoothened when finer spacing in term of wave numbers are used. Needless to say, this will incur more computation time. (9) Introducing better filtering and smoothening algorithm in cleaning up the random noise as acquired during wave cut measurements. (10) Database built on present experimental work can improve future numerical predictions using C F D tools. With the advance of numerical capability, CFD tools and model testing can then compliment each other. The work will likely see lead to great improvements in multi-hull designs with correct prediction of the degree of wave interference between hulls. This will improve efficiency of the optimization process during the design phase by cutting down costs, time and manpower. 137 References [1] Acheson, D.J., "Elementary Fluid Dynamics", Oxford University Press, 2000. [2] Akylas, T.R. and Mei, C.C., "Wave Resistance of a Two-Dimensional Obstacle", l-campus Project, School-wide Program on Fluid Mechanics, Module on Waves in Fluids, Chapter 4: Waves Due to Moving Disturbances, MIT, 2000. [3] Amromin, E .L , Lordkipanidze, A .N . and Timoshin, J .S. , "A New Interpretation of Linear Theory in the Calculation of Ship's Weave Resistance", Journal of Ship Research, Vol. 37, No. 1, March 1993, pp. 8-12. [4] Calisal, S . M . and McGreer, D., "A Resistance Study on a Systematic Series of Low L/B Vessels", Marine Technology, Vol. 30, No. 4, Oct. 1993, pp. 286-296. [5] Calisal, S .M . , Goren, O. and Danisman, D.B., "Resistance Reduction by Increased Beam for Displacement-Type Ships", Journal of Ship Research, Vol. 46, No. 3, Sept. 2002, pp. 208-213. [6] Couser, P.R., Molland, A.F. , Armstrong, N.A. and Utama, I.K.A.P., "Calm Water Powering Predictions for High Speed Catamarans", Fast 1997, Sydney, Australia, July 21-23, 1997. [7] Dipper, M.J., Junior, "Ship-Borne Wave Height Measurements", Marine Technology, Vol. 34, No. 4, Oct. 1997, pp. 267-275. [8] Dumez, F.X. and Cordier, S . , "Accuracy of Wave Pattern Analysis Methods in Towing Tanks", 21 s t - Symposium on Naval Hydrodynamics, 1997, pp. 147-160. [9] Eggers, K.W.H., "On the Determination of the Wave Resistance of a Ship Model by an Analysis of its Wave System", Proceedings of the International Seminar on Theoretical Wave Resistance, University of Michigan, Ann Arbor, Michigan, 1963, pp. 1313-1352. 138 [10] Eggers, K.W.H., Sharma, S.D. and Ward, L.W., "An Assessment of Some Experimental Methods for Determining the Wavemaking Characteristics of a Ship Form", Transaction SNAME, Vol. 75,1967, pp. 112-157. [11] Gertler, M., "A Reanalysis of the Original Test Data for Taylor Standard Series", David W. Taylor Model Basin, Report 806, Washington, 1954. [12] Gotman, A. Sh. , "Study of Michell's Integral and Influence of Viscosity and Ship Hull Form on Wave Resistance", Oceanic Engineering International, Vol.6, No.2, 2002, pp. 74-115. [13] Harvald, SV. AA., "Resistance and Propulsion of Ships", John Wiley & Sons, Inc. 1983. [14] Havelock, T.H., "Collected Papers", ed. C. Wigley, Office of Naval Research, Washington D.C., 1963. [15] Hughes, G. and Allan, J .F. , "Turbulence Stimulation on Ship Models", Transaction SNAME, Vol. 59, 1951, pp. 281-314. [16] Proceedings of International Seminar on Theoretical Wave Resistance, Proceedings of the 6 t h ' Symposium on Naval Hydrodynamics, Washington, D.C., Sept.-Oct. 1966 [17] International Towing Tank Conference (ITTC), "The Specialist Committee on Procedures for Resistance, Propulsion and Propeller Open Water Tests - Final Report and Recommendations to the 23rd ITTC", Proceedings of the 23rd ITTC, Vol. 11, 2002. [18] International Towing Tank Conference (ITTC), "Resistance Uncertainty Analysis, Example for Resistance Test", ITTC - Recommended Procedures 7.5-02-02-02, Revision 01, 2002, pp. 1-17. [19] Kent, J . L , "Model Experiments on the Effect of Beam on the Resistance of Mercantile Ship Forms", Transaction of Institute of Naval Architects, LXI, 1919, pp. 311-319. [20] Kent, J.L., "Model Experiments on the Effect of Beam on the Resistance of Mercantile Ship Forms", Transaction of Institute of Naval Architects, LXI, 1919, pp. 311-319. 139 [21] Lamb, Sir H., "Hydrodynamics", 5th ed., Dover Publications, New York, 1945. [22] Lewis, E.V., "Principles of Naval Architecture", Society of Naval Architects and Marine Engineers (SNAME), 2 n d - Rev., Vol. II, Jersey City, New Jersey, 1988. [23] Meadows, G., Lyzenga, D., Beck, R. and Lyden J . , "Nonintrusive, Multiple-Point Measurements of Water Surface Slope, Elevation and Velocity", 18 t h -Symposium on Naval Hydrodynamics, 1991, pp. 349-360. [24] Mei, C.C., "Wave Propagation", Lecture Notes 1.138J/2.062J, Chapter4: Waves in Water, MIT, Fall, 2000. [25] Michell, J .H. , "The Wave Resistance of a Ship", Phil. Mag. (5), Vol. 45, 1898, pp. 106-123. [26] Mikkelsen, J . , "Mech 351: Mechanical Engineering", Laboratory Manual, Department of Mechanical Engineering, University of British Columbia, 2003. [27] Mikkelsen, J . , "Mech 441: Experimental Naval Architecture", Laboratory Manual, Department of Mechanical Engineering, University of British Columbia, 2003. [28] Moran, D.D. and Landweber, L., "A longitudinal-Cut Method for Determining Wavemaking Resistance", Journal of Ship Research, Vol. 16, No. 1, March 1972, pp. 21-40. [29] Newman, J .N. , "Marine Hydrodynamics", MIT Press, 1977 [30] Proceedings, "International Seminar on Theoretical Wave Resistance", vol. I-II, University of Michigan, Ann Arbor, Michigan, Aug. 19-23, 1963 [31] Proceedings, " 2 n d DTNSRDC Workshop on Ship Wave-Resistance Computations", David W. Taylor Naval Ship Research and Development Center, Bethesda, Maryland, Nov. 16-17, 1983 [32] Proceedings, "6 t h - Symposium on Naval Hydrodynamics", Washington, D.C., Sept.-Oct. 1966 [33] Schneekluth, H., "Ship Design for Efficiency and Economy", Butterworths, London, 1987. 140 [34] Sharma, S.D. , "An Attempted Application of Wave Analysis Techniques To Achieve Bow-Wave Reduction", Preprint of the Proceedings of the 6th-Symposium on Naval Hydrodynamics, Washington, D.C., Sept.-Oct. 1966, pp. 30-1 to 30-35. [35] Sorensen, R.M., "Ship-Generated Waves", Advances in Hydroscience, Department of Civil Engineering, Texas A&M University, College Station, Texas. [36] Tan, B-Y, "On the Wave Loading Tests of Orlan Platform: Comparison Between Two Model Profiles", Internal Workterm Report, National Research Council - Canadian Hydraulics Centre, Ottawa, Ontario, Dec. 2001. [37] Tsai, C -E and Landweber, L., "Further Development of a Procedure for Determination of Wave Resistance from Longitudinal-Cut Surface-Profile Measurements", Journal of Ship Research, Vol. 19, No. 2, June 1975, pp. 65-75 [38] Tuck, E.O., "Wave Resistance of Thin Ships and Catamarans", Report T8701, Applied Mathematics Department The University of Adelaide, 1987. [39] Tuck, E.O., Lazauskas, L. and Scullen, D.C, "Sea Wave Pattern Evaluation, Part 1 Report: Primary Code and Test Results (Surface Vessels)", Applied Mathematics Department, The University of Adelaide, April 30, 1999. [40] Tuck, E.O., Scullen, D.C. and Lazauskas, L., "Ship-Wave Patterns in the Spirit of Michell", Proceedings of the IUTAM Symposium on Free Surface Flows, Birmingham, July 2000, ed. A .C . King and Y.D. Shikhmurzaev, Kluwer Academic Publishers, Dordrecht, 2001, pp. 311 -318. [41] Tuck, E.O., Scullen, D.C. and Lazauskas, L., "Wave Patterns and Minimum Wave Resistance for High Speed Vessels", 24th- Symposium on Naval Hydrodynamics, Fukuoka, Japan, July 8-13, 2002. 141 [42] Walpole, B., "An Investigation into Shallow Water Resistance Effects on the A M E C R C Systematic Series", Australian Maritime Engineering Cooperative Research Centre (AME CRC). [43] Ward, L.W., "The XY Method of Determination of Ship Wave Resistance from the Wave Pattern", Proceedings of the International Seminar on Theoretical Wave Resistance, University of Michigan, Ann Arbor, Michigan, 1963, pp. 383-410, 414. [44] Ward, L.W., "Direct Determination of Wave Resistance from the Wave Pattern, With Special Emphasis on the 'X-Y; method", Contribution to the 11th- International Towing Tank Conference, Tokyo, Japan, October 1966, 8 pages. [45] Wehausen, J .H. , "The Wave Resistance of Ships", Advances in Applied Mechanics, 1973, pp. 93-245. [46] Wehausen, J .H . and Laitone, E.H., "Surface Waves", Handbuch der Physik, ed. W. Flugge. Chapter 9, Springer-Verlag, Berlin, 1962, pp. 446-778. [47] Weinblum, G.P. , "Analysis of Wave Resistance", David W. Taylor Model Basin, TMB Report 710, Washington, 1950, pp. 27. [48] Zubaly, R.B., "Applied Naval Architecture", Cornell Maritime Press, May 1996. 142 APPENDIX A Ship Resistances & Extrapolation Methods A.1 Breakdown of Total Resistance A general resistance hierarchy showing the various components of hydrodynamic resistances is illustrated in Figure A-1: Induced Drag Total, Q Residuary, Cr Skin friction, Cfo (equivalent flat plate) Form effect on skin friction Pressure, Cp (normal stress) Friction, Cf (tangential stress) Viscous pressure, Cvp Wave, Cw Transom Drag Viscous, Cv Wave breaking and spray Wave pattern, Cwp I Wake resistance i Total, Q Figure A-1: Breakdown of Total Resistance (source: Couser et. al [6]). Resistance performance curves showing the relationship between resistance components with speeds are shown in Figure A -2 . The abscissa being the non-143 dimensionalized hull length based Froude number and the ordinate being the resistance coefficients: TOTAL RESISTANCE (GRAND TOTAL) STEERING RESISTANCE AIR RESISTANCE SPRAY RESISTANCE WAVEBREAKING RESISTANCE WAVE RESISTANCE WAVE PATTERN RESISTANCE PRESSURE RESISTANCE RESIDUARY RESISTANCE Figure A-2: Performance of specific resistance components with speed (source: Harvald [13]). A.2 Wave Interactions The resultant wave system of a ship is very often be considered to be built up of four components (see Figure A-3): 1. The bow wave system, owing to the high pressure in the vicinity when the incoming waves come into "shock" when colliding into the bow. 2. The forward shoulder wave system, owing to the low pressure at aft of the shoulder. 3. The aft shoulder wave system, owing to the low pressure at aft of the shoulder. 4. The stern wave system, owing to the high pressure at the stern. 144 These wave systems are constantly interfering each other more or less favorably. An example of such wave interactions was a work by Wigley (1930-1931) on a wedge-shaped body to represent a simplified hull form (see Figure A-3). Figure A-3: Interfering Wave Systems (source: Harvald [13]). The symmetrical surface disturbance is caused in accordance with Bernoulli's' theorem, and is also known as primary wave system. Whereas the bow, shoulder and stern wave systems are referred to as secondary wave system. The superposition of the five gives the resultant wave system. The constant wave interactions produce the noticeable humps and hollows as appeared on any typical resistance curves (see Figure A-4). At the humps, 145 resistance increased and the ship requires higher power to work towards the forward motion. While at the hollows, the resistance decreased and the ship forward motion requires lesser power supply. It is desired for a ship to propel within the hollow region in order to save power. 0,20 R 0,15 MN 0,10 0,05 ° 5 7,5 1 0 m / s 1 2 , 5 v 15 Figure A-4: Typical resistance exhibiting "humps" and "hollows" (source: Harvald [13]). 146 APPENDIX B General Pictorial Nomenclature of Ship Form Some typical and useful basic naval architectural terminologies being referred to are illustrated in Figure B-1: 1.0 A l.WL - LBP OWL SHEER AP •- CAMBER (a) FP DWL B / 2 — H BILGE R A D I U S — ^ L . - HALF SIDTNG J _ (b) FREEBOARD MOLDED DEPTH DESIGN DRAFT DEADRISE TUMBLEHOME CENTERLINE PLANE BASELINE PLANE \ -• \ MIDSHIP SECTION PLANE Figure B-1: Schematic illustration of basic naval architectural terminology (source: Zubaly [48]). 147 APPENDIX C Analysis Procedures C.1 ITTC 1957 Method Froude's hypothesis came to rescue for model testing as early as 1868: CT(RN,FN)=CF(RN)+CR(RN,FN) = CF(RN)+CW(FN)+ CFORM (RN,FN) For practical engineering purpose, the form resistance coefficient is treated as a constant and remains so for a geosim model, thus: ' F n ) ~ ) + {Fn ) + CFORM, constat! t In 1947, the American Towing Tank Conference adopted the Schoenherr formulation as the ATTC line for use in ship resistance computations (see Figure C-1:). But over the years, the introduction of larger ships resulted in comparatively smaller models. Problem arose was the deviation between results from experiments and actual ship resistance. To rectify this problem, the International Towing Tank Conference (ITTC) in 1957 adopted a new formula to improve the correlation between model-ship. It is known as the "ITTC 1957 model-ship correlation line" (see Figure C-1:). Though an interim solution for two-dimensional flow, it is still the current standard in use. Neither does it represent the frictional resistance of a planar nor a curved surface. It is important to appreciate the line as a model-ship correlation line rather than a frictional resistance line. 148 Figure C-1: Standard skin friction lines (source: PNA vol. II [22]). Combining the total resistance measured directly off the towing carriage during model testing and applying the ITTC-57 method on the model, the residuary resistance coefficients can be calculated by: CRM — CTM~CFOM By enforcing a model of identical geometry similitude with the ship at full-scale, we have the full-scale residuary resistance coefficients at the same Froude number as: CRM = CRS Applying the ITTC-57 method on the ship, the full-scale total resistance coefficients can be extrapolated by: CTs = CFOS + CRM + CA 149 where CA is the incremental resistance coefficient for model-ship correlation taking into account also the effect of the surface roughness of the ship. Typically, this value is in the order of 0.0004. There are various other references. For instance, by ship displacement (Harvald [13]): Displacement [tonne] CA 1000 10,000 100,000 1,000,000 0.0006 0.0004 0 -0.0006 And by ship length, N A V S E A (1982) recommendation based on sea trials Length [feet] L<190' 190'</.<960' /.>960' 0.0008 f 0.0083 ^ 0.0002 0.00064 Therefore, the full-scale total resistance of the ship can be predicted by: HTS~ 2 Ps^S^S where Vs is the ship speed, S s i s the wetted surface area of the ship, ps is the density of seawater. 150 A closer look of the extrapolating result as obtained by ITTC 1957 model-ship correlation line is illustrated in Figure C-2: T IT" C 1957 M ETH( )D \ \ \ j>DEL \ \ \ N F F F n J •v SHIP C T S / V r \ no 0 \ 1 b|c d i : • M i c R R O U G H / , cJ S M O O I H F J F n rtc9 be! I C R CF>> 0 075 RnM-2) 2 a b-d d — C F S 0.075 " ( l o g i o R n S - 2 ) 2 5 , c 2 i . i 2 5 C „ Q 2 5 Figure C-2: Schematic representation of ITTC 1957 method (source: PNA vol. II [22]). C.2 Hughes' Method In 1954, Hughes proposed a formulation for the model-ship correlation, i.e.: c 0.066 [ log(f ln) -2 .03] 2 Hughes suggested that "for a given body the mean specific resistance is a constant ratio of the specific resistance of plane surface of infinite aspect ratio at the same Reynolds number. The ratio is independent of Reynolds number and 151 depends only on the form of the body". He assumed that the total ship resistance could be regarded as a combination of two primary components, namely the viscous and wave-making resistance coefficients. The essence of Hughes' method is that: CFORM (Rn'Fn)^ (Rn ) Thus, CT (Rn, Fn)=CF (Rn)+ Cw (Fn)+ CFORM (Rn, Fn) = CF(Rn)+Cw(Fn)+kCF(Rn) = (l + k)CF(Rn)+Cw(Fn) = rCF(Rn)+Cw(Fn) = Cv{Rn)+Cw{Fn) where r is a resistance ratio that is constant for a given hull form and k is the hull form factor. The resistance ratio (7) can be determined by towing the model at slow speeds at which the wave making can be neglected, such that: 1 + fr = M y = r Until a certain speed, the total resistance curve of the model becomes parallel to the basic line (r = 0). Hughes referred to this very location as "run-in point" (see Figure C-3). The vertical difference between these two parallel curves represents the constant form resistance coefficients, i.e. kCF(Rn). Calculating 152 the CF(Rn) with reference to the Reynolds number at this point gives rise to the hull form factor k. Although being proposed in the International Towing Tank Conference (ITTC) in 1957, Hughes' method was not being adopted due to the difficulty involved in estimating the form factor k. Figure C-3: Schematic representation of Hughes's method (source: PNA vol. II [22]). 153 jde no 1 based) d=2.44m) 1 E c CM cn o o CO CM Is- LO CD CO 00 CM cn CM O CM CM o CM CM CD CM CM CO CO CM TJ-CM h~ LO CM CD CD CM LO r-CM Tf CO CM CO cn CM CM o CO CM CO CM CO o CO CO CD CO CO CO CO LO CO h-CD CO CD CO LO CO CO Tf CD CO CO o Tf CM Tf CM CM Tf CO Tf o Tf Tf CD Tf Tf oo LO Tf o d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d u. V T J . •o o E Is- Is- r- h- r- Is- Is- h- Is-~o fc, o o o o o O p o o O o p p p p p p p p p o o o o o p p p p p p o o o o o .c T J O E CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM O ) c 4 CM CD CM CD CM CD CM CD CM CD CM CD CM CD CM CD CM CD CM CD CM CD CM CD CM CD CM CD CM CD CM CD CM CD CM CD CM CO CM CD CM CD CM CD CM CD CM CD CM CO CM CD CM CD CM CD CM CD CM CD CM CO CM CD CM CD CM CD CM p CM CD Q> CD CD CD d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d rline o a. iE Tf CO Is-Tf CO Is-Tf CO N; Tf CO r-; Tf CO Is-CO CO r--. CO r--. CO CO r-; co r-. CO r-; TJ" CO r-. TJ-co Tf CO Tf CO N-Tf CO Tf CO Tf CO Tf CO Tf CO h-Tf CO h-Tf CO r-Tf CO r-Tf CO Tf CO t^ ; Tf CO r-; Tf CO h-; Tf CO r-; Tf CO h-. Tf CO Tf CO Tf CO h-Tf CO Is-Tf CO h-Tf CO h-V CO >. o CM 1^ CM 1^  CM CM 1^ CM CM CM CM CM CM 1^  CM CM CM CM CM CM CM CM CM CM r-^  CM CM 1^  CM t-^  CM CM CM r-^  CM r-! CM CM r-^  CM CM CM CM CM CM CM o cn cn cn cn cn CD CD CD CD CD CD CD CD CD CD CD CD CD cn cn CD CD cn cn O) cn cn cn CD cn a> cn CD cn cn cn cn cn cn cn cn CD cn cn CD CD CD CD CD CD CD CD CD CD CD cn CD cn CD CD CD cn CD CD CD CD cn cn cn CD cn cn Q . o cn d cn d cn d CD d cn d CD d CD d CD d CD d cn d CD d CD d cn d cn d CD d CD d CD d CD d CD d CD d CD d cn d CD d CD d CD d CD d CD d CD d CD d CD d CD d CD d CD d CD d CD d cn ~o T J O E E F E CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO CM LO area 1 CM r-CO CM Is-co CM Is-co CM Is-CO CM h-CO CM h-co CM h~ CO CM CO CM r » CO CM r-CO CM h-co CM r-co CM r-co CM r-co CM r-CO CM CO CM r-CO CM CO CM r--CO CM co CM r-CO CM co CM r-co CM r--co CM h-CO CM CO CM r-co CM CO CM r-CO CM co CM r-CO CM Is-CO CM r--C0 CM Is-CO CM Is-CO CM Is-C0 rface CD d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d rface h-co LO co LO Is-CO LO Is-CO LO Is-CO LO r-co to CO LO co LO co LO CO LO co LO r-CO LO co LO co LO CO LO CO LO CO LO r-CO LO h-CO LO CO LO co LO co LO h-CO LO h~ CO LO CO LO CO LO co LO h-CO LO h-CO LO CO LO CO LO h-co LO Is-CO LO 3 W LO LO LO T J Q) o a. >• Is-' CO CM r-" oo CM CO CM CO CM Is-' oo CM CO CM r-i CO CM r-! CO CM r-! CO CM CO CM 1^  CO CM r-! CO CM r-^  CO CM r-^  CO CM CO CM CO CM CO CM 00 CM r-^  00 CM r-^  OO CM CO CM CO CM CO CM CO CM CO CM 1^  CO CM 00 CM 00 CM CO CM 1^  CO CM CO CM r-^  CO CM r-^  CO CM r-^  CO CM CO CM r-^  oo CM O o o 1 Tf CM CO CM CO Tf CM CO Tf CM CO Tf CM CO TJ-CM CO CM CO CM CO CM CO CM CO Tf CM CO Tt CM CO TJ" CM CO TJ" CM CO Tf CM CO Tf CM CO Tf CM CO Tf CM CO Tf CM CO Tf CM CO Tf CM CO Tf CM CO Tf CM CO Tf CM CO Tf CM CO Tf CM CO Tf CM CO Tf CM CO Tf CM CO Tf CM CO Tf CM CO Tf CM CO Tf CM CO Tf CM CO Tf CM CO Tf CM CO a. LD cn LO cn LO CD LO CD LO CD d CD LO CD d CD d CD d CD d CD d CD d CD d CD d CD LO CD d cn d cn d CD d cn d cn d CD d CD d CD d CD d CD d CD d CD d CD d CD d cn d CD d cn d CD d CD d CD CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO E c cc CD LO o CM CO CO CM CM CD CM CO CO CO LO CO r--CO CO CO o TJ" CM TJ-CO Tf LO Tf CD Tf Tf cn Tf o LO LO CO LO Tf CO LO LO CD LO r-LO CO LO CD LO o CO CD CM CD CO CD Tf CD LO CD CD CD Is-CD "53 "5 o LO CD d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d o c « o E E c CC • LO O + LU CM Tf CD o + LU CO CD O + LU CO CD o + LU O LO CD o + LU CO CD CD o + LU cq CD O + LU CD cn CD o + LU LO O CD o + LU LO CD o + LU CM CD o + LU CO CO CD o + LU CM TT CD o + LU CM LO CD o + LU CD CD o + LU o h-CD o + LU cn h-CO o + LU CD CO CO o + LU CO CD CD o + LU r-o CD o + LU CD CD o + LU CD CM CD o + LU LO CO CD o + LU Tf Tf CD o + LU Tf LO CO o + LU CO CD CD o + LU CM CD o + LU CO CD o + LU CD CD o + LU o o CD o + LU CD o CD o + LU CO CO o + LU CO CM CD o + LU r-co CD a + LU CD Tf CD o + LU LO IO CD o + LU Tf CO •a cri CM CM CM CM" CM CM CM CM CM CM CM d d d d d d d d d d Tf" Tf Tf Tf Tf Tf Tf Tf o c 8T U) c cc CM CD o r--CD h~ CM CO Is-CO CM cn »fr cn CD CD CO CD CD CD o CO o LO o CD o CO o cn o CM CO LO CO h- CO cn o CM CM CM CO CM Tf CM LO CM CD CM r-CM CO CM CD CM cn CM o CO CO oc «D a. >. a o 1^ 1^  d d d d d d d d d d d d d d d d d d d d d d d d d d o o a in c CC • Is-O + LU CD Tf r-o + LU 00 cn Tf Is-o + LU cn Is-LO Is-o + LU CD d r-. o + LU CO Tf o + LU CM d o + LU LO CD d o + LU CD O d h~ o + LU Tt d 1^ o + LU CO CO d 00 o + LU CO p CO o + LU p CO o + LU CO o + LU LO CO o + LU CD CO o + LU CO CM CO o + LU CO CM CO o + LU CM CO CO o + LU CO CO 00 o + LU O Tf CO O + LU Tf Tf CO o + LU CO Tf oo o + LU CM LO CO o + LU CO LO CO o + LU o CD 00 o + LU Tf CD CO o + LU CO CD CO o + LU CM h-CO o + LU CO h-00 o + LU CO CO o + LU LO cq CO o + LU CD cq oo o + LU CO CD CO o + LU Is-CD CO o + LU o CM oo o + LU LO o CM o c LO CD 00 f -CO q o CM CO CO CO CO LO LO cn CM r-. LO CO CM O CD CO CO p LO h-o CO CO TJ-CM •<t CO CO CM Tt h~ o LO Tf cn LO o CO CD CO CD r--CO LO 00 CD CO cn CD CM O CM CD CD LO CO CM CM r-co oo m Tf LO Tf LO CO CO h- Tf o CO o CD 00 h~ Is-cn CO CD o o LO CO CO CM CO CM CO d CM CM CM CM CM CM CM CM CM CM CM d d d d d d d d d d d d Tf Tf Tf Tf a> T J O D (0 CO CO LO Is-Is-CD O cn LO CO LO CD o cn C0 p CM CO CM CD h~ CO CM LO r-CD CD CO CO CD CD 00 o CD LO CM o o Tf CD ^ J -LO CM CD CD o s CO 00 cn CO CD CM CO CM CO CO LO CO 00 LO CO Tf o o LO CO CD CM CM 00 CO LO Tf CO Is-Is-00 LO CM o CD CD CO CO E E > ^ CM CM' CM CM d d d d d d d TJ" Tf" Tf" Tf" Tf Tf Tf d d d d d d d d d d d d d r-i LO Tf Tf CO LO CO CM CO CM r-o o 00 CD CO 00 co CD CO CD CO CM O CD p CM CD LO o CM LO TJ-CM o cn CM Tf CO CO CD CO CO CM Tf 00 CO Tf CM LO CD LO LO o CD LO Tf CD o CD CD rf CO r-. CD N-r-. CO CM CO CO CD CO CM CD r-LO CD o o CO Tf o o cn o LO CO CD Is-Tf CM CM T J d d d d d d d d CM CM CM CM CM CM Spe< CD o CD o CM CO Is-Tf CO CO CO 00 Tf ^ CD CM LO o Is-Is-LO d CM CO h-d CO LO o CO r-^  CO r-" CD CD o d LO CO CO d LO LO CD d CD h-cn d r-cn CM d h-CD d CO CO cn d CO LO CM d cn LO d cn CD CO d o CM CM Tf LO CO CO CM CO CM CM o LO CM CO CM CO CM CO Tf d s Tf d LO CO d LO o Tf CO CM Tf Tf CO Tf Is-Tf Is-CD p d r-00 CO d CO o Is-d CD CM o d Q > O 0 > m i cn CD Tf CM CO LO LO CD CD 00 Tf o CO 00 o CD 00 CO p CO LO h-CD o CM CD cn TJ" LO CD CD CO CO CM CM CO LO CO CO CD CO 00 CD CO Tf cn Tf CO CM O CO CD LO CD s Tf CO CO CM N-Tf CD CM CD o CO h-Tf CO CO CO CO o CO Tf h-CD o LO CO Tf o CO CO LO cn o Q ui > LO d Is- d cri d CM d d Tf Tf d d d d d d d d CM d CM CM CM CM CM CM CM d CM d CM Tf" CM Tf" CM d CM d CM d CM Is-CM In cn Tf CD cn r-. cn CD o CO CO CO CO CO CD cn CO CD CM CO CD Tt CO CM CD CO cn CO LO CD CO CM CO CM CM LO Tj- CD CM 00 h-Tf CD CM r-r-CM CM r-O CD CM h-CD CO CD o CD CD CM CO CD LO CO CO CM CD CD o CO LO CM CM Tf CO CO LO o LO r-LO CD o CO o LO Tf CM CM CM CM CO CO d d d ^ Tf' Tf Tj1 Tf' d d d d d d d d d d d d d d de no i based) © T J o E de no i based) El £ O CM Tf CD 00 o CM CM CM CM CO CM CM LO CM CD CM CM CO CM cn CM o CO CO CM CO CO CO Tf CO LO CO CO CO CO CO CO cn CO o Tf CM Tf CO Tf L0 Tf CD Tf Is-Tf CO Tf CD Tf o LO Frou (length > § O Q E u. d d d d d d d O d d d d d d d d d d d d d d d d d d d d d d d d d d d d u ITT lO « cS X •_ JO TJ V o a. Ado Sal **- u o V £• a. .£ "a "5 w to , „ T3 n •o C 0 o _c V) £ °S Sji? o a i » o o t - t i S T » m H O « c - « M » H O i t - ' * N O ( - < » H a ) i o ( s m » M o t -- - J 6 I5 > o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o H r i H H H H r t H r t H H H H H H H H H H H H H H H H H H H H H H J>»J3 >r<^OJ COCOC*!a>Cnaja>OOC«l~'OlO^C» rHO< '* - t*^U'l^ eft a) *• -o fi 5. . 2° U3 _ . _ . _ .. , _ fx, 0*3 a> 0) -Q ro (0 c o c o o ro o »_ c .S » " * O r - o > u o. * t •»> o — C i E I I « a> _c — c *- i_ 2 ^ £ ^ . -o o • O TJ "fc D TP s * o a *— t o o o c r > c o o > o > w u 3 c q ^ r H e o t ^ c o o o o c c c x > T H ^ c o M < ^ t o _ - . l O M O O O f f l t j i N O a J ^ ^ 3 0 0 0 0 ) ^ 0 ) 0 9 0 ) 0 0 00 00 •<Wi-Hr4rHr-5 -^00OOO0OO O ^ e *-•s^  x ft W« I t B£B a ia°. l - T H O i n t - D O O t - t H O T H « i O C ) - t f N « H O W l O ^ " * « r t O M W C f t H C O H « ) o ^ u 5 U i a > t > y > t - « 5 - ^ c x 3 c n w ^ O ' ^ u * ) t o o i ^ o o o o i r - ^ a j ^ o t - o > C T i ... "3t™U3COr-tOSt—Ut>C0»HO 3 cn os cn a> oo So oo.co co 55 ' o d o o o o d o c i . o rt .5 •a •a N C O T j l l O t O t - O O O O r CM c i CO Q) .Q .CO CD CO **— o LU >< a z LU Q_ < T3 H— O CD d co CO CO _CD cz Z3 3 CO 03 CD CO c o co o T5 o E >* cz CO "3 o CD co CL CD O CO r- CM CM CO CD = CO i - CO CO . S O O O O O I ^ C O T - C O I -» § § § o § S ^ " i o ' « > S9°.°.°.R f;n(!inin C O O O O O O C M C M T - C M C M o o o CD o 6 o ci o ci ci cd o io o> co if) CM r- • f r i - (M 00 O Ifl (O o O C O O O C O C O i - t O mSS^cDiocnoicMcdLri ( O O Q i - C M C M C M C M C M C M C M CO r- CO O CM s O) m s o . m ^ l r i ^ O J T ^ C O - r ^ i O C X J i r i i ^ C O C O v - C O ^ T j - l O O C O t O COCMi-CMCMCMCMCMCMCMCM C O O > U ) ( D O N O ) r O ) ( D mcsir^o)c4^<6cd(6o>io ((Oi-CMCMCMCMCMCMCMCMCM (1) O • CD ' CD *t •«t CO CO h-CM CO O CO O o CO CD cn © CD o o> o o cri CO CO lO o o CO o> CN CO CM CO CM CO CM CM o CM CO CM CM r- ( O U ) C O « « 0 0 > ( O I O ,2 0 ) T - C D O N N C O O ) 0 1 t J -S N ^ N O ) O O I - C O O N CO o 2 CM CO CM LO CO* CM CO O) CM r--r-o C0 CO LO I LO CO CM CD co CM r- O O CO cri CO d CM If) CO CM CO LO CD LO S CO O O CO s s CO CD O) O) CO O J l O T - O j l O O l C M l O t ^^r^cocDCfiioco'io C J N O - r i - O J i - T - T t h - O - ^ T - T - T - T - T - T -L O L O C O L O C M C O L O C M T I -. ^ O c O N C M C O n c O C O • j r O C D C O h - C M C O C D O W O I - C O N K C O O C O CO O O t- _ -^ co cn o s o w M n N N c o n 3 <-» •* ^  2 ° O C O C d 6 d S T : CM CO CM O ( f l O C M C O n T t ^ f t LO ^ O i -CO LO T-" iri CD t CD .2 0 0 0 0 0 0 0 0 0 © CD D CQ j - c o d d d d d d d d d c o CO CO CO CO LO CO LO CO CO CO o> CO cn o> OJ eg CM CM M CM CM CM CO £ co s N <e .2 cn o in co ra iri ci <J OJ IO N CO CO -r- y~ i - CM E N W CO 5 in t co m .2 co co r*- cn re ^ CM Tt c i «- O) y- CO CD </> r-. i - T - CM CN h» o i CM CM O) J S ^ ^ r in N *-r a> o> o CO CO h- v- CM I S ! P co g § fo C> to SSI CO LO CO Tj-! o r- co co I jj O ^ (O K d£ 9 N O) CM •21(0 O CM LO CM 5 CD TJ- r- C7) .2 N CD O t TS ^ ^ ^ •5^  O CO CO Tf CO T- CO O CM CO •<* I O) i -< LO CM CM t CM CD > LO co to CO T- CN N O r i CO s o IN Cf) N > i - TJ- r-• i - t- CM CO c r- CM co 5 N CO Tt o .2 CO O N CO m O) 0> CO «~ i - CM CO CO CO CM f- i - CM LO c S CD N Jr co rr co T -° CM co co o ra t to t ^ <; <t CO O CO CM i - CO r- O) CO O) = CO CM O CM .2 rt CM i - CM TO °i CO CM CO S lO (O O r 40 tO T— CN CO <S CM CO d .2 rr ~ CM *2 LO CO o CO e .2 co — o CO < a. rr S co ci N ,2 o — *- o CD (0 o CO CD .2 •» E 2 8 .2.E.I,,, ol "D « CO jj c <o cn col o co d c £ i~ co c »,2ct $ 5 1= < co ci <2 co co ci 2 •* co ci CD <2 CM CO Ci CO 55 6 o CO o co ci 0) O ) C "55 l Q. O CQ CO CO Q. CA 0) CO CO to CM Oc5 UJ O n co CD CD to CD JZ CO CO o o CM co O O) CM r^ ! co co T - co co o LO o CM CM CM CM E. _ l D h 5>coS; o CO O) CM in LO CD CO o CM o CM O O CD E c M c o ^ - L o c o r ^ . _ i Q. .-i Q. CO [ => S d) CD .E .E ^ ^ c o O O 0) 0) »- t- Q •= CO 0> r-l i s a l = D. J 3 S • tr CM CN CO CD CO l - CO CD . 2 o o o o o r ^ c o i - c o i -S o o o o o s f e g g g C O O O O O O C N C M T - C M C N CO CO CO CO i - CD CD CO LO CO CO CO CO O O O S T f O O J r C O r ^ o o O T -ro  q q, . CO O LO O) CO LO ^ o f CN t 01 co in C O O O O T - I - C M C M T - C M C M C N r- T J T - T - C N C O O L O C D S O C O O O C O C O i - C O roSS^<Dio'cno>cM(DLO C O O O T - C N C N C N C N C M C M C N CO m 9 m * oi CD CO i - CO CO CM T- CM CM CO H= .2 o CD CO 3 K| rt r- CO cn in £ co co i -.2 rr q rr CD 1 ^ CM CO j - O T" W CO l - CM CM LO j - lO rT CD = CO CO o .2 T - co r--c3 CM cn *s CM T - co CO f- CM CM £ rt r? r? ,2 N <o CO <« 2 °* +£ O) i -CO CO T - CN T - CO CO rt rt m CM CM CM w rt in N y~ in o o> co co r-; q r - If) CO CO in in m co CM CM CM CM f- f- CM CM CD CD CD <D rt CO f1^ q 6 N r © in in co co CM CM CM CM f— LO CO CO LO r- -•* CD to CM CO CO r- CJ) CO O CO CO rt tO CM CM CM CM T3 CD O CD Q . CO CO CO CD #ri w ~ *J CM CO CN O CM i _ ra fc ro £ m CO D CO. H r- co in CO rr br i- CN m co l-,2 cn T co co s m ^ O CD CO CN • H S t S «7> O CO rr T - T - T - CN CO T -CD rt O rt CD CD O f~ CM CM CD CN rt rt CO oi cn v LO CO CO O CO LO CM CM CM i - cn CM CO O rt d O rt CN CN rr cn 1- © in CD cn o co CM CM o o rt CO cn o d d o co CM CM O) co r— rt LO LO cn cn cn CM T~ T~ CO O CN f— CM CM «2 CM CN CD cn CN h-f-o CO CO in O CO m CO rr LO CM CO CO CM r- rt o CO O CD oi CD d rt rt CM LO LO o> CM CO rr CO CO rr in r-tl O N W iJ J3 © ^  ^  p co o o co r- r~ co co cn cn co rt CM m cn CM m rt ^ R o S i r i c M i ^ d d - i - : - i - : T -C O o r t t o h - h - o o c n o o o o T -m 5 mi-cocDcorrcoo'i-° O W O r C O O C O O r r - o w s w c M s n c o c o i n S 9 6 r i c d 6 d s r r i o C O O C M C O C O r t r t r t r t r t r -O °> tr o o o o o o o o o o TOoooooqoqqcQ c o d d d d d d d d d c o r- rr O rT CO 5 CO N N © ,2 q q in cq "S LO rt d g CM LO I*- 00 CO i - T - i - CM t- N in co jx co rr co in .2 co co s cn (5 °i CM rt CM K O 1- CO CD CO r- i - i - CM CM I*- O j~ CO CM CM _. .2 co r- rr rr ro ui i*: O Cft © rt W CO S f CN •JS P- O) CO , ,s r- o r-o5 <o °? q cd o i CO CO CM CO i CO CO CM in c o 5 co rr rr ,2 o s co co •tr o rr co K -iS o i j * C jj* . r- cn CM CO O CM LO CM .2 rt h- rt CO * S O CD J CO CM • • CO *•* . Cn CO CO rt CO h- CM i - o in , .2 R . CM CD LO CO to cn T - CNJ OONO I ^ oi d h" ! h. i - « h-I 1- i - i - CM r** CM to co o 0 N C O 01 o> CO CM CD CO i - i - CM c .2 LO 5 ? 55 d .2 in <o to o> d CM c .2 CD * 5 CM 2 i-co d co c .2 r~ , 2 "> ! co d S a g 2 co © .2 o c o w © 1 g co r-"to c CO = c •- •= t e 2 °> o co d c £ i-eo c C/> O ro "3 in £ 2 S < co d C O 2 LO CO d 2 •* co d .2 LO <S CM CO d .2 ci S O N o> 1-cn co o to *<* ' J C O O T— T— CO #- cn co cn co CM o CM .2 * « T- « (S °1 CO CM CO •x in (O O r CO CO i - CM CO .a o IS ° CO oCO d E CO o .a T3 CO to CO CO l_ o c v ° c r -lO «t— o Q_ !E to •o CO n 3 n o c c75 O ) c 'to 3 C CO CO Q. to JQ CO CO to CD •4—> CD J E c CO •4—i 'c O CM CO o cn CM t-: •? "i i - CO CO o LO o CM CM CM 1 I CO O J O C O O O t - r C ) XRr'cMT'cMCMd TSOOJCMCMCMCMh-o CO LO CO *r LO o> CM LO CM •cf LO LO co CO CM CM tf d CO LO I--' d CM o CM o T- CO CD CO y— CM CU q> .5 £ * c r o O o «> t. %. O _CMCO^ -LOCDI>-_l®CDc £ _l_l_l_l-l-l " g S o ' S 0) d) — .s ^ 0 1 ) » CD t- i_ Q = 1 « i r CD C O O - 2 = Q . _l D S CM i LU O n co r- CM CN CO <D £ CO i - CO CD . 2 o O O O O h - C D i - C O i -C O O O O O O C M C M i - C M C M P- CO CO CO CD T - CO )Z CD CO lO CO CD CO CO .2oOOr - T t O > C M ^ - C O i -* - . . X C O r f C N r f O ) C O L O C O O O O T - T - C M C M I - C N C M CM — T t i - 1 - C M C O O L O C D = O t O O O C O C D i - t O .2 O O CO CM rt CM CO rt rt -i-; n S ^ t i o ' i d o i o i o i t t i i r i - 9 9 0 0 0 0 * 0 0 1 0 C O O O i - C M C M C M C M C M C M C M CO r- 00 O CM « O) Lf) S O .2 O) O CO CM (5 9 uj o> ^ CO CO -i- CO CO CM T- CM CM r CO O) m n cn co T -.2 rr o rr co CM CD • H O r CO CO T - CM CM Q_ .2 O I B ' cu I I (i CM S CD CO i - CM CM .2 r- co o "5" 9 o> *J h- O) T -CO 00 i - CM CO CO U) DO CM CO rj- O) CO 1^  rt rr rr in CM CM CM r O O N CD rf rr r-r- o CM io O r iri r lO (O (O N CM CM CM CM CM o m T -CM 01 r - CO t T - in s LfS ui i ~ cd in co s N CM CM CM CM CO CM CO O CO CD o w n CO O CO rT lO LO CM CM CM r- CO LO CO rf ^ - i - CM in co T -.2 CD T CO CO N m °? O CD CO CM « ; N t N O O (0 rr T - T - T - CM CM v-r- O) rT CM h- CD CM rr rr CO CM O) rt LO CO CO o co in CM CM CM i - O) CM CO O rt r-" d O rt CM CM rr CD i - o in r*-cd cn O CO CM CM o o rr CO o> o d d O CO CM CM co co r-rt LO LO 0 ) 0 ) 0 ) N T" T " CO O CM i - CM CM t O C M h - O O ^ O L O r t C M c n r ^ o c o L O c o c o L O ! r C M ( D C O C N N ^ i - O C O i £ 9 9 d d < d d r t r t c M i o ' ^ l O C M C O r t L O C O r t l O r * . I W O l r r r r r r - T - r m CD o o o h» r— : C O ( O C O t O C O O ) O ) C 0 r t : O C M L O ^ C M L O O ) C M L O r t j S h ^ N c o d o i i r i c d i r i 9 t N S O r r N r r t ) o r- o> T - i "o CD Q_ CO CO CO _(D c ZJ CD -«—' CD I e _c w ' c Z3 c .2 o CO o 55 c i L O L O C O L O C M C O L O C M ( O N c v o o n c o n o ) C D C O S O j C O O J O U l ^ C D N S O O C O t O C O CO I— C O C O T— ; r n o ID N r CM CO CM © .2 o • s§: co d i LO CO O) CD CO o T-CO CO CO CO 5 5 LO CM » ~ | _ O CO •C' c E m Zi m P ° o o o o o o o o o o • s o o o o o o o o o o c g o o o o o o o o O c o W d d d c i c i o d d d n f- CO T- io LO co ° ? o co co co co co n O j o o i n o i T - T - o i I ^ ^ N r c i c J l D T J O C M C M C M C M C M h -O CM CO O O) CM ^ CO CO O LO o CM CM CM E. i m CD LO o Ti-en CO CM LO CM LO LO CO CM CM CD CO r«i CO c i CO CM CO o CM o CM £ LO O O » >. w Q _ « n ^ i f l c o N j co co c f. - 1 - I - I - J - I - I S o CL'S 2 3 3 3 3 S 3 Q 3 D S r- 1- O t CO S CO N S CD .2 cn q LO co B II) ^ r 6 « ; CM lO I - CO CO T - T - 1 - CM C LO CO i co co LO .2 co co N. cn l o ° ! CM CM j ; O) i- CO CO CO I - T- 7- CM CM f- f . o rt CO CM CM cn .2 co r ~ . t t S <° T iri r-: B O) O) o » CO co r*. T- CM co r co S r- o co co .2 co o i - co IS I- o> <— ro o CO co «!" CM cn co co r*- o r*-co cn m o e « » o N co n g •* <°. 9 N O) N *! N O (O , 2 CM 5 *: 9 w •X O CO CO rt CO T - CO O) CM .2 £ tr co .2 S is f-co •>* *t Oi T-LO CM CM ^ CM CD LO CO CO CO T— OJ t- CO 1 -^ - c o o c o I - o CM 0) Ni T - I -T- T- CM CM CD o h- co oi co CD CO t - CM LO oi c ' ; n « .2 CM cn < n "* cd CO CM T-f- c o r o c o rt CO CM O CM .2 ^ CM T- CM (0 ° i CO CM CO S IO O O i -CO CD CM CO O O £ CD i - i _ Q — CU 0) r-§ s I £ | = a _i D S .2 o is °> 2 co c i IS °> 2 CO c i CM c .2 i -5S co ci .2 h-_ 2 °° CD CO O » = s 5 s fe CO d £ LO 3 CD 5 o - .2 o t - o CO CO o CO C CO CD .2 o o co d c £ t-co c < co d .2 o LO co d LO .2 co s° co d cn c 11= ro CM co d in oi c .2 T-ro? co d .2 o ro o 55 d E CO CO ja T3 CO CO CO CO o c 5^ o o a. !E to T3 CO 3 © O ) c "to O ) c "to 3 CO to CO I UJ aj n co to •o a) cn c to. 3^ +- CO CO CO a. to a> n co r- CM CM CO CD £ CO T - CO CD . S O O O O O h - C D i - C O i -• j i o o o o o c o , - ' • • E o _c E CO CD _Q LO >. "D CD N o . O co • CO D_ CO -*—* CO CO T3 CO DI c CD CO D. O _CD X 5 CO -*—> -f—' CD to 15 CD - C co CO .2 o o o 0 0 oo 0 0 LO CO CO CO CO CO CO CO o> CM i- 0 0 T -cd © in o> CO in T -CM CM CM o> CO CM LO CM CM t- ^ f r r C N t O O m i O = O C O O O C O C D T - C O S ° ° o o w r o T t o n i o W O O T - C M C M C M C M C M C M C M CO r- CO O CM S O) IO S O .2 O) O CO CM CO CO T - CO (0 CM i- CM CM 0- cn co !£ .2 O is ° §1 ^ co cn ul 5 co co T -<S CM co « o r n CO CM CM LO t- LO * o> = CO CO O .2 v-m co CO CM Is" OJ j ; CM T - CO CO » - CM CM i * * 5 .2 h- to cn « £ °* •r r- CT> T— 0) co i- CM r - CT) CD t ^ W CM CM CM CD CO CP CO o in r O OI t o w d o in (o s co CM CM CM CM if) N O r -CO O O ) r W N N O) CM CM CM CM O r N CO CO CD 00 CD i— W CO O in in cd co it m <o CM CM CM CM r- to in co t Jr T - CM w co i-.2 cn i- <o co r-fn °°. o oi co CM S h- h- O) O CO i- i- CM CD T -C0 CO Tf -r; CO O O CM CM CM r- o Tf CM <J> d co o co CM CM CM CO O Tf d o Tf CM CM Tf C» l - o m d o> O CO CM CM o o -d- co O) o d d O CO CM CM CD co in TJ - in CD CP CP Tf Cl r r CO O CM i- CM CM *2 CM CM CO O) CM o CO CO f - o m co in CO Tf m CM CD CO CM Tf 7- o CO O d o> d d Tf Tf CM* d in o> CM CO Tf m CD Tf u> CO <o m CO to o to o O) CO cn f -co Tf o CM in CM LO CM in Tf o h- m r-^  CO to o> iri CO in d CM h- o> o 1— CM * - CD CO 5 L O L O C O L O C M C O L O C M * . S o c o r - c M c o c o c o c o o r -• j j o c o c o h - C M c o c n o i o ^ j W o ^ o s s e o o i c o e o i -IO cri o CD o_ C0 CO CO _Q) C "CD | E c co ' c 3 * T- * T-• CO CO o 5 t « *: CM CO CM O LO CM L0 1- CO Oi CO CM CO o> f-0 0 CO CO o I-. io CO CO I O ci CM CO CO CO CO o o 5 I O f -CM .—, . 3 . S E. x ! x ro fc ra m ^ m P , 2 o o o o o o o o o o ~ o o o o o o o o o o c t j o o o o o o o o p ^ C O o o o o o o o o o c o CO CO CO CO LO CO LO CO CO CO cn CO o> T~ T— o> CM CM CM CM oi CM oi CM CO 1-I— < o O CM CO O O) CM I- n . y- CO CO O LO O CM CM CM E m _ O * CO t r i o c o * L O T - o > C M •g . r - . LO^ -CM'c r iO lO .J? cq CO CM CM ^ «J ri • i D K O M O N O O O £ _ C M C 0 f l O C 0 l - - _ | g J ' c S - l - l - l - l - J - l S n S ro J - Tf O Tf CO 5 CO S N <o . - <7) O I O CO m d Tf T-1 d *s CM in f- co CO 1- 1- t - CM r- S W CO CD Tf CO LO .2 co o s e» CO ^ CM Tf CM s cn 1- n <o CO h- r- 1- CM .2 CO N * ; CD 1-co c to » S O CO CO ,2 CO O r CO •J N O) CO , " •- Tf J ! CM o N w n *- o CO fj ro o i^ .' «: & *r . h>- O) CM = CM m .2 Tf h- Tf co *- h- o to ro CM _ ; • 2 •r • O) T - CO CO Tf co CM Jr CD Tf h- C7> ,2 r» o> o Tf X r P « CM CO CO CO T- CM m cd 5 s t .2 o co CD ^ CM to 1-o> o r- o h- CM to r*- co Tf © co 0 1^  cq I ^ O) O ) CO 1- CM CO CO CM -r- 1- CM N CO S Tf O 1-01 CO p CO ^ r CO CM »- cn co CD S CO CM O CM .2 Tf CM T - CM CD °J CO CM CO ( J l f l O O r CO tO T - CM CO CD CO .E .!= ^ ^ £ O O O o> a) 1- 1- Q ~ CD CO »-= Q. _l 3 S .2 ^ « °o i2 •* CO 0 CM co 0 CO c •2 CM CD CO S 55 o ih H ro &> E co 0 CD J D £1 LO N .2 O = S O ro ro o CO C CO 1.1 01 r SI? g g O 05 0 c £ co c ra j : Sj » 2 ™ < CO 0 co co 0 LO .2 * £ 0 1 ™ co CO.d o> c ° CM 2 CM CO 0 LO O) »l .if CO S co CO ^ CO 0 .H o 2§ CO 0 E CD a> n to CO CO o 0 s LO o Q. C0 X J (0 3 CD CL _ !E co «2 CO c to 3 3 CM 06 TJ" UJ QJ n co a> co c CD LO CL LO c 1: a> CQ k_ •—1 CO CL 10 a> ro cu to E CD i O _c E CO CD J D 5^ cy-O CM >^  X ! T J CD N ~o J O CO CO C L CO —< CO CO TJ CO _Q =5 J D _CD CJ) "co CD CO CL CD o _CD J D co -4—' •4—» <D CO CD J Z CO CO r- CM CM CO CD 5 co T - co co ,2ooooor -coT-coT-• J S C D O O O O C O f c J . X r ^ i r t W O O O O O N N r W C M r- CO CO CO CD i - CD = CO CO LO CO CO CO CO . S O O O S N - O I W r C O r raSSS^edddcjScoio C / J O O O r r C N C N r C N C M r- T f i - i - C M C O O L O C O S O l D O O r O i D r f f l O o o a c g f C M C o t ^ r m 2 2 ^ C D L O C D C D O J c d L O *<99oonn*tonio CO © © i - CM Oi CN Oi OJ CM CM e- CO O Oi t CD LO N O .2 CD p CO CM «S *P in W m 0-CD 3 H i t- CO CD LO 5 COCO r .2 TJ- © (3 1 ^ CM CO jg o CO CO T - CM CM r If) ^ O) CO CO o .2 r «D N TO CM 1^  CD *T" CM T - CO CO T - CM CM r- CD T J Jr, Tf Tf Tf .2 H <0 O) • S <P hJ T- O CO Tf LO LO CM CM CM CM CD CO CD i - © © *-O CM CO Tf tO CO LO CO in ic s co CM CM Oi CM « N r O r*» t o o r-o> © co p CD LO tO Tf L o c o o CM CM CM CO CO Tf CD CM CO © Tf © CD CO CO CO IO N O) (O CO rt CO h-CM CM CM CM r- CD LO CO Tf r l - CM LO CO i -,2 cn r co co s m *9 ©" O) co oi K S t S OJ O CO Tf i — i — T - CM to r-co in 1- CD CD © CM CM CM r- CD Tf Tf CM O) LO CO O CO CM CM T - CD CM CO O Tf N © © Tf CM CM Tf CD i - © to r-co CD © CO CM CM o o Tf CO CD O © © ' O CO CM CM CD CO Tf LO CD CD CM* T™ CO O l - CM *- C O C M f - C O i - O L O T f r C M C D N - O C D L O C O C O L O . S r C M C O C O W N ^ T O C O m S R d o S d d T f T f C M L O • ^ ^ • L O C M C O T f L O C O T f L O h -LO CO rt f ^ V VW «W V U l W J^T . 2 o C M L O T - C M L O C D C M L O T f fflSh^soo'ciroiocdio S 9 w S O r r N r i - » f O O O N - C D T - T - T - T - T - I - 1 -"O "o <D CL CO CO CO CD -4—' CD CO c =5 O S C S r J T - CO o J CO OJ r CO CO CM d CM CO C CO ffl ^ £0 H co 6 LO L O L O C O L O C M C O L O C M T f ( O S C N C O n i O C O O l N ( o c o N O i c o o o i n ^ o i l f i o i s O O r r r • t f t O S N C O C O C O C O i -co d LOi— C O C D C O T f C O O C M O I - C O C D C O O T -L q i ^ i q c M l ^ c o c o c o i d d d d © i s » T - " r ^ CM CO CO Tf Tf Tf Tf Tf h- < O CM CO O CD CM H m . m . T - CO CO O LO O CM CM CM E i m i . 2 o o o o o o o o o © • t j o o o o o o o o o o TOoooooppppcQ ( O d d o o d d d o d n •Jr. i - co i - in in co . H ' o c o c o c o c o c o c o O J O C D C O C D i - T - C D l ^ T ^ o i - r ' o i o i t o T J O C M C M C M C M C M h -_ O Tf CO i l O t O ' t l O t - C O C M iN- LO Tf CM Tf LO LO W C O C O C M C M ^ ^ ^ ^ L O S O C M O C N O I r C O I O C O r r W CU CD .E £ * J= j= o O O a> »- i_ Q _ C M C O T f L O t D h - _ | g g c t j j j j j j ? > Q" 'ro Tf o Tf CO CO H to CD © LO CO in Tf y-^ d CM m f- CO T- CM r- m CO CD Tf CO LO CO CO h- cn o> oi Tf oi O) CO CO r- CM H O CO CM CM CD CO h» Tf Tf r- to s s o co ro .2 0 0 O r CO o5 ^ °1 °9 oi « O i - T - O CO 1- LO CO CM in rt CO Tf Tf .2 © H CO CO «- O Tf CO ^ S ° - CO CM CO © CM LO CM c o E r- CM LO .2 Tf K- Tf co ^ N O CO n *- . O T - CO W Tf CO h- CM rt tO Tf CD .2 S CD O Tf - 9 « H O CO CO Tf CO T - CD CD CM o f f LO .2 S CO Tf Tf CD i -LO CM CM oi to LO CO CD CD i - CM M O r Tf CD © CO Is- o oi oi s 1- Tf i - i - Oi H CM tO CO Tf © O N CO CD CD CO CM CO CO ! i - i - CM .2 S M O N Tf CD T-cn co © CO ^ r Tf CO © T- T- CO i— CD CO 0> rt CO CM O CM .2 Tf CM •»- CM eg °* co oi co JS m to o T -C0 CD i - CM CO O O OJ oj l- i » Q = 0J CD (-r Q . J 3 E c .2 co 2 T" CO d 3$ co d CM c .2 CB co 2 CO c .2 r-1S CM g CO d != 2 S S E CO d H > o* ^ . « £ CO n _ 3 a> 5 O N -2 o — *- o CO CO CO Q CD C © C (D r-•B .1 .2 o. 8 ? S S co o co d c £ i -co c £ 2 P < co d E co CD rt T3 to CO CO >_ o c 5^ c5^  O CM O o. !E to T5 CO a> co c "55 co CO d .2 t -1S 1 0 co co co d o> c .2 <o IS "> ™ CM co d io cn 2 T -co d 2 § co d LO i LU 0) co CO TJ E E 0) CO c <o. o CO 1 c^ £ CM 0) i_ CO CL CO 0) rt co to to E co cu . Q LO . Q TJ CD _N O JD CO CO Q . CO CO cz o 13 - Q CD i _ O =5 JD 0) J O o T3 "co c CD CO C L CD O 0 ) J Q CO -I—• 0 *+— o CD -£Z CO CO f- CN CM CO 0 3 S CO i - CO C D . 2 O O O O O I " - C O T - < O I -^ O O O O O C O r J i n r - i L n C O O O O O O C M C M T - C M C M CM E .2 o o o o S q q CO CO CO CO CD CD CO I O CO CO CO CO o o f - TT O J CM r- CO T -o o T - CO d LO o> CO LO CO Tf CM Tf o> CO LO o o CM CM T - CM CM Tf ^_ CM CO O LO CO O CD O o CO CO T~ CO O J CM Tf CM CO Tf Tf i— Tt CD LO o i o i CM CD Ln o O CO CO Tf O CO LO T - CM CM CM CM CM CM CM co fc Q) O S i => f-CO o CM CO CO CM p- o> CO cn I O f - O CO CO f » Tf Tf CO OI o co CM 1- CO CM O) T— o LO Tf- oi co T - 10 CO LO CO CO CO Tf Tf LO o CO LO CM CM CM CM CM CM CM CM CM CO Ol LO CD o r~ a> ,_ O J CO CO CO T - o> f » a> CO CM CO CO Tf o Tf Tf CM CM ^— o Tf T— CM CO CM Tf CD CO c i LO o T— CO Tf Tf Tf LO o Tf LO T - CM CM CM CM CM CM CM CM CM LO CM CM CM CM CM CM CM a> CO CO O J C5) cn cn cn O O ) T— T - o CO CO CO CO <o cn CO Tf CM CM to CO CO CO CO o i CO LO CM CM LO LO LO LO LO Y- LO LO CM CM CM CM CM CM CM CM CM CO C O c .2 r i co ™ 15 Si r- a i co CO T -r- co LO t i - CM LO .2 o> T - co a m- d o i S r> Tf f> 0) » r r T- r- r- O r-O O O Tf O Tf Tf Tf cn Tf T- T-. T~ O T-O ) O O ) O O ) CM CM CN CM CM f » o o o cn o Tf O O O Tf o 1 - O O O OJ o io" d d d CM d T- Tf Tf Tf CO Tf CM CM CM CM T— CM CO c o ,_ °o iS CM CO CM IO CO IS ° 2 ° co d IO CM S CM OJ f » O CM CO CO CM O CD OJ CO LO CM CO Tf Ct r r r CO T- O LO Tf CD LO CO CO LO r— Tf T— o co io co co co co co CM LO T - CM h *i 1^ CO N S O r h- OJ T - T -O O CO f - I-CO OJ CD CO Tf LO OJ CM LO Tf cd o i LO CO LO T - CM T - T - Tf o CD C L CO CO CO _CD cz =3 i _ _CD "55 E CO c =5 C CD E CD i _ O c co o LO LO LO CO CO r - CM CO co f -oi LO CM Tf CO f-. CD CO LO CM Tf CO CO OJ OJ o LO T— d OJ CO CO T— r » T -CO CO o Tf CO 1 -CO CM d LO T— CO OJ CO Tf CO O T— o CM o T— CO OJ CO o T— T— o LO N LO CM CO cq CO LO o d CO CO d d r< T— T— LO d CM CO CO Tf Tf Tf Tf Tf f~ LN . — . |_ "x 2 K " g O J o c a . 2 o o o o o o o o o o E ~ P - £ 0 0 0 0 0 0 0 0 0 0 w c f l £ J j S o o o o o o o o o g j CQ _I CD r— C O d d d d d d d d d c o o O CM CO O OJ CM f-l CO LO T- CO CO O LO O CM CM CM .2>o CD O CO LO LO CO 0 0 CO OD CO CO CO OJ CO OJ T— ^ OJ T— CM T - CM CM CO CM CN CM CM CM f~ O Tf CO LO co Tf LO T— OJ CM LO Tf CM Tf LO LO co CO CM CM Tf CO CO LO 1 - d CM o CM O T - CO CD CO T - T— CM CM "Si E. _i m c C N C O T f L O C O f - - _ | g g= I I I I I 1 ^ > 0) £ * £ o O v Z. a CD c Q. • -o_ ca 3 S CO CO CO f -c o CO LO OJ c o 13 CO CO o Tf o Tf CO CO f - r- CO O J o LO CO LO Tf T— o CM LO f~ 0 0 T— T— T— CM f - LO CO CO Tf CO LO CO CO f - O J O J CM' Tf CM oi CO CO f~ T— T— CM f - O oo CM CM O J 0 0 r- Tf Tf CD T— LO f«-oi a i O Tf CO f~ T- CM CO f - O CO CO 0 0 O T— 0 0 r- O J CO CM o T - T— CO T— LO 0 0 CM o OJ 0 0 CO r » o f -CO OJ CD CO CO CM CO CO CM ™ CM f~ CM I- Tf O CO CO Tf r -l»- o> o T— o LO d CO CO T— CO O J CO Tf CM Tf OJ f - LO CM CM CD Tf CM CD LO LO CO CO T— OJ T— CM f~ CO T— Tf OJ o O CO f - o f - CM oi 1 -T— Tf t-. T~ T - T - CM l>- CM CO f - co Tf O CO o f - CO T— a i o i CO ^ CM CO 0 0 CM T— T— CM M O N CO Tf OJ T-CM OJ CO O OJ CO OJ CO CM O CM Tf CM r- CM OJ CO CM CO LO CD O T— CD CM CO CD CD ~ .£ ^ ^ £ o O O <» CD i - i _ Q = 2 <D c s I &| Q. _ l 3 S iS CO CO o c .2 co IS co iS LO co d CM c .2 »-IS LO jS K co d co c .2 CM 2 O J S w d h » B a s * g S co d co >, Si - ° LO C T 3 -3 CD 5 0 N -2 O co 15 co o 1 ss-_ CO CO C CO C 01 r S -i i -8 ? S S co o co d c £ f~ CO c < co d CO c .2 oo ™ LO co d io CO c .2 o 5 Tf co d OJ a co S CM W d LO oi c .2 K « ? +- . CO o c .2 o 2 ° c/> d E CO CD n •o CO CO CO CO ci^  LO o TJ o JO CO (0 n n o> c "co CO c 'co 3 c CD >_ CO CL CO n co CO CO CO £L CD LU o n co E 03 CD _ Q -° LO __ " O CD N O _ _ co co C L CO co c o - Q =3 - O .g Z3 _ _ _0 . O Z J O co J Z =3 SZ c CD CO C L 0 _0 __ CO -*-» CD CO O 0 SZ -4— 1 CO CO o c .2 o o o o o r~ •t; o o o o o GO 2 o o o o o r -W d 6 d 6 c i o i CM CM CO CO CO T— CO CO CO Y- CO T-in c i LO CO o> CO LO CM T"• CM CM CO CO CO CO T- CO CO CO LO CO CO CO CD o o o r - Ti- en CM CO r— o o o o o o T-CO ed TI-d CM LO TT o i O) CO CO ui LO c i c i c i - ' T- CM CM T-1 CM CM CO c o D L co CO I t 2 O SI 3 I -Tf 1— CM CO o in CD o CO o o CO CO CO o o o> CM Tf CM CO Tt 1— o o Tt cb in o i o i CM cb LO o o CO CO Tt o CO LO o o T- CM CM CM CM CM CM CM CO CO o o o o CO o CD en LO o o o o O) o CO O) o o o o o CM o T— o in ib d d d d cb d in cb CO Tf r - CO m CM T- CM CM CM CM <M CM CM CM CO CO CO _. r - CM _ CO CO r - o o o o o LO o to Tt r~ Tf Tt Tf Tt m Tt r -~t cb —^  cb —^  ib o CO CO cn cn o o> LO cn LO CM CM CM CM CM CM CM CM CM LO T~ CM CM CM CM CM CM CM cn CO CO cn cn CJ> CJ) cn o cn T— o CO CO CO CO CO cn CO Tt CM CM CO CO CO CO CO o i CO LO CM CM in LO m LO LO LO in CM CM CM CM CM CM CM CM CM O O i -CO — co T J - co r— — o s t T t T t c n c M c o c o _ S M O O) O » r • O OJ O O ^ ^ N o i ^ r i i o ' i o ' d d o * - r ^ - 0 > T - C 0 C 0 C 0 T t O C 0 T t W C O — • C M C M C M C M C M C M C M C M CO LO 00 Tt Tt o cn CO LO T- CM LO CO T- in cn Tt m CO cn T- CO CO 1- r - GO o> cn Tt 00 d o i CO CM ib in CM T- T-Tt r - cn O o T- CO o CM Tt CM CM CM CM CM > TD > CD CO CM i>- CO T- o LO Tt CM cn o CO LO CO CO in CM CO CO CM Tt Tt o CM CO CO CM O cb o i to d Tt in in CM CO Tt in CO Tt in CM cn in co o o co r» t»-C O t O C O t O t O O C A C O T I O C M L O i - C M L O C n C M L O T f S ^ ^ i ^ t b c b d i n c d i n S t V S O r r N r r * O S O r r r r r r r o CD Q . CO CO CO CD 0 •4—» 0 I E CO -*—' 'cz cz 0 E 0 o c cn n m m .2 O CD f» « o to co S o b u i oo o Tt co m o i c o to CO LO CM CO LO CM Tt CM CO CO CD CO O) Is* N W CO CO O LO ^ C J N d d r r r N N CO O) CO CO ^ C M C O C M O X c X co e ro _ _ _ h o O CM CO 0 0 )01 N " B ) T - CO CO o LO o CM CM CM CM a E. _i cat-LO 1— CO O) CO Tt CO o T-o CM o T— CO cn CO o T- T-o LO f~ LO CM t- CO CO CO LO o d CO 00 d d r~i T- T- in d CM CO CO Tt Tt Tt Tt Tt r -. 2 o o o o o o o o o o • j s o o o o o o o o o o ™ o o o o o o o o o a i C / j d d d d d d d d d c o —1 CO T— LO m CO CO CO CO CO CO CO cn CO cn T- cn CM' CM oi cb CM CM CM CM CM o Tt CO m CO Tt m T - 0) CM (-. LO Tt CM Tt m LO CO CO CM CM Tt CD CO in t< d CM o CM o T- CO CO CO T— T— CM CO CD ~ .- X o o • - _ Q — % a. c155355B % 3 B £ CM CO r t LO CD O- _ J CO .15 S5 Tt c O CD 15 <°. 53 °t LO c o Tt o Tt CO CO r - r - CO cn o LO CO m Tt T- d CM m r~ CO Y~ Y- CM l~~ LO CO CO Tt CO m CO CO !•» cn cn CM Tt o i d T- CO to T- T- CM r - o CO CM CM a> CO t- Tt Tt to T- ib i> d d o Tt CO I-. ~ • CM 1 o O CM CO o CO CO CM CO CO T" o oi CO CO CO m CO CM o CO r - CO o> CM r~ O) m cb CM CM Tt CM i> m CD o o l i o 0|CM U .2 53 CO J CD O c . o o rr f; ro 2 C M d I— C O Tt co o CM LO Tt CO c o 5 co Tt h- cn .2 r» o> o Tt O CO CO Tt oo T - co cn CM CO c CO Tt Tt cn LO CM CM ^ CM CD LO CO co cn T - CM 5 CO S O o i o i s cn § L 2 Z 53 CM LO d r** CM CD CO Tt o q N co cn cn co CM to CO T- T- CM N ID S n n f c n T -.2 CM O) CO o is r to Tt fi Tt Tt to o 00 C M T - T - C O f- en co cn S CO CM O CM .2 Tt CM 7- CM ro ° ? cd o i cd « m ib o T -C0 CO T — CM CO CO tD .£ .E ^ ^ J= o u o « o u i. Q — CD <D r-§ m j = D. _ l 3 E c .2 ro co C O d ™ m oo d CM c .2 T-•S m 2 ^  oo o CO c .2 CM ™ 53 d CO t: « .2 CM 5 ro g £ 53 d >> • ° m "O IT N .2 o ™ o ro co o § « -CO C CD C O ^ . I I T -CI T J « CO CD C 2 00 in\ o 00 O CO c rolC £ ro C < 00 d CO c .2 oo IS m ro m oo d LO CO c .2 o ro Tt oo d cn c .2 co IS cn ro C M oo d LO d c .2 r -2 -cn d o ca o 55 d £ CO CD JD TJ CD CA CO CD O c LO O < TJ O CO CO n 3 0 O) CO = CD cw c (75 3 C CD >_ CO Q. CO 0 CO T—• 0 CO CO 1^  LU 0 n co TJ O ! CQ CM CD "D CZ CO -(-— cz CD E CD O CO CO _CD c CD CD O O O O O O r» •5 O O O O O 0 0 <S O O O O O OJ CM CO CD CO CO CO CD —• CO —• ib o LO CO a CO LO CM CM CM CO CO CO CO —• CO CD CO LO CO CO CO CO O O Tf o OJ 1— CO T— O O ,— CO d LO cri CO LO CO TT OJ T* CO LO O o OJ CM OJ CNJ Tf o - 3 - CO CO r- CD C» o LO CO LO —^ d OJ LO CO -• l— OJ uo CO CO •"3- CO LO CO co OJ o> c\i •ST oi oS CO CD -' -• OJ O CD CO _• O T-CO TJ-55 d LO CVJ E ca CD _Q _Q T3 CD _N "o __ CO co C L CO co CO .g Z3 -Q _CD D ) CZ CO cz CD CO Q. CD o o __ CO CD CO o CD _z CO CO CO cz ZJ J2 ZJ _Q CD co CD a. CD __ £ CO -o o a) cz CD CD _cz 1_ CD "ca ci> LO CM ^_ O CZ o 'co cz £ CD O ) cz "ca CO CD tr o _z CO _z -t—' T3 T3 CD CD o Q_ CD CL Q_ CO CO CD .2 O ® o CO cz „ _? .2 m i ? 3 D h CO °J —• I-S. * 5. c§ E co -i.co.E_i m 2] m P LO , ? ,1' CO -2 B Q- „ 55_• CM CO o LO CO O CO o O CO CD CD o o o> CM OJ CO Tf Tr • o o Tf cd LO oi o i CM cb LO O o CO CO Tt o CO LO o o —• CM OJ CM CM OJ CM CM CO o CM CO CO r— O) CO cn LO O CO o o TT - 3 - o o> o co CM o CM o> l>-o LO Tt CO oi cb LO CO cb CD CO LO LO CO O CO LO OJ T ~ CM CM CM OJ CM CM CM CM CO O) LO cn CO _ o> i o> o> CO CO CO o LO OJ CO " 3 - o o O) Ti- o O " 3 - o oj CO LO •* ed d r-^  d d o CO LO CO CO o CO CM OJ OJ CM OJ CM OJ CM CM LO "3 - 05 LO r^ - o _ - 3 - o> _. CO CD O o o o O o CO - — LO cn Tf LO - 3 ; OJ d CO d o i CD d OJ CO LO F-— CD O CO O) OJ OJ OJ CM OJ OJ CM CM CM CD " 3 - o h~ CO O o CO •a- Tf CO CO ro Tt CO o> CO O) LO CO p o O CD r-^  oi LO LO LO cb d d CO cn CO •* LO CD o CO CD CO OJ OJ CM CM OJ CM CM OJ ' CO If) CO -3-C —• OJ LO CO —• O O) r ; CO CO S °? d oi co csi s S S CO o CO Tl- •- —• T- OJ CO o O CM CM OJ O) CO LO - 3 - LO CD O) O) - 3 -O J — : i -C 0 O CM - — CM O J CD O J r— CO o L O •ST O ) o CD LO CO CO LO CD CO CM r- -• o CO CO d cb d CM LO CM CO L O CO L O o> LO d LO CO o o CO CO CD CO CD CD 0 5 O ) CO o CM LO OJ L O O ) CM L O o LO 1 ^ CO CO d LO cb LO CM o CM o o> CO LO LO CO L O CM CO L O CM o CD r-~ CM CO CO CD CO O ) o CO CO CM CO o> o L O o d LO oj d d —^ 1-^  —• d - 3 - CO t-~ CO O ) CO CO LO CO O) CO - 3 - CO o T-o OJ o CO O) CO o o LO r~- LO CM CO CO CO LO o d cd cb d d Lfi d OJ CO CO " 3 - •a- •d-o o o o o o o o o o o ' • P O O O O O O O O O O M O O O O O O O O O J Q _• CO _• LO L O CO CO CO CO CO CO CO °> CO O) - — - — O) CM T~ oj CM d CM CM OJ CM OJ r— o CM CO O CO p cn CM LO CD ^ j- LO CM CO LO sz LO •a- CM LO LO cb CO leig co CO CM CM •^f d cb o LO o leig LO d CM o CM o CM CM CM _. CO CO CO CM E. CD CD ._ ._ —: —: u < _ < _ Q CAi CO ^ LO CO N _ J 0 <D C r- o CO CM CM o> CO CO ~• LO a> d O •cf CO r— -• CM CO o CO CO CO o CO cn CO CM d CO —• LO CO CM 13 «= 55 ^ LO o cn CO CO o CO O) d cb CO CM CO CO CM g 15 •it" o CO o CO CO o - 3 -  r--! o d CM d CM LO CM o ^ s co '•5 Is- o CD S CM d - 8 CO - 3 - CO CM C CO N O) O S O t O ^ 1 ^ °- ^ 1 ^ S O CO CO Tf CO —• CO O) CM CO CM Tt O) r-~ LO CM CM CO CM <6 LO LO CO CO o> •~ OJ CD T— O) o o CO o r~- CM d TT r-—• -' -• CM CM CO CO o CO o CO ~; d d cb CM CO CO CM CM LO °> NCON — CO Tf O) •— O CM O CO O '•(J ^ CD Tf ~ CO O CO CM —• CO C3> CO o> CO OJ o CM - 3 - OJ CM O) CO CM CO ib CD O CO —• CM CO CO CD ._ ._ _: —; o O O » oj _ _ D _ = 01 CD r-i | I a | CM cz O OJ '•5 cj> co co' 55 d o r-~ 1 0 ro co CO d co £ -< c .9--S 3 °> •S 55 d ±1 _ ° ° CD o ^ ro o co 55 T— c o ' « CD o CO c .9 p O - 3 -o •- '-5 o CD "o ro cn CO § CO d c CO O O J CO - J3 CM CD ro rV. < 55 d CO cz O O) S LO co d LO '•5 °) ro CO co d O ) c g CM ro CM co d LO 0 CO 1 2 co d .2 —1 ro o W d 5 E co CD £5 "D CD CO CO CD _. O c °^ m co ia 3 n cu c CO c "co 3 C CD _ . CO o. co 0) n (0 +-1 +•I CO CO CO I LU _CD __ CO CO SI Q . CO. O Q . !c CO TJ "I E « ..I - .9-!E CO TJ 0) CO c "35. _ Q . 5> m co CD CQ C o 'co c CO X CO CO c (0 T J c CO -*—' cz CD E CD i_ O _sz E co CD J D LO JD "D CD N O J D CO S -CO CL to to T J CO J D ZJ J D _CD CJ) C 'to J Z to to _CD c =5 0) "CD c to =5 cz CD CO Q. CD •)= O _CD J D CO CD to H— o CD J Z » CO "5 CO J Z !— J D ZJ J D CD J Z -*—• o CO CD L _ CD _ C -*—' o -*—' J Z O J cz _CD CD _tz "co o CO o cz o "co c 0) "x CD CJ) _c'co ' 1 -*—• to CD D) cz o T J c o o CD to J Z T J T J _CD CD o CL CD CL CL CO CO " CN CN CO CD C CO t - CO CO O O O O O O r ^ C D i - C O r -s q q o o q N n o , n m OTPPPPPCNCN*-CNCN CO CO CO CD T— CO C CO CO CO CO CO CO t o O O O O S ^ O C M T - C O I -S P P P c O T f c M T f c D c o L O 05 O O O -r- i~ CN CN T - C\J CN N Tf T - T -C O CO o 0 O O CD CN T f 1 S 8 § g » CO O O T - CM CM I I' CO 3Z .2 O CO ° c • J2 .2 ' -ID LJ I 3 I- -CM CO O CO CM CO CO o CN CO f~ CD CO LO O CO O o Tf Tf o o> o CO r~- Tf O CN O ) r--o LO Tf LO Tf CN ai LO CO <o CD CO Tf CO CO CO O CO LO CN CN CN CN CN CN CN CN CN CO Oi LO CD CO a> ^ CD cn CO CO CO LO CN CD Tf o Tf o CD Tf o q Tf o CN CO LO Tf CO CN c i d o CO LO CD CO o Tf CO T ~ CN CN CN CN <\J CN CN CN CN LO Tf CD LO h- o Tf CD CO CO O O O 5 o o T— CO f - y— LO CO Tf LO f~ Tf CN 1"^  c i CO d oi r~ CO oi -r^ CN CO LO f~ CD o CO CD CN CN CN CN CN CN CN CM CN CO Tf O CO O o CO Tf T f Tf CO S> CO CD Tf CO CD f - CO o> T~ LO CO O a> o O CD f^ oi LO LO LO CO d d CO CD CO Tf LO CO o CO CO CO T - CN CN CM CN C\i CN CN CN 1 CD LO CO Tfr C T— CM LO CO T— O O) T - to CO s aj c i a i cd CM ~ T J - a> o CO T f T— T— T— CM CO O O CM CM CM O LO CO CD i— CD Tf Tj" i -CM CD LO O CO LO CM CM CM O J CO LO Tt LO CD C l O) CM •»— i— CO O CM f- CM CM w C D C M N C O i - O l O ^ c c v j o s o c o m c o c o i r ) O i - C M C D C O t M S ' t r - O C O '•*= 00 o CO CM Oi LO Tj- T- ^ T-• CD CO O 5 * °° ^  CM CO CM O c§ E c§ CD 1^ CD h CO o CO O LO CO o o CO CO CO CO CD CD CN LO CM LO CD l-~ LO CO CD oi CN o CN 1^ CD LO LO CO LO CN CO CO r- CN CO CO CO CO CO r~ CN CO CD oi LO CM d d Tf CD f~ f~ CO CD LO CD CD CD Tf CN o CO CD CO LO h- LO CM CO d CO 00 d d t-^  CN CO CO Tf T f Tf o o o o o o o o o o o o o o o o o o d d d d d d CO LO LO CO 5 CO CO CO 00 CO CD CO CD *r- r- CD CN —^- CN CN CO CN CN CN CN CN q LO ^ o o o o o o P P cri O CN CO O CD CN r-: <*> T - CO CO O LO o CN CN CN E, CD 3 LO P Tf CD CO CN LO CN Tf LO LO CO CM CM Tf d CO r-^  CO d CO CN CO P CN P CN CD CO .E . £ ^ C N C O T f L O P f ~ _ l CD CD 5 -^ Tf O T f CO C CO S S CO O CD P LO 00 "rrt LO T f T— d CN LO co CO T— T— 1— CJ ^ I- LO CO C CD Tf CO LO O CO CO S O ) 'a ° ! CM T f CN ^ C I T - C O C O CO f~ T - i - CN C CO CN CN CD O CO f - Tf T f '•S CO T - I R ; K -CO CO c o CO CO o CO o CO "c5 Oi CO CM CO d CO LO CO OJ 't; CD w q , O O P - CO CO '•S O t (D ^ CO O I " CO o cz CM LO o Tt Tl- CO CM o oi tD CO CO CO Tf CO CM CZ CO T f f~ CD O CD O Tf --• °- "I ^ ~ O CO CO Tf CO T - CO O) OJ c; CM Tf cn T -O N I O CN CM "•g < ° ^ CM" d ~ LO LO CO CD CO r- CD T - CN LO » I - CO T -C Tf CD O o o co o ™ t ^  2 fc *; N T - ^ S CO T - T— T— CN r- CN CD C K CO Tf O o co q s q cn ^ cn o i co ^ T - CN CD CO CO CN i - T - CN LO 0 5 I— CO I— CZ CO Tf CD T -.O CN CD CO P to CO Tf r ~ Tf Tf 00 P CO CN T - T - CO CD CO CD C CO CN P CN O Tf CN i - CN •g ° ! CO CN CO jg LO CO P T -CO CD T - CN CO Q. CO 3 2 .i= ^ sz -C o O O » CD >— 1 — CD CU j— O CD '•S 0 0 CO T— O i -co CO Tf tfi d CN c O CD p ro 55 d co c O CO iS CO CO d CO < c .9-2 f ° •5 iS CD 2 5 5 °' > . LO CO tfi T— £ CD CO d CO o CM CO CN CD to S < 55 P O CD S LO co d CD c O CM CO CN CO d LO O 00 •B co CO T -.2 P CO o 55 d O CO 5 E co CD JD TJ CD CO CO CD i _ O c LO O C L (0 T3 CO JQ 3 JD CD C "55 c "to 3 C CD CO C L CO a> JD CO CD to T t 6> I LU o JD CO CM CO J= Q . CO. if- C L to TJ E I I co c "55. _ C L CO LO CD CQ c o "55 c CD 4-* X CD CO c '>Z ro "O cz co -t—' c CD E CD i _ O tz E co CD _Q LO _Q TD CD N "5 X ! CO CO C L CO CO CO "O CO -O _CD O ) cz 'CO o j D SZ> CO -I—» "CD U) CD CD CO CO CO CO _CD c Z3 0) "CD E CO _Q Z3 .O CD CO CD \ CD CZ ibi CD CD rz CD -t—' CO LO CO c CD i CO C L CD £ cz g co cz CD X CD O ) _cz 'co CO CD O ) c o •4—« > TD 73 CD CD t^z O C L CD C L C L CO CO ° CNJ CNJ CO CD C 00 i— CO CD o o o o o o r - . c D i — c o - r -C O O O O O O W W T - C J C M CO CO CO CD T— CD C CD CO LO CO CD CO <D O O O O N ^ C J C N J r C O r -' • s S S S ^ c o ' d L O o J c o L O S ° ° ° C O ^ C V t Q C O L O CO O O O i — CM CNJ T— CM CNJ CD ' CO CQ co 3 | - | ^ CO S o 5 * « ^  CM CO CM O CM ,—. i _ X C : <3 x~ nj E co E m £ j CD 13 CD H O CM CO O CT> CNJ N. w « T - CO CO O LO o CNJ CM CM CM I CD I « * T-C O CD O O O O O) CM f TO § § rj" CD LO • • o O CO CO O O T— CM CNJ w CO o C O) Lf) N O CB O CO S 9 LO ^ S CD CO i -C0 CM T - CM CO O) LO C CO CO r O ^ O i t CO Is*- CM CD CO i — CM CM LO ^ O) C CO CD O O T— CD O-CNI OO O CO r-. co co ai LO LO CM CM CM CO T~ CO O CO O O O LO CO r t CM LO CD N CM CM CM CM CD CO - ~ CD CO O LO T -O CD r t O LO r r r t LO LO CD S CO CM CM CM CM LO i>. O T -O O i - o i— LO CO r t c d d o i LO -N- CD CM CM CM CM O r - N CO' CO O I CO CD T - LO CO O LO LO LO CO CO r t LO CD CM CM CM CM] 1 CO LO 00 r t C CM LO CO T -O O) r - CD CO N ^ Q CD CO CM , ^ N t S O) O CO r t i — i — i — CM CO O O CM CM CM O LO CD CD i - CO r t r t i -CM CO LO O CO LO CM CM CM CM CD LO CO h- O O r f CM CM r t OI T- O LO h -CD OS O CO CM CM CD O O O O CO CM CM CD CO LO r t LO CD CD CD r t CM T~ T~ CO O CM T— CM CM CM cfM CD OJ N Cfl N O CO CO CM CO CD >>». 1- o LO CO r t i — LO r t CO LO O CO 95.0: CD CD CD CM CO r t O LO 3 3 CM LO LO Is-TO L O C O O O C O N N C C O C D C O C D C O C D C D C O r t O O C M L O i - C M L O O i C M L O r t ' • S ° h ^ N W C D o i L O ( X ) L r i ^ 9 W S O T - I - W T - I - N -C 0 O I ^ - 0 5 I - I - T - I - I - I - I -0 5 CO C L O L O C O L O C M C O L O C M r t O O C D M M C O C O C D C O O S ' • P O C D C O N C M C O O J O L O ^ S P o i L O ^ N O C i ^ ^ i -C O O T t C D N N C O C D C O C O T -LO oS C L O i - C D C D C D r t C O O T -O O C M O i - C 0 0 5 C O O t - t — • • P O L O N L O W N c o c o c q L q S P d c o r a o o N ^ ^ L O C 0 O C M C 0 C 0 r t r t r t r t r t i > -O O O O O O O O O O O " • E O O O O O O O O O O O O O O O O O O O O O c o d d d d d d d d d c o SZ _^ 00 ^ _ LO LO CO O) o CO CO CO CO CO CO "CD O CD CO CD CD X P CM CM CM CD TD O CM CM CM CM CM O r t CO - i - j L O C O r t L O T - C D C M • C N L O ^ C M ^ L O L O CTcqcqcNicM^^^ ^ L O N O C M O C M O X i — C O C D C O T — T - C M _ C M C O r t L O C O N - _ | CD CD £ _ I _ I _ I _ I _ ) _ I > 5 cu Q Q _ J 3 2 ™ CM , ca r t O r t CO CO 1^ CO CD O LO CO LO rj" T— O OJ LO N CO 1- i - i - CM S - LO CO CD r t CO LO CO CO S O) °? CM r t CM CD i — CO CD h - T - CM r-~ o CO CM CM CD CO r t r t ^ - L O N CD CD O r t CO N T - CM N O CO CO CO O i — CO ^ P °°. c\i CDT— •<— CO i — LO CO CM CD CO CO I s- o I s-, CO O) (fj CD r t r t O N CO CO O r t CD K J I s- CM LO r t N- r f CD " P CO « CD r f S O) CD O r t T— O LO ^ '•5 CO r t . Is- CO i -r t CD O • C O N O ^ C\i O) s h - i - r t I s-i - i - i - CM I s- CM CO r- CO r t o CD O Is- CO " ^ OS CD CO T - CNJ CO CO CM i - i - CM N CO N CO r t CD 1-CM CD CO O ^ ! CO r t i — r t r t CO O CM i - T - CO CD CO CD CO CM O CM r t CM T-; CM ° ? CO CM CO LO CD O T— CO i- CM CO e o CD CD S .£ ^ -c r: o O O a> dj i - v. Q Q. j ! D 2 O CD Co i— co »>. co d O CO i 3 co CO d CD CD ^ < C •S i5 CD -8 55 o " S S S = iS o ra 55 ^ «? CD OJ c c o • - ° a, -a ffl ro 03 § co o £ S S < 55 d .Q cn LO co d co c O '•5 ro i S co CO d .2 CM .23 c\j CO d o co ca o o '•B ° CO o Si d in co 5 E co CD n T3 CD CO CO CD o c ^ CO m co o Q . CO •o CO £} 3 £} CD C "Lo a> c w 3 C CD (0 D-C0 a> SZi CO CD CO •ct 0_ UJ CO Q . CO. CO T3 _CD c Q . LO m c o '55 c —> X CD LO CO O ) c ro CO o i CD E ZJ c "D c CO o o o ••5 o o <5 o o o o o LO o o o to o o o to c i c i c i CM L O C O Tf Tf Tf f- C O C M C D C O L O r-^  oi C M C O C M C M C M C M C M L O C O C O C M C M O C M C O Tf C O L O C D o> O cn o C O Tf C O C M L O O to C O C M L O L O 1 ^ C M Tf C O 55 C M C O O 1- T~ C M C M C M C M Tf O co r» CD o Tf C O r~ co LO to m LO Tf d s CM m s co CO T- i - T- CM h~ LO CO C to Tf CO LO O CO CO N O) 0J °? CM Tf CM ~ O) r- CO CD CO I— 1— T- CM O T-CO Tf 55 d CO -*—' c CD E CD CL X CD C o T3 CD CO CO •o CD CO > CD CO ID "a o E CO CD CD CO o 00 Z) "O CD T J C CD E E o o CD CD J Z CD Vi O CD J Z CO CO CM CD CM CO CO LO CM O LO CO CO Tf CO CM CO O CD r - CO to O f - CO CO f-^ CD LO CO o CD CM CO LO o> CM LO *~ T- CM CM CM T~ CM CM C O N C O i - C O C O O C O T - 0 ) 0 ) N W O r - T - N CO | « i to Zz\ •c ° j CO CO & s| CD CO 3 f - l CO Is-CO o "O CD O CD CL co co CO _CD cz Z5 i _ _CD "CD E o _CD J D CO = •4—» r m E CO -4—' 'cz Z5 to C O CO T-n OI LO O co « ^ -~. CM CO CM O CM co E co E m m JcoP o C O o C M C M o C M C O O C O Tf L O C D C O r -T ~ C M C M C M C M C M T— C M C M o C M _ C O L O C D C O C O C O o C O L O C D Tf C D C O L O C O C D C O r~ C O Tf C D r->; Tf r-^  C O L O Tf d oi o C M Tf L O to C O C D C O 00 C M C M C M C M C M C M T_ C M C M o C D C D C O C D _ o to ,_ C O C O C M L O C O o Tf C O T- Tf C O C M C D C D L O oi ^ — Tf C O d C M C O d Tf o C O Tf L O C O C D C O f~ C M C M C M C M C M C M 1— C M C M Tf f - Tf C O f - C O o o ,_ Tf C O C O C D L O C O C O C O C D cq Tf O to Tf L O C O q d L O C O ci C M C O •r-' d C D C M C O C O L O C D C M L O r_ C M C M C M C M C M 1— C M C M co f - C O C D C M O O f - O C O L O C O 1 ^ C M C O co C O o q C M T- C M L O 1 * -C M C O C D to C M C D Tf Tf Tf C O C D O f~ C D C M T~ T— 1 C M C M T_ ~^ C M C O to C O _ C O L O C M C O f - Tf C O C O C O C O C O L O Tf C M q r~ C O C M C O 00 Tf q T~ C M Tf d L O ci C O d d C O C M C O Tf Tf L O C O Tf C O C D C O f - C D C D C M Tf C O C O Tf o f - C D L O C O f -f - C M T- C M C D O C M C M C D C D C O d C M C O Tf C M C O C O o o C M L O C O C D C O C O Tf C O C D C O C M C O C O C O r-- L O Tf C D C D o ro C O C M C D o Tf C O C O T— L O d Tf C M d Tf 1 ^ Tf cd C O C M Tf C O f - f~ f - C O f~ r -Tf C O o C O r-- C D Tf C M C O t o o C D C O C O f - C O Tf T- cq C M C D C M C O oi o j cd C D C M ci d Tf C O C O C O C O Tf C O Tf C O C O o o o o O o o O Tf o o o o O o o o C O o o o o O o o o d d d d d d d d Tf C O L O L O C O 5 C O C O C O C O C O c> C O C D T- T- C D C M T- C M C M C D C M C M C M C M C M f -CO CD T J zs "co o C M co O C D C M <*? T- C O C O O LO o C M C M C M E. CQ O Tf CO *rLOCDTfLOT-CDCM ^h-LOTfCMTfLOLO CTcqcOCMCNI^fairi ' L O S O C N O C M O I T - C O C O C O * - T - C M CM CO —1 _ J L4 LO _ l L6 L7 g g g g g g CD CD .= . E * sz o CD CO _ i l l " h . o C CO CM CM CD O CO f~ Tf Tf '•g <R ^  iri ~ CD CD O Tf CO CO I— T- CM C r— O CO CO O CO O T- 00 5 N O) t o , CO • i i Tf 55 ^ CM O CD CO CO r-- o r-. M . ° ! C D C O C O C M C O C D C M CO o|55 o CO Tf Tf Of— CO CO O Tf CO ^  9 N oi CM O CM LO CM 5 N O CO CO CM ^ ?£. • C ) CO Tf CO CO CO r» CM C CO ^  N O) O CD O Tf ^ ° - ^ r--~ O CO CO T> CO i - CD CD CM >" CO Tf C CM Tf CD i -O I— LO CM CM ••g <°. * CM d LO LO CO CD CO T - CD CM LO °° I- CD i -C f- Tf CD O O O CO f- O •g ^  CM CD C-J' f» CM CO C f^ CO Tf o o t o q N co prj ^  CD CD CO S r - CM CO CO CO CM T- i - CM LO N CD I— C CO Tf CD O CM CD CO O '•g •* d Tf ~ Tf Tf CO O C/D CM T— T— CO C D C O C D C C O C M O C M O Tf C M T- C M •g ° J C O C M C O CD CD . E . E J C £ J = o ° ° Q — CD CD r-i 21 &| CM O Tf '•5 c^  £ f -55 d co c O f~ S co co d co £ Tf < c .Q--SS sz CO o S « -V. ° ° CD o .S » o co 55 T-o' « CD .2 g o o CD M5 « CD § 55 d c g ^ CO o CD CO "^ 3 Tf CD CO f-< 55 d & L O CO d O T-CO Tf C D C O C O £ C M CO d L O C D c O C D C O T -o o CO o CO o E 10 .2 'SZ CO CO o CQ •a a> T 3 c CD E E o o CD CO a> J D ro CD (0 T 3 CD •a c CD E E o u CD CC LU 0) JD CO APPENDIX F Records of Testing Conditions F.1 Model Testing Records The calculated and actual model testing conditions for the parent hull (BOp_single) are tabulated below: UBC SERIES MODEL TESTING TEST DATE COMMENTS PURPOSE July 18 & 21, 2003 PHASE 1 - UBC series model #3 (parent hull without blisters [B0p_single]) To establish baseline condition for comparison in Phase I tests to select which blister as subject of detail studies PRINCIPAL CHARACTERISTICS (SI units) ship model Lambda 13.75 1 LOA.ship - - [m] LWL (static) 27.734 2.017 [m] Initial Displacement (fw) @ Rho(standard)=1000 kg/m3 3202999.35 1232.1 [N] Wetted area (Iw) @ Rho(standard)=1000 kg/m4 287.54 1.5209 [mA2] ACTUAL TEST CONDITIONS (SI units) shlo (sw. fixed) model (fw) Water temperature (actual, measured) 15.0 21.5 [degC] Water density (actual, by interpolation) 998.88 997.68 [kg/mA3] Water kinematic viscosity (actual, by interpolation) 1.12E-06 9.68E-07 [mA2/s] Gravitational acceleration 9.81 9.81 [m/sA2] PRINCIPAL CHARACTERISTICS (Enalish units) ship model Lambda 13.75 1.00 -LOA.ship - - [«] LWL (static) 90.99 6.62 [ft] Initial displacement to DWL (fw) @ Rho(standard)=1.9406slug/ftA3 718307.70 276.31 [lb] Wetted area to DWL (fw) @ Rho(standard)=1.9406slug/ftA3 3095.32 16.37 [ftA2] REFERENCE CONDITIONS FOR INTERPOLATION (Enalish units) ship (sw ©59 Tdea Fl) model (fw &70.7 Tdea Fl) Lower reference -> Water temperature 70 70 [degF] Water density 1.9362 1.9362 [slug/ftA3] Water kinematic viscosity 1.05E-05 1.05E-05 [ftA2/s] Upper reference -> Water temperature 71 71 [degF] Water density 1.9360 1.9360 [slug/ftA3] Water kinematic viscosity 1.04E-05 1.04E-05 [ftA2/s] ACTUAL TEST CONDITIONS (Enalish units) ship (sw) model (fw) Water temperature (actual, measured) 59.0 70.7 [degF] Water density (actual, by interpolation) 1.9384 1.9361 [slug/ftA3] Water kinematic viscosity (actual, by interpolation) 1.20E-05 1.04E-05 [ftA2/s] Gravitational acceleration 32.17 32.17 . [ft/sA2] BALLAST CONDITIONS (Enalish units) ship (sw) model (fw) Corrected displacement to DWL (fw) @ Rho(tank)=1.9361slug/ftA3 - 275.67 [lb] Weight of bare hull (fixed, measured) - 73.56 [lb] Weight of blisters & screws (measured) 0.00 [lb] Weight of heave post (fixed, measured) - 12.40 [lb] Calculated ballast required to DWL (fw) @ Rho(tank)=1.9361slug/ftA3 - 189.71 [lb] Actual ballast used to DWL (fw) - 189.44 [lb] Table F-1: [P1 & 2] Model testing condition records for parent hull [B0p_single] 167 The calculated and actual model testing conditions for the parabolized parent hull with 5 % beam increment single midship bulb (B5p_single_midship) are tabulated below: UBC SERIES MODEL TESTING TEST DATE COMMENTS PURPOSE July 22, 2003 PHASE 1 - UBC series model #3 (parent hull with 5% single blister of 50% waterline length [B5p_slngle]) To study effects of single blister and compare to baseline thus select optimized configurations for further studies PRINCIPAL CHARACTERISTICS (SI units) ship model Lambda 13.75 1 LOA.ship - - [m] LWL (static) 27.734 2.017 [m] Initial Displacement (fw) @ Rho(standard)=1000 kg/m3 3244812.12 1248.2 [N] Wetted area (fw) @ Rho(standard)=1000 kg/m4 287.96 1.5231 [mA2] ACTUAL TEST CONDITIONS (SI units) ship (sw. fixed) model (fw) Water temperature (actual, measured) 15.0 21.5 [degC] Water density (actual, by interpolation) 998.88 997.68 [kg/mA3] Water kinematic viscosity (actual, by interpolation) 1.12E-06 9.68E-07 [mA2/s] Gravitational acceleration 9.81 9.81 [m/sA2] PRINCIPAL CHARACTERISTICS (Enalish units) ship model Lambda 13.75 1.00 -LOA.ship - - [ft] LWL (static) 90.99 6.62 [ft] Initial displacement to DWL (fw) @ Rho(standard)=1.9406slug/ftA3 727684.68 279.92 [lb] Wetted area to DWL (fw) @ Rho(standard)=1.9406slug/ftA3 3099.93 16.40 [ftA2] REFERENCE CONDITIONS FOR INTERPOLATION (Enalish units) ship (sw ©59 rdeo Fl) model (fw ©70.7 Tdea Fl) Lower reference -> Water temperature 70 70 [degF] Water density 1.9362 1.9362 [slug/ftA3 Water kinematic viscosity 1.05E-05 1.05E-05 [ftA2/s] Upper reference -> Water temperature 71 71 [degF] Water density 1.9360 1.9360 [slug/ftA3 Water kinematic viscosity 1.04E-05 1.04E-05 [ftA2/s] ACTUAL TEST CONDITIONS (Enalish units) ship (sw) model (fw) Water temperature (actual, measured) 59.0 70.7 [degF] Water density (actual, by interpolation) 1.9384 1.9361 [slug/ftA3 Water kinematic viscosity (actual, by interpolation) 1.20E-05 1.04E-05 [ftA2/s] Gravitational acceleration 32.17 32.17 [ft/sA2] BALLAST CONDITIONS (Enalish units) ship (sw) model (fw) Corrected displacement to DWL (fw) @ Rho(tank)=1.9361slug/ftA3 - 279.27 [lb] Weight of bare hull (fixed, measured) - 73.56 [lb] Weight of blisters & screws (measured) 3.32 [lb] Weight of heave post (fixed, measured) - 12.40 [lb] Calculated ballast required to DWL (fw) © Rho(tank)=1.9361slug/ftA3 - 189.99 [lb] Actual ballast used to DWL (fw) - 193.84 [lb] Table F-2: [P1] Model testing condition records for parent hull using single bulb at midship of 5% increased beam [B5p single_midship] 168 The calculated and actual model testing conditions for the parabolized parent hull with 10% beam increment single midship bulb (B10p_single_midship) are tabulated below: UBC SERIES MODEL TESTING TEST DATE COMMENTS PURPOSE July 23, 2003 PHASE 1 - UBC series model #3 (parent hull with 10% single blister of 50% waterline length [B10p_single]) To study effects of single blister and compare to baseline thus select optimized configurations for further studies PRINCIPAL CHARACTERISTICS (SI units) ship model Lambda 13.75 1 LOA.ship - - [m] LWL (static) 27.734 2.017 [m] Initial Displacement (fw) @ Rho(standard)=1000 kg/m3 3276900.54 1260.5 [N] Wetted area (fw) @ Rho(standard)=1000 kg/m4 288.53 1.5261 [mA2] ACTUAL TEST CONDITIONS (SI units) ship (sw. fixed) model (fw) Water temperature (actual, measured) 15.0 21.5 [degC] Water density (actual, by interpolation) 998.88 997.68 [kg/mA3] Water kinematic viscosity (actual, by interpolation) 1.12E-06 9.68E-07 [mA2/s] Gravitational acceleration 9.81 9.81 [m/sA2] PRINCIPAL CHARACTERISTICS (Enalish units) ship model Lambda 13.75 1.00 -LOA.ship - - [ft] LWL (static) 90.99 6.62 [ft] Initial displacement to DWL (fw) @ Rho(standard)=1.9406slug/ff>3 734880.86 282.69 [lb] Wetted area to DWL (fw) @ Rho(standard)=1.9406slug/ft"3 3106.03 16.43 [ftA2] REFERENCE CONDITIONS FOR INTERPOLATION (Enalish units) ship (sw ©59 rdea Fl) model (fw ©70.7 rdea Fl) Lower reference -> Water temperature 70 70 [degF] Water density 1.9362 1.9362 [slug/ftA3] Water kinematic viscosity 1.05E-05 1.05E-05 [ftA2/s] Upper reference -> Water temperature 71 71 [degF] Water density 1.9360 1.9360 [slug/ftA3] Water kinematic viscosity 1.04E-05 1.04E-05 [ftA2/s] ACTUAL TEST CONDITIONS (Enalish units) ship (sw) model (fw) Water temperature (actual, measured) 59.0 70.7 [degF] Water density (actual, by interpolation) 1.9384 1.9361 [slug/ftA3] Water kinematic viscosity (actual, by interpolation) 1.20E-05 1.04E-05 [ftA2/s] Gravitational acceleration 32.17 32.17 [ft/sA2] BALLAST CONDITIONS (Enalish units) ship (sw) model (fw) Corrected displacement to DWL (fw) @ Rho(tank)=1.9361slug/ftA3 - 282.03 [lb] Weight of bare hull (fixed, measured) - 73.56 [lb] Weight of blisters & screws (measured) 3.40 [lb] Weight of heave post (fixed, measured) - 12.40 [lb] Calculated ballast required to DWL (fw) @ Rho(tank)=1.9361slug/ftA3 - 192.67 [lb] Actual ballast used to DWL (fw) - 197.84 [lb] Table F-3: [Pi] Model testing condition records for parent hull using single bulb at midship of 10% increased beam [B10p_single_midship] 169 The calculated and actual model testing conditions for the parabolized parent hull with 1 5 % beam increment single midship bulb (B15p_single_midship) are tabulated below: UBC SERIES MODEL TESTING TEST DATE COMMENTS PURPOSE July 24 & 25, 2003 PHASE 1 - UBC series model #3 (parent hull with 15% single blister of 50% waterline length [B15p_single]) To study effects of single blister and compare to baseline thus select optimized configurations for further studies PRINCIPAL CHARACTERISTICS (SI units) Lambda LOA.ship LWL (static) Initial Displacement (fw) @ Rho(standard)=1000 kg/m3 Wetted area (iw) @ Rho(standard)=1000 kg/m4 ACTUAL TEST CONDITIONS (SI units) Water temperature (actual, measured) Water density (actual, by interpolation) Water kinematic viscosity (actual, by interpolation) Gravitational acceleration ship 13.75 27.734 3304136.42 289.17 ship (sw. fixed) 15.0 998.88 1.12E-06 9.81 model 1 2.017 1271.0 1.5295 model (fw) 22.0 997.59 9.57E-07 9.81 [m] [m] [N] [mA2] [degC] [kg/mA3] [mA2/s] [m/sA2] PRINCIPAL CHARACTERISTICS (English units) Lambda LOA.ship LWL (static) Initial displacement to DWL (fw) @ Rho(standard)=1.9406slug/ftA3 Wetted area to DWL (fw) @ Rho(standard)=1.9406slug/1tA3 ship 13.75 90.99 740988.80 3112.86 model 1.00 6.62 285.04 16.46 [Ib] [ftA2] REFERENCE CONDITIONS FOR INTERPOLATION (English units) Lower reference -> Water temperature Water density Water kinematic viscosity ship (sw @59 fdeo FI) 70 1.9362 1.05E-05 model (fw@71.6rdeg FI) 70 [degF] 1.9362 [slug/ftA3] 1.05E-05 [ftA2/s] Upper reference -> Water temperature Water density Water kinematic viscosity ACTUAL TEST CONDITIONS (English units) Water temperature (actual, measured) Water density (actual, by interpolation) Water kinematic viscosity (actual, by interpolation) Gravitational acceleration BALLAST CONDITIONS (English units) Corrected displacement to DWL (fw) @ Rho(tank)=1.9359slug/ftA3 Weight of bare hull (fixed, measured) Weight of blisters & screws (measured) Weight of heave post (fixed, measured) Calculated ballast required to DWL (fw) @ Rho(tank)=1.9359slug/ftA3 Actual ballast used to DWL (fw) 71 71 [degF) 1.9360 1.9360 [slug/ftA3] 1.04E-05 1.04E-05 [ftA2/s] ship (sw) model (fw) 59.0 71.6 [degF] 1.9384 1.9359 [slug/ftA3] 1.20E-05 1.03E-05 [ftA2/s] 32.17 32.17 [ft/sA2] ship (sw) model (fw) 284.35 [Ib] 73.56 [Ib] 3.60 [Ib] 12.40 [Ib] 194.79 [Ib] 200.84 [Ib] Table F-4: [P1 & 2] Model testing condition records for parent hull using single bulb at midship of 15% increased beam [B15p_single_midship] 170 The calculated and actual model testing conditions for the parabolized parent hull with 20% beam increment single midship bulb (B20p_single_midship) are tabulated below: UBC SERIES MODEL TESTING TEST DATE COMMENTS PURPOSE July 28, 2003 PHASE 1 - UBC series model #3 (parent hull with 20% single blister of 50% waterline length [B20p_single]) To study effects of single blister and compare to baseline thus select optimized configurations for further studies PRINCIPAL CHARACTERISTICS (SI units) ship model Lambda 13.75 1 LOA.ship - - [m] LWL (static) 27.734 2.017 [m] Initial Displacement (fw) @ Rho(standard)=1000 kg/m3 3329922.92 1280.9 [N] Wetted area (fw) @ Rho(standard)=1000 kg/m4 289.91 1.5334 [mA2] ACTUAL TEST CONDITIONS (SI units) ship (sw. fixed) model (fw) Water temperature (actual, measured) 15.0 22.0 [degC] Water density (actual, by interpolation) 998.88 997.59 [kg/mA3] Water kinematic viscosity (actual, by interpolation) 1.12E-06 9.57E-07 [mA2/s] Gravitational acceleration 9.81 9.81 [m/sA2] PRINCIPAL CHARACTERISTICS (Enalish units) ship model Lambda 13.75 1.00 -LOA.ship - - [ft] LWL (static) 90.99 6.62 [ft] Initial displacement to DWL (fw) @ Rho(standard)=1.9406slug/ftA3 746771.70 287.26 [lb] Wetted area to DWL (fw) @ Rho(standard)=1.9406slug/ftA3 3120.85 16.51 [ftA2] REFERENCE CONDITIONS FOR INTERPOLATION (Enalish units) ship (sw ©59 Tdea Fl) model (fw ®71.6 rdea Fl) Lower reference -> Water temperature 70 70 [degF] Water density 1.9362 1.9362 [slug/ftA3 Water kinematic viscosity 1.05E-05 1.05E-05 [ftA2/s] Upper reference -> Water temperature 71 71 [degF] Water density 1.9360 1.9360 [slug/ftA3 Water kinematic viscosity 1.04E-05 1.04E-05 [ftA2/s] ACTUAL TEST CONDITIONS (Enalish units) ship (sw) model (fw) Water temperature (actual, measured) 59.0 71.6 [degF] Water density (actual, by interpolation) 1.9384 1.9359 [slug/ftA3 Water kinematic viscosity (actual, by interpolation) 1.20E-05 1.03E-05 [ftA2/s] Gravitational acceleration 32.17 32.17 [ft/sA2] BALLAST CONDITIONS (Enalish units) ship (sw) model (fw) Corrected displacement to DWL (fw) © Rho(tank)=1,9359slug/ftA3 - 286.57 [lb] Weight of bare hull (fixed, measured) - 73.56 [lb] Weight of blisters & screws (measured) 4.46 [lb] Weight of heave post (fixed, measured) - 12.40 [lb] Calculated ballast required to DWL (fw) @ Rho(tank)=1.9359slug/ftA3 - 196.15 [lb] Actual ballast used to DWL (fw) - 202.84 [lb] Table F-5: [P1] Model testing condition records for parent hull using single bulb at midship of 20% increased beam [B20p_single_midship] 171 The parabolized parent hull with selected 15% beam increment integrated into forebody (F) is tested (B15p_single_forebody). The calculated and actual model testing conditions are tabulated below: UBC SERIES MODEL TESTING TEST DATE COMMENTS PURPOSE Sep 27, 2003 PHASE 3 - UBC series model #3 (parent hull with 15% beam increment double blisters of 50% waterline length cut into half and placed ONLY at FOREBODY [B15p_double_setE]) To study effects of single blisters at forebody location (6 sets to be.done for this configuration: set A, B, C, D, E, F where set A was in fact completed during Phase 2) PRINCIPAL CHARACTERISTICS (SI units) ship model Lambda 13.75 1 LOA.ship - - [m] LWL (static) 27.734 2.017 [m] Initial Displacement (fw) @ Rho(standard)=1000 kg/m3 3327416.84 1280.0 [N] Wetted area (fw) @ Rho(standard)=1000 kg/m4 294.35 1.5569 [mA2] ACTUAL TEST CONDITIONS (SI units) ship (sw. fixed) model (fw) Water temperature (actual, measured) 15.0 18.5 [degC] Water density (actual, by interpolation) 998.88 998.24 [kg/mA3] Water kinematic viscosity (actual, by interpolation) 1.12E-06 1.04E-06 [mA2/s] Gravitational acceleration 9.81 9.81 [m/sA2] PRINCIPAL CHARACTERISTICS (Enallsh units) ship model Lambda 13.75 1.00 -LOA.ship - - [«] LWL (static) • 90.99 6.62 [ft] Initial displacement to DWL (fw) @ Rho(standard)=1.9406slug/ftA3 746209.69 287.05 [lb] Wetted area to DWL (fw) @ Rho(standard)=1.9406slug/ft"3 3168.67 16.76 [ftA2] REFERENCE CONDITIONS FOR INTERPOLATION (Enallsh units) ship (sw (359 Tdea FI) model (fw ©65.3 Tdea FI) Lower reference -> Water temperature 70 70 [degF] Water density 1.9362 1.9362 [slug/ftA3] Water kinematic viscosity 1.05E-05 1.05E-05 [ftA2/s] Upper reference -> Water temperature 71 71 [degF] Water density 1.9360 1.9360 [slug/ft"3] Water kinematic viscosity 1.04E-05 1.04E-05 [ftA2/s] ACTUAL TEST CONDITIONS (Enallsh units) ship (sw) model (fw) Water temperature (actual, measured) 59.0 65.3 [degF] Water density (actual, by interpolation) 1.9384 1.9371 [slug/ftA3] Water kinematic viscosity (actual, by interpolation) 1.20E-05 1.12E-05 [ftA2/s] Gravitational acceleration 32.17 32.17 [ft/sA2] BALLAST CONDITIONS (Enalish units) ship (sw) model (fw) Corrected displacement to DWL (fw) @ Rho(tank)=1.9372slug/ft/l3 - 286.54 [lb] Weight of bare hull (fixed, measured) - 73.56 [lb] Weight of blisters & screws (measured) 0.00 [lb] Weight of heave post (fixed, measured) - 12.40 [lb] Calculated ballast required to DWL (fw) @ Rho(tank)=1.9372slug/ftA3 - 200.58 [lb] Actual ballast used to DWL (fw) - 202.42 [lb] Table F-6: [P3] Model testing condition records for parent hull using single bulb at forebody of 15% increased beam [B15p_single_forebody] 172 The parabolized parent hull with selected 1 5 % beam increment integrated into aftbody (A) is tested (B15p_single_aftbody). The calculated and actual model testing conditions are tabulated below: UBC SERIES MODEL TESTING TEST DATE COMMENTS Sep 27, 2003 PHASE 3 - UBC series model #3 (parent hull with 15% beam increment double blisters of 50% waterline length cut into half and placed ONLY at AFTBODY [B15p_double_setF]) PURPOSE To study effects of single blisters at aftbody location (6 sets to be done for this configuration: set A, B, C, D, E, F where set A was in fact completed during Phase 2) PRINCIPAL CHARACTERISTICS (SI units) Lambda LOA.ship LWL (static) Initial Displacement (fw) @ Rho(standard)=1000 kg/m3 Wetted area (fw) @ Rho(standard)=1000 kg/m4 ship 13.75 27.734 3328352.45 294.95 model 1 2.017 1280.3 1.5601 [m] [m] [N] [mA2] ACTUAL TEST CONDITIONS (SI units) Water temperature (actual, measured) Water density (actual, by interpolation) Water kinematic viscosity (actual, by interpolation) Gravitational acceleration ship (sw, fixed) 15.0 998.88 1.12E-06 9.81 model (fw) 18.5 998.24 1.04E-06 9.81 [degC] [kg/mA3] [mA2/s] [m/sA2] PRINCIPAL CHARACTERISTICS (English units) Lambda LOA.ship LWL (static) Initial displacement to DWL (fw) @ Rho(standard)=1.9406slug/ftA3 Wetted area to DWL (fw) @ Rho(standard)=1.9406slug/ftA3 ship 13.75 90.99 746419.51 3175.18 model 1.00 6.62 287.13 16.79 [lb] [ftA2] REFERENCE CONDITIONS FOR INTERPOLATION (Enalish units) Lower reference -> Water temperature Water density Water kinematic viscosity ship (sw (5)59 fdeg Fl) 70 1.9362 1.05E-05 model (fw @65.3 rdea Fl) 70 [degF] 1.9362 [slug/ftA3] 1.05E-05 [ftA2/s] Upper reference-> Water temperature 71 71 [degF] Water density 1.9360 1.9360 [slug/ftA3] Water kinematic viscosity 1.04E-05 1.04E-05 [ftA2/s] ACTUAL TEST CONDITIONS (English units) Water temperature (actual, measured) Water density (actual, by interpolation) Water kinematic viscosity (actual, by interpolation) Gravitational acceleration BALLAST CONDITIONS (English units) Corrected displacement to DWL (fw) @ Rho(tank)=1.9372slug/ftA3 Weight of bare hull (fixed, measured) Weight of blisters & screws (measured) Weight of heave post (fixed, measured) Calculated ballast required to DWL (fw) @ Rho(tank)=1.9372slug/ftA3 Actual ballast used to DWL (fw) ship (sw) model (fw) 59.0 65.3 [degF] 1.9384 1.9371 [slug/ftA3] 1.20E-05 1.12E-05 [ftA2/s] 32.17 32.17 [WsA2] ship (sw) model (fw) 286.62 [lb] 73.56 [lb] 0.00 [lb] 12.40 [lb] 200.66 [lb] 200.42 [lb] Table F-7: [P3] Model testing condition records for parent hull using single bulb at aftbody of 15% increased beam [B15p_single aftbody] 173 The parabolized parent hull with selected 1 5 % beam increment single midship bulb is tested with fairing extension of 2 5 % of waterline length applied to the rear of the bulb (B15p_single_midship_alpha1), i.e. equivalent to 490mm. The calculated and actual model testing conditions are tabulated below: UBC SERIES MODEL TESTING TEST DATE COMMENTS PURPOSE Nov 14, 2003 PHASE 4 - UBC series model #3 (parent hull with 15% single blister of 50% waterline length + blister aft fairing extension of shortest length 490mm for angle alphal [B15p_single_alpha1]) To study effects of application of blister end fairing extension PRINCIPAL CHARACTERISTICS (SI units) ship model Lambda 13.75 1 LOA.ship - - [m] LWL (static) 27.734 2.017 [m] Initial Displacement (fw) @ Rho(standard)=1000 kg/m3 3322396.50 1278.0 [N] Wetted area (fw) @ Rho(standard)=1000 kg/m4 291.17 1.5401 [mA2] ACTUAL TEST CONDITIONS (SI units) ship (sw. fixed) model (fw) Water temperature (actual, measured) 15.0 15.5 [degC] Water density (actual, by interpolation) 998.88 998.79 [kg/mA3] Water kinematic viscosity (actual, by interpolation) 1.12E-06 1.11E-06 [mA2/s] Gravitational acceleration 9.81 9.81 [m/sA2] PRINCIPAL CHARACTERISTICS (Enallsh units) ship model Lambda 13.75 1.00 -LOA.ship - - [ft] LWL (static) 90.99 6.62 [«] Initial displacement to DWL (fw) @ Rho(standard)=1.9406slug/ftA3 745083.82 286.61 [Ib] Wetted area to DWL (fw) @ Rho(standard)=1.9406slug/ftA3 3134.47 16.58 [ftA2] REFERENCE CONDITIONS FOR INTERPOLATION (Enallsh units) ship (sw @59 Tdea FI) model (fw (359.9 fdea FI) Lower reference -> Water temperature 70 70 [degF] Water density 1.9362 1.9362 [slug/ftA3] Water kinematic viscosity 1.05E-05 1.05E-05 [ftA2/s] Upper reference -> Water temperature 71 71 [degF] Water density 1.9360 1.9360 [slug/ftA3] Water kinematic viscosity 1.04E-05 1.04E-05 [ftA2/s] ACTUAL TEST CONDITIONS (Enalish units) ship (sw) model (fw) Water temperature (actual, measured) 59.0 59.9 [degF] Water density (actual, by interpolation) 1.9384 1.9382 [slug/ftA3] Water kinematic viscosity (actual, by interpolation) 1.20E-05 1.19E-05 [ftA2/s] Gravitational acceleration 32.17 32.17 [ft/sA2] BALLAST CONDITIONS (Enallsh units) ship (sw) model (fw) Corrected displacement to DWL (fw) @ Rho(tank)=1 .9383slug/ftA3 - 286.27 [Ib] Weight of bare hull (fixed, measured) - 73.56 [lb] Weight of blisters & screws (measured) 3.60 [Ib] Weight of heave post (fixed, measured) - 12.40 [lb] Calculated ballast required to DWL (fw) @ Rho(tank)=1.9383slug/(tA3 - 196.71 [lb] Actual ballast used to DWL (fw) - 200.84 [Ib] Table F-8: [P4] Model testing condition records for parent hull using single bulb at midship of 15% increased beam and 25% extended fairing length at rear of bulb [B15p_single_midship alphal] 174 The parabolized parent hull with selected 15% beam increment single midship bulb is tested with fairing extension of 3 0 % of waterline length applied to the rear of the bulb (B15p_single_midship_alpha2), i.e. equivalent to 610mm. The calculated and actual model testing conditions are tabulated below: UBC SERIES MODEL TESTING TEST DATE COMMENTS PURPOSE Dec 17, 2003 PHASE 4 - UBC series model #3 (parent hull with 15% single blister of 50% waterline length + blister aft fairing extension of second longest length 610mm for angle alpha2 [B15p_single_alpha2]) To study effects of application of blister end fairing extension PRINCIPAL CHARACTERISTICS (SI units) Lambda LOA.ship LWL (static) Initial Displacement (fw) @ Rho(standard)=1000 kg/m3 Wetted area (fw) @ Rho(standard)=1000 kg/m4 ship 13.75 27.734 3330252.46 291.49 model 1 2.017 1281.1 1.5418 [m] [m] [N] [m*2] ACTUAL TEST CONDITIONS (SI units) Water temperature (actual, measured) Water density (actual, by interpolation) Water kinematic viscosity (actual, by interpolation) Gravitational acceleration ship (sw. fixed) 15.0 998.88 1.12E-06 9.81 model (fw) 14.5 998.98 1.13E-06 9.81 [degC] [kg/m*3] [mA2/s] [m/sA2] PRINCIPAL CHARACTERISTICS (English units) Lambda LOA.ship LWL (static) Initial displacement to DWL (fw) @ Rho(standard)=1.9406slug/ft'l3 Wetted area to DWL (fw) @ Rho(standard)=1.9406slug/ftA3 ship 13.75 90.99 746845.61 3137.93 model 1.00 6.62 287.29 16.60 [lb] [ftA2] REFERENCE CONDITIONS FOR INTERPOLATION (English units) Lower reference -> Water temperature Water density Water kinematic viscosity ship (sw (359 Idea Fl) 70 1.9362 1.05E-05 model (fw (358.1 rdea Fl) 70 [degF] 1.9362 [slug/ft«3] 1.05E-05 [ft"2/s] Upper reference-> Water temperature 71 Water density 1.9360 Water kinematic viscosity 1.04E-05 ACTUAL TEST CONDITIONS (Enalish units) ship (sw) Water temperature (actual, measured) 59.0 Water density (actual, by interpolation) 1.9384 Water kinematic viscosity (actual, by interpolation) 1.20E-05 Gravitational acceleration 32.17 BALLAST CONDITIONS (English units) ship (sw) Corrected displacement to DWL (fw) @ Rho(tank)=1.9386slug/ftA3 Weight of bare hull (fixed, measured) Weight of blisters & screws (measured) Weight of heave post (fixed, measured) Calculated ballast required to DWL (fw) @ Rho(tank)=1.9386slug/ftA3 Actual ballast used to DWL (fw) 71 1.9360 1.04E-05 model (fw) 58.1 1.9386 1.22E-05 32.17 model (fw) 287.00 73.56 3.60 12.40 197.44 201.84 [degF] [slug/ftA3] [ft*2/s] [degF] [slug/ft^ ] [ftA2/s] [ft/s*2] [lb] [lb] [lb] [lb] [lb] [lb] Table F-9: [P4] Model testing condition records for parent hull using single bulb at midship of 15% increased beam and 30% extended fairing length at rear of bulb [B15p_single_midship_alpha2] 175 The parabolized parent hull with selected 15% beam increment single midship bulb is tested with fairing extension of 3 5 % of waterline length applied to the rear of the bulb (B15p_single_midship_alpha3), i.e. equivalent to 702mm. The calculated and actual model testing conditions are tabulated below: UBC SERIES MODEL TESTING TEST DATE Dec 16, 2003 COMMENTS PHASE 4 - UBC series model #3 (parent hull with 15% single PURPOSE To study effects of application of blister end fairing extension PRINCIPAL CHARACTERISTICS (SI units) ship model Lambda 13.75 1 LOA.ship - - [m] LWL (static) 27.734 2.017 [m] Initial Displacement (fw) @ Rho(standard)=1000 kg/m3 3335596.90 1283.1 [N] Wetted area (fw) @ Rho(standard)=1000 kg/m4 291.70 1.5429 [m*2] ACTUAL TEST CONDITIONS (SI units) ship (sw. fixed) model (fw) Water temperature (actual, measured) 15.0 14.5 [degC] Water density (actual, by interpolation) 998.88 998.98 [kg/mA3] Water kinematic viscosity (actual, by interpolation) 1.12E-06 1.13E-06 [mA2/s] Gravitational acceleration 9.81 9.81 [m/sA2] PRINCIPAL CHARACTERISTICS (Enallsh units) ship model Lambda 13.75 1.00 -LOA.ship - - [ft] LWL (static) 90.99 6.62 [ft] Initial displacement to DWL (fw) @ Rho(standard)=1.9406slug/ftA3 748044.16 287.75 [lb] Wetted area to DWL (fw) @ Rho(standard)=1.9406slug/ftA3 3140.12 16.61 [ft"2] REFERENCE CONDITIONS FOR INTERPOLATION (Enallsh units) ship (sw ©59 Tdea FI) model (fw (358.1 rdea FI) Lower reference -> Water temperature 70 70 [degF] Water density 1.9362 1.9362 [slug/ft"3 Water kinematic viscosity 1.05E-05 1.05E-05 [ft"2/s] Upper reference -> Water temperature 71 71 [degF] Water density 1.9360 1.9360 [slug/ft"3 Water kinematic viscosity 1.04E-05 1.04E-05 [ft*2/s] ACTUAL TEST CONDITIONS (Enalish units) ship (sw) model (fw) Water temperature (actual, measured) 59.0 58.1 [degF] Water density (actual, by interpolation) 1.9384 1.9386 [slug/ftA3 Water kinematic viscosity (actual, by interpolation) 1.20E-05 1.22E-05 [ftA2/s] Gravitational acceleration 32.17 32.17 [ft/s"2] BALLAST CONDITIONS (Enallsh units) Corrected displacement to DWL (fw) @ Rho(tank)=1.9386slug/ft"3 Weight of bare hull (fixed, measured) Weight of blisters & screws (measured) Weight of heave post (fixed, measured) Calculated ballast required to DWL (fw) @ Rho(tank)=1.9386slug/itA3 Actual ballast used to DWL (fw) ship (sw) model (fw) 287.46 73.56 3.60 12.40 197.90 202.84 [lb] [lb] [lb] [lb] [lb] [lb] Table F-10: [P4] Model testing condition records for parent hull using single bulb at midship of 15% increased beam and 35% extended fairing length at rear of bulb [B15p_single_midship_alpha3] 176 This is the offset table of the recommended UBC series model #3 revised based on experimental and numerical studies. The calculated and actual model testing conditions are tabulated below: UBC SERIES MODEL TESTING TEST DATE COMMENTS PURPOSE May 28, 31 2004 & June 1, 2004 (model #3 with revisionl#2, displacement scaled to same as parent hull, maximum beam increment from 15% down to 10.7%, entrance angle from 29.1deg down to 26.0degree, max beam location from station 5 to station 4) For comparison with previous results PRINCIPAL CHARACTERISTICS (SI units) ship . model Lambda 13.75 1 LOA.ship - - [m] LWL (static) 27.734 2.017 [m] Initial Displacement (fw) @ Rho(standard)=1000 kg/m3 3203413.86 1232.3 [N] Wetted area (fw) @ Rho(standard)=1000 kg/m4 285.84 1.5119 [mA2] ACTUAL TEST CONDITIONS (SI units) ship (sw. fixed) model (fw) Water temperature (actual, measured) 15.0 19.0 [degC] Water density (actual, by interpolation) 998.88 998.14 [kg/mA3] Water kinematic viscosity (actual, by interpolation) 1.12E-06 1.03E-06 [mA2/s] Gravitational acceleration 9.81 9.81 [m/sA2] PRINCIPAL CHARACTERISTICS (Enalish units) ship model Lambda 13.75 1.00 -LOA.ship - - [«] LWL (static) 90.99 6.62 [»] Initial displacement to DWL (fw) @ Rho(standard)=1.9406slug/ftA3 718400.66 276.35 [lb] Wetted area to DWL (fw) @ Rho(standard)=1.9406slug/ftA3 3077.01 16.28 [ftA2] REFERENCE CONDITIONS FOR INTERPOLATION (Enallsh units) ship (sw ©59 Tdea FI) model (fw (S66.2 Tdea FI) Lower reference -> Water temperature 70 70 [degF] Water density 1.9362 1.9362 [slug/ftA3] Water kinematic viscosity 1.05E-05 1.05E-05 [ftA2/s] Upper reference -> Water temperature 71 71 [degF] Water density 1.9360 1.9360 [slug/ftA3] Water kinematic viscosity 1.04E-05 1.04E-05 [ftA2/s] ACTUAL TEST CONDITIONS (Enalish units) ship (sw) model (fw) Water temperature (actual, measured) 59.0 66.2 [degF] Water density (actual, by interpolation) 1.9384 1.9370 [slug/ftA3] Water kinematic viscosity (actual, by interpolation) 1.20E-05 1.10E-05 [ftA2/s] Gravitational acceleration 32.17 32.17 [ft/sA2] BALLAST CONDITIONS (Enallsh units) ship (sw) model (fw) Corrected displacement to DWL (fw) @ Rho(tank)=1.937slug/ftA3 - 275.84 [lb] Weight of bare hull (fixed, measured) - 70.08 [lb] Weight of blisters & screws (measured) 0.00 [Ib] Weight of heave post (fixed, measured) - 12.20 [lb] Calculated ballast required to DWL (fw) @ Rho(tank)=1.937slug/ftA3 - 193.56 [Ib] Actual ballast used to DWL (fw) - [202.80 lb] Table F-11: [Recommended] Model testing condition records for the revised UBC series model #3 [B11p_single_midship_recommended] 177 to c o -4—' 'co "Z5 cr o co E o> to >^  cn CD CJ) c CO tr CD co 5 . I c S CD j< e |cg LU DC ICQ l < I o (0 TJ i_ O o CD DC E CD CO >» CO c oQ. c CD CO CL CD > CO CM • LL co c: CD co CO CD E co o "co =3 -4—' o CO CD CD CO CD CO CD f U m oj f ; CM , O ' CM CM O O O . O CD CD CD O O O < O "3 i II 1*1; i '0 ZJ c CD E JD _cg E CD CO CO i O CO CD E JD CO DC LU CO Z) 'CO c 0 E l -Q co m 41 ' o f-0) SPSS IOC X I — rr O x o ^ - O CO :1 f i : E Z3 F c i II: ll-H |0- I 3 C 0 E -Q CO -I—* l< DC o X ' 0 CO • 3 0 E CO LU CD < CD C CD § c g> 'c/> a CO V) o CO 0) o •o >» X o >< Q Z LU Q. Q_ < O E o E co cu JD TJ CU CO CO CD \ o _o CD JD cz o JZ CO CD CO CJ) — c CD JZ CO I— CD CD TJ O E o CO -*—' Z5 CO cu Q cu cz cu CO cz D) 'co CU TJ CU C L zs "cu cu E o TJ CU JO Z5 o CO o CO CO o '•4—1 CO CO o i, TJ 5^ JZ cu CM oc5 CD CO CO JZ Q_ o -— TJ CD - I — ' O Zi TJ C o o CO CO C L J Z CO T J "E "CO J D ZJ J D i C D CZ CO CJ) c CO zs o CM CO cq igle 1.25 CO o CM 00 !0p_sin 057411 80.93 1592.51 6933.4 41745. 9.66 83.63 C J 53.15 f - o f - 224231 7.23 6.89 3.53 CO CO CO CM LO f~-CO CO CO CM I— CM oo « . M CO OJ o CO LO o cd a> T f CD o CO CQ CO T— ( od OJ CM CO o d d d 00 CO 5p single 9562961.95 71.01 610.81 3607.01 37202.09 9.40 84.22 LO 55.40 00 CM N. LO 154110145. 0.12 9.51 3.53 a> cn CM T f LO LO T f CD CO T f i — o T f CO OJ C J cn LO o CO • 1^ CO T f LO o LO CQ cn 00 00 CM CM LO O d d d o p gle 9.21 33164.63 o i T f cn Opsin 849497 60.54 509.15 0251.1" 33164.63 9.13 84.84 r-- 57.72 o T f T f 155583 3.51 2.64 3.53 = CO LO CO LO CD LO CO a> LO CO eg C J r- co LO o cp f~ CO LO o O) m cn CO T— 1 00 T— -^C J CM T f o d o d CD CO gle 16.82 6872.81 CD cn 6849. ip_sin 72367 48.19 004.21 6872.81 29137 8.85 85.59 CJ CO 60.12 CM LO oo 22640 7.53 6.38 3.53 LO CO CM 00 LO T f CD CO O) CO 1— 00 C\l CM LO T f LO o CO 1 CD C J T f o CM m cn CO i CO CM CM T f O d d d CD LO I0p_single 5597139.48 32.11 538.64 4851.06 25590.00 8.54 86.03 T f 00 61.82 o CO o> 435490504. 2.90 1.44 3.53 cn T f o CO o 00 co 00 00 00 I0p_single Cvl CM CO LO o CO 1 LO CM T f o OJ cn CO ' — 1 co ~^ CM CM cr> o d d d 15 CO — cu o ]+, a CL !c x: CO CO T J • a E "E •c CO C J C J C J Q. Q. E T f E z E E E E E %LF E %LF E E E E E E E E E E E E %LF E %LF Z E E E E E , • t co ? ? o o H .o n E 0> o o E C c c ,— ter atio g to > BM) P KG) Wa CO w 15 + m 1 Wa •e •c "O "D c 5 . 5 . CD n II II O) a i 'S c a> 75 CO Q o .c o <§> 75 •*-" 2 o o CD >- C O T J MAR 75 2 ship, erpla etted, t needei 3 E o 73 •a CO etted, t DO LL needei o I— a - E m o o z hf (5 E CO > 5 < < < _ l _ l m O O O o o o CM I o T^ o CL !E co TJ CO JD 3 JD 0) O ) C '55 CJ) c "to 3 C CO Q. CD C E ro CO JD TJ CD (A ro CD B 2 ro — c O ) CO CD TJ » CO CO o T 3 CO CO o TJ >« X C\T 00 o 00 o JD CO " O CD -t—' o _0 CD CO -t—' CO E co CD _Q T3 0 CO CO 0 1 o c CO - I — ' 13 CO 0 1_ 0 0 _Q c o sz CO 0 CO CO _ C ' CO 0 -*—» 0 " O o E CO CL 0 5 . £ Q_ 0 "co O) 'co 0 ~o 0 CL =3 "0 0 E o T 3 0 _C0 « o CO o CO CO o -I—» CO - I — ' CO o J — >* •o 0 o zz TD cz o o CO CO > N TD O LL > v T 3 O _Q 0 0 CO -t—' c 0 E 0 o cz E CO 0 sz> ^° LO oo CO double 2529.42 CO CO 952.48 i5.01 J91.93 CNJ CO o • t f CM 1881535. I— O ) CO m a> LO o ci 00 952.48 L O •ST i—* o CD CO CO CO cn cn L O CD 1"- 00 00 CO CO O J CO CM LO CO CD r» CD LO a> 00 CD CM CM cn — co CO CNJ CO LO LO o CO CM o 00 03 cn CO • • CO CM CM ci ci O o u_ 1 CM 00 5p_double etE 0475842.21 79.97 952.48 9913.00 56894.50 8.86 73.32 CM 48.43 O ) 977454049. 0.12 8.98 3.53 LO L O L O 05 CO CD CM 00 O J cn TT CO CO CO LO LO o CO CD I CD CO o LO m cn CO CO O J CM ci O O o CO CO 5p_single 9562961.95 71.01 610.81 3607.01 37202.09 9.40 84.22 LO 1 ^ 55.40 00 CM L O 154110145. 0.12 9.51 3.53 ai O J L O L O CO OO o t 00 C\J O J cn L O o CO 1 CO L O o LO CQ a> CO • CO O J CM LO o ci o o CD LO ip_single 5597139.48 32.11 538.64 4851.06 25590.00 8.54 86.03 00 61.82 o CD cn 435490504. 2.90 1.44 3.53 5) •cf o CD o 00 CD I-. CO r-~ 00 00 C_J C\J O J CO L O o 00 LO CM o m CO 1— CO O J O J CO O o o C> -aft) -aft) - — . CD o O 2. 2 a Q 1c Ic CO CO T J •o 'E 'E •c •c 5 CO CNJ O J O J E 0 _ CL E E E E "E E E E E CL 1 0 - 1 .E E E z E E E E E -P E Z E E E E E So ? _i o o II J3 f<E t O O E r~ ter Iii c o to ation > m P KG) Wa w CO •aft B+ 2 Wa •c r T J T3 ian js 5. CD ca JI_ 10 c CJ "co CO Q .c ® o o S 2 o o CD >- c o X 3 tr CO 0J MA 2 ~S to a. CO led. need JS 2 o o T3 CO CD CD u_ need o r -CL 5 - 'E m o o •—• H CO A E co > < < < _l CO C3 O O o O o >» •o o CO T3 O JQ 0 CO n n 0 O ) c w Ui c "w E 3 CO = CD 3 -Q •= T J CD CO CO 0 c 0 CO a. 0 0 0 c TZ CO c O ) w 0 T3 •4-* CO CO o CO +^ CO o X CO 0. Csi 6 0 CO o c lO 3 J o —r 0 0 CO CD CD CZ CD -f—' CO c D) 'co CD T3 CD J Z o - »—' C L CD CD TJ 0 J? ZJ _o CO o CO CO o CO o TJ J Z CD LO CO II CO CO J Z _Q. co co E co CD JD TJ CD CO CO CD •E o « •§• CD CO J J 0 s LO CNJ CO J Z _g. 3. JD Z5 JD 0 CO CD -4—* CO c g 'to c _0 "x 0 CD CZ 5 JD Z5 JD CJ) c 'to CJ) c 'to Z5 ^ ° LO 5 o 0 JD CO 0 CO CJ) _c "•4—» CO 0 "0 TJ O E J Z CL 0 CO CO J Z CL T J 0 -4—' o Z5 TJ cz o o CO CO CO ro J= Q. CD to. T— _single_ 16601.56 00 2.88 42868.10 T f CM C D f -CM 137378. CO CO CO Q. LO o CO CO o CO CM CM 42868.10 CO O J 1^ CO CM cn CO L O CO T f L O CO T f f~-CM cd cn CM L O CO o T f f -T f L O L O CO o L O fs» o T f 00 CO CM O J 00 L O O cd 00 T f C D o cn' CO O J 00 T— 1 00 -^CM CM L O c i d d d CM CO JZ Q. O ) to. CM 5p_single_ 0587033.49 81.06 610.81 0330.35 41788.56 9.59 87.03 cn 00 57.67 o T f T f C D 607814706. 2.49 2.08 '3.53 L O L O C D T f L O d L O CO cn T f o T f CO CO CM C D CO L O o CO i 00 T f CO o CO m cn CO I - 1 00 CM CM L O d d d d T— CO JZ Q. f -CO cn 5p_single_ 0278982.78 78.04 610.81 8985.68 40091.38 9.52 85.95 T f CO 56.97 CO CO CM C D 522530157. 2.18 1.69 3.53 r - L O T f L O 00 T f cq 00 T f o T f cq CO CM cn L O o CO 1 I^ J 00 T f CO o co CQ O J co T— CO ^— CM CM L O d d d d 00 CO 5p_single 9562961.95 71.01 610.81 3607.01 37202.09 9.40 84.22 L O 55.40 CO CM f -L O 154110145. 0.12 9.51 '3.53 O J CJJ CM T f L O L O T f CD CO T f o T f oq CM CM cn r- L O o cd • f^I 00 T f L O o L O CQ O J co T— 1 — 00 T— CM C M L O d d d d C D L O CO T f ip_single 5597139.4 32.11 538.64 4851.06 25590.00 8.54 86.03 T f 00 61.82 O C D O J 43549050. 2.90 1.44 '3.53 O J T f o CO o 00 C D CO f -00 oq CM CM CO L O o cd L O CM T f o m O J CO t 00 CM CM CO d d d d £ " ro ro — CD o O 5, Q . Q . j= J Z CO CO T J T> E E g g CO CM CM CM "E Q -"E Q . B T f E" "E "E "E E E E E E Q_ 1 Q -1 E E E i z E E E E E E z E E E E E co g g g o o II n n "E CD o o E, f— ter lh ation c o ro >. JZl BM) P KG) Wa 55 55 ro + CQ raft Wa tr •c T J T J c g. g. CD CT JI_ o> "55 CZ CD "co CO Q jr ® o o eg ro o o CD >- c o T J CC ro CD MA ra ~£ JlL|S a CD ted, sed Is o o T J ro 4 - " CD CQ u_ c o a g - E g g CQ o o Z CD E to > 5 < < < vr _ l _ i H m (D O O O O •o CO CO CO CO o c o a. JZ CO r° JO JD O 3> co O) CO c *-'co To cn jz tz *± •| ? I f Q. 0 O g CM CO CO c m CO CO c D) '55 CO •a co CO o CO ' CO o •o >. I r ™ - i a. CO 6 0 JD CO X CO ^° cr-IO CO I LO CM E CO CO JD APPENDIX H Hull Form Factors H.1 Experimental Based - Hughes-Prohaska's Method Definition: fr A r c \ C y FOM J F n N + (1 + /f) where N-4 The calculated and actual model testing conditions for the parent hull (BOp_single) are tabulated below: Froude no Model Scale Resistance Coefficients Fnm=Fns CTM CFOM CRM (FnmA4)/CFOM CTM/CFOM selected data points . - (ITTC '57) CRM=CTM-CFOM x-axIs y-axis x-axls y-axis 0.10 0.00723 0.00475 0.00248 0.02245 1.52227 0.12 0.00693 0.00457 0.00236 0.04790 1.51686 0.14 0.00654 0.00442 0.00212 0.09098 1.47928 0.16 0.00625 0.00430 0.00195 0.15868 1.45226 0.18 0.00600 0.00420 0.00180 0.25925 1.42818 0.25925 1.42818 0.20 0.00626 0.00411 0.00215 0.40040 1.52363 0.40040 1.52363 0.21 0.00650 0.00407 0.00243 0.49197 1.59682 0.49197 1.59682 0.22 0.00680 0.00403 0.00276 0.59611 1.68505 0.59611 1.68505 Slope = 0.762 _ y_intercept = (1+k) = hull form factor = 1.226 3.5 3.0 2.5 c? 2.0 O O 1.5 1.0 0.5 0.0 o o o y = 0.762x + 1.226 o overall x selected — linear fitting (selected) 0.0 0.2 0.4 0.6 0.8 Fn / C F 0 M T a b l e H -1: [P1 & 2] H u g h e s - P r o h a s k a ' s f o r m f a c t o r for p a r e n t h u l l [ B 0 p _ s i n g l e ] 183 The calculated and actual model testing conditions for the parabolized parent hull with 5 % beam increment single midship bulb (B5p_single_midship) are tabulated below: Froude no Model Scale Resistance Coefficients Fnm=Fns CTM CFOM CRM (FnmM)/CFOM CTM/CFOM selected data points - - (ITTC '57) CRM=CTM-CFOM x-axis y-axis x-axis y-axis 0.10 0.00685 0.00475 0.00210 0.02245 1.44160 0.12 0.00608 0.00457 0.00151 0.04790 1.33075 0.04790 1.33075 0.14 0.00596 0.00442 0.00153 0.09098 1.34705 0.09098 1.34705 0.16 0.00650 0.00430 0.00220 0.15868 1.51063 0.18 0.00655 0.00420 0.00235 0.25925 1.56055 0.20 0.00622 0.00411 0.00211 0.40040 1.51250 0.40040 1.51250 0.21 0.00679 0.00407 0.00272 0.49197 1.66914 0.22 0.00655 0.00403 0.00251 0.59611 1.62272 0.59611 1.62272 S l o p e = 0.535 y_intercept = (1+k) = hull form factor = 1.301 3.5 3.0 -2.5 -§ 2.0 LL o o 1.5 1.0 0.5 0.0 y = 0.535x + 1.301 o overall x selected — linear fitting (selected) 0.0 0.2 0.4 Fn / C F O M 0.6 0.8 Table H-2: [P1] Hughes-Prohaska's form factor for parent hull using single bulb at midship of 5% increased beam [B5p_single_midship] 184 The calculated and actual model testing conditions for the parabolized parent hull with 10% beam increment single midship bulb (B10p_single_midship) are tabulated below: Froude no Model Scale Resistance Coefficients Fnm=Fns CTM CFOM CRM (FnmA4)/CFOM CTM/CFOM selected data points - - (ITTC '57) CRM=CTM-CFOM x-axis y-axis x-axis y-axis 0.10 0.00850 0.00475 0.00375 0.02224 1.78894 0.12 0.00703 0.00457 0.00246 0.04752 1.53855 0.14 0.00662 0.00442 0.00220 0.09098 1.49748 0.09098 1.49748 0.16 0.00647 0.00430 0.00217 0.15775 1.50353 0.15775 1.50353 0.18 0.00647 0.00420 0.00227 0.25925 1.54044 0.25925 1.54044 0.20 0.00658 0.00411 0.00247 0.40040 1.60155 0.40040 1.60155 0.21 0.00678 0.00407 0.00271 0.49197 1.66586 0.49197 1.66586 0.22 0.00633 0.00403 0.00229 0.59611 1.56878 S l o pe = 0.421 yintercept = (1+k) = hull form factor = 1.444 o LL o o 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 o x X— *" y = 0.421 x + 1.444 o overall x selected — linear fitting (selected) 0.0 0.2 0.4 0.6 0.8 Fn / C F O M Table H-3: [P1] Hughes-Prohaska's form factor for parent hull using single bulb at midship of 10% increased beam [B10p_single_midship] 185 The calculated and actual model testing conditions for the parabolized parent hull with 1 5 % beam increment single midship bulb (B15p_single_midship) are tabulated below: Froude no Model Scale Resistance Coefficients Fnm=Fns CTM CFOM CRM (FnmM)/CFOM CTM/CFOM selected data points - - (ITTC '57) CRM=CTM-CFOM x-axis y-axis x-axis y-axis 0.10 0.00656 0.00474 0.00182 0.02230 1.38473 0.02230 1.38473 0.12 0.00648 0.00456 0.00192 0.04765 1.42086 0.04765 1.42086 0.14 0.00688 0.00441 0.00247 0.09060 1.55913 0.16 0.00630 0.00429 0.00201 0.15815 1.46723 0.15815 1.46723 0.18 0.00630 0.00419 0.00211 0.25854 1.50384 0.25854 1.50384 0.20 0.00613 0.00410 0.00203 0.39950 1.49533 0.39950 1.49533 0.21 0.00633 0.00406 0.00226 0.49096 1.55752 0.49096 1.55752 0.22 0.00647 0.00402 0.00244 0.59755 1.60734 0.59755 1.60734 S l o p e = 0.331 y_intercept = (1+k) = hull form factor = 1.398 3.5 -j 3.0 --2.5 E 2.0 LL o overall x selected — linear fitting (selected) y = 0.331 x + 1.398 o.o -I • i ^ i 1 H 1 1 0.0 0.2 0.4 0.6 0.8 Fn / C F O M Table H-4: [P1 & 2] Hughes-Prohaska's form factor for parent hull using single bulb at midship of 15% increased beam [B15p_single_midship] 186 The calculated and actual model testing conditions for the parabolized parent hull with 2 0 % beam increment single midship bulb (B20p_single_midship) are tabulated below: Froude no Model Scale Resistance Coefficients Fnm=Fns C T M C F O M C R M (Fnm*4)/CFOM CTM/CFOM selected data points - - (ITTC '57) CRM=CTM-CFOM x-axis y-axis x-axis y-axis 0.10 0.00726 0.00474 0.00252 0.02230 1.53197 0.12 0.00724 0.00456 0.00268 0.04765 1.58782 0.14 0.00685 0.00441 0.00244 0.09060 1.55181 0.16 0.00638 0.00429 0.00209 0.15907 1.48604 0.15907 1.48604 0.18 0.00653 0.00419 0.00234 0.25854 1.55751 0.25854 1.55751 0.20 0.00666 0.00410 0.00255 0.39950 1.62242 0.21 0.00665 0.00406 0.00258 0.49096 1.63615 0.49096 1.63615 0.22 0.00652 0.00402 0.00250 0.59501 1.62033 0.59501 1.62033 Slope = 0.314 y_intercept = (1+k) = hull form factor = 1.457 3.5 3.0 2.5 c ! 2.0 LL o 1.5 1.0 0.5 0.0 o ° o v -X-y = 0.314x+ 1.457 o overall x se lected — linear fitting (selected) 0.0 0.2 0.4 0.6 0.8 Fn / C F O M Table H-5: [P1] Hughes-Prohaska's form factor for parent hull using single bulb at midship of 20% increased beam [B20p_single_midship] 187 The parabolized parent hull with selected 15% beam increment at forebody (F) is tested (B15p_single_forebody). The calculated and actual model testing conditions are tabulated below: Froude no Model Scale Resistance Coefficients Fnm=Fns CTM CFOM CRM (FnmM)/CFOM CTM/CFOM selected data points - - (ITTC '57) CRM=CTM-CFOM x-axis y-axis x-axis y-axis 0.10 0.00712 0.00449 0.00263 0.08856 1.58575 0.08856 1.58575 0.12 0.00757 0.00443 0.00314 0.11793 1.71022 0.11793 1.71022 0.14 0.00818 0.00437 0.00381 0.15473 1.87283 0.15473 1.87283 0.16 0.00863 0.00431 0.00431 0.19868 1.99996 0.19868 1.99996 0.18 0.00911 0.00426 0.00485 0.25274 2.13796 0.25274 2.13796 0.20 0.00932 0.00422 0.00510 0.31543 2.21050 0.31543 2.21050 0.21 0.00973 0.00417 0.00556 0.39042 2.33250 0.39042 2.33250 0.22 0.01120 0.00409 0.00711 0.58116 2.73741 0.58116 2.73741 s l o p e = 2.212 y_intercept = (1+k) = hull form factor = 1.493 3.5 3.0 2.5 i 2.0 + LL. o o 1.5 1.0 0.5 0.0 y = 2.212x+ 1.493 o o v e r a l l x s e l e c t e d — l i n e a r f itt ing ( s e l e c t e d ) 0.0 0.2 0.4 0.6 0.8 Fn / C F O M Table H-6: [P3] Hughes-Prohaska's form factor for parent hull using single bulb at forebody of 15% increased beam B15p_single_forebody] 188 The parabolized parent hull with selected 1 5 % beam increment at aftbody (A) is tested (B15p_single_aftbody). The calculated and actual model testing conditions are tabulated below: Froude no Model Scale Resistance Coefficients Fnm=Fns CTM CFOM CRM (FnmM)/CFOM CTM/CFOM selected data points . - (ITTC ' 5 7 ) CRM=CTM-CFOM x-axis y-axis x-axis . y-axis 0.10 0.00556 0.00449 0.00107 0.08892 1.23871 0.12 0.00577 0.00443 0.00135 0.11793 1.30430 0.14 0.00688 0.00437 0.00251 0.15501 1.57536 0.15501 1.57536 0.16 0.00765 0.00431 0.00334 0.19968 1.77485 0.19968 1.77485 0.18 0.00798 0.00426 0.00372 0.25355 1.87311 0.25355 1.87311 0.20 0.00815 0.00421 0.00394 0.31686 1.93376 0.31686 1.93376 0.21 0.00808 0.00417 0.00391 0.39098 1.93744 0.39098 1.93744 0.22 0.00936 0.00409 0.00526 0.58192 2.28656 0.58192 2.28656 S l ope = 1.455 yintercept = (1+k) = hull form factor = 1.437 0.0 0.2 0.4 0.6 0.8 Fn / C F 0 M Table H-7: [P3] Hughes-Prohaska's form factor for parent hull using single bulb at aftbody of 15% increased beam [B15p_single_aftbody] 189 The parabolized parent hull with selected 1 5 % beam increment single midship bulb is tested with fairing extension of 2 5 % of waterline length applied to the rear of the bulb (B15p_single_midship_alpha1), i.e. equivalent to ~490mm. The calculated and actual model testing conditions are tabulated below: Froude no Model Scale Resistance Coefficients Fnm=Fns CTM CFOM CRM (FnmM)/CFOM CTM/CFOM selected data points - . (ITTC'57) CRM=CTM-CFOM x-axis y-axis x-axis y-axis 0.10 0.00662 0.00455 0.00207 0.08735 1.45472 0.08735 1.45472 0.12 0.00695 0.00449 0.00246 0.11610 1.54894 0.14 0.00670 0.00443 0.00227 0.15209 1.51342 0.15209 1.51342 0.16 0.00658 0.00437 0.00221 0.19568 1.50595 0.19568 1.50595 0.18 0.00658 0.00432 0.00226 0.24898 1.52427 0.24898 1.52427 0.20 0.00700 0.00427 0.00273 0.31125 1.63922 0.31125 1.63922 0.21 0.00681 0.00423 0.00258 0.38472 1.61072 0.38472 1.61072 0.22 0.00747 0.00415 0.00332 0.57357 1.80047 0.57357 1.80047 Slope = 0J691 y_intercept = (1+k) = hull form factor = 1.385 o LL o h -o 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 y = 0.691 x+ 1.385 o overall x se lected — linear fitting (selected) 0.0 0.2 0.4 Fn / C F O M 0.6 0.8 Table H-8: [P4] Hughes-Prohaska's form factor for parent hull using single bulb at midship of 15% increased beam and 25% extended fairing length at rear of bulb [B15p_single_midship_alpha1] 190 The parabolized parent hull with selected 1 5 % beam increment single midship bulb is tested with fairing extension of 3 0 % of waterline length applied to the rear of the bulb (B15p_single_midship_alpha2), i.e. equivalent to - 6 1 0 m m . The calculated and actual model testing conditions are tabulated below: Froude no Model Scale Resistance Coefficients Fnm=Fns CTM CFOM CRM (FnmMJ/CFOM CTM/CFOM selected data points - - (ITTC ' 5 7 ) CRM=CTM-CFOM x-axis y-axis x-axis y-axis 0.10 0.00681 0.00457 0.00225 0.08912 1.49170 0.08912 1.49170 0.12 0.00679 0.00450 0.00229 0.11827 1.50845 0.11827 1.50845 0.14 0.00772 0.00444 0.00328 0.15527 1.73925 0.16 0.00665 0.00439 0.00227 0.19981 1.51751 0.19981 1.51751 0.18 0.00687 0.00433 0.00254 0.25388 1.58576 0.25388 1.58576 0.20 0.00707 0.00429 0.00279 0.31747 1.65040 0.31747 1.65040 0.21 0.00699 0.00424 0.00275 0.39196 1.64869 0.39196 1.64869 0.22 0.00716 0.00416 0.00301 0.58393 1.72250 0.58393 1.72250 S l o p e = 0.494 yjntercept = (1+k) = hull form factor = 1.451 3.5 o LL o I— o 3.0 2.5 + 2.0 1.5 1.0 0.5 0.0 y = 0.494x + 1.451 o o v e r a l l x s e l e c t e d — l i n e a r f i t t ing ( s e l e c t e d ) 0.0 0.2 0.4 0.6 0.8 Fn / C F O M Table H-9: [P4] Hughes-Prohaska's form factor for parent hull using single bulb at midship of 15% increased beam and 30% extended fairing length at rear of bulb [B15p_single_midship_alpha2] 191 The parabolized parent hull with selected 1 5 % beam increment single midship bulb is tested with fairing extension of 3 5 % of waterline length applied to the rear of the bulb (B15p_single_midship_alpha3), i.e. equivalent to ~720mm. The calculated and actual model testing conditions are tabulated below: Froude no Model Scale Resistance Coefficients Fnm=Fns CTM CFOM CRM (FnmMJ/CFOM CTM/CFOM selected data points - - (ITTC ' 5 7 ) CRM=CTM-CFOM x-axis y-axis x-axis y-axis 0.10 0.00679 0.00457 0.00222 0.08876 1.48688 0.12 0.00646 0.00450 0.00196 0.11827 1.43441 0.11827 1.43441 0.14 0.00641 0.00444 0.00197 0.15638 1.44358 0.15638 1.44358 0.16 0.00678 0.00439 0.00240 0.19948 1.54676 0.18 0.00298 0.00433 -0.00135 0.25388 0.68783 0.20 0.00687 0.00429 0.00258 0.31747 1.60313 0.21 0.00679 0.00424 0.00255 0.39196 1.60032 0.39196 1.60032 0.22 0.00699 0.00416 0.00283 0.58393 1.68005 0.58393 1.68005 S l o p e = 0.552 y_intercept = (1+k) = hull form factor = 1.367 o o r -o 3.5 3.0 2.5 2.0 1.5 1.0 0.5 + 0.0 y = 0.552x + 1.367 o o v e r a l l x s e l e c t e d — l i n e a r f itt ing ( s e l e c t e d ) 0.0 0.2 0.4 0.6 0.8 Fn / C F O M Table H-10: [P4] Hughes-Prohaska's form factor for parent hull using single bulb at midship of 15% increased beam and 35% extended fairing length at rear of bulb [B15p_single_midship_alpha3] 192 This is the offset table of the recommended UBC series model #3 revised based on experimental and numerical studies. The calculated and actual model testing conditions are tabulated below: Froude no Resistance coefficients model Fnm=Fns CTM CFOM CRM (FnmM)/CFOM CTM/CFOM selected data points - (ITTC ' 5 7 ) CRM=CTM-CFOM x-axis y-axis x-axis y-axis 0.10 0.00626 0.00483 0.00143 0.02069 1.29642 0.02069 1.29642 0.12 0.00621 0.00464 0.00157 0.04465 1.33832 0.04465 1.33832 0.14 0.00642 0.00449 0.00193 0.08550 1.42956 0.16 0.00701 0.00436 0.00264 0.15003 1.60597 0.18 0.00648 0.00426 0.00223 0.24629 1.52292 0.24629 1.52292 0.20 0.00637 0.00417 0.00220 0.38362 1.52882 0.11 0.00675 0.00473 0.00202 0.03094 1.42747 0.13 0.00630 0.00456 0.00174 0.06257 1.38058 0.06257 1.38058 0.15 0.00636 0.00442 0.00193 0.11434 1.43730 0.11434 1.43730 0.17 0.00632 0.00431 0.00201 0.19364 1.46704 0.19364 1.46704 0.19 0.00579 0.00421 0.00158 0.30919 1.37432 slope = 0.914 y_intercept = (1+k) = hull form factor = "| .303 1.8 1.5 1.3 O 1.0 + u_ O $ 0.8 o 0.5 0.3 0.0 y = 0.914x + 1.303 o overall X selected Linear (selected) -+-0.0 0.1 0.2 0.3 Fn4/CFOM 0.4 0.5 Table H-11: [Recommended] Hughes-Prohaska's form factor for the revised UBC series model #3 [B11 p_single_midship_recommended] 193 0) •> c ro co O O CO T-M~ o CB c O ) o I CO CO 11 0 ci) cn co 1 a E 5 CO cp Q . > LO O CM T -CD C 3 E ^ — CD d) £ : £ ; CD i I O ) CO c CD o o + o CD O ) B c CD o CD Q . m o _ i m i - - i ^ CO 3 C -Q O CD co CO E CO C L II > o .!= £ X c a) 0° CD O CO p O o 5 .9 d + a. o E o CD > O CD > o a) a3 CD C o ca " O c CD a . a . co o o o m E o CD CD CO II CD cn co •a c CD Q . CD-CO T J CD cn CD C o co T J c CD Q . a . CO o3 CO T J > T J CZ "o o o « E o CD O ) CO T J c CD Q . Q . CO "CO ZJ T J II II j - , co co j i co < z CL CD _C0 3 E ro o ™ '5. E LU • •D CD (/> CO CQ 75 o CO CO CM CD CD CM 3 eg -CZ — CD ca o -Q > H— < o Z o CL o CD O F ZJ L _ 1 o o CO LL o CD O S I i > CO CO o N 1 tr —1 _ l \. y CO c i 00 CO o /• CO -1 X o 00 o + CO d + CO CD D) CO T3 cz CD CL CL CO o3 CD CO .O o o -4—* o E o CO CO J S C + LO O c\i o °° o • ^ C M - ^ C O O v J i - C M C M C M CO LO o o -ci- oo o CO g_ cn 5 co CD Q -O £ 03 CO CO CO c _co t=: co 0) Jr. 6)*- co 0 !5 5 0 CD CO CO D ) ( 0 0 3 c W CO DO DO CO CO CO d . Q CD E o T3 CO i _ CD CD CJ) CD O CD CO CO CJ, CD O CO C\j + o o CO M — E o "co CD > o CD .> "•4—» O CD CD CD CD O 0 o o o o CD JD E ^ " CD CD * E cz CD _C "iZ CD CO 5 0 E CZ c n co cz 0 0 J D T J 0 T3 Z5 o E 0 0 E c .-4—' oj "O T3 0 TJ ZS o E 0 CJ) CO 0 > CO ^ O J CD i — o to LO CO o E o E & >—• CD 0 is iz 0 1 l CD LO 0 > c CO 6 0 o ZJ o CO 00^  O c d + CO 1_ 0 > o 0 > 0 'o _ TJ CJ) cp 0 o o o * 0 o ; j Z ° 0 J D CZ _0 0 cr i _ 0 03 0 JD "O 0 TJ ZS o E ^ O - I CD 0 0 APPENDIX I Repeatability Analysis of Model Scale Total Resistance 1.1 Parent Hull (BOp_single) Achieved conditions P1-B0P_single (baseline) P2-B0P_single_setA P2-B0P_single_setB Froude no Total resistance Total resistance Total resistance prototype=model model model model Fnm=Fns R T M ave %RT M, ave R T M ave %R T M > ave (%RTM,ave)2 R T Mave %R T M, ave (%RTM,ave)2 - [lb] - [lb] - - [lb] - -0.10 0.252 0.000 0.252 0.000 0.000 0.252 0.000 0.000 0.12 0.346 0.000 0.346 0.000 0.000 0.346 0.000 0.000 0.14 0.443 0.000 0.443 0.000 0.000 0.443 0.000 0.000 0.16 0.551 0.000 0.551 0.000 0.000 0.551 0.000 0.000 0.18 0.668 0.000 0.668 0.000 0.000 0.668 0.000 0.000 0.20 0.858 0.000 0.858 0.000 0.000 0.858 0.000 0.000 0.21 0.982 0.000 0.982 0.000 0.000 0.982 0.000 0.000 0.22 1.125 0.000 1.125 0.000 0.000 1.125 0.000 0.000 0.25 1.479 0.000 1.505 1.758 3.090 1.472 -0.473 0.224 0.26 1.631 0.000 1.653 1.380 1.904 1.612 -1.135 1.287 0.27 1.805 0.000 1.823 0.997 0.994 1.829 1.330 1.768 0.28 1.873 0.000 1.900 1.442 2.078 1.877 0.214 0.046 0.29 2.236 0.000 2.235 -0.045 0.002 2.287 2.281 5.202 0.30 2.599 0.000 2.697 3.771 14.218 2.630 1.193 1.423 0.31 3.315 0.000 3.224 -2.745 7.536 3.362 1.418 2.010 0.32 3.677 0.000 3.846 4.596 21.124 3.828 4.107 16.864 0.33 4.360 0.000 4.515 3.544 12.557 4.531 3.922 15.382 0.34 4.904 0.000 4.967 1.285 1.650 4.943 0.785 0.616 0.35 5.599 0.000 5.429 -3.028 9.166 5.527 -1.277 1.631 0.36 5.987 0.000 5.986 -0.008 0.000 5.841 -2.430 5.907 0.37 6.396 0.000 6.270 -1.962 3.851 6.437 0.649 0.421 0.38 6.622 0.000 6.826 3.081 9.490 6.880 3.896 15.180 0.39 7.403 0.000 7.209 -2.621 6.867 7.413 0.135 0.018 0.40 8.250 0.000 8.061 -2.291 5.248 8.244 -0.073 0.005 0.41 9.123 0.000 8.978 -1.589 2.526 9.023 -1.096 1.202 0.42 10.135 0.000 10.123 -0.118 0.014 10.282 1.450 2.104 0.43 11.805 0.000 11.798 -0.059 0.004 11.771 -0.288 0.083 0.44 13.765 0.000 13.449 -2.296 5.271 13.326 -3.186 10.149 0.45 14.792 0.000 15.025 1.575 2.481 14.991 1.345 1.810 (%RTM,ave)mln= -3.0 (%RTM ,ave) m l n = -3.2 (%RTM, ave)max = 4.6 (%RTM , ave)max = 4.1 (%RTM,ave)rms= 1.9 (%R T H iave)m s= 1.7 Table 1-1: [P2] Repeatability analysis of model scale total resistance on parent hull: R T M , % R T M 196 16 14 12 10 8 6 4 2 - o — P1-B0P_single (baseline) -+- . . P2-B0P_single_setA - A - - - P2-B0P_single_setB 0.10 0.15 0.20 0.25 0.30 0.35 Froude number (achieved) 0.40 0.45 — o — P 1 - B 0 P _ s i n g l e (baseline) . . . + . . . P2-B0P_single_setA — it— P2-B0P_single_setB :: I: \ 4U V Is x i <i (a) R, TM (b) %Rr, (RTM\, baseline x100 0.10 0.15 0.20 0.25 0.30 0.35 Froude number (achieved) 0.40 0.45 F i g u r e 1-1: [P2] R e p e a t a b i l i t y a n a l y s i s of m o d e l s c a l e tota l r e s i s t a n c e o n p a r e n t h u l l : R T M , % R T M 197 1.2 Parent Hull with 15% Increased beam (B15p single) Achieved conditions P1-B15P_single (baseline) P2-B15P_single_setA P2-B15P_single_setB Froude no Total resistance Total resistance Total resistance prototype=model model model model Fnm=Fns RTM,ave %R T M, ave R T Mave %R T M i ave (%RTM. ave)2 R™,ave %RTM, ave (%RTM.ave)2 - [lb] - [lb] - - [lb] - -0.10 0.23 0.000 0.229 0.000 0.000 0.229 0.000 0.000 0.12 0.32 0.000 0.324 0.000 0.000 0.324 0.000 0.000 0.14 0.47 0.000 0.467 0.000 0.000 0.467 0.000 0.000 0.16 0.56 0.000 0.557 0.000 0.000 0.557 0.000 0.000 0.18 0.70 0.000 0.704 0.000 0.000 0.704 0.000 0.000 0.20 0.84 0.000 0.843 0.000 0.000 0.843 0.000 0.000 0.21 0.96 0.000 0.959 0.000 0.000 0.959 0.000 0.000 0.22 1.08 0.000 1.077 0.000 0.000 1.077 0.000 0.000 0.25 1.59 0.000 1.595 0.409 0.167 1.601 0.787 0.619 0.26 1.90 0.000 1.942 2.292 5.253 1.935 1.949 3.800 0.27 2.15 0.000 2.143 -0.326 0.106 2.102 -2.233 4.984 0.28 2.35 0.000 2.401 2.127 4.523 2.389 1.616 2.613 0.29 2.61 0.000 2.598 -0.593 0.352 2.698 3.253 10.582 0.30 2.84 0.000 2.967 4.656 21.679 2.902 2.363 5.585 0.31 3.28 0.000 3.205 -2.272 5.161 3.344 1.967 3.868 0.32 3.78 0.000 3.734 -1.217 1.481 3.893 2.989 8.937 0.33 4.24 0.000 4.121 -2.875 8.267 4.244 0.024 0.001 0.34 4.62 0.000 4.534 -1.840 3.386 4.427 -4.157 17.279 0.35 5.31 0.000 5.155 -2.956 8.735 5.330 0.339 0.115 0.36 5.77 0.000 5.702 -1.247 1.555 5.871 1.680 2.822 0.37 6.20 0.000 6.163 -0.661 0.437 6.334 2.095 4.391 0.38 6.77 0.000 6.707 -0.872 0.760 7.034 3.961 15.689 0.39 7.54 0.000 7.317 -2.906 8.445 7.616 1.062 1.127 0.40 8.37 0.000 8.147 -2.664 7.098 8.576 2.461 6.057 0.41 8.80 0.000 8.915 1.295 1.678 9.331 6.022 36.265 0.42 10.57 0.000 10.125 -4.183 17.496 10.706 1.315 1.730 0.43 12.01 0.000 11.397 -5.104 26.052 11.989 -0.175 0.031 0.44 13.74 0.000 13.099 -4.644 21.570 13.913 1.281 1.642 0.45 15.60 0.000 15.438 -1.045 1.092 15.935 2.141 4.583 (%RTM,ave)mln= -5.1 (%R T M ,ave) m i„= -4.2 (%RTMiave)max= 4.7 (%RTM, ave)max = 6.0 (%RTMiave)mls= 2.2 (%RTM, ave)m s = 2.1 Table I-2: [ P 2 ] Repeatability analysis of model scale total resistance on parent hull with single bulb at midship of 1 5 % increased beam: R T M , % R T M 198 18 16 c C3 m 12 = 10 co 2 8 + 4 4 8 8 .22 6 TO 4 o ^ 9 a) 2 - o — P1-B15P_single (baseline) •+-•• P2-B15P_single_setA -A--- P2-B15P_single_setB 0.10 0.15 0.20 0.25 0.30 0.35 Froude number (achieved) 0.40 - o — P1-B15P_single (baseline) .+... P2-B15P_single_setA •tr- P2-B15P_single_setB 0 "A h - A A-A-A CD cB -4 !- tt •i +-(a) 0.45 (b) (RTM ), ' j^TM Xaseline {^TM \aseline X 1 0 0 .10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Froude number (achieved) Figure I-2: [P2] Repeatability analysis of model scale total resistance on parent hull with single bulb at midship of 15% increased beam: R T M , % R T M 199 1.3 Recommended Hull with 11% Increased beam (B11p_single) Achieved conditions B11P_single_set1 (baseline) B11P_single_set2 Froude no Total resistance Total resistance prototype=model model model Fnm=Fns RTM , ave % R T M , ave RTM , ave % R T M , ave (%RTM.ave)2 - [Ib] - [Ib] - -0.10 0.210 0.000 0.210 0.000 0.000 0.12 0.300 0.000 0.300 0.000 0.000 0.14 0.422 0.000 0.422 0.000 0.000 0.16 0.602 0.000 0.602 0.000 0.000 0.18 0.705 0.000 0.705 0.000 0.000 0.20 0.855 0.000 0.855 0.000 0.000 0.21 0.963 0.000 0.951 -1.246 1.553 0.22 1.058 0.000 1.066 0.756 0.572 0.25 1.548 0.000 1.534 -0.904 0.818 0.26 1.755 0.000 1.765 0.570 0.325 0.27 1.977 0.000 1.951 -1.315 1.730 0.28 2.014 0.000 2.042 1.390 1.933 0.29 2.161 0.000 2.182 0.972 0.944 0.30 2.387 0.000 2.400 0.545 0.297 0.31 2.798 0.000 2.794 -0.143 0.020 0.32 3.249 0.000 3.241 -0.246 0.061 0.33 3.771 0.000 3.822 1.352 1.829 0.34 4.219 0.000 4.258 0.924 0.854 0.35 4.623 0.000 4.688 1.406 1.977 0.36 5.026 0.000 5.045 0.378 0.143 0.37 5.365 0.000 5.443 1.454 2.114 0.38 5.932 0.000 5.927 -0.084 0.007 0.39 6.572 0.000 6.513 -0.898 0.806 0.40 7.382 0.000 7.332 -0.677 0.459 0.41 8.165 0.000 8.146 -0.233 0.054 0.42 9.433 0.000 9.413 -0.212 0.045 0.43 11.078 0.000 11.061 -0.153 0.024 0.44 12.797 0.000 12.851 0.422 0.178 0.45 14.775 0.000 14.550 -1.523 2.319 (%R T M,ave) mi n= -1.5 (%RTM,ave)max= 1.5 (%RTM,ave)rms= 0.8 T a b l e I-3: [ R e c o m m e n d e d ] R e p e a t a b i l i t y a n a l y s i s of m o d e l s c a l e tota l r e s i s t a n c e o n r e c o m m e n d e d h u l l w i t h 11% i n c r e a s e d b e a m : R T M , % R T M 200 (a) 16 -, , , 1 1 . • 1 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Froude number (achieved) F i g u r e I-3: [ R e c o m m e n d e d ] R e p e a t a b i l i t y a n a l y s i s of m o d e l s c a l e tota l r e s i s t a n c e o n r e c o m m e n d e d h u l l w i t h 11% i n c r e a s e d b e a m : R T M , % R T M 201 APPENDIX J Matlab Script Files & Execution Manual The analyses were performed using Matlab version 6.5. Any exterior file formats are converted to Matlab compatible file format such as TXT , .M, .MAT and J P G . Postprocessing of some results in Microsoft Excel are necessary. Some important script files written for analysis are listed below: J.1 STEP 1: Spatial Transformation Matrix (transform.M) This is the very first step of the wave pattern profiling analysis. This is originated from the spatial transformation matrix as computed by the camera acquisition software, Ranger system, upon completion of camera calibration. The matrix is used to correspond screen pixel positions to real-life spatial coordinates. % RUN THIS M-FILE IN THE HOME DIRECTORY FIRSTLY, % ITS OUTPUT .MAT IS TO BE CALLED UP BY spatial.M % Author: J . TAN (UBC, Naval Lab) % Purpose: Redefine spatial transformation matrix in compatible format % Input : User input by pasting the spatial transformation matrix % Output: Spatial transformation matrix in .MAT Matlab workspace file clear all close all % Open transformation matrix output from .TRF in Excel and paste numbers here % in the brackets: % note: TRF is a spatial transformation file self-generated by the camera % software Ranger upon calibration temp= 202 [ % open bracket % begin of cutting & pasting, in same sequence but include only numbers here— 0.00001400874 0.00162055905 -1.02788494971 -931.34736651859 -0.00163597339 0.00001389911 0.42472786080 0.00001053835 -0.00113546850 -1.19105179329 -0.00001088836 -0.00126278589 1.67265187958 417.73761657611 -2266.42074522016 2224.22032151508 1.00000000000 % end of cutting & pasting ]; % closing bracket W(1) = temp(1); W(2) = temp(2); W(3) = temp(3); W(4) = temp(4); P(1,1) = temp(5); P(1,2) = temp(6); P(1,3) = temp(7); P(2,1) = temp(8); P(2,2) = temp(9); P(2,3) = temp(10) P(3,1) = temp(11) P(3,2) = temp(12) P(3,3) = temp(13) T(1) = temp(14); T(2) = tempO 5); 203 T(3) = temp(16); Y SCALE = temp(17); save transform_output % output to a .MAT file format J.2 STEP 2: Spatial Transformation & Noise Filtering This is the second step of the wave pattern profiling analysis. Upon establishing the spatial transformation matrix, the wave pattern acquisition can be performed. The acquisition software, Ranger system, offers options for output file format, one being text format (good for replaying) and the other being Matlab format (good for analysis). J.2.1 STEP 2a: Spatial Transformation (spatial.M) This step requires the output to be loaded specifically in Matlab format and then apply the transformation matrix to determine the actual wave elevations. The main purpose here is to filter out the random noise contained in the acquired wave elevations. % RUN THIS M-FILE IN THE HOME DIRECTORY SECONDLY, % THE HOME DIRECTORY SHOULD ALREADY CONTAINS: transform.MAT % splinefit.M % splinefit_output.MAT % ITS OUTPUT MAT-FILE TO BE CALLED UP BY sharma_multiple_cut.M % Note: splinefit.M and splinefit_output.MAT are intermediate files % required during this analysis process, they require the curvefit % toolbox of the Matlab to be already installed % Author: J . TAN (UBC, Naval Lab) % Purpose: To clean up the entire acquired wave profiles 204 % Input : The already transformed but noisy wave profiles in .M format % Output: Cleaned, smoothened and fitted wave profiles save in .MAT workspace file clear all close all % — open captured laser profiles [filename] = uigetfile('*.m') % select laser profiling datafile filename=filename(1 :length(filename)-2); % to exclude '.m' in the string Fn=input('Froude no (refer to filename)= '); % Froude number run(filename) % — test conditions rho=1000; % tank water density [kg/mA3] g=9.81; % gravitational acceleration [m/sA2] 1=2.017; % model length [m] b=3.660; % total tank width [m] Vm=Fn*sqrt(g*l); % model speed [m/s] ko=g/(VmA2); % wave number [1/m] % — camera settings Yoffset=675; integrationtime=40000; inttime=integrationtime*1 e-6; horpoints=512; vertpoints=512; linelength=585; camera horres=linelength/horpoints; verres=linelength/vertpoints; longres=inttime*Vm*1000; [row column]=size(Word); % — perform spatial coordinate transformation to captured wave profiles load transform_output.mat % load camera spatial transformation matrix for i=1 :row for j=1:column v=Offset+((Resolution - Word(i,j))/2); % [mm] wrt tank sidewall % [micro s] % [s] % number of horizontal pixels used % number of vertical pixels used % [mm] length of laser line (approx) captured by % [mm/pixel] horizontal resolution % [mm/pixel] vertical resolution % [mm] longitudinal resolution 205 u=j; w=-W(4)/(W(1)*u + W(2)*v + W(3)); Y(i,j)=w*(P(1,1)*u + P(1,2)*v+ P(1,3)) + T(1); % [mm] transverse location Z(i,j)=w*(P(3,1)*u + P(3,2)*v + P(3,3)) + T(3); % [mm] vertical height S(i,j)=(i-1)*integrationtime*1E-6; % [s] elapse time end end % — redefine parameters and save to workspace Z_original=-Z; % due to activation of "FLIP Y" function in Ranger camera system Z=Z_original; figure(1); mesh(Z); title('3D view before any filtering'); xlabel('Y [pixel*]"); ylabel('X [profile*]'); zlabel('Z [mm]'); % coarse elimination of erroneous data points (primary pass) figure(2); plot(Z) title('2D view before any filtering'); xlabel('X [profile*]'); ylabel('Z [mm]'); [xa,za]=ginput(1); % cut-off if above the height forycut=1:column; zz=Z(1:row,ycut); [xxa,zza]=find((zz>abs(za)) I (abs(zz)obs(za))); zz(xxa)=0; [xxb,zzb]=find(zz~=0); swl=mean(zz(xxb)); zz(xxb)=zz(xxb)-swl; Z(1 :row,ycut)=zz; end figure(3); plot(Z) title('2D view after 1 coarse filtering'); xlabel('X [profile*]'); ylabel('Z [mm]'); % — fine elimination of erroneous data points (secondary pass) filter=1; % to initiate WHILE loop 206 count=1; % to initiate filtering count while (filter==1) figure(3); disp(['Left or right click for zooming. Press Enter when done...']) zoom input('') disp(['Click 1 time to select left boundary & 1 time to select right boundary...']) [ax1 ,bz1]=ginput(1); % bounded left cut-off region [ax2,bz2]=ginput(1); % bounded right cut-off region for ycut=1 :column zz=Z((round(min([ax1 ,ax2])):round(max([ax1 ,ax2]))),ycut); if (mean([bz1,bz2])>0) [xax,zbz]=find(zz>mean([bz1 ,bz2])); % upper bound cut-off (above x-axis) else [xax,zbz]=find(zz<mean([bz1 ,bz2])); % lower bound cut-off (below x-axis) end zz(xax)=0; Z((round(min([ax1 ,ax2])):round(max([ax1 ,ax2]))),ycut)=zz; end figure(3); plot(Z) title(['2D view after' num2str(count)' fine filtering"]); xlabel('X [profile*]'); ylabel('Z [mm]'); filter=input('Continue filtering [Yes=1 / No=2] ? '); count=count+1; % update filtering counts end % — 3D plots figure(4); mesh(Z); view([-112.5,30]); colorbar('vert'); axis equal; title(['3D view after' num2str(count)' filtering (not adjusted to proper spatial coordinates)']); xlabel('Y [pixel#]'); ylabel('X [profile*]'); zlabel('Z [mm]'); % — define the XY-planar working grid space for filtered data for i=1 :row xx(i)=-S(i,1)*Vm*1000; end for j=1:column yy(j)=horres*(j-1 )+Yoffset; 207 end % [YI,XI]=meshgrid(yy,xx); % figure(5); mesh(XI,YI,Z); colorbar(Vert') % title(['3D view after' num2str(count)' filtering (adjusted to proper spatial coordinates, not to scale)']); % xlabel('X [mm]'); ylabel('Y [mm]'); zlabel('Z [mm]'); % % figure(6); mesh(XI,YI,Z); view(2); colorbar('vert') % V=axis; %axis([V(1) V(2) 0 1830]) % title(['2D view after' num2str(count)' filtering (adjusted to proper spatial coordinates, not to scale)']); % xlabel('away from model <— X [mm] - -> approaching model'); % ylabel('to tank sidewall <— Y [mm] —> to tank centerline'); Z_filtered=Z; zz=Q; for j=1:column for i=1:row xaa(i)=xx(i); % original longitudinal locations [mm] zaa(i)=Z_filtered(i,j); % filtered wave heights [mm] end xaa_interp1=linspace(xaa(1),xaa(length(xaa)),1000); % interpolated longitudinal locations [mm] zaa_interp1 =interp1 (xaa,zaa,xaa_interp1,'spline'); % interpolated wave heights [mm] splinefit(xaa_interp1 ,zaa_interp1) % call subroutine (require curve fit toolbox) load splinefit_output % load fitted result from subroutine x_fit=xa; % interpolated longitudinal locations [mm] z_fit=z_SGfit; % wave heights of fitted signal [mm] ZZ(:,j)=z_fit; % reconstruct matrix of fitted wave heights [mm] fprintf(1,'Now processing column %3g out of %3g columns. Please wait further \n',j,column) end % — define the XY-planar working grid space for fitted data 208 [YI,XI]=meshgrid(yy,xJit); % 3D plots close all figure(7); mesh(XI,YI,ZZ); view(2); colorbar('vert') V=axis; axis([V(1) V(2) 0 1800]) title(['3D view after filtering & fitting (adjusted to proper spatial coordinates, not to scale)']); xlabelfaway from model <— X [mm] —> approaching model'); ylabel('to tank sidewall <— Y [mm] —> to tank centerline'); % — save data save([filename '_spatial_output']) % warning off % delete([filename '_XYZ.txt1]) % warning on % diary([filename '_XYZ.txt']) % fprintf(1 ,'X[mm] wrt stern\tY[mm] wrt tank sidewall\tZ[mm] wrt SWLAn') %for j=1:column % for i=1: length (xjit) % fprintf(1,'%8.3f\t%8.3f\t%8.3f\n',xJit(i),yy(j),ZZ(i,j)); % end % disp(['"]) % disp(['— separate to next transverse cut —']) % disp([' 1) % end % diary off J.2.2 STEP 2b: Data smoothing & fitting subroutine (splinefit.M) No user intervention is required at all for this subroutine. It is called up internally when executing spatial transformation (spatial.M) and generates output file to be used for smoothing and fitting of each individual longitudinal-209 wise acquired wave profiles. Important thing to note is that the execution of this subroutine requires the curve fit toolbox of Matlab already installed. % THIS M-FILE WILL BE CALLED UP INTERNALLY IN spatial.M % IT IS TO BE SAVED IN THE HOME DIRECTORY % Note: splinefit.M and splinefit_output.MAT are intermediate files % required during this analysis process, they require the curvefit % toolbox of the Matlab to be already installed % Author: J . TAN (UBC, Naval Lab) % Purpose: To smoothen and fit the acquired individual longitudinal-wise wave profile % Input : No need user intervention (input is self-generated upon executing spatial.M) % Output: Cleaned, smoothened and fitted wave profile save in .MAT workspace file function splinefit(xa,za) % % GENERAL COMMAND PROMPT ISSUING: % S_G_splinefit(xa,Z_filtered(1 :row,300)) % figure;plot(xa,cf_(xa)) % xdl=xa; % zdl=cf_(xa); % GENERAL PLOTTING FOR DESIRED INPUTS: % load(uigetfile('*.mat')) % col=300; % xa=XI(1:row,col); % za=Z_filtered(1 :row,200); %fit(xa,za,ft_,foJ; % figure;plot(xa,za,'k.',xa,ans(xa),'b') % legend('originaP,'fitted') % TO PLOT FITTED DATA & RESIDUALS: % col=300; % xa=XI(1:row,col); 210 % za=ZJiltered(1 :row,200); %fit1=fit(xa,za,ft_,foJ; %subplot(2,1,1) %plot(fit1,'k-',xa,za,'b.'); %subplot(2,1,2) % plot(fit1 ,'k-',xa,za,'b.','residuals'); % TO SHOW ERROR BAND: % E = std(za)*ones(size(xa)); % errorbar(xa,za,E) % % S_G_SPLINEFIT Create plot of datasets and fits % S_G_SPLINEFIT(XA,ZA) % Creates a plot, similar to the plot in the main curve fitting % window, using the data that you provide as input. You can % apply this function to the same data you used with cftool % or with different data. You may want to edit the function to % customize the code and this help message. % % Number of datasets: 2 % Number of fits: 2 % Data from dataset "za vs. xa": % X = xa: % Y = za: % Unweighted % Data from dataset "za vs. xa (S-G smoothen)": % X = xa: % Y = za: % Unweighted % % This function was automatically generated 211 % Set up figure to receive datasets and fits f_ = elf; f igure(fj; legh_ = 0; legt_ = {}; % handles and text for legend xlim_ = [Inf -Inf]; % limits of x axis ax_ = subplot(2,1,1); ax2_ = subplot(2,1,2); set(ax2_,'Box','on'); legrh_ = Q; legrt_ = {}; set(ax_,'Box','on'); grid(ax_,'on'); grid(ax2_,'on'); axes(ax_); hold on; % — Plot data originally in dataset "za vs. xa" xa = xa(:); za = za(:); h_ = line(xa>za,'Parent',ax_, ,Color',[0.333333 0 0.666667],... 'LineStyle','none', 'LineWidth',1 'Marker','.', 'MarkerSize',6); xlim_(1) = min(xlim_(1),min(xa)); xlim_(2) = max(xlim_(2),max(xa)); legh_(end+1) = h_; legt_{end+1} = 'original points'; % — Plot data originally in dataset "za vs. xa (S-G smoothen)" sm_.y2 = smooth(xa,za,5,'sgolay',2); h_= line(xa,sm_.y2,'Parent',ax_,'Color',[0 1 0],... 'LineStyle','none', 'LineWidth',1 'Marker'.'x', 'MarkerSize',6); xlim_(1) = min(xlim_(1),min(xa)); xlim_(2) = max(xlim_(2),max(xa)); legh_(end+1) = h_; legt_{end+1} = 'S-G filtering smoothen points'; % Nudge axis limits beyond data limits if all(isfinite(xlim_)) 212 xlim_ = xlim_ + [-1 1] * 0.01 * diff(xl imj; set(ax_,'XLim',xlim_) set(ax2_,'XLim',xlim_) end % — Create fit "fit original points using spline" fo_ = fitoptions('method', ,SmoothingSpline' I'SmoothingParam',9.3412638e-007); ft_ = fittype('smoothingspline'); % Fit this model using new data cf_ = fit(xa,za,ft_ ,fo_); % Plot this fit h_= plot(cf_,'fit',0.95); legend off; % turn off legend from plot method call set(h_(1),'Color',[1 0 0],... 'LineStyle','-', 'LineWidth',2,... 'Marker','none', 'MarkerSize',6); legh_(end+1) = h_(1); legt_{end+1} = 'spline fit original points'; res_ = za-cf_(xa); [x_, i j = sort(xa); axes(ax2J; hold on; h_= line(x_,res_(iJ,'Parent',ax2_,'Color',[1 0 0],... 'LineStyle','-', 'LineWidth',1.... 'Marker','.', 'MarkerSize',6); axes(ax_); hold on; legrh_(end+1) = h_; legrt_{end+1} = 'spline fit original points'; % — Create fit "fit S-G points using spline" fo_ = fitoptions('method','SmoothingSpline','SmoothingParam',9.3412638e-007); ft_ = fittype('smoothingspline'); % Fit this model using new data cf_ = fit(xa,sm_.y2,ft_ , fo J ; % Plot this fit 213 h_ = plot(cf_,'fit',0.95); legend off; % turn off legend from plot method call set(h_(1),'Color',[0 0 1],... 'LineStyle','-', 'LineWidth',2,... 'Marker'.'none', 'MarkerSize',6); legh_(end+1) = h_(1); legt_{end+1} = 'spline fit S-G points'; res_ = sm_.y2 - cf_(xa); [x_, i j = sort(xa); axes(ax2_); hold on; h_= line(x_,res_(iJ,'Parent' Iax2_,'Color',[0 0 1],... 'LineStyle','-', 'LineWidth',1 'Marker','.', 'MarkerSize',6); axes(ax_); hold on; legrh_(end+1) = h_; legrt_{end+1} = 'spline fit S-G points'; hold off; legend(ax_,legh_, l eg t j ; Iegend(ax2_,legrh_, legr t j ; xa=xa; % [mm] z_SGfit=cf_(xa); % [mm] save splinefit_output xa z_SGfit J.3 STEP 3: Processed Wave Patch & Wave Cuts Analysis This is the third step of the wave pattern profiling analysis. Upon establishing the spatial transformation matrix, the wave pattern acquisition can J.3.1 STEP 3a: Processed Wave Patch (Z_patch.M) This processes the wave patch into a matrix containing only the necessary information sufficient for multiple wave cut analysis. 214 % RUN THIS M-FILE IN THE HOME DIRECTORY THIRDLY, % IT IS TO BE SAVED IN THE HOME DIRECTORY % Author: J . TAN (UBC, Naval Lab) % Purpose: To contain only the necessary information (wave elevations, lateral % positions and longitudinal positions all contained in a single matrix % Input : The output .MAT file from spatial.M containing information on the % entire wave patch % Output: The processed and detailed wave patch in millimetre save in .M file clear all cla reset close all % — load & define filename of the processed laser profiles [filename1,path]=uigetfile('*.mat','Open a MAT-file for processed laser profiles'); load(filenamel) f ilename=filename1 (1 :length(f i lenamel )-19); ko=9.81/(VmA2); % wave number [1/m] B=3.660; % full tank width [m] fprintf (1 ,* -\n',Fn); fprintf(1,'Froude no = %5.3f, i.e. model speed = %5.3f m/s\n',Fn,Vm) fprintf(1,' -\n',Fn); % — select lateral cut position %y_cut_CL=input('y_cut location [mm], wrt tank centerline (between 570-1155 mm) = '); y_cut_CL=1155; % this corresponds to column 1 of longitudinal cut that is closest to tank sidewall y_cut_SW=((B*1000)/2) - y_cut_CL; % lateral cut position wrt tank sidewall [mm] y_cut=y_cut_SW; % rename working lateral cut position [mm] % — transpose lateral cut position into column number & define working longitudinal cut vectors (xa.za) [i_cut_nearest,j_cut_nearest]=max((YI(1,1:512) > (y_cut-horres)) & (Yl(1,1:512) < (y_cut+horres))); 215 xa=XI(1 :size(XI,1),j_cut_nearest); % length of untruncated signal [mm] za=ZZ(1:size(XI,1),j_cut_nearest); % amplitude of untruncated signal [mm] % — interpolate untruncated filtered & fitted (xdl.zdl) to higher resolution to precisely locate bow wave peak location interp_points=1000; % by default xa_interp1 =linspace(xa(1 ),xa(length(xa)),interp_points); za_interp1 =interp1 (xa,za,xa_interp1,'spline'); x_fit=xa_interp1; % length of fitted signal [mm] z_fit=za_interp1; % amplitude of fitted signal [mm] figure(1) plot(xa_interp1 ,za_interp1 ,'g+'); grid on; hold on; plot(x_fit,zJit,'b','LineWidth',2); hold on titleU'LONGITUDINAL CUT WAVE PROFILE: Vm=',num2str(Vm),'m/s @ y=',num2str(y_cut_CL),'mm (wrt tank centerline)']); xlabel('downstream <- LONGITUDINAL DISTANCE, X [mm] -> upstream'); ylabel('WAVE HEIGHT, Z [mm]'); % — setup left & right vertical boundary to find bow wave height & location (xbow.zbow) [ibow,jbow]=ginput(2); % click two points (left & right) for i=1: length (x_f it) if ((abs(x_fit(i)) > (min(abs(ibow))-longres)) & (abs(x_fit(i)) < (min(abs(ibow))+longres))) ibow_lower=i; % transpose boundary location to column no. break end end for i=1 :length(x_fit) if ((abs(x_fit(i)) > (max(abs(ibow))-longres)) & (abs(x_fit(i)) < (max(abs(ibow))+longres))) ibow_upper=i; % transpose boundary location to column no. break end end [zbow,ibow_temp]=max(z_fit(ibowJower:ibow_upper)); % bow wave height [mm] ibow=ibow_temp+ibow_lower; xbow=abs(round(x_fit(ibow))); % bow wave peak location [mm] 216 disp(['Bow wave of height ',num2str(round(zbow)),' mm is detected astern at ',num2str(xbow),' mm']); figure(1) plot(x_fit(ibow),z_fit(ibow),'ko','LineWidth',2); hold on % mark out bow wave peak location % — define effective working signal length (xdl.zdl) Xo=xbow; % signal starting location astern [mm] L_signal=2*(y_cut_SW/tan(19.5/180*pi)); % wrt bow wave location to first wall reflection [mm] Xt=Xo + L_signal; % truncated signal location astern [mm] for i=1:length(x_fit) if ((abs(x_fit(i)) > (Xt-longres)) & (abs(x_fit(i)) < (Xt+longres))) i_truncated=i; % transpose boundary location to column no. break end end x_cut=x_fit(i_truncated); % longitudinal cut-off location wrt column 1 profile that is closest to tank sidewall [mm] i_begin_patch=1; % fprintf(1,'Patch of %3g columns begins at row %1g out of %4g rows (away from tank centerline)\n',size(XI,2),i_begin_patch,size(XI,1)); i_end_patch=i_truncated; % fprintf(1,'Patch of %3g columns ends at row %4g out of %4g rows (closest to tank centerline)\n',size(XI,2),i_endjDatch,size(XI,1)); disp(['Longitudinal cut is started upstream astern at ',num2str(round(abs(x_fit(ibow)))),' mm"]); disp(['Longitudinal cut is truncated downstream astern at ',num2str(round(abs(x_fit(i_truncated)))),' mm"]); disp(['Total signal length used is approx. ',num2str(round(L_signal)),' mm']); disp(['']) %x_trunc=x_fit(ibow:i_truncated); % filtered & fitted truncated signal [mm] %z_trunc=z_fit(ibow:i_truncated); % filtered & fitted truncated signal [mm] x_trunc=x_fit(ibow:i_end_patch); % filtered & fitted truncated signal [mm] z_trunc=z_fit(ibow:i_end_patch); % filtered & fitted truncated signal [mm] figure(1) plot(x_trunc,z_trunc,'r','LineWidth',2); hold on 217 Iegend('filtered (untruncated)Vfiltered + fitted (untruncated)','bow wave peak','filtered + fitted (truncated)') % save wave patch in text file (.TXT) X_patch=xa_interp1 (1 :i_end_patch)'; for j=1:column % note: all units in [mm] % wrt bow [mm] yy(j)=horres*(j-1 )+Yoffset; end Y_patch=((B*1000)/2) - yy(1:column); % wrt tank centerline [mm] deltaX=X_patch(2)-X_patch(1); deltaY=horres; fid=fopen([filename '_Z_patch_mm.TXT'],'w'); % if open in Matlab, use LOAD command to read the text file (eg. load XXX.txt) fprintf(fid,'%% Longitudinal position [mm], wrt bow: X(row1)=%8.3f, X(last row)=%8.3f, deltaX=%3.3f\n', X_patch(1),X_patch(i_end_patch),deltaX); fprintf(fid,'%% Transverse position [mm], wrt centerline: Y(column1)=%8.3f, Y(last column)=%8.3f, deltaY=%3.3f\n\n', YjDatch(1),Y_patch(column),deltaY); fprintf(fid,'%% deltaX = %3.3f\n',deltaX); fprintf(fid,'%% deltaY = %3.3f\n',deltaY); fprintf(fid,'%% x_cut = %8.3f\n',x_cut); % longitudinal cut-off location wrt column 1 profile that is closest to tank sidewall [mm] fprintf(fid,'%% Froude_no = %5.3f\n',Fn); fprintf(fid,'%% Vm_metre_per_sec = %5.3f\n\n\n',Vm); fprintf(fid,'%% Z_patch_mm=[\n'); Z_patch=0; for i=1:size(x_fit,2) z_patch=ZZ(i,1 :column); Z_patch=[Z_patch ; z_patch]; end fprintf(fid,'%%];\n'); fclose(fid); % save wave patch in matlab file (.M) % note: all units in [mm] fid=fopen([filename '_ZjDatch_mm.m'],'w'); % a matrix name is needed for data in .M format to be read by Matlab. To open the file, type only the filename XXX and excluding .M 218 fprintf(fid,'%% All lengths in unit [mm]\n'); fprintf(fid,'%% Longitudinal position [mm], wrt bow: X(row1)=%8.3f, X(last row)=%8.3f, deltaX=%3.3f\n', X_patch(1),X_patch(i_end_patch),deltaX); fprintf(fid,'%% Transverse position [mm], wrt centerline: Y(column1)=%8.3f, Y(last column)=%8.3f, deltaY=%3.3f\n\n', Y_patch(1 ),Yj)atch(column),deltaY); fprintf(fid,'deltaX=%3.3f; % longitudinal spacing [mm]\n',deltaX); fprintf(fid,'deltaY=%3.3f; % transverse spacing [mm]\n',deltaY); fprintf(fid,'x_cut=%8.3f; % truncatated location of first reflection for long-cut closest to sidewall [mm]\n',x_cut); % longitudinal cut-off location wrt column 1 profile that is closest to tank sidewall [mm] fprintf(fid,'Froude_no=%5.3f;\n',Fn); fprintf(fid,'Vm_metrejDer_sec=%5.3f;\n\n\n',Vm); fprintf(fid,'ZjDatch_mm=[\n'); Z_patch=Q; for i=1:size(x_fit,2) z_patch=ZZ(i,1:column); Z_patch=[Z_patch ; z_patch]; end fprintf(fid,'];\n'); fclose(fid); J.3.2 STEP 3b: Multiple Cut Analysis (sharma multiplecut.M) This will perform longitudinal wave cut analysis using Sharma's method. This process will be looped over and over to process the entire wave patch for multiple wave cuts. The least-square method used to approximate the asymptotic behaviour for the truncation correction is incorporated within this code. The output is the wave resistance at each lateral position. % RUN THIS M-FILE IN THE HOME DIRECTORY THIRDLY, % IT IS TO BE SAVED IN THE HOME DIRECTORY % Author: J . TAN (UBC, Naval Lab) 219 % Purpose: To apply Sharma's method of longitudinal wave cut method on the % entire acquired wave profile (single longitudinal cut) % Input : The output .MAT file from processed Z_patch.M containing the wave % patch detailing the wave elevations, lateral positions and longitudinal % positions all contained in a matrix % Output: Wave numbers (u & s) % Wave amplitude functions (C & S , F & G) % Wave resistance (Rw) & wave resistance coefficients (Cw) % * output save in J P G / .M / .MAT file diary workspace.txt % clear all % cla reset % close all % % % — load wave patch profiles % [filename,path]=uigetfile('*.M','Open a M-file for processed wave patch'); % filename=filename(1 :length(filename)-2); % to discard the file extension % fprintf(1,'filename = %s\n',filename); % Fn=input('Froude no (refer to filename)= '); %Froude number % fprintf(1 ,M[please wait, data loading in progress ]\n'); run(filename); % — looping to find wave resistance at all transverse location in the patch results=Q; for ii=1 :size(Z_patch_mm,2) close all % % define working parameters (remember to change wetted area & waterline length for different hull)!!! A_wetted=1.526; % parent hull model scale wetted surface area [mA2] g=9.81; % gravitational acceleration [m/sA2] rho=1000; % water density [kg/mA3] B=3.660; % full tank width [m] 220 L=2.017; Vm=Fn*sqrt(g*L); ko=9.81/(VmA2); % % full model length [m] % model speed [m/s] % wave number [1/m] % — define longitudinal (xa) and transverse locations (ya) for i=1:size(Z_patch_mm,1) xa(i)=(i-1)*deltaX; end for j=1 :size(Z_patch_mm,2) ya(j)=1155- (j-1)*deltaY; end % % — define lateral cut position (y) to define the working wave height % y_cut_CL=ya(ii); % y_cut location [mm], wrt tank centerline (between 570~1155 mm) fprintf(1,'please wait, now processing long-cut %3g of %3g\n',ii,size(Z_patch_mm,2)) % y_cut_SW=((B*1000)/2) - y_cut_CL; % lateral cut position wrt tank sidewall [mm] % y_cut=y_cut_SW; % rename working lateral cut position % % — transpose lateral cut position into column number (j) to define working long-cut % [i_cut_nearest,j_cut_nearest]=max((ya > (y_cut_CL-deltaY)) & (ya < (y_cut_CL+deltaY))); % wrt tank centerline [mm] % j=j_cut_nearest; [mm] % % — define lateral cut position & define the working wave height (za) y_cut_SWJixed=((B/2)*1000) - 1155; % distance of long-cut closest to sidewall wrt sidewall [mm] y_cut_CL=ya(j); % y_cut location wrt tank centerline [mm] y_cut_SW=((B*1000)/2) - y_cut_CL; % lateral cut position wrt tank sidewall [mm] 221 y_cut=y_cut_SW; still wrt tank sidewall[mm] za=Z_patch_m m (1 :size(Z_patch_m m, 1) J); za=za'; % rename working lateral cut position, % — interpolate untruncated filtered & fitted (xdl.zdl) to higher resolution to precisely locate bow wave peak location interp_points=1000; % by default xa_interp1 =linspace(xa(1 ),xa(length(xa)),interp_points); za_interp1 =interp1 (xa,za,xa_interp1,'spline'); x_fit=xa_interp1; % length of fitted signal [mm] z_fit=za_interp1; % amplitude of fitted signal [mm] figure(1) plot(xa_interp1,za_interp1,'g+'); grid on; hold on; plot(x_fit,z_fit,'b'); hold on title(['LONGITUDINAL CUT WAVE PROFILE: Vm=',num2str(Vm),'m/s @ y=',num2str(y_cut_CL),'mm (wrt tank centerline)']); xlabel('downstream <- LONGITUDINAL DISTANCE, X [mm] -> upstream'); ylabel('WAVE HEIGHT, Z [mm]'); % — define effective working signal length (xdl.zdl) half_L_signal=(y_cut_SWJixed/tan(19.5/180*pi)); % half length of long-cut signal column 1 profile that is closest to sidewall [mm] L_signal=half_L_signal + (y_cut_SW/tan(19.5/180*pi)); % full signal length wrt longitudinal cut-off location x_cut [mm] Xo=x_cut + L_signal; % signal starting location astern wrt longitudinal cut-off x_cut [mm] % signal ending location astern wrt bow from column 1 long-cut closest to sidewall [mm] %Xt=x_fit(length(x_fit)); % used for Dr. Calisal's wave patch format Xt=x_cut - ((y_cut_SW/tan(19.5/180*pi)) - half_L_signal); % used for Jeff's wave patch format [row,col]=max(find((x_fit < (Xo-deltaX)) & (x j i t > (Xo+deltaX)))); 222 i_cut_begin=row; [row,col]=max(find((xJit < (Xt-deltaX)) & (x j i t > (Xt+deltaX)))); i_cut_end=row; figure(1) plot(x_fit(i_cut_begin),zJit(i_cut_begin),'ko','LineWidth',2); hold on; % mark out bow wave peak location x_trunc=x_fit(i_cut_begin:i_cut_end); % filtered & fitted truncated signal [mm] z_trunc=z_fit(i_cut_begin:i_cut_end); % filtered & fitted truncated signal [mm] figure(1) plot(x_trunc,z_trunc,'r','LineWidth',3); hold on legend('filtered (untruncated)Vfiltered + fitted (untruncated)Vstarting point'.'filtered + fitted (truncated)') % re-define control parameters x_mm=x_trunc; [mm] y_cut_C L_m m=y_cut_C L; centerline [mm] z_mm=z_trunc; transverse location [mm] % sternward coordination of waveheights % lateral cut position wrt tank % long-cut waveheights at corresponding % — non-dimensionalize all units x_trunc=x_trunc/1000*ko; z_trunc=z_trunc/1000*ko; Xo=Xo/1000*ko; Xt=Xtyi000*ko; y_cut_nearest=y_cut_CI_/1000*ko; B=B*ko; % wrt tank centerline % — compute wave resistance using Sharma's single longitudinal wave cut method s=1:0.05:10; s3=45; % default cut-off point upto u=10 only for plot visualization, not used for any calculation purpose u=s.*sqrt(s.A2 -1); 223 for i=1:length(s) %Func(i)=trapz(x_trunc,z_trunc.*exp(sqrt(-1)*s(i).*x_trunc)); % Sharma's equation (12), go with equation (14) Func(i)=trapz(x_trunc,sqrt(s(i).A2 - 1).*z_trunc.*exp(sqrt(-1)*s(i).*x_trunc)); % Sharma's equation (17), go with equation (25) C(i)=real(Func(i)); iS(i)=imag(Func(i)); end %f_s=((s.A2 - 1)./(s A2.*(2*s.A2 - 1)).*(C A 2 + iS.A2)); % weighting function as in Sharma's equation (14), go with equation (12) f_s=(1 ./(s.A2.*(2*s.A2 - 1))).*(C A 2 + iS. A2); % weighting function as in Sharma's equation (25), go with equation (17) figure(2) subplot(2,1,1) [s1,s2]=find(s==5); plot(s(1:s2),f_s(1:s2),'r-.','linewidth',2); grid on; hold on title(['WAVE ENERGY S P E C T R U M (Sharma, eq.25): Vm=',num2str(Vm),'m/s @ y=',num2str(y_cut_CL),'mm (wrt tank centerline)']); xlabel(['LONGITUDINAL WAVE NUMBER, s (truncated at s=',num2str(s(length(s))),')']); ylabel('WAVE ENERGY SPECTRUM') ; figure(2) subplot(2,1,2) plot(u(1:s3),f_s(1:s3),'r-.','linewidth',2); grid on; hold on title(['WAVE ENERGY S P E C T R U M (Sharma, eq.25): Vm=',num2str(Vm),'m/s @ y=',num2str(y_cut_CL),'mm (wrt tank centerline)']); xlabel([ 'TRANSVERSE WAVE NUMBER, u (truncated at s=',num2str(round(s(length(s)))),', i.e. u=',num2str(round(u(length(u)))),')']); ylabel('WAVE E N E R G Y SPECTRUM') ; Rw_nondim=1/pi*trapz(u,f_s); % Rw with truncation correction, non-dim, Sharma's equation (14) or (25) Rw_dim_N=Rw_nondim*1000*Vm A2/ko A2; % Rw without truncation correction [N] 224 Rw_dimJb=Rw_dim_N/g*2.2; % Rw without truncation correction [Ib] s % — apply asymptotic truncation correction with least square fitting c1=100; % initial guess for asymptotic function c2=100; % initial guess for asymptotic function c3=0; % follow assumption from paper of Eggers,Sharma, Ward zeta=(c1*cos(x_trunc) + c2*sin(x_trunc))./sqrt(c3 - x j runc); % Sharma's equation (18) for i=1:20 % convergence checking for least square Residual=(z_trunc - zeta); At=[(cos(x_trunc)./(c3 - x_trunc).A(1/2))' (sin(x_trunc)./(c3 - x_trunc).A(1/2))' ]';% (-1/2*(crcos(Xfit)+c2*sin(Xfit))./(c3-Xfit).A(3/2))' ]; a=At*At'; b=At*Residual'; dc=a\b; c1=c1+dc(1); % update c1 c2=c2+dc(2); % update c2 %c3=c3+dc(3); c3=0; zeta=(c1 *cos(x_trunc)+c2*sin(x_trunc))./sqrt(c3-x_trunc); % Sharma's equation (18) end x_extend=-500:0.25:min(x_trunc); % extend wave profile astern beyond truncation z_extend=(crcos(x_extend) + c2*sin(x_extend))./sqrt(c3 - x_extend); figure(3) plot(x_trunc,z_trunc,'r'); grid on; hold on plot(x_extend,z_extend,'b'); title(['LONGITUDINAL CUT WAVE PROFILE: Vm=',num2str(Vm),'m/s @ y=',num2str(y_cut_CL),'mm (wrt tank centerline)']); xlabel('downstream <- LONGITUDINAL DISTANCE, X [non-dim] -> upstream'); ylabel('WAVE HEIGHT, Z [non-dim]'); legend('without truncation correction'.'with truncation correction') % — evaluate fresnel integrals between [a,b;c] to correct wave resistance to account for truncation 225 % where a=z , lower bound of integrating range % b=100, default upper bound of integrating range % c=100, default integrating points Xe=min(x_extend); % should be negative value zF_upper=sqrt(2*(c3-Xe)*(s+1)/pi); % Sharma's equation (23) zFJower=sqrt(2*(c3-Xe)*(s-1)/pi); % Sharma's equation (23) CF_upper=rj; SF_upper=Q; CF_lower=D; SF_lower=D; for i=1:length(zF_upper) %fprintf(1,'please wait, now processing fresnel integrals, %3g of %3g\n',i,length(zF_upper)) [cf_upper,sf_upper] = fresnelCS(zF_upper(i), 100,10); CF_upper=[CF_upper cf_upper]; SF_upper=[SF_upper sf_upper]; [cf_lower,sf_lower] = fresnelCS(zF_lower(i), 100,10); CF_lower=[CF_lower cfjower]; SF_lower=[SF_lower sfjower]; end % — compute truncation correction for wave amplitude functions: dC_star, idS_star for i=1:length(s) d1(i)=(sqrt(s(i)-1))*(c1*cos(c3*(s(i)+1)) + c2*sin(c3*(s(i)+1))); % Sharma's equation (21a) d2(i)=(sqrt(s(i)-1 ))*(c1 *sin(c3*(s(i)+1)) - c2*cos(c3*(s(i)+1))); % Sharma's equation (21b) d3(i)=(sqrt(s(i)+1))*(c1*cos(c3*(s(i)-1)) - c2*sin(c3*(s(i)-1))); % Sharma's equation (21c) d4(i)=(sqrt(s(i)+1))*(d*sin(c3*(s(i)-1)) + c2*cos(c3*(s(i)-1))); % Sharma's equation (21 d) end for i=1:length(s) dC_star(i) =(sqrt(pi/2))*( d1 (i).*CF_upper(i) + d2(i).*SF_upper(i) + d3(i).*CF_lower(i) + d4(i).*SF_lower(i)); % Sharma's equation (20a) idS_star(i)=(sqrt(pi/2))*(-d1(i).*SF_upper(i) + d2(i).*CF_upper(i) - d3(i).*SF_lower(i) + d4(i).*CF_lower(i)); % Sharma's equation (20b end 226 df_s=(1 ./(s.A2.*(2*s.A2 - 1))).*(dC_star A 2 + idS_star. A2); % weighting function as in Sharma's equation (25) %f_plus_df_s=(1./(s.A2.*(2*s.A2 - 1))).*((C A 2 + iS. A2) + (dC_star. A2 + idS_star A2)); % weighting function as in Sharma's equation (25) f_plus_df_s=(1 ./(s.A2.*(2*s.A2 -.1))).*((C + dC_star). A2 + (iS + idS_star).A2); % weighting function as in Sharma's equation (25) figure(2) subplot(2,1,1) plot(s(1 :s2),f_plus_df_s(1 :s2),'k','linewidth',1 );hold on legend('without truncation correction: (CA2+SA2)','with truncation correction: (C A2+dC A2) + (S A2+dS A2)') figure(2) subplot(2,1,2) plot(u(1:s3),fjilus_df_s(1:s3)>'k','linewidth',1);hold on legend('without truncation correction: (CA2+SA2)','with truncation correction: (C A2+dC A2) + (S A2+dS A2)') Rw_trunc_nondim=1/pi*trapz(u,f_plus_df_s); % Rw with truncation correction, non-dim, Sharma's equation (14) or (25) Rw_trunc_dim_N=Rw_trunc_nondim*1000*VmA2/koA2; % Rw with truncation correction [N] Rw_trunc_dim_lb=Rw_trunc_dim_N/g*2.2; % Rw with truncation correction [Ib] CwO=Rw_dim_N / (0.5*rho*Vm*Vm*A_wetted); % Cw without truncation correction Cw1 =Rw_trunc_dim_N / (0.5*rho*Vm*Vm*A_wetted); % Cw with truncation correction results_temp=[[results]; [y_cut_CL ; Rw_dim_lb ; CwO ; Rw_trunc_dim_lb ; Cw1]T; results=results_temp'; % — display wave resistance fprintf(1,'\n \n',Fn); fprintf(1,'Froude no = %5.3f, i.e. model speed = %5.3f m/s\n',Fn,Vm) fprintf (1 An'.Fn); fprintf(1,'Y = %8.3f [mm], (transverse location wrt tank centerline)\n',y_cut_CL); fprintf(1,'Rw = %8.3f [Ib] , (wave resistance excluding truncation correction)\n',Rw_dimJb); fprintf(1 ,'Cw = %8.5f , (wave resistance coefficient excluding truncation correction)\n\n',CwO); 227 f printf (1 ,'Rw = %8.3f [Ib] , (wave resistance including truncation correction)\n',Rw_trunc_dimJb); fprintf(1 ,'Cw = %8.5f , (wave resistance coefficient including truncation correction)\n',Cw1); fprintf(1,' -\n\n',Fn); % — calculate wave spectrum (F & G) and plot resultant wave spectrum (F A2 + G A 2) fid=fopen([filename '_Y' num2str([num2str(y_cut_CL-mod(y_cut_CL,1)) 'p' num2str(round(mod(y_cut_CL,1)*10))]) '_wave_spectrums.M'],'w'); fprintf(fid,'%% Model_speed=%7.3f;\t\t%% model speed [m/s]\n',Vm); fprintf(fid,'%% Froude_no=%5.3f;\t\t%% Froude number\n\n\n',Fn); fprintf(fid,'%% u\ts\tRtG\t(FA2+GA2)\tdRtdG\t[(F+dF)A2+(G+dG)A2]\tC\tS\t(CA2+SA2)\tdC\tdS\t[(C+dC)A2+(S +dS)A2]\n\n\n'); fprintf (fid,'wave_spectrums=[\n'); for i=1:length(s) % without truncation correction F(i)=(1/(4*pi))*(4./(2*(s(i)A2) - 1)) * (C(i)*sin(u(i)*(y_cut_nearest)) + iS(i)*cos(u(i)*(y_cut_nearest))); % Sharma's equation (24a) G(i)=(1/(4*pi))*(4./(2*(s(i)A2) - 1)) * (C(i)*cos(u(i)*(y_cut_nearest)) iS(i)*sin(u(i)*(y_cut_nearest))); % Sharma's equation (24b) F2G2(i)=(F(i)).A2 + (G(i)).A2; % with truncation correction dF(i)=(1/(4*pi))*(4./(2*(s(i)A2) - 1)) * (dC_star(i)*sin(u(i)*(y_cut_nearest)) + idS_star(i)*cos(u(i)*(y_cut_nearest))); % Sharma's equation (24a) dG(i)=(1/(4*pi))*(4./(2*(s(i)A2) - 1)) * (dC_star(i)*cos(u(i)*(y_cut_nearest)) -idS_star(i)*sin(u(i)*(y_cut_nearest))); % Sharma's equation (24b) FdF2_GdG2(i)=(F(i) + dF(i)) A 2 + (G(i) + dG(i)).A2; fprintf(fid,'%8.5nt%8.5f\t%8.5W%8.5f\t%8.5f\t%8.5At%8.5f\t%8.5nt%8.5m \t%8.5f\t%8.5f\n',u(i),s(i),F(i),G(i),F2G2(i),dF(i),dG(i),FdF2_GdG2(i),C(i),iS(i),f_s(i),dC_star(i),i dS_star(i),f_plus_df_s(i)); end 228 fprintf(fid,1;\n'); fclose(fid); figure(4) subplot(2,1,1) [a,b]=find(s==5); plot(s(1:b),F2G2(1:b),'r-.','linewidth',2) ; grid on ; hold on % without truncation correction plot(s(1:b),FdF2_GdG2(1:b),'k','linewidth ,,1); grid on % with truncation correction title(['WAVE ENERGY S P E C T R U M (Sharma, eq.24): Vm=',num2str(Vm),'m/s @ y=',num2str(y_cut_CL),'mm (wrt tank centerline)']); xlabel(['LONGITUDINAL WAVE NUMBER, s (truncated at s=',num2str(s(length(s))),')']); ylabel('WAVE ENERGY SPECTRUM') ; legend('without truncation correction: (F A2+G A2)','with truncation correction: (F+dF) A2 + (G+dG)A2') figure(4) subplot(2,1,2) plot(u(1:s3),F2G2(1:s3),'r-.','linewidth',2) ; grid on ; hold on % without truncation correction plot(u(1:s3),FdF2_GdG2(1:s3),'k','linewidth',1); grid on % with truncation correction title(['WAVE ENERGY S P E C T R U M (Sharma, eq.24): Vm=',num2str(Vm),'m/s @ y=',num2str(y_cut_CL),'mm (wrt tank centerline)']); xlabel([ 'TRANSVERSE WAVE NUMBER, u (truncated at s=',num2str(round(s(length(s)))),', i.e. u=',num2str(round(u(length(u)))),')']); ylabel('WAVE ENERGY SPECTRUM') ; legendfwithout truncation correction: (F A2+G A2)','with truncation correction: (F+dF) A2 + (G+dG)A2') end % — overall resistances output and result interpretations format long results 229 [RwJb_max_corrected,cutjndex_Rw_max_corrected]=max(results(:,4)) y_cut_mm_wrtJank_centerline=ya(cutjndex_Rw_max_corrected) % y_cut location [mm], wrt tank centerline (between 570-1155 mm) figure(5) subplot(2,1,1) plot(results(:,1),results(:,2),'b:','LineWidth',2); grid on ; hold on plot(results(:,1),results(:,4),'r','LineWidth',2); hold on plot(results(cut_index_Rw_max_corrected,1),results(cut_index_Rw_max_corrected,4),'ko ','LineWidth',2); title(['MULTIPLE LONG-CUT @ Vm=',num2str(Vm),' [m/s] , Fn=' num2str(Fn) ' - > max(Rw)=',num2str(max(results(:,2))),' [Ib] max(Rw,corrected)=',num2str(max(results(:,4))),' [lb]']); xlabel(['(x=0) to tank centerline <- TRANSVERSE LOCATION, X [mm] -> to tank sidewall (x=' num2str((B/ko)/2*1000)')']); ylabel('Rw [lb]'); legend('uncorrected','corrected','maximum',2) figure(5) subplot(2,1,2) plot(results(:,1)/((B/ko)/2*1000),results(:,2),'b:','LineWidth',1); grid on ; hold on plot(results(:,1 )/((B/ko)/2*1000),results(:,4),'r','LineWidth',2); hold on; plot(results(cut_index_Rw_max_corrected,1)/((B/ko)/2*1000),results(cut_index_Rw_max _corrected,4),'ko','LineWidth',2); titletf'MULTIPLE LONG-CUT @ Vm=',num2str(Vm),' [m/s] , Fn=' num2str(Fn) ' - > max(Rw)=',num2str(max(results(:,2))),' [Ib] max(Rw,corrected)=',num2str(max(results(:,4))),' [lb]']); xlabel('(x=0) to tank centerline <- TRANSVERSE LOCATION, X [non-dim] -> to tank sidewall (x=1)'); ylabel('Rw [lb]'); legend('uncorrected','corrected','maximum',2) figure(6) subplot(2,1,1) plot(results(:,1),results(:,3),'b:','LineWidth',1); grid on ; hold on plot(results(:,1),results(:,5),'r','LineWidth',2); hold on 230 plot(results(cutjndex_Rw_max_correctedJ),results(cutjndex_Rw_max_corrected,5),'to VLineWidth',2); title(['MULTIPLE LONG-CUT @ Vm=\num2str(Vm),' [m/s] , Fn=' num2str(Fn) ' - > max(Cw)=',num2str(max(results(:,3))),', max(Cw,corrected)=',num2str(max(results(:,5)))]); xlabel(['(x=0) to tank centerline <- TRANSVERSE LOCATION, X [mm] -> to tank sidewall (x=' num2str((B/ko)/2*1000) ')'])•, ylabel('Cw'); legend('uncorrected','corrected','maximum',2) figure(6) subplot(2,1,2) plot(results(:>1)/((B/ko)/2*1000),results(:,3),'b:','LineWidth',1); grid on ; hold on plot(results(:,1)/((B/ko)/2*1000),results(:,5),'r','LineWidth',2); hold on; plot(results(cut_index_Rw_max_corrected,1)/((B/ko)/2*1000),results(cut_index_Rw_max _corrected,5),'ko','LineWidth',2); title(['MULTIPLE LONG-CUT @ Vm=',num2str(Vm),' [m/s] , Fn=' num2str(Fn) ' - > max(Cw)=',num2str(max(results(:,3))),', max(Cw,corrected)=',num2str(max(results(:,5)))]); xlabel('(x=0) to tank centerline <- TRANSVERSE LOCATION, X [non-dim] -> to tank sidewall (x=1)'); ylabel('Cw'); legend('uncorrected','corrected','maximum',2) savep lename '_final_results_Rw_Cw'],'results') save workspace diary off % — to reproduce plottings for post-visualization reproduce_plottings_T25 J.3.3 STEP 3c: Fresnel's Integrals (fresnelCS.M) This will perform computations of the Fresnel's integrals to obtain the wave amplitude functions (C and S) that are compatible with Sharma's longitudinal wave cut method in order to avoid the numerical ambiguity at s=1. 231 % RUN THIS SCRIPT IN HOME DIRECTORY IN ORDER TO REPRODUCE THE PLOTTINGS % Author: J . TAN (UBC, Naval Lab) % Purpose: To calculate the fresnel's integrals for used with Sharma's longitudinal wave cut method to avoid the numerical ambiguity at s=1 % Input : lower (a) & upper bound (b) of integral, and spacing (np) % Output: Wave amplitude functions (C & S) function [C,S]=fresnelCS(a,b,n) % [C,S]=fresnelCS(a,b) % % C->Fresnel cosine Integral lnt(a,b,cos(pi/2*xA2),dx) % S->Fresnel sine Integral lnt(a,b,sin(pi/2*xA2),dx) % % n is the number of points to perform the numerical integration wrt % the smallest repetion interval of the functions % to be integrated, i.e. min {l_n=2*[sqrt(n+1) - sqrt(n)]}, % depending on the values of a and b % % Integration is performed with the classical rectangular formula if (a>b) error('a is greater than b!') elseif (a==b) C=0; S=0; else n=0; if (abs(a) > abs(b)) while (2*sqrt(n) <= abs(a)) n=n+1; end if (a<0) n=-n+1; 232 end else while (2*sqrt(n) < abs(b)) n=n+1; end if (b>=0) n=n-1; else n=-n; end end l=2*abs(sqrt(abs(n+1)) - sqrt(abs(n))); p=l/n; x=a:p:b; C=sum(cos(pi/2*x A2))*p; S=sum(sin(pi/2*x.A2))*p; end J.3.4 STEP 3d: Plottings (reproduce_plottings_T25.M) This will perform plottings of the results from multiple cuts. The output is the wave resistance, wave coefficients at each lateral position in J P G format. This script file acts as subroutine to be called up by sharma_multiple_cut.M, but it can also be run individually for post processing of results. % RUN THIS SCRIPT IN HOME DIRECTORY IN ORDER TO REPRODUCE THE PLOTTINGS % Author: J . TAN (UBC, Naval Lab) % Purpose: To reproduce the plottings for post-visualization % Input : The output .MAT file from processed Z_patch.M containing the wave % patch detailing the wave elevations, lateral positions and longitudinal % positions all contained in a matrix % Output: Wave numbers (u & s) 233 % % Wave amplitude functions (C & S , F & G) Wave resistance (Rw) & wave resistance coefficients (Cw) % * output J P G file clear all cla reset close all % — replot results at longitudinal cut when wave resistance is at the maximum load workspace figure(1) subplot(2,1,1) za=Z_patch_m m (1 :size(Z_patch_m m, 1) ,cutJndex_Rw_max_corrected); za=za'; interp_points=1000; % by default xaJnterp1=linspace(xa(1),xa(length(xa)),interp_points); za jn te rp l =interp1 (xa,za,xajnterp1,'spline'); x_fit=xajnterp1; % length of fitted signal [mm] z_fit=zajnterp1; % amplitude of fitted signal [mm] plot(xaJnterp1 ,zajnterp1 ,'g+'); grid on; hold on; plot(xJit,zJit,'b'); hold on title(['LONGITUDINAL CUT WAVE PROFILE: Fn=' num2str(Fn)', Vm=',num2str(Vm),'m/s @ y=',num2str(y_cut_mm_wrt_tank_centerline),'mm (wrt tank centerline)']); xlabel('downstream <- LONGITUDINAL DISTANCE, X [mm] -> upstream'); ylabel('WAVE HEIGHT, Z [mm]'); plot(x_fit(i_cut_begin),z_fit(i_cut_begin),'ko','LineWidth',2); hold on; % mark out bow wave peak location x_trunc=xjit(i_cut_begin:i_cut_end); % filtered & fitted truncated signal [mm] plot(x_trunc,z_trunc,'r','LineWidth',3); hold on legend('filtered (untruncated)','filtered + fitted (untruncated)Vstarting point'.'filtered + fitted (truncated)') V=axis; [mm] z_trunc=z_fit(i_cut_begin:i_cut_end); % filtered & fitted truncated signal 234 subplot(2,1,2) za=Z_patch_mm(1 :size(Z_patch_mm,1 ),cut_index_Rw_max_corrected); za=za; interp_points=1000; % by default xa_interp1=linspace(xa(1),xa(length(xa)),interpjDoints); za_interp1 =interp1 (xa,za,xa_interp1,'spline'); x_fit=xa_interp1; % length of fitted signal z_fit=za_interp1; % amplitude of fitted signal plot(xa_interp1/1000*ko,za_interp1/1000*ko,'g+'); grid on; hold on; plot(x_fit/1000*ko,z_fit/1000*ko,'b'); hold on title(['LONGITUDINAL CUT WAVE PROFILE: Fn=' num2str(Fn) ', Vm=',num2str(Vm),'m/s @ y=',num2str(y_cut_mm_wrt_tank_centerline),'mm (wrt tank centerline)"]); xlabel('downstream <- LONGITUDINAL DISTANCE, (X/1000*ko) [non-dim] -> upstream'); ylabel('WAVE HEIGHT, (Z/1000*ko) [non-dim]'); plot(x_fit(Lcut_begin)/1000*ko,z_fit(i_cut_begin)/1000*ko,'ko','LineWidth',2); hold on; % mark out bow wave peak location xjrunc=xjit(i_cut_begin:i_cut_end); % filtered & fitted truncated signal z_trunc=z_fit(i_cut_begin:i_cut_end); % filtered & fitted truncated signal plot(x_trunc/1000*ko,z_trunc/1000*ko,'r','LineWidth',3); hold on legend('filtered (untruncated)','filtered + fitted (untruncated)','starting point'.'filtered + fitted (truncated)') axis([V(1 )/1000*ko V(2)/1000*ko V(3)/1000*ko V(4)/1000*ko]) saveas(gcf,[filename '_Y' num2str([num2str(y_cut_mm_wrt_tank_centerline-mod(y_cut_mm_wrt_tank_centerline,1)) 'p' num2str(round(mod(y_cut_mm_wrt_tank_centerline,1 )*10))]) '_waveheighf],'jpg') num2str(round(mod(y_cut_mm_wrt_tank_centerline,1 )*10))]) '_wave_spectrums']) % — uncomment and use the 4 lines below for plotting results for any user-selected longitudinal cut % [filename,path]=uigetfile('*wave_spectrums.M','Open a M-file for processed wave patch'); figure(2) runp lename '_Y' mod(y_cut_mm_wrt_tank_centerline,1)) num2str([num2str(y_cut_mm_wrt_tank_centerline-'p' 235 % filename=filename(1 :length(filename)-2); % fprintf(1,'filename = %s\n',filename); % run(filename); % to discard the file extension subplot(2,1,1) [a,b]=find(s==5); plot(wave_spectrums(1:b,2),wave_spectrums(1:b,11),'r-.','linewidth',2) ; grid on ; hold on % without truncation correction plot(wave_spectrums(1:b,2),wave_spectrums(1:b,14),'k','linewidth',1); grid on % with truncation correction title(['WAVE ENERGY S P E C T R U M (Sharma, eq.24): Vm=',num2str(Vm),'m/s @ y=',num2str(y_cut_mm_wrt_tank_centerline),'mm (wrt tank centerline)"]); xlabel(['LONGITUDINAL WAVE NUMBER, s (truncated at s=',num2str(s(length(s))),')']); ylabel('WAVE ENERGY SPECTRUM') ; legend('without truncation correction: (C A2+S A2)','with truncation correction: (C+dC) A2 + (S+dS)A2') subplot(2,1,2) s3=28; plot(wave_spectrums(1:s3,1),wave_spectrums(1:s3,11),'r-.','linewidth',2) ; grid on ; hold on % without truncation correction plot(wave_spectrums(1:s3,1),wave_spectrums(1:s3,14),'k','linewidth',1); grid on % with truncation correction title(['WAVE ENERGY S P E C T R U M (Sharma, eq.24): Vm=',num2str(Vm),'m/s @ y=',num2str(y_cut_mm_wrt_tank_centerline),'mm (wrt tank centerline)1]); xlabel([ 'TRANSVERSE WAVE NUMBER, u (truncated at s=',num2str(round(s(length(s)))),', i.e. u=',num2str(round(u(length(u)))),')']); ylabel('WAVE ENERGY SPECTRUM') ; legend('without truncation correction: (C A2+S A2)','with truncation correction: (C+dC) A2 + (S+dS)A2') saveas(gcf,[filename '_Y' num2str([num2str(y_cut_mm_wrt_tank_centerline-mod(y_cut_mm_wrt_tank_centerline,1)) 'p' num2str(round(mod(y_cut_mm_wrt_tank_centerline,1 )*10))]) '_wave_spectrums_C_and_S'],'jpg') figure(3) plot(x_trunc/1000*ko,z_trunc/1000*ko,'r','linewidth',2); grid on; hold on 236 plot(x_extend,z_extend,'b'); title(['LONGITUDINAL CUT WAVE PROFILE: Fn=' num2str(Fn)', Vm=',num2str(Vm),'m/s @ y=',num2str(y_cut_mm_wrMank_centerline),'mm (wrt tank centerline)']); xlabel('downstream <- LONGITUDINAL DISTANCE, (X/1000*ko) [non-dim] -> upstream'); ylabel('WAVE HEIGHT, (Z/1000*ko) [non-dim]'); legend('without truncation correction'.'with truncation correction',2) saveas(gcf,[filename '_Y' num2str([num2str(y_cut_mm_wrt_tank_centerline-mod(y_cut_mm_wrt_tank_centerline,1)) 'p' num2str(round(mod(y_cut_mm_wrt_tank_centerline,1)*10))]) '_asymptotic_freewaves'],'jpg') num2str(round(mod(y_cut_mm_wrt_tank_centerline,1)*10))]) '_wave_spectrums']) % uncomment and use the 4 lines below for plotting results for any user-selected longitudinal cut % [filename,path]=uigetfile('*wave_spectrums.M','Open a M-file for processed wave patch'); % filename=filename(1 :length(filename)-2); % to discard the file extension % fprintf(1,'filename = %s\n',filename); % run(filename); subplot(2,1,1) [a,b]=find(s==5); plot(wave_spectrums(1:b,2),wave_spectrums(1:b,5),'r-.','linewidth',2) ; grid on ; hold on % without truncation correction plot(wave_spectrums(1:b,2),wave_spectrums(1:b,8),'k','linewidth',1); grid on % with truncation correction title(['WAVE ENERGY S P E C T R U M (Sharma, eq.24): Vm=',num2str(Vm),'m/s @ y=',num2str(y_cut_mm_wrt_tank_centerline),'mm (wrt tank centerline)']); xlabel(['LONGITUDINAL WAVE NUMBER, s (truncated at s=',num2str(s(length(s))),')']); ylabel('WAVE ENERGY SPECTRUM') ; legend('without truncation correction: (F A2+G A2)','with truncation correction: (F+dF) A2 + (G+dG)A2') subplot(2,1,2) figure(4) run([filename '_Y' mod (y_cut_m m_wrt_tank_ce nterl i ne, 1)) num2str([num2str(y_cut_mm_wrt_tank_centerline-•p' 237 S3=28; plot(wave_spectrums(1:s3,1),wave_spectrums(1:s3,5),'r-.','linewidth',2) ; grid on ; hold on % without truncation correction plot(wave_spectrums(1:s3,1),wave_spectrums(1:s3,8),'k','linewidth',1); grid on % with truncation correction title(['WAVE ENERGY S P E C T R U M (Sharma, eq.24): Vm=',num2str(Vm),'m/s @ y=',num2str(y_cut_mm_wrt_tank_centerline),'mm (wrt tank centerline)']); xlabel([ 'TRANSVERSE WAVE NUMBER, u (truncated at s=',num2str(round(s(length(s)))),', i.e. u=',num2str(round(u(length(u)))),')']); ylabel('WAVE ENERGY SPECTRUM') ; legend('without truncation correction: (F A2+G A2)','with truncation correction: (F+dF) A2 + (G+dG)A2') saveas(gcf,[filename '_Y' num2str([num2str(y_cut_mm_wrt_tank_centerline-mod(y_cut_mm_wrt_tank_centerline,1)) 'p' num2str(round(mod(y_cut_mm_wrt_tank_centerline,1)*10))]) '_wave_spect ru m s_F_and_G'], 'j pg') figure(5) subplot(2,1,1) plot(results(:,1),results(:,2),'b-.','LineWidth',1); grid on ; hold on plot(results(:,1),results(:,4),'r','LineWidth',2); hold on plot(results(cut_index_Rw_max_corrected,1),results(cut_index_Rw_max_corrected,4),'ko ','LineWidth',2); title(['MULTIPLE LONG-CUT @ Fn=' num2str(Fn) ',Vm=',num2str(Vm),'[m/s]->max(Rw,uncorrected)=',num2str(max(results(:,2))),'[lb],max(Rw,corrected)=',num2str(max(re sults(:,4))),'[lb]']); xlabel(['(y=0) to tank centerline <- TRANSVERSE LOCATION, Y [mm] -> to tank sidewall (y=' num2str((B/ko)/2*1000)')']); ylabel('Rw [lb]'); legend('uncorrected','corrected','maximum',2) V=axis; axis([V(1) V(2)0V(4)]) subplot(2,1,2) plot(results(:,1)/1000*ko,results(:,3),'b-.','LineWidth',1); grid on ; hold on plot(results(:,1)/1000*ko,results(:,5),'r','LineWidth',2); hold on; 238 plot(results(cutjndex_Rw_max_correctedJ)/1000*ko,results(cutjndex_Rw_max_correc ted,5),'ko','LineWidth',2); title(['MULTIPLE LONG-CUT @ Fn=' num2str(Fn) ' , Vm=',num2str(Vm),' [m/s] - > max(Cw,uncorrected)=',num2str(max(results(:,3))),' max(Cw,corrected)=',num2str(max(results(:,5)))]); title(['max(Cw,uncorrected)=',num2str(max(results(:,3))),' max(Cw,corrected)=',num2str(max(results(:,5)))]); xlabel(['(y=0) to tank centerline <- TRANSVERSE LOCATION, (Y/1000*ko) [non-dim] ->to tank sidewall (y=' num2str(((B/ko)/2*1000)/1000*ko) ')D; ylabel('Cw'); legend('uncorrected','corrected','maximum',2) axis([V(1 )/1000*ko V(2)/1000*ko 0 (V(4)/2.2*g)/(0.5*rho*Vm*Vm*A_wetted)]) saveas(gcf,[filename '_final_results_Rw_Cw'],'jpg') num2str(round(mod(y_cut_mm_wrt_tank_centerline,1)*10))]) '_wave_spectrums']) % — uncomment and use the 4 lines below for plotting results for any user-selected longitudinal cut % [filename,path]=uigetfile('*wave_spectrums.M','Open a M-file for processed wave patch'); % filename=filename(1 :length(filename)-2); % to discard the file extension % fprintf(1,'filename = %s\n',filename); % run(filename); subplot(2,1,1) [a,b]=find(s==5); plot(wave_spectrums(1:b,2),wave_spectrums(1:b,11),'r-.','linewidth',2) ; grid on ; hold on % without truncation correction plot(wave_spectrums(1:b,2),wave_spectrums(1:b,14),'k','linewidth',1); grid on % with truncation correction title(['WAVE ENERGY SPECTRUM: Fn=' num2str(Fn) ', Vm=',num2str(Vm),'m/s @ y=',num2str(y_cut_mm_wrt_tank_centerline),'mm (wrt tank centerline)']); ylabeK'Sharma eq.17-23: f(C,S)'); figure(6) run([filename '_Y' mod(y_cut_mm_wrt_tank_centerline,1)) num2str([num2str(y_cut_mm_wrt_tank_centerline-'p' 239 legend('without truncation correction: (C A2+S A2)','with truncation correction: (C+dC) A2 + (S+dS)A2') subplot(2,1,2) s3=28; plot(wave_spectrums(1:s3,1),wave_spectrums(1:s3,5) I'r-.','linewidth',2) ; grid on ; hold on % without truncation correction plot(wave_spectrums(1:s3,1),wave_spectrums(1:s3,8),'k','linewidth',1); grid on % with truncation correction x labe l([TRANSVERSE WAVE NUMBER, u (computations truncated at u=',num2str(round(u(length(u)))),', i.e. s=',num2str(round(s(length(s)))),')']); ylabel('Sharma eq.24: f(F,G)'); legend('without truncation correction: (F A2+G A2)','with truncation correction: (F+dF) A2 + (G+dG)A2') saveas(gcf,[filename '_Y' num2str([num2str(y_cut_mm_wrt_tank_centerline-mod(y_cut_mm_wrt_tank_centerline,1)) 'p' num2str(round(mod(y_cut_mm_wrt_tank_centerline,1)*10))]) '_wave_spectru m s_CS_and_FG'], 'j pg') 240 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0080733/manifest

Comment

Related Items