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A numerical procedure for the prediction of the flow field and resistance of fishing nets Chu, Franky K.Y. 1989

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A N U M E R I C A L P R O C E D U R E FOR T H E PRED ICT ION O F T H E F L O W F IELD A N D R E S I S T A N C E O F FISHING N E T S BY FRANKY K.Y. CHU B.Eng.(Hon.), Brighton Polytechnic, 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of Mechanical Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1989 © FRANKY K.Y. CHU, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada DE-6 (2/88) " A B S T R A C T This thesis presents a numerical model for the c a l c u l a t i o n of the flow f i e l d and resistance of a f i s h i n g net. This numerical method i s based on the p o t e n t i a l flow theory and an empirical formula to p r e d i c t the force acting on a mesh of the netting. An experiment to determine the shape, flow f i e l d , and the resistance of a c o n i c a l net has been c a r r i e d out i n the flume tank at the Marine I n s t i t u t e located at St. John's, Newfoundland, and the experimental r e s u l t s are presented as well. The net drag force obtained by th i s numerical procedure i s compared with the flume tank experiments as well as with methods developed by other researchers. Although the r e s u l t from the numerical model does not agree well with some methods developed by other researchers, i t does have better agreement with the r e s u l t s obtained from the flume tank experiments. The flow f i e l d around the f i s h i n g net ca l c u l a t e d by the numerical model at various incoming v e l o c i t i e s i s obtained. i i T A B L E O F C O N T E N T S ABSTRACT i i TABLE OF CONTENTS i i i LIST OF FIGURES v i NOMENCLATURE x ACKNOWLEDGEMENT x i i CHAPTER 1 INTRODUCTION 1 1.1 METHOD OF FISHING 1.2 LITERATURE REVIEW 1.3 PURPOSE OF THE PRESENT RESEARCH-CHAPTER 2 NUMERICAL MODELLING OF A FISHING NET 6 2.1 INTRODUCTION 2.2 MOTION OF THE FLUID THROUGH THE NET 2.3 NUMERICAL MODEL 2.4 BOUNDARY CONDITIONS CHAPTER 3 MATHEMATICAL FORMULATION OF BOUNDARY ELEMENT METHOD 9 3.1 SOLUTION METHODS OF THE LAPLACE EQUATION 3.2 ASSUMPTIONS 3.3 GENERAL FORMULATION OF THE BOUNDARY ELEMENT METHOD CHAPTER 4 MATHEMATICAL FORMULATION OF AN IRROTATIONAL WAKE MODEL . 16 4.1 WAKE MODEL 4.2 BOUNDARY CONDITIONS 4.3 MATHEMATICAL FORMULATION 4.4 MATCHING TECHNIQUE i i i CHAPTER 5 RESULTS OF IRROTATIONAL WAKE MODEL 23 5.1 INTRODUCTION 5.2 RESULTS WITHOUT POTENTIAL DISCONTINUITY 5.3 RESULTS WITH POTENTIAL DISCONTINUITY 5.4 DISCUSSION CHAPTER 6 MATHEMATICAL FORMULATION OF THE NET MODEL 26 6.1 THE DETERMINATION OF FORCES ACTING ON THE NET 6.2 ALGORITHM OF PRESSURE DROP ACROSS A NET 6.3 DETERMINATION OF THE VELOCITY COMPONENTS 6.4 ALGORITHM FOR REPRESENTING POTENTIAL DISCONTINUITY CHAPTER 7 CONICAL NET EXPERIMENT 33 7.1 INTRODUCTION 7.2 MEASUREMENT OF THE CONICAL NET PROFILE 7.3 MEASUREMENT OF THE VELOCITY INSIDE THE NET 7.4 OBSERVATION OF THE FLOW FIELD AROUND THE NET 7.5 MEASUREMENT OF THE DRAG FORCE CHAPTER 8 RESULTS OF THE CONICAL NET EXPERIMENT 36 8.1 PROFILE OF THE CONICAL NET 8.2 FLOW FIELD INSIDE THE NET 8.3 FLOW FIELD AROUND THE NET 8.4 DRAG FORCE OF THE CONICAL NET CHAPTER 9 RESULTS OF THE NUMERICAL MODEL 39 9.1 INTRODUCTION 9.2 RESULTS CHAPTER 10 DISCUSSION AND CONCLUSION 41 i v BIBLIOGRAPHY 47 APPENDIX 49 FIGURES 75 v L I S T O F F I G U R E S FIGURE 1.1.1 TRAWLING ..75 FIGURE 1.1.2 MID-WATER TRAWLING 75 FIGURE 1.1.3 TYPICAL FISHING NET 76 FIGURE 1.1.4 SAMPLE DRAWING OF THE TRAWLING NET 77 FIGURE 2.1.1 MESH PARAMETERS 77 FIGURE 2.1.2 FISHING NET IN FLUME TANK 78 FIGURE 3.3.1 COORDINATE SYSTEM FOR RING ELEMENT 79 FIGURE 3.3.2 CONTROL DOMAIN FOR THE SPHERE 79 FIGURE 3.3.3 FLOW FIELD AROUND A SPHERE 80 upstream v e l o c i t y = 1.0 m/s FIGURE 3.3.4 VELOCITY ALONG THE DOWNSTREAM BOUNDARY OF THE CONTROL DOMAIN 81 upstream v e l o c i t y = 1.0 m/s FIGURE 3.3.5 ELEMENTS AT CORNER 82 FIGURE 3.3.6 COEFFICIENT OF PRESSURE C , ALONG THE SURFACE p OF A SPHERE 83 upstream v e l o c i t y = 1.0 m/s FIGURE 4.1.1 CONTROL DOMAINS FOR THE NUMERICAL MODEL OF FISHING NET 84 FIGURE 5.2.1 FLOW FIELD OF A VERTICAL BOUNDARY 85 without p o t e n t i a l difference upstream v e l o c i t y = 2.0 m/s FIGURE 5.2.2 NORMAL VELOCITY ALONG VERTICAL BOUNDARY .86 without p o t e n t i a l difference upstream v e l o c i t y = 2.0 m/s v i FIGURE 5.2.3 VELOCITY ALONG DOWNSTREAM BOUNDARY 87 without p o t e n t i a l difference upstream v e l o c i t y = 2.0 m/s FIGURE 5.2.4 FLOW FIELD OF A 90 DEGREES ARC BOUNDARY 88 without p o t e n t i a l difference upstream v e l o c i t y = 2.0 m/s FIGURE 5.2.5 NORMAL VELOCITY ALONG 90 DEGREES ARC 89 without p o t e n t i a l difference upstream v e l o c i t y =2.0 m/s FIGURE 5.2.6 VELOCITY ALONG DOWNSTREAM BOUNDARY 90 without p o t e n t i a l d i f f e r e n c e upstream v e l o c i t y =2.0 m/s FIGURE 5.2.7 FLOW FIELD OF A 60 DEGREES BOUNDARY 91 without p o t e n t i a l difference-upstream v e l o c i t y = 2.0 m/s FIGURE 5.2.8 NORMAL VELOCITY ALONG 60 DEGREES BOUNDARY 92 without p o t e n t i a l difference upstream v e l o c i t y = 2.0 m/s FIGURE 5.2.9 VELOCITY ALONG DOWNSTREAM BOUNDARY 93 without p o t e n t i a l difference upstream v e l o c i t y =2.0 m/s FIGURE 5.3.1 FLOW FIELD OF A VERTICAL BOUNDARY 94 with p o t e n t i a l d i f f e r e n c e upstream v e l o c i t y = 2.0 m/s FIGURE 5.3.2 FLOW FIELD OF A 90 DEGREES ARC BOUNDARY 95 with p o t e n t i a l difference upstream v e l o c i t y = 2.0 m/s v i i FIGURE 5.3.3 FLOW FIELD OF 60 DEGREES BOUNDARY 96 with p o t e n t i a l d i f f e r e n c e upstream v e l o c i t y = 2.0 m/s FIGURE 6.1.1 AN UNIT NETTING 97 FIGURE 6.4.1 AN ELEMENT ALONG THE NET 97 FIGURE 6.4.2 VELOCITY COMPONENTS ON THE SURFACE OF THE NET ..98 FIGURE 7.1.1 FLUME TANK AT MARINE INSTITUTE 99 FIGURE 7.1.2 GEOMETRY OF CONICAL NET PANEL 100 FIGURE 7.3.1 PROPELLER TYPE CURRENT METER 101 FIGURE 7.3.2 CALIBRATION GRAPH OF CURRENT METER 102 FIGURE 7.4.1 TELL-TAILS ATTACHED TO THE CONICAL NET .103 FIGURE 8.1.1 PROFILE OF THE CONICAL NET AT VARIOUS WATER SPEED 104 FIGURE 8.1.2 MESH OPENING ALONG THE CONICAL NET 105 FIGURE 8.3.1 TELL-TAILS AT VARIOUS POSITION OF THE CONICAL NET 106 FIGURE 9.2.1 FLOW FIELD AROUND THE CONICAL NET AT 0.4 M/S ...107 FIGURE 9.2.2 FLOW FIELD AROUND THE CONICAL NET AT 0.6 M/S ... 108 FIGURE 9.2.3 FLOW FIELD AROUND THE CONICAL NET AT 0.8 M/S ...109 FIGURE 9.2.4 FLOW FIELD AROUND THE CONICAL NET AT 1.0 M/S ...110 FIGURE 9.2.5 NORMAL VELOCITY ALONG THE NET AT 0.4 M/S I l l FIGURE 9.2.6 NORMAL VELOCITY ALONG THE NET AT 0.6 M/S 112 FIGURE 9.2.7 NORMAL VELOCITY ALONG THE NET AT 0.8 M/S 113 FIGURE 9.2.8 NORMAL VELOCITY ALONG THE NET AT 1.0 M/S 114 FIGURE 9.2.9 TANGENTIAL VELOCITY ALONG THE NET AT 0.4 M/S ...115 FIGURE 9.2.10 TANGENTIAL VELOCITY ALONG THE NET AT 0.6 M/S ..116 FIGURE 9.2.11 TANGENTIAL VELOCITY ALONG THE NET AT 0.8 M/S ..117 v i i i FIGURE 9.2.12 TANGENTIAL VELOCITY ALONG THE NET AT 1.0 M/S ..118 FIGURE 9.2.13 PRESSURE DROP ACROSS THE NET AT 0.4 M/S 119 FIGURE 9.2.14 PRESSURE DROP ACROSS THE NET AT 0.6 M/S 120 FIGURE 9.2.15 PRESSURE DROP ACROSS THE NET AT 0.8 M/S 121 FIGURE 9.2.16 PRESSURE DROP ACROSS THE NET AT 1.0 M/S 122 FIGURE 9.2.17 COMPARISON OF DRAG FORCES 123 FIGURE 10.1 SURFACE OF CONICAL NET UNDER TEST 124 FIGURE E . l MESH PLANE PARAMETERS 125 FIGURE E.2 DEFINITION OF THE AXIS 126 FIGURE G.l REFERENCE FRAME OF COORDINATE FOR PROFILE MEASUREMENT 127 FIGURE H.l REFERENCE FRAME OF COORDINATE FOR VELOCITY MEASUREMENT 128 FIGURE H.2 WATER VELOCITY INSIDE THE NET AT 0.4 M/S 129 FIGURE H.3 WATER VELOCITY INSIDE THE NET AT 0.6 M/S 130 FIGURE H.4 WATER VELOCITY INSIDE THE NET AT 0.8 M/S 131 FIGURE H.5 WATER VELOCITY INSIDE THE NET AT 1.0 M/S 132 ix N O M E N C L A T U R E Bar length Area of an u n i t n e t t i n g Non - dimensional r e f r a c t i o n c o e f f i c i e n t Constant Non - dimensional force c o e f f i c i e n t Pressure c o e f f i c i e n t Knot diameter Twine diameter Complete E l l i p t i c Function of f i r s t and second kind Force a r i s i n g from the bars alone Force a r i s i n g from the bars alone i n i d i r e c t i o n Force i n i d i r e c t i o n Green's function Pressure drop c o e f f i c i e n t Normal u n i t vector on the surface of the c o n t r o l domain,-which i s pointing outward. Component of normal u n i t vector i n h o r i z o n t a l and r a d i a l d i r e c t i o n Pressure drop i n i d i r e c t i o n Pressure drop i n normal d i r e c t i o n Undisturbed s t a t i c pressure Radius of the control domain Distance between point P and Q V e l o c i t y component i n X and Y d i r e c t i o n V e l o c i t y on the surface x V Free stream v e l o c i t y V V e l o c i t y i n normal d i r e c t i o n n V V Tangential v e l o c i t y on the outside and inside surface of the net V , V V e l o c i t y component i n X d i r e c t i o n on the outside and X 1 X2 J inside surface of the net V , V V e l o c i t y component i n Y d i r e c t i o n on the outside and Y 1 Y 2 J r i n s i d e surface of the net X, Y, Z coordinate axes x, R Horizontal and r a d i a l measurement of point Q x , R Horizontal and r a d i a l measurement of point P p p v a Angle of incidence Q Half s e t t i n g angle $ V e l o c i t y p o t e n t i a l $, V e l o c i t y p o t e n t i a l d e r i v a t i v e n $ V e l o c i t y p o t e n t i a l on the outside surface of the net R e g i o n 1 $ V e l o c i t y p o t e n t i a l on the inside surface of the net R e g i o n 2 $ V e l o c i t y p o t e n t i a l difference across the net D i f f J r 9 Angular measurement from Y axis ft Constant p Water density T C i r c u l a t i o n •8 Skew angle 8 Kronecker d e l t a i 3 x i A C K N O W L E D G E M E N T In the course of conducting t h i s research, I have drawn upon the knowledge and resources of many i n d i v i d u a l s , too numerous to mention here. I wish to thank a l l of them for t h e i r input and assistance. However, I must single out a few people whose contributions have been p a r t i c u l a r s i g n i f i c a n t to t h i s work. In p a r t i c u l a r , I wish to express my sincere gratitude to Dr. S.M. C a l i s a l , my supervisor for h i s patient guidance and support throughout these years. I also wish to thank the Department of F i s h e r i e s and Oceans of Canada and National Science and Engineering Research Council ( NSERC ) of Canada for funding t h i s p r o ject. I would l i k e to thank the s t a f f , at the flume tank of the Marine I n s t i t u t e of Newfoundland, and the towing tank of the B r i t i s h Columbia Research, for t h e i r help during the experiments and the use of the f a c i l i t y . I am deeply indebted to my parents and my brother, Dr. K.M. Chu and other family members for t h e i r support and encouragement throughout the years of my study. Thanks to my fiancee, V i v i a n f or her understanding and support during these years. I would l i k e to thank Dr. J.L.K. Chan for h i s advise on the a p p l i c a t i o n of the Boundary Element Method. x i i F i n a l l y , thanks to a l l of my colleagues i n the Naval Architecture group for the help they have provided. Thanks to Jon Mikkelsen for h i s patient help i n e d i t i n g of t h i s t h e s i s . x i i i C H A P T E R 1 I N T R O D U C T I O N 1.1 METHOD OF FISHING The ocean i s a giant resource which provides v i t a l l y important supplies such as food, minerals and petroleum. Mankind has developed a heavy dependence on the f i s h i n g industry to meet i t s d a i l y food requirements from the ocean. Although there are many d i f f e r e n t means of harvesting f i s h , trawling i s one of the p r i n c i p a l methods used by f i s h i n g f l e e t s throughout the world. Large quantities of ground f i s h and pelagic species are harvested on both coast of Canada by bottom and mid-water trawling methods described as follows: i ) Bottom Trawling Bottom trawling consists of towing a cone shaped net along the ocean bottom. "Doors" or "Otterboards" attached to cables between the vessel and the net serve to keep the mouth of the net h o r i z o n t a l l y open while the net i s making i t s tow along the ocean bottom. (Figure 1.1.1) i i ) Mid-Water Trawling Mid-water trawling i s s i m i l a r to bottom trawling, but generally involves towing a much larger net at a selected depth above the ocean f l o o r . (Figure 1.1.2) A t y p i c a l f i s h i n g net i s made up of several components. These are: (Figure 1.1.3) i ) the net bag 1 i i ) the codend i i i ) l i n e s i v ) ground warps v) f l o a t s v i ) doors v i i ) towing warps Also included with the net bag are wing b r i d l e s , b r i d l e l i n e s , r i b l i n e s , a footrope, a headline and a hanging l i n e . In Figure 1.1.4, a sample drawing of the trawling net i s shown. Only recently, a f t e r the o i l c r i s i s of 1973 pushed the p r i c e of f u e l up, have researchers started to work on the methods to improve the cost e f f i c i e n c y of the f i s h i n g operation. Since about 80% of the e n t i r e f u e l consumption during a f i s h i n g operation i s used during the towing of the net, i t i s b e n e f i c i a l to cut down the f u e l consumption during t h i s phase i n order to reduce the p r i c e of f i s h i n g . 1.2 LITERATURE REVIEW Most of the research work to determine the drag of the f i s h i n g net i s done experimentally. These experiments are conducted e i t h e r i n towing tanks, flume tanks, or f u l l scale t e s t i n g at sea. The r e s u l t s are then used to develop empirical formulas. Some of the early research to determine the drag of sheet n e t t i n g p a r a l l e l to water flow was done by Konagaya and Kawakami (1971) [1]. From t h i s data, these researchers developed an 2 empirical formula to determine drag on a sheet net. A seri e s of tank tests on c o n i c a l nets were c a r r i e d out by Yinggi Zhou [2]. Zhou also developed a method to p r e d i c t the drag force of the c o n i c a l net. A study of the water v e l o c i t y d i s t r i b u t i o n inside a c o n i c a l net body was conducted by Higo and Mouri (1975) [3]. The water v e l o c i t y inside the net was defined as a percentage of the upstream v e l o c i t y of water entering the net. The researchers found that the water v e l o c i t y inside the net v a r i e d from 94% to 111% upstream v e l o c i t y . Ferro and Stewart (1981) [4], reported that the drag of a net can be estimated by the simple formula below: Drag = m V n (1.2.1) In the equation above, V i s the upstream water speed, m and n are constants, which are a function of the net geometry. The values of n are s i g n i f i c a n t l y less than 2 i n a l l cases. Similar formulas were derived by Iraai and Marin (1978) [5], Mangunsukarto and Fuwa (1978) [6] and also Higo and Mouri (1975) [3]. Dickson (1980) [7] developed an a l t e r n a t i v e method to p r e d i c t the drag of the f i s h i n g net. His method includes the c a l c u l a t i o n of drag caused by the n e t t i n g bars and knots i n a towing net. His method provides a good agreement with measurements of the drag of f u l l scale trawls at sea. Similar work has been done by Wileman and Hansen (1988) [8]. 3 Dudko, Swiniarski, Przybyszewski, Kwidzinski, Nowakowski and Sendlak (1982) [9], reported the resistance of a net i s p r a c t i c a l l y independent of the e l l i p t i c a l base oblateness at the mouth wit h i n a range of 0.25 to 1.0. In other words, the drag of a net with e l l i p t i c a l mouth shape should have a s i m i l a r resistance value compared to a net of c i r c u l a r mouth shape having the same perimeter. Extensive research work has been c a r r i e d out at the Department of Mechanical Engineering of U.B.C.. The thrust of t h i s work has been i n the design of f i s h i n g nets, optimization of f i s h i n g gear, and the p r e d i c t i o n of drag on a trawl net. A computer algorithm which predicts the drag of a net based on the method developed by Kowalski and Giannotti (1974) [10] was w r i t t e n by Wang. The predicted r e s u l t s were confirmed experimentally by C a l i s a l , Mcllwaine, Wang and Fung (1984) [11]. I t has been determined that f or a t y p i c a l commercial f i s h i n g net, the drag of the codend may account for approximately 15% of the drag of the en t i r e net. Also, since the codend has a c y l i n d r i c a l shape, each succeeding mesh along the codend l i e s i n the wake of the previous mesh. This geometry causes the flow f i e l d s near and around the codend section to be much more complicated than any other section of the net. 1.3 PURPOSE OF THE PRESENT RESEARCH A more d e t a i l e d examination of the flow condition i n and around the f i s h i n g net i s necessary to understand the mechanism 4 associated with the drag force. The equations used i n the p r e d i c t i o n of the drag require the speed of the upstream flow, but not the v e l o c i t y of the f l u i d around the net. The previously developed drag equations suggest that the net drag i s proportional to the square of the upstream v e l o c i t y . A 10% error i n v e l o c i t y w i l l mean about 20% error i n the p r e d i c t i o n of drag. The goal of t h i s thesis i s to develop a numerical procedure which can p r e d i c t the flow f i e l d surrounding the net. 5 CHAPTER 2 NUMERICAL MODELLING OF A FISHING NET 2.1 INTRODUCTION A f i s h i n g net i s a very porous, f l e x i b l e structure. I t i s us u a l l y made up of several panels with d i f f e r e n t mesh geometry. The mesh geometry can be described by the following four parameters: (Figure 2.1.1) i ) twine diameter, d^. i i ) knot diameter, d . k i i i ) bar length, a. iv) h a l f s e t t i n g angle, a , which i s defined as the h a l f angle between the adjacent bars of an undeformed mesh. The materials used to make the twine are usually polyethylene, polyester, nylon or new nylon. Each material has d i f f e r e n t material properties associated with i t , e.g. s t i f f n e s s , water absorption a b i l i t y , etc. The twine i s eit h e r braided or twisted. Hence, the surface texture of the twine i s extremely i r r e g u l a r . In other words, the surface i s very rough. During trawling operation, the f i s h i n g net i s towed by one or two f i s h i n g vessels at a speed of two to four knots. The shape of the net while i t i s being towed under water i s more or less c o n i c a l with an e l l i p t i c a l mouth opening. The trawling operation causes the net to be put under a stress which cause meshes to 6 s t r e t c h lengthwise. This mesh str e t c h i n g causes the twine diameter to reduce. Also, because of the geometry of the net, the n e t t i n g s o l i d i t y increases towards the a f t end. Netting s o l i d i t y i s defined as the r a t i o of the open area to the t o t a l area of the nett i n g . Figure 2.1.2 shows the shape of a t y p i c a l trawl net model being tested i n the flume tank. 2.2 MOTION OF THE FLUID THROUGH THE NET While a net i s being towed, water flows through the meshes. Since the net or i t s elements acts as disturbance, a wake i s formed downstream while the upstream flow remains uniform. As the water passes through the net, the s t a t i c pressure i s reduced and the streamlines are deflected. While the component of v e l o c i t y i n the d i r e c t i o n normal to the net's surface i s kept constant, a d i s c o n t i n u i t y of the v e l o c i t y i n the d i r e c t i o n tangential to the net's p r o f i l e on the two sides i s expected. 2.3 NUMERICAL MODEL In order to numerically model the f l u i d flow through a net, a number of assumptions are made. Generally, the flow upstream of the net can be assumed to be uniform and i r r o t a t i o n a l , while i n the region surrounding the net and downstream of the net, the flow i s assumed r o t a t i o n a l because of the vortex shedding from the c i r c u l a r twines. One can model the net and f l u i d i n t e r a c t i o n by imagining the net to act as a continuous d i s t r i b u t i o n of sources and v o r t i c e s of very weak strength i n the approaching uniform flow. 7 Hence, the ent i r e flow f i e l d can be v i s u a l i z e d as made up of i r r o t a t i o n a l and r o t a t i o n a l flow. An ad d i t i o n a l assumption can also be made here. The r o t a t i o n a l flow i n the downstream region can be ignored because of low s o l i d i t y of f i s h i n g nets. Therefore, the ent i r e flow f i e l d can be represented by a p o t e n t i a l function, $. This p o t e n t i a l function must s a t i s f y the co n t i n u i t y equation for incompressible flow: V 2$ = 0 (2.3.1) which i s known as the Laplace Equation. Where, ax 2 ay 2 a z 2 $ = P o t e n t i a l value at any point The Laplace Equation can be solved with the appropriate boundary conditions. Any function which s a t i s f i e s the Laplace Equation i s a harmonic function. 2.4 BOUNDARY CONDITIONS The conditions that the numerical model must s a t i s f y are l i s t e d as below: i ) Continuity equation across the net and the boundaries of the co n t r o l domains. i i ) The v e l o c i t y along the net's p r o f i l e i n the d i r e c t i o n normal to the surfaces on two sides must be equal, i i i ) D i s c o n t i n u i t y of the v e l o c i t y p o t e n t i a l across the net, or di s c o n t i n u i t y of the v e l o c i t y component tangential to the net geometry i s permitted. 8 CHAPTER 3 M A T H E M A T I C A L F O R M U L A T I O N OF BOUNDARY E L E M E N T METHOD 3.1 SOLUTION METHODS OF THE LAPLACE EQUATION There are numerous methods a v a i l a b l e to solve the Laplace Equation with the appropriate boundary conditions. For a simple geometry, a n a l y t i c a l methods can be applied without many d i f f i c u l t i e s . The F i n i t e Element or F i n i t e Difference Method i s normally used when a more complicated geometry i s encountered. However, the computational cost involved i n these two methods are very expensive due to the large matrix s i z e involved. Besides these methods, the Boundary Element Method (BEM) i s an a d d i t i o n a l choice. This method has an advantage because of the considerably smaller matrix si z e i t requires compared to the F i n i t e Element Method or F i n i t e Difference Method. However, a disadvantage of the method i s that the BEM r e s u l t s i n a f u l l matrix to be solved. 3.2 ASSUMPTIONS Before the Boundary Element Method can be applied to the f i s h i n g net problem, c e r t a i n assumptions have to be made. These assumptions are l i s t e d below: i ) The p o t e n t i a l flow theory can be applied by neglecting the r o t a t i o n a l flow f i e l d i n the region surrounding the net, as well as downstream of the net, because of the weak r o t a t i o n a l solenoid f i e l d caused by the net. 9 i i ) According to Dudko, Swiniarski, Przybyszewski, Kwidzinski, Nowakowski and Sendlak (1982) [9], although the net has an e l l i p t i c a l cone shape when i t i s towed underwater, i t can be treated as an axisymetric cone shape having the same perimeter at the mouth. 3.3 GENERAL FORMULATION OF THE BOUNDARY ELEMENT METHOD Due to the axisymetric property of the problem, r i n g elements are chosen to represent the geometry of the net. The v a r i a t i o n of the p o t e n t i a l value along each element i s independent of the geometric angle of the element. Consider a c y l i n d r i c a l control domain with radius, R^, and the coordinate system as shown i n Figure 3.3.1. Any a r b i t r a r y point can be defined by (x, y, z) or (x, R, 8), where angle, 8, i s measured from the Y axis. The coordinate systems are r e l a t e d by. y = R cos 8 z = R s i n 8 (3.3.1) Point P i s c a l l e d the "point of i n t e r e s t " , i t i s the point where the p o t e n t i a l i s to be calculated. Point Q i s the "running point", i t i s a running parameter when computing the p o t e n t i a l value at P. While point P can be located inside, outside, or on the boundary of the cont r o l domain, point Q i s always located on the surface of the control domain. Let point P be described by (x p, Rp, 0) and point Q be defined by (x, Rcos 8, Rsin 8). The distance, r, between P and Q 10 i s obtained by: r - / ( X - x p ) 2 + (R cos 9 - R p ) 2 + (R s i n 6)2 (3.3.2) The normal unit vector pointing out of the con t r o l domain surface, n, i s described by: n = (n ,'n cos 6, n s i n 9) (3.3.3) x R R Brebbia (1978) [12], suggests a s o l u t i o n to the Laplace Equation using Green's i d e n t i t y , which i s given below: Cft$(P) + f S ( Q ) ^ ^ dS - f G ( P , Q ) ^ f ^ > dS (3.3.4) J s an J s an t t Where C i s a constant equal to 1 i n three dimensional cases, and i s equal to 1/2 i n two dimensional problems. When P i s located inside the con t r o l domain, ft i s set to 47r and when P i s outside the c o n t r o l domain, ft i s set to zero. I f P i s at the boundary, ft i s equal to 2n. G i s known as the Green's Function, and i s defined as 1/r i n three dimensional problems and l n ( l / r ) i n two -dimensional problems. i s the t o t a l surface area of the control domain. To perform the area i n t e g r a t i o n of Equation (3.3.4), a numerical computation i s generally used. A d i s c r e t i z a t i o n procedure i s employed to d i s c r e t i z e the en t i r e boundary into a series of r i n g elements. There are a v a r i e t y of elements that can be chosen, they are: i ) Constant element 11 The p o t e n t i a l and the p o t e n t i a l d e r i v a t i v e are assumed to be constant over the element, i i ) Linear element The v a r i a t i o n of the p o t e n t i a l and the p o t e n t i a l d e r i v a t i v e are assumed to be l i n e a r within each element. In the analysis conducted i n t h i s t h e s i s , a constant element i s chosen i n order to s i m p l i f y the computation. The mid-point of each element i s defined as the node point. For a three dimensional a p p l i c a t i o n with point P located on the boundary, Equation (3.3.4) can be rewritten as: 2TT$(P) + J " * ( Q ) - | ^ C-^-) dS = J dS (3.3.5) s s t t Suppose the e n t i r e surface of the c o n t r o l domain i s composed of N r i n g elements. There i s one equation s i m i l a r to the above equation for each element where P i s located. The whole system can be represented i n a form of a matrix equation with the siz e of N x N elements on the l e f t hand side as well as the r i g h t hand side. This matrix equation can be represented as: (see Appendix A) K ] h ] - [ B i J [ • • . J c 3 - 3 - 6 ) Where, i = 1 N j = 1. . . .N Note that only $ or has to be prescribed as a boundary 12 condition on the surface of the con t r o l domain. Hence, there are N unknowns i n the matrix equation mentioned above. Reordering the equations i n such a way that a l l the unknowns are on the l e f t hand side, Equation (3.3.6) can be rewritten i n the form shown below: A X = B Where X i - s the vector of unknowns $ and $, (3.3.7) Using the r e l a t i o n s h i p of r i n Equation (3.3.2), Equation (3.3.5) can be rewritten as: (see Appendix B) -R 2TT$(P) + * ( Q ) 2 n - 4 E ( — , 6 ) ((x - x )R n + R 2n ) 2 P x R (a - b) / a + b / 1 1 r + b L a E ( — ,6) a - b F( d l 1 a*( Q ) 4 R F ( — > 5 ) an ( a + b ) 1/2 d l (3.3.8) Where, E ( — — ,5) = Complete E l l i p t i c Function of f i r s t kind F(—^—,5) = Complete E l l i p t i c Function of second kind 2 a a + b Using the above formulation with the a p p l i c a t i o n of the constant element, the flow f i e l d around a sphere with a radius of 1.0 m inside a c y l i n d r i c a l c ontrol domain i s solved. Due to the axisymetric shape of the sphere and the con t r o l volume, only a quarter of the domain i s needed i n the c a l c u l a t i o n , as shown i n 13 Figure 3.3.2. For t h i s analysis, the v e l o c i t y at upstream boundary i s taken to be 1 m/s. The v e l o c i t y f i e l d inside the control domain as computed by the BEM i s g r a p h i c a l l y i l l u s t r a t e d as i n Figure 3.3.3. The v e l o c i t y along the downstream boundary of the control domain, shown as l i n e CD on Figure 3.3.2, has been c a l c u l a t e d using the BEM, and i s g r a p h i c a l l y i l l u s t r a t e d i n Figure 3.3.4. The r e s u l t s are compared with r e s u l t s obtained from the general a n a l y t i c a l formula which i s derived from the p o t e n t i a l flow theory of uniform flow past a sphere: U x (3.3.9) Where, U = Horizontal v e l o c i t y at the node point along CD r = Radius of the node point l y r = Radius of the sphere From the fi g u r e , the maximum difference of the v e l o c i t y along CD between the r e s u l t s obtained from the BEM and Equation (3.3.9) i s around 7%, except at the elements near the corners of the c o n t r o l domain. This discrepancy at the corners i s believed to be caused by the numerical inaccuracy occurring as the boundary experiences a sudden change of geometry. Brebbia (1978) [12] suggests t h i s problem can be improved by assuming there are two points very close to each other which belongs to d i f f e r e n t boundaries as shown i n Figure 3.3.5. By doing so, each element can have a d i f f e r e n t p o t e n t i a l or p o t e n t i a l d e r i v a t i v e . 14 Figure 3.3.6 shows the pressure c o e f f i c i e n t , C p, along the surface of the sphere. The r e s u l t s obtained from the BEM are compared with the r e s u l t s from the following a n a l y t i c a l formula, which i s also derived from the p o t e n t i a l theory: C = 1 - — s i n 2 ^ (3.3.10) P A Where, i/> = Angular measurement of the node point on the surface of the sphere with the X axis. In Figure 3.3.6, the maximum difference of C between the p two methods i s around 23%. This i s due to the v e l o c i t i e s along the surface of the sphere being over predicted by the BEM. In order to improve the accuracy of the r e s u l t s , a smaller element s i z e i s recommended by Chan (1984) [13]. 15 CHAPTER 4 MATHEMATICAL FORMULATION OF AN IRROTATIONAL WAKE MODEL 4.1 WAKE MODEL Due to the disturbance of the net, a d i s c o n t i n u i t y of p o t e n t i a l occurs across the net. This d i s c o n t i n u i t y leads to a wake being formed at the downstream flow f i e l d . The d i s c o n t i n u i t y of the p o t e n t i a l generally depends on the geometry of the mesh, the net material, the f l u i d properties, and the v e l o c i t y as we l l . However, i n order to si m p l i f y the problem, the in v e s t i g a t i o n of the dependence of t h i s d i s c o n t i n u i t y along the net's p r o f i l e w i l l not be c a r r i e d out. A l t e r n a t i v e l y , the di s c o n t i n u i t y of the p o t e n t i a l across the net w i l l be represented by a function which i s based on the coordinates of the net i t s e l f . The objectives here are to v e r i f y the a p p l i c a b i l i t y of the Boundary Element Method to t h i s problem, and to investigate whether an i r r o t a t i o n a l wake model can be constructed using t h i s procedure. I t i s convenient to represent the flow f i e l d s i n and outside the net as two d i s t i n c t regions. As shown i n Figure 4.1.1, the region outside the net i s named Region 1, and the region inside the net i s named Region 2. A c y l i n d r i c a l outer boundary, with radius R^, i s chosen to define the global control volume. This global control volume contains a smaller control domain, Region 2, which represents the i n t e r n a l region of the net. The net i s represented by the cont r o l boundary, AB, d i v i d i n g 16 the two regions. A d i s c o n t i n u i t y of the v e l o c i t y i n tangential d i r e c t i o n , as well as a d i s c o n t i n u i t y of pressure on the two sides of the net i s permitted. 4.2 BOUNDARY CONDITIONS Before s o l v i n g the Laplace Equation, the boundary conditions enforced i n the problem are: i ) Assume the p o t e n t i a l value along the downstream boundary, S , i s constant, and i t can be represented by any a r b i t r a r y constant value; <E> - 1.0 i i ) Assume the order of magnitude of the v e l o c i t y i n the d i r e c t i o n normal to the surface of the outer co n t r o l domain, S^, i s small compared with the v e l o c i t y i n the tangential d i r e c t i o n . Therefore, a non-permeable boundary condition can be assumed, i . e . $ = 0.0 i i i ) Assume the p o t e n t i a l d e r i v a t i v e along the upstream boundary, S 3, i s a negative incoming v e l o c i t y , i . e . $ = - V iv) There i s no d i s c o n t i n u i t y of p o t e n t i a l along the a r t i f i c i a l boundary, S^  , i . e . $ - $ = 0.0 R e g i o n 1 Reg ion 2 v) The boundary condition along the net, S^, can be represented by the p o t e n t i a l difference across the net. This p o t e n t i a l d i f f e r e n c e can be represented by a function, i . e . $ - $ * 0.0 R e g i o n 1 Reg ion 2 17 4.3 MATHEMATICAL FORMULATION In t h i s problem, there are two d i f f e r e n t c o n t r o l domains, where the common boundaries of these two regions are AB and BC. One equation, s i m i l a r to Equation (3.3.5), can be written for each region. For Region 1, the following equation i s obtained: (See Appendix C) 2TT$(P ) + * ( Q ) 0 - 4 E ( — ,6) ((x - x )R n + R 2n ) 2 P x R (a - b) v^~a~T" 2 n r i + b L a E(-=-,fi) -T-h n^r,s) a - b 2 ] dy * ( Q ) 2 n 4 E ( — , 5 ) ((x - x )R n + R 2n ) 2 P x R (a - b) / a + b ' u r a E ( — — ,6) -i d l * ( Q ) 2 n 4 E ( — , 5 ) ((x - x )R n + R 2n ) 2 P x R (a - b) / a + b ' r a E ( — ,6) -i FF~ [ - r * r - - «-i-'> ] dy * ( Q ) 2 n 4 E ( — , 5 ) ((x - x )R n + R 2n ) 2 P x R (a - b) / a + b  r a E( ,6) a - b • ^ • « ) + b L d l 18 * ( Q ) 4 E(-^-,6) ((x - x )R n + R 2n ) 2 P x B. (a - b) / a + b ' 2 n r a E( ,6) n d l ^ a» ( Q) 4 R F(Hh6) 3n 0 .0 L y /.X ( a + b ) 1/2 a * ( Q ) , « ) an ( a + b ) 1 / 2 dy d l 1 a*(Q) 4 R F ( — ' 5 ) 3n ( a + b ) 1/2 dy 1 a*( Q) 4 R F ( — - 5 ) an ( a + b ) 1 / 2 d l * 2 a<KQ) 4 R F H b f i > an ( a + b ) 1/2 d l (4.3.1) S i m i l a r l y , the following equation i s obtained f o r Region 2. 2TT$(P ) + * ( Q ) x 4 E(—,6") ((x - x )R n + R 2n ) 2 P x R (a - b) / a + b 2 n r i + b L a E ( — , 6 ) a T ^ b F ^ ' 5 ) d l ,0 * ( Q ) 4 E(-^-,S) ((x - x )R n + R 2n ) Z P X R (a - b) / a + b 19 2 n * ( Q ) 2 n a - b d l 4 E ( — ,5) <<x - x )R n + R 2n ) 2 P x R (a - b) / a + b 1 1 r + b L a E(-^-,S) a - b 2 dy .0 .0 3n ( a + b ) 1 / 2 . a*(Q) 4 R F(-^-,5) an ( a + b ) 1 / 2 a*(Q) 4 R F ( ^ - , 6 ) 3n ( a + b ) 1 / 2 d l d l dy (4.3.2) Equations (4.3.1) and (4.3.2) can be written i n a matrix equation form s i m i l a r to Equation (3.3.6). These matrix equations are l i s t e d below: (4.3.3) (4.3.4) Since there are common boundaries between the two regions, a method of matching the $ and $, must be employed i n order to solve the above matrix equations. 20 4.4 MATCHING TECHNIQUE The c o n t r o l domain i s d i s c r e t i z e d into a t o t a l of N T 1 elements on Region 1, and a t o t a l of N elements on Region 2, while along the common boundaries, AC, there are a t o t a l of N^ elements. This set up i s i l l u s t r a t e d on Figure 4.1.1. For Region 1, there are N plus N unknowns, and for ° ' T l r c Region 2, there are N plus N unknowns. However, the N & T2 r c c unknowns are common to both regions. Therefore, the t o t a l number of unknowns of $ and i s equal to the sum of N , N and 2N . n T 1 T2 c Af t e r combining the two matrix equations, (4.3.3) and (4.3.4), the combined matrix equation i s reordered i n such a way that the $ and values along the common boundaries are matched. The n following matrix equation i s obtained: (see Appendix D) N T 1 N T 2 N T 1 C i j -D i j j N T2 E i j -F i j $ j N 1 1 - 1 - 1 n N"~" 5 1 1 - 1 - 1 N 5 1 1 1 1 n N l l 1 1 n N 4 N 5 N 5 N 4 N it N 5 N 5 N 4 N + N + N 1 2 3 N 1J 0 Di f f 0 0 n N l n N 2 -n N 3 n N 6 (4.3.5) According to the boundary conditions set out i n 4.2, the column vector on the l e f t hand side of Equation (4.3.5) consists of the prescribed p o t e n t i a l values as well as the unknowns. Hence, i t i s required that the matrix be rearranged i n such a way that a l l the unknown values are located on the column vector of 21 the l e f t hand side of Equation (4.3.5). Therefore, an equation s i m i l a r to Equation (3.3.7) can be obtained: AX = B 22 CHAPTER 5 RESULTS OF IRROTATTONAL WAKE MODEL 5.1 INTRODUCTION In t h i s chapter, the r e s u l t s of the s i x tes t cases are discussed. Three of the tes t cases are without d i s c o n t i n u i t y of p o t e n t i a l across the boundary AB, shown i n Figure 4.1.1. The other three t e s t cases have a d i s c o n t i n u i t y i n p o t e n t i a l across the boundary. Three p r o f i l e s of AB are investigated, these are: i ) v e r t i c a l i i ) 90 degrees arc i i i ) 60 degrees with X axis Each of the p r o f i l e s mentioned above i s tested with and without a p o t e n t i a l d i s c o n t i n u i t y across the boundary AB. The upstream v e l o c i t y i s taken to be 2 m/s for a l l t e s t cases. The reason to include the tes t cases without the p o t e n t i a l d i f f e r e n c e across the boundary AB i s to ensure that the computer program i s correct before i t i s applied to the l a t e r analysis. 5.2 RESULTS WITHOUT POTENTIAL DISCONTINUITY The r e s u l t s of $ and computed by the Boundary Element n Method for the v e r t i c a l , 90 degrees arc, and 60 degrees boundary p r o f i l e s are gr a p h i c a l l y i l l u s t r a t e d i n Figures 5.2.1, 5.2.4 and 5.2.7 r e s p e c t i v e l y . Also included i n these figures, are the v e l o c i t y vectors inside the cont r o l domains; Comparison of the normal v e l o c i t y along the net with the a n a l y t i c a l r e s u l t s of the tested cases are shown i n Figures 5.2.2, 5.2.5, and 5.2.8. While comparison of the v e l o c i t y at the downstream boundary with the a n a l y t i c a l r e s u l t s are shown i n Figures 5.2.3, 5.2.6 and 5.2.9. A l l of the r e s u l t s have a very good agreement with the t h e o r e t i c a l values, except at the turning points of the boundaries where some numerical inaccuracy has occurred. 5.3 RESULTS WITH POTENTIAL DISCONTINUITY The functions to represent the p o t e n t i a l d i s c o n t i n u i t y across the boundary AB, are shown below: i ) V e r t i c a l $ = 0 . 7 y 2 + 0 . 3 y + 0 . 5 D IFF J J i i ) 90 degrees arc $ = 0.2 x 2 + 0.03 x - 0.01 DIFF i i i ) 60 degrees with X axis S> = 1 . 2 x 2 + 0 . 6 x DIFF Where x and y are the X and Y coordinates of the node point of the elements along the boundary. There i s no s p e c i f i c reason to represent the p o t e n t i a l d i s c o n t i n u i t y across the boundary AB i n the above formats. As mentioned i n the previous chapter, the d i s c o n t i n u i t y of the p o t e n t i a l across the net depends on a large number of parameters. Therefore, i t i s not impossible to represent the p o t e n t i a l d i f f e r e n c e across the net by the formats stated above or other 24 formats. Using these functions for the p o t e n t i a l d i s c o n t i n u i t y across the boundary, the BEM was used to determine the $ values, the $, values, and the v e l o c i t y vectors for the c o n t r o l domains with a v e r t i c a l , 90 degrees arc, and 60 degrees domain boundary. These r e s u l t s are p l o t t e d and shown i n Figures 5.3.1, 5.3.2 and 5.3.3 r e s p e c t i v e l y . From each of the three cases, one can notice the formation of a wake at the downstream end as shown i n each f i g u r e . Also i n Figures 5.3.1 and 5.3.3, one can notice that vortex l i k e flows are formed i n the flow f i e l d i n a d d i t i o n to the formation of a wake. 5.4 DISCUSSION Even though there i s a lack of a n a l y t i c a l or experimental r e s u l t s to compare with, one can conclude from the graphical r e s u l t s that i f a proper d e s c r i p t i o n of the p o t e n t i a l d i s c o n t i n u i t y along the net i s known, then the BEM can be used to s u c c e s s f u l l y simulate the flow f i e l d s near and around the net. In the next chapter, the representation of the p o t e n t i a l d i s c o n t i n u i t y across the net i s investigated. 25 CHAPTER 6 MATHEMATICAL FORMULATION OF THE NET MODEL 6.1 THE DETERMINATION OF FORCES ACTING ON THE NET Through pri v a t e communication between Crewe, P. of B r i t i s h Hovercraft Corporation and Dr. S.M. C a l i s a l of the Mechanical Engineering Department, U.B.C., the forces acting on a u n i t of netting, ( i . e . 1 knot and 4 h a l f bars) as shown i n Figure 6.1.1, are given below: F = p a d V 2 C (6.1.1) i t i where, F = Force i n the i d i r e c t i o n ( i = 1, 2, 3) i p = Density of water C = Non-dimensional force c o e f f i c i e n t s . i From the analysis of Crewe, the following equation can be derived from Equation (6.1.1): (See Appendix E) F = 1.03 F p a d V 2 1 c t t F = 1.03 (F cos a - F s i n a) p a d V 2 2 c n c s t F = 1.03 UF cos a + F s i n a) 3 I C S CD {d s i n al /• d \ 2 t 1 k l „ 2 p a d V (6.1.2) t Where, a = Angle of incidence F = Force a r i s i n g from the bars alone i n t d i r e c t i o n c t F = Force a r i s i n g from the bars alone i n n d i r e c t i o n c n 26 F = Force a r i s i n g from the bars alone i n s d i r e c t i o n c s 6.2 ALGORITHM OF PRESSURE DROP ACROSS A NET Aft e r the forces acting on the net are obtained, the pressure drop across the net can be ca l c u l a t e d by: F. A P = ——-— (6.2.1) i Area Where, A P = Pressure drop i n the i d i r e c t i o n ( i = 1, 2, 3) i Area = Area of a unit n e t t i n g ( i . e . 1 knot and 4 h a l f bars) The pressure drop across the net i n the normal d i r e c t i o n , A P , can then be obtained as: (See Appendix F) n A P = A P s i n a + AP cos o (6.2.2) n 3 2 Also, the t o t a l drag of the net can be represented by the summation of the forces i n d i r e c t i o n i = 3, i . e . : To t a l Drag - E F (6.2.3) 6.3 DETERMINATION OF THE VELOCITY COMPONENTS The reduction i n s t a t i c pressure when water passes through the net i s usu a l l y expressed by a dimensionless pressure drop c o e f f i c i e n t , K, which i s defined as: A P K (6.3.1) p V 2 n Where V i s the v e l o c i t y i n the normal d i r e c t i o n to the net n geometry. 27 Since the net i s a very porous structure, to a f i r s t degree of approximation, the normal v e l o c i t y , V , can be written as: ii V = V s i n a (6.3.2) n Hence, Equation (6.3.1) can then be rewritten as: AP K = (6.3.3) 1 -.2 . 2 p V s i n a 2 A f t e r the pressure drop across the net i n the normal d i r e c t i o n i s obtained from Equation (6.2.2), the pressure drop c o e f f i c i e n t , K, can be c a l c u l a t e d from Equation (6.3.3). McCarthy (1964) [14] introduced a non-dimensional r e f r a c t i o n c o e f f i c i e n t , B, to describe the r e l a t i o n s h i p between the tangential v e l o c i t i e s on both sides of the net. This r e f r a c t i o n c o e f f i c i e n t i s expressed as: V B (6.3.4) s i Where, V = Tangential v e l o c i t y i n front of the net V = Tangential v e l o c i t y behind the net He also showed that the pressure drop c o e f f i c i e n t , K, and the r e f r a c t i o n c o e f f i c i e n t , B, has a r e l a t i o n s h i p expressed as: B — (6.3.5) A T K I f ones assumes that the tangential v e l o c i t y i n front of the net can be represented by: 28 V s i = V cos a (6.3.6) Then, from Equation (6.3.4), one obtains, V S2 = B V s 1 (6.3.7) Once the r e f r a c t i o n c o e f f i c i e n t , B, i s obtained from Equation (6.3.5), the tangential v e l o c i t y behind the net i s r e a d i l y c a l c u l a t e d by Equation (6.3.7). 6.4 ALGORITHM FOR REPRESENTING POTENTIAL DISCONTINUITY Consider a t y p i c a l element along the net, as shown i n Figure 6.4.1. Due to the difference i n tangential v e l o c i t y on the two sides, a c i r c u l a t i o n can be expected on the element. This c i r c u l a t i o n , T, can be represented by: Where c i s the contour of the surface, and u can be expressed as the gradient of the p o t e n t i a l , Using the c i r c u l a t i o n theorem, the i n t e g r a l between any two points, A and B, on the surface can be written as: In t h i s problem, the l a s t term on the r i g h t hand side of Equation (6.4.2) can be ignored due to axisymetry of the net, and so Equation (6.4.2) can be expressed as: u dc (6.4.1) (6.4.2) r B - $ A (6.4.3) 29 Where $ i s the p o t e n t i a l at point A, and $ i s the p o t e n t i a l at A B point B. I f point A i s the node point of the element on the i n s i d e surface, and point B i s the node point on the outside surface of the same element,then the p o t e n t i a l d i f f e r e n c e , $ r DIFF across the element can be r e l a t e d by the c i r c u l a t i o n , T, on the surfaces of the element as shown i n Equation (6.4.3). From Equations (6.4.2) and (6.4.3), the following equation can be obtained: (6.4.4) $ = (V - V ) Ax + ( V - V ) Ay DIFF X2 XI Y2 Y l J Where, DIFF B A and V and V are the v e l o c i t y components on the outside X 1 Y1 J r surface, and V and V are the v e l o c i t y components on the X2 Y2 J f inside surface of the element, as shown i n Figure 6.4.2. Hence, the d i s c o n t i n u i t y of the p o t e n t i a l across the net on each element i s r e a d i l y obtained from Equation (6.4.4). A f t e r the v e l o c i t y i n the normal d i r e c t i o n along the net, V , and the tangential v e l o c i t i e s on both sides of the net, V n SI and V ^ , are obtained from the Equation (6.3.2), (6.3.6) and (6.3.7) r e s p e c t i v e l y . Then the v e l o c i t y components i n the normal d i r e c t i o n and the tangential d i r e c t i o n are r e l a t e d to the v e l o c i t y components i n the X and Y d i r e c t i o n by the following equations: ' V X 1 V Y 1 J c o s a s i n a ' V S 1 - s i n a cos a V n (6.4.5) 30 ' V X2 V L Y2 J cos a s i n a " V S2 - s i n a cos a V n (6.4.6) By s u b s t i t u t i n g the p o t e n t i a l difference, $ , into the J b DIFF combined matrix Equation (4.3.5) with the boundary conditions ( i ) , ( i i ) , ( i i i ) and (iv) found i n Section 4.2, a numerical model of the net can be set up. When the attack angle, a, i s equal to zero degrees, then the above formulation w i l l break down, as no c i r c u l a t i o n w i l l form on the element. Hence, the attack angle, a, must always be d i f f e r e n t from zero degrees. The t o t a l drag of the net must be balanced by an equal and opposite force acting on the f l u i d . This drag force r e s u l t s from the pressure on the upstream and downstream boundaries plus the net rate of f l u x of momentum across the boundaries. The t o t a l drag of the numerical model can be ca l c u l a t e d based on the wake model formulation, and i s written i n the following form: To t a l drag = J ( P q + p V 2) dA - J - J p U y V dA ( P + p U ) dA o x (6.4.7) Where, P = Undisturbed s t a t i c pressure V = Incoming v e l o c i t y at upstream U^ = V e l o c i t y component i n X d i r e c t i o n U^ = V e l o c i t y component i n Y d i r e c t i o n 31 However, the continuity equation across the that: j" p V d A + J p U x d A = | p U y dA Hence, Equation (6.4.7) can be rewritten as Tot a l Drag = f p U (V - U ) dA boundaries requires (6.4.8) (6.4.9) 32 CHAPTER 7 CONICAL NET EXPERIMENT 7.1 INTRODUCTION An experiment to determine the flow f i e l d inside a c o n i c a l net, as well as the shape of the net under flowing water has been c a r r i e d out i n the flume tank of the Marine I n s t i t u t e located at St. John's, Newfoundland. The i n t e r i o r view of the establishment i s shown i n Figure 7.1.1. The flume tank has a working section 21.5 metres long, 8.0 metres wide, and 4.0 metres i n depth. The maximum water v e l o c i t y of the flume tank i s 1.0 metre per second. The tested c o n i c a l net i s constructed of 4 panels made with braided nylon knotless meshes. The net was mounted on a c i r c u l a r hoop with a diameter of 1.354 metre. Floats were attached to the hoop i n order to ensure a neutral buoyancy. As shown i n Figure 7.1.2, each panel had 41 meshes at the mouth, 23 meshes at the end, and 72 meshes along the length. The meshes had a bar length of 28.5 mm and the twine diameter was 1.85 mm. The c o n i c a l net was tested with the water v e l o c i t y set at 0.4, 0.6, 0.8 and 1.0 m/s. A f t e r the net was tested i n the flume tank, i t was tested also i n the towing tank of B r i t i s h Columbia Research i n order to confirm the r e s u l t s of the drag force obtained from the o r i g i n a l flume tank t e s t s . 33 7.2 MEASUREMENT OF THE CONICAL NET PROFILE During tests i n the flume tank at the Marine I n s t i t u t e , the net p r o f i l e was measured for the various flow v e l o c i t i e s . In order to obtain the p r o f i l e of the c o n i c a l net at the d i f f e r e n t t e s t v e l o c i t i e s , the net was divided into 10 stat i o n s , each s t a t i o n being 8 meshes apart. Then, the coordinates of each s t a t i o n were measured by a coordinate measurement device i n s t a l l e d i n the flume tank. This device consists of 3 cameras with a cross h a i r moving along the flume tank. 7.3 MEASUREMENT OF THE VELOCITY INSIDE THE NET The water v e l o c i t y inside the net was measured by a pr o p e l l e r type current meter (see Figure 7.3.1). The p o s i t i o n of the current meter was measured by the coordinate " measurement device. The p r o p e l l e r speed reading of the current meter can be converted to f l u i d v e l o c i t y by using the c a l i b r a t i o n graph of the current meter, which i s shown i n Figure 7.3.2. 7.4 OBSERVATION OF THE FLOW FIELD AROUND THE NET In order to observe the flow f i e l d close to the netting, " T e l l - T a i l s " , which are j u s t pieces of wool t i e d to the net, were attached to the ne t t i n g at various p o s i t i o n s along the net as shown i n Figure 7.4.1. The r e l a t i v e angle of the " T e l l - T a i l s " , as well as t h e i r r e l a t i v e motions, would indicate what type of flow f i e l d e x i s t s around the net. 34 7.5 MEASUREMENT OF THE DRAG FORCE For the d i f f e r e n t flow v e l o c i t i e s , the t o t a l drag force of the c o n i c a l net, hoop, f l o a t s , and the tow l i n e was measured by a force transducer attached to the tow l i n e . The angle of the tow l i n e to the v e r t i c a l d i r e c t i o n was also measured. In order to obtain the drag force of the net t i n g i t s e l f , the drag force of the hoop, f l o a t s and the tow l i n e must f i r s t be measured. This was done by dismounted the netting from the hoop and measuring the drag force on the e x i s t i n g equipment without the net, as well as the angle of the tow l i n e , at the water v e l o c i t i e s of 0.4, 0.6, 0.8 and 1.0 m/s. Then the drag force of the net t i n g alone i s ca l c u l a t e d by subtracted the measured drag force without n e t t i n g from the t o t a l drag force measured with the netti n g . I t i s assumed that the interference from the hoop, f l o a t s and the tow l i n e i s n e g l i g i b l e . 35 CHAPTER 8 RESULTS OF THE CONICAL NET EXPERIMENT 8.1 PROFILE OF THE CONICAL NET For each of the flow v e l o c i t i e s , the coordinates of each s t a t i o n were measured and are l i s t e d i n Appendix G. Also, the p r o f i l e of the net at the various tested speeds are g r a p h i c a l l y p l o t t e d i n Figure 8.1.1. These plo t s have been smoothed by a f i f t h order parametric curve smoothing function to give a smooth p r o f i l e of the net. From Figure 8.1.1, one can see that the length of the net becomes longer at a higher v e l o c i t y , as well as the net becomes more symmetrical. I t i s believed that at low speeds, the gravity force p u l l s the net downward, causing the net to become less symmetrical. From the photographs shown i n Figure 8.1.2, the meshes appear more open at the entry of the net and close towards the end of the net. 8.2 FLOW FIELD INSIDE THE NET The measured water v e l o c i t i e s inside the net, as well as the p o s i t i o n s of the current meter, are shown i n Appendix H. From the r e s u l t s , i t seems that the magnitude of the water v e l o c i t y inside the net are randomly d i s t r i b u t e d but c o n s i s t e n t l y le s s than the incoming v e l o c i t y by 8.5% to 18.9%. This i s 36 contrary to the report by Higo and Mouri (1975) [3], which states that the v e l o c i t y inside a co n i c a l net ranges from 94% to 111%. These researchers also concluded that the water v e l o c i t y inside the net i s influenced by the r e l a t i o n s h i p between the diameter of the twine and the mesh s i z e . Hence, d i f f e r e n t mesh sizes and twines w i l l produce d i f f e r e n t r e s u l t s i n the flow f i e l d inside the net. This may account for the difference i n the r e s u l t s found i n the experiment compared to the r e s u l t s of Higo and Mouri. 8.3 FLOW FIELD AROUND THE NET During the experimental t e s t s , i t was observed that the flow was not s i g n i f i c a n t l y disturbed by the net t i n g as the water passes through the meshes. From the photographs shown i n Figure 8.3.1, the " T e l l - T a i l s " maintain a h o r i z o n t a l p o s i t i o n except at the rear part of the net. At t h i s part of the net, a larger degree of turbulence i s expected due to the smaller mesh openings. 8.4 DRAG FORCE OF THE CONICAL NET For the various tested v e l o c i t i e s i n the flume tank, as well as i n the towing tank, the t o t a l drag forces of the net, incl u d i n g the netting, hoop, f l o a t s and tow l i n e , were measured and l i s t e d i n Appendix I. Also included i n the Appendix I, are the measured drag forces excluding the netting. The angles of the tow l i n e to the v e r t i c a l d i r e c t i o n of both cases f o r the flume tank experiment, and the ca l c u l a t e d drag force of the netting alone from the flume tank, and the towing tank, are also 37 included. 38 CHAPTER 9 RESULTS OF THE NUMERICAL NET MODEL 9.1 INTRODUCTION In t h i s chapter, the numerical net model obtained from the Boundary Element Method i s tested by using the p r o f i l e s obtained during the flume tank t e s t i n g . The p r o f i l e s of the net were d i s c r e t i z e d and used for the input of the net model program. The mesh angle of each element was then c a l c u l a t e d by the program i t s e l f . The p o t e n t i a l difference of each element along the net can then be ca l c u l a t e d by Equation (6.4.4). 9.2 RESULTS The c a l c u l a t e d $ and $, values, as well as the v e l o c i t y n vectors inside the control domains at upstream v e l o c i t i e s equal to 0.4, 0.6, 0.8 and 1.0 m/s are shown i n Figures 9.2.1, 9.2.2, 9.2.3 and 9.2.4 res p e c t i v e l y . From the figures, one can observe that the flow f i e l d near the region close to the net and at the end of the net i s i r r e g u l a r , while i n the regions further . from the net, the flow f i e l d i s r e l a t i v e l y uniform. The v e l o c i t y i n the normal d i r e c t i o n along the net at the tested speeds, obtained by both Equation (6.3.2) and the BEM, are p l o t t e d i n Figures 9.2.5, 9.2.6, 9.2.7 and 9.2.8. The tangential v e l o c i t y on the inside and outside surfaces of the net obtained by the BEM are compared with the values c a l c u l a t e d by Equations (6.3.6) and (6.3.7). These r e s u l t s , 39 computed at various flow v e l o c i t i e s are shown i n Figures 9.2.9, 9.2.10, 9.2.11 and 9.2.12. The pressure drop across the net cal c u l a t e d by the BEM i s also compared with the r e s u l t s obtained from Crewe's formulation, and are shown i n Figures 9.2.13, 9.2.14, 9.2.15 and 9.2.16. The drag force obtained from the various empirical methods, the BEM, and the experimental r e s u l t s are shown i n Figure 9.2.17. 40 CHAPTER 10 DISCUSSION AND CONCLUSION A numerical model to p r e d i c t the drag force of a c o n i c a l net and also to obtain the flow f i e l d around the net based on the p o t e n t i a l flow theory i s developed. This numerical method requires the knowledge of the shape of the net underwater so i t can be input into the program. Then the p o t e n t i a l d i f f e r e n c e across the net can be obtained from Equation (6.4.4), which was derived from the forces acting on a u n i t of netting. A p r i n c i p a l assumption i s used i n t h i s numerical procedure, whereby the normal v e l o c i t y along the net, and the tangential v e l o c i t y i n f r o n t of the net, are approximated by the v e l o c i t y components of the u n d i s t r i b u t e d flow as stated i n Equation (6.3.2) and Equation (6.3.6). This assumption i s p r a c t i c a l i f the net has a very coarse structure. Hence, as the water passes through the net, the water v e l o c i t y w i l l not increase by a s i g n i f i c a n t amount due to the blockage e f f e c t of the meshes. In Figure 9.2.5, 9.2.6, 9.2.7 and 9.2.8, the normal v e l o c i t y obtained by BEM along the net i s observed to fl u c t u a t e , e s p e c i a l l y near the end of the c o n i c a l net. Yet i t seems that the normal v e l o c i t y follows the trend of the value from Equation (6.3.2). The tangential v e l o c i t i e s on the two surfaces of the net c a l c u l a t e d by the BEM are shown i n Figure 9.2.9, 9.2.10, 9.2.11 and 9.2.12. From the figures, the tangential v e l o c i t i e s e x h i b i t a s i m i l a r f l u c t u a t i o n problem. The f l u c t u a t i o n behavior of the 41 pressure drop across the net i s shown i n Figure 9.2.13, 9.2.14, 9.2.15 and 9.2.16. This f l u c t u a t i o n behavior i s the r e s u l t from the f l u c t u a t i o n phenomenon of the v e l o c i t y components. Since the pressure drop across the net i s the function of the v e l o c i t y components on the two surfaces of an element along the net, the pressure w i l l f l u c t u a t e as the v e l o c i t y f l u c t u a t e s . The f l u c t u a t i o n behavior of the s o l u t i o n i s more pronounced at the end of the net, and i s a r e s u l t of the high s o l i d i t y at that region. At high s o l i d i t y , the assumption that the water v e l o c i t y w i l l not change by a s i g n i f i c a n t amount as i t passes through the net may not be v a l i d . However, the f l u c t u a t i o n behavior of the s o l u t i o n at the other region may be a t t r i b u t e d to the si z e of the elements or the type of element being used. This problem has also been encountered by Chan (1984) [13]. He suggested a possible explanation which i s r e l a t e d to the size and the type of element. When the element s i z e becomes larger, the constant p o t e n t i a l or p o t e n t i a l d e r i v a t i v e can no longer represent the actual value on the surface of the body. This i s e s p e c i a l l y true f o r an object with complicated geometry. Therefore, i n order to suppress the f l u c t u a t i o n phenomenon, a f i n e r element siz e or more sophisticated type of element i s recommended..However, t h i s w i l l r e s u l t i n a longer computation time and require more memory space of the computer. The drag forces computed by the various methods are shown i n Figure 9.2.17. One can observe a large discrepancy between the 42 r e s u l t s of the BEM and the other methods. I t i s i n t e r e s t i n g to point out that the drag value of the co n i c a l net tested i n the flume tank and the towing tank has a difference of around 30%. For t h i s studied c o n i c a l net, the Reynolds Number based on a smooth c i r c u l a r c y l i n d e r having the same diameter as the twine diameter, f o r the tested speed range, i s about 700 to 1800. From published Reynolds charts, the c o e f f i c i e n t of drag f o r the twine can be assumed constant through out t h i s speed range. Hence, the drag force of the ne t t i n g i s a function of the surface area of the c o n i c a l net alone. Based on t h i s argument, the shape of the co n i c a l net i n the flume tank i s d i f f e r e n t from the towing tank. The only explanation i s the difference of the flow f i e l d between the towing tank and the flume tank. The flow f i e l d of the flume tank i s not uniform, the v e l o c i t y gradient i s changing throughout the flume tank. Hence, the shape of the net may change as we l l . This might a f f e c t the flow around the co n i c a l net, and lead to a large discrepancy between the r e s u l t s of the drag value between the two tanks. I t would be valuable to record the shape of the net i n the towing tank and then compare i t with the p r o f i l e from the flume tank. The reasons of the large discrepancy of the drag value compared with the other methods may be quite complicated. Usually, the researchers carry out a serie s of experiments f o r a p a r t i c u l a r type of f i s h i n g net, and then develop a formula or method to p r e d i c t the drag force based on the experimental data. Since the drag force of the n e t t i n g and the flow f i e l d around a 43 f i s h i n g net are dependent on a large number of parameters, t h i s method of formulation may r e s t r i c t the a p p l i c a b i l i t y of the method to other kinds of f i s h i n g nets. The most important parameters are: water v e l o c i t y , type of material, method of net making, type of knot i n the net panel, si z e of mesh, h a l f s e t t i n g angle, i n c l i n a t i o n of net panel to the water flow, and twine diameter. The e f f e c t of some of the parameters on the drag force and the flow f i e l d around a net i s not quite understood by the researchers. A comprehensive method to p r e d i c t the drag force of the d i f f e r e n t types of f i s h i n g net i s almost impossible before the e f f e c t of a l l these parameters i s well understood. For example, nets made out of d i f f e r e n t materials may have d i f f e r e n t p r o f i l e s underwater, even though the nets may have the same mesh geometry and same panel geometry. The p r o f i l e of a net made from a s t i f f e r material may not change the net geometry dramatically compared to a net with a les s s t i f f e r material when increasing the towing speed. I t i s known that the shape of the net d e f i n i t e l y w i l l a f f e c t the value of the drag force and the flow f i e l d around the net. This i s because the surface area of the netting, the mesh opening, and the attack angle of each panel are d i f f e r e n t . In most cases, the mesh opening and the attack angle along the same section are d i f f e r e n t from place to place. A less s t i f f e r material and a smaller twine diameter may increase the v i b r a t i o n of the twine, and could lead to a higher turbulence l e v e l i n the flow f i e l d . I t has s i m i l a r e f f e c t when the mesh size i s increased. The l e v e l of v i b r a t i o n i s dependent on the v e l o c i t y 44 as w e l l . The surface roughness of the twine w i l l have an e f f e c t on the drag force. As the surface roughness increases, a higher value of the skin f r i c t i o n i s usually expected, and a higher l e v e l of turbulence generated i n the flow f i e l d i s also expected. Wileman and Hansen (1988) [8] found that an extremely high drag c o e f f i c i e n t of the netting when one set of bars l i e s at a r i g h t angle to the water flow. An increase i n the drag value i s then expected when the meshes are more open i n the water since the bars l i e more across the water flow. I f the material has a higher degree of water absorption a b i l i t y , the absorbed water may increase the twine diameter s i g n i f i c a n t l y . This w i l l r e s u l t i n a higher drag value, and at the same time, may change the net geometry as w e l l . In addition, the surface of the f i s h i n g net i n the numerical model i s smooth and axisymetric, the attack angle and the mesh opening are constant along the section. However, i n r e a l i t y , the surface of the net i s i r r e g u l a r , the angle of attack and the mesh opening are d i f f e r e n t through out the surface of the netting, as shown i n Figure 10.1. This may cause a more complicated flow f i e l d around the net, and generate a higher turbulence l e v e l i n the flow f i e l d . Due to the i r r e g u l a r i t y of the surface, a larger surface area of the netting compared with the numerical model i s expected. Also, i n the c a l c u l a t i o n of the forces acting on an u n i t n e tting for the numerical model, the skew angle, •d, i s assumed to be zero degrees. However, i n a r e a l i s t i c case, the skew angle w i l l vary continuously through out 45 the surface of the net. The skew angle of each mesh i s e n t i r e l y dependent on how the net takes shape underwater, and t h i s makes the c a l c u l a t i o n of the skew angle impossible. I f the skew angle i s included i n the c a l c u l a t i o n , i t could give a higher drag force. Another error may come from the measurement of the twine diameter. Because of the extremely i r r e g u l a r surface of the twine, a standard methodology to measure the diameter of the twine i s not a v a i l a b l e . The numerical procedure presented, serves as a means to p r e d i c t the drag force of a f i s h i n g net and the flow f i e l d around i t from a t h e o r e t i c a l point of view. Although i t does not include a l l of the parameters that a f f e c t the drag force, i t o f f e r s a numerical procedure for the c a l c u l a t i o n of the flow inside, and around, a f i s h i n g net. The v e l o c i t y f i e l d obtained from t h i s numerical procedure seems to simulate an actual flow f i e l d . Also, the c a l c u l a t e d drag values of the net are promising compared to the flume tank r e s u l t s . 46 BIBLIOGRAPHY 1. Konagaya, T. and Kawakami, T., Resistance Of A Plane Net Set Parallel TO The Stream II. B u l l e t i n Of The Japanese Society Of S c i e n t i f i c F i s h e r i e s Vol. 37, No. 10, 1971 2. Yinggi Zhou, Study On Conical Nets With Reference To Drag Coefficients, Geometry And Modelling Rules. Fishery Engineering Department, Shanghai F i s h e r i e s U n i v e r s i t y . 3. Higo, N. and Mouri, K., Studies On The Drag Net IV. Mem. Fac. Fish . , Kagoshima U n i v e r s i t y Vol. 24, pp 57 - 63, 1975 4. Ferro, R.S.T. and Stewart, P.A.M., The Drag Of Cod-Ends. International Council For The Exploration Of The Sea, 1981 5. Imai, T. and Marin, H.R., Model Experiments On Double Rigged Shrimp Trawl Gear. Mem. Fac. Fis h . , Kagoshima U n i v e r s i t y Vol. 27, No. 1, pp. 139 - 146, 1978 6. Mangunsukarto, K. and Fuwa, S., Studies On Trawl Net Model - Experiment Of Three Types Of Trawl Net. Mem. Fac. Fis h . , Kagoshima U n i v e r s i t y Vol. 27, No. 1, pp. 155 - 165, 1978 7. Dickson, W., Trawl Drag Area And Netting Geometry. The I n s t i t u t e Of Fishery Technology Research, Bergen, 1980 8. Wileman, D. and Hansen, K., Estimation Of The Drag Of Trawls Of Known Geometry. Danish F i s h e r i e s Technology I n s t i t u t e , 1988 9. Dudko, S., Swiniarski, J . , Przybyszewski, Z., Kwidzinski, Z., Nowakowski, P. and Sendlak, H., Effects Of Mouth Shape Of Conical Netting Constructions On Properties Of Their Resistance. Acta Ichthyologica Et P i s c a t o r i a , V o l. XII, Fasc. 2, Szczecin, 1982 10. Kowalski, T. and Giannotti, J . , Calculation Of Trawling Gear Drag. Marine Technical Report No. 16, Ocean Engineering, U n i v e r s i t y Of Rhode Island, Kingston, 1974 47 11. C a l l s a l , S.M., Mcllwaine, R.H., Wang, K. and Fung, 0., A Comparative Research On Trawl Net Drag. U.B.C., Mechanical Engineering Department, Naval Architecture, Series: MENA 84-3 December 1984 12. Brebbia, C.A., The Boundary Element Method For Engineering, Pentech Press, 1978 13. Chan, J.L.K., Hydrodynamic Coefficients For Axisymmetric Bodies At Finite Depth. Master's Thesis, U.B.C., 1984 14. McCarthy, J.H., Steady Flow Past Non-uniform Wire Grids. Journal Of F l u i d Mechanic, Vol. 19, pp 491, 1964 48 APPENDIX A TRANSFORMATION OF EQUATION (3.3.5) INTO EQUATION (3.3.6) Equation (3.3.5): ( — ) dS r t t The boundary has been d i s c r e t i z e d into N elements. The values of the p o t e n t i a l and i t s d e r i v a t i v e are assumed to be constant on each element, and are equal to the value at the node point of the element. Equation (3.3.5) rewritten f or a given i point i n d i s c r e t i z e d form i s shown below: Hence, one equation s i m i l a r to Equation (A.l) can be w r i t t e n for each i node, obtaining N equations. Therefore, Equation (3.3.5) can be expressed i n matrix form as: 2**(i) + I *<j)-|—<-±-) dS = I ( ~ ) dS (A.l) j=i j=i (3.3.6) Where, i = 1, N N 49 APPENDIX B S i m p l i f i c a t i o n of Equation (3.3.5) into Equation (3.3.8) Equation (3.3.5) , 2TT$(P) + J " $ ( Q ) | - ( J - ) dS - | 5 $ ( Q ) <-*-) dS s - ' *s 5 n r Where, | _ ( J _ ) L vr-ri" (B.l) 3n r 2 r From Equation (3.3,2) the distance, r, between P and Q i s : r = J (x - x p ) 2 + (R cos 8 - R p ) 2 + (R s i n 0 ) 2 and, (x - x ), (R cos 5 - R ), (R s i n B) V r P ? (B.2) J (x - x p ) 2 + (R cos 9 - R p ) 2 + (R s i n 6)Z From Equation (3.3.3) a normal u n i t vector, n, i s shown as: n = (n , n cos 8, n s i n 6) x R R So Equation (B.2) becomes: (x - x )n + (R cos 6 - R )n cos 6 + (R s i n 0)n s i n 9 _ -> P x P R R I Z I Z Z Z Z Z Z Z I Z ^ Z Z Z Z Z I Z Z I Z ^ Z Z I Z Z Z Z Z Z Z Z Z I Z Z Z y (x - x p ) 2 + (R cos 6 - R p ) 2 + (R s i n 9 ) 2 (B.3) and, 3n k r ' (x - x )n + (R cos 8 - R )n cos 8 + (R s i n 0)n s i n P x P R R [ (x - x p ) 2 + (R cos 8 - R p ) 2 + (R s i n 8 ) Z ]3/2 50 ( x - x ) n + R n - R n cos P x R P R [ ( x - x ) 2 + R 2 + R 2 - 2 R R cos 8 } 1 p p p ' Let a = (x - x ) 2 + R 2 + R2 and b = 2 R R p p p 3/2 (B.4) Subs t i t u t i n g a and b into Equation (B.4), i t can be rewritten as: (B.5) dnK r ; (x - x )n + R n - R n cos P x R P R (a - b cos 8) 3/2 The second term of Equation (3.3.5) i s : • J s * ( Q ) fs ( -r> d S t r 1 1 -27T 1 *<Q) -J o J 0 ( x - x ) n + R n - R n cos P x R P R (a - b cos 6) 3 12 R dd d l -27T i -J o J 0 ( x - x ) R n + R n - R R n cos P x R P R (a - b cos 8) 3 / 2 d9 d l pR -27T i * ( Q ) -J o J 0 (x - x )R n + R n P x R (a - b cos 8) 3 / 2 (a - b cos 8 - a)n 2 (a - b cos 8) 3 / 2 68 d l rR -27T 1 * ( Q ) -J o J 0 (x - x )R n + R n p x R_ (a - b cos 8) (a - b cos 8)n + 2 (a - b cos 8) 3/2 51 a n 2 (a - b cos 8) 3/2 d8 d l -27T *(Q) (x - x )R n + R n P x R (a - b cos 6 ) 3 ' 2 n 2 (a - b cos 8) 1/2 a n 2 (a - b cos 8) 3/2 d6 d l J l 2 *(Q) (x - x )R n + R n P x R (a - b cos 8) 3/2 n a n 2 (a - b cos 8) 1/2 2 (a - b cos 8) 3/2 d0 d l (B.6) (a - b . c o s ' 0 ) 1 / Z y a + b ' 2 d ^ 2 E(-=- S) 3/2 , , . i 2 (a - b cos 8) ' (a - b ) y a + — 6) = Complete E l l i p t i c Function of f i r s t kind — 5) = Complete E l l i p t i c Function of second kind / 2 * / a + b 52 Equation (B.6) becomes, 1 - 2 *(Q) 2 E ( — 5) ((x - x )R n + R 2n ) 2 , P x R (a - b) / a + b 2 F ( — 5) n 2 E ( — 5) a n 2 , R 2 , R 2 y a + b ' 2 (a - b>y a + b d l *(Q) 4 E ( — 5) <(x - x )R n + R 2n ) 2 , P x R (a " b) / a + b 2 F ( — S) n 2 E ( — 6) a n 2 , R + v 2 , R / a + b (a - b ) ^ a + b ' d l *(Q) 4 E ( — 5) ((x - x )R n + R 2n ) 2 , p ' x R (a - b) / a + b 2 n r i + b L a E ( — 6) n — 6) a - b 2 , d l The r i g h t hand side of Equation (3.3.5) i s : - J . -**<9> ( i ) d s 3n v r i o i 0 a*(Q) an R d0 d l y (x - x p ) 2 + (R cos 9 - R p ) 2 + (R s i n 9)2 a$(Q) an R d6 d l (a - b cos 9) 1/2 1 a*(Q) 4 R F ( — ,*> a n ( a + b ) 1 / 2 d l 53 Therefore, Equation (3.3.5) can be rewritten as: 2 T T $ ( P ) + *(Q) 4 E ( — 5) ((x - x )R n + R 2n ) 2 p' x r ' (a - b) / a + b / 2 n a + b r a E(-^- 5) a - b F(-=- 5) d l ^ 8*(Q) 4 R F<"T /> 3n ( a + b ) 1/2 d l (3.3.8) 54 APPENDIX C DERIVATION OF EQUATION ( 4 . 3 . 1 ) AND ( 4 . 3 . 2 ) From Equation ( 3 . 3 . 5 ) 2 T T $ ( P ) + j * ( Q ) | ^ ( - ^ ) dS s t J . » M (4-, as The e n t i r e boundary of Region 1 i s made up by S , and S^, hence, Equation ( 3 . 3 . 5 ) can be written as: 2* # ( p ) + 1 * C Q ) | ^ ( - ^ - ) D S + J ' ^ i n ^ - r ) d s S S 1 2 3 A + 1 * « > f e < 4 - > d s I, J . ^ < 4 ' -• M 2 ) ( 4 ) as ( C I ) From Appendix B, the terms on the l e f t hand side of the above equation with an i n t e g r a l sign has a form of: *(Q) 4 E ( — , 6 ) ((x - x )R n + R 2n ) 2 P x R a + b 2 n r =^ i + b L a E ( — , 5 ) —r J4 F H r ' 5> a - b 2 d l 5 5 And the terms on the r i g h t hand side has a form of: a*( Q) 5n , , , .1/2 ( a + b ) d l With the appropriate l i m i t s , Equation (C.l) can be written as: 2 T T $ ( P ) + *(Q) 4 E ( - | - ,6) ((x - x p)R n x + R 2n R) (a - b)^ a + b ' 2 n r i + b L a E ( — ,5) dy "° r - 4 E ( — ,6) ((x - x )R n + R 2n ) 2 P x R *(Q) (a - b ) / a + b 2 n r i + b L 3 E H r - 5 ) ii F(-=- ,5) a - d 2 d l *(Q) R - 4 E ( — ,5) ((x - x )R n + R 2n ) 2 ?' x R (a - b ) / a + b 2 n r i + b L a E(-=- ,6) dy -x *(Q) - 4 E ( — ,6) ((x - x )R n + R 2n ) 2 V x R (a - b ) / a + b ' 2 n r i + b L a E(-=- ,5) a - b " F<"f- • « d l 56 *(Q) x - 4 E ( — ,6) ((x - x )R n + R 2n ) 2 p' x R y (a - b ) / a + b / 2 n r = -i + b L a E<^- ,6) - - F ( - f - ,6) a - b d l 1 a»(Q) 4 R F ( — •«) 0 ,0 L ] y X ] 0 X an ( a + b ) 1 / 2 a*(Q) 4 R an ( a + b ) 1 / 2 3*(Q) 4 R F<-=- ,5) an ( a + b ) 1 / 2 a$(Q) 4 R F(-=- .«) an ( a + b ) 1 / 2 a«(Q) 4 R F(-=- .«> an ( a + b ) 1 / 2 dy d l dy d l d l (4.3.1) S i m i l a r l y , Region 2 i s represented by S , S and S , and 4 5 6 the following equation i s obtained: / . X 2 T T $ ( P ) + *(Q) 4 E ( — ,6) ((x - x )R n + R 2n ) 2 V x R ' (a - b)/~a~T" 2 n r i + b L a E(-=- ,5) 2-r- F ( - ^ - ,5) a - b 2 d l .0 *(Q) 4 E ( — ,5) ((x - x )R n + R 2n ) 2 P x R__ (a - b ) / a + b ' 57 2 n a E ( - ^ - ,6) / T T b " - F(- , 6 ) d l *(Q) 4 E ( — ,6) ((x - x )R n + R 2n ) 2 P x R (a - b ) / a + b + 2 n r i + b L a E ( — ,5) a ! b - F < ~ r dy ^ 3*(Q) 4 R F H h ' 5 ) 3n ( a + b ) 1 / 2 d l "° a*(Q> 4 R F H r > 5 > an ( a + b ) 1 / 2 d l "° a»(Q) 4 R F H r . 5 > 3n ( a + b ) 1/2 dy (4.3.2) 58 APPENDIX D COMBINATION OF THE EQUATION (4.3.3) AND (4.3.4) From Equation (4.3.3) and (4.3.4): K ] h ] - K ] [ * • » * ] ( 4 - 3 - 3 ) [ E i J h ] - [ F i J [ • • , , ] ( 4 - 3 - 4 ) Suppose there are N , N , N , N , N and N elements on the 1 2 3 A 5 6 boundaries S , S , S , S , S and S re s p e c t i v e l y . 1 2 3 A 5 6 Let N = N + N + N + N + N II 1 2 3 A 5 N = N + N + N T2 A 5 6 Therefore, matrix C and D have a size of (N x N ) and 11 11 matrix E and F have a size of (N x N ). In Equation (4.3.3), i T 2 T2 n and j are from 1 to N , where i n Equation (4.3.4), they are 1 to N . T2 According to boundary conditions (iv) and (v) i n 4.2, one can move the N + N + N + l t o N + N + N + N + N columns of 1 2 3 1 2 3 A 5 matrix D, and 1 to N + N columns of matrix F, to the l e f t hand A 5 side, one can also move the corresponding rows of the $, of the n two regions. Then the two matrix are combined as: 59 N 11 T 2 N T 1 c i j -D i j j N T 2 E i j - F i j j N 4 1 1 - 1 - 1 n 5 1 1 - 1 - 1 * r n N 5 1 1 1 1 n 4 1 1 1 1 n N 4 N 5 N 5 N 4 N 4 N 5 N 5 N 4 N + N + N 1 2 3 i J 0 D i f f 0 0 N n N 1 *, n N 2 ®, n N 3 *. n N 6 (4.3.5) Where $ = $ - $ D i f f R e g i o n 1 R e g i o n 2 60 APPENDIX E HYDRODYNAMIC LOADING FORMULATION FROM PETER CREWE The forces F i n the i d i r e c t i o n ( i = 1, 2, 3) on a un i t of i netting, ( i . e . one knot and four h a l f bars) as shown i n Figure 6.1.1, are given by: p a d V C t i (6.1.1) The force c o e f f i c i e n t s C^ depend on the ad d i t i o n a l parameters: (Figure E.l) i ) The i n c l i n a t i o n of mesh plane to the flow, a. i i ) The h a l f s e t t i n g angle of the mesh, a . s i i i ) The skew angle of the mesh, i9. iv) Diameter of the knot, d . following The force c o e f f i c i e n t s C are given as: C = 1.03 F + 8 i c i i 3 d^ |sin a| t d \ 2 k t ' (E.l) Where, F i s the i ' t h component of F , the force a r i s i n g from c i c the bars alone, and 5 i s the Kronecker d e l t a (8 = 1, i f i = 13 13 3, otherwise, i t i s set to zero). The second term a r i s e s from the knots. The d i r e c t i o n of axes 1, 2 and 3 are shown i n Figure E.2, and are oriented so that 1 i s i n the transverse d i r e c t i o n , 2 being the v e r t i c a l d i r e c t i o n and 3 i s the drag d i r e c t i o n . The force F i s most simply defined i n terms of i t s c components r e l a t i v e to the net plane axes, n, s, and t, as shown 61 i n Figure E.2. Angles B and B2 are defined as the angles of the bars to the d i r e c t i o n of the motion of the net and are given by: cos B = cos(i9 + a )cos a 1 s s i n B •= + 1 - cos2/3 ^1 ' l cos B2 ~ cos(i9 - a )cos a s i n Bz = + / 1 - cos2B2 (E.2) The components of F are then given by: 2 2 F = - cos a -<"sin (•& + a ) s i n 5 + s i n (i9 - a ) s i n 5 o s 2 I s 1 s H2 F = - s i n a - | s i n f l + s i n 5 >• c n 2 [ 1 2) / • = - cos a •< s n 2 \ b [cos 2(i9 + a ) |cos /S^  | + cos 2(i9 - a ) | cos ^2|]j" ^ cos a -|sin (i9 + a )cos(i9 + a ) [s i n B1 " b | cos / ^ l ] (# - o s)cos(i3 - a g) [s i n B2 - b | cos / 9 2 l ] | F c t 2 + sine (E.3) Where b i s equal to 0.035, which i s determined e m p i r i c a l l y . F , F and F have to be transformed into the d i r e c t i o n c n c s c t of drag, v e r t i c a l and transverse, which are l i s t e d below: F = F c 1 c t F = F cos a - F s i n a c 2 c n c s F = F cos a + F s i n a (E.4) c 3 c s c n 62 Equation (E.l) can be rewritten as: C = 1.03 F 1 ct C = 1.03 (F cos a - F s i n a) 2 c n c s C = 1.03 (F cos a + F s i n a) + fc d s i n a 3 c s c n 3 a d -v 2 k (E.5) A f t e r C , C and C are calculated, the forces i n i 1 2 3 d i r e c t i o n can be obtained as: F = 1.03 F p a d V 2 1 c t t F = 1.03 (F cos a - F s i n a) p a d V 2 2 c n c s t F = 1.03 UF cos a + F s i n a) 3 I c s c n d s i n a t ' + ^ { ^ d t d A 2> k 1 1 ' V 2 (6.1.2) 63 APPENDIX F FORMULATION OF THE PRESSURE DIFFERENCE ACROSS THE NET From Equation (6.2.1): AP i Area Where, Area = Area of u n i t netting ( i . e . 1 knot and 4 h a l f bars) and. Area = 2 a s i n a cos (F.l ) S u b s t i t u t i n g Equations (6.1.2) and ( F . l ) into Equation (6.2.1), the pressure drop i n i d i r e c t i o n can be rewritten as: 1.03 F p d V ct t  i 2 a s i n a cos a s s A P = A P = 2 1.03 (F cos a - F s i n a) p d V en c s t 2 a s i n a cos a AP 1.03 p d V 2 a s i n U V f { a cos a I (F cos a + F s i n a) cs cn d |sin aI r d \2 t 1 1 f k 3 a (F.2) Due to the impossible nature i n the p r e d i c t i o n of the skew angle, i3, i t i s assumed to be zero degrees. According to Equation (E.2), and S,^ must be equal. Therefore, from Equation (E.3), the force i n the " t " d i r e c t i o n , F , must be zero. Hence, the c t 64 pressure drop across the net i n the d i r e c t i o n normal to. the n e t t i n g plane, A P , i s : n AP = AP s i n Q + A P cos a ( 6 . 2 . 2 ) 65 APPENDIX G PROFILE OF CONICAL NET The coordinates of each s t a t i o n of the c o n i c a l net at various water v e l o c i t y are l i s t e d below. The reference frame of coordinates i s shown i n Figure G.l. G.l WATER VELOCITY -0.4 m/s STATION # X (mm) Y (mm) LOWER Y (mm) UPPER 1 0 240 1594 2 345 354 1487 3 724 427 1384 4 1090 492 1273 5 1440 544 1157 6 1773 576 1075 7 2157 602 1002 8 2555 614 993 9 2856 582 970 10 3166 776 66 G.2 WATER VELOCITY =0.6 m/s STATION # X (mm) Y (mm) LOWER Y (mm) UPPER 1 0 1370 . 2724 2 387 1431 2644 3 757 1522 2565 4 1129 1632 2504 5 1521 1742 2429 6 1903 1813 2373 7 2325 1860 2327 8 2702 1880 2299 9 3031 1905 2275 10 3371 2099 G.3 WATER VELOCITY =0.8 m/s STATION # X (mm) Y (mm) LOWER Y (mm) UPPER 1 0 1203 2557 2 413 1282 2481 3 813 1370 2418 4 1200 1449 2326 5 1580 1513 2207 6 1983 1581 2135 7 2384 1633 2091 8 2796 1659 2059 9 3200 1686 2050 10 3495 1880 67 G.4 WATER VELOCITY =1.0 m/s STATION # X (mm) Y (mm) LOWER Y (mm) UPPER 1 0 1263 2617 2 431 1316 2548 3 822 1412 2449 4 1210 1493 2361 5 1598 1590 2275 6 2000 1655 2210 7 2413 1708 2166 8 2824 1724 2132 9 3231 1759 2116 10 3549 1940 68 A P P E N D I X H VELOCITY MEASUREMENT INSIDE THE CONICAL NET The water v e l o c i t i e s inside the co n i c a l net at various v e l o c i t i e s are l i s t e d below. The conversion of the current meter's reading to v e l o c i t y i s according to the c a l i b r a t i o n graph shown i n Figure 7.3.2. The reference frame of the coordinates i s shown i n Figure H.l. Also, the water v e l o c i t i e s inside the co n i c a l net along each section at the d i f f e r e n t tested speeds are p l o t t e d and shown i n Figures H.2, H.3, H.4 and H.5. H.l INCOMING VELOCITY =0.4 m/s X Y Meter Reading V e l o c i t y % Decrease (mm) (mm) RPM (m/s) 320 152 74 0.3416 14.6 320 623 73 0.3373 15.7 320 1036 75 0.3459 13.5 1341 257 73 0.3373 15.7 1341 373 73 0.3373 15.7 1341 518 72 0.3330 16.8 1341 711 70 0.3244 18.9 1341 874 73 0.3373 15.7 2737 241 74 0.3416 14.6 2737 458 74 0.3416 14.6 2737 580 73 0.3373 15.7 69 H.2 INCOMING VELOCITY =0.6 m/s X Y Meter Reading V e l o c i t y % Decrease (mm) (mm) RPM (m/s) 303 367 118 0.5312 11.5 303 569 116 0.5226 12.9 303 843 115 0.5183 13.6 303 1280 115 0.5183 13.6 1444 526 116 0.5226 12.9 1444 782 113 0.5096 15.1 1444 912 115 0.5183 13.6 1444 1080 116 0.5226 12.9 2630 614 115 0.5183 13.6 2630 783 115 0.5183 13.6 2630 994 118 0.5312 11.5 70 H.3 INCOMING VELOCITY =0.8 m/s X Y Meter Reading V e l o c i t y % Decrease (mm) (mm) RPM (m/s) 388 202 153 0.6820 14.7 388 635 154 0.6863 14.2 388 966 155 0.6906 13.7 388 1069 148 0.6604 17.5 1300 381 160 0.7122 11.0 1300 601 157 0.6992 12.6 1300 716 159 0.7078 11.5 1300 894 161 0.7165 10.4 2537 463 153 0.6820 14.7 2537 658 158 0.7035 12.1 2537 810 163 0.7251 9.4 71 H.4 INCOMING VELOCITY =1.0 m/s X Y Meter Reading V e l o c i t y % Decrease (mm) (mm) RPM (m/s) 394 209 197 0.8716 12.8 394 347 193 0.8543 14.6 394 813 193 0.8543 14.6 394 971 187 0.8285 17.2 1234 407 199 0.8802 12.0 1234 610 207 0.9147 8.5 1234 839 200 0.8845 11.6 2705 503 204 0.9017 9.8 2705 585 205 0.9061 9.4 2705 721 202 0.8931 10.7 2705 813 204 0.9017 9.8 72 APPENDIX 1 DRAG FORCE FROM EXPERIMENT 1.1 RESULTS FROM THE FLUME TANK EXPERIMENT The drag forces and angles of the tow l i n e to the v e r t i c a l d i r e c t i o n obtained from the experiments conducted i n the flume tank are l i s t e d below: Case 1 Case 2 Measured Vel. Drag Force Angle Drag Force Angle Drag Force (m/s) (Kgf) (Kgf) (Kgf) 0.4 3.114 75.64° 0.920 72.96° 2.194 0.6 6.705 82.30° 1.382 74.16° 5.324 0.8 11.690 84.15° 2.017 74.57° 9.673 1.0 19.635 85.64° 2.793 76.23° 16.842 Where the drag forces and the angles of Case 1 include the co n i c a l netting, hoop, f l o a t s and tow l i n e , while Case 2 excludes,the net t i n g . The measured drag forces are the drag of the n e t t i n g i t s e l f at d i f f e r e n t tested v e l o c i t i e s and i s obtained from the following equation: Drag = Drag s i n (Angle ) M e a s u r e d C a s e 1 C a s e l Drag s i n (Angle ) (1.1) " C a s e 2 b C a s e 2 73 1.2 RESULTS FROM THE TOWING TANK EXPERIMENT The drag forces obtained from the experiments conducted i n the towing tank are l i s t e d below: Case 1 Case 2 Measured V e l . Drag Force Drag Force Drag Force (m/s) (Kgf) (Kgf) (Kgf) 0.4 4.280 0.490 3.790 0.6 9.050 1.450 7.600 0.8 15.850 2.430 13.420 1.0 23.900 3.545 20.355 Where the drag forces of Case 1 include the c o n i c a l netting, hoop, f l o a t s and tow l i n e , while Case 2 i s excludes the netting. The measured drag forces are the drag of the n e t t i n g i t s e l f at d i f f e r e n t tested v e l o c i t i e s and i s obtained from the following equation: Drag = Drag - Drag (1.2) M e a s u r e d ° C a s e 1 ° C a s e 2 74 FIGURE 1.1.1 TRAWLING FIGURE 1.1.2 MID-WATER TRAWLING 75 Towing Warp Door v Wing Bridles Ground Warp Foot Rope. Float Cod End v i Rib Line FIGURE 1.1.3 TYPICAL FISHING NET 76 FIGURE 3.3.2 CONTROL DOMAIN FOR THE SPHERE VELOCITY POTENTIAL FIGURE 3.3.3 FLOW FIELD AROUND A SPHERE UPSTREAM VELOCITY = 1.0 M/S 80 1 l 1 1 1 1 1 r 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 D O W N S T R E A M V E L O C I T Y ( M / S ) FIGURE 3.3.4 VELOCITY ALONG THE DOWNSTREAM BOUNDARY OF THE CONTROL DOMAIN FIGURE 3.3.5 ELEMENTS AT CORNER 82 -2 0 0.5 1 1.5 X COORDINATE ALONG SPHERE ( M ) FIGURE 3.3.6 COEFFICIENT OF PRESSURE C p , ALONG THE SURFACE OF A SPHERE 83 FIGURE 4.1.1 CONTROL DOMAINS FOR THE NUMERICAL MODEL OF FISHING NET 84 VELOCITY POTENTIAL Flow Direction < NORMAL VELOCITY VELOCITY POTENTIAL VELOCITY POTENTIAL ON THE OUTSIDE SURFACE NORMAL VELOCITY ON THE OUTSIDE SURFACE VELOCITY POTENTIAL ON THE INSIDE SURFACE NORMAL VELOCITY ON THE INSIDE SURFACE FIGURE 5.2.1 FLOW FIELD OF A VERTICAL BOUNDARY WITHOUT POTENTIAL DIFFERENCE UPSTREAM VELOCITY = 2.0 M/S 85 Q 1.8 1.6 H 5 1-4 ct: Ld 1.2 CJ o cr: w 0.8-1 \ \ \ \ \ 1 IE e> U J x 0.6H 0.4-1 0.2 H U P S T R E A M V E L O C I T Y = 2.0 M / S Legend O ANALYTICAL X BEM 1.9 2.1 2.2 2.3 2.4 2.5 NORMAL VELOCITY (M/S) 2.6 2.7 FIGURE 5.2.2 NORMAL VELOCITY ALONG VERTICAL BOUNDARY WITHOUT P O T E N T I A L D I FFERENCE 86 Q 1.85 7 x 7 \ f U P S T R E A M V E L O C I T Y = 2.0 M / S Legend O ANALYTICAL X BEM i i i i i I I 1.9 1.95 2 2.05 2.1 2.15 2.2 NORMAL VELOCITY (M/S) i i 2.25 2.3 FIGURE 5.2.3 VELOCITY ALONG DOWNSTREAM BOUNDARY WITHOUT P O T E N T I A L D I FFERENCE 87 VELOCITY POTENTIAL Flow Direction NORMAL VELOCITY VELOCITY POTENTIAL VELOCITY POTENTIAL ON THE OUTSIDE SURFACE NORMAL VELOCITY ON THE OUTSIDE SURFACE VELOCITY POTENTIAL ON THE INSIDE SURFACE NORMAL VELOCITY ON THE INSIDE SURFACE FIGURE 5.2.4 FLOW FIELD OF A 90 DEGREES ARC BOUNDARY WITHOUT POTENTIAL DIFFERENCE UPSTREAM VELOCITY = 2.0 M/S 88 UPSTREAM VELOCITY = 2.0 M/S Legend O ANALYTICAL X BEM i 1 1 1 0 0.5 1 1.5 2 NORMAL VELOCITY (M/S) FIGURE 5.2.5 NORMAL VELOCITY ALONG 90 DEGREES ARC WITHOUT POTENTIAL DIFFERENCE 89 3-1 2.5H 1.5 H H 0 . 5 0 + 1.85 Q UPSTREAM VELOCITY = 2.0 M/S Legend O A N A L Y T I C A L X B E M 1.9 1.95 2 2.05 2.1 2.15 2.2 NORMAL VELOCITY (M/S) 2.25 2.3 FIGURE 5.2.6 VELOCITY ALONG DOWNSTREAM BOUNDARY WITHOUT POTENTIAL DIFFERENCE 90 Q x VELOCITY POTENTIAL Flow Direction NORMAL VELOCITY VELOCITY POTENTIAL VELOCITY POTENTIAL ON THE OUTSIDE SURFACE NORMAL VELOCITY ON THE OUTSIDE SURFACE VELOCITY POTENTIAL ON THE INSIDE SURFACE NORMAL VELOCITY ON THE INSIDE SURFACE FIGURE 5.2.7 FLOW FIELD OF A 60 DEGREES BOUNDARY WITHOUT POTENTIAL DIFFERENCE UPSTREAM VELOCITY = 2.0 M/S 91 1.8 1.6H 1.4 Cr: 1-2H Ld o o cr: H ^ 0.8-1 LU n: »— i— o X 0.4 H 0.2-UPSTREAM VELOCITY = 2.0 M/S G) X I Legend O A N A L Y T I C A L X B E M i r 1.3 1.4 1.5 1.6 1.7 1.8 NORMAL VELOCITY (M/S) 1.9 FIGURE 5.2.8 NORMAL VELOCITY ALONG 60 DEGREES BOUNDARY WITHOUT POTENTIAL DIFFERENCE 92 >-en < O m < L J Crl I— (Jl O Q o L d •z. Q or O O O >-2.5 2H 1.5 H 0.5 0 + 1.85 t f —K UPSTREAM VELOCITY = 2.0 M/S Legend O A N A L Y T I C A L X B E M 1.9 1.95 2 2.05 2.1 2.15 2.2 NORMAL VELOCITY (M/S) 2.25 2.3 FIGURE 5.2.9 VELOCITY ALONG DOWNSTREAM BOUNDARY WITHOUT POTENTIAL DIFFERENCE 93 O n . VELOCITY POTENTIAL Flow Direction NORMAL VELOCITY VELOCITY POTENTIAL VELOCITY POTENTIAL ON THE OUTSIDE SURFACE || II MIl.lLJ-U-u-'-NORMAL VELOCITY ON THE OUTSIDE SURFACE VELOCITY POTENTIAL ON THE INSIDE SURFACE JL_ rrrm NORMAL VELOCITY ON THE INSIDE SURFACE FIGURE 5.3.1 FLOW FIELD OF A VERTICAL BOUNDARY WITH POTENTIAL DIFFERENCE UPSTREAM VELOCITY = 2.0 M/S 94 H O T VELOCITY POTENTIAL Flow Direction NORMAL VELOCITY VELOCITY POTENTIAL VELOCITY POTENTIAL ON THE OUTSIDE SURFACE njjj 11111 NORMAL VELOCITY ON THE OUTSIDE SURFACE «1 VELOCITY POTENTIAL ON THE INSIDE SURFACE flttbtt NORMAL VELOCITY ON THE INSIDE SURFACE FIGURE 5.3.2 FLOW FIELD OF A 90 DEGREES ARC BOUNDARY WITH POTENTIAL DIFFERENCE UPSTREAM VELOCITY = 2.0 M/S 95 "1„. VELOCITY POTENTIAL NORMAL VELOCITY Flow Direction VELOCITY POTENTIAL ON THE OUTSIDE SURFACE JJJJ M i l l NORMAL VELOCITY ON THE OUTSIDE SURFACE VELOCITY POTENTIAL ON THE INSIDE SURFACE NORMAL VELOCITY ON THE INSIDE SURFACE FIGURE 5.3.3 FLOW FIELD OF 60 DEGREES BOUNDARY WITH POTENTIAL DIFFERENCE UPSTREAM VELOCITY = 2.0 M/S 96 FIGURE 6.4.1 AN ELEMENT ALONG THE NET 97 FIGURE 6.4.2 VELOCITY COMPONENTS ON THE SURFACE OF THE NET 98 FIGURE 7.1.1 FLUME TANK AT MARINE INSTITUTE 99 FIGURE 7.1.2 GEOMETRY OF CONICAL NET PANEL 100 FIGURE 7.3.1 PROPELLER TYPE CURRENT METER 101 Current Meter Calibration Graph 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Veloclty(m/8) FIGURE 7.3.2 CALIBRATION GRAPH OF CURRENT METER 102 FIGURE 7.4.1 TELL-TAILS ATTACHED TO THE CONICAL NET 103 4000 X COORDINATE ALONG NET (MM) FIGURE 8.1.1 PROFILE OF THE CONICAL NET AT VARIOUS WATER SPEED REAR PORTION OF THE NET FIGURE 8.1.2 MESH OPENING ALONG THE CONICAL NET 105 REAR PORTION OF THE NET FIGURE 8.3.1 TELL-TAILS AT VARIOUS POSITION OF THE CONICAL NET 106 VELOCITY POTENTIAL FLOW DIRECTION NORMAL VELOCITY VELOCITY POTENTIAL nnrnrrw^ •^^iLLLLLLLLLLi VELOCITY POTENTIAL ON THE OUTSIDE SURFACE — .0 flfl - u u 111 NORMAL VELOCITY ON THE OUTSIDE SURFACE lirfnTTTTrrrfn^ —^^J-u-UiijjjLijjj^ VELOCITY POTENTIAL ON THE INSIDE SURFACE ftfV NORMAL VELOCITY ON THE INSIDE SURFACE FIGURE 9.2.1 FLOW FIELD AROUND THE CONICAL NET AT 0.4 .M/S 107 VELOCITY POTENTIAL FLOW DIRECTION NORMAL VELOCITY VELOCITY POTENTIAL ^maLLlJi VELOCITY POTENTIAL ON THE OUTSIDE SURFACE •y-j U J ijT-H-^nJlj1 NORMAL VELOCITY ON THE OUTSIDE SURFACE InTTTTTTTT .^ ^uiiiLLLlI' VELOCITY POTENTIAL ON THE INSIDE SURFACE NORMAL VELOCITY ON THE INSIDE SURFACE FIGURE 9.2.2 FLOW FIELD AROUND THE CONICAL NET AT 0.6 M/S 108 VELOCITY POTENTIAL FLOW DIRECTION NORMAL VELOCITY VELOCITY POTENTIAL fTTTn-i. VELOCITY POTENTIAL ON THE OUTSIDE SURFACE NORMAL VELOCITY ON THE OUTSIDE SURFACE L ^ U J _ L L L L L L ' VELOCITY POTENTIAL ON THE INSIDE SURFACE NORMAL VELOCITY ON THE INSIDE SURFACE FIGURE 9.2.3 FLOW FIELD AROUND THE CONICAL NET AT 0.8 M/S 109 VELOCITY POTENTIAL 1 1 ' 1 * • "" . FLOW DIRECTION NORMAL VELOCITY VELOCITY POTENTIAL VELOCITY POTENTIAL ON THE OUTSIDE SURFACE NORMAL VELOCITY ON THE OUTSIDE SURFACE VELOCITY POTENTIAL ON THE INSIDE SURFACE NORMAL VELOCITY ON THE INSIDE SURFACE FIGURE 9.2.4 FLOW FIELD AROUND THE CONICAL NET AT 1.0 M/S n o 2.5 1.5 £ H o 3 0.5H LJ > < § 0 -0.5 -1 -— 1-5 - f 1 1 1 1 1 1 1 0 0.5 1 1.5 2 2.5 3 3.5 X COORDINATE ALONG NET (M) FIGURE 9.2.5 NORMAL VELOCITY ALONG THE NET AT 0.4 M/S i n 1 0.8 0.6H CO >- 0.4 O o > _l < O 0.2 -\ OH -0.2 -0.4 0.5 1 1.5 2 2.5 X COORDINATE ALONG NET (M) -r 3 3.5 FIGURE 9.2.6 NORMAL VELOCITY ALONG THE NET AT 0.6 M /S 112 0.8 FIGURE 9 . 2 .7 NORMAL VELOCITY ALONG THE NET AT 0.8 M /S 113 1.6 1.4 H -0 .6 -J 1 1 1 1 1 1 1 1 0 0.5 1 1.5 2 2.5 3 3.5 4 X COORDINATE ALONG NET (M) FIGURE 9.2.8 NORMAL VELOCITY ALONG THE NET AT 1.0 M/S 114 1.6 1.4-1.2-1-0.8 0.6 0.4 0.2-0 --0 .2 -0 .4 / rj ^ . Legend • EQUATION 6.3.6 (INSIDE) • BEM (INSIDE)  • |QyATL°_N_ (QUTSipE) O BEM_(p_UTSJDEl 0.5 1 1.5 2 2.5 3 X COORDINATE ALONG NET (M) 3.5 FIGURE 9.2.9 TANGENTIAL VELOCITY ALONG THE NET AT 0.4 M/S 115 1.4 1.2 1-00 >-o 3 LxJ > 0.8-O < 0.6-0.4 0.2-Legend • E Q U A T I O N 6.3.6 ( INSIDE) • B E M ( INSIDE)  • E Q U A T I O N _6,3._7_(OUTSIDE) ° ?£M_C°.u_!sMl -0.2 0.5 1 1.5 2 2.5 3 X COORDINATE ALONG NET (M) 3.5 FIGURE 9.2.10 TANGENTIAL VELOCITY ALONG THE NET AT 0.6 M /S 116 1.1 I I I 1 1 1 1 r 0 0.5 1 1.5 2 2.5 3 3.5 X COORDINATE ALONG NET (M) FIGURE 9.2.11 TANGENTIAL VELOCITY ALONG THE NET AT 0.8 M/S 117 1.8 0 0.5 1 1.5 2 2.5 3 3.5 4 X COORDINATE ALONG NET (M) FIGURE 9.2.12 TANGENTIAL VELOCITY ALONG THE NET AT 1.0 M/S 1 1 8 600 400H 200-0H (N * Q_ O CC Q UJ c r ZD CO CO UJ en -200--400H -600-- 8 0 0 ^ - 1 0 0 0 H -1200-0.5 1 1.5 2 2.5 3 3.5 X COORDINATE ALONG NET (M) FIGURE 9.2.13 PRESSURE DROP ACROSS THE NET AT 0.4 M/S 119 400 300 200 -IOOH -200 0.5 1 1.5 2 2.5 X COORDINATE ALONG NET (M) 3.5 FIGURE 9.2.14 PRESSURE DROP ACROSS THE NET AT 0.6 M/S 120 500-1 u u l I 1 1 1 1 1 1 0 0.5 1 1.5 2 2.5 3 3.5 X COORDINATE ALONG NET (M) FIGURE 9.2.15 PRESSURE DROP ACROSS THE NET AT 0.8 M/S 121 1500 X COORDINATE ALONG NET (M) FIGURE 9.2.16 PRESSURE DROP ACROSS THE NET AT 1.0 M/S 122 300 250-200 150-< cr Q 100-Legend • CREWE • S I N E S Q U A R E • ZHOLJ O B E M A F L U M E T A N K E X P E R I M E N T X T O W I N G T A N K E X P E R I M E N T / 0.6 0.7 0.8 VELOCITY (M/S) FIGURE 9.2.17 COMPARISON OF DRAG FORCES 123 FIGURE 10.1 SURFACE OF CONICAL NET UNDER TEST 124 Netting FIGURE E.1 MESH PLANE PARAMETERS 125 FIGURE E.2 DEFINITION OF THE AXIS 126 Y X FIGURE G.1 REFERENCE FRAME OF COORDINATE FOR PROFILE MEASUREMENT 127 FIGURE H.1 REFERENCE FRAME OF COORDINATE FOR VELOCITY MEASUREMENT 128 1100-1 1000 900H ^ 800 Z o O 700 L J CO L J IE H 600H o L J Z 500 Q O O <-> 400-1 y-300 200H 100 • Legend • X = 320 MM • X = 1341 MM • X = 2737 MM • • • • 0.320 T 0.325 0.330 0.335 0.340 0.345 VELOCITY INSIDE THE NET (M/S) 0.350 F IGURE H.2 W A T E R V E L O C I T Y INSIDE T H E N E T A T 0.4 M/S 1 2 9 o UJ CO LLJ X 1300 1200 1100-1 1000 900 800 _ 700-1 Q O O O >_ 600 500 400 300 Legend • X = 303 MM • X = 1444 MM • X = 2630 MM • • • • 0.505 0.510 0.515 0.520 0.525 0.530 VELOCITY INSIDE THE NET (M/S) 0.535 FIGURE H.3 WATER VELOCITY INSIDE THE NET AT 0.6 M/S 130 1100 1000-900 Legend 800-o I— o LLJ CO • X = 388 MM • X = 1300 MM • X = 2537 MM 700H • • O L J 600 <£ -z. Q 8 5 0 0 CJ >-• 4 0 0 H • 300 200 0.66 0.67 0.68 0.69 0.70 0.71 0.72 0.73 VELOCITY INSIDE THE NET (M/S) FIGURE H.4 WATER VELOCITY INSIDE THE NET AT 0.8 M/S 131 1000 9 0 0 H 800-• z o o U J 700H Q O O CJ >-600H 500-400-Legend • X = 394 MM • X = 1234 MM • X = 2705 MM • • 300-200-0 . 8 2 0 . 8 4 0 . 8 6 0 . 8 8 0 . 9 VELOCITY INSIDE THE NET (M/S) 0 . 9 2 FIGURE H.5 WATER VELOCITY INSIDE THE NET AT 1.0 M/S 132 

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