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Using parabolic waterlines to reduce the resistance of a trimaran Vyselaar, Daniel P. 2006

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USING PARABOLIC WATERLINES TO REDUCE THE RESISTANCE OF A TRIMARAN by DANIEL P. VYSELAAR BASc, University of British Columbia, 2002  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in  THE FACULTY OF GRADUATE STUDIES (Mechanical Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA March 2006 © Daniel P. Vyselaar, 2006  ABSTRACT Traditional ship design practices suggest that the optimum hull shape minimizes the beam of a vessel. However, recent studies on coaster tankers and typical west coast fishing vessels show that the addition of parabolic bulbs to a vessel's parallel mid body can significantly reduce the wave making resistance. These parabolic bulbs are added at the waterline and create their own wave pattern. The waves created by these bulbs can interact with the shoulder wave system of the base hull in a beneficial manner for certain speed ranges. Previous parabolization studies have been primarily experimental in nature, and have focused on vessels with small length to beam ratios, typical of workboats. This thesis uses both numeric tools and experimental approaches to alter the waterlines of a slender, high speed trimaran.  A Michell's integral based solver and a Rankine source panel method were used to predict the wave making characteristics of the trimaran. A parametric study, varying the size and location of bulbs, was performed first on only the centre hull to identify beneficial arrangements. The study has then been extended to the trimaran to evaluate the additional wave interactions from the outriggers.  Calculations done by both numeric methods predicted reductions in wave resistance due to the increase of the trimaran's beam at the waterline. Experimental work confirmed the differing wave interactions seen as a result of hull form parabolization, and a new trimaran form was designed that reduced the total resistance of the vessel by up to 6%.  ii  TABLE OF CONTENTS Abstract Table of Contents List of Tables List of Figures List of Symbols and Abbreviations Acknowledgements Chapter 1: Project Motivation & Background 1.1  Previous Work  1.2  Scope for this Thesis  1 2 :  Chapter 2: Numeric Approaches  7 8  2.1  A Review of Wave Resistance Theory  2.2  Solution of the Wave Resistance Problem Using Thin Ship Theory  12  2.3  Boundary Element Methods  15  Chapter 3: Implementation of Parabolization  8  21  3.1  Implications of Parabolization on Complete Vessel Design  21  3.2  Motivation for Studying the Trimaran Hullform  22  3.3  Trimaran Hullform Particulars  23  3.4  Feasibility of Parabolization Study  24  3.5  Implementation of Parabolization  26  Chapter 4: Analysis by Michell's Integral  28  4.1  Parabolization for Monohull  30  4.2  Parabolization of the Trimaran Hullform  34  4.3  Conclusions on Analysis by Michell's Integral  41  Chapter 5: Wave resistance Analysis by Dawson's Method  45  5.1  Mesh Generation for the NPL Monohull  46  5.2  Mesh Generation for the NPL Trimaran  55  5.3  Parabolization Study of the NPL Monohull  58  5.4  Parabolization Study of the NPL Trimaran  66  5.5  Conclusions on Dawson's Method  74  iii  Chapter 6: Experimental Approaches  75  6.1  Theoretical Background for Experimental Work  76  6.2  Model Construction and Particulars  77  6.3  Test Procedures & Experimental Results  79  6.3.1  Form Factor Determination  80  6.3.2  Monohull Test Results and Full Scale Powering Predictions  81  6.3.3  Trimaran Test Results and Full Scale Powering Predictions  84  6.4  Comparison to Numeric Work  89  Chapter 7: Conclusions  91  7.1  Economic Implications of Parabolization  93  7.2  Recommendations for Future Work  94  References  95  Appendix A: Offset Table for the NPL trimaran  99  Appendix B: Offset Table for the Parabolized Trimaran (hull #7)  111  Appendix C: A guide to pre processing for Michell's Integral Analysis by Michlet  123  Appendix D: A guide to pre processing for the Panel Method Trawson  134  iv  LIST OF TABLES Table 3-1 - Main Particulars for the NPL Trimaran  24  Table 3-2 - Overall Test Matrix  27  Table 6-1 - Model scale particulars for the NPL trimaran  77  Table 6-2 - Summary of calculated form factors  80  Table 6-3 - Comparison of form factors for different NPL forms  81  v  LIST OF FIGURES Figure 1-1- Resistance Components for a Typical Fishing Hull  2  Figure 1-2 - Hull forms for studying the effect of parabolic bulbs  3  Figure 2-1 - Definitions of terms  10  Figure 2-2 - Open transom  19  Figure 3-1 - Reduction in entrance angle and transom area caused by parabolic bulbs.. 22 Figure 3-2 - Lines plan for the NPL trimaran  24  Figure 3-3 - Resistance Components for the NPL monohull estimated from thin ship theory and the ITTC '57 line  26  Figure 3-4 - The initial hull forms for study: Hull 1 (top), Hull 2 (middle) Hull 3 (bottom)  27  Figure 4-1 - Michlet data file structure  28  Figure 4-2 - Wave resistance for hulls 1-3 vs base hull  31  Figure 4-3 - Percentage change for hulls 1-3 relative to base hull  31  Figure 4-4 - Powering requirements for hulls 1-3 vs base hull  32  Figure 4-5 - Hull configuration 4  33  Figure 4-6 - Wave resistance for hull 4 vs base hull  33  Figure 4-7 - Percentage change in total resistance between hull 4 and base hull  34  Figure 4-8 - Total resistance for trimaran, centre hull and equivalent monohull  35  Figure 4-9 - Viscous resistance, trimaran, centre hull and equivalent monohull  36  Figure 4-10 - Wave resistance for trimaran and equivalent monohull, showing effect of LWL  37  Figure 4-11 - Wave resistance for the parabolized trimaran  38  Figure 4-12 - Percent change in RT for the parabolized trimaran  39  Figure 4-13 - Wave resistance for hull 4 vs base hull in trimaran configuration  40  Figure 4-14 - Percentage change in RT for hull 4 in the trimaran configuration  40  Figure 4-15 - Wave profile created by hull 3 at Fn = 0.32  42  Figure 4-16 - Wave profile created by hull 4 at Fn= 0.32  43  Figure 5-1 - Flow chart for analysis by Dawson's method  46  Figure 5-2 - Mesh for the NPL centre hull  48  vi  Figure 5-3 - General parameters to describe free surface discretization  49  Figure 5-4 - Numeric reflection resulting from too narrow a domain  49  Figure 5-5 - Visible reflection (green circle) due to abrupt grid transition (red circle)... 50 Figure 5-6 - Refined mesh for the NPL monohull  52  Figure 5-7 - Wave elevations for Fn = 0.5 for the low resolution mesh (left) and high resolution mesh (right)  53  Figure 5-8 - Wave elevations for Fn = 0.31 for the low resolution mesh (left) and high resolution mesh (right)  54  Figure 5-9 - Effect of mesh refinement on the calculation of Cw  55  Figure 5-10 - Mesh of the NPL trimaran. Complete vessel shown for clarity  56  Figure 5-11 - Mesh of the NPL trimaran and free surface  57  Figure 5-12 - The effect of mesh refinement on wave resistance for the NPL trimaran . 58 Figure 5-13 - Comparison of Cw for base hull and hulls 1 - 4  59  Figure 5-14 - Wave elevations at Fn = 0.35 for base hull (a), hull 1 (b) and hull 4 (c)... 60 Figure 5-15-Hull 5  63  Figure 5-16 - Wave resistance for the base hull, and hulls 4 & 5 (Dawson's method)... 64 Figure 5-17 - Percentage change in RT between base hull and hulls 4 and 5  64  Figure 5-18-Hull 6  65  Figure 5-19 - Percentage change in Rw between base hull and hulls 5 and 6  65  Figure 5-20 - (a) = base hull (b) = hull 6 Fn =0.33, note the reduced transverse waves in (b)  66  Figure 5-21 -Wave resistance predictions for hulls 1-3 (a), and percent change from base trimaran (b)  68  Figure 5-22 - Wave resistance predictions for hull 4 and 6 (a), and percent change from base trimaran (b)  69  Figure 5-23 - Wave elevations for the base trimaran (a) and hull 4 (b) at Fn = 0.39  71  Figure 5-24 - Hull configuration 7, note the more abrupt bulb shape  71  Figure 5-25 - Total wave resistance predictions for hulls 4 & 7 (a), and percent change from base trimaran (b)  73  Figure 6-1 - The Ocean Engineering Centre Tow Tank Facility  75  Figure 6-2 - The NPL trimaran model attached to the heave post  78  vii  Figure 6-3 - Bulb 6 (top) and bulb 7 (bottom). Note the different sizes of the two bulbs  79  Figure 6-4 - Determination of the form factor from slow speed tests for trimaran hull 7  80  Figure 6-5 - Full scale resistance plots for monohull configurations  82  Figure 6-6 - A comparison of resistance data for the NPL base monohull  83  Figure 6-7 - Percent change in total resistance between hull designs  84  Figure 6-8 - Full scale trimaran resistance  85  Figure 6-9 - Percentage change in total resistance between trimaran hull designs  86  Figure 6-10 - Wave resistance components for the three trimaran hulls  86  Figure 6-11 - Percentage change in wave resistance  87  Figure 6-12 - Wave patterns created by the base hull (left) and hull 7 (right) at Fn = 0.36  88  Figure 6-13 - Comparisons of wave resistance predictions for hull 7 with experimental results  90  Figure 6-14 - Predictions of changes in wave resistance by hull 7  90  Figure A - l - The pre processor launchpad  130  Figure A-2 - User input for the mesh generation process  131  Figure A-3 - Projected points ready forexport  132  Figure A-4 - Options for exporting points file  132  Figure A-5 - Entering mesh parameters to generate output files  139  viii  LIST OF SYMBOLS AND ABBREVIATIONS V  Displacement volume or gradient operator depending on context  A  Displacement mass or Laplacian operator depending on context  V  Wave elevations  \  Integration constant for Michell's Integral  H  Viscosity of water  P  Density of water Velocity potential function, subscripts denote differentiation  o  Velocity potential for the double body flow  A, S  Area or wetted surface area  B  Beam  BM  Distance from the centre of buoyancy to the ship's metacentre  c  Advance speed of ship  C  A  Model-Ship Correlation allowance, accounts for differences in surface roughness between full scale and model scale. Typically 0.0004  C  D  Drag coefficient  C  F  Skin friction coefficient  CFO  Skin friction coefficient calculated by the ITTC '57 line  CR  Residuary drag coefficient  C  Viscous drag coefficient =(i + k) C  V  F  Cw  Wave resistance coefficient  D  Drag force  EHP  Effective horsepower  Fn  Froude number, Fn = -¥=, L = LWL unless otherwise noted  fx  Slope of the hull along the waterline  g  Gravitational acceleration  GM  Metacentric height  IT  Transverse moment of inertia of the waterplane  k  Form factor ix  KB  Vertical distance from the keel to the centre of buoyancy  KG  Vertical distance from the keel to the centre of gravity  L  Length  LWL  Waterline length of the hull  L/B  Length to beam ratio  ITTC  International Towing Tank Conference  ITTC '57  The friction correlation line adopted from the ITT conference of 1957  n  Normal vector to a surface  NPL  National Physics Laboratory, U.K.  OEC  Ocean Engineering Centre, operated by Vizon SciTec  p  Pressure  Re  Reynolds number, Re =  RF  Skin friction resistance  R  Total Resistance  T  Rw  Wave resistance  T  Draft  U  Velocity  v  Velocity vector  x  ACKNOWLEDGEMENTS The author would like to thank Dr. Sander M. Calisal for his supervision and direction during the course of studies, and for providing the opportunity to work on this project. Throughout this project his knowledge and expertise has been invaluable in guiding the numeric and experimental research, and without his tireless patience and many revisions, this thesis could not have been written.  The author would also like to thank Dr. Omer Goren, who was influential in guiding the initial research of the trimaran form while visiting UBC, and who provided his implementation of Dawson's method for multihull vessels with transom sterns. His support in both providing this program, and providing the technical support to help generate the input and output files, was essential for the numeric studies of parabolization.  Special thanks are also due to Dr. Bruce Dunwoody, Jon Mikkelson, Dan McGreer, Dr. Dan Walker and Dr. Dave Hally for sitting on the author's thesis committee. Their time, effort, and many useful suggestions for thesis revisions are very much appreciated.  Gary Novlesky at the Ocean Engineering Centre deserves special recognition, as he built the models of the NPL trimaran, and provided many hours of help with the tow tank carriage. His contributions were often made on short notice of deadlines, and have been very much appreciated.  Voytek Klaptocz and Kevin Gould have been close personal friends and colleagues since the start of this thesis, and their thoughts and efforts have been a great help in many aspects of this work.  Thanks are also extended to Tim Waung for his help with experimental work while he worked in the naval lab as a work term student.  xi  1.0  PROJECT MOTIVATION & BACKGROUND  Minimizing a ship's resistance has always been of importance to the naval architect. In the past this has been either to improve the performance of a vessel or to reduce the capital cost of installed horsepower. In today's environment, with escalating fuel costs and concerns over climate change, minimizing powering requirements becomes increasingly important in order to reduce both operational costs and environmental impact.  Traditionally, one of the fundamental approaches to minimizing resistance has been to minimize a vessel's beam, often through the use of a parallel mid section [26]. Increases to a vessel's beam increase the total wetted surface of a vessel and can increase the possibility of flow separation near the stern of a vessel.  Kent [23] conducted a systematic study on the effect of beam increases, beam to draft ratio, and the use of a parallel mid body. His study was limited to Froude numbers of 0.21 and lower, and concluded that the use of a parallel mid body with a decreased beam would reduce ship resistance. Studies by Harvald [15] and Schneekluth [33] also showed that increases in beam will lead to increases in resistance, particularly at moderate and high speeds. These studies have shaped the fundamental approaches of naval architects; however, in general the studies have been limited to Froude numbers < 0.25.  Many fishing boats and workboats seen in Canada's coastal waters find themselves operating above these low Froude numbers due to their short lengths. Additionally, with the advent of high speed sea transportation for cargo vessels and for today's high speed ferry lines, many longer ships also now operate at higher Froude numbers. At these Froude numbers a different vessel shape may be more beneficial than one with a long parallel mid body.  1  A typical resistance plot for a fishing hull, showing the components associated with wave making and with the viscous drag of the ship, is shown in Figure 1-1. As seen in this figure, when ships operate at higher Froude numbers, the wave making drag becomes the dominant resistance component. At these higher speeds, the use of appendages that act as wave cancellation devices is often beneficial. An example of one such appendage is the bulbous bow, which is seen on many large vessels.  0.10  0.15  0.20  0.25  0.30  0.35  0.40  0.45  0.50  Froude Number Figure 1-1- Resistance Components for a Typical Fishing Hull  1.1 Previous Work  Calisal, Goren and Danisman (2002) first observed the potential benefits of parabolization when conducting a resistance study on a retrofit for an oceanographic survey vessel.  The vessel was to be fitted with sponsons to  improve stability, and the widening of the vessel was expected to increase the resistance.  A numeric analysis was conducted to study the effect on EHP  requirements, and suggested the opposite conclusion, that replacing the parallel mid body of a vessel with parabolic lines at the waterline in fact lowered the wave resistance by ~ 22% at a design speed of 15 knots. 2  This spumed a collaborative research project between the University of British Columbia and Istanbul Technical University. In order to better understand the wave interactions caused by beam increases, a mathematical investigation into the differences between parallel mid bodies and parabolic waterlines was conducted within the limits of thin ship theory. This work was supported by a case study involving both experimental and numeric modeling of a coaster tanker.  In this investigation, two simplified hull forms are considered, as depicted in Figure 1-2. The first vessel is a wall sided vessel, with parallel mid body and parabolic and symmetric bow and stern sections. The second vessel shares the same bow and stern sections, but a bulb with parabolic waterlines is added in lieu of the parallel midbody.  Figure 1-2 - Hull forms for studying the effect of parabolic bulbs  Both hulls are analysed using a form of Michell's Integral derived by Wehausen & Laitone [39], that gives the wave resistance of a thin ship based on the slope of the waterline,/^: R  w  -  (1-1)  71C  where P{X) = \\dxdyf (x,^)exp  cos  (1-2)  1  x  J  V  J  and Q W = Wdxdyfx ( > y) x  e x  P  I  C  2 y sin — c  )  l  (1-3)  x  )  Since both hulls are symmetric about their transverse axis, the contribution from P is zero, and we need only consider the effect of the Q term on our wave resistance. 3  For the hull with the parallel midbody, the function Q will see contributions from the bow and stern. Since the parallel midbody has zero slope, it will have no contribution to the Q term. Our expression for Q then becomes: Q = Qb + Q ,en,OW  (1-4)  s  Since the bow and stern are identical, this reduces (1-4) to Q = 2Q  (1-5)  b0W  In the second hull, where the parabolic bulb adds curvature to the vessel, there is an additional contribution to the Q term. The equation for this hull becomes Q = 2Q +Q . b0W  (1-6)  bulb  Expanding theQ term in the integral of equation (1-1) gives 2  Q =4QL +^Q o Q u, +Qlif  (1-7)  2  b  W  b  b  Since the curvature at the bow and stern is significantly larger than the curvature in the bulb, Q > Qbuw  (1-8)  bow  and therefore QiL»QLb-  (1-9)  Neglecting the contribution of the Q  2  add the term Q Q bow  bulb  bulb  term, the effect of our parabolic bulb is to  to our wave resistance expression. Depending on the sign  of this term, the wave resistance will either increase or decrease, and the possibility for wave resistance reduction by hullform parabolization exists.  In the numeric study on the coaster tanker, the addition of a parabolic bulb increased the beam by 20%, while holding the length and draft constant. This resulted in an increase in displacement of 7%, but at the vessel's design speed of Fn = 0.275, the wave resistance was reduced by more than 15%. This numeric study was supported by experimental work conducted at the A. Nutku Ship Model Testing Laboratory, which showed the EHP requirements of the vessel were reduced by 10% at the same corresponding speed [3].  4  Tan and Sireli [34] performed a systematic experimental investigation into the effects of parabolic bulbs on the wave resistance of a typical west coast fishing vessel. As previous parabolization studies were conducted on tanker hull forms, the focus of this study was to examine the application of parabolic bulbs to smaller work boats, as well as to establish guidelines on appropriate sizes and locations for beam increases. Their study first fixed the beam increases amidships, and tested several bulbs that varied the size of the beam increase from 5% to 20%. Between Fn = 0.32 and Fn = 0.38 all bulbs tested showed an improvement in performance compared with the base hull. An increase in beam of 15% showed the largest decrease in resistance in this speed range, while exhibiting no substantial increase in resistance at higher speeds. Further increases in beam showed diminishing results, possibly due to adverse pressure gradients on the aft side of the bulb.  Following this work, the location of the 15% beam increment was varied both fore and aft of midships. Placing the bulb near the shoulder of the forebody resulted in too rapid a change in cross sectional area, and substantial increases were seen in the form factor. This increase was assumed to have been caused by flow separation. Moving the bulb aft of midships altered the wave pattern seen around the hull, shifting the improvements caused by parabolic waterlines to higher speeds.  The increase in beam of the vessel has the added benefit of  increasing the metacentric height, GM, of the vessel, and the conclusion of the study was that parabolic bulbs fitted at the waterline of the vessel could decrease total resistance by ~ 10%, while increasing payload capacity, and increasing the GM of the vessel by 47% [34].  The study by Tan and Sireli demonstrated the effectiveness of parabolic bulbs as a retro fit to existing vessels. However, for a new vessel design, the increase in displacement caused by the bulb would not be required, and it is more appropriate to compare vessels of both equal length and displacement in evaluating changes to hull form. With this in mind, Tan designed a new hull that matched the length 5  and displacement of the original fishing hull. The new hull increased the beam by 11%, and allowed the entrance angle and transom area to be reduced. The combination of these effects produced a hull with a decrease in EHP requirements of about 15%, while increasing the GM by 38% and maintaining the same payload capacity as the original baseline.  There have been several other studies based on applied theoretical work that have come to the conclusion that wave resistance can be minimized by replacing the parallel mid body of a vessel with some form of midship bulb.  Gotman [13] performed a systematic evaluation of Michell's Integral to investigate the discrepancies between calculations of the wave resistance at low Froude numbers and experimental results. Her study produced a new form of Michell's Integral that better accounted for wave interactions at low speeds. The complexity of this Integral does not lend itself to accurate numeric solutions for real hull forms, however, by representing generic hull forms with a series of 8 polynomial equations, Gotman is able to use her form of Michell's Integral to produce forms of minimum wave resistance.  None of the hull forms found by Gotman's numeric study exhibited a parallel mid body [13]. In each case, the waterlines are found to be continuously varying, and the hull form exhibiting the lowest wave resistance had a midship side bulb.  Hsuing [17] developed an optimization routine for finding ships of minimum wave resistance according to Michell's Integral.  Using this optimization  procedure, three optimum hullforms were designed, two of the hulls had bulbous bows, and the third had a parabolic midship bulb. This procedure developed bulbs while holding constant both the displacement and the beam of the vessel during the optimization.  6  Tow tank tests of these three hulls compared to a base hull with no bulbs revealed that for Fn > 0.28, the midship bulb reduced the residuary resistance, and in the range from 0.28 < Fn < 0.36, the midship bulb outperformed the two different bulbous bow designs.  1.2 Scope for this Thesis Previous parabolization studies have been focused on ships with low L/B ratios operating at moderate speeds. Tan's study focused on a fishing hull with a L/B ratio of 3.6 [34], and the first published work on parabolization by Calisal, Goren and Danisman studied a coaster tanker with a L/B ratio of 6.6 [3]. By extending the parabolization study to slender, multi hull vessels, it is hoped to gain an understanding of how the bulbs modify the wave interactions between hulls, and to use this knowledge to produce a trimaran with reduced resistance.  7  2.0  NUMERIC APPROACHES  The previous studies by Calisal, Goren, and Danisman, and Tan and Sirelli have primarily evaluated parabolization directly at the experimental level.  Systematic  experimental studies, such as that done by Tan, are both expensive and time consuming. In order to streamline the hull design process, and to rapidly evaluate the suitability o f parabolization for a given hullform early in the design cycle, the use o f numeric tools is necessary.  Consequentially, there is a need to assess different numeric tools to determine their effectiveness at predicting changes in resistance caused by increases in beam. While the accurate prediction o f a ship's total resistance is still a topic o f much research today, many useful numeric tools exist that can help a designer select the hull shape with minimum resistance.  To predict the effects o f parabolic bulbs, two different approaches to calculating ship resistance are implemented: M i c h e l l ' s Integral based approaches, and Rankine Source Panel methods.  2.1 A review of wave resistance theory To model the resistance o f a ship moving forward with constant velocity, the dimensional analysis must include the drag, D , length, L , velocity, U , density, p, and the viscosity o f water, ju. This situation is frequently seen in aerodynamics, where the Reynolds number and drag coefficient are sufficient to model the flow. However, when a ship moves forward across an otherwise undisturbed free surface, it also generates a characteristic set o f waves that behave as small amplitude gravity waves. Because o f this, the waves produced by a ship can be described by the Froude number, and the resistance o f a ship scales with the drag coefficient, Reynolds number and Froude number:  ^ L _ = c  (2-1)  M  8  In practice, the dependency of the drag coefficient on both the Reynolds number and the Froude number complicates model testing, and it is customary to use Froude's hypothesis that the total resistance can be decomposed into Reynolds and Froude based components: (2-2)  C (Re, Fn)» C (Re)+ C (Fn) D  F  R  The skin friction drag is the result of the viscous shear stress acting over the entire hull, while the residuary drag is composed of the wave making resistance and the viscous form drag.  The turbulent boundary layer is the primary contributor to the skin friction drag, and it acts over most of the hull's wetted surface. Since small changes in vessel dimensions result in only small changes in the wetted surface area, the nature of the skin friction resistance is such that one does not expect to be able to alter it much by subtle changes in design. In wave resistance, the emphasis is quite different. Here one expects subtle changes in the form of the design to affect the wave pattern produced by a vessel, and hence to change the overall wave resistance appreciably. There is obviously then a desire for the designer to be able to estimate the effect on the wave making of changes in hull shape.  The modern state of potential flow theory provides a useful tool to the design engineer in predicting the effects of changes in design on the overall wave resistance. While the theory does not allow for accurate numeric values for the overall resistance, as required for powering estimates, it does allow for qualitative assessments of hull forms to be made, enabling the evaluation of different hull design alternatives.  In order to use potential flow to determine the flow around our hull, we must first make a few important simplifications to our problem and define exactly the determining characteristics of our boundaries.  9  We consider our ship to be moving forward with a constant speed, c in the x direction. The ship produces a wave pattern, the elevations of which are given by V = f(x,y).  x, y  c Figure 2-1 - Definitions of terms  For the speeds involved in marine flows, water can be regarded as incompressible. Since we are separating the resistance components according to Froude's hypothesis, and analyzing the wave resistance as independent from the effects of viscosity, we make the assumptions that the fluid is inviscid and the flow irrotational, that is, Vxv = 0.  (2-3)  By making these assumptions, the velocity components are no longer independent of each other, and we can now represent the velocity field by a single equation, the velocity potential: b£" dj>_ V-(/> = i+ j+ bx by bz r  (2-4)  The continuity equation for an incompressible fluid is: „ _ bu bv bw V-v = — + — + — = 0 dx by dz n  (2-5)  Substituting the velocity potential into (2-5) simplifies the continuity equation to Laplace's equation: A ^ =&,+^+4=0  (2-6)  And Bernoulli's equation for a frame of reference fixed with the ship becomes:  10  ^ [ ( & - c ) +<f) + $ 2  2  2  y  + gz + -p = const. P  (2-7)  Since we have assumed the fluid to be inviscid, the no-slip condition of a viscous fluid no longer applies on the surface of our hull. Instead, unless our hull were to have a hole in it, the flow must be tangential to the hull. This is expressed as: v-« = 0onS  (2-8)  Where n is the outward normal vector for any given point on the wetted hull surface, S. This condition, which must be satisfied everywhere on the hull, is the impermeable boundary condition for the wetted surface of our hull.  Since the wave produced by our ship is steady in time with respect to our hull, we know that the velocity at the free surface must be tangential to the free surface. This means (2-8) applies to our free surface as well. This is the impermeable boundary condition for the free surface at z = n(x,y).  Sufficiently far upstream of the boat, the disturbance the boat creates must vanish, leaving us with a radiation condition for the velocities: limU +^ +^ ] = 0 2  2  (2-9)  2  z  And also for the elevations of the free surface: limfV-^l = 0  (2-10)  Further, the pressure at the water's surface is constant, leading to an expression for the free surface elevations in terms of Bernoulli's equation: g"-c<f> +^[tf+</> +ti] 2  x  y  =0  on z = n(x,y)  (2-11)  This is the dynamic boundary condition on our free surface.  Also, if we consider deep water, we have the boundary condition on the bottom: \\md{x,y,z) = Q  (2-12)  Z-»-oo  If we were considering the case of shallow water we would replace the limit of z ->• -oo with some finite distance, -h. 11  This leaves us with the mathematical problem of finding a solution for the velocity potential, <j>{x,y,z), and for the free surface elevations, n(x, y), subject to boundary conditions (2-8), (2-9), (2-10), and (2-12).  2.2 Solution of the Wave Resistance Problem Using Thin Ship Theory  In order to obtain a solution for the velocity potential it is first necessary to linearize the problem. While Laplace's equation is of course linear, an inspection of the boundary conditions shows that only (2-9), (2-10) and (2-12) are linear. In addition, the location of the free surface where the boundary conditions are to be satisfied is unknown. Linearizing the problem will yield a solution that is only an approximation to the real solution, therefore it is necessary to ensure the steps that we take to linearize our solution still provide an accurate representation of the actual problem. A fairly general approach to seek a solution to the velocity potential is based on the concept of systematic perturbation. In a perturbation analysis, we express the total potential as a sum of two different potential functions: </) = <p +8<l>.  (2-13)  {0)  Here ^  (0)  is a known or easily computed basic potential, and 8</> is a small  perturbation potential that perturbs the basic potential by only a small amount. In the limiting case when our perturbation parameter approaches zero, we recover our basic potential as our total potential solution.  In practice our perturbing  parameter cannot be described so simply as just 8<f>, but rather consists of a Taylor series expansion involving some small dimensionless parameter, s.  Since the fluid we are considering is inviscid, in the limit that our ship becomes vanishingly thin, it would pass through the water with no disturbance. In this case our known solution is the undisturbed uniform stream flow. Assuming that our ship is narrow, that is the beam to length ratio is small, we use the beam to length 12  ratio as our perturbing parameter, e, and our perturbation of the uniform stream flow becomes: * = 0 +fi* +*V +... (o)  (,)  (2-14)  (2)  Also in this limiting case of a vanishingly thin ship, we simplify the ship hull boundary condition to apply on the ship's centreplane, rather than on the hull itself.  In this case, the velocity now becomes tangent to the ship at the  centreplane, rather than on the hull itself, so one must remember the thin ship limitations in order to expect accurate results.  Using this simplified form of the boundary conditions and the velocity potential, we can now arrive at a solution for the wave resistance. This is done either by integrating the pressure distribution around our hull to find the resistance force, or by considering the free surface elevations downstream of the hull, and finding the wave resistance based on energy considerations.  The former approach is the original approach taken by Michell in 1898, and the solution is (1-1), the well known Michell's Integral. The latter approach yields the same equation and was developed by Havelock. This approach uses Havelock sources distributed along the centreplane of the vessel in order to simulate the flow around the vessel.  Tuck implements an equivalent form of Michell's Integral in his "Michlet" software package: (2-15) Where P and Q are functions of  F(X,6),  P(0)= JF(x,d)cos(>ccsecd)dx  (2-16)  Q(e) = \F(x, 9) sin(/ec sec 6)dx  (2-17)  and  13  The function  F{X,6)  is determined by the hull offsets, y = ±Y(x,z)rather than the  slopes of the hull surface, as were used in (1-1): F(x,e)= JY(x,z)e'  dz  (2-18)  asec20  In order to obtain this new form of Michell's Integral, an integration by parts is used to express the hull in terms of the hull offsets rather than the longitudinal derivative, and the triple integration required in the original form of the equation has been replaced by three separate integrals.  These manipulations allow the  wave resistance to be calculated rapidly using an ordinary desktop computer. By utilizing the hull offsets, rather than the slopes of the hull, the time required to develop an appropriate representation of the hull is also substantially reduced.  While Michell's Integral provides a quick method for calculating the wave resistance based on the hull offsets of a vessel, the final expression is the result of several approximations which limit the applicability of the theory, and are summarized below: •  The fluid is assumed to be inviscid and the flow irrotational, neglecting flow separation and viscous forces. The hull is slender, that is the length to beam ratio is large.  This is  important for two reasons: o  Since the method is developed on a perturbation analysis using the undisturbed flow as the known potential, the disturbance of the ship must be small,  o  The hull boundary conditions are approximated as occurring on the vessel's centreplane. This requires the slope of the hull surface relative to the centreplane to be small everywhere to ensure the directions of the velocity vectors along the hull are approximately representative of the true flow conditions.  •  The waves generated by the ship have small wave heights compared with their wave lengths. This is necessary to ensure that the waves are properly modeled as gravity waves. Phenomena such as wave breaking are also neglected in this assumption. 14  The ship does not experience sinkage or trim.  The first approximation is fundamental to establishing a potential flow, but has some important implications.  Vessel's with a transom stern introduce a  singularity in the potential flow formulation and of course introduce flow separation in the real vessel.  The transom therefore requires some form of  correction in the Michell's Integral formulation to ensure continuity. In the case of Tuck's software, the transom is regarded as dry and the wake is represented as a straight impermeable cylinder which continues to infinity downstream with the cross section of the wake identical to the shape of the transom [36].  The formulation also does not take into account the sinkage and trim experienced by a real vessel, which will change the wetted surface of the hull and the actual wave making form of the vessel. A simple solution to account for this is to calculate the trim moment and heave force and adjust the underwater portion of the hull in an iterative manner until the trim moments and heave forces approach zero for a given speed. This is of course a time consuming procedure, and for the case of long vessels that trim little, the change in trim and heave can generally be neglected without an appreciable loss in the accuracy of the solution.  2.3 Boundary Element Methods In the preceding section the uniform stream flow was taken as the starting solution to a perturbation analysis.  An alternative approach developed by  Dawson is to combine the underwater portion of the hull with its mirror image reflected about the undisturbed free surface. In this approach, the actual free surface is approximated by a rigid wall, and the method of images can then be used to construct the streamlines around the hull. In the limiting case of zero Froude number, the rigid wall approximation becomes exact, and the streamline flow around the double model becomes the known solution for a perturbation analysis:  15  0 = <D + <5#  (2-19)  Here O is the double model flow at zero Froude number and the actual free surface elevations provide the perturbance.  With this approach Dawson  developed the free surface boundary condition by using the double body linearization about the fixed free surface:  (ofal+gf^ltfQ,,.  (2-20)  The subscript / denotes differentiation along the streamlines at z = 0, and the streamlines of the free surface flow are approximated as having the same flow direction as the double model streamlines in order to obtain a solution.  Without implementing the upstream radiation condition (2-9), there is not a unique solution to this perturbation approach, as waves which propagate upstream can satisfy the free surface and hull boundary conditions as easily as those that propagate downstream. However, in a 3-D flow, the upstream radiation condition becomes numerically expensive to implement. Dawson's approach avoids the numeric complexity of this boundary condition by not explicitly enforcing it. Instead, in the numeric calculation of derivatives along the double model streamlines, a four point upstream finite differencing scheme is used tofilterout upstream propagating disturbances.  The Green function used in the double model flow that satisfies Laplace's equation is the simple Rankine source, which has the velocity potential: G{x,y,z;£)  (2-21)  = —1— 4m-  This function is much simpler to evaluate than the source function used in thin ship theory, which is the Kelvin or Havelock source, with velocity potential: G(x,y,z;C) =  (\ *"A * dB  4K  [ •dk  ik(xcos8+ysind)  e  -k\z-(\  k + k sec 9 2  a  ^  k-k„ sec 9 2  (2-22)  16  (z+(r)  This simpler Rankine's source function could be seen as an advantage of the double model flow, however, implementation of the Rankine source requires discretizing both the hull surface and a portion of the free surface into panels. Since the double model linearization is based on streamwise differentiation, the free surface mesh is required to follow the streamlines of the double model flow. This streamwise mesh can be determined first from a separate pre processor. Then, after discretizing the hull and free surface, a constant source strength distribution is considered over each panel. It is important to note that since the hull boundary conditions are imposed on the panels of the hull surface, and the perturbation is of the double model flow rather than the undisturbed free stream flow around a thin body, this approach is not limited to thin ships. This has the obvious advantage that many real ship geometries can be modeled by Dawson's method.  By integrating the contributions from each panel, the potential flow around the double model can be calculated: j(x,y,z) = -\\-±-  ' r(x,y,z;£,n,Q  (2-23)  v  JJs  Where a is the source strength, (x,y,z) denotes a field point on the free surface, and (&n,Q denotes the location of the source point.  The velocity field at any  point, /, is then determined by differentiation of the potential equation: u. =  v, =  w, =  (8^  N  ( 80 7=1  f 8<^  N  Yo-.C. 4-1  J  lj  Here Ay, By, and Qj, are the influence coefficients for the components of the velocity at point /, induced by a unit source distribution over the panel with its centroid at point j. To determine the source strengths on the panels, the boundary conditions are imposed on the rigid body and on the undisturbed free surface. 17  The resulting problem is a linear set of M equations with M unknowns for the source strengths. This matrix is full, and not diagonally dominant, so it is solved by Gaussian elimination. Once the source strengths are determined, the wave elevation is determined from: 7 = ^ [ c  2  + ^ - 2 ^ , ]  (2-25)  and the pressure by: /? = - ^ [ c + V ^ - V ^ - 2 V ^ - V O ]  (2-26)  2  The wave resistance can be determined by integrating the pressure on each panel of the hull, with area Ac. * * = i > i » „ 4  (2-27)  ;=i  Where L is the number of panels on the wetted surface of the hull, and n is the x x  component of the normal vector. The heave force and trim moments are determined in a similar fashion:  1=1  M  y  = J PA 1=1  [n z, - n X/  Zi  (x, - LCF)]  (2-29)  As was the case for thin ship theory, the effects of sinkage and trim are not accounted for in Dawson's approach. Since the heave force and trim moment are calculated though, an iterative approach can be used to calculate the exact wetted surface of the hull. However, this requires generating a new mesh for each speed and is very time consuming. A n estimate of the wave resistance suitable for engineering studies can usually be achieved by neglecting the effects of sinkage and trim.  In the panel method, continuity is required of the normal vector along the hull surface. The presence of a transom stern violates this condition, and the existence  18  of a solution is no longer assured.  Cheng [5] first solved this problem by  removing all panels from the transom and leaving it open as depicted in Figure 2-2.  Figure 2-2 - Open transom  With this approach, the flow around the hull is assumed to detach from the hull at the stern, and continue in a straight path downstream to infinity.  The open  transom is then treated as a fictitious inflow, where water begins at the transom and flows in a straight path aft of the transom. Thefictitiousinflow begins at a boundary panel located inside the hull, with its downstream edge lying on the transom itself.  The flow behind the transom is handled in two different states, depending on the speed of the ship. For low and moderate speed flows, that is Fn < 0.35, the transom is considered wet, and for high speeds, Fn > 0.45, the flow is considered to clear the transom and it is regarded as dry. The region from 0.35 < Fn < 0.45 can be considered to be a transition region, where neither state is exactly representative of the actualflowconditions.  At the transom, the flow underneath the boat clears the surrounding water and is exposed to the air. This means the static pressure can be equated to atmospheric pressure: PT -  P*>  at  X  = T X  A  N  Q  L  Z  = T Z  f°  ra  given y.  19  (2-30)  Bernoulli's equation can then be written as a balance of the total kinetic and potential energy, and rearranged as Cheng first did to give the first boundary condition for the flow at the transom: l + l + f ? . , . ^  c  ( 2  .  3 1 )  c  In this equation z is the height of the transom, which will lie below the mean T  water level, and have a negative value.  The second boundary condition is that the flow must leave the transom tangential to the hull, and is enforced using the upstream edge panels along the hull to calculate the normal vector.  The wet transom boundary condition is enforced by assuming the fluid behind the transom moves with the advance speed of the ship. Then the usual free surface boundary conditions can be imposed by including the boundary panels as part of the calculation.  20  3.0  IMPLEMENTATION OF PARABOLIZATION  Previous parabolization studies have focused on ships with a moderate or low length to beam ratio, where the wave making resistance represents a large percentage of overall resistance. In order to examine the benefits of parabolization as wave making decreases, a high speed, slender hullform was sought. Such a hullform can show the benefit of parabolic waterlines on some of the latest concepts in high speed, high performance ships.  The focus of this thesis is to apply the concepts of waterline parabolization to a trimaran design representative of recent research work in high speed ferries and military applications. The parabolization procedure will be applied to the lines of the centre hull only, since the shorter outriggers operate at too high a Froude number for wave resistance reduction by this method. The use of a multihull form in this study allows investigation into what role reduced wave heightsfromthe centre hull might play in wave interactions. 3.1 Implications of Parabolization on Complete Vessel Design  The concept of using waterline parabolizationfirstbegan with the addition of sponsons to an existing vessel. Several studies since then have shown the benefit of adding such shapes to current hull forms. However, integrating parabolic waterlines into the complete hull design, rather than as a retrofit, offers an even greater potential for resistance reduction. Placing additional displacement in the form of a parabolic bulb near midships allows the naval architect to remove itfromthe two areas that influence the wave making the most: the bow and stern. By increasing the beam of the vessel by removing parallel mid sections and instead parabolizing the waterlines, the GM of the vessel will be increased, and both the entrance angle at the bow and the transom area at the stern can be decreased (see Figure 3-1).  If we recall our symmetric ship from chapter 1, we can quickly examine the benefit that parabolization can offer the naval architect.  21  Figure 3-1 - Reduction in entrance angle and transom area caused by parabolic bulbs  If we assume that by adding a parabolic bulb, we are able to reduce the curvature in the bow and stern of the symmetric ship by 10%, we expect a 10% reduction in the Q term in Michell's Integral: Q{X)= j\dxdy0.9f (x,y)exp  f  v  ' sin  x  J  Q(A) = 0.9 \\dxdyf {x,y)ex4>  sin  s V  x  V  V  ^x  (  „2  X  )  (3-1)  g£  J  Since the Q term is then squared in Michell's Integral, this 10% reduction in the slope produces a 19% reduction in the wave resistance. Tan and Sireli reduced the entrance angle and transom area of the UBC series fishing hull in this manner, and succeeded in substantially reducing the total resistance of this vessel. 3.2 Motivation for Studying the Trimaran Hullform  All vessels must possess adequate stability to ensure safety of crew and cargo while at sea. The stability of a ship inclined by a small amount is dependant on the GM, which can be determined from: (3-2)  BM = -3-, GM = KB + BM - KG V  For a monohull, the transverse moment of inertia is increased by increasing the beam of a vessel. Therefore, a minimum beam is required for safety of the vessel and for sea going operations. For the majority of cargo carrying vessels, this is not generally a concern, as the economics of carrying a large payload at a slow speed already dictates a full form with wide beam.  However, for vessels  intending to travel at high speeds, the implication of having a wide beam is that the vessel must push a large volume of water aside in order to make way. This imparts significant energy into the wave system, results in pronounced wave making, and therefore increases the wave making resistance of the vessel. 22  An effective way to overcome this stability limit is through the use of a multihull form. The outriggers of a trimaran stabilize a slender centre hull that would on its own lack adequate stability. For high speed monohulls, L/B ratios are limited to around 9.0 [29], as more slender hulls cannot achieve adequate stability. Using outriggers to create a trimaran hull form, the L/B ratio can be increased without risking an unstable hull.  As a result, trimaran hulls are characterized by  extremely slender hulls that minimize the energy lost to wave generation. The outriggers may also act as wave cancellation devices. Depending on the intended operational speed, the transverse and fore and aft spacing of the outriggers can be optimized to provide different wave cancellations.  For slow speed operations, the benefits of the trimaran form may not be realized, as the increased wetted surface area of the multihull configuration results in an increase to the viscous drag. In this region, where viscous forces are dominant, a monohull form that minimizes wetted surface area shows superior performance. 3.3 Trimaran Hullform Particulars The centre hull of the trimaran for this study is an NPL standard series hull form, with an L/B ratio of 13.5. The waterline length of the vessel is 160 metres, with a draft of 6 metres. Each outrigger is also an NPL hull, with a length of 57.7m and an L/B ratio of 22.5. The longitudinal axis of the outriggers lie 14.2 m from the centre of the vessel, and the outriggers are positioned in an orientation ideal for high speed operations. The vessel particulars are summarized in Table 3-1 and the lines plan is shown infigure3-2. A table of offsets for the centre hull and outriggers is given in appendix A.  23  BOW  WATERLINE ELEVATION Figure 3-2 - Lines plan for the NPL trimaran  Hull Form Trimaran  Centre Hull  Port Outrigger  Stbd Outrigger  L\\i |m|  160  160  57.7  57.7  A[LT]  4980  4700  140  140  B[m]  31  12  2.6  2.6  T[m]  6  6  2.4  2.4  Transverse Hull Spacing (x), [m]  -  -  14.2  14.2  Table 3-1 - Main Particulars for the NPL Trimaran  3.4 Feasibility of Parabolization Study To provide a systematic approach for applying the parabolization principles to the high speed trimaran, the centre hull is isolated for analysis first, before including the effects of the outriggers and their wave interactions. While this centre hull would be unstable on its own, analyzing the complete trimaran alone would make it difficult to distinguish the effects of the outriggers from the parabolic bulbs.  24  In order to justify a parabolization study, the magnitudes of the resistance components were estimated to ensure that such a slender ship still produced a significant wave worth optimizing. The wave resistance was estimated using thin ship theory, and the viscous resistance was estimated using the ITTC '57 friction correlation line. The results of the resistance predictions are shown in figure 3-3. For most speeds, the viscous and wave making components are of nearly identical magnitudes. This is quite different from Figure 1-1, which shows the resistance components for a fishing vessel. For the fishing hull, it is evident that for all moderate to high speeds, any significant reduction in wave resistance will greatly improve the powering characteristics of the fishing vessel. Figure 3-3 shows that for the high speed monohull a delicate balance exists between the wave and viscous components. As any increase in beam will likely result in larger viscous drag through both an increase in wetted surface area and a potential increase in the form factor, real reductions in total resistance become more difficult for this hull since the wave making resistance grows slowly with speed.  For the moderate and lower speed ranges, there are two significant humps in the wave resistance predicted by Michell's Integral. The first is centred near Fn = 0.25 and the second is centred near Fn = 0.32. These moderate speed humps are usually associated with wave interactions between the bow wave and the shoulder wave system, and previous parabolization studies have shown reduction in wave resistance at these Froude numbers [3]. It is also important to note however, that at the lower speeds (Fn < 0.30), the assumptions of thin ship theory can often over predict wave interactions.  25  Estimated Resistance C o m p o n e n t s 1400  i  -i  0.20  0.25  0.30  0.35  0.40  0.45  0.50  Froude Number  Figure 3-3 - Resistance Components for the NPL monohull estimated from thin ship theory and the ITTC '57 line  3.5 Implementation of Parabolization  While the preliminary analysis by Michell's Integral is not conclusive in the low to moderate speed ranges, there are two specific regions that may offer room for substantial optimization. Even if Michell's Integral has overpredicted the effects of wave interaction in this region, we expect the wave resistance to be at least of the same order as the viscous resistance.  In this case, if we are able to, for  example, reduce the wave resistance by 10% while increasing our viscous drag by only 5%, we would still produce a hull with reduced resistance.  A parametric study, varying the length and position of beam increases, is performed on the centre hull to identify side bulb shapes that offer potential drag reductions. A matrix, showing the overall scheme for the numeric testing is given in Table 3-2. In each instance, the beam of the vessel is increased by 11%. Tan's study on the UBC series fishing hull found that an 11% beam increase produced an ideal balance of decreased wave making and increased viscous resistance[34]. Due to the length of the NPL hull, and the scale factors involved in experimental 26  work, which will be discussed later in further detail, the beam increment was not varied as it was felt it would be too difficult to discern the effect of small changes in beam.  Monohull Phase  Trimaran Phase  Base hull  Base hull  Hull #1  Small bulb, centred midships  Small bulb, centred midships  Hull #2  Small bulb, fwd of midships  Small bulb, fwd of midships  Hull #3  Small bulb, aft of midships  Small bulb, aft of midships  Hull #4  Large bulb, location determined from tests w. small bulb  Large bulb, location determined from tests w. small bulb  Table 3-2 - Overall Test Matrix  Figure 3-4 - The initial hull forms for study: Hull 1 (top), Hull 2 (middle) Hull 3 (bottom)  27  4.0  ANALYSIS BY MICHELL'S INTEGRAL  Michell's Integral is implemented as described in section 2.2 by Tuck et al. [35] in the software package Michlet. Michlet requires two sets of input files: one to describe the wetted portion of the hull geometry and another to describe the parameters of the analysis. Following analysis, Michlet outputs the results of the calculations into several output files. The details of the I/O process are summarized in Figure 4-1.  Analysis Parameters and Vessel Particulars: in.mlt  Hull Geometry Files: useroffl .csv useroff2.csv <more hull files if necessary>  Michlet  Categorized Output Files: Same output as main files, but output separated into Main Output Files: individual files out.mlt ship_output_by_speed. mlt  Free Surface Elevation Files <user specified name>.mlt  Figure 4-1 - Michlet datafilestructure  The hull geometry is described in the useroff.csv input files, by a non dimensional table of offsets. This must be input as a rectangular grid of points, regardless of the actual hull shape. For most ships this results in several of the offset values near the bow and stern having half breadths of zero. However, if these zero half breadths are not entered, a program error results.  28  The parameters of the analysis, including the overall vessel dimensions, water properties, speeds for analysis, trim and heel angles, as well as calculation methods and various other parameters are described in the main input fde, in.mlt.  The hull geometry can  alternatively be described in the in.mlt file, but this is not a recommended practice, since it leads to impractically large text fdes, and complicates the fde automation process.  Once the appropriate input files are generated, the Michlet processor can be started in order to analyze the resistance. After calculating the resistance, Michlet automatically generates a series of output files, two of which, the main output fdes, contain all of the analysis results.  For flexibility, the same information also outputs into several  categorized files. Michlet can also calculate the free surface disturbance downstream of the ship. The views of the free surface are controlled by the user, and output can be saved to fdes for analysis by different post processing programs if required.  In order to analyse the trimaran hull, a pre processor has been developed to discretize the below water portion of the CAD model for both the centre hull and the outriggers. In order to generate the hull offsets in an appropriate manner for Tuck's software, the hulls are discretized by first creating a rectangular grid of points beside the hull, and then projecting that grid onto the wetted portion of each hull. A total of 21 equally spaced points are used per station with 53 evenly spaced stations. All of the offsets are non dimensionalized using the waterline length, and the hull is later scaled to full scale by the inputs from the parameter fde, using the displacement and waterline length of each hull. This results in a mesh for each hull consisting of 1113 points.  In order to calculate the viscous drag, and to estimate the powering requirements of the hull, the viscous resistance coefficient is calculated by using the ITTC '57 correlation line: C „ = - ^ « (log Re-2)  (4-1)  10  The viscous drag is then calculated by multiplying by the dimensions of the flow: R =\c V S  (4-2)  2  F  FoP  29  The ITTC '57 line is an empirically derived correlation line that does not account for the physics of the flow. The resistance coefficient is based on the length of the hull, and is similar to flat plate resistance coefficients used in experimental approaches. The only variable that is changed between different hull configurations is the wetted surface, and so it is to be expected that the viscous resistance predictions change very little between hull shapes.  In reality the different bulbs may cause larger changes in the viscous resistance, as each bulb will alter the pressure distribution in a unique manner, causing the boundary layer growth along the hull to change between designs. Flow separation may occur if the bulb ends too abruptly [24],[34].  Despite the shortcomings of this empirical approach, the ITTC '77 method provides a quick means to account for the viscous resistance. Assuming that changes to the viscous resistance are small relative to the wave cancellations achieved by bulb design, quick estimates of the impact on horsepower requirements can be obtained.  4.1 Parabolization for Monohull  The total resistance for the base hull, and for hull configurations 1 - 3, is calculated for the speed range from Fn = 0.2 to Fn = 0.5. For the full scale hull, this Froude number range corresponds to speeds of 7.92 m/s to 19.81m/s (15.4 kn to 38.5 kn). Plots of the wave resistance and changes in the total resistance are shown for the different hull configurations in Figure 4-2 and Figure 4-3. The plots are presented in this manner to illustrate both the bulbs impact on wave interactions, and also the overall impact of these interactions on the total resistance.  30  700  W a v e R e s i s t a n c e - B a s e hull v s S m a l l B u l b Configurations Base Hull  600  Hull 1 500  Hull 2 Hull 3  2  400  I  300 200 100 0 0.20  0.25  0.30  0.35  0.40  0.45  0.50  Froude Number Figure 4-2 - Wave resistance for hulls 1-3 vs base hull  Percentage C h a n g e in R  T  10%  i a)  5%  V)  to  5  <D >  | © D) CO  o  0% 0.50 -5% * • Hull 1 ---Hull 2 -10%  — Hull 3  -15%  Froude Number Figure 4-3 - Percentage change for hulls 1 - 3 relative to base hull  31  Figure 4-2 shows the predicted wave interactions caused by the parabolic bulbs. Improvements in resistance are seen in the two regions identified for optimization in Figure 3-3.  Hull configuration 1, with a parabolic bulb placed aft,  demonstrates improved wave making compared with all other designs at the Fn = 0.27 hump, while all three hull variations reduce the wave resistance at the Fn = 0.32 hump by a comparable margin relative to the baseline.  The reduction in  wave resistance in this region is greatest for hull 3 at approximately 8.5%, which results in a predicted reduction in powering requirements of about 5%.  Keeping in mind that changes to the viscous resistance not predicted by the ITTC '57 line may offset a portion of the gains predicted in the wave resistance, the initial study does show that beneficial wave interactions should at least be possible. The predicted reductions in EHP at full scale, shown in Figure 4-4 are appreciable, at nearly 500 hp for speeds near Fn = 0.32.  Effective Horsepower - Base hull vs Small Bulb Configurations 14 000 Base Hull  12 000 H 10 000 a. £ 8 000 Q.  5  6 000 4 000 2 000 0 0.20  0.22  0.24  0.26  0.28  Froude Number  Figure 4-4 - Powering requirements for hulls 1-3 vs base hull  32  0.30  0.32  0.34  The most successful hulls at the moderate speed hump near Fn = 0.32 are hulls 2 and 3. The parametric study is expanded by blending these two hulls to create a hull with a larger bulb. This fourth hull configuration, hull 4, is shown in Figure 4-5. Repeating the above resistance calculation procedure, the wave resistance and total resistance predicted for hull 4 are compared to that of the base hull in figures 4.6 and 4.7. More substantial wave interactions are seen in the moderate speed range, with a maximum reduction in wave resistance of about 13% occurring at Fn = 0.32. After factoring in the viscous resistance component, a reduction of about 7% is anticipated for the total resistance at this stage. In terms of powering requirements, this amounts to a predicted savings of about 750 hp at Fn = 0.32.  Figure 4-5 - Hull configuration 4  Comparison of Wave Resistance - Base hull vs Large Bulb Configuration 700 i  0.20  0.25  0.30  0.35 Froude N u m b e r  Figure 4-6 - Wave resistance for hull 4 vs base hull  33  0.40  0.45  0.50  Percentage Change in R  T  0.45  0.50  Froude Number Figure 4-7 -  Percentage change in total resistance between hull 4 and base hull  4.2 Parabolization of the Trimaran Hullform  As earlier mentioned, while the monohull case does allow full investigation into the effect of the bulbs on the wave making, ultimately, the centre hull is too slender to be stable without the addition of outriggers. The trimaran case studied represents a realistic hullform, similar to that used in recent military concepts and car ferry designs. The resistance for the base trimaran is calculated in the same manner as in section 4.1. In order to give a comparison of the benefits of the trimaran form at high speeds, the resistance of the trimaran is compared to an equivalent monohull, as well as to the base centre hull in figures 4-8, 4-9 and 4-10. The equivalent monohull is derived by matching the displacement of the trimaran with an NPL monohull, and scaling the overall dimensions of this monohull to have an L/B ratio of 9.0 in order to ensure adequate GM. The resulting hull has a length of 138.3 m, and a beam of 14.7 m. While experimental data exists in the public domain for such an NPL hull [1], in order to ensure a fair comparison between designs, the resistance shown in the following figures has been 34  calculated using the method of section 4.1. It should be noted that the difference in length between the equivalent monohull and the trimaran results in the two vessels operating at different Froude numbers, as seen in the humps and hollows of the wave resistance plot. The difference in lengths also means a different Reynold's number, so the viscous resistance coefficients will be different.  Total R e s i s t a n c e - Trimaran v s M o n o h u l l 2000 i  o  -I  1  1  :  1  i  i  8  10  12  14  16  18  20  U [m/s]  Figure 4-8 - Total resistance for trimaran, centre hull and equivalent monohull  35  Viscous Resistance - Trimaran vs Monohull 900  T  0 -I 8  ,  ;  ;  ;  ,  ;  10  12  14  16  18  20  U [m/s]  Figure 4-9 - Viscous resistance, trimaran, centre hull and equivalent monohull  36  W a v e R e s i s t a n c e - Trimaran v s M o n o h u l l 1200  8  10  12  14  16  18  20  U [m/s]  Figure 4-10 - Wave resistance for trimaran and equivalent monohull, showing effect of L  W L  The additional wetted surface area of the outriggers increases the viscous resistance by about 26%, as calculated by the ITTC '57 line. The outriggers cause substantial changes to the wave pattern of the vessel, resulting in beneficial wave cancellations at higher speeds, seen in Figure 4-10. These wave cancellations are to be expected, as Yang [41] identifies the outrigger configuration used in this study as a beneficial orientation for high speed operation. At moderate to high speeds, the wave resistance is substantially reduced by the placement of the outriggers, more than offsetting the increases in the viscous resistance.  This  produces a hull with superior powering characteristics.  The parabolization study of section 4.1 is repeated for the trimaran, now including the effects of the outriggers. Plots of the wave resistance and percentage change in the total resistance are given in Figure 4-11 and Figure 4-12 for the base trimaran compared with hull configurations 1-3. The resistance predictions are  37  similar to those seen for the monohull case. The different bulb orientations are again predicted to reduce the wave resistance at moderate speeds, with hull 3 predicted to reduce wave making by 7% at Fn = 0.32.  This corresponds to a  reduction in EHP of about 3.5%, or 500 hp. W a v e R e s i s t a n c e - B a s e Trimaran v s S m a l l B u l b s 700 i  100 ^  0  -I  0.20  j  0.25  i  j  0.30  0.35 Froude Number  Figure 4-11 - Wave resistance for the parabolized trimaran  38  j  0.40  i  i  0.45  0.50  Percentage Change in R - Trimaran T  -6%  J  Froude Number Figure 4-12 - Percent change in R for the parabolized trimaran T  Hull 4 is also tested in the trimaran configuration.  This hull shows improved  resistance characteristics in comparison to the base hull and the three small bulb hull configurations. The wave resistance is reduced by 12% at Fn = 0.32, which translates into a reduction in EHP of 6%, or 800 hp. Plots of the wave resistance, and change in the total resistance are given in Figure 4-13 and Figure 4-14. Meaningful wave interactions for hull 4 extend further into the high speed region in the trimaran configuration than was seen for the monohull. Significant wave cancellation is seen up to Fn = 0.38, whereas the addition of the bulb to the centre hull alone gave improved wave cancellations until Fn = 0.37, with appreciable cancellations dropping off around Fn = 0.34.  39  700 600 500  Wave Resistance - B a s e Trimaran v s Large B u l b  —  Base Trimaran  — Hull 4  400 * 300 200 100 0 0.20  0.25  0.30  0.35  0.40  0.45  0.50  Froude Number Figure 4-13 - Wave resistance for hull 4 vs base hull in trimaran configuration  Percentage Change in R for Hull 4 T  0.50  Froude Number Figure 4-14 - Percentage change in R for hull 4 in the trimaran configuration T  40  4.3 Conclusions on Analysis by Michell's Integral  Implementation of the thin ship theory of Michell provides an extremely quick numeric method for calculating the wave resistance.  Calculating the far field  energy of the wave profde provides an efficient computational method, and the determination of a ship's powering curve takes less than 30 seconds using a standard desktop PC. The use of hull offsets to discretize the hullform provides a rapid method to calculate the wave resistance using either a lines plan or 3D CAD model as a starting point. The use of semi-automated macros to generate the input files from the CAD geometry requires on average less than 15 minutes to generate powering estimates from the hull geometry.  Another excellent advantage of Tuck's implementation of Michell's Integral is that it allows the transverse and longitudinal separation of multihulls to be changed with a single variable in the program input files.  This provides an  excellent tool for determining optimal spacings of catamarans and trimarans.  Despite these advantages, there are some very real limitations on the applicability of using thin ship theory and far field wave energy considerations to calculate the resistance of a ship. As discussed in section 2.2, the essential assumption of thin ship theory is that the beam of the ship is small compared to the other principal dimensions. This limitation arises since the potential flow of the undisturbed free stream flow is taken as the basis for a perturbation analysis in this approach. While it is difficult to give an exact definition of when a ship can truly be considered thin, the slender hulls of a multihull vessel certainly fall into this category.  A drawback of the far field calculation of the wave resistance, in terms of bulb design, is the lack of intuitive feedback to the designer. Figure 4-15 and Figure 4-16 show plots of the wave patterns created at a Froude number of 0.32 by hull 3 and hull 4 respectively. While the plots do show differences in wave heights, it is difficult to gain insight as to how the differences in bulb shape between the two 41  designs have caused these changes in wave height. From this perspective, it is desirable to be able to visualize the pressure and velocity contours along the hull, as well as to visualize the free surface contours in the immediate vicinity of a bulb.  Figure 4-15 - Wave profile created by hull 3 at Fn = 0.32  42  Figure 4-16 - Wave profile created by hull 4 at Fn= 0.32  Another limitation o f M i c h e l l ' s Integral is its calculation o f the wave resistance at lower Froude numbers. Newman [30] shows that below Froude numbers o f about 0.32 for a destroyer hull form, the thin ship theory exaggerates the effects o f interference on the wave resistance.  The actions o f viscosity are likely to play a role i n these exaggerations, as it is possible that the viscosity dampens wave interactions when wave heights are small and the speed o f the ship is slow.  Another possible cause for these  exaggerations is the thin ship limitation. The thin ship limitation requires that the beam is small compared to not just the length, but for all length scales o f the problem.  The transverse wavelength o f the ship's wave becomes o f the same  order as the beam as speed decreases, since X^U . 1  Because o f this it becomes  inevitable that M i c h e l l ' s approximation breaks down as the Froude number becomes small.  However, it is also worth while noting that at these speeds the  viscous terms become more important that the wave resistance terms.  43  For the trimaran under study the significance of this is that the hump in the resistance curve centred around Fn = 0.25 should be treated with caution, and one should not be too optimistic about increases or decreases predicted in this region.  44  5.0  WAVE RESISTANCE ANALYSIS BY DAWSON'S METHOD  Dawson's method [7] overcomes a few of the shortcomings of Michell's Integral. The thin ship limitation is removed by taking the low speed flow as the basis for a perturbation analysis, which allows the impermeable boundary condition to be satisfied on the hull panels, rather than on the vessel's centreplane.  Additionally, the large  oscillations in wave resistance coefficient seen in thin ship theory at low speeds are not encountered in results calculated by a panel method. Perhaps the most important benefit of implementing Dawson's method though, is that the velocity and pressure distribution are obtained along the hull, as well as the wave pattern around the hull. This information provides direct feedback to the designer that can help hull and appendage design.  To implement Dawson's method, the hull and a portion of the free surface must be discretized into quadrilateral panels. Various macros have been written to streamline this process, however, a complete preprocessor does not currently exist for generating the panels from a 3D CAD model. As a result, the overall input file creation process is still quite tedious and requires a lot of time and patience.  Since the method of Dawson is based on differentiation of the velocity potential along the streamlines, it is practical to first determine the streamlines from a low speed theory in order to develop an appropriate mesh of the free surface. To accomplish this, a separate program is used, taking the discretized hull and various field points on the free surface as input.  This program calculates the double model flow characteristics and creates  streamlines around the body at the undisturbed free surface. The output of this program creates the free surface mesh, from which the wave potential is calculated. An overview of the process is illustrated in Figure 5-1.  45  Existing C A D Geometry  Generate Hull Panels  Generate Free Surface Field Points  Input file for Streamline calculation  Streamline Calculation  Is Transom Wet?  Yes  No  1 Dry Transom Solver  Wet Transom Solver  Post Processing Macros  Output Data to Excel Spreadsheets Data analysis of results  Tecplot Files Visualization of free surface, and speed and pressure contours on hull  Figure 5-1 - Flow chart for analysis by Dawson's method  5.1 Mesh Generation for the NPL Monohull Since the speed range of interest is from Fn = 0.2 to Fn = 0.45, the treatment of the transom stern, discussed in section 2.3, becomes important. At lower speeds, 46  typically for Fn < 0.35, flow recirculates into the transom region, the transom is wet, and the water level at the transom is close to the still water level. At higher speeds, generally regarded as Fn > 0.45, the flow clears the transom, leaving it dry, while for speeds in between the two regions, the transom will be partially wetted.  For the NPL trimaran, the transom has been regarded as wet over the entire speed range. This approach is justified since in the upper range of Froude numbers, neither boundary condition is exact.  It is worth noting, however, that the trimaran outriggers operate at an even higher Froude number than the centre hull, and for Froude numbers based on the centre hull length greater than 0.275, the transoms of the outriggers are expected to be dry. A combined solution method for multihulls that allows implementation of different transom boundary conditions based on the local Froude number of each hull may provide an interesting future research topic to improve wave resistance predictions.  To calculate the wave resistance using Dawson's algorithm, the NPL hull is discretized using 247 panels as shown in Figure 5-2.  Since Dawson's method  uses the method of images to take a reflection about the centreplane of the vessel, only half of the vessel needs to modeled. The mesh density is increased near the bow and stern to help capture the curvature of the hull in this region. The majority of the panels are quadrilaterals, with two triangular panels required to blend the panels into the bow.  47  Figure 5-2 - Mesh for the N P L centre hull  The mesh of the free surface is more sensitive to panel size and distribution than the hull surface. To obtain a free surface mesh that provides accurate resistance predictions, the user should consider the impact of the following parameters on the free surface discretization: •  Domain of discretization,  •  Panel size,  •  Panel distribution and transitions between different panel sizes,  •  Speed dependency of mesh.  To ensure that an adequately large portion of the free surface is captured to model the wave making around the hull, three general parameters are important to characterize the domain of the mesh: the width, mw, upstream length, ul, and downstream length, dl.  These parameters are depicted in Figure 5-3.  An  improper selection of the upstream or downstream lengths can have an adverse effect on the wave resistance predictions. Experience has shown that taking ul = 0.3 L L and dl - 1.25 LWL provides an appropriate mesh, mw should be taken W  such that the divergent waves from the bow leave the domain at the ends of the mesh, rather than the sides. Using the Kelvin angle (9 = 19.5°), this becomes a geometric property of the mesh based on the chosen downstream length of the mesh.  If waves are permitted to leave the sides of the domain due to an  inadequate choice of mw, the sides of the domain can act as a "numeric wall,"  48  causing reflection o f waves back into the domain. predictions for the  wave resistance.  This results i n inaccurate  A n example o f numeric reflection  encountered from too narrow a domain is shown i n Figure 5-4.  ul  dl  mw  Figure 5-3 - General parameters to describe free surface discretization  Wave Elevations |mj  -0.7  -0.46 -0.22  0.02  0.26  0.5  Figure 5-4 - Numeric reflection resulting from too narrow a domain.  Once an appropriate domain has been set by the user, it is then discretized into panels. The wavelength o f the transverse waves generated by a ship, X =  2xU /g, 2  vary with velocity, and as a general guideline, one should use approximately 20 panels per wavelength in order to obtain an adequately resolved mesh [12]. For estimates o f the wave resistance at low speeds, this requires a suitably refined mesh, whereas for higher speeds a coarser mesh can provide accurate results.  49  The region of the free surface immediately adjacent to the hull, as well as the first crest and trough of the bow wave, and also the stern region require the highest resolution, and special care should be taken to visually examine the output wave patterns in these areas. If the panels here are too large, the resolution of the free surface will be insufficient to capture both the divergent and transverse arms of the wave pattern, and the wave resistance will be poorly predicted. Additionally, if the panel size grows too quickly, numeric reflection can occur within the grid itself, causing fictitious wave patterns. An example of this reflection due to a poor grid transition is shown in Figure 5-5.  Figure 5-5 - Visible reflection (green circle) due to abrupt grid transition (red circle)  Experience has shown that abrupt changes in the aspect ratios of panels produce this reflection, and areas where the slope of the wave surface is steepest tend to be the most susceptible to this problem. In practice this means that, especially for finer meshes, the spacing of the streamsheets must be nearly constant near the first crest and trough of the bow wave.  In general, growth factors for the  streamsheet spacing should not exceed 5%, except far outboard from the hull, where growth factors of up to 10% may be acceptable depending on the exact mesh shape. The longitudinal length of the panels should be taken such that the panels are as square as practical near the bow, and stern, and along the primary crest and trough of the bow wave.  50  For low and moderate speeds, the implications of these requirements are that the resulting structured grids of the free surface are computationally expensive. This leads to problems with most Windows based Fortran compilers, as they can usually only allow memory allocations of up to 256 MB. As the number of panels on the free surface mesh nears 2500 panels (the exact number depending on the hull mesh as well), this limit is reached, and in order to solve the wave resistance problem, a high performance compiler suitable for numeric work is required. Such compilers usually require either a Linux environment, or at least a Linux emulator for Windows.  A highly refined mesh for low to moderate speeds also substantially increases computational time. A high speed mesh with 1000 free surface panels may take less than 30 seconds to compute the wave resistance for a given speed, while a low speed mesh, with roughly 3500 panels takes on the order of 20 minutes for the wave resistance calculation for a single speed.  The long, slender NPL hull requires a very wide mesh in order to capture enough of the wave profile downstream and avoid reflectionfromthe edge of the domain. To obtain an acceptable view of the free surface, the domain is 134 metres (0.84 LWL) wide. The upstream portion of the mesh extends 50 metres (0.31 LWL) forward of the bow, and the downstream portion continues 200 metres (1.25 LWL) aft of the transom.  This region can be adequately resolved using a mesh of 3244 panels. This mesh is shown in Figure 5-6, and consists of 28 streamsheets around the hull, and 4 streamsheets behind the transom.  51  Figure 5-6 - Refined mesh for the NPL monohull  In order to ascertain the suitability of this mesh resolution, a great deal of time and effort has been spent varying the different mesh parameters, including the total panel count, number of streamsheets, spacing of streamsheets near the hull, and the transitionfromtightly spaced streamsheets near the hull to wider spacings further outboard.  Since the wave pattern a ship produces is well known and well studied, one of the best methods of judging the appropriate level of mesh refinement is a visual inspection of the resulting wave profile.  One should expect to see the  characteristic divergent and transverse waves associated with the bow and stern of the vessel.  The wave elevations are calculated using a coarse mesh of the free surface with 704 panels, and the refined mesh with 3244 panels. Results are presented side by 52  side for the low and high resolution grids for Froude numbers of 0.5 and 0.31 in Figure 5-7 and Figure 5-8. The benefit of the high resolution grid is clear, as the crests and troughs of the bow wave system can be seen to fall primarily within the Kelvin envelope (solid black lines on all figures). For the low resolution grid, the pictures are not resolved enough to confirm that the waves are constrained by the Kelvin angle.  Wave Elevations [m]:  -0.75 -0.5 -0,25  0  0.25 0.5 0.75  1  Wave Elevations [ml:  -0.75 -0.5 -0.25  0  0.25 0.5 0.75  1  Figure 5-7 - Wave elevations for Fn = 0.5 for the low resolution mesh (left) and high resolution mesh (right)  At lower Froude numbers (Figure 5-8), the inadequacy of the low resolution mesh becomes more apparent, as the divergent and transverse wave systems blend together. At this speed the high resolution grid is necessary to clearly distinguish both wave systems.  53  Wave Elevations[m]:  -0.6 -0.4 -0.2 0  0.2 0.5  1  Wave Elevations [m]:  -0.6 -0.4 -0.2 0 0.2 0.5 1  Figure 5-8 - Wave elevations for Fn = 0.31 for the low resolution mesh (left) and high resolution mesh (right)  The impact that the different meshes have on the wave resistance predictions is shown in Figure 5-9. The high resolution of the second mesh appears more than sufficient for the high speed calculations, as the disceraable differences seen visually between the high and low resolution meshes do not generate a significant change in the wave resistance predictions. For calculations in this speed range, it appears that sufficient accuracy could be achieved with fewer panels than the high resolution mesh, reducing computational demands.  In the low to moderate speeds, the mesh refinement has a significant impact on wave resistance predictions, and clearly the low resolution mesh does an inadequate job of predicting wave interactions here. For all work reported in this thesis, the refined mesh of 3244 panels on the free surface has been used.  54  Mesh Refinement Study - Base Hull 0.60 -,  0.10 0.00 -I— 0.20  i  1  1  0.25  0.30  0.35  1  1  1  0.40  0.45  0.50  Froude Number Figure 5-9 - Effect of mesh refinement on the calculation of C  w  An investigation by Hally [14] has examined the sensitivity of Dawson's method to further mesh refinements. Hally reports that further levels of mesh refinement still result in some level of numeric dispersion, and that the divergent arms of the wave pattern always slightly exceed the Kelvin wave pattern as a result of this dispersion.  Hally also examined the change in wave resistance values with  further mesh refinement, and the difference reported in wave resistance values in doubling the mesh density from 20 panels per wavelength to 40 panels per wavelength is only 0.4% at Fn = 0.3[14]. At a Froude number of 0.30, the mesh of 3244 panels used in this study has 29 panels per wavelength.  5.2 Mesh Generation for the N P L Trimaran  In order to generate the appropriate mesh to study the NPL trimaran, the hull mesh is expanded to include the outriggers, and the existing free surface mesh is modified to accommodate them. Only one outrigger needs to be modeled, and 216 panels are used to define this geometry. This results in a hull mesh for the 55  complete trimaran of 463 panels, as shown in Figure 5-10. The free surface mesh has been refined between the two hulls to properly capture wave interactions, and a total of 3436 panels are used. The complete mesh of the hull and free surface is shown in Figure 5-11.  Figure 5-10 - Mesh of the NPL trimaran. Complete vessel shown for clarity.  56  Figure 5-11 - Mesh of the NPL trimaran and free surface The mesh refinement level has again been justified by an extensive study o f the effects o f panel size on wave interactions. O f particular concern to the trimaran hullform is the level o f panel refinement between the two hulls.  The effect o f  varying this parameter is seen clearly in Figure 5-12 , in a comparison o f a finely refined and a coarser mesh.  The coarser mesh with  1148 panels is still  substantially more refined than the coarse mesh used i n the monohull mesh study. However, in the new coarse mesh there are only 3 streamsheets between the two hulls, whereas the refined mesh has 6 streamsheets between hulls. In the Froude number range from 0.23 to 0.32, the coarse mesh appears nearly converged, as the mesh refinement only causes slight changes to Cw-  A b o v e F n = 0.32, a  substantial difference exists in the predictions by the two meshes, as the coarse mesh does a poor job o f predicting the wave interactions here. 57  M e s h Refinement Study - Trimaran 0.45  s o  0.10  0.05 -  o.oo -I 0.20  i  i  0.25  0.30  ! 0.35  i 0.40  1  i  0.45  0.50  Froude Number  Figure 5-12 - The effect of mesh refinement on wave resistance for the NPL trimaran  5.3 Parabolization Study of the NPL Monohull Each of the four hull configurations analyzed in chapter 4 by Michell's Integral are discretized for analysis by Dawson's method. In each instance this required stitching the free surface mesh for the base hull to fit the modified hull form, and then ensuring that the streamlines were accurately captured by the new mesh. Once the resistance had been calculated, the output was sorted using macros. The resistance data was then input into Excel for analysis and the free surface elevations, along with the flow velocities and pressure contours were input into Tecplot for visualization. The wave resistance coefficients for each of the four hull configurations are shown in Figure 5-13. The predicted results do not predict the same trends as predicted by the Michell's Integral analysis. All four hull configurations are predicted to cause an increase in the wave resistance coefficient for all speeds.  58  Wave Resistance C o m p a r i s o n  Base Hull -*—Hull 1 - * — Hull 2 -e—Hull 3 *- - • Hull 4  0.20  0.25  0.30  0.35  0.40  0.45  0.50  Froude Number Figure 5-13 - Comparison of C  w  for base hull and hulls 1-4  It is important to note that the displacement of each hull has increased by approximately 2% for hulls 1-3, and by close to 4% for hull 4. This was not the case with the thin ship theory, as the hull shape was scaled by the solver to match the displacement given in the input file. For the analysis by Dawson's method, the results shown in Figure 5-13 should be interpreted as the effect of retrofitting a bulb to the base hull. The results show that hull 2, with a parabolic bulb placed starting at the vessel's shoulder, and hull 4, with a larger bulb, also starting at the vessel's shoulder, show the most promising results. For these hull configurations, the wave resistance is comparable to that of the base hull in the speed range from Fn = 0.32 to 0.36.  An examination of the wave elevations for the different hull configurations gives excellent insight into why these bulbs see comparable wave interactions at these 59  speeds. Figure 5-14 compares the wave elevations for the base hull, hull 1 and hull 4 at Fn = 0.35.  WaveElevations[m]: -0.8 -0.4 -0.1  0.2  06  0.8  £  Figure 5-14- Wave elevations at Fn = 0.35 for base hull (a), hull 1 (b) and hull 4 (c)  The impact of hull configuration 1 on the wave making is evident in Figure 5-14 (b). A n additional wave crest is created at the start of this bulb, and a wave  60  trough created at the bulb's aft end. It is clear from the figure that neither element of the wave system created by this bulb is placed appropriately to interact with the ship's waves. Since the bulb in hull configuration 1 occurs so far aft, the primary wave trough from the bow wave will not interact with this bulb until the wave length of the bow wave is large enough to reach it (ie. at a higher Fn number). Presumably, to produce wave cancellation in the bow wave at this larger wavelength, one would need a much longer bulb, such that it create a single wave crest to cancel the long trough of the bow wave at this speed. In general, since the bulb in hull configuration 1 is so short, it would likely only be beneficial for very low Froude numbers, where the ship's wave system has a short wave length comparable to the separation of the wave elements created by this bulb.  The effects of wave cancellations can be seen in hull 4, as Figure 5-14 (c) shows a reduction in the primary wave trough of the bow wave. Here the bulb starts at the shoulder, creating a wave crest that cancels a portion of the transverse wave trough.  The aft end of the bulb appears to occur too quickly though, as a  significant wave trough occurs at the aft end of the bulb, persisting down stream as an additional wave system.  In general, one can conjecture that the small bulbs of hulls 1-3 cause local crests and troughs that are too closely spaced to adequately interact with the hull wave systems. The fourth bulb is much larger, and separates the wave systems it produces. Near Fn = 0.34, the local crest produced by the bulb begins to cancel the primary trough of the ship. However the aft end of the bulb produces a large trough that would otherwise not be present.  The net effect of these wave interactions results in the wave resistance being similar to that of the base hull. If the wave trough created by hull 4 were not present, we should expect a decrease here in wave resistance, since the bulb has partially cancelled the ship's wave system.  61  The predictions for the initial hull configurations do not agree well with those from the thin ship theory. However, since the beneficial resistance predictions seen in chapter 4 primarily occurred at or below Fn = 0.32, those predictions by thin ship theory have been disregarded, and hulls 1-3 have been considered unsuccessful.  The initial hull configurations were chosen based on the results of parabolization studies of fishing hulls, with little emphasis based on the different wave profiles of the NPL hull. In order to produce a hull with reduced resistance, it appears a different approach should be taken.  The free surface profiles of the different hull configurations suggest an approach be taken that treats the bulbs as wave generation devices. In the forward end of the bulbs, where the beam is increasing, the pressure is increased on the hull, creating a local wave crest. The pressure decreases in the aft end of the bulb, where the beam is decreasing, resulting in a local wave trough. Approaching hull design in this way provides criteria for bulb shape based on the free surface elevations of the base hull.  By properly placing a bulb, with its beam increasing in the region of the primary trough, and decreasing in the secondary wave crest, it could be possible to reduce the wave heights in these regions.  A fifth hull, shown in Figure 5-15, is generated by stretching the bulb of hull 4 forward to place the local crest it creates more directly in the ship's primary wave trough at Fn = 0.35. The aft end of the bulb is also extended several stations to separate the wave making elements of the bulb such that they can effectively interact with the ship's wave system.  62  Figure 5-15-Hull 5  The predicted resistance for this hull is shown in Figure 5-16, along with the resistance curve for hull 4 and the base hull. Significant reductions in the wave resistance are seen with hull 5 from Fn = 0.29 to Fn = 0.36. The maximum wave resistance reduction compared to the base hull is 7%, occurring at Fn = 0.32. In order to estimate the impact on the total resistance, the viscous resistance is again calculated using the ITTC '57 friction correlation line, and the percentage change in the total resistance between hulls is given in Figure 5-17. It is important to note that Dawson's method predicts the wave resistance to be smaller in magnitude than thin ship theory predicted. As a result the wave interactions caused by the bulb have less of an impact on the total resistance, and the total resistance savings are only predicted to be around 3%, despite the 7% decrease in wave resistance.  63  Wave Resistance by Dawson's Method - Base Monohull vs Hulls 4 & 5  300 250  Base Hull --"--Hull 4  200 c  —•—Hull 5 S,  150  a: 100 50  0.2  0.25  0.3  0.35  0.4  0.45  0.5  Froude Number Figure 5-16 - Wave resistance for the base hull, and hulls 4 & 5 (Dawson's method)  Percentage Change in R  T  Froude Number Figure 5-17 - Percentage change in R between base hull and hulls 4 and 5 T  In order to produce a hull that better balances potential decreases in wave resistance with minimal increases in viscous resistance, the fore and aft end of the fifth bulb are faired considerably into the hull, creating hull 6, which is shown in 64  Figure 5-18. The predicted change in Rw for this hull is shown in Figure 5-19. The wave resistance reductions are comparable to the predictions for hull 5, and since the bulb transitions are properly faired into the hull, it is expected that this hull will have reduced resistance compared to the base hull. The maximum wave resistance reduction predicted is 5%, occurring at Fn = 0.33.  Klaptocz [24]  reports that this hull shows minimal increases in viscous drag compared to the base NPL hull, so it is hoped that this hull can achieve a net decrease in resistance. The plot of change in RT is omitted here since the ITTC '57 line used to calculate the viscous term does not properly account for the differences between hull 5 and hull 6.  Figure 5-18 - Hull 6  Percentage Change in R  w  10% 8% I  6%  fi  4%  o  2%  <D (0  — Hull 5 — h -•Hull 6 /  /  0) O) c  v  -2%  re O -4%  2  / / *  /+'  * ' \ ]  \  \  f i  0.25  •  Vo:3 \%  /  o.35y  +t  •  0:4  \  V\  -6% -8%  Froude Number  Figure 5-19 - Percentage change in R between base hull and hulls 5 and 6 w  65  •  V45 J  ^ 0  An examination of the free surface contours for hull 6 and the base hull at Fn = 0.33, shown in Figure 5-20, clearly shows the wave interactions caused by the new hull. The transverse wave system in the vicinity of the hull has been substantially reduced by the addition of the bulb.  Wave Elevations [m]:  -0.8 -0.4 -0.1 0.2 0.6  0.8 Wave Elevations [m]: -0.8 -0.4 -0.1 0.2 0.6 0.8  (a)  (b)  Figure 5-20 - (a) = base hull (b) = hull 6 Fn =0.33, note the reduced transverse waves in (b)  5.4 Parabolization study of the NPL Trimaran For the monohull study, minimizing the wave resistance amounts to minimizing the waves produced by the hull. As was seen for hull configurations 1-4, bulbs that produced additional wave systems resulted in increases in the wave resistance. Reductions to the wave resistance were achieved by "tuning" the parabolic bulbs to produce wave elements that directly interacted with the base hull's wave system.  In the case of multi hull ships, the approach is somewhat different, as now the wave resistance problem becomes one of minimizing the wave pattern produced by a system of hulls. Hull shapes that on their own do not minimize wave resistance may interact with one another to produce a multihull shape with 66  reduced wave resistance.  In order then to investigate the effect of wave  interactions on the wave resistance, the parametric study is repeated to test hulls 1-4 in the trimaran configuration using Dawson's method. Hull configuration 6 is also tested in the trimaran configuration, since this hull shows promising results from the monohull analysis.  The results of the analysis are presented in Figure 5-21 and Figure 5-22. The data are presented as resistance data and also in terms of percentage change relative to the base NPL trimaran for comparison. The percentage change is expressed in terms of wave resistance only in order to illustrate the significant wave interactions predicted for hull 4.  67  T r i m a r a n Wave R e s i s t a n c e P r e d i c t i o n s  68  Trimaran Wave Resistance Predictions  250 -*—Hull4 Hull 6 Base Trimaran  200  150 2  100  50  0.20  0.30  0.25  0.35  0.40  0.45  0.50  Froude Number  (a)  Percentage Change in R  w  40%  =  30% -a- - - Hull 6  0)  <§ 20% m |  -x--- Hull 4  10%  •4—'  TO 0)  K  0) o> c  0% 0.20  G----a  .'"a. 0.25  ^ ^ '  Q  3  "B-B-E3  0.40  .-X6.45  . -x P0  OTO -10% X  -20%  x-x-x'  Froude N u m b e r  (b) Figure 5-22 - Wave resistance predictions for hull 4 and 6 (a), and percent change from base trimaran (b)  For the monohull analysis, hull 6 had shown a decrease in wave resistance of about 5 % at Fn = 0.33, and due to the gentle curvature of its bulb, was considered 69  the most promising hull form. In the trimaran configuration this hull shows similar characteristics, with a maximum reduction in wave resistance of about 6%, again predicted to occur at Fn = 0.33, and reductions in wave resistance in the Froude number rangefrom0.34 < Fn < 0.45.  The results for hull 4 are somewhat surprising, as the monohull study indicated no benefits to the wave resistance for this configuration.  In the trimaran  configuration the wave making is predicted to be substantially reduced. The predicted savings reach 15% at Fn = 0.39 and extend over the range from 0.34 < Fn < 0.45.  The mechanism for the reduction in drag appears to be much different between the two hulls. Hull 6 shows wave cancellations similar to the monohull case, where interaction takes place between the bulb and the transverse wave system of the centre hull. The effect of the outriggers is simply superimposed onto the wave pattern seen around the centre hull, and the wave interactions are not noticeably different than that seen with the base trimaran. For hull 4, the wave interactions are much different. The additional waves created by the bulb interact with the bow wave of the outriggers, resulting in greatly reduced wave amplitudes past the outriggers. The wave profiles are compared for the base trimaran and trimaran 4 at a Froude number of 0.39 in Figure 5-23.  70  Wave Elevations [m]: -0.8 -0.4 -0.1  0.2  0.6  0.8 Wave Elevations [m]: -0.8 -0.4 -0.1 0.2  0.6  0.8  Figure 5-23 - Wave elevations for the base trimaran (a) and hull 4 (b) at Fn = 0.39  The results seen for hull 4 are very encouraging, and to develop a suitable hull for tow tank testing, hull 4 is further faired to reduce the viscous drag and form factor. The beneficial wave making component o f hull 4 appears to be the wave trough generated over the aft end o f this bulb. Care is taken i n designing the fairing to ensure a fairly dramatic beam transition is kept, while ensuring continuity o f the hull slope. In this manner, the fairing is actually created b y reducing the size o f the bulb until the waterline shape is continuous.  The new hull generated by this process is named hull 7. It is shown below i n Figure 5-24. A table o f offsets for hull 7 is given in appendix B .  Figure 5-24 - Hull configuration 7, note the more abrupt bulb shape 71  The analysis of hull 7 by Dawson's method is presented, along with the results for hull 4 in Figure 5-25. The design of hull 7 has successfully captured almost the same wave interactions seen by hull 4, while minimizing the potential increase in the form factor.  One of the most encouraging aspects of the wave interactions seen by both hull 4 and hull 7 is the speed range in which they occur. Previous parabolization studies have examined the effect of waterline shape on wave resistance reduction for monohulls, and have mainly reported resistance reductions around Froude numbers of 0.27 to 0.35.  While reductions at these speeds are beneficial to  vessels operating at moderate to moderately high speeds, the trimaran hull form tends to operate at even higher Froude numbers. Yang [41] identifies the aft outrigger placement used in this NPL trimaran as an ideal location for trimarans operating near Froude numbers of 0.45.  This in particular makes it especially  encouraging that wave reductions are seen at these higher speeds, as it combines the benefits of both the outriggers and the parabolic bulbs near the operational speed of the vessel.  72  T r i m a r a n Wave R e s i s t a n c e P r e d i c t i o n s  73  5.5 Conclusions on Dawson's Method Dawson's method has been used to generate two different hulls, hull 6 and hull 7, that are both expected to reduce the total resistance of the trimaran. Estimates of the resistance components in section 3.3 show that the viscous and wave resistance terms are of approximately equal magnitudes. This suggests that if the viscous drag penalty is minimized, a reduction of about 3% in total resistance could be seen with hull 6 near Fn = 0.33, and for hull 7 a reduction of about 7.5% may be possible near Fn = 0.39. It should also be noted that these reductions are expected to be seen while increasing the displacement of the vessel. Scaling the vessel dimensions down to match the original displacement of the base hull should reduce the resistance even further, since if the length were held constant, the addition of the bulb would allow the entrance angle and the transom area to be reduced. However, because this approach would require several models to be produced for tow tank validation, it has not been explored.  The hull configurations originally deemed suitable by Michell's integral for drag reduction at lower Froude numbers have not shown the same trends with Dawson's method. This is not unexpected, as explained in section 4.3, and bulb shapes predicted by Michell's Integral to produce wave resistance reductions only at the lower Froude numbers have been in general discarded.  The near field wave elevations and pressure contours along the hull provided by Dawson's method provide the user with appropriate information to design wave cancellation bulbs.  74  6.0  EXPERIMENTAL APPROACHES  To validate the numeric work done in sections 4 and 5, a model o f the base N P L trimaran is constructed. The two most promising hull configurations, hull 6 and hull 7, are created by constructing bulbs that are attached to the base hull. Resistance tests are conducted in the towing tank facility at the V i z o n SciTec Ocean Engineering Centre ( O E C ) , which is located at the University o f B C Vancouver campus. A view o f the tow tank showing the test carriage, with model attached, is shown in Figure 6-1  The tow tank is 67m long, 3.7m wide and 2.4m deep. This size facility permits the use o f models o f up to about 3 m i n length, and carriage speeds can reach a maximum o f 6m/s.  Models mount to the tow tank carriage v i a a heave post, which leaves the model free to squat and trim, but constrains the other 4 degrees o f motion for the vessel. This set o f constraints is typical for tow tank testing, and allows the user to collect resistance, sinkage and trim data for a given speed.  The resistance o f the model is measured by a load cell mounted on the heave post, and the squat and trim o f the vessel are measured by optical encoders which are also attached here. 75  6.1 Theoretical Background for Experimental Work In tow tank testing, the total resistance of the ship model is measured. As outlined in section 2.1, this resistance term is composed of Reynolds dependant and Froude dependant components. This prevents direct scaling from model scale to mil scale, as dynamic similitude does not exist between the model and the full scale ship.  It is customary to overcome this problem by using Froude's  hypothesis (2-2), that the Reynolds and Froude dependant terms act independently of each other.  In this manner, once one has measured the total resistance  coefficient, the viscous resistance coefficient, which is assumed to be a function of the Reynolds number only, is determined from slow speed tests and empirical correlations. This viscous coefficient is then deducted from the total resistance coefficient, and the remaining term, the residuary resistance coefficient, is assumed to be dependant on only the Froude number: Cf -Cy(Re) = C%'(Fn)  (6-1)  Here the superscripts M and FS represent model scale and full scale, respectively.  Since Froude numbers are matched from model scale to full scale, the residuary resistance coefficient then remains the same at full scale.  A new viscous  coefficient is calculated based on the different Reynolds number at full scale, and a correlation allowance is added to give the full scale resistance predictions: Cy  FS 4- Cyj + C£ — Cj  (6-2)  At low speeds, the wave making resistance depends on the fourth power of the Froude number, and hence becomes very small. Where it can be neglected, the curve of Cj vs F« will run approximately parallel to the friction line for a two4  dimensional plate [19]. At this point, called the run in point, the form factor, which is due to the three-dimensional shape of the hull, is defined: l+)t =  C"(*"o)  (6-3)  76  C  F0  is the equivalent flat plate resistance coefficient, and is determined from the  ITTC '57 model ship correlation line (4-1). The viscous resistance coefficient is defined as: C f =(l + k)C&(Xe)  (6-4)  6.2 Model Construction and Particulars  A 3 metre model of the NPL trimaran is constructed by shaping layered up foam to match templates of the hull offsets. The foam shape is undersized to allow for a layer offiberglass,which provides the rigidity of the hull. Wood inserts are used to provide strength as required for mounting points. Due to the length and fine hull form of the NPL trimaran, this construction method is required to minimize the weight of the model, as the scaled displacement of the 3m trimaran is only 71 lbs. The particulars for the model are summarized in Table 6-1, and the model is shown in Figure 6-2.  Turbulence inducing pins are used to trigger  turbulence along the model. The pins are 3 mm in diameter, and extend 3 mm above the surface of the hull. They are placed 10% of the length from the bow, and are spaced approximately every 2.5 cm.  [m] B [m] T[m] A[lbs]  LWL  Centre Hull  Outrigger (each)  Total  3.00 0.22 0.11 66.6  1.08 0.05 0.05 2.0  3.00 0.58 0.11 70.6  Table 6-1 - Model scale particulars for the NPL trimaran  77  Two different methods have been used to construct the bulbs for hull 6 and hull 7. Hull 6 is created in a similar manner as the hull, with fiberglass laid onto the hull and foam used as necessary to form the shape. A layer of aluminum foil placed between the bulb and the hull acts as a mold release, and double sided tape is used to secure the bulbs to the hull. This allows the bulbs to be removed after testing. Due to time constraints in the model shop, a different approach had to be used for hull 7. The bulbs for hull 7 were built using the Stratasys FDM Titan, a rapid prototyping machine owned by the Department of Mechanical Engineering at UBC. This machine is able to create three-dimensional shapes directly from CAD surfaces. Once the user generates the part using CAD software, an IGES or STL fde is exported to the pre-processing software of the rapid prototyping machine for tool path generation. The rapid prototyping machine "prints" the bulb by using the Fused Deposition Modeling process. This process involves depositing liquefied ABS plastic in layers 10 thousandths of an inch thick. A supporting plastic is also printed as needed by the machine to stabilize the part as it is created. This supporting plastic later dissolves in a solvent wash.  78  The rapid prototyping machine can accommodate parts approximately 12" x 12" xl6", and since the bulb is longer than these dimensions, the machine prints the bulb in a series of connecting pieces. After the part is created, it is hand finished and wet sanded to remove ridges from the printing method. Both bulb fabrication methods work out to be comparable in cost, the cost of bulb 7 primarily consists of the cost of the ABS plastic itself.  Bulb 6 and bulb 7 are shown in Figure 6-3.  Figure 6-3 - Bulb 6 (top) and bulb 7 (bottom). Note the different sizes of the two bulbs  6.3  Test Procedures & Experimental Results  The three different hulls are tested in both monohull and trimaran configurations. The displacement is increased by adding the bulbs, to validate the numeric predictions by Dawson's method.  In order to determine the form factor of each  hull setup, low speed tests are conducted, with the total speed range tested ranging from Fn = 0.1 to Fn = 0.5. This corresponds to carriage speeds from 0.542 m/s to 2.712 m/s.  79  6.3.1 Form Factor Determination The value of the form factor is determined from extrapolating the gradient of the  cp/cg,  vs.  Fn /Cp 4  0  plot as the Froude number approaches zero. The plot  used to determine the form factor for trimaran hull 7 is given in Figure 6-4, and the rest of the form factors are summarized in Table 6-2. At very slow speeds laminar flow can stabilize downstream of the turbulence pins, and vibrations transmitted from the rails and motor to the test equipment can also affect data. At these speeds the user must carefully examine each carriage run as it happens and only keep data sets that are repeatable and shows no signs of transition back to laminar flow (typified by lower resistance values). In the determination of the form factor, carriage speeds below 0.705 m/s have in general been neglected.  o o  0.5  0.3  o  overall  X  selected  — L i n e a r (selected)  0.0  0.0  0.1  0.2  0.3  0.4  0.5  Fn /C 4  0.6  0.7  0.8  0.9  M F O  Figure 6-4 - Determination of the form factor from slow speed tests for trimaran hull 7  Base Hull  Hull 6  Hull 7  Monohull  1.156  1.214  1.234  Trimaran  1.209  1.236  1.237  Table 6-2 - Summary of calculated form factors  80  1.0  The calculation of the form factor for the base hull has been repeated in May 2005 and January 2006 to check for repeatability of the data. The data points have been adjusted to account for differences in the viscosity and density of the water, based on the different seasonal temperatures recorded in the tank. The form factors reported in Table 6-2 are taken as the average of those calculated on each date, but the individual form factor calculations differ by only 0.51% for the monohull data, and 0.49% for the trimaran. Repeatability studies conducted by Vyselaar [38] have shown that the repeatability of resistance results in the OEC tow tank is typically about 1.5%.  The results of the form factor testing correlates well with predicted results by Klaptocz [24].  The increases to the form factors indicate that the beam  increase in hull 6 and hull 7 has caused an increase in the viscous drag of the vessel. The addition of the outriggers in the trimaran form dilutes this affect, due to the larger wetted surface of the vessel.  The form factor calculated for the base hull is compared to form factors reported by Couser et. al. [6] for high speed, slender NPL forms in Table 6-3, and found to be in good agreement.  L/V  Couser  Current  1 / J  6.3 7.4 8.5 9.5 9.63  1+k 1.45 1.30 1.26 1.22 1.156  Table 6-3 - Comparison of form factors for different NPL forms  6.3.2 Monohull Test Results and Full Scale Powering Predictions The resistance data for the monohull is scaled up to full scale as outlined in section 6.1. Plots of total resistance and of powering requirements for the three hull configurations are shown in Figure 6-5.  81  Full Scale Total Resistance 1600  1400  1200  1000 z" =£. 800  it£  600  400  —  Base Hull  -»  Hull 6  — - Hull 7 200  0.20  0.25  0.30  0.35  0.40  0.45  0.50  Froude Number Figure 6-5(a) -  Full scale resistance plots for monohull configurations Effective Horsepower  40000  35000  30000 — 25000  Base Hull Hull 6  —  Hull 7  I 20000 UJ 15000  10000  5000  0.20  0.25  0.30  0.35  Froude Number Figure 6-5(b) -  Full scale monohull powering requirements 82  0.40  0.45  0.50  Figures 6.5 (a) and (b) indicate that little advantage is gained with hull 6, and in the higher speeds this bulb produces a distinguishable increase in resistance. Hull 7 shows a decrease in resistance in the moderate speed ranges, and at higher speeds, the resistance appears unchanged from the base hull.  The base hull shows an oscillating resistance value inconsistent with the wave pattern typically seen from the interactions of a vessel's wave pattern. At first this oscillation was thought to be potentially erroneous experimental data, however, a re-evaluation of the base hull, shown in Figure 6-6, shows that the oscillating pattern is repeatable. It is thought that this oscillation may be due to the extreme L/B ratio of the monohull causing a yawing motion around the heave post.  This oscillation was not seen in the multihull configuration,  which has substantially increased directional stability due to its outriggers. Comparison of Baseline Resistance Oscillation 1600 -, 1400 -  200 4 0 A 0.34  ,  i  ,  ,  i  0.36  0.38  0.40  0.42  0.44  Froude Number Figure 6-6 - A comparison of resistance data for the NPL base monohull  In order to compare the different hulls, the oscillating component has been averaged, and the percentage change in R p is plotted in Figure 6-7. For hull 6, small decreases are seen in the total resistance. Unfortunately, the magnitude of these decreases places them close to the error margins of the extrapolated 83  data, and little confidence can be had that any real reduction is seen. Hull 7 shows a decrease in resistance of more than 5% between Fn = 0.3 and Fn = 0.35. This reduction is substantial, and indicates that the beam increase has produced beneficial wave interactions for this speed.  Percentage Change in RT 20%  -1  15% -a-Hull 6 (avg) I  CO CS  -x- Hull 7 (avg)  0.50  -10%  J  Froude Number  Figure 6-7  - Percent change in total resistance between hull designs  6.3.3 Trimaran Test Results and Full Scale Powering Predictions The trimaran is tested by attaching the outriggers to the monohull, and adjusting the ballast weight of the model. Results of the full scale resistance and the powering requirements are presented in Figure 6-8 and a comparison of the percentage change in total resistance is given in Figure 6-9. The tests show that hull 6 has not improved the resistance characteristics of the trimaran. Hull configuration 7 shows a reduction in resistance for speeds from Fn = 0.32 to Fn = 0.45. Throughout this speed range, the resistance reduction is around 5%, with a maximum reduction of 6.5% occurring at Fn = 0.36. This correlates well with the numeric predictions of chapter 5.  84  Trimaran Full Scale Total Resistance  0.20  0.25  0.30  0.35  0.40  0.45  0.50  Froude Number Figure 6-8(a) - Full scale trimaran resistance Trimaran Effective Horsepower  0.20  0.25  0.30  0.35 Froude Number  Figure 6-8(b) - Full scale trimaran powering predictions  85  0.40  0.45  0.50  Percentage Change in RT  0.50  Froude Number  Figure 6-9 - Percentage change in total resistance between trimaran hull designs  The wave resistance component is separated from the total resistance and plotted in Figure 6-10 and Figure 6-11 for the three hull configurations. Hull 7 shows substantial decreases in wave resistance above Fn = 0.33, with a maximum decrease of about 21% occurring at a Froude number of 0.36. Trimaran Full Scale Wave Resistance 600  0 -I 0.20  -  1  0.25  .  1  0.30  0.35  1  0.40  Froude Number  Figure 6-10 - Wave resistance components for the three trimaran hulls  86  1  0.45  '  0.50  Percentage Change in Rw 20% 1  0.50  Froude Number Figure 6-11 - Percentage change in wave resistance  A comparison in Figure 6-12 of the wave patterns seen at a Froude number of 0.36 for the base hull and hull 7 illustrates the wave resistance reduction. As predicted by Dawson's method in section 5.4, the bulb creates a wave system that interacts with the bow wave of the outriggers, reducing wave amplitudes downstream of the outrigger.  A careful visual examination of the wave  heights shows that the wave height along the outrigger has been significantly reduced by the addition of the bulb. Attempts were made to quantify this wave height reduction using capacitance probes and the methods derived by Eggers [8], [9] for calculating the wave resistance from the free surface wave heights. However, the combination of the small wave making of a high speed multihull form, and the scale factors of the tow tank used prevented the capacitance probesfromproviding useful data.  87  Figure 6-12 - Wave patterns created by the base hull (left) and hull 7 (right) at Fn = 0.36 88  6.4 Comparison to Numeric Work In order to evaluate the accuracy of numeric predictions, the wave resistance is plotted in Figure 6-13 as predicted by thin ship theory and by Dawson's method, alongside the experimental results. The percentage change to the wave resistance caused by hull 7 is plotted for each method in Figure 6-14.  Both Dawson's  method and the thin ship theory are able to predict the wave interactions between the hull and the outriggers, although each method predicts the wave interactions to occur at slightly different speeds. The magnitude of the wave resistance is over predicted by thin ship theory and underpredicted by Dawson's method. This is not uncommon at the Froude numbers of interest, and it is reasonable to suspect that this is at least in part due to the selection of the transom boundary conditions for each program. The thin ship theory implements a transom boundary condition equivalent to a fully dry transom, whereas a fully wet transom boundary condition has been implemented for Dawson's method. Since the NPL hull generates such a small wave, the transom depression represents a fairly large portion of the overall wave resistance. Treating the transom as wet would tend to underpredict the wave resistance and a fully dry transom would tend to overpredict the wave resistance.  89  Numeric and Experimental Determination of Rw 700 -i -Base Hull(exp) - Hull 7 (exp) -Base Hull (thin ship) -Hull 7 (thinship) - Base Hull (Dawson) • Hull 7 (Dawson)  600 500 „  z  400  y—  K  300 200 100 0.20  0.25  0.30  0.35  0.40  0.45  0.50  Froude Number  Figure 6-13 - Comparisons of wave resistance predictions for hull 7 with experimental results  Percentage Change in Rw  * — Experimental Thin Ship Theory Dawson's Method  0.50  Froude Number Figure 6-14 - Predictions of changes in wave resistance by hull 7  90  7.0  CONCLUSIONS  A common simplified model of the wave making of a ship is that the bow and stern sections create a pattern of divergent and transverse waves which interact with each other as a function of Froude number [11]. In reality, although the bow and stern have the most significant effect on the wave making, each transverse section of the ship contributes to the overall wave pattern a ship produces. Previous studies by Calisal, Goren and Danisman, Tan, and Hsiung have all demonstrated the importance of the vessel's entire waterline shape on wave making resistance[3], [34], [17].  They have  demonstrated in particular the importance of the shoulder regions on the wave resistance of vessels operating at Froude numbers at and above 0.27.  Tan's study was used as a basis to implement parabolic bulbs at the waterline for the centre hull of a trimaran configuration. Starting with an initial parametric study, the hull design process was eventually guided by the free surface elevations generated by Dawson's method, and a new hull was designed. Expectations were that this new hull, hull 6, would achieve a reduction in overall resistance of between 2-3% near Fn = 0.33. Unfortunately, experimental tests could not prove that this reduction had been achieved, and an increase in resistance was seen above the target speed range.  The slender NPL hull produces very small wave amplitudes, and a study using wave capacitance probes was unable to determine what the exact impact of hull 6 was on wave cancellations. However, a new methodology for designing wave cancellation bulbs for a trimaran was discovered as a result of the above study. By considering the complete trimaran as a system of hulls, the waterlines of the centre hull were adjusted to produce wave cancellations downstream of the outriggers in hull 7. This effect was predicted by both Dawson's method and thin ship theory, and experimental work confirmed these predictions.  91  The wave cancellations seen by this method were substantial, and the wave resistance of the trimaran was reduced by 21% at Fn = 0.36.  For the full scale 160 metre ship,  powering predictions show that savings of just over 6%, or 1200 hp are possible.  The two tools primarily evaluated for determining the effect of parabolization on the wave resistance were Michell's Integral and Dawson's method. Cursory work was done to evaluate parabolization using RANS solvers, however, the current lead times required for mesh generation and the computational demands for solution of the wave resistance problem make optimization by this method a very challenging task.  For the trimaran ship, thin ship theory can provide accurate predictions of the wave resistance at higher Froude numbers. For optimization studies in the moderate Froude numbers (below 0.30) though, this method can lead to accepting bulbs that may not actually reduce the wave resistance. In general, the applicability of thin ship theory to further parabolization projects depends on two factors: whether the vessel in study is indeed "thin," and whether or not the speed range of interest is appropriate.  The thin ship solver that was evaluated in this parabolization study, Michlet, was unable to resolve the wave patterns and flow velocities in the near field. This is not a limitation for thin ship theory though, and implementations do exist that resolve the nearfieldwave pattern[37].  Dawson's method is though, in general, applicable to a wider range of hull forms, and it is not prone to the low speed oscillations encountered in thin ship theory. This makes it more useful from an optimization perspective, as it is more likely to correctly identify beneficial bulb configurations.  The solution of the near field wave pattern by Dawson's method was the single most significant tool for the bulb design in this study, and obtaining the wave elevations for a parent hull is the recommended starting point for future parabolization studies. A new  92  bulb can then be designed by trying to "tune" the elements of the bulb to interact with the ship's existing wave pattern. 7.1 Economic Implications of Parabolization For the trimaran, a parametric study of parabolic bulbs at the waterline was able to produce a hull which reduced the wave resistance by 21%. At full scale, this translated into a reduction in total resistance of about 6%.  It is also very  encouraging that the wave resistance reductions occur up to Fn = 0.45. At this Froude number, the outriggers already produce beneficial wave interactions, and further decreases to the wave resistance result in a hull with very low resistance.  For a high speed ship, a reduction to the required horsepower of over 6% is quite substantial, as the powering requirements have been reduced by 24 000 hp at 26 knots (Fn = 0.36 for 160m full scale vessel). To give a rough order of magnitude for the cost savings this could amount to, the cost of installed horsepower has been estimated as $350/ kW, and fuel has been estimated at 30cents/L for marine diesel fuel. The fuel consumption has been estimated at 17g /kW hr, operating on a ferry route that requires 15 h per day, for 320 days per year (this allows time for service withdrawls, while otherwise maintaining a regular year round schedule). This would indicate savings in the initial capital cost of the engines would be approximately $600 000, and operational costs would be reduced by at least $50 000 per year.  Since the midship bulbs do not involve complex curvature, it is unlikely that they would increase fabrication costs, and the potential economic savings they offer appear substantial.  93  7.2 Recommendations for Future Work Previous parabolization studies have been successful in showing that the overall resistance of a monohull can be significantly reduced at moderate Froude numbers by changing the shoulder wave system. The wave interactions seen for hull 7 in the trimaran case are of a different nature, where the protruding bulb creates a new wave which interacts with the wave system of the outriggers. Future studies into wave cancellation devices for multihulls may provide much different hull forms that are able to reduce the total resistance. The current hull form was derived by increasing the displacement of the parent vessel through the use of a parabolic bulb. It should be possible to achieve a further reduction in resistance by reducing the entrance angle and transom area as suggested in chapter 3, in order to produce a hullform with the same displacement as the parent hull. In order to validate the wave cancellations for a multihull ship, it is recommended that time be spent on measuring the wave surface using laser profiling methods. Capacitance probes were unable to get sufficient resolution to provide meaningful data for the small wave amplitudes produced by the NPL trimaran. Seeding the surface of the water with a reflectant such that it can reflect laser light has proven an effective method for calculating the wave resistance and it is recommended that such an approach be taken for future multihull projects.  To improve the numeric capabilities of the naval lab, it is recommended that a complete pre-processor be written for Dawson's method.  This would allow  integration of this program and boundary layer programs into an optimization routine.  94  References [I]  Bailey, D. "The NPL High Speed Round Bilge Displacement Hull Series: Resistance, Propulsion, Manoeuvering and Seakeeping data", Maritime Technology Monograph #4, 1976  [2]  Bertram, V. "Practical Ship Hydrodynamics" Butterworth-Heinemann, 2000  [3]  Calisal, S. M., Goren, O. and Danisman, D. B. "Resistance Reduction by Increased Beam for Displacement-Type Ships", Journal of Ship Research, Vol. 46 No. 3, Sept. 2002  [4]  Calisal, S.M., McGreer, D., "A Resistance Study on a Systematic Series of Low L/B Vesserls", Marine Technology, Vol. 30, No.4, Oct. 1993, pp. 286-296.  [5]  Cheng, B. "Computations of 3D Transom Stern Flows" Fifth International Conference on Numerical Ship Hydrodynamics, Hiroshima, 1989  [6]  Couser, P.R., Molland, A.F., Armstrong, N.A., Utama, I.K.A.P. "Calm Water Powering Predictions for High Speed Catamarans" Proceedings, Conference on Fast Sea Transportation (FAST '97), Sydney, 1997  [7]  Dawson, C W . "A Practical Computer Method for Solving Ship Wave Problems," Proceedings of the 2 International Conference on Numerical Ship nd  Hydrodynamics, University of California, Berkeley, 1977 [8]  Eggers, K.W.H., "On the Determination of the Wave Resistance of a Ship Model by an Analysis of its Wave System", Proceedings of the Int'l Seminar on Theoretical Wave Resistance", University of Michigan, Ann Arbor, Michigan, 1963, pg 1313-1352  [9]  Eggers, K.W.H., Sharma, S.D. and Ward, L.W., "An Assessment of Some Experimental Methods for Determining the Wave Making Characteristics of a Ship Form", SNAME Transactions, Vol. 75, 1967, pg 112-157  [10]  Gertler, M. "A Reanalysis of the Original Test Data for Taylor Standard Series", David W. Taylor Model Basin, Report 806, Washington, 1954  [II]  Gillmer, T., Johnson, B. "Introduction to Naval Architecture", Naval Institute Press, 1982  95  [12]  Goren, O., Atlar, M., "A Computational Study for the Wave Resistance Analysis of Multi Hull Forms", Report Number MT-1998-031, Department of Marine Technology, The University of Newcastle Upon Tyne, 1998  [13]  Gotman, A.Sh. "Study of Michell's Integral and Influence of Viscosity and Ship Hull Form on Wave Resistance", Ocean Engineering International, Vol.6, No.2, 2002  [14]  Hally, D. "Tests of the Sensitivity of Dawson's Panel Method to the Free Surface Panels", Etude No. 2522, Piece No. 6, Bassin d'Essais des Carenes, Val de Reuil, France, Dec. 1995  [15]  Harvald, Sv. Aa. "Resistance and Propulsion of Ships", Krieger Publ., Malabar, 1991  [16]  Hess, J.L., Smith, A.M.O., "Calculation of Potential Flow Around Arbitrary Bodies", Progress in Aeronautical Sciences, Vol. 8, 1967  [17]  Hsiung, C.C., "Optimal Ship Forms for Minimum Wave Resistance", Journal of Ship Research, Vol 25, No.2, June 1981  [18]  Hsiung, C.C., Shenyan, D. "Optimal Ship Forms for Minimum Total Resistance", Journal of Ship Research, Vol. 28, No. 3, Sept 1984  [19]  Hughes, G. "Friction and Form Resistance in Turbulent Flow and a Proposed Formulation for Use in Model and Ship Correlation" Transactions, INA, Vol. 96  [20]  Hughes, G., Allan, J.F., Turbulence Stimulation on Ship Models", SNAME Transactions, Vol 59, 1951  [21]  International Towing Tank Conference (ITTC), "The Specialist Committee on Powering Performance Prediction - Final Report and Recommendations to the 24 ITTC", Proceedings of the 24 ITTC, 2005 th  [22]  th  Janson, C , Larsson, L., "A method for the Optimization of Ship Hulls from a Resistance Point of View," Twenty-First Symposium on Naval Hydrodynamics, 1997  [23]  Kent, J.L. "Model Experiments on the Effect of Beam on the Resistance of Mercantile Ship Forms", Transactions, Institute of Naval Architects, LXI, 311319, 1919  96  [24]  Klaptocz, V. "Effect of Parabolization on Viscous Resistance of Displacement Vessels" Masters Thesis, Univ. of British Columbia, 2006  [25]  Larsson, L., Eliasson, R. "Principles of Yacht Design", The McGraw-Hill Companies, 2000  [26]  Lewis, E. V. (Ed.), "Principles of Naval Architecture", Vol. II, SNAME, Jersey City, New Jersey, 1988  [27]  Maissonneuve, J.J., "Resolution du problemede la resistance de vagues des navire par une metode de singularites de Rankine", Doctoral Thesis, Univ. of Nantes, 1989  [28]  Michell, J.H. "The Wave Resistance of a Ship", Phil. Mag. (5), Vol. 45, 1898, pg 106-123  [29]  Migali, A., Miranda, S., Pensa, C , "Experimental Study on the Efficiency of Trimaran Configuration for High Speed, Very Large Ships" Proceedings, Conference on Fast Sea Transportation (FAST 2001), Southampton, UK, 2001  [30]  Newman, J.N. "Marine Hydrodynamics", MIT Press, 1977  [31]  Papanikolaou, A., Kaklis, P., Koskinas, C , Spanos, D. "Hydrodynamic Optimization of Fast- Displacement Catamarans", Proceedings, 21 Symposium st  on Naval Hydrodynamics, The National Academy of Sciences, 1997 [32]  Rice, J.R., Walker, D., Fu, T . 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"Experimental Validation of Computational Models for the Platform Supply Vessel 'El Pionero'" Report submitted to the Department of Mechanical Engineering, 2005  [39]  Wehausen, J.H., Laitone, E.H. "Surface Waves", Handbuch der Physik, ed. W. Flugge, Chapter 9, Springer-Verlag, Berlin, 1962  [40]  White, F. M. "Fluid Mechanics", McGraw-Hill, 2003  [41]  Yang, C , "Practical CFD Applications to Design of a Wave Cancellation Multihull Ship", Proceedings, 23 Symposium on Naval Hydrodynamics, Val de rd  Reuil, France, 2000  98  APPENDIX A: OFFSET TABLE FOR THE NPL TRIMARAN  99  NPL T r i m a r a n c e n t r e (x,y,z)  = 0,0,0  Format o f o f f s e t XI (Yl,Y2,...YN) (Zl,Z2, ...ZN)  Hull  Offset  a t bow, c e n t r e l i n e , table  T a b l e - A l l d i m e n s i o n s i n m. waterline  i s as f o l l o w s :  X2 (Y1.Y2,...YN) (Zl,Z2,...ZN)  XN (Yl,Y2,...YN) (Zl,Z2,...ZN)  = 0.0 ( Y , Z ) (0.0) (0.0)  X  = 3.08 (0.48,0.432,0.384,0.336,0.288,0.24,0.208,0.16,0.128,0.08,0.048,0.016,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48)  X  = 6.15 (0.944,0.88,0.8,0.736,0.672,0.608,0.56,0.496,0.432,0.384,0.32,0.272, 0.208,0.144,0.08,0.016,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64)  X  = 9.23 (1.36,1.264,1.184,1.104,1.024,0.96,0.88,0.8,0.736,0.656,0.592,0.512, 0.448,0.368,0.288,0.192,0.096,0.016,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22)  X  = 12.31 (1.728,1.616,1.52,1.44,1.344,1.248,1.168,1.072,0.992,0.896,0.816,0.72, 0.64,0.544,0.448,0.352,0.256,0.144,0.032,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51)  X  = 15.38 (2.048,1.936,1.84,1.728,1.616,1.52,1.424,1.312,1.216,1.12,1.024,0.912, 0.816,0.704,0.592,0.48,0.352,0.24,0.112,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51)  X  = 18.46 (2.352,2.24,2.112,2,1.888,1.776,1.648,1.536,1.424,1.312,1.2,1.088,0.96, 0.832,0.72,0.592,0.448,0.32,0.176,0.032,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8)  X  = 21.54 (2.64,2.496,2.384,2.256,2.128,2,1.872,1.744,1.616,1.504,1.36,1.232, 1.104,0.96,0.816,0.688,0.544,0.384,0.24,0.08,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8)  X  100  X = 24.62 (2.896,2.752,2.624,2.496,2.352,2.224,2.08,1.952,1.808,1.664,1.52,1-376, 1.232,1.088,0.928,0.768,0.608;0.448,0.288,0.112,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) X = 27.69 (3.136,2.992,2.864,2.72,2.576,2.432,2.288,2.144,1.984,1.84,1.68,1.52, 1.36,1.184,1.024,0.848,0.688,0.512,0.32,0.144,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) X = 30.77 (3.376,3.232,3.072,2.928,2.784,2.64,2.48,2.32,2.16,2,1.824,1.648,1.472, 1.296,1.12,0.928,0.752,0.56,0.368,0.16,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) X = 33.85 ( 3 . 6 , 3 . 4 4 , 3 .296, 3 . 1 3 6 , 2 . 9 9 2 , 2 . 8 3 2 , 2 . 6 7 2 , 2 . 4 9 6 , 2 . 3 3 6 , 2 . 1 6 , 1 . 9 6 8 , 1 . 7 9 2 , 1.6,1.408,1.216,1.008,0.816,0.608,0.4,0.192,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) X = 36.92 (3.808,3.648,3.504,3.344,3.184,3.024,2.848,2.672,2.496,2.32,2.128,1.92, 1.712,1.504,1.296,1.088,0.864,0.656,0.432,0.208,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) X = 40 (4,3.856,3.696,3.536,3.376,3.2,3.04,2.848,2.656,2.464,2.256,2.048,1.84, 1.616,1.392,1.168,0.928,0.688,0.448,0.208,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) X = 43.08 (4.192,4.048,3.888,3.728,3.552,3.392,3.2,3.024,2.816,2.624,2.4,2.176, 1.952,1.712,1.472,1.232,0.992,0.736,0.48,0.224,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) X = 46.15 (4.368,4.224,4.064,3.904,3.728,3.552,3.376,3.184,2.976,2.768,2.544, 2.304,2.064,1.824,1.568,1.296,1.04,0.768,0.496,0.224,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) X = 49.23 (4.544,4.4,4.24,4.08,3.904,3.728,3.536,3.344,3.136,2.912,2.672,2.432, 2.176,1.904,1.648,1.36,1.088,0.8,0.512,0.208,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) X = 52.31 (4.704,4.56,4.4,4.24,4.064,3.888,3.696,3.488,3.28,3.056,2.8,2.544,2.288 2,1.712,1.424,1.12,0.816,0.512,0.192,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06  4.35,4.64,4.93,5.22,3.51,5.8) X = 55.38 (4.848,4.704,4.56,4.384,4.224,4.032,3.84,3.648,3.424,3.184,2.928,2.656, 2.384,2.096,1.792,1.472,1.152,0.832,0.496,0.16,0)  101  (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) X = 58.46 (4.992,4.848,4.704,4.528,4.368,4.176,3.984,3.776,3.552,3.312,3.056, 2.768,2.48,2.176,1.856,1.52,1.184,0.848,0.48,0.128,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) X = 61.54 (5.12,4.976,4.832,4.672,4.496,4.32,4.128,3.92,3.696,3.44,3.168,2.88, 2.576,2.256,1.92,1.568,1.2,0.832,0.464,0.08,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) X = 64.62 (5.232,5.104,4.96,4.8,4.624,4.448,4.256,4.048,3.808,3.552,3.28,2.976, 2.656,2.32,1.968,1.6,1.216,0.832,0.432,0.016,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) X = 67.69 (5.344,5.216,5.072,4.912,4.752,4.576,4.368,4.16,3.936,3.664,3.392,3.072 2.736,2.384,2.016,1.632,1.232,0.816,0.384,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51) X = 70.77 (5.44,5.312,5.168,5.024,4.864,4.688,4.48,4.272,4.032,3.776,3.488,3.168, 2.816,2.448,2.064,1.648,1.216,0.784,0.32,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51) X = 73.85 (5.52,5.408,5.264,5.12,4.96,4.784,4.592,4.384,4.144,3.872,3.568,3.248, 2 . 8 8 , 2 . 4 9 6 , 2 . 0 8 , 1 . 6 4 8 , 1 . 2 , 0 . 7 3 6 , 0 . 2 56,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51) X = 76.92 (5.6,5.488,5.36,5.216,5.056,4.88,4.688,4.48,4.24,3.968,3.664,3.312, 2.944,2.544,2.112,1.648,1.184,0.688,0.176,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51) X = 80 (5.664,5.552,5.44,5.296,5.136,4.96,4.768,4.56,4.32,4.048,3.728,3.376, 2.992,2.576,2.128,1.648,1.152,0.624,0.08,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51) X = 83.08 (5.728,5.616,5.504,5.376,5.216,5.04,4.848,4.64,4.4,4.112,3.792,3.44, 3.024,2.592,2.128,1.616,1.104,0.544,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22) X = 86.15 (5.792,5.68,5.568,5.44,5.28,5.12,4.928,4.704,4.464,4.176,3.856,3.472, 3.056,2.608,2.112,1.584,1.04,0.464,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22)  102  X = 89.23 (5.824,5.728,5.616,5.488,5.344,5.184,4.992,4.768,4.528,4.224,3.888, 3.504,3.072,2.592,2.08,1.536,0.96,0.352,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22) X = 92.31 (5.872,5.776,5.664,5.536,5.392,5.232,5.04,4.832,4.576,4.272,3.92,3.52, 3.072,2.576,2.048,1.472,0.864,0.24,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22) X = 95.38 (5 .904, 5.808, 5.696,5.584,5.44,5.28,5.088,4.864,4.608,4.304,3.952,3.536, 3.072,2.544,1.984,1.392,0.768,0.112,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22) X = 98.46 (5.92,5.84,5.728,5.616,5.472,5.312,5.136,4.912,4.64,4.336,3.952,3.52, 3.04,2.496,1.92,1.296,0.64,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93) X = 101.54 (5.936,5.856,5.76,5.632,5.504,5.344,5.152,4.928,4.656,4.336,3.952,3.504 2.992,2.432,1.824,1.184,0.512,0) (0,0.29^0. 58,6.87]l". 16,1.45,l!74,2.03,2. 32,2.61,2.9,3.19, 3.48,3.77,4.06 4.35,4.64,4.93) X = 104.62 (5.952,5.872,5.776,5.664,5.52,5.376,5.184,4.96,4.672,4.336,3.936,3.472, 2.944,2.368,1.728,1.072,0.368,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93) X = 107.69 (5.952,5.872,5.776,5.664,5.536,5.392,5.2,4.96,4.672,4.336,3.92,3.424, 2.88,2.272,1.616,0.928,0.208,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93) X = 110.77 (5.952,5.872,5.776,5.68,5.552,5.392,5.2,4.96,4.672,4.304,3.872,3.36, 2.784,2.16,1.472,0.768,0.032,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93) X = 113.85 (5.936,5.872,5.776,5.664,5.536,5.392,5.184,4.944,4.64,4.272,3.808,3.28, 2.688,2.032,1.328,0.608,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64) X = 116.92 (5.92,5.856,5.76,5.664,5.536,5.376,5.168,4.928,4.608,4.224,3.744,3.184, 2.56,1.888,1.168,0.416,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64) = 120 (5.904,5.824,5.744,5.648,5.504,5.344,5.152,4.88,4.56,4.144,3.648,3.072, 2.416,1.728,0.992,0.224,0)  X  103  (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64) X = 123.08 (5.872,5.808,5.712,5.616,5.488,5.312,5.104,4.848,4.496,4.064,3.536, 2.944,2.272,1.552,0.8,0.016,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64) X = 126.15 (5.84,5.76,5.68,5.568,5.44,5.28,5.056,4.784,4.416,3.968,3.424,2.784, 2.096,1.36,0.576,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35) X = 129.23 (5.792,5.728,5.632,5.52,5.392,5.216,4.992,4.704,4.336,3.856,3.28,2.624, 1.904,1.152,0.352,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35) X = 132.31 (5.744,5.664,5.584,5.472,5.328,5.152,4.928,4.624,4.224,3.712,3.104, 2.432,1.696,0.912,0.112,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35) X = 135.38 (5.68,5.6,5.52,5.408,5.264,5.072,4.832,4.512,4.08,3.552,2.928,2.224, 1.456,0.672,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06) X = 138.46 (5.6,5.536,5.44,5.328,5.168,4.976,4.72,4.384,3.936,3.376,2.72,1.984, 1.216,0.4,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06) X = 141.54 (5.52,5.44,5.36,5.232,5.072,4.864,4.592,4.24,3.76,3.168,2.48,1.728, 0.928,0.096,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06) X = 144.62 (5.424,5.36,5.264,5.136,4.96,4.736,4.448,4.064,3.552,2.944,2.224,1.44, 0.608,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77) X = 147.69 (5.328,5.248,5.152,5.024,4.848,4.608,4.288,3.872,3.328,2.672,1.92,1.104 0.256,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77) X = 150.77 (5.216,5.152,5.04,4.896,4.704,4.448,4.112,3.648,3.072,2.384,1.584,0.736 0.0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48) X = 153.85 (5.104,5.04,4.928,4.784,4.576,4.288,3.904,3.424,2.784,2.048,1.2,0.288,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48) 104 X = 156.92 (4.992,4.912,4.816,4.656,4.432,4.128,3.712,3.152,2.48,1.664,0.752,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19)  X = 160 (4.864,4.8,4.688,4.56,4.32,4,3.552,2.976,2.256,1.312,0.48,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19)  NPL Baseline Trimaran Outrigger Offset Table - A l l dimensions in m. (Port and starboard outriggers i d e n t i c a l and symmetric about centreline.) ( x , y , z ) = 0 , 0 , 0 at outrigger bow, c e n t r e l i n e , waterline Transverse spacing of Outriggers (centreline of centre hull to centreline of outrigger) = 14.2 m  Longitudinal Spacing of Outriggers (Transom of outrigger to transom of centre h u l l , p o s i t i v e distance = outrigger transom fwd of centre hull transom) = 0.0 m  Format of o f f s e t table i s as follows: xl (Yl,Y2,...YN) (Zl,Z2,...ZN) X2 (Yl,Y2,...YN) (Z1.Z2,...ZN)  XN (Yl,Y2,...YN) (Zl,Z2,...ZN) X = 0 (0.096,0.096,0.096,0.08,0.08,0.064,0.064,0.048,0.048,0.032,0.032,0.016, 0.0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446) X = 1.131 (0.208,0.192,0.192,0.176,0.16,0.144,0.144,0.128,0.112,0.096,0.096,0.08, 0.064,0.048,0.032,0.016,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928) X = 2.263 (0.288,0.272,0.256,0.24,0.224,0.208,0.192,0.176,0.16,0.144,0.128,0.112, 0.096,0.08,0.064,0.048,0.032,0.016,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169) X = 3.394 (0.368,0.352,0.336,0.32,0.288,0.272,0.256,0.24,0.224,0.192,0.176,0.16, 0.144,0.128,0.096,0.08,0.064,0.032,0.016,0) 105  (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29) X = 4.525 (0.432,0.416,0.4,0.368,0.352,0.336,0.304,0.288,0.272,0.24,0.224,0.208, 0.176,0.16,0.128,0.112,0.08,0.048,0.032,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29) X = 5.657 (0.496,0.48,0.448,0.432,0.4,0.384,0.352,0.336,0.304,0.288,0.256,0.24, 0.208,0.176,0.16,0.128,0.096,0.064,0.048,0.016,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 6.788 (0.56,0.528,0.496,0.48,0.448,0.416,0.4,0.368,0.336,0.32,0.288,0.256, 0.24,0.208,0.176,0.144,0.112,0.08,0.048,0.016,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 7.92 (0.608,0.576,0.544,0.528,0.496,0.464,0.448,0.416,0.384,0.352,0.32,0.288 0.256,0.224,0.192,0.16,0.128,0.096,0.064,0.032,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 9.051 (0.656,0.624,0.592,0.56,0.528,0.496,0.48,0.448,0.416,0.384,0.352,0.32, 0.288,0.24,0.208,0.176,0.144,0.112,0.064,0.032,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 10.182 ( 0 . 6 8 8 , 0 . 6 7 2 , 0 . 6 4 , 0 . 6 0 8 , 0 . 5 7 6 , 0 . 5 4 4 , 0 . 5 1 2 , 0 . 4 8 , 0 . 4 4 8 , 0 . 4 1 6 , 0 . 3 8 4 , 0 . 3 52, 0.304,0.272,0.24,0.192,0.16,0.112,0.08,0.032,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 11.314 (0.736,0.704,0.672,0.64,0.624,0.592,0.56,0.512,0.48,0.448,0.416,0.368, 0.336,0.288,0.256,0.208,0.176,0.128,0.08,0.048,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 12.445 (0.768,0.752,0.72,0.688,0.656,0.608,0.576,0.544,0.512,0.464,0.432,0.384 0.352,0.304,0.272,0.224,0.176,0.128,0.096,0.048,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 13.576 (0.816,0.784,0.752,0.72,0.688,0.656,0.624,0.576,0.544,0.512,0.464,0.416 0.384,0.336,0.288,0.24,0.192,0.144,0.096,0.048,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 14.708 (0.848,0.816,0.784,0.752,0.72,0.688,0.656,0.624,0.576,0.544,0.496,0.448 0.4,0.352,0.304,0.256,0.208,0.16,0.096,0.048,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41)  106  X = 15.839 (0.88,0.848,0.816,0.784,0.752,0.72,0.672,0.64,0.592,0.56,0.512,0.464, 0.416,0.368,0.304,0.256,0.208,0.16,0.096,0.048,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 16.971 (0.912,0.88,0.848,0.816,0.784,0.752,0.72,0.672,0.64,0.592,0.544,0.496, 0.448,0.384,0.336,0.272,0.224,0.16,0.112,0.048,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 18.102 (0.944,0.912,0.88,0.848,0.816,0.784,0.752,0.704,0.656,0.608,0.56,0.512, 0.464,0.4,0.3 52,0.288,0.224,0.176,0.112,0.048,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 19.233 (0.976,0.944,0.912,0.88,0.832,0.8,0.768,0.72,0.672,0.624,0.576,0.528, 0.464,0.416,0.352,0.288,0.224,0.16,0.096,0.032,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 20.365 (0.992,0.976,0.944,0.912,0.88,0.832,0.8,0.752,0.72,0.672,0.608,0.56, 0.496,0.432,0.368,0.304,0.24,0.176,0.096,0.032,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 21.496 (1.024,0.992,0.96,0.928,0.896,0.864,0.832,0.784,0.736,0.688,0.64,0.576, 0.512,0.448,0.384,0.32,0.24,0.176,0.096,0.016,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 22.627 (1.04,1.024,0.992,0.944,0.912,0.88,0.832,0.8,0.752,0.688,0.64,0.576, 0.512,0.448,0.384,0.32,0.24,0.16,0.096,0.016,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 23.759 (1.072,1.04,1.008,0.976,0.944,0.912,0.88,0.832,0.784,0.736,0.672,0.624, 0.544,0.48,0.4,0.336,0.256,0.176,0.08,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29) X = 24.89 (1.088,1.056,1.024,0.992,0.96,0.928,0.896,0.848,0.8,0.752,0.688,0.624, 0.56,0.48,0.416,0.336,0.24,0.16,0.08,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29) X = 26.022 (1.104,1.072,1.04,1.008,0.976,0.944,0.896,0.864,0.816,0.752,0.688,0.624 0.56,0.48,0.4,0.32,0.24,0.144,0.064,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29) X = 27.153 (1.12,1.088,1.072,1.04,1.008,0.976,0.928,0.896,0.848,0.784,0.736,0.656, 0.592,0.512,0.432,0.336,0.24,0.144,0.048,0)  107  (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1-205,1-326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29) X = 28.284 (1.136,1.104,1.088,1.056,1.024,0.976,0.944,0.896,0.848,0.8,0.736,0.672, 0.592,0.512,0.416,0.336,0.24,0.128,0.032,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29) X = 29.416 (1.152,1.12,1.088,1.056,1.024,0.992,0.96,0.912,0.864,0.8,0.736,0.672, 0.592,0.512,0.416,0.32,0.224,0.128,0.016,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29) X = 30.547 (1.152,1.136,1.104,1.088,1.056,1.024,0.976,0.944,0.896,0.832,0.768, 0.688,0.608,0.528,0.432,0.32,0.224,0.112,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169) X = 31.678 (1.168,1.152,1.12,1.088,1.056,1.024,0.992,0.944,0.896,0.832,0.768,0.688 0.608,0.512,0.416,0.32,0.208,0.096,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169) X = 32.81 (1.168,1.152,1.136,1.104,1.072,1.04,0.992,0.944,0.896,0.832,0.768,0.688 0.608,0.512,0.416,0.304,0.192,0.064,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169) X = 33.941 (1.184,1.168,1.136,1.12,1.088,1.056,1.024,0.976,0.928,0.864,0.784,0.704 0.624,0.512,0.416,0.304,0.176,0.048,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169) X = 35.073 (1.184,1.168,1.152,1.12,1.088,1.056,1.024,0.976,0.912,0.848,0.784,0.704 0.608,0.496,0.4,0.272,0.16,0.032,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169) X = 36.204 (1.2,1.168,1.152,1.12,1.104,1.056,1.024,0.976,0.928,0.864,0.784,0.704, 0.592,0.496,0.384,0.256,0.128,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049) X = 37.335 (1.2,1.184,1.152,1.136,1.104,1.072,1.04,0.992,0.944,0.864,0.784,0.704, 0.592,0.496,0.368,0.24,0.112,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049) X = 38.467 (1.2,1.184,1.168,1.136,1.104,1.072,1.04,0.992,0.928,0.864,0.784,0.688, 0.576,0.464,0.352,0.208,0.08,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049)  108  X = 39.598 (1.2,1.184,1.168,1.136,1.104,1.072,1.04,0.992,0.928,0.864,0.768,0.672, 0.576,0.448,0.32,0.192,0.048,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049) X = 40.729 (1.2,1.184,1.168,1.152,1.12,1.088,1.04,0.992,0.928,0.864,0.768,0.672, 0.56,0.432,0.304,0.16,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928) X = 41.861 (1.2,1.184,1.168,1.136,1.104,1.072,1.04,0.992,0.928,0.848,0.752,0.656, 0.528,0.4,0.272,0.128,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928) X = 42.992 (1.2,1.184,1.168,1.136,1.12,1.072,1.04,0.976,0.912,0.832,0.736,0.624, 0.512,0.384,0.24,0.096,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928) X = 44.124 (1.2,1.184,1.168,1.136,1.104,1.072,1.04,0.976,0.912,0.832,0.72,0.608, 0.48,0.352,0.208,0.048,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928) X = 45.255 (1.184,1.168,1.152,1.136,1.104,1.072,1.024,0.96,0.896,0.8,0.704,0.576, 0.448,0.304,0.16,0.016,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928) X = 46.386 (1.184,1.168,1.152,1.12,1.104,1.056,1.008,0.96,0.88,0.784,0.672,0.56, 0.416,0.272,0.128,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808) X = 47.518 (1.168,1.152,1.136,1.12,1.088,1.056,1.008,0.944,0.864,0.768,0.64,0.512, 0.384,0.224,0.08,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808) X = 48.649 (1.168,1.152,1.12,1.104,1.072,1.04,0.976,0.912,0.832,0.736,0.608,0.48, 0.336,0.176,0.032,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808) X = 49.78 (1.152,1.136,1.12,1.088,1.056,1.024,0.96,0.896,0.8,0.704,0.576,0.432, 0.288,0.128,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687) X = 50.912 (1.136,1.12,1.104,1.072,1.04,0.992,0.944,0.864,0.768,0.656,0.528,0.384, 0.224,0.064,0)  109  (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1-085,1-205,1.326, 1.446,1.567,1.687) X = 52.043 (1.12,1.104,1.072,1.056,1.008,0.976,0.912,0.832,0.736,0.608,0.48,0.32, 0.16,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567) X = 53.175 (1.088,1.072,1.056,1.024,0.992,0.944,0.88,0.8,0.688,0.56,0.416,0.256, 0.096,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567) X = 54.306 (1.072,1.056,1.024,0.992,0.96,0.912,0.832,0.736,0.624,0.496,0.336,0.176 0.0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446) X = 55.437 (1.04,1.024,1.008,0.976,0.928,0.864,0.784,0.688,0.56,0.416,0.256,0.08,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446) X = 56.569 (1.008,0.992,0.976,0.944,0.896,0.832,0.736,0.624,0.48,0.32,0.144,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326) X = 57.7 (0.976,0.96,0.944,0.912,0.848,0.768,0.672,0.544,0.4,0.208,0.016,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326)  110  APPENDIX B: OFFSET TABLE FOR THE PARABOLIZED TRIMARAN (HULL #7)  111  Parabolized Trimaran Centre Hull (Hull #7) Offset Table - A l l dimensions i n m. ( x , y , z ) = 0,0,0  at bow, c e n t r e l i n e ,  waterline  Format of o f f s e t table i s as follows: XI (Yl,Y2,...YN) (Z1.Z2,...ZN)  X2 (Yl,Y2,...YN) (Zl,Z2,...ZN)  XN (Yl,Y2,...YN) (Z1.Z2,...ZN) X = 0.0 ( Y , Z ) 0 0 X = 3.08 (0.496,0.448,0.4,0.352,0.304,0.256,0.224,0.176,0.144,0.112,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9) X = 6.15 (0.96,0.896,0.816,0.752,0.704,0.64,0.576,0.528,0.464,0.416,0.352,0.304, 0.24,0.176,0.112,0.048,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64) X = 9.23 (1.36,1.28,1.2,1.12,1.056,0.976,0.896,0.832,0.752,0.688,0.608,0.544, 0.464,0.384,0.304,0.224,0.144,0.048,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22) X = 12.31 (1.728,1.632,1.536,1.44,1.36,1.264,1.184,1.088,1.008,0.912,0.832,0.752, 0.656,0.576,0.48,0.384,0.288,0.176,0.08,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51) X = 15.38 (2.048,1.952,1.84,1.744,1.632,1.536,1.44,1.344,1.232,1.136,1.04,0.928, 0.832,0.72,0.608,0.496,0.384,0.272,0.144,0.032,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) X = 18.46 (2.352,2.24,2.128,2.016,1.904,1.792,1.68,1.568,1.456,1.328,1.216,1.104, 0.992,0.864,0.736,0.608,0.48,0.352,0.224,0.08,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) X = 21.54 (2.64,2.512,2.384,2.256,2.144,2.016,1.888,1.76,1.648,1.52,1.392,1.264, 1.12,0.992,0.848,0.72,0.576,0.432,0.288,0.128,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8)  112  X = 24.62 (2.896,2.768,2.624,2.496,2.368,2.224,2.096,1-968,1.824,1.696,1.552, 1.408,1.264,1.104,0.96,0.816,0.656,0.496,0.336,0.176,0.016,0) (0,0.29,0.58,0.87,1-16,1.45,1.74,2.03,2.32,2.61,2.9,3.'19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8,6) X = 27.69 (3.136,3.008,2.864,2.72,2.576,2.448,2.304,2.16,2,1.856,1.696,1.552, 1.392,1.232,1.056,0.896,0.72,0.56,0.384,0.208,0.032,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8,6) X = 30.77 (3.376,3.232,3.088,2.944,2.8,2.64,2.496,2.336,2.176,2.016,1.856,1.68, 1.52,1.344,1.152,0.976,0.8,0.608,0.416,0.24,0.048,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8,6) X = 33.85 (3.6,3.44,3.296,3.152,2.992,2.832,2.672,2.512,2.352,2.176,2,1.824,1.632 1.44,1.248,1.056,0.864,0.672,0.464,0.2 56,0.048,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8,6) X = 36.92 (3.808,3.648,3.504,3.344,3.184,3.024,2.864,2.688,2.512,2.336,2.144, 1.952,1.76,1.552,1.344,1.136,0.928,0.72,0.496,0.272,0.064,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8,6) X = 40 (4,3.856,3.696,3.536,3.376,3.216,3.04,2.864,2.672,2.48,2.288,2.08,1.872 1.664,1.44,1.216,0.992,0.768,0.528,0.288,0.064,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8,6) X = 43.08 (4.192,4.048,3.888,3.728,3.552,3.392,3.216,3.024,2.832,2.64,2.432,2.208 1.984,1.76,1.536,1.296,1.056,0.8,0.56,0.304,0.048,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8,6) X = 46.15 (4.368,4.224,4.064,3.904,3.728,3.552,3.376,3.184,2.992,2.784,2.56,2.336 2.112,1.872,1.616,1.36,1.104,0.848,0.576,0.32,0.048,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8,6) X = 49.23 (4.544,4.4,4.24,4.064,3.904,3.728,3.536,3.344,3.136,2.928,2.704,2.464, 2.224,1.968,1.696,1.44,1.152,0.88,0.592,0.304,0.016,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8,6) X = 52.31 (4.72,4.56,4.4,4.224,4.064,3.888,3.696,3.488,3.28,3.056,2.832,2.576, 2.32,2.048,1.776,1.488,1.2,0.912,0.608,0.304,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) X = 55.38 (4.848,4.688,4.544,4.384,4.208,4.032,3.84,3.632,3.424,3.2,2.944,2.688, 2.432,2.144,1.856,1.552,1.248,0.928,0.608,0.272,0)  113  (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) X = 58.46 (4.96,4.816,4.672,4.528,4.352,4.176,3.984,3.776,3.552,3.328,3.072,2.8, 2.528,2.224,1.92,1.6,1.28,0.944,0.592,0.256,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) = 61.54 (5.168,4.992,4.832,4.656,4.48,4.304,4.112,3.904,3.68,3.44,3.184,2.912, 2.608,2.304,1.984,1.648,1.296,0.944,0.576,0.208,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8)  X  X = 64.62 (5.44,5.232,5.04,4.848,4.656,4.448,4.24,4.032,3.808,3.552,3.296,3.008, 2.704,2.384,2.032,1.68,1.312,0.944,0.544,0.16,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) X = 67.69 (5.696,5.488,5.296,5.072,4.864,4.64,4.4,4.16,3.92,3.664,3.392,3.104, 2.784,2.448,2.096,1.712,1.328,0.928,0.512,0.096,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) X = 70.77 (5.92,5.728,5.504,5.28,5.04,4.8,4.544,4.288,4.032,3.76,3.488,3.184, 2.848,2.496,2.128,1.744,1.328,0.896,0.464,0.016,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51,5.8) X = 73.85 (6.16,5.968,5.744,5.504,5.248,4.976,4.704,4.416,4.144,3.872,3.584,3.264 2.928,2.56,2.16,1.76,1.328,0.864,0.416,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51) X = 76.92 (6.448,6.24,6.016,5.76,5.504,5.216,4.928,4.624,4.32,3.984,3.664,3.328, 2.976,2.592,2.176,1.76,1.296,0.832,0.336,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51) = 80 (6.64,6.448,6.208,5.952,5.68,5.376,5.072,4.752,4.416,4.08,3.744,3.392, 3.024,2.624,2.208,1.744,1.28,0.768,0.256,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51)  X  X = 83.08 (6.768,6.56,6.32,6.064,5.776,5.488,5.168,4.832,4.496,4.16,3.808,3.44, 3.056,2.64,2.208,1.728,1.232,0.704,0.16,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51) = 86.15 (6.768,6.56,6.336,6.08,5.808,5.52,5.2,4.88,4.56,4.208,3.856,3.488,3.088 2.656,2.208,1.712,1.184,0.624,0.064,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22,5.51)  X  114  X = 89.23 (6.656,6.464,6.2 56,6.016,5.76,5.472,5.184,4.88,4.56,4.224,3.888,3.504, 3.104,2.656,2.176,1.664,1.104(0.528,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22) X = 92.31 (6.48,6.304,6.112,5.888,5.648,5.392,5.136,4.848,4.56,4.24,3.904,3.536, 3.12,2.656,2.144,1.6,1.024,0.432,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22) X = 95.38 (6.352,6.192,6.016,5.824,5.6,5.376,5.12,4.864,4.576,4.272,3.936,3.552, 3.104,2.624,2.096,1.536,0.928,0.304,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22) X = 98.46 (6.24,6.096,5.92,5.744,5.568,5.36,5.12,4.88,4.608,4.304,3.952,3.552, 3.104,2.592,2.048,1.456,0.832,0.176,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93,5.22) X = 101.54 (6.112,5.968,5.808,5.648,5.488,5.312,5.104,4.88,4.608,4.304,3.952,3.536 3.056,2.528,1.952,1.344,0.704,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93) X = 104.62 (5.968,5.872,5.744,5.616,5.472,5.312,5.12,4.896,4.624,4.304,3.936,3.504 3.008,2.464,1.872,1.232,0.56,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93) X = 107.69 (5.952,5.872,5.776,5.664,5.536,5.392,5.2,4.96,4.672,4.336,3.92,3.424, 2.88,2.272,1.616,0.928,0.208,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93) X = 110.77 (5.952,5.872,5.776,5.68,5.552,5.392,5.2,4.96,4.672,4.304,3.872,3.36, 2.784,2.16,1.472,0.768,0.032,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64,4.93) X = 113.85 (5.936,5.872,5.776,5.664,5.536,5.392,5.184,4.944,4.64,4.272,3.808,3.28, 2.688,2.032,1.328,0.608,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64) X = 116.92 (5.92,5.856,5.76,5.664,5.536,5.376,5.168,4.928,4.608,4.224,3.744,3.184, 2.56,1.888,1.168,0.416,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64) X = 120 (5.904,5.824,5.744,5.648,5.504,5.344,5.152,4.88,4.56,4.144,3.648,3.072, 2.416,1.728,0.992,0.224,0)  115  (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64) X = 123.08 (5.872,5.808,5.712,5.616,5.488,5.312,5.104,4.848,4.496,4.064,3.536, 2.944,2.272,1.552,0.8,0.016,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35,4.64) X = 126.15 (5.84,5.76,5.68,5.568,5.44,5.28,5.056,4.784,4.416,3.968,3.424,2.784, 2.096,1.36,0.576,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35) X = 129.23 (5.792,5.728,5.632,5.52,5.392,5.216,4.992,4.704,4.336,3.856,3.28,2.624, 1.904,1.152,0.352,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35) X = 132.31 (5.744,5.664,5.584,5.472,5.328,5.152,4.928,4.624,4.224,3.712,3.104, 2.432,1.696,0.912,0.112,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06 4.35) X = 135.38 (5.68,5.6,5.52,5.408,5.264,5.072,4.832,4.512,4.08,3.552,2.928,2.224, 1.456,0.672,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06) X = 138.46 (5.6,5.536,5.44,5.328,5.168,4.976,4.72,4.384,3.936,3.376,2.72,1.984, 1.216,0.4,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06) X = 141.54 (5.52,5.44,5.36,5.232,5.072,4.864,4.592,4.24,3.76,3.168,2.48,1.728, 0.928,0.096,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77,4.06) X = 144.62 (5.424,5.36,5.264,5.136,4.96,4.736,4.448,4.064,3.552,2.944,2.224,1.44, 0.608,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77) X = 147.69 (5.328,5.248,5.152,5.024,4.848,4.608,4.288,3.872,3.328,2.672,1.92,1.104 0.256,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48,3.77) X = 150.77 (5.216,5.152,5.04,4.896,4.704,4.448,4.112,3.648,3.072,2.384,1.584,0.736 0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48) X = 153.85 (5.104,5.04,4.928,4.784,4.576,4.288,3.904,3.424,2.784,2.048,1.2,0.288,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19,3.48) X = 156.92 (4.992,4.912,4.816,4.656,4.432,4.128,3.712,3.152,2.48,1.664,0.752,0) 116 (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19)  = 160 (4.864,4.8,4.688,4.56,4.32,4,3.552,2.976,2.256,1.312,0.48,0) (0,0.29,0.58,0.87,1.16,1.45,1.74,2.03,2.32,2.61,2.9,3.19)  X  Parabolized Trimaran (Hull # 7") Outrigger Offset Table - A l l dimensions in m. (Port and starboard outriggers i d e n t i c a l and symmetric about c e n t e r l i n e , outrigger offsets i d e n t i c a l to base hull outriggers) (x,y,z) = 0,0,0  at outrigger bow, c e n t r e l i n e , waterline  Transverse spacing of Outriggers (Centreline of centre hull to centreline of outrigger) =  14.2 m Longitudinal Spacing of Outriggers (Transom of outrigger to transom of centre h u l l , p o s i t i v e distance = outrigger transom fwa of centre hull transom) =  0.0 m Format of o f f s e t table i s as follows:  XI (Yl,Y2,...YN) (Z1.Z2,...ZN) X2 (Yl,Y2,...YN) (Zl,Z2,...ZN)  XN  (Yl,Y2,...YN)  (Z1.Z2....ZN) X =0 (0.096,0.096,0.096,0.08,0.08,0.064,0.064,0.048,0.048,0.032,0.032,0.016, 0.0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446) X = 1.131 (0.208,0.192,0.192,0.176,0.16,0.144,0.144,0.128,0.112,0.096,0.096,0.08, 0.064,0.048,0.032,0.016,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928) X = 2.263 (0.288,0.272,0.256,0.24,0.224,0.208,0.192,0.176,0.16,0.144,0.128,0.112, 0.096,0.08,0.064,0.048,0.032,0.016,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169) X = 3.394 (0.368,0.352,0.336,0.32,0.288,0.272,0.256,0.24,0.224,0.192,0.176,0.16, 0.144,0.128,0.096,0.08,0.064,0.032,0.016,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29)  117  X = 4.525 (0.432,0.416,0.4,0.368,0.352,0.336,0.304,0.288,0.272,0.24,0.224,0.208, 0.176,0.16,0.128,0.112,0.08,0.048,0.032,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29) X = 5.657 (0.496,0.48,0.448,0.432,0.4,0.384,0.352,0.336,0.304,0.288,0.256,0.24, 0.208,0.176,0.16,0.128,0.096,0.064,0.048,0.016,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 6.788 (0.56,0.528,0.496,0.48,0.448,0.416,0.4,0.368,0.336,0.32,0.288,0.256, 0.24,0.208,0.176,0.144,0.112,0.08,0.048,0.016,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 7.92 (0.608,0.576,0.544,0.528,0.496,0.464,0.448,0.416,0.384,0.352,0.32,0.288 0.256,0.224,0.192,0.16,0.128,0.096,0.064,0.032,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 9.051 (0.656,0.624,0.592,0.56,0.528,0.496,0.48,0.448,0.416,0.384,0.352,0.32, 0.288,0.24,0.208,0.176,0.144,0.112,0.064,0.032,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 10.182 ( 0 . 6 8 8 , 0 . 6 7 2 , 0 . 6 4 , 0 . 6 0 8 , 0 . 5 7 6 , 0 . 5 4 4 , 0 . 5 1 2 , 0 . 4 8 , 0 . 4 4 8 , 0 . 4 1 6 , 0 . 3 8 4 , 0 . 3 52, 0.304,0.272,0.24,0.192,0.16,0.112,0.08,0.032,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 11.314 (0.736,0.704,0.672,0.64,0.624,0.592,0.56,0.512,0.48,0.448,0.416,0.368, 0.336,0.288,0.256,0.208,0.176,0.128,0.08,0.048,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 12.445 (0.768,0.752,0.72,0.688,0.656,0.608,0.576,0.544,0.512,0.464,0.432,0.384 0.352,0.304,0.272,0.224,0.176,0.128,0.096,0.048,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 13.576 (0.816,0.784,0.752,0.72,0.688,0.656,0.624,0.576,0.544,0.512,0.464,0.416 0.384,0.336,0.288,0.24,0.192,0.144,0.096,0.048,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 14.708 (0.848,0.816,0.784,0.752,0.72,0.688,0.656,0.624,0.576,0.544,0.496,0.448 0.4,0.352,0.304,0.256,0.208,0.16,0.096,0.048,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41)  118  X = 15.839 (0.88,0.848,0.816,0.784,0.752,0.72,0.672,0.64,0.592,0.56,0.512,0.464, 0.416,0.368,0.304,0.256,0.208,0.16,0.096,0.048,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 16.971 (0.912,0.88,0.848,0.816,0.784,0.752,0.72,0.672,0.64,0.592,0.544,0.496, 0.448,0.384,0.336,0.272,0.224,0.16,0.112,0.048,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 18.102 (0.944,0.912,0.88,0.848,0.816,0.784,0.752,0.704,0.656,0.608,0.56,0.512, 0.464,0.4,0.352,0.288,0.224,0.176,0.112,0.048,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 19.233 (0.976,0.944,0.912,0.88,0.832,0.8,0.768,0.72,0.672,0.624,0.576,0.528, 0.464,0.416,0.352,0.288,0.224,0.16,0.096,0.032,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 20.365 (0.992,0.976,0.944,0.912,0.88,0.832,0.8,0.752,0.72,0.672,0.608,0.56, 0.496,0.432,0.368,0.304,0.24,0.176,0.096,0.032,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 21.496 (1.024,0.992,0.96,0.928,0.896,0.864,0.832,0.784,0.736,0.688,0.64,0.576, 0.512,0.448,0.384,0.32,0.24,0.176,0.096,0.016,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 22.627 (1.04,1.024,0.992,0.944,0.912,0.88,0.832,0.8,0.752,0.688,0.64,0.576, 0.512,0.448,0.384,0.32,0.24,0.16,0.096,0.016,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29,2.41) X = 23.759 (1.072,1.04,1.008,0.976,0.944,0.912,0.88,0.832,0.784,0.736,0.672,0.624, 0.544,0.48,0.4,0.3 36,0.2 56,0.176,0.08,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29) X = 24.89 (1.088,1.056,1.024,0.992,0.96,0.928,0.896,0.848,0.8,0.752,0.688,0.624, 0.56,0.48,0.416,0.336,0.24,0.16,0.08,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29) X = 26.022 (1.104,1.072,1.04,1.008,0.976,0.944,0.896,0.864,0.816,0.752,0.688,0.624 0.56,0.48,0.4,0.32,0.24,0.144,0.064,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29) X = 27.153 (1.12,1.088,1.072,1.04,1.008,0.976,0.928,0.896,0.848,0.784,0.736,0.656, 0.592,0.512,0.432,0.336,0.24,0.144,0.048,0) 119  (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29) X = 28.284 (1.136,1.104,1.088,1.056,1.024,0.976,0.944,0.896,0.848,0.8,0.736,0.672, 0.592,0.512,0.416,0.336,0.24,0.128,0.032,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29) X = 29.416 (1.152,1.12,1.088,1.056,1.024,0.992,0.96,0.912,0.864,0.8,0.736,0.672, 0.592,0.512,0.416,0.32,0.224,0.128,0.016,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169,2.29) X = 30.547 (1.152,1.136,1.104,1.088,1.056,1.024,0.976,0.944,0.896,0.832,0.768, 0.688,0.608,0.528,0.432,0.32,0.224,0.112,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169) X = 31.678 (1.168,1.152,1.12,1.088,1.056,1.024,0.992,0.944,0.896,0.832,0.768,0.688 0.608,0.512,0.416,0.32,0.208,0.096,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169) X = 32.81 (1.168,1.152,1.136,1.104,1.072,1.04,0.992,0.944,0.896,0.832,0.768,0.688 0.608,0.512,0.416,0.304,0.192,0.064,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169) X = 33.941 (1.184,1.168,1.136,1.12,1.088,1.056,1.024,0.976,0.928,0.864,0.784,0.704 0.624,0.512,0.416,0.304,0.176,0.048,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169) X = 35.073 (1.184,1.168,1.152,1.12,1.088,1.056,1.024,0.976,0.912,0.848,0.784,0.704 0.608,0.496,0.4,0.272,0.16,0.032,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049,2.169) X = 36.204 (1.2,1.168,1.152,1.12,1.104,1.056,1.024,0.976,0.928,0.864,0.784,0.704, 0.592,0.496,0.384,0.256,0.128,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049) X = 37.335 (1.2,1.184,1.152,1.136,1.104,1.072,1.04,0.992,0.944,0.864,0.784,0.704, 0.592,0.496,0.368,0.24,0.112,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049) X = 38.467 (1.2,1.184,1.168,1.136,1.104,1.072,1.04,0.992,0.928,0.864,0.784,0.688, 0.576,0.464,0.352,0.208,0.08,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049)  120  X = 39.598 (1.2,1-184,1.168,1.136,1.104,1.072,1.04,0.992,0.928,0.864,0.768,0.672, 0.576,0.448,0.32,0.192,0.048,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928,2.049) X = 40.729 (1.2,1.184,1.168,1.152,1.12,1.088,1.04,0.992,0.928,0.864,0.768,0.672, 0.56,0.432,0.304,0.16,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928) X = 41.861 (1.2,1.184,1.168,1.136,1.104,1.072,1.04,0.992,0.928,0.848,0.752,0.656, 0.528,0.4,0.272,0.128,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928) X = 42.992 (1.2, l!184,1.168,1.136,1.12,1.072,1.04,0.976,0.912,0.832,0.736,0.624, 0.512,0.384,0.24,0.096,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928) X = 44.124 (1.2,1.184,1.168,1.136,1.104,1.072,1.04,0.976,0.912,0.832,0.72,0.608, 0.48,0.352,0.208,0.048,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928) X = 45.255 (1.184,1.168,1.152,1.136,1.104,1.072,1.024,0.96,0.896,0.8,0.704,0.576, 0.448,0.304,0.16,0.016,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808,1.928) X = 46.386 (1.184,1.168,1.152,1.12,1.104,1.056,1.008,0.96,0.88,0.784,0.672,0.56, 0.416,0.272,0.128,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808) X = 47.518 (1.168,1.152,1.136,1.12,1.088,1.056,1.008,0.944,0.864,0.768,0.64,0.512, 0.384,0.224,0.08,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808) X = 48.649 (1.168,1.152,1.12,1.104,1.072,1.04,0.976,0.912,0.832,0.736,0.608,0.48, 0.336,0.176,0.032,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687,1.808) X = 49.78 (1.152,1.136,1.12,1.088,1.056,1.024,0.96,0.896,0.8,0.704,0.576,0.432, 0.288,0.128,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567,1.687) X = 50.912 (1.136,1.12,1.104,1.072,1.04,0.992,0.944,0.864,0.768,0.656,0.528,0.384, 0.224,0.064,0)  121  (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1-085,1.205,1.326, 1.446,1.567,1.687) X = 52.043 (1.12,1.104,1.072,1.056,1.008,0.976,0.912,0.832,0.736,0.608,0.48,0.32, 0.16,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567) X = 53.175 (1.088,1.072,1.056,1.024,0.992,0.944,0.88,0.8,0.688,0.56,0.416,0.256, 0.096,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446,1.567) X = 54.306 (1.072,1.056,1.024,0.992,0.96,0.912,0.832,0.736,0.624,0.496,0.336,0.176 0.0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446) X = 55.437 (1.04,1.024,1.008,0.976,0.928,0.864,0.784,0.688,0.56,0.416,0.2 56,0.08,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326, 1.446) X = 56.569 (1.008,0.992,0.976,0.944,0.896,0.832,0.736,0.624,0.48,0.32,0.144,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326) X = 57.7 (0.976,0.96,0.944,0.912,0.848,0.768,0.672,0.544,0.4,0.208,0.016,0) (0,0.121,0.241,0.362,0.482,0.603,0.723,0.844,0.964,1.085,1.205,1.326)  122  APPENDIX C: A GUIDE TO PRE PROCESSING FOR MICHELL'S INTEGRAL ANALYSIS BY MICHLET  123  Overall Flow Chart for Analyis by Michlet:  Analysis Parameters and Vessel Particulars: in.mlt  Hull Geometry Files: useroffl.csv userofQ.csv <more hull files if necessary>  Michlet  Categorized Output Files: Same output as main files, but output separated into Main Output Files: individual files out.mlt ship_output_by_speed. mlt  Free Surface Elevation Files <user specified name>.mlt  In order to analyse hull forms by the software program Michlet, the user must supply two inputs, the analysis parameters and vessel particulars, and the hull geometry files. The analysis parameters and vessel particulars are entered in the input file in.mlt. A sample in.mlt file is given on the following page. The generation of this file is straightforward. In order to generate the hull geometry points, two macros are written. Thefirstgenerates a rectangular grid of points in Rhino and projects them onto the hull surface. It then exports these points to a text file. The second macro sorts the points using excel and generates the file useroff#.csv, where the hull # is supplied by the user. These two macros are detailed following the sample in.mlt file.  124  # ================================================ # NPL TRIMARAN hull 7 #  # =================== INPUT FILE TYPE AND SUBTYPE • # input File Type (0=standard) 0  # input File Subtype (0=Standard) 0  # =================== OUTPUT FILE TYPE AND SUBTYPE # Output File Type (0=Standard) 0  # Output File Subtype (0=standard) 0  # ====================== COURSE AND VESSEL TYPE == # Course Particulars (0=None) 0  # Number of Hulls ( 1 , 2  or 5)  3  # ======================== PHYSICAL QUANTITIES ============ # Gravitational Acceleration (m/sec/sec) (min 9 . 6 , max 9 . 9 ) 9.80665  # ========================= WATER PROPERTIES ============== # water Density (kg/cubic metre) (min 9 9 5 . 0 , max 1030.0) 1025.9  # Water Kin. Viscosity (sq. m/sec * 10A-6) (min 0 . 8 , max 1.31) 1.18831  # Base Eddy Kin. viscosity (non-dimensional, min 1.0) 10.0  # water Depth (metres) (max=10000.0) 10000.0  # ========================== AIR PROPERTIES ================== # Air Density (kg/cubic metre) (min 0 . 9 , max 2 . 0 ) 1.26  # Air Kin. viscosity (sq. m/sec * 10A-6) (min 1 0 . 0 , max 2 0 . 0 ) 14.4  # wind Speed (m/sec) 0.0  # wind Direction  (degrees)  0.0  # ======================= CALCULATION PARAMETERS ============= # Minimum Speed (m/sec) (min 0 . 0 1 , max 3 9 . 9 ) 7.924  # Maximum speed (m/sec) (max 4 0 . 0 ) 19.809  # # Number of Speeds (min 2 , max 101) 31  # Leeway Parameters (0=None) 0  125  # wave Drag Ntheta 960  # Skin Friction Method (0=None, 1=ITTC1957) 1  # viscous Drag Form Factor 1.237  # wave Drag Form Factor 1.0  # Pressure Signature Method (0=None,l=slender body) 1  # ==================== SHIP CALCULATION PARAMETERS ================= # Number of offset stations (rows) (odd integer: min 5, max 81) 53  # Number of Offset water!ines (columns) (odd integer: min 5, max 81) 21  # Ship Loading Type 3  # ship Loading Formula Parameters 1.0,0.0,0.0  # ===================== WAVE ELEVATION PARAMETERS ================== # sectorial cuts and Patches # R0 5.0 # Rl 20.0  # Beta 22.5  # Nr 101  # Nbeta 101  # Rectangular Cuts and Patches # xO 5.0 # xl 20.0 # yO -7.5 #yl 7.5  # NWX 101  # Nwy 101  # Beaches and walls # xO 5.0 # xl  126  20.0 # yO 7.5 # zO -5.0 # z l 0.0  # slope 90.0 # Nbx 2 # Nbz 2  # ============================ FIRST HULL ================= # offsets -1  # Displacement volume (cubic metres) 4587.0  # Length (metres) 160.0  # Draft (metres) 5.8  # Longitudinal Separation (metres) ( 0 . 0 for a monohull) 0.0  # Lateral separation Distance (metres) ( 0 . 0 for a monohull) 0.0  # Loading Type for this hull 3 # Loading Formula Parameters 1.0,0.0,0.0  # Trim Method 0  # Trim: Number of speeds ( >= 2) 2  # Trim: speed, angle 0.0,0.0 40.0,0.0  # Sinkage Method 0  # sinkage: Number of speeds ( >= 2) 2  # sinkage: speed, amount 0.0,0.0 40.0,0.0  # Heel Method 0  # Heel: Number of speeds ( >= 2) 2  127  # Heel: speed, angle 0.0,0.0 40.0,0.0 # Appendages (0=None) 0 # other Particulars (0=None) 0 # ============================ SECOND HULL # offsets -1 # Displacement volume (cubic metres) 138.4 # Length (metres) 57.7 # Draft (metres) 2.41 # Longitudinal Separation (metres) 51.5 # Lateral Separation Distance (metres) 14.24 # Loading Type for this hull 3 # Loading Formula Parameters 1.0,0.0,0.0 # Trim Method 0 # Trim: Number of speeds ( >= 2) 2 # Trim: speed, angle 0.0,0.0 40.0,0.0 # Sinkage Method 0 # Sinkage: Number of speeds ( >= 2) 2 # Sinkage: speed, amount 0.0,0.0 40.0,0.0 # Heel Method 0 # Heel: Number of speeds ( >= 2) 2 # Heel: speed, angle 0.0,0.0 40.0,0.0 # Appendages (0=None) 0 # Other Particulars (0=None) 128  0  # ============================ THIRD HULL # Offsets -1  # Displacement volume (cubic metres) 138.4  # Length (metres) 57.7  # Draft (metres) 2.41  # Longitudinal separation (metres) 51.5  # Lateral Separation Distance (metres) -14.24  # Loading Type for this hull 3  # Loading Formula Parameters 1.0,0.0,0.0  # Trim Method 0  # Trim: Number of speeds ( >= 2) 2  # Trim: speed, angle 0.0,0.0 40.0,0.0  # sinkage Method 0  # sinkage: Number of speeds ( >= 2) 2  # sinkage: speed, amount 0.0,0.0 40.0,0.0  # Heel Method 0  # Heel: Number of speeds ( >= 2) 2  # Heel: speed, angle 0.0,0.0 40.0,0.0  # Appendages (0=None) 0  # Other Particulars (0=None) 0  129  The process of discretizing the hull for analysis by Michlet is started by launching the program "MichletPreProcessor.exe". The pre processor written requires an existing 3-D half model of the hull, and executes a series of commands using Rhino to extract the offset data. This pre processor has been written using Rhinoceros version 3.0, it has not been tested with different versions of Rhino.  The program starts by launching the menu shown in Figure A-1. If the user hasn't yet started Rhino or loaded a ship model, thefirsttwo buttons guide him or her through these steps. The hull surface should be faired, and a flat surface should also be created along the centreplane of the hull, with the same length and draft as the hull, as shown in Figure A-2. This surface is trimmed to remove the interior portion intersecting the hull.  Launch Rhino  Load a Ship Model  Non Dimensionalize Model  Generate Mesh  Quit  Figure A - l - The pre processor launchpad Once a hull model has been loaded, the dimensions of the hull are read by clicking the third button, "Non Dimensionalize Model". This prompts the user to select the bow and stern of the vessel and enter the draft. The bow and stern points must both lie on the centerline of the vessel at the design waterline. The draft entered must be the exact draft of the model. If the surfaces created make this information difficult to extract, launching the BoundingBox command within Rhino will create a box around the entire surface extremities.  This creates end points at the bow and stern on the centerline, and the 130  bottom o f the box created w i l l be the lowest point on the hull surface.  Displaying the  height o f this box w i l l then give the user the exact draft, and the non dimensionalize model command can be re-run.  The last step of the pre processing process is to create points i n a rectangular grid, project them onto the hull surface, and export the points to a text file. This is done by invoking the Generate M e s h command. A t this point the user is prompted for the required number of waterlines and stations to describe the hull. The user's screen should now look like Figure A-2. After this information is entered, a grid of points is created by the program to one side o f the hull.  D L5 U 0  * to a ^ fi  £> £J [o] p <p ffl * >* & «u «« ss 0 O , O  <»U  0  •  ^ ^ D ^ ^ I ^ K ^ A I  1118 points added to selection Command Delete  0  r . sfart  End NM> CPkme x 20.894  '  •••• ' •••  V  Point P Mid  |  y 23.349  <9  * B ''  ......... PCen zO  r (nt  0  R Perp r  _6-Mluo...  Tan P Quad i'" Knot Jlonesurfaceonly  • 9* Maecs  Pro|ect [Oteabtej Snap Ortho 'V fjWJMjftje'fcfi'ttmar*,...  Planar  .  Osnap  • LaunrhPad  Upgrade Now,..  4 ,8§fv, !.. PM  Figure A-2 - User input for the mesh generation process. Note the trimmed centreplane surface After creating the point grid, the project command is launched. The points are selected by using the Edit-> Select Objects-> Select Points command. Then, the hull surface and centreplane surface are selected to project the points onto. F o r this step to work properly, the user must select the surfaces in a viewport that shows an outboard elevation o f the 131  hull surface. The successfully projected points are shown i n Figure A - 3 . These points are exported by selecting a l l points, and then exporting selected objects as a points fde. The dialogue box for this operation is shown in Figure A - 4 . V N P l b u l b f i for Trimaran He  Edit  •  3  View  Curve  H H  Surface  x *  Rhinoceros  [Perspective]  Sold Transform  e  «- f)  Tools  ^jjw  Dtnenslon  Analyze  0> £3 'p] 0  Render  Help  J& ffl.«* &  £ • «LL HIDE  SJ  '0  :  Q  A  0  \_f  (?|  ; •  ,  Command: _Undo Undoing Delete  i V  sldJjJ  Command: [  CHane  xZ3,eot  y 8,646  10  0  f|onesurfecoonry  I Snap : Ortho  ' 1» M H O  |  V  Planar  Qinap '  UpgradoNow...  {  *  WibufoBforTrmafa...  Figure A-3 - Projected points ready for export  Points File Export Options Delimeler -  m  ~~~  (* Comma f Semicolon C Space  r Tab r  Other  f~  Geneial P  "ij  significant digts  r* Surround values with double-quotes OK  Cancel  Figure A-4 - Options for exporting points file  Once the text file is created, Rhino can be closed by quitting the pre-processor launch pad. The text file is then sorted i n Microsoft Excel i n the spreadsheet MichletPreProcessor.xls. 132  This spreadsheet contains a single macro that first imports the text fde containing the points, and then sorts these points and exports a formatted table of offsets to a csv fde. Once this csv file is created (multiple csv fdes are required for a multihull analysis), Michlet can be run using the in.mlt file and the hull offset file. Complete details on how to use the program Michlet, including how to view and save output, are available from the web page www.cyberiad.net/michlet.htm.  133  APPENDIX D: A GUIDE T O P R E PROCESSING F O R T H E P A N E L METHOD TRAWSON  134  Overall Flow Chart for Analysis by Dawson's Method:  Existing CAD Geometry  Generate Hull Panels  Generate Free Surface Field Points  Input file for Streamline calculation  Streamline Calculation Is Transom Wet? Yes  No Dry Transom Solver  Wet Transom Solver  Post Processing Macros  Output Data to Excel Spreadsheets  Tecplot Files  Visualization of free surface, and speed and pressure contours on hull  Data analysis of results  In order to analyse hull forms by Dawson's method, the user must generate a mesh of panels on both the wetted portion of the hull and a portion of the undisturbedfreesurface. Guidelines for mesh generation on the undisturbed free surface are given in chapter 5. 135  While no single pre processor exists for the overall process, a series of macros have been written to automate much of the file manipulation process. The initial mesh generation was the most time consuming process, as this was done using CAD software.  Different tools, such as the hull mesh generation tools in PRECAL, or the Fluent preprocessor GAMBIT, can generate meshes quickly. Using this type of approach would be the recommended next step for the development of numeric tools for the parabolization project. If this is done, software would only need to be developed to export the data from these meshes and convert them into the appropriate input data.  The main tools used in the study of parabolization have been the solver Trawson, the visualization program Tecplot, and Microsoft Excel.  Macros have been written to  convert data files between these programs, and to extract data from the results of Trawson.  Building the Input Files for Trawson:  A complete listing of all the required inputs for Trawson is given in [12], and only the information as it pertains to the usage of newly created macros is presented here.  In order to generate the input files for Trawson, the file can be split into 4 sections. The first section consists of the first two lines of the file, and supplies information such as the number of points in the mesh, number of panels, longitudinal centre of flotation, etc. This information is compiled by the user at the end of the mesh creation process. The next section contains the points of the mesh, followed by a section detailing the connectivity of points. Thefinalsection gives the speeds at which the wave resistance is to be calculated. The middle two sections, extracting the individual points of the mesh, and generating the connectivity points between the mesh, have been automated with macros.  136  Generating the points for the free surface mesh is straightforward. A script was written to extract the points from Rhino directly, "RhinoPoints.rvb", and the user simply provides an output file name and selects the points to be exported. It is important that the points are selected in the manner in which they are referenced in the connectivity list, or else the connectivity will produce incorrect panels. The connectivity is generated by running the macro "connectivity.dvb" and supplying the requested information of the mesh parameters.  Once the user has compiled both the mesh points and connectivity, the Excel sheet "SortAndOuputScripts.xls" is used to convert the file to the correct format for Trawson. In  order to  visualize  the hull  surface  using  Tecplot,  the Excel  sheet  "HullVisualization.xls" is run. Example files created by each program are given below: Example Hull Visualization File for Tecplot created by "HullVisualization.xls": ZONE T="HullandFreeSurfMesh"  N=3388, E=3212, ZONETYPE=FEQuadri1ateral DATAPACKING=POINT DT=(SINGLE SINGLE SINGLE )  -126.4438 -120.627 -114.8101 -109.1915 -103.5729 -98.5354 -93.4979 -89.3848 -85.2716 -82.3632 -79.4548  0 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0  2 3 4 5 6 7 8 9  114 115 116 117 118 119 120 121  113 114 115 116 117 118 119 120  3 4 5 6 7 8 9 10  ( a l l hull and free surface points here, format =x,y,z, total of 3388 points req'd)  (connectivity, f o r sign conventions and zone d e f i n i t i o n s , see Ref. 7, total of 3212 panel def'ns req'd)  137  Example Hull Input File for Trawson created by "SortAndOutputScripts.xls": 3673 24733273491 28 -16.500 -70 7014 0. 8973 -61 0603 2. 0555 3. 6512 -41 7781 -22 4959 4. 7883 -3 2137 5. 5103 16 0685 5. 8691 •  -126 -109 -93 -82 -76 -72 -67  0 0 0 0 0 0  0  0000 0000 0000 0000 0000 0000  0  ( L i s t of parameters such as # of points, panel count on h u l l , total panels,  -67 -54 -35 -16  4877 6329 3507 0685 3 2137 22 4959  1 2 4 5 5 5  3284 6585 0762 0737 6473 9227  0 0 0 0 0 0  0000 -64 2740 0000 -48 2055 0000 -28 9233 0000 - 9 6411 0000 9 6411 0000 28 9233  (This section l i s t s the hull  •  4438 1915 4979 3632 4438 4523 3588  4  6.0000  _ 120 6270  - 103 -89 -79 -75 -70 -65  0. 0000 0. 0000 0. 0000 0. 0000 0. 6315 I. 3390  5729 3848 4548 2058 9368 3028  0 0 0 0 0 0 1  1.7125 3.1849 4.4539 5.3165 5.7843 5.9488  0 0 0 0 0 0  0000 0000 0000 0000 0000 0000  points organized X l Y l Z l X2 Y2  0000 0000 0000 0000 1987 8560 5869  114 -98 -85 -77 -73 -69 -63  8101 5354 2716 9493 9678 1478 2468  0.0000 0.0000 0.0000 0.0000 0.3975 1.1197 1.8312  (This section l i s t s the free surface points organized x l Y l 2  8  14 20 26  1 7 13 19 25  3l' 37 43 49 55  32 38 44 50 56  3 9  15 21 27  2  8  14 20 26  32 38 44 50 56  33  39 45 51 57  4 10 16 22 28  3 9  15 21 27  33  39 45 51 57  34 40 46 52 58  5 11 17 23 29  4 10 16 22 28  34 40 46 52 58  35 41 47 53 59  6  12 18 24 30  5  11 17 23 29  35 41 47 53 59  36 42 48 54 60  x2 Y2 7  13 19 25 32  6  12 18 24 31  X3 Y3) 36 42 48 54 61  37 43 49 55 62  (This section l i s t s the connectivity for a l l panels, organized lA IB 1C ID 2A 2B 2c 2D) 7.924 8.716 9.508 9.905 10.301  (This section l i s t s the speeds for analysis in m/s)  Page 138  etc)  Sorting the Output from Trawson and Visualizing With Tecplot:  Once Trawson has been run, the outputfromthis program is sorted using the Excel sheet "OutputSort.xls." The new data is loaded into the sheet by refreshing the data, and inputting it as a tab delimited fde. The embedded macros then sort the data and generate the appropriate output files based on user input to describe the mesh parameters. The user form detailing the output fdes is shown in Figure A-5.  1]  Lrt  f-  r  - t  tfT»t  look fiata !£rtow  aeip  ^  T  • *• .i-sj  K  I  B Ji 2=  I  "!)AlA  ji _4  NORMAL 5 Bi ELEMENT :NX:.  I  J  'I Kit  L  «nl  M  |Dan:.:. 'R  ^COMPONENTS CENTROID IN GLOBAL COORDINATES: . :NZ . :Y: :Z. 0 1249 0.9618! -0:2437! -69.039: 1:0478 -0.28511 -0 1086 ~ 09513' "-0"2885: -65.8584: 1.425 -0.3233: -0.0963 : 0.9396I -0.3283: -62.6545: 1:7664 '-0.3403:  7  I  jl  §£ >  3a  Insert output data f la as la tab detmnted File here.  12:  ii 16 J718? 19: 20 2«  J?  ;  23' 2£  27 » 30 3l| 32J 33: 34: » 35 37' 38: i?<  i >r\sheetlXsheet2^sheet3/shaet4/  Claw  • <AytoshapBS 7  ,F  Figure A-5 - Entering mesh parameters to generate output files  Once the user has completed the user form, the output is sorted and output files are generated. These files can then be directly opened by Tecplot, and the resistance values are sorted for plotting with Excel.  Page 139  I  Sorting the Output from Trawson and Visualizing With Tecplot: Once Trawson has been run, the output from this program is sorted using the Excel sheet "OutputSort.xls."  The new data is loaded into the sheet by refreshing the data, and  inputting it as a tab delimited file. The embedded macros then sort the data and generate the appropriate output files based on user input to describe the mesh parameters.  The  user form detailing the output files is shown in Figure A - 5 .  Tyiw- .i question fc< help . . . a x  -  HI A  1 2  JL  _« a  e  "OUTPUT  4 5 6 ELEMENT 7  NORMAL NX  a  2 3  '..?  11 12 13 14 15 16 17 18  -0.1249 -0.1086 -0.0963  H  11  21 22 23 24 "25 26 228 9"  T f1610 "—.— 470  | 1626 1 » r r  _ ——  freeSurface Data i enterftstRow* wtih free Surface Ee l vato i ns: Enter # of mesh pon i ts: ' ] Entet separato i n bew t een speed runs: Locao tin of Res-stance Info: Start coulmn for Free Surface: ! Start Coulmn for Wave Ressitance  30 31 32  3w>!2 / AytoShapes*  • »• \ Minel l,(  M  1 1  ast"" X Staet4  3t»t3  f 21:1* r i24 field  •  [ 2108 •  j  • OK  33 34 3 8 39  L  1• 11  10 11 32  37  J |K  Dan: Insert output data fie as a tab dem il n i ted file here  COMPONENTS C ENTROIO IN GLOBAL COORDINATES NZ X Y ill... 0.9618 -69 039 -0.2437 •0 2851 1.0478 0.9513 •0.2B85 -65.8564 -0 3233 1.425 0 9396 -0.3283 •62 6545 -0 3403 1.7664  j 1632| '  Enter Coulmn wtih Pressure Data: Enter End Coulmn Enter Destn i ato i n Coulmn  27  j:  ||!]  HUI Output Data Enter Frst Row* v*th speed, pressure data Enter / of hut parts. Enter separato i n bew t een speed runs: location of speed Wo; Total Nurfcer of Speeds:  20  4  I  Enter file Particulars:  19  14  G  '  VECTOR NY  1  F  E  i  *TjATA"~—~  '-'  B  -0.11G5 -0.0968 -0.0866 -Tin7fi1  Cancel  09658 0 9559 09444' 09779  / ^ ' mm  ' A•  -0.2315 •0.2765 •03155 -h 365  =  ^  4  J  J  -69.0516 -65.8594 -62 6545 -57 8096  0.9037 1.2327 1 5328 1 9336  "jigBl..,,,.,  1  -0.8618 •0.9712 -1.0224 .1 0639  .. :• . ..  > V1  |^jg||j|yg^  Figure A-5 - Entering mesh parameters to generate output files Once the user has completed the user form, the output is sorted and output files are generated. These files can then be directly opened by Tecplot, and the resistance values are sorted for plotting w i t h E x c e l .  Page 139  

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