Study of Semi-Empirical Methods for Ship Resistance Calculations

UPTEC F 20024 Examensarbete 30 hp Juni 2020

Study of semi-empirical methods for ship resistance calculations

Emil Petersson Abstract Study of semi-empirical methods for ship resistance calculations Emil Petersson

Teknisk- naturvetenskaplig fakultet UTH-enheten In the early ship design process a quick overview of which ship design that could be the optimal choice for the intended usage needs Besöksadress: to be investigated. Therefore the feasibility and accuracy of Ångströmlaboratoriet Lägerhyddsvägen 1 interpolating between measurement data from model resistance series Hus 4, Plan 0 when estimating unknown hulls were conducted. A parametric study was undertaken in order to investigate which parameters carry the most Postadress: importance in regard to calm water resistance for semi-displacing Box 536 751 21 Uppsala hulls. In order to asses the whole estimation process one semi- displacing ship (FDS-5) and one bulk carrier with a bulbous bow (JBC) Telefon: were estimated in regard to calm water resistance by using semi- 018 – 471 30 03 empirical methods and were later compared with CFD results. The CFD

Telefax: results came from a in part parallel conducted work. The results 018 – 471 30 00 showed that it is possible to estimate the total resistance with semi-empirical methods to an unknown hull by linear interpolation Hemsida: with an accuracy of below 5% in the designed speed interval both for http://www.teknat.uu.se/student FDS-5 and JBC. The CFD simulations achieved a lower accuracy compared to the semi-empirical approach, however by furhter calibrating the models, the accuracy could potentially be improved. Linear interpolation between two hulls in order to estimate an unknown hull, is only advised when the hulls are nearly identical. Meaning that the hulls must be of the same ship type and that only one parameter is allowed to differ compared to the unknown hull. The parametric study resulted in parameter importance in falling order: Slenderness ratio, length-beam ratio, longitudinal prismatic coefficient, block coefficient and beam-draught ratio. Even though the CFD approach not yet is completely reliable, it could still be a useful complement to the semi-empirical approach by calculating parameters such as a dynamic wetted surface, resistance due to appendages or air resistance of the full-scale ship. Simply by incrementally increasing the accuracy of individual resistance components an overall improvement could potentially be achieved.

Handledare: Love Frimodig Ämnesgranskare: Per Isaksson Examinator: Tomas Nyberg ISSN: 1401-5757, UPTEC F 20024 Tryckt av: Uppsala Sammanfattning

I den tidiga skeppdesignprocessen så är det användbart att ha förmågan att snabbt kunna undersöka vilken skeppsdesign som potentiellt är den optimala utifrån tilltänkt användningsområde. FOI utvecklar för tillfäl- let sin egen programvara ShipDes för skeppsdesign och behöver därför undersöka några nyckelaspekter i den tidiga designprocessen för att potentiellt kunna förbättra nuvarande kapacitet av programvaran. För att kunna undersöka vilken skeppsdesign som potentiellt är optimal krävs en databas med motståndsdata från testade skepp i modelskala. Historiskt sett har modellserier testats och motståndsdatan uppmätts, där beskrivande skrovparameterar systematiskt har varierats. Denna modelldata har därefter används i semi- empiriska metoder, för att skala upp uppmätt motstånd från modelskala till fullskala. Därav innefattade en del av detta arbete att undersöka och sammanställa tillgänliga testresultat i form av en databas. Ibland är det skepp som önskas undersökas inte testat, men dess design kan ligga mittemellan två testade skeppsmodeller. Därför har möjligheten och precisionen att interpolera mellan känd modelldata undersökts, för att på så viss uppskatta okänd modelldata genomförts. För att undersöka vikten av vanligt förekommande beskrivande parameterar av skrovformen och nyttja dessa parametrar optimalt under interpolering utfördes en param- eterstudie på den sammanställda databasen med motståndsdata från modelltester. Beräkningsprecisionen av skrovgeometrin undersöktes genom att använda ShipDes samtidigt som den deﬁnerand skrovgeometrin varierade i upplösning. Detta gjordes för två väldokumenterade fall och resultaten kunde på så vis jämföras med kända resultat.

Delvis parallellt till detta projekt utfördes ett CFD (Computational ﬂuid dynamics) projekt [1]. Tillsammans mellan projekten beslutades det att undersöka två skrov, ett semi-deplacerande (FDS-5) och ett bulkfartyg med bulb (JBC). Detta genomfördes för att få insikt kring hur den semi-empiriska estimeringsprocessen gentemot CFD simuleringarna presterar och potentiellt ﬁnna sätt som de båda metoderna kan komplettera varandra.

Resultaten visade att FDS-5 kan bli linjärt interpolerad och estimerad i fullskala under 4 % fel under mer- parten av hastighetsintervallet med hjälp av modelldata från samma modellserie och modelldata från en annan modellserier. Även då inga lämliga set av modelldata från bulkfartyg tillät för interpolering, så går det att välja liknande fartyg och använda dess motståndsdata i kombination med semi-empiriska metoder. Detta resulterade i mindre än −5% fel över hela hastighetsintervallet. CFD simuleringarna resulterarde konsekvent i sämre precision, dock är simuleringsprogramvaran på FOI nyligen installerade och i behov av ytterliggare kalibrering. Parameterstudien påvisar i fallande ordning vikten av de olika parametrarna som undersöktes: Slankhetstalet λ, längd-bredd ratio L/B, longitudinell prismatisk koeﬃcient CP , block koeﬃcient CB och bredd-djup ratio B/T . Parameterstudien indikerar även att L/B, CP , CB och B/T är mer praktiska att nyttja vid linjär interpolering och att variationen av dessa parameterar mer konsekvent resulterar i det förväntade beteendet. Beräkningarna av skrovgeometrin för FDS-5 och JBC med hjälp av ShipDes resulterade i tillfredställande noggrannhet, t.ex under 1 % fel i våt yta S och djup T.

För att kunna utföra en estimering av motståndsdata för ett okänt skrov med precision och tillförlitlighet krävs det att att skroven som används vid interpolering är mycket lika det okända skrovet. Endast en parameter bör skilja och dessutom vara nära önskad parameter på det okända skrovet. Annars är det troligtvis bättre att nyttja en regressionsfunktion för att uppskatta motståndsdatan. Skrovgeometrin uppskattas generellt sätt väldigt bra, men för bulbfartyg kan precisionsförluster uppstå när skärningspunkten för vattenlinjen vid fören ska beräknas. Vidare så är de semi-empiriska metoderna uppbyggda kring experimentell data och ger oss endast ett försök till korrelation mellan model- och fullskala. Speciellt en term som bör uppdateras när möjligheten ges är ytråhetstillägget, efterom denna formel är baserad på exerimentell data där numera gamla bottenmålningsfärger nyttjats. Moderna bottenmålningsfärger är bättre på att motverka tillväxt på skrovet, vilket innebär att ytråhetstillägget bör minskas. Trots att CFD simuleringar ännu inte är fullt tillförlitliga vid beräkningar av det totala motståndet för fartyg, så är det ett mycket användbart verktyg. Att stegvis beräkna olika delar av det totala motståndet, skulle kunna förbättra noggranheten i nyttjandet av de semi- empiriska metoderna. Ett exempel på användningsområden där CFD simuleringar skulle kunna erhålla goda resultat vore t.ex beräkningen av motståndet från bihang (fenor, roder, anod, slingerköl etc) och hur olika placeringar av bihangen påverkar motståndet. Vidare om objekten ovan vattenlinjen är kända, så skulle luftmotståndet kunna beräknas i detalj. Även en dynamisk våt yta S kan beräknas vid olika hastigheter, vilket är en viktig parameter för de semi-empiriska metoderna och en förbättring gentemot att nyttja en statisk våt yta beräknad vid stillaliggande läge. Table of contents

1 Nomenclature 1

2 Introduction 2

3 Background 2 3.1 Historical background...... 2 3.2 Present situation...... 2 3.3 Scope of the report...... 3

4 Theory 4 4.1 Physical background...... 4 4.2 Computational fluid dynamics...... 4 4.3 Laminar and turbulent flow...... 5 4.4 Model testing procedure...... 6 4.5 Regression functions...... 7 4.6 ITTC-78 performance prediction method...... 8 4.6.1 ITTC-78 method steps...... 8 4.6.2 Coefficient explanation...... 10 4.7 Introduction of the Form factor...... 12 4.7.1 Empirical form factor formulas...... 13 4.7.2 ITTC form factor estimation procedure...... 13 4.8 Hull form design...... 15 4.8.1 Bulk carrier with bulbous bow...... 15 4.8.2 Semi-displacing vessel with transom stern...... 16 4.8.3 Hull parameters...... 17 4.9 Marin parametric study...... 18

5 Methodology 20 5.1 Literature study...... 20 5.2 Database compilation...... 20 5.3 Evaluation of linear interpolation...... 20 5.4 Geometry calculation study...... 21 5.5 Comparative study...... 23 5.6 Parametric study...... 23

6 Model series 25 6.1 The MARIN Systematic Series...... 25 6.2 Series-64...... 26 6.3 NTUA series...... 27 6.4 NPL high speed round bilge displacement series...... 27 6.5 Southampton Catamaran series...... 27 6.6 Taylor-Gertler Standard Series...... 27 6.7 Harvald & Guldhammer...... 27

7 Selected hulls 28 7.1 FDS-5 summary...... 28 7.2 JBC summary...... 29 8 Results 30 8.1 Interpolations for the FDS-5 model...... 30 8.2 FDS-5 resistance estimation...... 32 8.2.1 Model-scale FDS-5 estimation...... 32 8.2.2 Full-scale FDS-5 estimation...... 34 8.3 JBC resistance estimation...... 36 8.3.1 Model-scale JBC estimation...... 37 8.3.2 Full-scale JBC estimation...... 39 8.4 Parameter study...... 40 8.5 Hull geometry results...... 45 8.5.1 FDS-5 geometry...... 45 8.5.2 JBC geometry...... 45

9 Discussion 47 9.1 Interpolation...... 47 9.2 FDS-5 resistance estimations...... 48 9.3 JBC resistance estimations...... 49 9.4 Parameter study...... 50 9.5 Adaptation of the ITTC-78 performance prediction method...... 52 9.6 Complimenting discussion to CFD and semi-empirical co-usage...... 53 9.7 Hull geometry...... 53 9.7.1 FDS-5 geometry...... 53 9.7.2 JBC geometry...... 54

10 Conclusions 55

11 References 56

A Appendix 59 A.1 FDS-5 model resistance...... 59 A.2 JBC model resistance...... 60 1 Nomenclature Symbol Unit Description B m2 Maximum beam (breadth) at waterline 2 BT m Beam (breadth) at transom stern T m Draught amidship S m2 Wetted surface 2 AM m Submerged sectional area amidship 2 AT m Submerged sectional area of the transom 2 AA m Maximum transverse area of ship above waterline g m/s2 Acceleration due to gravity (9.81 m/s2) LCF m Longitudinal centre of flotation LCB % Longitudinal centre of buoyancy. X%L fwd. rel. to amidship. LCG m Longitudinal centre of gravity LPP m Length between perpendiculars LWL m Length on designed waterline (v = 0 m/s) m kg Mass ◦ ie Half angle of entrance CF - Frictional resistance coefficient ∆CF - Additional surface roughness coefficient CR - Residual resistance coefficient CT - Total resistance coefficient CA - Correlation allowance coefficient CAA - Air resistance coefficient CDA - Drag coefficient of the ship above waterline CP - Prismatic coefficient CB - Block coefficient RF N Frictional resistance RR N Residual resistance RV N Viscous form - and friction resistance RT N Total resistance RAP P N Appendage resistance V m/s Ship speed ρ kg/m3 Density of fluid ν m2/s Kinematic viscosity p kg/ms2 Pressure ∇ m3 Displaced volume ∆ t Displacement mass (tonnes) λ - Slenderness ratio or length-displacement ratio Fn - Froude number Fn∇ - Volumetric froude number Re - Reynolds number

1 2 Introduction

In the early ship design process it is useful to have the capability to quickly investigate which ship design that could be the optimal choice for the intended usage. Historically, model resistance test series on various ship types have been carried out in order to gain insights in how a ship design performs on model-scale. This data have thereafter been used in semi-empirical methods when scaling up the results to full-scale. Naturally, conducting a complete model series or just one model test on its own, is a time consuming task, as well as expensive. Therefore, a regression analysis can be performed on the model series test data in an attempt to create a useful regression function. The regression function can then be used in order to estimate the resistance for hulls with an intermediate design. However, it might be possible to gain some accuracy if just two hulls are used in order to estimate an intermediate design. Another approach that is gaining popularity and usability is Computational Fluid Dynamics (CFD), it uses the fundamental Navier-Stokes equations as basis for solving the simulations numerically. However, model tests still have to be performed in order to conﬁrm that the CFD catches the physical phenomenons correctly [2]. Therefore partly in parallel to this work a CFD work was conducted [1]. Jointly between both works it was decided to investigate one semi- displacing hull (FDS-5) and one bulk carrier with bulbous bow (JBC) with semi-empirical methods as well as CFD. This was conducted in an attempt to gain insights in regards to how both approaches compare and potentially could complement each other. Also a parametric study was undertaken in order to investigate which parameters carry the most importance, how varying the parameters aﬀect the resistance and how the parameters investigated optimally can be used for linear interpolation for semi-displacing hulls.

3 Background 3.1 Historical background Until the early 1860’s, the underlying understanding of ship resistances and the ideas surrounding the powering at that time was lacking. I.e the design procedure of the propeller was based on trial and error, which was both costly and time consuming. Furthermore the installed power in the ships often ended up non-optimal, thereof the necessity of developing prediction methods that could assist in the early ship design stages. One of the forerunners of this work was Froude, in 1870 he initiated an investigation into ship resistance with help of models. By noting that similar wave configurations around geometrically similar forms were created at corresponding speeds, implying on the fact that the speed was proportional to the square root of the model length. From his work he suggested that the total resistance could be divided up into skin friction and residual resistance. Froude derived estimates of frictional resistance lines by performing series of measurements on planks with varying lengths and surface roughness. Furthermore, Froude also introduced the idea that the residual resistance coefficient would remain constant at corresponding speed between model and ship. At around the 1890’s the full potential of utilising model resistance tests had been realised. Testing was performed for specific ships and model test series was carried out. Notable early work on this have been performed by Taylor and Baker. The next era of model testing mainly focused on investigating the effects of changing the hull parameters, the effect of shallow water and the applicability and correctness of Froude friction values [3].

3.2 Present situation FOI is currently developing a program for ship design, where some key aspects of the early ship design process are in need of investigation in order to allow for further enhancements of current capability. The program under development for early ship design estimation at FOI is called ShipDes. The aim of this program is to enable the user to quickly and as accurately as possible define the characteristics of the ship of interest. By inputting the hull geometry, displacement, longitudinal center of gravity LCG and transverse center of gravity TCG. The program should be able to as accurately as possible calculate the total resistance of the ship and the needed installed power. By inputting LCG, TCG, weight of the ship and a file defining

2 the hull geometry, several parameters such as S, LWL, B and T can be calculated. The current accuracy of calculating these parameters with ShipDes have to be assessed with a couple of well known documented cases. By knowing S, LWL, B, and T, several dimensionless parameters usually used to describe hulls such as λ, CB, B/T and L/B can be derived. After properly defining the geometry of the hull with help of several dimensionless parameters, the aim is to perform an as accurate estimation of the total resistance of the ship as possible. The ITTC (International Towing Tank Conference) is an organizations that provides a basis to how predictions of the hydrodynamic performance of ships should be undertaken. Many of the research organization do use some variation of ITTC-78 performance prediction method[4], this is also the method utilised in the ShipDes program. In order to perform this prediction method a suitable set of residual resistance coefficient data must be available, meaning either the exact ship have been tested or a few similar ships have been tested. A full-scale ship is rarely tested, therefore a resistance towing test are performed on the ship, but in model-scale. Currently, no database with model resistance measurement data exists at FOI. Utilising the database optimally is a key in order to allow for an as accurate estimation of the total resistance and the needed installation power of the ship. Therefore the optimal way of utilising a larger database was investigated. The choice of empirical and semi-empirical formulas utilised in this calculation process also had to be revised in order to be optimal for any given situation. By performing the above investigations, a better understanding of the current performance of the ShipDes program will be achieved as well as finding concrete ways of improving the current ship resistance estimation process.

3.3 Scope of the report The computer program for ship design estimation written in C# at FOI is currently under development. One of the key elements of the program is to as accurately as possible predict the total resistance of the ship of interest. This work will therefore focus on the investigation and optimization process regarding the calm water resistance calculations during the early ship design process. This thesis will therefore contain a summary of the literature study and a brief description of the underlying physics. The tasks during the project have been narrowed down to the following points:

• Within current implementation and method choices for residual resistance calculations several sources of errors exists. How do they manifest themselves and how do they aﬀect the overall result? • What limiting parameters exist within the currently recommended ITTC process, in FOI’s current method application and how does lack of accuracy from model test aﬀect the results?

• How can the current implementation of the program be improved? • How could CFD be utilised in order to asses the implementation and how well does the semi-empirical models compare to the CFD calculations? • How could CFD be used in order to complement the semi-empirical method calculations?

3 4 Theory 4.1 Physical background The fundamental equations used within fluid dynamics are the Navier-Stokes equations. The Navier-Stokes equations are a set of coupled differential equations representing the conservation of mass, momentum and energy in an infinitesimal particle. Within naval fluid dynamics there are two fluids that needs to be considered, air and water. These fluids most often can be considered as fully incompressible fluids, which simplifies the equations significantly. By making this assumption the mass and momentum equations uncouples from the energy equation. The equations that needs to be considered is then the momentum and mass equation [5]. The mass equation, often named the continuity equations, can be written on the differential form shown in Equation1.

δρ δ(ρu ) + i = 0 (1) δt δxj Since the ﬂuid is assumed to be incompressible, Equation1 simpliﬁes to Equation2.

δ(ρu ) i = 0 (2) δxj This means that the continuity equation have simplified to a volume continuity equation, which in this case clearly states that the divergence of the speed field is zero everywhere. Which translate to that the the flow of water through a converging pipe, solely is dependant on the speed. The momentum equation can be written on the following form shown in Equation3.

2 δui δuiuj δ ui 1 δp + = ν 2 − (3) δt δxj δxj ρ δxi Where i,j = 1, 2 and 3, kinematic viscosity ν, density of ﬂuid ρ and pressure p. The incompressible Navier- Stokes equations needs boundary conditions in order to be solvable. The no slip condition is assumed for solid surfaces, meaning that the velocity ui is zero at the surface. [5]

4.2 Computational fluid dynamics Analytically solving Navier-Stokes equations with turbulence have proven to be a substantially difficult task. So notoriously difficult that a large part of modern fluid mechanics research is conducted within the field computational fluid mechanics (CFD). With the nowadays available computational power the CFD discretize the governing equations and solve them numerically [5].

The method of discretizing the governing equations and directly solving them numerically is called Direct Nu- merical Simulation (DNS). DNS could potentially solve the entire spectra of turbulent ﬂuctuations, however require ﬁne grids and are therefore very computationally expensive. Furthermore, the number of operations increases cubically with the Reynolds number (Re). This means that the DNS only is feasible up to moderate Re and almost exclusively is used for research [5].

In many practical applications the majority of the flow is significantly greater than the turbulent fluctu- ations, which means that studying the average flow is sufficient. This is performed by decomposing the flow into an average parts and a fluctuating part in a process called Reynolds decomposition. By using Reynolds-Averaged Navier-Stokes equations (RANS) the complexity of the problem is reduced and the required computational power decreases [5].

4 In the master thesis conducted partly in parallel with this work the approach of using RANS and the software OpenFOAM have been used. The extension Naval Hydro Pack was also used for the simulations [1].

4.3 Laminar and turbulent flow In Figure1 a smooth plate is subjected to a homogeneous flow of a non-viscous and viscous fluid fluid. For the viscous fluid it is clear that the adhesion condition is present. Further out from the plate the fluid particles remain unaffected by the presence of the plate. Between these extremes, in the boundary layer, the particle speeds vary drastically as can be seen in Figure1 to the right. The δ, boundary layer thickness is defined as the distance along a surface normal vector to the point where the particles have 99% of the potential flow speed [2].

Figure 1: Velocity profile for non-viscous (left) and respective viscous fluid when subjected to flow along a smooth plate.

This flow is governed by the viscosity, which is strongly related to the Reynolds number. However, the boundary layer thickness differs along the plate and therefore it is not clear which characteristic length that should define the Reynolds number. One could for example define a local Reynolds number Rnx with the distance from the plates front edge as the characteristic length. At low Reynolds numbers the flow in the boundary layer is laminar, however when the Reynolds number increases it transition into a turbulent flow. The shift from laminar to turbulent flow also depends on the surface roughness. The shift does not occur at the same Re over the whole surface, but starts at local roughnesses [2]. The actual shift from laminar to turbulent flow is quite sudden and is illustrated in Figure2 below.

5 Figure 2: Principle of boundary layer thickness δ as a function of local Rnx and the characteristics of the boundary layer ﬂow.

The resistance increases for a turbulent boundary layer, partly due to the shear forces on the surface, but also due to that a thick boundary layer increases the pressure drag. Ships are long enough in order for the laminar part of the flow to be neglected and only the turbulent flow needs to be considered. However, for models ships this is not the case. In order to simulate an as accurate situation as possible the turbulent flow have to be tripped. Models have such a smooth surface that a large proportion of the flow otherwise would be laminar, which clearly is not representative to what the full-scale ship would experience [2].

4.4 Model testing procedure The sole purpose of performing model towing tank tests, is to gain insights regarding the characteristics of the specific models. The model towing tests are performed in large test basins, which allows for a more controlled testing process. I.e the MARIN Deepwater Towing Tank have the dimensions (250 m x 10.5 m x 5.5m). By performing the test in large inside basins a greater control of the ambient conditions, such as wind, waves, water salinity and water temperature is achieved. Most often the model is towed without any appendages. However, sometimes a rudder is attached, in order to maintain course stability during the towing test, the rudder is then viewed upon as an integrated part of the hull. The air resistance is considered as negligible, due to that the transverse projected area AA above the waterline is very small for a model. The model hull surface is very smooth, therefore most often turbulence tripping is implemented. Turbulence tripping is used in order for the boundary layer along the model to be as similar to the boundary layer along the ship as possible. The turbulence can be tripped by placing a strip of "sand paper" or small spikes on each side of the bow, just under the waterline. The towing force should be applied in the line where the longitudinal center of gravity is located, and if possible directly at the center of gravity. The speed is varied through the speed range that the ship is designed to operate within. If the ship type allows for extrapolation of the form factor, then the slow speed measurements (0.1 < Fn < 0.2) can be used for this procedure. This form factor extrapolation process is described in detail in Section 4.7. Values that most often are measured at modern towing tank tests are, the total resistance, trim, sinkage forward and aft and water temperature. The water temperature can later be used to calculate the water viscosity and density. The density of the water is of importance, therefore it should be specified if the test was conducted in salt water or fresh water. Preferably the models tested in the basin, should be of suitable size, in order for the shallow water effect to be negligible. The shallow water effect can be attempted to be corrected for, however it introduces some uncertainty into the measurement data [2][3]. Stated in [6] was the accuracy error expected from performing a calm water towing experiment to be about 2 %.

6 4.5 Regression functions The sole purpose of performing a regression analysis is to ﬁnd relations between a number of independent variables and the dependent variables. The typical application within ship estimations is to perform a regression analysis on model test series data. The regression functions is obtained by minimising the residuals, usually by utilising a least square method on the data. The advantage of performing this procedure is that a consistent performance predication estimation both for the tested hulls and intermediate designs can be achieved [6]. Out of the semi-displacement hull series encountered in this work all have undergone a regression analysis of some sort. Additionally there are several fast semi-displacement hull series with transom stern of varying relevance that have undergone regression analysis, of which a few of these are mentioned in [7]. The Marin series data underwent several regression analyzes that resulted in functions that allows for predicting residual resistance, trim, sinkage and wetted surface [6]. Furthermore, a regression analysis was also performed on an extended database which contained several transom stern model series [6]. Figure3 show the relative standard error from predicting the residual resistance over a range of volumetric froude numbers Fn∇. It should be mentioned that the relative standard error is calculated by using the regression function to estimate hulls, which the regression function is based upon and then comparing the results with the actual measured residual resistance data [6].

30 Ext. DB Marin series 25

10 Rel. Standard Error [%] 5

0 1 2 3 4 Fn [-]

Figure 3: Relative standard error from estimating the residual resistance with the regression function for the Marin series and extended database. Reproduction of graph in [6].

Figure3 shows that the extended database regression function consistently result in worse predictions compared to only the regression function of the Marin series. However the extended database regression function is still viewed as satisfactory when applied in an early design stage. Losing accuracy when making a regression function for a collection of model series is a well known fact [8]. The regression function for the Marin series result in an error below 5 % over a large proportion of the speed interval and the accuracy is considered as very satisfactory [6]. It should be stressed that the accuracy of predicting resistance data for intermediate designs can not be stated for the regression function. The NTUA series also underwent a regression analysis, which result in an overall standard error for the residual resistance below 5 % over 62.5 % of the interval, standard error between 5 - 10 % over 28.4 % of the interval and the remainder have over 10 % error [7]. And lastly mentioned is that the NPL regression function of the total resistance consistently deliver a standard error below 0.3 % over its entire speed range [8].

7 When using a speciﬁc regression function it is paramount that it is only used to estimate a hull with similar hull characteristics and secondary parameters as the regression function is based upon, i.e type of hull (round bilge, hard-chine and double-chine), CB, CP , AT /AM and LCB. The regression function itself only use for example independent parameters such as L/B, B/T , λ and Froude number Fn as input, however secondary parameters are held constant during the entire series. Exactly which parameters that were viewed as secondary, should be stated in the regression analysis. To summarize, it is not advised to prospect a hull if the secondary parameters and the hull type does not correlate with what was used for the regression function [8].

4.6 ITTC-78 performance prediction method In order to predict the total power needed to enable the full-scale ship to achieve the desired ship speed, the total resistance have to be estimated. It should be mentioned that the resistance due to wave and wind conditions have not been investigated in this work, simply due to that it was outside the scope of the report. However, the wind and wave conditions that the ship is intended to operate in should be taken into account when choosing the design of the ship. Over the years it have been shown through trial and error, that suitable correlation factors are needed in order to obtain realistic estimations of ship resistance and power requirements [3]. A proven and often utilised method is to perform model towing tank resistance tests in order to gain insights on the speciﬁc ship behaviour and characteristics on model-scale. Thereafter, most ship builders utilise some variation of the recommended ITTC-78 performance prediction method to estimate the full-scale resistance [4][9]. A questionnaire was carried out in [4], which showed that the testing facilities use a signiﬁcant variation in the details of the powering prediction process. The ITTC-78 performance prediction method attempts to separate out and correct various of elements of the prediction process. One of the reason for the ITTC-78 performance prediction method being the recommended procedure is that it attempts to scale the individual components of the power estimate [3]. Furthermore, it allows for updates to be made to the individual components as new data become available.

Within naval architecture both Fn and Re are signiﬁcant ﬁgures used when determining the resistance of partially submerged objects moving through water. The Froude number is a dimensionless quantity and can be seen in the formula described in Equation4 below.

V F n = √ (4) g · L Where the speed V, gravity g and submerged object length L are used. Dynamics of vessels that have the same Froude number are easily compared as they produce a similar wake, even if their size or geometry are otherwise different [3]. The Reynolds number is a dimensionless quantity within fluid mechanics. which is used to help predict flow patterns in different fluid flow situations [5]. The formula describing the Reynolds number is shown in Equation5

V · L Re = (5) ν Where the speed V, submerged object length L and kinematic viscousity ν. Tabulated values of water properties for the kinematic viscousity, water density, dynamic viscousity can be found at [10]. At low Reynolds numbers, the flows tend to be dominated by laminar flow. The turbulent flow occur at high Reynolds numbers, which is dominated by inertial forces, that produce chaotic eddies, vortices and instabilities in the flow [5].

4.6.1 ITTC-78 method steps Firstly, the step-by-step scaling process will be covered and thereafter the individual coeﬃcients will be explained more thoroughly in Section 4.6.2. When performing a towing resistance test on a model without

8 any appendages in a test tank, the following resistance components are present. The residual resistance RR and viscous form- and friction resistance RV m. Residual resistance is almost entirely wavemaking resistance that refers to the energy loss caused by waves created by the vessel and a small part viscous pressure resistance. The frictional resistance is the net forces upon the ship due to tangential shear forces. The total resistance RT m can be written with its sub-components on the form shown in Equation6[3][9]. The denotation m stands for model-scale.

RT m = RV m + RRm = (1 + k)RF m + RRm (6) The form factor k came to be as an attempt to better estimate the viscous resistance due to the shape of the hull. The form factor can be obtained by following the procedures described in detail in Section 4.7[3]. After measuring RT m over a speed range, the idea is to transform the data into dimensionless coeﬃcients. This approach have been adapted in order to assist in the process of re-scaling model to full-scale ship [3]. Equation6 can be expressed on the following form shown in Equation7 below.

1 1 R = C ρ V 2 S = ((1 + k)C + C ) ρV 2 S (7) T m T m 2 m m m F m Rm 2 m m

Where total resistance coefficient CT m, water density ρm, speed Vm, wetted hull surface Sm, form factor k, frictional resistance coefficient CF m and residual resistance coefficient CRm have been used. The term (1 + k)CF m together make out the viscous form - and friction resistance coefficient. The frictional resistance coefficient CF is defined by a formula covered in Section 4.6.2. It should be noted that a running wetted surface area could be used and is fundamentally more physically correct, however for practical powering purposes using the static wetted surface is sufficient [3]. For practical reasons the static wetted area was therefore consistently used throughout this work. By re-formulating Equation7 on the form shown in Equation8 the residual resistance coefficient can be obtained.

CRm = CT m − (1 + k)CF m (8) It was derived by Froude that the residual resistance coeﬃcient is the same between model and ship for equal Froude numbers. Meaning that CRm = CRs, where the denomination s stands for ship. For the full-scale ship additional components to the total resistance can be added, i.e air resistance and resistance due to appendages. Now a formula for the total resistance coeﬃcient CT s for the full-scale ship can be formulated, this formula is shown in Equation9 below.

SBK + Ss CT s = ((1 + k)CF s + ∆CF + CA) + CAA + CRs + CApp (9) Ss

Where the roughness allowance ∆CF , correlation allowance CA , air resistance coefficient CAA, appendage resistance coefficient CApp, wetted hull surface area Ss and bilge keel area SBK . If the ship is fitted with a bilge keel of modest size, then the multiplication factor SBK Ss can be used [9]. Ss

The total ship resistance can now be estimated by using Equation 10 below.

1 R = C ρ V 2S (10) T s T s 2 s s s

By estimating the total resistance RT s over a speed range, the eﬀective power needed for the ship to maintain the desired ship speed can be calculated [2]. The eﬀective power is described by Equation 11 below.

PE = RT · Vs (11)

9 The effective power PE is under ideal conditions the needed power to maintain the designed ship speed. However, the situation is never ideal in reality, therefore extra consideration in respect to i.e shaft efficiency, propulsion efficiency, hull efficiency and a sea margin, have to be taken into account [2].

4.6.2 Coeﬃcient explanation ITTC-57 friction correlation line Originally in the early 1920 Von Karman derived a friction law for ﬂat plates based on a two-dimensional analysis of turbulent boundary layers. The theoretical "smooth turbulent" friction law can be seen below in Equation 12[3].

1 √ = A + B · log10(Re · CF ) (12) CF Following Von Karman’s publication Schoenherr analysed all available data from plank experiments both in air and water in order to determine the constants A and B to ﬁt the available data [3]. The Schoenherr formula can be seen in Equation 13 below.

1 √ = 4.13 · log10(Re · CF ) (13) CF Since the the Schoenherr formula was viewed as inconvenient to use, the work was continued by Hughes [3]. Based on earlier work and plank friction experiments he suggested the following more convenient friction formula shown in Equation 14.

0.066 CF = 2 (14) (log10Re − 2.03) However, since hulls does not have the same geometry as a plank the ITTC-57 correlation coeﬃcient have adopted the Hughes line, but with a 12 % form eﬀect built in [3]. The ITTC-57 correlation line can be seen in Equation 15 below.

0.075 CF = 2 (15) (log10Re − 2) The ITTC-78 performance prediction method recommends the usage of the ITTC-57 correlation line and this formula was therefore used throughout this work. It should be emphasised that the ITTC-57 correlation line should not be considered to represent the drag of a ﬂat plate or the skin friction of an actual ship form. However it may be a tolerable approximation of the skin friction of actual ships for most forms. It should rather be viewed as solely a correlation line from which to judge the scaling allowance to be made between model and ship [3]. There are several other correlation formulas available, some even slightly better than the ITTC correlation line, however the improvements appear to be so small that the test tank community generally have not decided to adopt any of the other correlation lines [3].

Roughness allowance The roughness allowance is an attempt to correct for the difference in surface roughness between the model and the ship. The model hull surface is smoother compared to the ship, especially due to that no fouling have been allowed to occur. Due to fouling the frictional resistance could have an annual increase of about 10 %, which would mean a total annual resistance increase of about 5 % [3]. In order to correlate the added resistance with hull roughness and incorporate the effect of the Reynolds number the two works [11] and [12] was carried out. These works studied the detailed experimental data concerning fluid flow over rough

10 painted surfaces. The work resulted in an empirical formula called the Townsin formula, which is shown in Equation 16. This is also the formula recommended by ITTC [9] for estimating the roughness allowance.

1/3 ks −1/3 ∆CF = 0.044 − 10 · Re + 0.000125 (16) LPP

If the surface roughness ks of the ship hull is unknown, then the recommended default value by ITTC is ks = −6 150 · 10 m. The LPP can be substituted with LWL depending on the ship type. Furthermore, it is worth noting that the situation is as such that since 1985 no new formula or relevant data proposed to estimate the ∆CF . Which have resulted in that the formulas currently used no longer correlate with modern coatings and ﬁnishes as well as desired. Therefore it have been suggested that this work should be undertaken, however this have not been carried out at present time of this work[13].

Another formula derived to be used as a correlation allowance that includes the roughness allowance between model and ship is the Bowden-Davison formula. This formula should be viewed as an empirical addition, derived from analyses of the correlation between extrapolated power predictions and trial measurements. It accounts for hull resistance due to surface roughness, paint roughness, corrosion, and fouling of the hull surface. The formula was derived from an analysis of trial data from 10 full-scale tankers. The Bowden- Davison formula can be seen below in Equation 17.

1/3 ks ∆CF = 0.105 + 0.00064 (17) LPP Correlation allowance

Furthermore, the correlation allowance CA is a model-ship correlation factor. This correlation allowance, should be used together with the roughness allowance shown in Equation 16[9]. If possible it should be based on a comparison between full scale ship trial results and model results. It should be viewed as a correction for systematic errors in the model test and powering prediction procedures, i.e including any facility biases. Since no comparison between ship and model can be made in this work, the recommended formula by ITTC have been used throughout this report. It should also be mentioned that the correlation allowance formula have been derived by taking the diﬀerence between Equation 16 and Equation17. Equation 18 shows the recommended correlation allowance formula stated by ITTC [9].

−3 CA = [∆CF ]Bowden − [∆CF ]T ownsin = (5.68 − 0.6log10Re) · 10 (18) Air resistance coeﬃcient

The air resistance coeﬃcient CAA can be approximated with the Equation 19

A C ≈ 0.001 A (19) AA S

Where the transverse projected area above the waterline AA and wetted hull area S have been used. Fur- thermore, a more precise calculation of the air resistance can be calculated with help of Equation 20.

ρA · AA CAA = CDA (20) ρW · S

Where air density ρA, water density ρW and drag coeﬃcient of the ship above waterline CDA have been used. The drag coeﬃcient can be determined with either wind model tests, empirical calculation formulas

11 or CFD. CDA is typically between 0.5 - 1.0, where 0.8 can be used as a default value [9]. The air resistance is a relatively small part of the total resistance at calm water, for this case it is about approximately 2-5% of the total resistance. However, if the wind is considered then the air resistance can be approximately as high as 10 % for bulkier vessels. [2].

Appendage resistance coeﬃcient

The appendages resistance coefficient corrects for all the additional appendages being fitted to the ship compared to the model. The appendage could i.e be rudders, fins, skeg and anode. In [14] an approximate formula for various appendages was suggested. This is approximate formula is shown in Equation 21 below.

S C = (1 + k )C App (21) App 2 F S

Here the wetted area of the appendage SApp, wetted hull surface S, frictional resistance coeﬃcient CF and the appendage form factor k2 have been used. If multiple of appendages are mounted on the ship hull, then a summation of several CApp can be performed. The appendage form factor values for various generic appendages were tabulated in [14]. The tabulated values can be viewed in Table1 below.

Table 1: Approximate 1 + k2 values. Further details can be found in [14]

rudder behind skeg 1.5 - 2.0 rudder behind stern 1.3 - 1.5 twin-screw balance rudder 2.8 shaft brackets 3.0 skeg 1.5 - 2.0 strut bossings 3.0 hull bossings 2.0 shafts 2.0 - 4.0 stabilizer ﬁns 2.8 dome 2.7 bilge keel 1.4

For high-speed craft, the appendage drag usually represents a larger proportion of the total resistance compared to conventional fully displacing ships. The appendage resistance can be estimated in several more ways, one of which would be to test the model with and without appendages [15].

4.7 Introduction of the Form factor During the 1950’s Hughes proposed the the idea of taking the form factor of the hull into account in the extrapolation process. Previously explained was the reasoning behind the frictional coefficient CF for a flat plate, however a ship hull is not flat. The viscous resistance is higher on the hull compared to a flat plate of the same length and wet area as the ship hull. Therefore the idea of introducing the form factor as a correcting factor in order to better estimate the viscous resistance came to be. The form factor can be estimated by either an empirical formulae developed based on results from previous models tests. Or by performing low speed towing tank tests for the model. Due to practical purposes the form factor is assumed to be constant over different speeds and between model and ship. This assumption is inherently incorrect, due to a number of reasons. In example, viscous effects changes depending on the Reynolds number and the full-scale ship have slightly different characteristics compared to a model [3]. However, it should be mentioned that larger

12 models have proven to consistently result in both increased accuracy in the extrapolation of the form factor compared to smaller models with the same geometry, when tested at corresponding Froude numbers. [16]

4.7.1 Empirical form factor formulas Multiple of empirical formulae for the form factor k have been developed based on results from models tests. An empirical method can be suitable to use, when no suitable resistance data from model tests are available and the ship type is suitable for usage of the formula. Unfortunately, no single formula is universal nor does their use signiﬁcantly improve the resistance predictions. The formulas mentioned below can be found in [3].

Watanabe’s formula in Equation 22 below.

CB k = −0.095 + 25.6 q (22) LPP 2 B [ B ] T

Grigson’s formula in Equation 23 below.

λ r B k = 0.028 + 3.30 2 CB (23) LPP LPP Conn and Ferguson’s formula shown in Equation 24 below.

B 2 k = 18.7 cB (24) LPP

4.7.2 ITTC form factor estimation procedure The form factor can also be determined by performing low speed towing tests for the actual model ship. Testing at low Froude number (Fn < 0.2) leads to the CT running parallel to CF , which means that the residual coeﬃcient CR approaches zero. This is expected since the residual resistance is very small for low speeds. However, since it is challenging to measure CT at low speeds accurately, just one single measurement does not give the best result. Therefore, following procedure was developed by Prohaska’s. By assuming that 4 CR ∝ AF n for low speeds, where A is a constant and Fn the froude number. This leads to the Equation 25.[3]

C AF n4 T = (1 + k) + (25) CF CF However, for all ships the points may not plot on a straight line and a power of Fn between 4 to 6, may be more suitable to the situation. Therefore, ITTC recommends a slightly modiﬁed formula to Prohaska’s, this is shown below in Equation 26.[17]

C AF nn T = (1 + k) + (26) CF CF By performing a 1st, 2nd or 3rd order polynomial least square fitting to the data, the constant term become (1+k). Where n is varied between 4, 5 and 6 in order to obtain the best linearization of the data. And the polynomial fitting is performed in order to achieve the best fit of the data. However, the final choice comes down to personal judgement [18]. When F n −→ 0 then CT /CF = (1+k). The (1+k) can be found by plotting Equation 26 as shown in Figure4. Figure4 shows an example of a trend that allows for utilising ITTC form

13 factor estimation procedure on the JBC data. The JBC data is speciﬁed to have (1+k) = 1.314, so performing the ITTC form factor estimation procedure only serve as a conﬁrmation that it is a reasonable choice [19].

1.42

1.4

1.38 [-] F

/C 1.36 T C

1.34 Measured data 1st order polyfit 2nd order polyfit 1.32 3rd order polyfit

0 0.05 0.1 0.15 0.2 Fn4/C [-] F

Figure 4: Prohaska’s method performed on the JBC data. 1st, 2nd and 3rd order least square polyﬁt performed on the measured data.

As can be observed in Figure4 the intersections along the y-axis is where (1+k) is found. Utilising Prohaska’s method introduces several inaccuracies. Accurately measuring at low froude numbers are difficult and the assumption that the form factor is speed independent is incorrect. The form factor is in fact dependent on the Reynolds number, which means that the form factor can not be considered to be constant for varying speeds. Also worth noting is that, for several ship types, i.e those with an immersed transom stern with a large area, the form factor can not be determined at slow speeds. Since for ships with an immersed transom stern an important region of so called dead water aft of the stern is created, which at low speeds have a strong effect on the resistance. An additional resistance at low speeds impact the determination of the form factor, since for higher speeds the flow of the fluid have a drastically different effect on the resistance for such a hull geometry. Proposed in [20] was that the form factor is speed dependent and that it approaches zero at higher froude numbers. This can be observed in Figure5 shown below.

14 1.5

1 1+k [-]

0.5 0.2 0.4 0.6 0.8 Froude number [-]

Figure 5: Example of dynamic form factor dependent on Froude number. Reproduction of graph shown in [20].

This fact have resulted in that many high speed semi-displacing hull series present their CR where a static k = 0 have been used. The semi-displacing crafts are intended for service speeds at higher froude numbers, which means that a lack of accuracy at lower froude numbers are deemed as acceptable [6]. Furthermore, stated by ITTC Resistance Committee, is that modern bulbous bows hamper the determination of a form factor, even if it is fully submerged. Also a partly submerged transom area distort the determination of a form factor, not only in the case for high speed vessels. Despite all of these drawback and limitations, the Prohaska’s method is still the recommended procedure by the ITTC, simply because it is currently the best alternative available [6][21].

4.8 Hull form design Firstly, the hydrodynamic behaviour of a hull can be categorised into three stages over the the total speed range. Generally it can either be displacing (0 < Fn <0.5), semi-displacement (0.5 < Fn < 1.0) or planning (1.0 < Fn). A displacement vessel is supported entirely of buoyant forces, a semi-displacement vessel is supported by a mixture of buoyant forces and dynamic lift and a planning vessel is fully supported by dynamic lift. Naturally in order to achieve an optimal vessel for each respective category the basic hull geometry changes. There a several dimensionless parameters that attempt to describe the somewhat complicated hull geometry. These parameters can in turn be coupled to the resistance of the ship, stability and sea fairing characteristics [2]. The two ship types primarily investigated in this work was of two drastically diﬀerent types, one bulk carrier with a bulbous bow and a faster semi-displacement vessel with a transom stern.

4.8.1 Bulk carrier with bulbous bow A bulk carrier is specially designed to transport goods long distances. It is primarily optimized in regards to capacity, eﬃciency and durability. A bulk carrier is a fully displacing ship and generally have a designed service speed at low froude numbers [3]. Figure 7a shows an example of a bulk carrier moored at a harbour.

15 (a) Bulk carrier named Nyon (Public domain)[22]. (b) General example of bulbous bow in proﬁle.

Figure 6: Typical bulk carrier and bulbous bow in proﬁle.

The most prominent features of a bulk ship are its length, the blocky shape and its bulbous bow. The purpose of utilising a bulbous bow for finer faster vessels tend to entail a reduction in wavemaking resistance, meanwhile for fuller slower vessel it tend to entail a reduction in viscous resistance [3]. The bulb also tend to realign the the flow around the fore end, which later is carried downstream. The bulb also have an effect on the wake fraction, thrust deduction and hull efficiency. In a ballast condition, which is where the bulbous bow tend to have the most positive impact, the total resistance have been reported to be reduced with up to 15 % for fuller slower vessels. Figure 6b shows an example of a bulbous bow in profile. The geometry of the bulb should be optimized for the vessel and the intended usage of the vessel. In general there are three types of bulbs, the ∇-type, the ∆-type and the O-type. The advantage of using either one of them depends on the intended usage of the ship [3].

4.8.2 Semi-displacing vessel with transom stern A semi-displacing ship with a transom stern is frequently used together with waterjets as propulsion. Wa- terjets as propulsion for vessels intended for higher speeds are often the preferred choice. The transom stern is ideal for the design and mounting of waterjets. The advantage of waterjets as propulsion is multiple, increased eﬃciency compared to propellers, reduction of draught of the ship, protection of the propulsion unit and improved manoeuvrability. Figure7 shows a military semi-displacing ship with transom stern and waterjets as well as a semi-displacing ship in proﬁle showing some key parameters.

(a) Drawing of the Swedish Visby corvette (Public (b) Semi-displacing ship with transom stern in proﬁle. domain)[23]. Where τ is the trim and AT is the wetted transom area.

Figure 7: Typical example of corvette and semi-displacing ship in proﬁle.

Figure 7b shows the trim τ of the vessel. The trim angle comes from the height difference between the forward and aft perpendicular. Meaning at rest the vessel have zero trim, the waterline intersection point forward and aft is noted, thereafter these are used in order to determine the trim at varying speeds. It should be noted that the trim is present, regardless of the ship type. The trim and sinkage are related to the main hump in the residual resistance [6]. In Figure 7b the trim is positive, this is due to that the bow have lifted and the stern lowered in respect position at rest. Also pointed out in the figure is the parameter AT , which is the submerged area of the transom stern. The size of AT have a strong effect on the resistance at

16 low speeds. Aft of the submerged transom stern an important region called dead water is created. The dead water increases the resistance at low speeds, however at higher speeds the flow of the fluid have a drastically different effect on the resistance for such a hull geometry [6]. Figure8 shows three general cross sections of different hull types that frequently are used for semi-displacing crafts.

Figure 8: General hull types for semi-displacing crafts. Round bilge (left), hard-chine (middle) and double- chine (right)

4.8.3 Hull parameters The hull parameters can give a strong indication of what type of ship is being described and how it may look, without ever looking at the ship itself. In order illustrate a couple of key dimensions and units of ships Figure9 was made.

(a) Ship in proﬁle below waterline. (b) Ship in 3D below waterline.

Figure 9: Two examples of ships below waterline showing important parameters and units.

Firstly, the waterline length LWL is defined as the length between the waterline intersection points at the bow and stern. The length between perpendiculars is usually defined as the distance between the rudder shaft and the waterline intersection point at the bow. The draught T is usually defined amidship, which most often is half of LWL or LPP . The waterline beam BWL is defined between the waterline intersection points amidship on both sides of the hull. Area amidship AM is the transverse area below the waterline amidship.

The slenderness ratio (length-displacement ratio) usually has an important influence on the hull resistance for the majority of the ships. For the resistance point of view for high speed vessels, it is well known that the slenderness is the most influential parameter [7]. With increasing slenderness, but with constant displacement the residuary resistance RR decreases. Also if the volume displacement decreases and the length held fixed the slenderness ratio, increases and the RR decreases. Typical slenderness ratio values for cargo ships vessels are 5.0 - 7.0, for tankers and bulk carriers 5.5 - 6.5, for passenger ships 7.0 - 8.0 and for semi-displacement crafts 6.0 - 9.0. The expression for this dimensionless unit is shown in Equation 27 below [3].

L λ = WL (27) ∇1/3 Another important parameter is the length-breadth ratio, L/B. The length-breadth ratio increase if the slenderness increases and the other parameters are held constant. When the length increases the RR decreases.

17 Generally a large length-beam ratio is favourable for faster ships, furthermore a longer ship normally result in better seakeaping performance. The standard range of L/B is 6.0 - 7.0 for cargo ships, 5.5 - 6.5 for tankers and bulk carriers, 6.0 - 8.0 for passenger vessels and 5.0 - 7.0 for semi-displacement ships. [3]

Beam-draught ratio, B/T is an important parameter in regards to the wave resistance. For slower vessels if B/T increases the wave resistance increases, due to that the displacement is brought closer to the surface. This note is based on observations from the BSRA series and the Taylor-Gertler series, which are series conducted for cruiser ships and therefore tested at lower Froude numbers [3]. However, for faster vessels the opposite behaviour can be observed. For example in a parametric study conducted for semi-displacing ships with respect to the Marin Series, which is covered in detail in Section6.1, the opposite relation for B/T can be noted. For higher B/T the resistance decreases and lower B/T increases the resistance [6]. Generally beam-draught ratio for cargo vessel is about 2.5 and for stability-sensitive vessels such as passenger ships beam-draught ratio can be as high as 5.0 [3].

The block coefficient CB defines the overall fullness of the vessel. By looking at Equation 28 below it is clear that the block coefficient is a ratio that shows how much of the displaced volume take up of the rectangular cuboid mapped out by the ships outer dimensions [3]. For some ship types i.e those with a transom stern the Lpp can be substituted with LWL.

∇ CB = (28) BTLPP

The longitudinal prismatic coefficient CP also give a measure of were the displacement is distributed. A high CP indicate that the displacement is fairly evenly distributed over the ship length, meanwhile a lower value indicate that the displacement is fairly centered around amidship. A few typical prismatic coefficient values for various ship types would be, for bulk carrier and tankers 0.80 - 0.95, for frigates 0.56 - 0.64 and for passenger ships 0.6 - 0.7. The Equation 29 for the prismatic coefficient is shown below. For some ship types i.e those with a transom stern the Lpp can be substituted with LWL.

∇ CP = (29) AM LPP

The longitudinal centre of buoyancy LCB is usually expressed as percentage of length from amidship. I.e LCB = 2%L , would indicate that the LCB is located 2% of length L forward of amidship. The length L could either be LPP or LWL depending on the ship type. The afterbody of a symmetrical hull produces less wavemaking resistance than the forebody, due to the boundary layer suppression of the afterbody waves. By shifting the LCB aft, the wavemaking of the forebody decreases more than the increase in the afterbody, however the pressure resistance of the afterbody will increase. The pressure resistance for fine forms are low, a fine form is indicated by a low CP , for a fine form vessel it is advantageous to move LCB aft. The limitation will be in the end be due to pressure drag and propulsion implication. However, for a fuller slower vessel the optimal position of LCB is about 2 - 2.5 %L forward of amidship [3].

4.9 Marin parametric study Depending on the situation, varying a parameter could affect the overall result differently. One of the methods for investigating the effect from varying a parameter is by conducting a parametric study on the available data. In the Marin material a parametric study was carried out on the tested models, which were of a semi-displacing type. This parametric study was carried out on the hulls tested in the Marin series with respect to four parameters, L/B, B/T , CP and CB. One parameter was allowed to vary meanwhile the three other parameters were held constant [6]. The slenderness ratio λ value was not considered, however by plotting the data on the y-axis on the form RR/∆ [-] the potential change in displaced weight is handled. The main points to take from this parametric study was the following:

18 • L/B is the most important parameter and increasing L/B decreases the resistance. • Changing between L/B = 4 to L/B = 8 is an enormous change in resistance compared to changing between L/B = 8 to L/B = 12. • Decreasing B/T increases the resistance.

• Changing B/T aﬀect the resistance the most when CB = 0.5, which is high for the Marin series.

• Changing CB aﬀect the resistance the least compared to the other parameters.

• Changing CB when L/B is low have the strongest eﬀect on the resistance. Changing CB when L/B is high have the lowest eﬀect on the resistance.

• CP = 0.626 was conﬁrmed to be optimal, provided that a design speed near the main hump is selected.

• The resistance humps are more pronounced for low L/B, low B/T and high CB

19 5 Methodology

By assessing the current version of the program at FOI with regard to the resistance calculation process further iterative improvements can be discovered. These findings will lay as basis to what and how future methods can be implemented in order to gain more accuracy in predicting the ship resistance. By carefully selecting two well documented hulls during the literature study with different hull geometry, the estimation process could be evaluated when using these as test cases. By systematically going through the process of implementing the selected hull forms of interest into the program, potential inaccuracies and problems can be discovered in the current implementation. Carefully mapping out current strengths and weaknesses within the current implementation will help to define where and how the program potentially can be improved.

5.1 Literature study In order to understand the process of how ship design traditionally have been carried out and how the process could be performed in the future, a literature study was conducted. The aim of conducting the literature study was to gain insights in ship building concepts, as well as understand where and how the resistance estimation process potentially could be further improved.

5.2 Database compilation No database of model series test result was present at FOI at the start of this work. Therefore an important task was to establish a database of test results in order to be used when performing ship prediction estimations. This was performed by searching through literature and other freely available sources. The primary focus was on collecting model series with semi-displacing hulls with a transom stern. When a potentially useful series was found the data had to be extrapolated, either by reading of tables or graphs. Furthermore, key pieces of useful information such as λ, L/B, B/T , CB, CP , LCB, LWL, hull type, testing temperature, water density, mass and displacement were recorded. Each individual model was saved in a ﬁle format with all its belonging information and the derived or given residual resistance coeﬃcients together with corresponding Froude number.

5.3 Evaluation of linear interpolation The usefulness of regression functions and what one could expect in accuracy have already been mentioned in Section 4.5. However, a potential improvement could be gained if the residual resistance coeﬃcient CR of a hull could be estimated by data from two other similar hulls directly. Simply based on an initial assessment on parameters that appeared to carry the most importance it was decided that λ, B/T , CB, CP and the ship type should describe the hulls. Using even more parameters to describe the hulls would risk that not enough hulls could form pairs that would allow for interpolation. The simple idea was that the residual resistance coeﬃcient CR of an unknown hull with i.e parameter B/T = 3 should be located linearly in the middle between two known hulls with B/T = 4 and B/T = 2. Provided that the other parameters (λ, CB and CP ) remained equal between the hulls. This scenario would allow for a simple linear interpolation to be performed with help of the two hulls. Additionally a special case for interpolation called two-folded interpolation was proposed. The idea was that a two-folded interpolation could be performed if two hulls are of the same ship type, two parameters are identical between the hulls and the other two parameters are equidistantly close to desired hull parameters. Table2 shows an example where a two-folded interpolation could happen.

20 Table 2: The four main parameters for three models. Model-X is the unknown hull, Model-A is known and Model-B is known.

Model λ B/T CP CB Model-X 7 4 0.65 0.40 Model-A 6.5 4 0.65 0.35 Model-B 8 4 0.65 0.50

In Table2 Model-X and its residual resistance which is unknown, could potentially be estimated with Model- A and Model-B by making a two-folded integration. The reason for attempting this kind of interpolation is that λ and B/T are equidistantly located to each respective parameter value and performing this sort of operation would allow for interpolation in special cases only when two parameters are identical to the unknown hull. The equidistant relation can be observed by inputting the values of λ and CB from Table2 in Equation 30 below.

• Case 1: All models in the database are available, but the known model (FDS-5) that one wishes to estimate is excluded from being used. • Case 2: The known model (FDS-5) and its entire series (Marin Series, see Section 6.1) where it came from is excluded from usage.

No suitable model series data was found to be used for estimating the bulk carrier vessel JBC by interpolation. Therefore the following cases was tested instead:

• Case 1: The most similar ship in regards to the four parameters deﬁning the JBC vessel was chosen. • Case 2: The most similar ship in regards to the four parameters deﬁning the JBC vessel was chosen as well as the characteristics of the residual resistance was considered.

AC# script was created in order to select and interpolate files in the database. The script had access to the database, it selected the files with hull parameters most similar to the desired hull and performed a simple or two-folded interpolation. The most similar hulls are based on that three or two parameters matches the desired hull’s parameters and that the parameter(s) that does not match differ as little in percent as possible. Furthermore, the two selected files also have to cover a interval, so that the unmatched parameter(s) is located in the interval.

The results from estimating the FDS-5 vessel by interpolation and estimating the JBC was later used in the comparative study described in Section 5.5.

5.4 Geometry calculation study The current implementation of calculating the dimensions and parameters of a hull is performed mostly in the ShipDes program. The ShipDes program uses a ﬁle that contains a set of points specifying the (x,y,z) coordinates that together make out a 3-D representation of the hull. The set of points is then used in order to calculate the wetted surface S at rest, draught T, beam B, trim τ and waterline length LWL. In order

21 to do this procedure accurately the displaced mass m, water density ρ, vertical centre of gravity VCG and longitudinal centre of gravity LCG have to be known. The files containing the set of points was created by taking a CAD-file of the hull and simply running a short Python script in a program called Rhinoceros 6 that placed out points on the hull surface. The ship was divided up into X amount of ribs equidistantly placed along the ship length. And the number of points Y were equidistantly placed along each rib length. (For the interested more details about the calculation process can be found in [24]). Since the wetted surface S, draught T, LWL and LPP for the FDS-5 and JBC are specified, see Section7. As well as both vessels being available in a CAD format. The situation was exemplary to investigate to what accuracy the ShipDes program combined with Rhinoceros 6 could calculate S, T, LWL and LPP . This was done on model-scale for both the FDS-5 and JBC. Figure 10 shows an example where the FDS-5 have been divided up in ribs and points places along the ribs.

(a) FDS-5 viewed from the side.

(b) FDS-5 viewed from above.

Figure 10: Example of set of points representing the hull in Rhinoceros 6 program. The FDS-5 model viewed from the side and from above.

By varying X amount of ribs and Y amount of points per rib, it was studied how the ShipDes program performed when the point resolution varied. The results from the study could then be compared to the documented dimensions and parameters of the hulls.

The calculated wetted surface S from the FDS-5 and JBC models were later used for predicting the total resistance RT at model-scale and full-scale of the FDS-5 and JBC vessel in Section 8.2 and 8.3. The exact values used was the one’s with the highest resolution, which can be seen in Section 9.7. The wetted surface S was only calculated at model-scale, which means that the surface had to be scaled up with geometrical uniformity. This was done by following the two formulas below.

L L = s (31) m α

S S = s (32) m α2

22 In Equation 31 and 32 the denotation m stands for model and s stands for full-scale ship. The length L is either LWL or LPP . S stands for wetted surface and α is the scale factor. The model-scale and the full-scale lengths for the FDS-5 and JBC can be seen in Section7.

5.5 Comparative study By carefully selecting two hulls of diﬀerent ship type to be investigated both by semi-empirical methods and CFD simulations, several insights could be gained. By estimating the total resistance of the selected hulls from the comparison regarding the total resistance and its composites insights can potentially be gained. Furthermore, a working protocol of how the semi-empirical methods and the CFD simulations can complement each other at FOI can potentially be established.

5.6 Parametric study

In Section 5.3 it was decided that λ, B/T , CB, CP should be the main parameters utilized when attempting to estimate intermediate hull residual resistance data. Therefore it was viewed as fitting to primarily base the parametric study on these parameters as well. Furthermore, this also means that not exactly the same parameters are being studied as in the Marin parametric study described in Section 4.9. The parametric study was conducted on the entire database with transom hulls. The process was performed by creating file pairs, where three parameters were identical and one parameter differed. An average percentage value XP was used for estimating which parameter affect the residual resistance coefficient CR the most. For each file, an Akima spline fit [25] was performed on the CR data. The CR data was thereafter calculated in 1000 points over its entire speed interval. It should be mentioned that the parameter residual resistance divided by volume displacement RR/∇ was used in the parametric study conducted in Marin [6]. However, in order to get a somewhat different angle on this study XP was used. In order to get a broader overview the XP was calculated over the entire speed range, in the displacing interval (0 < Fn < 0.5), in the semi-displacing interval (0.5 < Fn < 1.0) and in the planning interval (Fn < 1.0). In order to calculate XP below equations were used.

|CRA −CRB | 2 CR% = 100 · (33) CRA +CRB 2

|PA−PB | P = 100 · 2 (34) % PA+PB 2

N 1 X CR%(i) X = (35) P N P i=1 % In Equation 33 and 34 the denominator A and B stands for file-A and file-B. The parameter P stands for the parameter (λ, B/T , CP or CB) that is different between the two files. By calculating CR% an average percentage in difference between the residual resistance coefficient CR between file-A and file-B is calculated. By calculating P% an average percentage in difference between the parameter value between file-A and file-B is calculated. In Equation 35 the XP value itself, indicate how many percent the CR value changes if the parameter P changes 1 % over the speed range. The XP can be viewed as the importance of the parameter. The N indicate how many calculations are made in the summation and the i indicate which CR out of the 1 to 1000 that is being calculated. This study was performed over the entire speed interval, to investigate the XP of each parameter and to asses if any specific speed interval is of specific importance. In order to perform this parametric study efficiently a C# script was created that paired files, performed the Akima spline fit and performed the calculations from using Equation 33, 34, 35, 36, 37 and 38. To show an example

23 of how Equation 33, 34 and 35 can be used in one single measurement point, picture the following scenario. Two files are paired, file-A and file-B, the parameter that is varying is CB. The following is known, file-A −3 −3 have CR = 3 · 10 and CB = 0.55 and file-B have CR = 2.75 · 10 and CB = 0.45. This would result in

CR% = 4.35 %, P% = 10.00 % and XCB =0.435 %.

Furthermore, the correlation and trend between the λ and L/B was studied. This was done by plotting the λ vs L/B for entire transom stern database. In order to estimate the correlation between the two parameters the population Pearson correlation coefficient was calculated. The Pearson correlation coefficient result in a value between -1 to 1, where -1 or 1 indicate complete correlation and 0 would indicate zero correlation [26]. The Pearson correlation coefficient is shown in Equation 36 below.

cov(X,Y ) ρX,Y = (36) σX σY The covariance was calculated by using Equation 37 below.

N 1 X cov(X,Y ) = (x − E(X))(y − E(Y )) (37) N i i i=1

In Equation 37 the E(X) and E(Y) are the mean values for variable X and Y. Given (x1,y1),...,(xi,yi) consisting of N pairs. The standard deviation was calculated by using the Equation 38 below.

v u N u 1 X σ = t (x − E(X))2 (38) N i i=1

24 6 Model series

Model test experiments have formed the basis of understanding naval architecture. The accumulation of this data have been used as basis to the development of the semi-empirical methods that are used at present day. Naturally in order to build and develop a powerful early ship design tool the underlying model test data have to be readily available to be utilised when needed. Therefore multiple of model series, where the residual resistance coefficient CR data can be extrapolated, have been thoroughly studied and transferred to FOI’s database. Also hull defining parameters, temperature, water density was saved for each model if the information was available. In order to cover a broad range of naval ship types, from large bulk carriers to fast semi-displacing ships, the total number of models collected ended up being 249 pcs. The total number of fast semi-displacing models was 117 pcs. The total number of passenger ships/cargo ships/cruiser ships was 126 pcs. And lastly the total number of bulk carriers was 5 pcs. Most of the models come from model tests series, the series have therefore been described briefly in order to give context to how, why and what the test data could be used for.

Table 3: Parameters held ﬁxed and varied during the series

Model series λ B/T CP CB L/B Froude number Marin series 4.31 - 12.07 2.5 - 5.5 0.561 - 0.685 0.35 - 0.55 4 -12 0.14 - 1.30 Series-64 8.60 - 12.40 2.0 - 4.0 0.630 0.35 - 0.55 8.45 - 18.26 0.06 - 1.50 NPL 4.47 - 8.30 1.7 - 6.9 0.693 0.39 3.33 - 7.5 0.30 - 1.20 NTUA 6.20 - 10.00 3.2 - 6.2 0.582 - 0.742 0.34 - 0.54 4.30 - 7.50 0.20 - 1.10 Southampton 6.30 - 9.50 1.50 -2.50 0.693 0.397 7 - 15.1 0.20 - 1.00 Taylor-Gertler 5.50 -10.00 2.25 - 3.75 0.50 - 0.80 0.463 - 0.740 - 0.16 - 0.58 Harvald & 4.00 - 8.00 2.5 0.50 - 0.80 - - 0.15 - 0.50 Guldhammer

6.1 The MARIN Systematic Series The Royal Netherlands Navy and the Netherlands Ship Model Basin jointly carried out a systematic series in order to meet the changing demands of the naval design criteria during the end of the 1970’s. The MARIN Systematic Series consist of in total 35 models of fast displacement ships with a transom stern with varying length between 3 to 6 meters. It was viewed as such that investigations regarding monohull displacement ships for higher speeds needed to be performed in order to achieve improved seakeeping quality. The main purpose of the project was to define the best hull form with respect to optimal resistance and powering performance, as well as eminent seakeeping characteristics. The speed range of the series was set to Fn = 0.14 - 1.3. Where a PHF(parent hull form) preferably should be optimal over the entire range. But when trade-off between low and high speed was present, then the speed range Fn = 0.7 - 1.0 should be prioritized. All the models where designed to have the prismatic coefficient close to CP = 0.626, except for FDS-21 where CP = 0.685 and FDS-22 where CP = 0.561. FDS-21 and FDS-22 was built with identical dimensions as FDS-5, but with different prismatic coefficients. This was done to assert the belief that CP = 0.626 indeed was the optimal choice for the PHF. The prismatic coefficient CP = 0.626 was held constant for the majority of the MARIN series, due to the fact that enough knowledge about the prismatic coefficient was considered to be known and the chosen value close to the optimum. For the Marin series the block coefficient CB varied between four values 0.35, 0.4, 0.50 and 0.55. The longitudinal centre of buoyancy held between LCB = 5.05 - 5.19 % LWL aft of amidship for the entire series. The submerged area at transom divided by the submerged area amidship was held almost constant at AT /AM = 0.31 for the entire Marin series.

The parent hull form parameters L/B = 8, B/T = 4 and CB = 0.4, was based out of the MARIN Legacy data. The Marin Legacy data was the current database of actually built ships available at MARIN. The

25 specific hull form was defined by analysing the body plans and model test results related to the legacy data. From the "best models" it was clear that a round bilge design with a V-shape section forward and nearly flat bottom aft was optimal. A hard-chine form was disregarded, since it was viewed as not optimal when transforming the hull into different shapes. With the three chosen parameters in mind Sub-series 1 also called the forerunner series, consisting of 6 models was constructed. The Sub-series 1 tests was performed, due to that an optimal shape of the waterline and all related parameters was desired to be found for the PHF. The six models all had the same curve of sectional areas, however with varying LCF , Cwp, Cvp and ie. Also the BT /B was varied along LCF . The six models where all built with LWL = 5.0 meters and thereafter tested in calm water, regular head waves and in two sea states identified as "low" and "high". (More details about the specific hull forms for the six models and experimental results are available in [6]). From the experimental data it was clear that Model 5 was the overall best choice. It had slightly higher resistance, 2.2% compared to model 6, however far better seakeeping characteristics such as heave, pitch and acceleration.

All models in the Marin series were constructed by wood, hand-ﬁnished and spray-painted. The tow point for all models were situated at a point 0.169 m above the keel line. The models were completely bare, without any kind of appendages or skeg ﬁtted to the models. For the turbulence stimulation when measuring the calm-water resistance a sand strip with width 25 mm and surface roughness 50 µm was used. These strips were also present for the wave experiments.

The accuracy of the experiments conducted in calm water depend on the data acquisition system and the model error. The data acquisition system performs the measurement, which consist of three steps. • Calibration

• Zero measurement • Dynamic measurement This measurement system was formally evaluated during an experimental programme carried out between 1986 to 1990. It concluded that the measurement accuracy due to random error was approximately 1.2%. Furthermore, the model error(hull form, draught and loading condition) aﬀect the total resistance and components of the resistance. The total accuracy error for both the measurement and the model error was found to be about 2%. [6]

6.2 Series-64 During the early 1960’s knowledge about speed-length ratios of displacement-type surface ships were limited to under 2.0. However, in order to allow for tactical missions the American navy was interested in speed- length ratios above 2.0. Therefore, the model series Series-64 was conduced, which consist of 27 models of round bilge hull forms. The longitudinal centre of buoyancy was held constant at LCB = -6.56 % LW l for all models. The submerged area at transom divided by the submerged area amidship was held almost constant at AT /AM = 0.40 for the entire series. All the models were constructed of wood with designed waterline length 10 feet = 3.048 m [27].

All the models were tested at the David Taylor Model Basin deep-water model basin (B = 15.545 m x D = 6.7056 m) in Washingtion D.C. The tests were conducted by towing the models at the center of ﬂota- tion. The models were free to pitch, heave and roll, but restricted in yaw. The resistance was measured by using towing dynamometer for the calm water tests. Due to past experiences of models with similar hull forms, the experiments were conducted without turbulence stimulation [27]. The Series 64 residual resistance data were tabulated in [3].

26 6.3 NTUA series The NTUA series focused on investigating the resistance and seakeeping characteristics for a double-chine hull. The series consist of 6 models that have been tested under diﬀerent loading conditions. The primary use of the series is to be applied in the design stages of large high-speed vessels mainly operating between 0.5 < Fn < 0.9. The most similar series to the NTUA series are the NPL series and the SKLAD series. The turbulence tripping was performed by mounting strips along the lower chine [7]. Furthermore, it should be noted that the longitudinal prismatic coeﬃcient CP was estimated by looking at the lines of the model hulls and with help of the data given in [7]

6.4 NPL high speed round bilge displacement series The NPL high speed round bilge displacement series was performed in order to add information about this ship type. LCB = - 2 to - 6.4%L and AT /AM = 0.52. All the models were constructed by either wood or polyurethane with a painted surface and a waterline length of 2.54 m. They were all tested without keels and any kind of appendages. The turbulence stimulation was created by studs of 3 mm in diameter, length 2.5 mm and spaced 25 mm apart, ﬁtted near the bow proﬁle [8].

6.5 Southampton Catamaran series The Southampthon Catamaran series was carried out in order improve the understanding of resistance components and to provide design data for high speed displacing catamarans.

The series consist of in total 10 tested monohulls and 40 tests with variations of catamaran conﬁgurations. Added to FOI’s database was only the monohull test results. LCB = -6.4%L. All the model had waterline length 1.6 m. Test facilities Southampton Institute of Higher Education test tank, which have the the dimensions, Length: 60 m, Breadth: 3.7 m and depth 1.85 m.

6.6 Taylor-Gertler Standard Series The parent hull form of the original Taylor Standard Series was based on the British armoured cruiser Leviathan and Taylor’s mathematical lines. The models used when testing all had approximately LWL= 20 feet = 6.096074 meter and constructed by wood. The original results and tests conducted from the Taylor Standard Series was completely revised and re-analysed by M.Gertler [28]. The reason for this was that the original Taylor Standard Series was subjected to several errors, such as failure to compensate in basin water temperature, for transitional ﬂow at lower Reynolds numbers and for restricted channel eﬀects. One of the largest errors in the original Taylor series was that the basin water temperature was not accounted for. Estimations of the test temperatures was therefore based on later water temperature records from the basin collected during 1913-1918. Thereby, the temperature prediction could in most cases be predicted within 1 Farenheit. (For more details the entire correction process is described in detail in [28]). The Taylor-Gertler Standard Series residual resistance data were tabulated in [3] and consist of 64 model ships.[29][30].

6.7 Harvald & Guldhammer Numerous of model test series have been compiled for merchant/cruiser type ships in [31]. It is unfortunately not speciﬁed which waterline length each model have. This introduces an inaccuracy, since a mean waterline length had to be assumed for the entire compendium. The assumed mean length was chosen to 4 meters, since some of the series are public and their waterline length known. It should also be noted that the data graphs where the prismatic coeﬃcient Cp is higher than 0.7 have a higher uncertainty. These graph are based on fewer or little data and are therefore less accurate compared to the other graphs. [31]

27 7 Selected hulls

It was jointly decided between this work and [1], to focus the investigation on two drastically diﬀerent hull types. One semi-displacing ship called the FDS-5 originating from the Marin Series [6] and one bulk carrier with a bulbous bow called JBC [19].

7.1 FDS-5 summary The FDS-5 model(Fast Displacement Ship Model 5) was chosen as parent hull form for the entire MARIN Series, mainly due to the fact that it is a well documented semi-displacement hull with good sea characteristics. Its application could be used for vessels with approximately length 30-100 meters. Some applicable ship types would be e.g a yacht, patrol vessel or corvette. It have been covered extensively in several reports originating from the MARIN Fast Displacement Ship Series project[32][33][34]. Figure 11 shows the FDS-5 model being tested at the Marin deep water tank facility.

Figure 11: FDS-5 tested at the Marin deep water tank. Published with authors permission [6].

Table4 shows a compilation of key parameters in regard to the FDS-5 model and the calm water conditions it was tested in [6]. Table4 also contains a full-scale example of the FDS-5 where its waterline length is 85 meters. For the full-scale example the FDS-5 have simply been scaled up and subjected to salt water conditions.

28 Table 4: Model-scale and full-scale information. Based on information from [6].

Symbol Model-scale Full-scale Unit Description LWL 5 85 m Designed waterline (v = 0 m/s) BWL 0.625 10.625 m Beam at designed waterline (v = 0 m/s) T 0.15625 2.65625 m Draught amidship (v = 0 m/s) LWL/BWL 8 8 - Length-beam ratio BWL/T 4 4 - Beam-draught ratio S 2.9288 846.4232 m2 Wetted surface (v = 0 m/s) LcF -8.68 -8.68 % Distance centre of flotation aft to amidship/LWL LcB -5.11 -5.11 % Distance centre of buoyancy aft to amidship/LWL LcB.Dist 2.2445 38.1565 m Distance from aft perpendicular Vcg 0.169 2.873 m Dist. vert. center of gravity amidship from hull bottom cP 0.626 0.626 - Prismatic coefficient cB 0.396 0.396 - Block coefficient ie 11 11 deg Half entrance of angle of the waterline ∇ 0.195 958.035 m3 Displaced volume m 194.985 981985.875 kg Mass of model ρ 998.8968 1025 kg/m3 Water density ν 1.1007 · 10−6 1.1394 · 10−6 m/s2 Kinematic viscousity t 16.3 16.3 C◦ Water temperature 1/3 LWL/∇ 8.68 8.68 - Slenderness (length-displacement ratio)

7.2 JBC summary

Figure 12: Ship type: Capesize Bulk Carrier

The Japanse bulk carrier was initially designed to undergo EFD(experimental ﬂuid dynamic) tests, which later were compared to CFD.

29 Table 5: Model-scale and full-scale information

Symbol Model-scale Full-scale Unit Description LPP 7 280 m Length between perpendiculars LWL 7.125 285 m Designed waterline (v = 0 m/s) BWL 1.125 45 m Beam at designed waterline (v = 0 m/s) T 0.4125 16.5 m Draught amidship (v = 0 m/s) LWL/BWL 6.333 6.333 - Length-beam ratio BWL/T 2.727 2.727 - Beam-draught ratio S 12.222 19556.1 m2 Wetted surface (v = 0 m/s) ∇ 2.787 178369.9 m3 Displaced volume m 2782.0 178048834.2 kg Mass ρ 998.2 998.2 kg/m3 Water density LcB 2.5475% of LPP 2.5475% of LPP m Long. dist. centre of buoyancy fwd to amidship cP 0.860 0.860 - Longitudinal prismatic coeﬃcient cB 0.858 0.858 - Block coeﬃcient Vs 0.142 0.142 - Service speed in Froude number k 0.314 0.314 - Form factor 1/3 LWL/∇ 5.063 5.063 - Slenderness (length-displacement ratio)

8 Results 8.1 Interpolations for the FDS-5 model Given the data files available with transom sterns the interpolation script choose different files to estimate the FDS-5 measurement data with. For case 1, where all files was available to choose from the interpolation script chose the data files of the models tabulated in Table6. The table contain a few important parameters of the model hulls.

Table 6: Important parameters for the data ﬁles used when estimating FDS-5 for case 1. For comparison the FDS-5 and its parameters were included as well.

Model λ B/T CP CB L/B FDS-5 8.68 4 0.626 0.40 8 FDS-11 9.03 4 0.626 0.35 8 FDS-12 7.98 4 0.626 0.50 8 FDS-21 8.68 4 0.561 0.40 8 FDS-22 8.68 4 0.685 0.40 8

The FDS-11 and FDS-12 was used for the interpolation shown in Figure 13a. The FDS-21 and FDS- 22 was used for the interpolation shown in Figure 13b. Shown in purple in both ﬁgures are the estimated values created by interpolation at corresponding measurement Froude number as for the FDS-5. The residual resistance coeﬃcient CR for the FDS-5 have been included in both Figure 13a and Figure 13b for comparison.

30 10 -3 10 -3 4 2.8 FDS-5 FDS-5 FDS-11 2.6 FDS-21 3.5 FDS-12 FDS-22 Interpolated 2.4 Interpolated 3 2.2

2 2.5 1.8

CR [-] 2 CR [-] 1.6

1.5 1.4 1.2 1 1

0.5 0.8 0.1 0.3 0.5 0.7 0.9 1.1 1.3 0.1 0.3 0.5 0.7 0.9 1.1 1.3 Froude number [-] Froude number [-] (a) Interpolation with FDS-11 and FDS-12 in an eﬀort to (b) Interpolation with FDS-21 and FDS-22 in an eﬀort to estimate FDS-5’s measurement data. estimate FDS-5’s measurement data.

Figure 13: Interpolation estimations of FDS-5 measurement with use of files from the Marin series. The files used for interpolation have been Akima spline fitted and are therefore shown with a solid line [25].

The combined result from Figure 13a and Figure 13b was used in Section 8.2. For case 2, where the entire Marin Series is excluded for usage, the following files was selected by the interpolation script. The two files selected both came from Series-64, for convenience the files have been called Model-1 and Model-2, a few important parameters of the files have been tabulated below in Table7.

Table 7: Important parameters for the test ﬁles used when estimating FDS-5 for case 2. For comparison the FDS-5 and its parameters were included as well.

Model λ B/T CP CB L/B FDS-5 8.68 4 0.626 0.40 8.000 Model-1 8.60 4 0.630 0.45 8.454 Model-2 9.60 4 0.630 0.45 9.948

31 10 -3 2.8 FDS-5 2.6 Model-1 Model-2 2.4 Interpolated 2.2

1.8 CR [-] 1.6

1.4

1.2

0.8 0.1 0.3 0.5 0.7 0.9 1.1 1.3 Froude number [-]

Figure 14: Interpolation with Model-1 and Model-2 in an effort to estimate FDS-5’s measurement data. The files used for interpolation have been Akima spline fitted and are therefore shown with a solid line [25].

The interpolated result in Figure 14 was later used in Section 8.2.

8.2 FDS-5 resistance estimation The FDS-5 calm water bare hull resistance estimations were performed on model-scale and full-scale. There- fore these results were divided up into two sections. It should be noted that the CFD simulations were only performed on model-scale for the FDS-5, these results were therefore only included in Section 8.2.1.

8.2.1 Model-scale FDS-5 estimation The model-scale estimations resulted in below plots. For several plots the FDS-5 measurement were included in order to more easily compare the results. When the FDS-5 values were included they were simply called "FDS-5" in the plots. The interpolated estimation from using the Marin series shown in Figure 13 was called "Marin" in all plots. The interpolated estimation from using Series-64 shown in Figure 14 was called "Series-64" in all plots. For the "Marin" and "Series-64" results the calculated wetted surface S = 2.95440 m2 was used, this result is tabulated in Table 12. And for the "FDS-5" and "CFD" the exact given wetted surface S = 2.9288 m2 was used, this is tabulated in Table4. Lastly in some of the plots at model-scale the CFD results from simulating the FDS-5 were included as well [1]. Figure 15 shows the total resistance RT in calm water and the relative RT error for the Marin, Series-64 and CFD estimations.

32 25 20 Marin 400 Series-64 15 CFD 10 300 5 0 200 -5 -10 FDS-5 Total resistance [N] 100 Marin Relative RT error [%] -15 Series-64 CFD -20 0 -25 0.1 0.3 0.5 0.7 0.9 1.1 1.3 0.1 0.3 0.5 0.7 0.9 1.1 1.3 Froude number [-] Froude number [-]

(a) RT measurement and estimations of FDS-5. (b) Relative total resistance error of model estimations made with Marin, Series-64 and CFD compared to FDS-5 measurement.

Figure 15: Total resistance RT and relative error from estimations.

The relative total resistance error was calculated under the assumption that the RT values from FDS-5 measurement are accurate. Figure 16 shows the CR data from FDS-5, Marin, Series-64 and CFD as well as the relative CR error from the Marin and Series-64 estimations.

10 -3 3 50 FDS-5 Marin Marin 40 Series-64 2.5 Series-64 CFD 30

20 2 10

CR [-] 0 1.5 -10

1 Relative CR error [%] -20

-30

0.5 -40 0.1 0.3 0.5 0.7 0.9 1.1 1.3 0.1 0.3 0.5 0.7 0.9 1.1 1.3 Froude number [-] Froude number [-]

(a) CR measurement and estimations of FDS-5. (b) Relative residual resistance coeﬃcient error of model estimations made with Marin Series and Series-64 compared to FDS-5 measurement.

Figure 16: CR measurement and estimations and relative CR error

Figure 17 shows the resistance coeﬃcients distribution in percent on the form CF /CT and CR/CT for the FDS-5 model measurement.

33 75

55 C /C 50 F T C /C R T 45

Resistance coefficients distribution [%] 25 0.1 0.3 0.5 0.7 0.9 1.1 1.3 Froude number [-]

Figure 17: Resistance coeﬃcients distribution from FDS-5 measurement.

8.2.2 Full-scale FDS-5 estimation The full-scale estimations have been performed by following the ITTC-78 performance prediction method. The full-scale estimations was performed by using the CR data from the FDS-5 measurement and by using the Marin and Series-64 estimation of the CR data. For the "Marin" and "Series-64" results the calculated wetted surface S = 853.8216 m2 have been used, this was obtained by using Equation 31 and 32. And for the "FDS-5" the exact given wetted surface S = 846.4232 m2 have been used, this is tabulated in Table4.

The full-scale total resistance prediction was performed by using the FDS-5 measured CR data, the Marin series and the Series-64. Also the relative RT error was calculated with the assumption that the FDS-5 estimations was accurate. These results are illustrated below in Figure 20.

34 10 6 2 30 Marin Series-64 20 1.5

10 1 0

0.5 Total resistance [N] FDS-5 Relative RT error [%] -10 Marin Series-64 0 -20 0.1 0.3 0.5 0.7 0.9 1.1 1.3 0.1 0.3 0.5 0.7 0.9 1.1 1.3 Froude number [-] Froude number [-] (a) Total resistance of full-scale estimations made with (b) Relative total resistance error of full-scale estima- FDS-5, Marin Series and Series-64 tions made with Marin Series and Series-64 compared to FDS-5 full-scale estimation.

Figure 18: RT measurement and estimations and relative RT error

Figure 19 shows the distribution in percent of the CF /CT , CR/CT and ∆CF /CT for the full-scale FDS-5 estimation.

40 C /C F T C /C 30 R T C /C F T 20

Resistance coeffcients distribution [%] 0.1 0.3 0.5 0.7 0.9 1.1 1.3 Froude number [-]

Figure 19: Resistance coeﬃcients distribution from FDS-5 full-scale estimation.

Figure 20a was plotted in order to visualize at which eﬀective power PE the desired speed is achieved. This was done for the FDS-5, the Marin series and the Series-64 estimation. Furthermore, Figure 20b shows the relative speed error for the Marin and Series-64 estimation plotted against the eﬀective power required for the FDS-5.

35 40 15 Marin 35 Series-64 10 30

25 5 20

15 0 Speed [m/s] 10 FDS-5 -5 5 Marin Relative speed error [%] Series-64 0 -10 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 P [W] 7 P [W] 7 E 10 E 10 (a) Speed vs total resistance graph, made with results (b) Relative speed error for Marin series and Series-64 from FDS-5, Marin series and Series-64 estimations. estimation compared to FDS-5 estimation.

Figure 20: RT measurement and estimations and relative RT error

8.3 JBC resistance estimation Since no suitable model series hulls were found for estimating and interpolating the JBC’s residual resistance coeﬃcient CR, the most similar model hull and its CR was chosen to be used directly. The JBC vessel was estimated for two cases. For case 1 the best ship was selected by the interpolation script in regards to the four parameters. The chosen model hull ﬁle contained the Galati vessel [35]. For case 2, the best ship was selected based on both the four parameters and the characteristics of the data. For case 2 the model data for a ship called KVLCC2 was chosen. A few important parameters are tabulated below in Table8 in regards to the JBC, Galati and KVLCC2 vessels.

Table 8: Important parameters for the test ﬁles used when estimating JBC

Model λ B/T CP CB L/B JBC 5.063 2.727 0.860 0.858 6.333 Galati 5.073 2.33 0.841 0.837 5.517 KVLCC2 4.710 2.79 0.810 0.811 6.740

Firstly, the ITTC form factor prediction process described in Section 4.7.2 was followed in order to check if the form factor can be utilised on the Galati and KVLCC2 data. Figure21 below shows the plots where the form factor is attempted to be determined. It should be noted that the KVLCC2 was speciﬁed to have k = 0.16 - 0.18, which means that Figure 21b only serves as control of this statement [36].

36 1.22 1.7 Measured data 1.21

1.6 1.2 [-]

F 1.19 1.5 [-] F /C /C T T 1.18 C C 1.4 1.17 Measured data 1.3 1st order polyfit 1.16 2nd order polyfit 3rd order polyfit 0 0.05 0.1 0.15 0.2 0.25 1.15 0 0.05 0.1 0.15 4 Fn /C [-] Fn4/C [-] F F (a) Prohaska’s method utilised on the Galati data. (b) Prohaska’s method utilised on the KVLCC2 data.

Figure 21: Form factor estimations of the Galati and KVLCC2 data.

It was decided that the Prohaska’s method could not be applied on the Galati vessel. The short motivation to this is simply that if a curve ﬁt was performed on the measured data in Figure 21a, the (1+k) would be unreasonably high. Therefore a mean value from the semi-empirical form factor estimation methods shown in Equation 22, 23 and 24 was used in order to estimate a form factor k for the Galati vessel. Thereafter Equation8 was used in order to calculate the residual resistance coeﬃcients CR for the Galati vessel. When estimating RT on model-scale for the JBC vessel with the Galati and KVLCC2 data, an estimate of the JBC form factor had to be made. Therefore a mean value from the semi-empirical form factor estimation methods in Equation 22, 23 and 24 was used once again, but with the scenario of inputting the JBC’s parameters LPP , CB, B and T. By then using the estimated form factor for JBC and the CR data from the Galati and KVLCC2, two total resistance prediction RT could be made for the JBC vessel.

The JBC calm water bare hull resistance predictions have been performed on model-scale and full-scale. Therefore these results have been divided up into two sections. It should be noted that the CFD simulations was only performed on model-scale for the JBC, these results have therefore only been included in Section 8.3.1.

8.3.1 Model-scale JBC estimation The model-scale estimations resulted in below plots. For several plots the measured JBC values were included in order to more easily compare the results. The JBC values were simply called "JBC" when they were included in the plots. Furthermore, when available at model-scale the CFD results from simulating the JBC were included as well [1]. Figure 22 shows the total resistance RT in calm water and the relative RT error for the Galati and KVLCC2 estimations. For the "Galati" and "KVLCC2" results the calculated wetted surface S = 12.159423 m2 was used, this result is tabulated in Table 13. And for the "JBC" and "CFD" the exact given wetted surface S = 12.222 m2 was used, this is tabulated in Table5.

37 50 10 Galati KVLCC2 8 45 CFD

6 40 4

35 2

JBC 0 Relative RT error [%] Total resistance [N] 30 Galati KVLCC2 -2 CFD 25 -4 0.115 0.125 0.135 0.145 0.155 0.165 0.115 0.125 0.135 0.145 0.155 0.165 Froude number [-] Froude number [-] (a) Total resistance of model measured for JBC and es- (b) Relative total resistance error of model estimations timated with Galati, KVLCC2 and CFD. made with Galati, KVLCC2 and CFD compared to JBC measurement values.

Figure 22: Total resistance and relative error from measurement and estimations of the JBC vessel.

The relative total resistance error was calculated under the assumption that the measured RT values from JBC was accurate. The relative CR error was calculated under the assumption that the CR values from the JBC measurement were accurate. It should also be mentioned that for the CFD graph the CR values were derived from the RT data by following the ITTC performance prediction method in Section 4.6. Figure 23a below shows the relative error from using the Galati, KVLCC2 and CFD. Figure 23b shows the distribution in percent of the CF /CT and CR/CT for the JBC measurement. In Figure 23b two y-axes were implemented in an eﬀort to more clearly show the trends of the data.

600 100 10 550 Galati (1+k)CF/CT KVLCC2 500 CR/CT CFD 450 98 8 400 350 96 6 300 250 200 94 4 150 CR/CT [%]

100 (1+k)CF/CT [%] Relative CR error [%] 50 92 2 0 -50 -100 90 0 0.115 0.125 0.135 0.145 0.155 0.165 0.115 0.125 0.135 0.145 0.155 0.165 Froude number [-] Froude number [-] (a) Relative residual resistance coeﬃcient error of model (b) Resistance coeﬃcients distribution derived from JBC estimations made with Galati and KVLCC2 compared to measurement. JBC measurement.

Figure 23: Information about the relative error for CR and the distribution of CR and (1 + k)CF relative to CT .

38 8.3.2 Full-scale JBC estimation The full-scale estimations were performed by following the ITTC-78 performance prediction procedure described in Section 4.6. The full-scale estimations was performed by using the CR data from the JBC measurement and by using the CR data from the Galati and KVLCC2 vessel. For the "Galati" and "KVLCC2" results the calculated wetted surface S = 19455.1 m2 have been used, this was obtained by using Equation 31 and 32. And for the "JBC" and "CFD" the exact given wetted surface S = 19556.1 m2 have been used, this is tabulated in Table5.

Figure 24 shows the total full-scale resistance RT in calm water and the relative RT error for the Galati and KVLCC2 estimations.

10 6 1.8 25 Galati 1.7 20 KVLCC2 1.6

1.5 15

1.4 10 1.3 5 1.2

1.1 0 Total resistance [N]

1 Relative RT error [%] JBC -5 0.9 Galati KVLCC2 0.8 -10 0.115 0.125 0.135 0.145 0.155 0.165 0.115 0.125 0.135 0.145 0.155 0.165 Froude number [-] Froude number [-] (a) Total resistance of full-scale estimations made with (b) Relative total resistance error of full-scale estima- JBC, Galati and KVLCC2. tions made with Galati and KVLCC2 compared to JBC’s full-scale estimation.

Figure 24: Total resistance for the JBC, Galati and KVLCC2 estimation. Relative RT error is also shown.

Figure 25 shows the distribution in percent of the (1 + k)CF /CT , CR/CT , ∆CF /CT and CA/CT for the full-scale JBC estimation. The (1+k)CF /CT was plotted against the right y-axis and the other distributions were plotted against the left y-axis. This was done in order to better visualize the data in one plot.

39 20 85 CR/CT 18 CA/CT 84 CF/CT 16 (1+k)CF/CT 83 14 82 12 81 10 80 8 (1+k)CF/CT [%] 79 6

4 78 Resistance coeffcients distribution [%] 2 77 0.115 0.125 0.135 0.145 0.155 0.165 Froude number [-]

Figure 25: Resistance coeﬃcients distribution from JBC full-scale estimation.

Figure 26a was plotted in order to visualize at which resistance the desired speed is achieved. This was done for the JBC, Galati and KVLCC2 estimations. Furthermore, Figure 26b shows the relative speed error for the Galati and KVLCC2 estimation plotted against the eﬀective power required for the JBC.

8.5 10 Galati 8 KVLCC2 8 6

7.5 4

2 7 Speed [m/s] 0 6.5 JBC

Galati Relative speed error [%] -2 KVLCC2 6 -4 0.5 1 1.5 0.5 1 1.5 P [W] 7 P [W] 7 E 10 E 10 (a) Speed vs eﬀective power, graph made with results from (b) Relative speed error for Galati and KVLCC2 estima- JBC, Galati and KVLCC2 estimations. tion compared to JBC’s estimation.

Figure 26: speed vs eﬀective power for the JBC, Galati and KVLCC2 estimations. Also the relative speed error is shown.

8.4 Parameter study The purpose of the parameter study was to assess the importance of the parameters that potentially could deﬁne a ship and how they best could be used when performing interpolations. The main parameters to

40 investigate was slenderness ratio λ, beam-draught ratio B/T , prismatic coefficient CP and block coefficient CB. Furthermore, in order to gain further insight in how different parameters affect the CR data the slenderness λ was substituted with the length-beam ratio L/B, the result by doing this is shown in Table 11. By forming pairs where three parameters were identical or within the 0.04% of each other, the importance of the parameters could be investigated.

One important aspect to investigate was the relation between the slenderness ratio λ and the length-beam ratio L/B. These two parameters are most likely strongly coupled, however to what degree is an important aspect to be aware of when making ship resistance estimations. Firstly in order to visualize the trend the λ value and L/B value was plotted against each other for the entire database with transom sterns hulls. This plot is shown below in Figure 27

Slenderness [-] 6

( , L/B) 4 Linear regression

5 10 15 20 L/B [-]

Figure 27: Database of transom stern ships plotted with respect to slenderness ratio vs length-beam ratio. The least square linear regression ﬁtting of the data have also been included in the graph.

Furthermore, in an effort to investigate the correlation between the slenderness λ and the length-beam ratio L/B Equation 36 called the population Pearson correlation coefficient was used. This resulted in the correlation coefficient, ρX,Y = 0.8565.

In order to calculate the importance of varying one parameter at a time the three Equations 33, 34 and 35 was used on all the pairs that were created in the parameter study.

41 Table 9: Tabulated importance of parameter λ, B/T, CP and CB over the speed range. The values Xλ,

XB/T , XCP and XCB show how many percent the CR value changes if the parameter changes 1 %. The standard deviation σ have been included as well. Pairs were only created if the ﬁles came from the same series.

Xλ σλ XB/T σB/T XCP σCP XCB σCB Mean value 2.447 0.628 0.210 0.119 0.749 0.182 0.342 0.093 Fn < 0.5 2.165 0.773 0.286 0.232 1.259 0.376 0.419 0.124 0.5 1.0 2.247 0.836 0.192 0.151 0.580 0.157 0.230 0.085 # pairs 65 58 3 3

In order to expand the study to include pairing between diﬀerent series, an allowance of 0.04% was used. Meaning that three parameters have to be identical within 0.04% of one another. Furthermore, pairing between the Series-64 and Marin series was not allowed. The reasoning behind this is lifted in Section 9.4.

Table 10: Tabulated importance of parameter λ, B/T, CP and CB over the speed range. The values Xλ,

XB/T , XCP and XCB show how many percent the CR value changes if the parameter changes 1 %. The standard deviation σ have been included as well. Pairs between diﬀerent series were allowed.

Xλ σλ XB/T σB/T XCP σCP XCB σCB Mean value 2.514 0.701 0.255 0.310 1.045 0.535 0.985 0.610 Fn < 0.5 2.190 1.319 0.422 0.846 1.439 0.451 1.720 1.374 0.5 1.0 2.418 0.827 0.196 0.174 0.580 0.157 0.531 0.329 # pairs 107 83 4 14

In order to gain further insights in how diﬀerent sets of parameters aﬀect the CR Table 11 was created. The slenderness ratio was substituted with the length-beam ratio. Meaning that in this case the four parameters considered was L/B, B/T , CP and CB.

Table 11: Tabulated importance of parameter L/B, B/T, CP and CB over the speed range. The values Xλ,

XL/B σλ XB/T σB/T XCP σCP XCB σCB Mean value 1.691 0.437 1.005 0.146 0.749 0.180 1.081 0.287 Fn < 0.5 1.549 0.540 0.806 0.244 1.259 0.350 1.168 0.457 0.51.0 1.528 0.560 0.902 0.240 0.580 0.157 0.839 0.394 # pairs 65 53 3 41

Figure 28, 29, 30 and 31 have been created in an eﬀort to illustrate individually how varying the parameter aﬀect the CR measurement over the speed interval.

42 9 = 9.50 = 8.50 = 7.41 7 = 6.27

5 CR x 1000 [-] 3

1 0.2 0.4 0.6 0.8 1 Froude number [-]

Figure 28: Example from the Southampton series of CR data when the slenderness ratio λ is being varied. The points show the measurement values and the line represent the Akima spline ﬁtting [25].

Figure 28 contain four files from the Southampton series with varying slenderness ratio. The other three parameters were fixed at B/T = 2, CP = 0.693 and CB =0.397. Even though it was not primarely considered it should be noted that L/B = 13.1, 11, 9 and 7 for the different files.

3.5 C = 0.55 B C = 0.45 B 3 C = 0.35 B

2.5

CR x 1000 [-] 1.5

0.5 0.2 0.4 0.6 0.8 1 1.2 1.4 Froude number [-]

Figure 29: Example from Series-64 of CR data when the block coeﬃcient CB is being varied. The points show the measurement values and the line represent the Akima spline ﬁtting [25].

Figure 29 contain four files from Series-64 with varying the block coefficient CB. The other three parameters were tried to be held fixed at B/T = 4, CP = 0.63 and λ = 8, 8.6 and 9.3. Even though it was not primarily considered it should be noted that L/B = 8.454 was constant for the different files.

43 2.6 C = 0.685 P C = 0.626 2.4 P C = 0.561 P 2.2

1.8

1.6

CR x 1000 [-] 1.4

1.2

0.8

0.2 0.4 0.6 0.8 1 1.2 Froude number [-]

Figure 30: Example of CR data when the slenderness ratio λ is being varied. The points show the measurement values and the line represent the Akima spline ﬁtting [25].

Figure 30 contain four files from the Marin series with varying the prismatic coefficient CP . The other three parameters were held fixed at λ = 8.68, B/T = 4 and CB = 0.4. Even though it was not primarily considered it should be noted that L/B = 8 was constant for the different files.

2.2 27 B/T = 4 B/T = 3.19 25 2 B/T = 3 B/T = 2.43 B/T = 2 23 B/T = 1.72 1.8 21 19 1.6 17 1.4 15 13 1.2 CR x 1000 [-] CR x 1000 [-] 11 1 9 7 0.8 5 0.6 3 0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8 1 Froude number [-] Froude number [-]

(a) CR data from Series-64 where the B/T ratio is being (b) CR data from the NPL series where the B/T ratio is varied. The points show the measurement values and the being varied. The points show the measurement values line represent the Akima spline ﬁtting [25]. and the line represent the Akima spline ﬁtting [25].

Figure 31: Two examples from diﬀerent series where the B/T ratio is being varied.

Figure 31a contain three files from Series-64 when varying the beam-draught ratio B/T . The other three parameters were held fixed at λ = 10.5, CP = 0.63 and CB = 0.55. Even though it was not primarily considered it should be noted that L/B = 17.734, 14.479 and 12.54. Figure 31b shows a second example of varying the B/T ratio, in this case the files come from the NPL series. The other three parameters were held fixed at λ = 4.47, CP = 0.693 and CB = 0.39. Worth mentioning is that the length-beam ratio varied between L/B = 3.33, 3.849 and 4.54 for the NPL files.

44 8.5 Hull geometry results 8.5.1 FDS-5 geometry Results from estimating the FDS-5 model geometry with the ShipDes and Rhinoceros 6 programs, when varying the point resolution of the hull. Figure 32 shows an example of how the point cloud representation of the FDS-5 model hull looks like in the Rhinoceros 6 program. The exact values for S, LWL and T are given in Table4. The results from this study have been tabulated below in Table 12.

Figure 32: Example of point cloud generated in Rhinoceros 6 representing the FDS-5 model.

Table 12: Various results by extrapolating and calculating several hull key parameters.

2 Tot. points Points/rib Ribs S [m ] Serr[%] LWL [m] LWLerr [%] T [m] Terr[%] 17160 33 520 2.95440 +0.874 5.000 0.00 0.157707 +0.525 11451 33 347 2.95437 +0.873 4.995 -0.10 0.157707 +0.525 5742 33 174 2.95378 +0.853 4.980 -0.40 0.157705 +0.512 2871 33 87 2.95343 +0.841 4.980 -0.40 0.157705 +0.512 3306 19 174 2.94712 +0.626 4.980 -0.40 0.158244 +1.276 2958 17 174 2.94461 +0.540 4.980 -0.40 0.158433 +1.397 1118 13 86 2.94268 +0.474 4.980 -0.40 0.158450 +1.408 549 9 61 2.90510 -0.809 4.971 -0.58 0.162395 +3.933

8.5.2 JBC geometry Results from estimating the JBC model geometry with the ShipDes and Rhinoceros 6 programs, when varying the point resolution of the hull. Figure 33 shows an example of how the point cloud representation of the JBC model hull looks like in the Rhinoceros 6 program. The exact values for S, LWL and T are given in Table5. The results from this study have been tabulated below in Table 13.

Figure 33: Example of point cloud generated in Rhinoceros 6 representing the JBC model.

45 Table 13: Various results by extrapolating and calculating several hull key parameters.

2 Tot. points Points/rib Ribs S [m ] Serr[%] LWL [m] LWLerr [%] T [m] Terr[%] 24552 33 744 12.159423 -0.496 7.2600 +1.895 0.4100249 -0.600 14817 33 449 12.165397 -0.447 7.2666 +1.987 0.4125656 +0.016 7392 33 224 12.162844 -0.468 7.2666 +1.987 0.4125663 +0.016 1521 9 169 12.061903 -1.294 7.2160 +1.277 0.4086549 -0.932 810 9 90 12.091326 -1.053 7.2500 +1.754 0.4135394 +0.252 450 5 90 12.140580 -0.650 7.2500 +1.754 0.4371165 +5.968

46 9 Discussion

The discussion have been divided up into several parts, one for each segment in the result section.

9.1 Interpolation For case 1 any files were allowed to be used, the interpolation script exclusively selected files from the Marin series. Since the FDS-5 model was the parent hull that all the other models in the Marin series was based on, several models are very similar to it. This situation makes it ideal from the perspective of attempting to estimate the FDS-5’s CR data with different hulls by linear interpolation. In Figure 13a the two models FDS-11 and FDS-12 were used to estimate the FDS-5. In Table6 we can see that FDS-11 and FDS-12 have different λ and CB compared to FDS-5, however the situation appears to be equidistant, which allows for a "twofolded" interpolation. As can be observed in Figure 13a the interpolation gives a surprisingly good estimation of FDS-5’s CR data. In general the interpolation appear to overestimate the CR at higher Froude numbers and underestimate the CR at lower Froude numbers. The overall impression is that the estimation closely resemble the FDS-5 between 0.35 < Fn < 1.0. At both ends of the interval, it appears to be difficult to catch the sudden changes for CR and the accuracy therefore decrease in these regions. The rather sharp change of the residual resistance coefficient for the FDS-5 at about Fn = 1.25 can be observed in Figure 13b, the reason for this is unfortunately not specified in any of the Marin related material [6],[34], [33]. The trim τ might have allowed for an explanation, however FDS-22 have almost identical trim, so this is probably not the explanation [6]. The FDS-21 and FDS-22 might have the same sharp increase in CR as FDS-5, but at a slightly higher Fn than they were tested at. Another reason could be that there is some spray from the water arising, which increases the residual resistance coefficient just when the hull was about to start planning. Lastly it could also be due to a simple measurement error, however this seems unlikely since this should probably have been noticed and the measurements should have been repeated with corrections for the error at fault.

In Figure 13b the two models FDS-21 and FDS-22 were used to estimate the FDS-5. In Table6 we can see that FDS-21 and FDS-22 have diﬀerent CP compared to FDS-5, which allows for a simple linear interpolation in regards to one parameter. As can be observed in Figure 13b the interpolation appear to closely estimate FDS-5’s CR data. In general the interpolation appear to overestimate the CR at higher Froude numbers and underestimate the CR at lower Froude numbers. Once again the overall impression is that the estimation closely resemble the FDS-5 between 0.35 < Fn < 1.0. Also at both ends of the speed interval, it appears to be diﬃcult to catch the sudden changes for CR and the accuracy is therefore decrease in these regions.

In Figure 14 the two models Model-1 and Model-2 were used to estimate the FDS-5. In Table7 we can see that Model-1 and Model-2 have different λ, CP , CB and L/B to FDS-5. Firstly, Model-1 and Model-2 have a CP very close to the one for FDS-5, therefore they were viewed to be equivalent. Secondly, CB = 0.45 both for Model-1 and Model-2, which means that it is not possible to integrate to CB = 0.40, which is the parameter value that the FDS-5 have. Therefore, the CB = 0.45 was considered as sufficiently close and left the way it is. The interpolation had to be performed in regard to the slenderness ratio. It should be noted that L/B was not considered at all for Model-1 and Model-2 when doing the interpolation, however we can see in Table7 that L/B is somewhat higher both for Model-1 and Model-2 compared to FDS-5’s value. Figure 14 shows the interpolation results, the results are fairly good at estimating the height of the CR curve, however the large peak for the integration and the FDS-5 do not align. With accuracy pinpointing the specific reason for the peaks not being aligned is difficult, however one could speculate that it probably could be due to the differences in L/B, CB and in LCB. All of these values are slightly higher for Model-1 and Model-2 compared to FDS-5. However, the estimation could still be considered as sufficient for usage at an early design stage.

Even though a two-folded interpolation does appear to work, it is probably safer to rely on a regression function in this case. Furthermore, performing multiple consecutive interpolations in order to estimate

47 unknown residual resistance coeﬃcient data is probably not advised, the risk of accumulating a large error appear to be high. So yet again using a regression function is probably preferable in that case. Performing one single linear interpolation between two well known similar hulls appear to have the capability of resulting in a slightly better estimations than a regression function.

9.2 FDS-5 resistance estimations Model-scale Firstly by observing Figure 15a it is clear that the total resistance estimations at model-scale overlap measured RT for the FDS-5 model to a high extent. Most notably is that the difference in the graphs become more apparent at the end of the speed interval (Fn > 1.0), however still closely resemble the measured RT values for FDS-5. This observation is especially true for the Marin estimation. Figure 18b was created in an effort to highlight the relative difference of the total resistance for the Marin and Series-64 estimations compared to the FDS-5 estimation. The relative RT error shows that both the Marin series and Series-64 estimations perform the best at the middle of the speed interval (0.5 < Fn < 1.0), which is the semi-displacing region. In the middle of the speed interval both estimations perform consistently below 3.4% in error. Fur- thermore, at both ends of the speed interval the estimations lose accuracy, especially for Series-64 at low Froude numbers. The Series-64 estimation error is at highest 16.4% at about Fn = 0.35 and the largest error for Marin series estimation is −5.5% at about Fn = 0.14. Shown in Figure 15a and Figure 15b is also the CFD simulation that were performed on the FDS-5 hull on model-scale. The relative RT error shows that the CFD estimation is about −14% to −21%. The main reason for this rather large underestimation of the total resistance is probably due to a common phenomena called numerical ventilation [37][1]. In Figure 16b the estimated relative error for CR was plotted for Marin and Series-64. The same behaviour observed in Figure 15b is repeated, but the relative error is higher. Most notably is the high relative error for Series-64 at low froude numbers. In Figure 17 the resistance coefficient distribution in percent was plotted. This figure clearly illustrate what weight CF and CR have on the total CT at different speed. From Figure 17 it is clear that the frictional coefficient is dominant over the whole speed interval, which in turn means that the estimated CR values are allowed to deviate from the measured CR values without any larger punishment in the estimation of total resistance at model-scale.

Full-scale In Figure 18a the full-scale total resistance RT estimations have been plotted for FDS-5, Marin and Series-64. The graphs appear to overlap to a large extent, however at Fn > 1.0 the RT values differ slightly more. In Figure 18b relative RT error shows that both the Marin series and Series-64 estimations perform the best at the middle of the speed interval (0.5 < Fn < 1.0). In the middle of the speed interval both estimations perform consistently below 4% in error. Furthermore, at both ends of the speed interval the estimations lose accuracy, especially at the low Froude numbers. The Series-64 estimation error is at highest 22% at about Fn = 0.35 and the largest error for Marin series estimation is −9% at about Fn = 0.14. Worth mentioning is that the relative error for Marin is below 4% between 0.25 < Fn < 1.2, which could be viewed as very good. In Figure 19 the resistance coefficient distribution in percent was plotted for the full-scale FDS-5 estimation. This figure clearly illustrate what weight CF , CR and ∆CF have on the total CT at different speed. Firstly ∆CF /CT increases almost linearly over the speed range and is at the end almost 11%. Furthermore, now being at a full-scale ship the frictional coefficient becomes lower since the Reynolds number is higher. This result in that the CR/CT and CF /CT alternate in which term is the most dominant. Generally CR/CT is dominant at the middle of the speed interval (0.4 < Fn < 0.9) and CF /CT is dominant at both ends of the speed interval. This combination is helpful from the perspective of the estimated CR from the Marin series and Series-64 having the lowest accuracy at both ends of the speed interval. Figure 20b illustrate at at which effective power the desired speed is achieved for the FDS-5, Marin and Series-64 estimations. The typical pattern of very close estimations in the middle of the speed interval and some deviations at both ends of the speed interval is repeated. Figure 20b shows the relative speed error for Marin and Series-64 with the assumption that the FDS-5 speed estimation is accurate. The relative speed error for Marin is consistently below 5% and below 2% over the majority of the interval. The Series-64 relative speed error is at maximum

48 10.43% and below 3.3% over the majority of the interval.

The overall impression is that the FDS-5 can be suﬃciently accurately estimated both with measurement data from other models in the Marin series and with a diﬀerent series, in this case Series-64. Both full-scale RT estimations are below 4.1% over the middle of the speed interval. A bit surprisingly, the CR estimation from Series-64 was almost as good as the Marin estimation in the interval 0.5 < Fn < 1.0. However, the overall preference would be to use the Marin estimation, since it produces the most consistently accurate results over the entire speed interval. Furthermore, by trying to estimate a model with other models from the same series, less unknown variables are introduced.

9.3 JBC resistance estimations Firstly by observing Figure 22a it is clear that the model-scale total resistance estimations are fairly close to the measured RT for the JBC. Most notable is that the Galati estimation is the worst at the lowest speeds, but then approaches the JBC measurement values. Furthermore, the KVLCC2 estimation is almost completely parallel to the JBC measurement values. The KVLCC2 estimation are consistently lower than the JBC values, which indicate that the CR values have somewhat similar trends, but are lower in amplitude for the KVLCC2. The CFD simulation consistently have the least accurate estimate, except for the very lowest froude number. In an effort to highlight the relative difference between the measured and estimated RT values Figure 22b was created. The relative RT error clearly shows that the estimated error for the Galati vessel, initially is about 10%, but then rapidly decreases to about 2% at about Fn = 0.130 and holds low over the remainder of the speed interval. The KVLCC2 error is below 4% over its entire speed interval, the best performance is achieved between 0.12 < Fn < 0.140. Furthermore, the CFD is overall the simulation that appear to perform the worst. However, it is the only estimation that consistently gets better with an increasing Froude number. In Figure 23a the estimated relative error for the CR values have been plotted for the Galati, KVLCC2 and CFD simulation. The same behaviour as observed in Figure 22b is repeated, but the relative error is much higher. For the Galati vessel the relative CR error is initially as high as 587%, which should be viewed as a strong indication that the data from the Galati vessel is unsuitable to use when estimating the JBC. Furthermore, the KVLCC2 have a relative CR error around −40% over its whole speed range. However, this is rather expected since the KVLCC2 is a less bulky vessel compared to the JBC. In Figure 23b the resistance coefficient distribution in percent have have been plotted. This figure clearly illustrate what weight (1 + k)CF and CR have on the total CT at different speeds. From Figure 23b it is clear that the frictional coefficient is very dominant over the whole speed interval, which in turn result in that the CR values are allowed to deviate from the measured CR values without any larger punishment in the estimation of total resistance at model-scale.

In Figure 24a the full-scale total resistance RT estimations have been plotted for the JBC, Galati and KVLCC2. Notable is that the Galati vessel now clearly estimate the JBC’s RT values poorly at both ends of the speed interval. This was not as evident in Figure 22b. This is due to that the Reynolds number have increased. Figure 24b shows that the relative RT error have increased for both the Galati and the KVLCC2 compared to the model-scale relative RT error. Furthermore, for Galati at Fn = 0.12 at model-scale the RT error was about 10%, however at full-scale it is about 20%. The KVLCC2 relative RT error consistently holds just under −5% over its entire speed interval. In Figure 25 the resistance coeﬃcient distribution in percent have have been plotted for the full-scale JBC estimation. This ﬁgure visualize what weight CF , CR, ∆CF and CA have on the total CT over the entire speed interval. Firstly ∆CF /CT increases almost linearly over the speed range and is at maximum about 5.5%. The CA/CT decreases linearly over the speed interval and is at maximum about 7%. The CR/CT appear to increase exponentially and the (1 + k)CF /CT appear to decrease exponentially. The (1 + k)CF /CT is at its minimum 77.5%, which clearly makes it the dominant resistance over the entire speed interval. By comparing the (1 + k)CF /CT at model-scale and full-scale, it can be concluded that the percentage have lowered with approximately 20% over the entire speed range. Even though neither of Galati and KVLCC2 appear to be well suited for predicting the JBC total resistance, the results are not so bad since the large majority of the total resistance come from form and frictional

49 resistance. Figure 26b illustrate at at which resistance the desired speed is achieved for the JBC, Galati and KVLCC2 estimations. Unsurprisingly, the KVLCC2 estimations performs overall more consistently and the Galati performs the best at the middle of the interval. Figure 26b shows the relative speed error for the Galati and KVLCC2 with the assumption that the JBC speed estimation is accurate. Once again the Galati vessel performs poorly at the ends of the interval and good in the middle. And the KVLCC2 holds consistently just under 3% over the entire interval.

The overall impression is that the JBC can be estimated with the KVLCC2 vessel at a very early design stage. The CFD simulation of the JBC did not perform as well as the other estimation, however the simulation model could probably be calibrated to produce reliable and consistent result for bulky ships, if given enough time. The Galati vessel should purely be considered as a type example of when the parameters correlate well with the desired vessel, but the data does not resemble the expected trend at all. One should be expected to know if the ITTC form factor prediction method can be utilized or not on the desired vessel. If the data one would like to use when estimating the desired vessel does not allow for ITTC form factor prediction method, then the data should not be advised to be used. The reason for the Galati CR data having an unexpected appearances could potentially be due to the effect of the bulbous bow. Covered in Section 4.7.2 was the potentially disturbing effect from modern bulbous bows. Another reason for the CR data behaving the way it does may be due to that the Galati vessel have a large enough transom stern that causes this, however this seems more unlikely. Naturally a vessel more closely resembling the JBC would be preferable to use when performing estimation predictions, however finding closer matching vessel proved to be unfruitful. From the literature study not that many extremely bulky vessels with a bulbous bow similar to the JBC, appear to have been tested at all. The element of a bulbous bow, clearly appear to add complexity to the task of estimating the total resistance of an unknown bulbous ship.

9.4 Parameter study The parameter study was carried out in order get a deeper understanding of how different parameters affect the CR data when being systematically varied. Given that the series encountered when compiling a database of transom stern hulls with CR data, used different parameters to vary, it is of interest to investigate which parameters carry the most importance. Due to that the slenderness ratio λ and the length-beam ratio L/B when observing the series appear to carry the most importance, but at the same time being strongly related, it is interesting to asses the relation and correlation between the parameters. Therefore, the overall trend of the database with respect to the λ vs L/B was plotted in Figure 27. The data appear to be normally distributed along a linear line and a linear regression was therefore added. This linear distribution is useful to be aware of, especially if one of the parameters would be unknown. Furthermore the population Pear- son correlation coefficient was calculated to be ρX,Y = 0.8565, which means that the slenderness ratio and length-beam ratio are linearly related to a degree of 0.8565. It should be noted that 1 or -1 indicate complete linearity, which means that 0.8565 indicate a significant correlation.

Table9 show the tabulated importance of each parameter and its standard deviation. Only pairs between files within the same series were allowed to be created. It should be noted that this study was primarily performed in respect to the four parameters, λ, B/T , CP and CB. An attempt to expand the study with L/B to include five parameters in total was made, however not enough pairs could be created in order to allow for a fair assessment. The most obvious thing to note from Table9 is that λ clearly is the dominant parameter. Overall Xλ = 2.447 % which is significantly higher than any other parameter. Varying the B/T ratio appear to have the smallest impact on the CR value, on average only XB/T = 0.210. However in the database the B/T ratio varies between values 1.5 - 7.324, so the B/T parameter can not be exempted from being one of the parameters when defining a hull. The relevance of varying CP have an average value of

XCP = 0.749. The effect of varying CP is therefore significant, however all semi-displacing ships encountered when collecting the database have a prismatic coefficient between approximately 0.561 - 0.74 and the majority of the ships in the database are very close to CP =0.63, simply because many of the series have viewed it as an optimal value. So even though CP indeed is an important parameter, it should not be expected

50 to vary that much when making ship resistance predictions for semi-displacing ships. One drawback with Table9 is that only three pairs could be created when varying CB and CP .

In an effort to somewhat remedy this issue, pairing between the series was allowed with 0.04% allowance. This means that three parameters must match within 0.04% of one another between the files. This naturally introduces inaccuracies, since the hull forms between the series are not identical, the secondary parameters (LCB and AT /AM ) are not identical and the test themselves have been conducted under slightly different conditions. Furthermore, pairs between the Marin series and Series-64 was not allowed, since for some hull files the CR data was significantly different at low froude numbers (Fn < 0.25), but then as expected at higher speeds. Including the Marin and Series-64 pairs appeared to somewhat skew the results in a non proportional amount. The reason for this difference could potentially be due to that the wetted area at the transom stern is significantly different between some of the models in the series or that the Series-64 was tested without turbulence tripping. Table 10 shows the results from making the parameter study with an allowance of 0.04%. All values appear to have increased somewhat, which is expected since pairing between models from different series were allowed. The λ and B/T appear to have almost the same relevance as before. The prismatic coefficient have increased to average XCP = 1.045. The most notable change is the importance of CB, the XCB have almost increased threefold compared to Table9. This makes it hard to draw any definite conclusion in regards to varying the CB parameter. The best would naturally be if Table 9 had more than 3 pairs when varying CB. However what could be said is that the CB most appear to be the third most important parameter out of the four. Its relevance is most likely somewhere between 0.342 to 0.985.

Table 11 shows a parameter study of varying L/B, B/T , CP and CB. Unsurprisingly varying L/B effect the CR data the most, however it carry less importance than λ did in both Table9 and Table 10. On the other hand XB/T and XCB have increased, are very close in value to each other and are both just above 1. Stated in the parametric study [6], was that varying B/T effected the resistance the least out of L/B, B/T , CP and CB. In Table 11 the result on the other hand indicate that varying B/T and CB have almost the same importance. The CP have the same importance as in Table9, this is due to that the exactly the same files were used for making the three pairs.

In an effort to visualize how varying the parameters λ, B/T , CP and CB affect the behaviour of the residual resistance coefficient CR several figures were made. These figures contain either four or three CR data measurements that illustrate how varying each respective parameter systematically can effect the CR data. In all the figures as many of the parameters except from the one that is supposed to vary were tried to be held as constant as possible.

Firstly Figure 28 shows how varying the slenderness ratio affect the CR data. The obvious conclusion that can be drawn is that a higher λ result in a lower CR. Another thing worth noting is that, i.e varying λ = 6.27 to λ = 9.50 result in approximately a threefold decrease in resistance. One can conclude that the earlier mentioned relation in Section 4.8.3, that a higher slenderness result in a lower CR data clearly can be observed in this parameter study. Furthermore, varying λ = 6.27 to 7.41 compared to varying λ = 7.41 to 8.50 result in a significantly higher absolute difference. The conclusion that can be made is that changing the slenderness ratio at lower values result in a larger absolute change in CR value compared to changing the slenderness ratio at higher values.

In order to find three similar files to illustrate how varying the CB affect the CR data, the λ was allowed to vary slightly between them. From Figure 29 the following can be observed. A higher CB results in higher CR data. Looking at Equation 28 which describes the block coefficient it is clear that a higher volume displacement must occur if the parameters B, T and L are fixed. Travelling through the water and displacing more volume naturally require more energy, which in this case means that the CR data increases when the block coefficient increases. Furthermore, using CB =0.35 appear in this specific case to completely have cancelled

51 out the ﬁrst hump at about Fn = 0.3. Instead the CB =0.35 graph have only one elongated hump.

The prismatic coefficient CP had very few pairs. The reason for this is simply that the series collected in this database for transom sterns hold the prismatic coefficient fixed and vary other parameters i.e Cb, B/T , L/B or λ. Naturally, more pairs would be desirable in order to asses the prismatic coefficient to a further extent. Figure 30 shows three models with varying prismatic coefficient from the Marin series, FDS-21 with CP = 0.561, FDS-22 with CP = 0.685 and FDS-5 with CP = 0.626. The situation for comparing these three models is ideal from the perspective of having virtually exactly the same dimensions and values between the models expect for the prismatic coefficient. From the book covering the large majority of the work concluded in relation to the Marin series, it was concluded that CP =0.626 appeared to be the optimal choice simply because of its overall superiority in resistance performance, in the designed speed range 0.5 < Fn <1.0 [6]. A higher CP appear in this case to result in a CR data with higher and lower peaks. A lower CP appear to almost over the entire speed interval result in higher CR data. When choosing CP =0.626 an optimal value appear to be achieved almost over the whole interval, except for the very lowest and highest froude numbers. The conclusion that can be made is that the CP does indeed have a noticeable effect on the CR data, most prominently on how the curve is shaped rather than the absolute CR value.

From both Table9 and 10 it was shown that B/T appeared to have the lowest effect on the CR data. Looking at Figure 31a and 31b this indeed appear to be the case. Noticeable is that Figure 31a and 31b show different behaviour when B/T is being varied. For Figure 31a a higher B/T result in a lower CR curve, but in Figure 31b the three curves with different B/T intersect each other multiple of times. The reason for the the curves intersecting each other could be due to that the L/B of the files in Figure 31b vary between 3.33 to 4.54, which indeed should carry enough effect in order to skew the results. In the parametric study conducted in [6] it was consistently observed that a higher B/T result in a lower CR data and a lower B/T result in a higher CR data.

Overall varying a parameter either lowers the curve or increases it over the entire speed range. The exception from this appear to be the parameter CP , in this case the shape of the curve appear to vary more than the absolute CR value. When varying a parameter the strongest effect is usually noticed around the humps at Fn = 0.3 and Fn = 0.5, they either become more pronounced, the first hump could disappear or one elongated hump could be formed. Variations of the parameters near the ends of their intervals appear to result in a lower effect, i.e varying the slenderness ratio at higher numbers i.e between 8 to 9, have a smaller effect than varying the slenderness ratio between 4 to 5. Varying B/T for different scenarios can result in different behaviour. The reason for this appear to be that at low λ the L/B can change enough to skew the result in a somewhat non-predictive manner. This non-predictive behaviour was not observed in the parametric study in [6]. From the parameter study conducted in this work the conclusion that can be made is that primarily the L/B, B/T , CP and CB appear to be slightly more useful as primary main parameters. The reason for this is simply that varying these parameters appear to consistently behave according to the expected result. Naturally, the slenderness ratio is useful as well, however primarily varying L/B, B/T , CP and CB appear to be more practical. Of course as many of the parameters defining a hull should be matched with what is desired when possible, however primarily considering L/B, B/T , CP and CB appear to a higher degree exclude the possibility of making very wrong predictions. Especially where the L/B = 3-5, which was the case in Figure 31b.

9.5 Adaptation of the ITTC-78 performance prediction method It is evidently clear that the ITTC-78 performance method is flawed in multiple of ways. Dividing the total resistance into sub-components accurately is notoriously difficult. Determining a form factor accurately at low speeds is difficult and then utilising it over the entire speed range is fundamentally incorrect. The form factor tend to change with varying speed. Furthermore, the roughness allowance ∆CF is approximate and out-dated for hulls with modern hull coating. This also leads to the CA potentially being out-dated and in need of a rework. Therefore, when and if a new proposed formula for ∆CF and CA is introduced, the

52 adaptation of those should deﬁnitely be considered. Despite all this, the ITTC-78 formula is still the recommended and the best procedure available.

It should be mentioned that when utilising the ITTC-78 performance prediction method the more details available the better. I.e knowing the water density, temperature and the experimental testing error, from the model test where the used residual resistance coefficient data come from is paramount in order to estimate the total error of the estimation. When estimating a large ship with a bulbous bow, extra consideration have to be taken into account of how the bulbous bow actually affect the flow around the hull and in the end what residual resistance coefficient data that seems reasonable. Observed in Section 8.3 was that the viscous form - and frictional resistance was dominant for the JBC vessel, however in order to achieve better accuracy the effect of the bulbous bow needs to be properly considered.

9.6 Complimenting discussion to CFD and semi-empirical co-usage Even though the CFD simulations are a bit off desired accuracy the technique still shows great potential. The software OpenFOAM and the extension package Naval Hydro Pack, was simply very recently introduced as a tool available at FOI and are therefore in need of a calibration process before it can yield reliable, accurate and consistent results. As can be seen in Section 4.6, the ITTC-78 performance prediction method contains many terms that separately can be updated when more data is available. This is something the CFD simulations can be very useful for. One usage could be to exactly calculate the air resistance originating from everything above waterline at the full-scale ship. I.e if the approximate position and shape of the structure to be mounted on the deck of the ship are known, then use CFD simulations to obtain an as accurate result as possible. Another usage could be to separately calculate a high fidelity estimate of the resistance originating from all appendages such as skeg, anode, fins, rudder and so on. Calculating a dynamic wetted surface would also further improve the accuracy of semi-empirical models. When the numerical ventilation issue have been handled properly, then larger model simulations series with varying scale could be conducted on the hulls of interest. Varying the model simulation size would possibly enable a derivation of a function that allows for scaling the ship to full-scale. It would also help in the assistance to derive a dynamic form factor to be used. In the end when the CFD have come to the point that it is deemed as reliable, consistent and more computational power is available, then complete full-scale ship series could be calculated, which would be a great asset. The usage the CFD simulations can have from the traditional ship prospecting process is probably the model series measurement data. The CFD is in need of high quality measured data to be bench-marked against, therefore precisely performed model series with detailed reports are needed. This would preferably entail precisely specified water density, temperature, the trim and sinkage of the models, center of gravity (COG) of the models and so on. Furthermore, when calculating appendages of different sorts, it could be useful to have a database on such measurements. This would allow for the CFD simulations to be bench-marked against actual measurement data, during the process of calibrating the CFD simulations for such appendages.

9.7 Hull geometry By creating several ﬁles with various resolution of the FDS-5 and JBC models an estimation of the process accuracy could be achieved. The study in itself was quite small, but gives an indication to what accuracy one can expect. Since CAD drawings was available for both FDS-5 and JBC, as well as all their key parameters already being well known, the situation was ideal to benchmark the process against already known data.

9.7.1 FDS-5 geometry

Looking at Table 12 it is clear that the estimation of LWL becomes better with an increasing resolution. The draught T also become better with an increasing resolution, especially when the number of points per

53 rib increases. The wetted surface does not show any clear trend, however the error is below 0.9% for all files. An interesting observation is that the worst estimation of the wetted surface is achieved at the highest resolution. It should be mentioned that in Section 6.1 it was noted that the actual model error, meaning the hull form, draught and loading condition is about 0.8 %. So the comparison and the error when comparing measured surface area on the actual model and the calculated surface area from the CAD geometry could differ slightly. The four files were the points per rib is 33 result in the best and almost equal performance. The key observation that can be drawn is that it appear to be preferable to increase the number of points per rib instead of number of ribs.

9.7.2 JBC geometry Looking at Table 13 it hard to observe any clear trends in regards to accuracy when the resolution varies. Noticeable is that the surface error improvement only is 0.15% if the total number of points is changed between 450 points to 24552 points. This is most likely due to that the form of the hull is very blocky. Worth noting is that the error for LWL consistently is around 1.3 - 2.0 %. A potential explanation to this could be that the ShipDes ﬁnds it diﬃcult to determine the intersection point at the bulbous bow or that the inputted LCB is slightly incorrect.

The overall impression is that the process and ShipDes program result in suﬃciently accurate estimations.

54 10 Conclusions

The FDS-5 can be linearly interpolated and estimated by hulls within its own series as well as from hulls outside of the Marin series. When making a linear interpolation between hulls, great care should be taken in order to ensure that all main parameters and secondary parameters are very similar to the desired hull. One single linear interpolation appear to be able to give a slightly better performance compared to a regression analysis. However, the great advantages with using regression functions is that the data have been carefully processed and that it is easy to use.

The resistance estimations performed for both FDS-5 and JBC both resulted in suﬃciently accurate results to be used for an early design stage. The Marin estimation gave a RT error below 4% for the full-scale between 0.25 < Fn < 1.2, which should be viewed as good. Even though the CR estimate made with KVLCC2 was about −40% wrong over the whole interval, the RT full-scale estimate was only about −5% wrong. This was due to the strong dominance of the viscous and form drag at such low froude numbers.

The parameter study confirmed that the λ and L/B are strongly correlated and that their trend can be linearized. The parametric study resulted in parameter importance in falling order: Slenderness ratio λ, length-beam ratio L/B, longitudinal prismatic coefficient CP , block coefficient CB and beam-draught B/T . The parametric study indicate that L/B, B/T , CP and CB appear to be slightly more practical and useful as primary main parameters. Varying these parameters appear to consistently yield the expected behaviour. Naturally as many as possible of the parameters defining a hull should be matched with what is desired when possible, however primarily considering L/B, B/T , CP and CB appear to a higher degree exclude the possibility of making very wrong predictions.

Calculations of the main dimensions and parameters appear to give suﬃciently good result by using the Rhinoceros 6 program in combination with ShipDes. Calculating the wetted surface S and draught T below 1 % should be viewed as satisfactory. The improvements that could be done, is probably better handling of hulls with a more complex geometry, i.e a ship with a bulbous bow.

When using the ITTC-78 performance prediction method, multiple of careful consideration have to be made if the best result is to be achieved. As earlier mentioned the method in itself have multiple of flaws, however is still considered to be the best option available at the moment. The more information about the ship that is being investigated the better. This would entail all describing parameters λ, L/B, B/T , CP , CB, LCB and AT /AM . Also how and under what conditions the model test was conducted. Furthermore, if for example details such as which size, position and shape of a fin being mounted to the hull is known, then Table1 might not be ideal. Then it could be better to investigate if this specific case have been studied and use that appendage resistance instead. The possibility of incrementally improving an estimation is always present, ultimately the amount of information and time are the governing factor in regards to how precise an estimate can become.

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58 A Appendix A.1 FDS-5 model resistance

Table 14: Measurement values for RT , CT , CR, CF , SAP P and SFPP at diﬀerent speeds. SAP P and SFPP is the sinkage at aft and forward perpendicular, both positive for displacement downwards. The table is based on tabulated values in [6]

3 3 3 Froude nr [-] Speed [m/s] RT [N] CT ·10 [-] CR ·10 [-] CF ·10 [-] SAP P [mm] SFPP [mm] 0.143 1.000 6.982 4.7733 1.3156 3.4577 2 2.5 0.214 1.502 15.985 4.8438 1.6342 3.2096 4 1.5 0.285 2.000 27.920 4.7716 1.7210 3.0506 7 0.5 0.356 2.496 41.286 4.5304 1.5948 2.9356 15 4.4 0.427 2.992 67.068 5.1216 2.2754 2.8463 - - 0.500 3.502 95.262 5.3101 2.5382 2.7719 54 -22.2 0.571 4.002 117.533 5.0168 2.3056 2.7112 53 -44.4 0.642 4.502 138.195 4.6612 2.0020 2.6592 54 -53.0 0.714 5.002 162.045 4.4276 1.8136 2.6140 54 -55.0 0.786 5.509 187.758 4.2293 1.6558 2.5736 55 -57.8 0.858 6.012 216.227 4.0897 1.5519 2.5378 57 -60.6 0.927 6.499 244.342 3.9548 1.4483 2.5065 58 -65.4 1.000 7.006 276.391 3.8495 1.3726 2.4768 59 -70.2 1.071 7.502 312.479 3.7957 1.3453 2.4503 59 -71.2 1.142 8.000 352.216 3.7623 1.3365 2.4258 57 -69.3 1.213 8.503 399.101 3.7736 1.3708 2.4028 58 -75.1 1.256 8.803 460.893 4.0659 1.6760 2.3899 60 -79.8 1.260 8.830 461.795 4.0490 1.6602 2.3888 - - 1.267 8.876 454.823 3.9466 1.5598 2.3869 60 -80.0

59 A.2 JBC model resistance

Table 15: RT , CT , CR, CF at diﬀerent speeds. The table have been calculated from information given in [19].

3 3 3 Froude nr [-] Speed [m/s] RT [N] CT ·10 [-] CF ·10 [-] CR ·10 [-] 0.1199 0.993 26.265 4.365 3.258 0.084 0.1200 0.994 26.294 4.364 3.257 0.084 0.1229 1.018 27.531 4.352 3.243 0.091 0.1266 1.049 29.091 4.340 3.226 0.101 0.1293 1.071 30.308 4.331 3.214 0.108 0.1294 1.072 30.367 4.331 3.213 0.109 0.1325 1.097 31.768 4.320 3.199 0.117 0.1356 1.123 33.221 4.315 3.186 0.129 0.1357 1.124 33.254 4.315 3.186 0.129 0.1396 1.156 35.125 4.309 3.170 0.144 0.1425 1.180 36.623 4.309 3.158 0.159 0.1426 1.181 36.659 4.309 3.158 0.159 0.1452 1.203 38.000 4.309 3.148 0.173 0.1473 1.220 39.184 4.312 3.139 0.187 0.1496 1.239 40.432 4.317 3.131 0.203 0.1497 1.240 40.470 4.317 3.131 0.203 0.1518 1.260 41.686 4.324 3.123 0.220 0.1519 1.258 41.762 4.325 3.123 0.221 0.1538 1.274 42.872 4.331 3.116 0.237 0.1540 1.276 42.988 4.332 3.115 0.239 0.1556 1.289 43.982 4.341 3.109 0.256 0.1559 1.291 44.181 4.343 3.108 0.259 0.1569 1.300 44.829 4.350 3.105 0.270 0.1574 1.304 45.162 4.353 3.103 0.276 0.1582 1.310 45.663 4.358 3.100 0.285 0.1599 1.324 46.766 4.370 3.094 0.305

Table 15 have been created by extrapolating values from a graph representing CR data over the speed range given in [19]. By using the CR values and corresponding speeds the other values have been calculated. The remaining values in Table 15 have been calculated by following the ITTC-78 performance prediction method described in Section 4.6. This method was also speciﬁed to have been followed in [19].