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Electric Boat Hull Resistance Methods

This document summarizes a degree project that evaluated methods for predicting hull resistance in the early design of electric boats. The project applied computational fluid dynamics (CFD), Holtrop & Mennen, Savitsky, and model test methods to predict resistance of a test hull. CFD and Savitsky provided the most accurate resistance predictions, with Savitsky recommended for planing speeds and CFD for lower speeds where Savitsky is less applicable. The results aim to help electric boat companies like X Shore systematically predict resistance early in the design process to decrease energy consumption.

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0% found this document useful (0 votes)

141 views74 pages

Electric Boat Hull Resistance Methods

Uploaded by

Sri Nurhumairoh

We take content rights seriously. If you suspect this is your content, claim it here.

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DEGREE PROJECT IN MECHANICAL ENGINEERING,

SECOND CYCLE, 30 CREDITS

STOCKHOLM, SWEDEN 2020

Methods to Predict Hull Resistance

in the Process of Designing
Electric Boats

ELIN LINDBERGH

FELICIA AHLSTRAND

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ENGINEERING SCIENCES
Methods to Predict Hull Resistance in the Process of Designing Electric Boats
Swedish title: Metoder för att Uppskatta Skrovmotståndet i Designprocessen för Elektriska Båtar

ELIN LINDBERGH, FELICIA AHLSTRAND

TRITA: TRITA-SCI-GRU 2020:185

Degree Project in Naval Architecture, Second Cycle, 30 credits

Course SD271X
Stockholm, Sweden, 2020

Supervisors: Hans Liwång, KTH

Johannes Roselius, X Shore
Examiner: Jakob Kuttenkeuler, KTH

Center for Naval Architecture

School of Engineering Sciences
KTH Royal Institute of Technology
SE-100 44, Stockholm
Sweden
Telephone: +46 8 790 60 00

Host company: X Shore

Abstract
Combustion engines in boats cause several environmental problems, such as greenhouse gas emis-
sions and acidification of oceans. Most of these problems can be reduced by replacing the com-
bustion engines with electric boats. The limited range is one of the main constraints for electric
boats, and in order to decrease the energy consumption, applicable resistance prediction methods
are necessary in the hull design process. X Shore, which is a start-up company in the electric boat
sector, lacks a systematic way of predicting resistance in an early design phase.

In this study, four well-known methods - CFD, Holtrop & Mennen, the Savitsky method and
model test - have been applied in order to predict resistance for a test hull. The study is limited
to bare hull resistance and calm water conditions. CFD simulations are applied using the software
ANSYS FLUENT 19.0. The simulations were based on the Reynolds Average Navier-Stokes equa-
tions with SST k-ω as turbulence model together with the volume of fluid method describing the
two-phased flow of both water and air surrounding the hull. The semi-empirical methods, Holtrop
& Mennen and the Savitsky method, are applied through a program in Python 3, developed by
the authors. The results from each method have been compared and since model tests have been
conducted outside of this study, the model test results will serve as reference. To evaluate the
methods, a number of evaluation criteria are identified and evaluated through a Pugh Matrix, a
systems engineering tool.

Holtrop & Mennen predicts the resistance with low accuracy and consistency, and the error varies
between 2.2% and 70.6%. The CFD simulations result in acceptable resistance predictions with
good precision for the speeds 4 − 6 knots, with an average deviation of the absolute values as
12.28% which is slightly higher than the errors found in previous studies. However, the method
shows inconsistency for the higher speeds where the deviation varies between 1.77% and −43.39%.
The Savitsky method predicts accurate results with good precision for planing speeds, but also for
the speeds 7 and 8 knots. The method is under-predicting the resistance for all speeds except for
7 knots, where the total resistance is 10.7% higher than for model tests. In the speed range 8 − 32
knots, the average error is an under-estimation of 17.58%. Furthermore, the trim angles predicted
by the Savitsky method correspond well with the trim angles from the model test.

In conclusion, the recommendation to X Shore is to apply the Savitsky method when its ap-
plicability criteria are fulfilled, and CFD for the lowest speeds, where the Savitsky method is not
applicable.
Sammanfattning
Förbränningsmotorer i båtar orsakar flera miljöproblem, som exempelvis utsläpp av växthusgaser
och försurning av hav. De flesta av dessa problem kan minskas genom att ersätta båtar med
förbränningsmotorer med elbåtar. Den begränsade körsträckan är en av de största begränsningarna
för elbåtar, och för att minska energiförbrukningen behövs metoder för att uppskatta motståndet
under designstadiet. X Shore, ett startup-företag i elbåtsbranchen, saknar ett systematiskt tillväga-
gångssätt för att uppskatta motstånd i tidiga skeden i designprocessen.

I den här studien har fyra välkända metoder - CFD, Holtrop & Mennen, Savitsky-metoden och
modelltester - applicerats för att uppskatta motståndet hos ett testskrov. Studien är begränsad till
ett skrov utan bihang och lugnvattenmotstånd. CFD-simuleringar har gjorts i mjukvaran ANSYS
FLUENT 19.0. Simuleringarna är baserade på Reynolds Average Navier-Stokes ekvationer och tur-
bulensmodellen SST k − ω har använts tillsammans med metoden volume of fluid som beskriver
flödet av både vatten och luft runt skrovet. De semi-empiriska metoderna, Holtrop & Mennen och
Savitsky-metoden, har applicerats genom ett program i Python 3 som utvecklats av författarna.
Resultaten från alla metoder har jämförts, och eftersom modelltester genomförts på detta skrov
tidigare har de resultaten använts som referensvärden. Ett antal kriterier har identifierats och en
Pugh-matris har använts för utvärdering av metoderna.

Holtrop & Mennen uppskattar motståndet med låg noggrannhet och precision, felen varierar mel-
lan 2.2% och 70.6%. CFD-simuleringarna ger acceptabla resultat av motståndsberäkningarna för
hastigheterna 4 − 6 knop, med ett genomsnittligt absolut fel på 12.28% vilket är något högre än
avvikelserna presenterade i tidigare studier. För högre hastigheter uppvisar metoden lägre pre-
cision där avvikelsen varierar mellan 1.77% och −43.39%. Savitsky-metoden ger resultat med
hög noggrannhet och god precision för planingshastigheter, men även för hastigheterna 7 och 8
knop. Metoden underskattar motståndet för alla hastigheter förutom för 7 knop där motståndet
är 10.7% högre än för modelltesterna. I hastighetsintervallet 8 − 32 knop är det genomsnittliga
felet en underskattning på 17.58%. Vidare överensstämmer trimvinkeln från Savitsky-metoden bra
med resultaten från modelltesterna.

Sammanfattningsvis rekommenderas X Shore att använda Savitsky-metoden när dess kriterier

för tillämplighet är uppfyllda och CFD för de lägsta hastigheterna när Savitsky-metoden inte är
tillämpbar.
Acknowledgements
This Master’s thesis is the final project of the Master’s program in Naval Architecture at KTH
Royal Institute of Technology and has been carried out at X Shore. We would like to thank the
entire X Shore team, and especially our supervisor Johannes Roselius, for helping us, giving us
advice and making us feel welcome at the company. Secondly, we want to thank Adam Persson at
SSPA for giving us useful advice regarding CFD simulation of planing hulls.

We would also like to express our gratitude to our supervisor at KTH, Hans Liwång, for con-
tinuous feedback and support during this semester. Finally, special thanks to Matz Ahlstrand for
providing us with a computer that made it possible to finalize our CFD simulations, and to Michael
Dellstad for the never-ending IT support.

Thank you.

Elin Lindbergh and Felicia Ahlstrand

Stockholm, June 2020
Work distribution
The authors have contributed equally to the project and worked both independently and in collab-
oration. The authors have shared the responsibility for the report and its structure and content.
Both authors have contributed equally to the sections: 1.2 Purpose and objectives, 1.3 Scope, 1.4
Method, 1.5 Report structure, 2. Ship theory, 5. Evaluation of methods, 6.5 Comparison, 6.6
Evaluation, 7. Discussion and Conclusion and 8. Future work.

Elin Lindbergh has written section 1. Introduction, 1.1 Background, 3. Semi-empirical and
empirical methods for ship design and 6.1 Results from previous studies. Elin Lindbergh was
responsible for the Savitsky method, including developing the Python script, determining input
values, obtaining results and writing section 6.3 Savitsky in the report and associated appendices.
Elin Lindbergh converted all Python scripts in this project to executable programs and has also
performed CFD simulations.

Felicia Ahlstrand was responsible for Holtrop & Mennen, including developing the Python script,
determining input values, obtaining results and writing section 6.2 Holtrop & Mennen in the report
and associated appendices. Felicia Ahlstrand has performed most of the CFD simulations and has
written section 4. Computational fluid dynamics for ship design and 6.4 CFD.
Contents
Acronyms i

Nomenclature ii

1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 The design process at X Shore . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Methods to predict resistance . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Purpose and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Report structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Ship theory 5
2.1 Hull resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Semi-empirical and empirical methods for ship design 9

3.1 Holtrop & Mennen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Savitsky method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 Model tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Computational fluid dynamics for ship design 12

4.1 Geometry and computational domain . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.1.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2.1 Boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.3 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.4 Two-phased flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.4.1 Numerical ventilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.5 Fluid-structure interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.6 Dynamic mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.7 Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.8 Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.9 Visualisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Evaluation of methods 19
5.1 Evaluation tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 Results 21
6.1 Results from previous studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.2 Holtrop & Mennen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.2.1 How to apply Holtrop & Mennen . . . . . . . . . . . . . . . . . . . . . . . . 22
6.2.2 Holtrop & Mennen applied on Eelex 2020 . . . . . . . . . . . . . . . . . . . 23
6.3 Savitsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.3.1 How to apply the Savitsky method . . . . . . . . . . . . . . . . . . . . . . . 26
6.3.2 The Savitsky method applied on Eelex 2020 . . . . . . . . . . . . . . . . . . 27
6.4 CFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.4.1 Assumptions and limitations . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.4.2 Geometry and computational domain . . . . . . . . . . . . . . . . . . . . . 29
6.4.3 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.4.4 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.4.5 Mesh convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.4.6 Results from the CFD simulations . . . . . . . . . . . . . . . . . . . . . . . 31
6.5 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6.5.1 Resistance predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6.5.2 Power predictions and efficiency . . . . . . . . . . . . . . . . . . . . . . . . 38
6.6 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

7 Discussion and Conclusion 42

7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

8 Future work 46

References 48

A Holtrop & Mennen I

B Savitsky method VII

C Program interface for Holtrop & Mennen X

D Program interface for the Savitsky method XI

E Results XII

F Modifying and combining methods XIII

Acronyms

Acronyms Expansions
CAD Computer aided design
CFD Computational fluid dynamics
DNS Direct numerical simulation
DOF Degrees of freedom
HIRC High resolution interface capturing scheme
LCB Longitudinal center of buoyancy
LCG Longitudinal center of gravity
NM Nautical miles
RANS Reynolds Average Navier-Stokes
SST Shear stress transport
UDF User defined function
VCG Vertical center of gravity
VOF Volume of fluid

i
Nomenclature
Dimensionless Numbers
Symbol Description
(1 + k) Form factor

∆CF Roughness allowance coefficient

Cf Friction coefficient
CV Speed coefficient

CAA Air resistance coefficient

CF Viscous friction coefficient
CT Total resistance coefficient
CW Wave-making and wave-breaking resistance coefficient

Fn Froude number
ReL Reynold’s number based on the length between perpendiculars
ReΛ Reynold’s number based on the mean wetted length

Greek Symbols
Symbol Description
α Scale factor

β Deadrise angle
∆ Displacement mass
Angle between keel and propeller shaft
ηD Propulsion efficiency

ηG Gearbox efficiency
ηH Hull efficiency
ηM Motor efficiency
ηO Open water propeller efficiency

ηR Relative rotary efficiency

Λ Wetted length-beam ratio
∇ Displacement volume

ν Kinematic viscosity
ρ Density
τ Trim angle
τw Wall shear stress

ii
Roman Symbols
Symbol Description
∆R Resistance increase from propeller
∆t Time step
AM Midships cross sectional area

B Beam
Bm Model beam
D Draught at LP P /2
d Water depth

Dm Model draught
EB Battery capacity
f Distance between propeller shaft and center of gravity

FT Propeller thrust force

g Gravitational acceleration
kSM Sea margin
LC Wetted chine length

LK Wetted keel length

LOA Length overall
LP P m Model length between perpendiculars

LP P Length between perpendiculars

LW L Length on waterline
m Weight of the boat
PE Bare hull power

PM Motor power
PS Required power
r Range

RA Model-ship correlation resistance

RB Additional pressure resistance of bulbous bow
RF Frictional resistance
RT Total resistance

RW Wave-making and wave-breaking resistance

RAP P Resistance of appendages
RT R Additional pressure resistance of immersed transom stern

S Wetted surface

iii
td Thrust deduction factor

U Ship velocity
UA Incident propeller velocity
u∗ Friction velocity
UM Model velocity

W Towing tank width

w Wake fraction
y Distance
y+ Dimensionless wall distance

iv
1. Introduction
In 2018, Swedish leisure boats emitted 176 200 tonnes of carbon dioxide equivalent, which cor-
responds to approximately a third of the emissions from domestic flights in Sweden [1]. Sweden
aims to have net-zero greenhouse gas emissions by 2045 at the latest [2], and to meet that goal, a
transition from fossil fuels to more sustainable options in the boating sector is inevitable.

Besides the greenhouse gas emissions, boats with traditional combustion engines cause under-
water noise, which is known to have a negative impact on marine life, and emit other pollutants,
e.g. nitrogen and sulfur oxides that contribute to eutrophication and acidification [3]. By using
electric boats, both the emissions and underwater noise pollution can be reduced.

One of the main issues with electric boats is the range, which is limited by the battery capac-
ity. In order to increase the range there are two options: add more batteries, or decrease the
energy consumption when driving the boat. Since adding batteries increases both weight and cost,
lowering the energy consumption is the more desirable option. This can be achieved by designing
a hull with lower resistance.

1.1 Background
The process of designing a ship can be divided into three phases; concept design, contract design
and detailed design. How long time that is spent on each phase, as well as how deeply different
aspects are studied, depends on ship type and size and varies for each case. However, having a
systematic approach when designing a ship is of importance, since it ensures that no aspect is
forgotten or dealt with at the wrong time. [4] Moreover, it is important to have a clear priority of
the goals of the design. The ship owner must decide if the ship should be optimized with respect
to functionality, efficiency, maintenance, environment, or another design aspect, as well as the
importance of each aspect. [5]

The first phase, the concept design, is considered the most important phase. The naval archi-
tect, together with the project owner, will clarify the environment that the ship will operate in
based on the intended route. A basic design, including speed requirements, size and type of hull
will also be set to ensure that the owner’s requirements will be met. Another purpose of the
concept phase is to determine if it is possible to meet the owner’s objectives without exceeding
the budget. It is therefore important to spend a sufficient amount of time in this design stage, so
that a realistic cost estimation can be made and expensive changes in later phases can be avoided.
[4] This design phase is an iterative process, resulting in a chosen hull design that will be further
developed in the next phase [5]. Figure 1 presents Molland’s [5] flowchart of this stage.

Figure 1: The iterative process of the first design phase [5].

1
The next phase is the contract design. This phase can either be performed by the same engineers
as in the first phase, or by a shipbuilder or a third party. During this stage, performance character-
istics will be predicted, for example with CFD simulations. The systems and subsystems necessary
in order for the ship to function as intended are identified, sized and positioned. The layout of
the ship is determined, in such detail that it can be guaranteed that all necessary equipment and
functions fit and that the cargo capacity is sufficient. [4] Since most design aspects of a ship, e.g
dimensions, speed and stability, are dependent on each other, the contract design phase is an iter-
ative process [5]. Decisions that are made during this phase should be documented and it should
be controlled that the one who made the decision had the authority to do so [4].

The outputs from the contract design will be the input to the final phase, which is the detailed
design. The purpose of this phase is to provide the ship builder with enough information, and on
such a detailed level, so that the ship can be produced. Additionally, the detailed design should
specify the tests necessary in order to determine if the ship meets regulations and the owner’s
requirements. [4]

The traditional way of designing a ship is to base the design on an already existing ship of the
same type. The naval architect can then improve the design, based on known issues in the already
existing ship. Identifying differences between the new and the existing ship, such as an increased
need of cargo capacity or higher speed requirements, will also affect the design, as well as new
regulations that are to be met. If the new design deviates too much from already existing ship
types, the design process will be more iterative. With no existing ship to base the design on, it
is important to identify the ship’s functions, capabilities and attributes. The functions can be to
float and move, and the capabilities will ensure that the functions are achieved, for instance by the
capability to operate in a certain speed. The attributes are the features that the naval architect
designs into the ship in order to support the capabilities. Stability, manoeuvrability and strength
are examples of attributes, and several attributes might be necessary for one capability. [4]

1.1.1 The design process at X Shore

X Shore is a Swedish company that designs and manufactures fully electric leisure boats. The
current models in the X Shore family are the 6 m long Eelord and the 8 m long Eelex. X Shore em-
braces the long tradition of maritime craftsmanship and with new technology, innovative research,
smart design and sustainable materials their aim is to discard the usage of fossil fuels [6].

The design process at a company like X Shore is affected by the fact that it is a small start-
up company with limited resources and competence. The main design objectives for X Shore are
to combine sufficient range with the silence from an electric motor. As a start-up company, X
Shore is still in the progress of defining the typical X Shore hull, and consequently, major design
changes are made when designing new hulls. Hence, X Shore has no existing hulls to base the
designs on. The innovative and nontraditional design process result in a more iterative process
where the functions, capabilities and attributes need to be identified.

During the concept design, an external hull designer works closely together with a design team at
X Shore. The hull designer possesses valuable knowledge and experience, and the X Shore team
ensures that the brand identity is kept as well as requirements are met. The process is iterative,
with design reviews where the hull designer presents a draft design, and the X Shore team gives
feedback and suggests modifications.

In the contract design phase, engineers at X Shore identify, size and position necessary systems
and subsystems. Previously, external consultants have been involved to a small extent, in order
to predict performance characteristics. However, X Shore has no systematic way of predicting
performance and thus, no methods to evaluate the impact of design changes.

The detailed design is carried out by X Shore’s engineers. X Shore also have the competence
to specify what tests that need to be conducted and how to ensure that regulations and require-

2
ments are met. While both the concept design and detailed design work out well, there are some
deficiencies in the contract design regarding performance prediction. For future designs, X Shore
wants to develop their design process so that performance predictions can be made in-house to the
greatest extent possible.

As energy consumption is one of the main constraints for the range of the boats, one of the
biggest challenges when designing a new hull for the X Shore family is to reduce the resistance for
both displacing and planing speeds.

1.1.2 Methods to predict resistance

The resistance forces acting on a hull are caused by characteristics of both the water and the hull.
There are different approaches to predict the resistance in the design phase. Well-known methods
are computational fluid dynamics (CFD), model tests, the Savitsky method and Holtrop & Men-
nen. CFD is applied in most disciplines involving a flowing fluid, including the ship industry. The
tool is especially useful when there is a complex fluid flow involved [7]. Since the time and cost
required for CFD are lower than for model test, the use of the tool in ship design is increasing.
The difficulties with CFD in ship design have previously been to accurately model a free surface.
However, the models have improved significantly and the tool can now yield good results. [8]

Model tests are empirical tests using scale models in towing tanks. Model tests provide more
accurate results than CFD, but to a much higher cost [7]. The Savitsky method and Holtrop &
Mennen are semi-empirical methods for planing and displacement hulls, respectively. These two
methods are easily implemented [9] and less time and cost consuming in comparison to CFD and
model tests. The drawback is however that these two methods are based on simple hull shapes
and therefore restricted to shapes similar to these [10]. For more complex hull shapes another tool
is required.

1.2 Purpose and objectives

Since methods to predict performance characteristics are absent in X Shore’s design phase, the aim
of this project is to provide X Shore with a systematic approach to predict resistance of new hull
designs. Four well-known methods - CFD, Holtrop & Mennen, the Savitsky method and model
tests - will be evaluated and compared with each other in order to find the methods that are most
suitable for X Shore. To enable comparisons, a set of evaluation criteria will be identified. It will
also be clarified when and how to apply the suggested methods. As the goal for X Shore is to
lower the energy consumption, the results from the resistance predictions will be presented as the
corresponding range for each speed.

1.3 Scope
The study is limited to evaluating and performing CFD simulations, Holtrop & Mennen and the
Savitsky method on the test hull Eelex 2020 developed by X Shore, see figure 2.

Figure 2: Eelex 2020.

3
Model tests will be investigated, but not conducted. Instead, the results from the other methods
will be compared to existing results from model tests made on the test hull. To enable comparisons
between the methods, resistance predictions are made on a bare hull, meaning that no appendages,
e.g. propellers or rudders, are included. Moreover, the predictions are made for calm water
conditions. The main characteristics of Eelex 2020 are presented in table 1.

Table 1: Main characteristics of Eelex 2020.

Symbol Value Unit

Length overall LOA 7.955 m
Length between perpendiculars LP P 7.489 m
Breadth B 2.550 m
Draught at LP P /2 D 0.399 m
Weight m 2600 kg
Battery capacity EB 120 kWh

1.4 Method
Initially, a literature study is performed in order to get better understanding of the characteristics
that will be analyzed, the methods for resistance predictions and the guidelines on how to apply
these methods. To evaluate the methods, the methods are applied on Eelex 2020 and compared
considering a set of identified criteria. The evaluation and comparison study is the basis of the
conclusion regarding when and how to systematically apply the suggested approach.

1.5 Report structure

In chapter 2, planing hull theory and the resistance forces acting on a hull are explained together
with theory regarding power requirements and efficiency. The semi-empirical Savitsky method
and Holtrop & Mennen are presented in chapter 3. The empirical model test are presented in the
same chapter. Chapter 4 explains the theory behind CFD and guidelines in how to perform a
simulation on a hull. In chapter 5, the evaluation criteria are identified and the results presented
in chapter 6 contains of a step by step summary in how to apply the suggested methods followed
by an evaluation based on the identified criteria. Finally, when to apply the different approaches
are discussed and concluded in chapter 7, and suggestions for future work can be found in chapter
8.

4
2. Ship theory
The purpose of this chapter is to provide the reader with necessary ship theory. The first section
introduces the theory of hull resistance and planing hulls, and is followed by a section describing
efficiency and power requirements.

2.1 Hull resistance

According to Almeter [11] hulls can be divided into three different types: pre-planing, semi-planing
and fully planing hulls. The Froude number can be used as an indication of the type of a hull, and
is defined as
U
Fn = √ (1)
gLP P

where U is the speed of the boat, LP P is the length between perpendiculars and g is the gravi-
tational acceleration. While there is no clear limit [12], a general rule is that Fn < 0.4 indicate
a pre-planing hull, also known as displacement hull, and Fn > 1.0 indicate a fully planing hull.
Between 0.4 and 1.0 the hull is assumed to be semi-planing. [13] Thus, all hulls are displacing in
low speeds.

Figure 3 presents both a displacement and a planing hull and the main difference between these is
the pressure generated forces acting on the hull. There are two different types of pressures involved;
hydrostatic and hydrodynamic pressure, acting through a point called center of pressure. For a
displacement hull the hydrostatic pressure, i.e. buoyancy, is predominant [13]. The buoyancy is
equal to the weight of the volume of fluid that a hull displaces. As the speed increases, the impor-
tance of the buoyancy decreases and the hull is mainly supported by hydrodynamic pressure. The
hydrodynamic pressure is generated by the flow around the hull and is proportional to the speed
squared. [12] The vertical components of the hydrodynamic pressure lift the boat out of the water,
which results in planing, while the longitudinal components cause resistance [14].

Figure 3: The hull to the left is displacing, while the hull to the right has reached planing.

Resistance is a force acting on the hull in the opposite direction of the motion, i.e. in the horizontal
direction. The components of the total resistance can be divided in different ways, whereof one is
presented in figure 4. Wave resistance and viscous pressure resistance are caused by the longitudinal
components of the hydrodynamic pressure, and is thus increasing with the speed [14]. The viscous
friction, also called drag force, is caused by the friction between the hull and the water as the boat
is moving [15]. For low speeds, where the hull is mainly supported by hydrostatic pressure, the
total resistance is dominated by the frictional resistance. The pressure resistance increases with
increasing speeds, and wave resistance and friction resistance are dominating for planing hulls. [9]

5
Figure 4: The components of the total resistance.

The total resistance is defined as

1 2
RT = ρU CT S (2)
2

where ρ is the density of the fluid, U is the speed of the boat and S is the wetted surface of the
hull. CT is a dimensionless total resistance coefficient defined as

CT = (1 + k)CF + CW + ∆CF + CAA (3)

where (1 + k)CF is the viscous friction and form drag coefficient, CW is the wave-making and
wave-breaking resistance coefficient, ∆CF is the roughness allowance coefficient and CAA is the air
resistance coefficient. Resistance from appendages are not included in equations 2 and 3, but can
be added as separate terms. [9]

In order to predict resistance for a boat, variables that define the basic dimensions and loading
of the hull are of interest. According to Almeter [11] these variables are speed and displacement,
length and beam, deadrise angle and longitudinal center of gravity (LCG). The deadrise angle is the
angle between the keel and the horizontal plane, see figure 5a. A large deadrise angle reduces the
lift force, which in turn increases the wetted surface, resulting in higher resistance. Consequently,
a small deadrise is desirable in order to minimize the resistance. However, reducing the deadrise
angle results in increased slamming, which is undesirable. [14] Thus, determining the appropriate
deadrise when designing a hull is a balance between resistance and slamming forces.

(a) The deadrise angle β. (b) The trim angle τ .

Figure 5: The deadrise angle β is shown to the left and the trim angle τ is shown to the right.

LCG is the longitudinal position of the center of gravity in the boat, which affects the trim angle,
i.e. the angle between hull and the water surface, see figure 5b. The trim angle affects the wetted
surface and therefore also the resistance [14]. For displacement hulls and semi-planing hulls, a
typical position for the LCG is around 40% − 45% and 33% − 45% of the chine length forward of
the transom, respectively. Planing hulls, designed to be lifted up from the water to decrease the
displacement volume, tend to have the LCG placed at 25% − 35% of the chine length from the
transom. [11]

6
In displacement speeds, the draught and trim of the hull, i.e. the running attitude, is assumed to
be the same as in zero speed, since the hull is mainly supported by hydrostatic pressure and both
LCG and the center of pressure is constant. As the speed increases, the hydrodynamic pressure
will lead to a decrease in submerged volume and draught, and consequently, the center of pressure
will move. Since LCG is constant, the change of center of pressure will cause varying trim angles.
The running attitude influences the wetted surface, which in turn affect the resistance, and it
is therefore important to capture the effects from changes in trim and draught when calculating
resistance in semi-planing and planing speeds.

2.2 Efficiency
The power requirements for a ship in calm water can be expressed as
PE
PS = (4)
ηD

where PE is the bare hull power and ηD the propulsion efficiency. The propulsion efficiency can
be expressed as

ηD = ηO ηR ηH (5)

where ηO is the open water propeller efficiency, ηR is relative rotary efficiency, approximately equal
to 1. The hull efficiency, ηH , is defined as
1 − td
ηH = (6)
1−w

where td is the thrust deduction factor and w is the wake fraction:

∆R
td = (7)
FT
UA
w =1− (8)
U

∆R is the increased resistance due to the propeller, FT is the propeller thrust force, UA is the
incident propeller velocity and U is the ship velocity. For vessels with one propeller, the thrust
deduction factor td and the wake fraction w can be estimated as
w = 0.5CB − 0.05 (9)

td = 0.6w (10)

and for vessels with two propellers as

w = 0.55CB − 0.20 (11)

td = 1.25w (12)

CB is the block coefficient, defined as

∇
CB = (13)
BDLP P

where ∇ is the displacement volume, B is the beam, D is the draught and LP P is the length
between perpendiculars. It should be noted that the propulsion efficiency ηD can exceed 1, since
ηH usually is larger than 1. To capture the effects from waves and winds, a sea margin of 15 % is
usually added to the power requirements in equation 4. [9]

7
If there is a gearbox between the motor and the propeller shaft, further energy losses occur. The
required output power from the motor, PM can then be expressed as
kSM PS
PM = (14)
ηG

where kSM is the sea margin and ηG is the efficiency of the gearbox. The electric motor used in
the X Shore boats has a nominal speed of 4400 rpm and a maximum speed of 12000 rpm [16], but
since that speed is too high for the propeller, a gearbox is needed. When the motor power PM is
known, the range r can be calculated by
EB
r = ηM U (15)
PM

where ηM is the motor efficiency and EB is the energy in the battery, i.e. the battery capacity.

8
3. Semi-empirical and empirical methods
for ship design
This chapter presents three widely used methods for resistance prediction in ship design; the semi-
empirical methods by Savitsky and Holtrop & Mennen, and empirical model tests. If used, these
methods should be applied in the contract design phase.

3.1 Holtrop & Mennen

Holtrop & Mennen is a method based on empirical equations, derived from a large number of
model test results. It is considered one of the most accurate methods for predicting resistance
in the design phase. However, since the equations are derived from model tests performed in the
1970s and 1980s, the hulls might differ from today’s designs and the equations might give less
accurate results. [17] The method is intended for resistance prediction for displacement hulls and
the range of applicability is limited by following criteria [15]:

Fn < 0.45
0.55 ≤ CP ≤ 0.85
L
3.9 ≤ ≤ 9.5
B

CP is the prismatic coefficient defined as [9]

∇
CP = (16)
LP P AM

where ∇ is the displacement volume, LP P is the length between perpendiculars and AM is the
midship cross sectional area.

The total resistance RT is defined as

RT = (1 + k)RF + RAP P + RW + RB + RT R + RA (17)

where RF is the frictional resistance according to the ITTC formulation [18], (1 + k) is a form
factor describing the viscous resistance of the hull form in relation to RF , RAP P is the resistance
of appendages, RW is the wave-making and wave-breaking resistance, RB is additional pressure
resistance of bulbous bow near the water surface, RT R is additional pressure resistance of immersed
transom stern and RA is the model-ship correlation resistance. Detailed descriptions on how to
determine the different resistances can be found in Appendix A. [19]

3.2 Savitsky method

The Savitsky method is widely used to evaluate resistance for planing hulls. It is based on empirical
prismatic equations and assumes that the part of the hull in contact with water when planing
has a constant cross-section. [11] By determining the equilibrium trim angle through iteration,
the method takes the running attitude of the hull into account when prediction the resistance.
According to Savitsky [20] the total hydrodynamic drag of a planing hull can be described as

ϕU 2 Cf ΛB 2
D = ∆tan(τ ) + (18)
2cos(β)cos(τ )

where ∆ is the displacement mass, τ is the trim angle, U is the hull speed, B is the beam and β
is the deadrise angle.

9
ϕ is defined as
γ
ϕ= (19)
g

where γ is the specific weight of water and g is the gravitational acceleration. Λ is the mean wetted
length-beam ratio, defined as
LK + LC
Λ= (20)
2B

where LK is the wetted keel length, LC the wetted chine length and Cf is a friction coefficient.
Savitsky [20] refers to the friction coefficient formulated by Schoenherr, which was the standard
friction coefficient from 1947. However, this formulation had certain deficiencies and ITTC devel-
oped an improved formulation in 1957. [21] The ITTC formulation [22] will be used in this project
and is expressed as
0.075
Cf = (21)
[log10 (ReΛ ) − 2]2

where ReΛ is the Reynold’s number based on the mean wetted length, defined in Savitsky [20] as
U ΛB
ReΛ = (22)
ν
where ν is the kinematic viscosity of the water. Even though the friction coefficient Cf was
developed for traditional hulls, and consequently may not give accurate results for unconventional
hull shapes, it is widely used. [23] The method is applicable when following criteria are met [20]:

0.6 ≤ CV ≤ 13

2◦ ≤ τ ≤ 15◦
Λ≤4

where CV is a speed coefficient defined as

U
CV = √ (23)
gB

Detailed descriptions on how to calculate the resistance with the Savitsky method can be found in
Appendix B.

3.3 Model tests

Model tests are performed with a scale model of the hull in a towing tank. The model should have
the same Froude number as the full scale hull in order to obtain useful results, which means that
the model speed should be adjusted according to Froude’s model law:
U
UM = √ (24)
α

where UM is the model velocity, U is the ship velocity and α is the scale factor. When the model
is towed in the towing tank, resistance, trim and sinkage are measured. Moreover, the water tem-
perature is measured in order to calculate the water’s density and viscosity. [9] By measuring the
towing force, the resistance can be determined [24].

The model can be tested in two conditions; with and without appendages. When testing without
appendages, also known as bare hull, the resistance due to the hull shape is determined. Sometimes
a rudder is included in the bare hull test. The purpose of tests with appendages is to determine how

10
much the appendages affect the total resistance. [24] To obtain as accurate trim measurements as
possible, the direction of the towing force should be applied in the longitudinal center of buoyancy
(LCB) and in the line of the propeller shaft. [24]

Generally, the bigger the model the better. However, if the model gets too big in comparison
to the towing tank, it will affect the velocity field and thus the results. Hence, the following cri-
teria for model length between perpendiculars LP P m , model draught Dm and model beam Bm
should be fulfilled:

LP P m < d

LP P m < W

Wd
Dm Bm <
200

where d is the water depth and W the width of the towing tank. [9] Model tests have played an
important role in traditional ship design, but it is a costly method [8].

11
4. Computational fluid dynamics for ship
design
Computational fluid dynamics (CFD) is the analysis of fluid flows, using numerical methods. CFD
can be used to simulate a flow and analyse the interaction between a fluid and an object, e.g.
to determine lift or drag, and it can therefore be used to simulate ship hydrodynamics. It is an
efficient supplementary technique to costly model tests [8] and it has been demonstrated that a
CFD simulation give acceptable accuracy compared to experimental data such as the Savitsky
method [25]. If CFD is used in the design process, the simulations are done during the contract
design. One of the advantages with CFD for resistance predictions in the early design phase is
that modifications of the hull can be done in both a relatively short time and to a relatively low
cost, which enables comparisons of results for different hull forms [7]. In this project, ANSYS
FLUENT 19.0 has been used and the following chapter will focus on theory relevant to perform a
CFD simulation for resistance prediction on a hull using that software.

4.1 Geometry and computational domain

Geometries used in CFD simulations are defined in a computer aided design (CAD) software and
then converted to a file format compatible with the CFD software [23].

The orientation of the reference coordinate system, as well as the origin, should be chosen carefully.
In order to minimise the risk of errors, the aim is to have the coordinate system aligned with the
forces of interest and recommendations are to use the same coordinate system in both the CAD
and the CFD software. [23]

For simulations of ship hydrodynamics, the computational domain can be built in several ways.
It is important to make it big enough to capture the wake behind the ship, but small enough to
save computational cost. According to ITTC guidelines for ship CFD applications [23] the do-
main is built as a rectangular block around the imported hull surface. To reduce the grid size
by half, the port-starboard symmetry can be taken into advantage and only half of the full do-
main is then computed, using a symmetry boundary condition on the ship’s center plane [8][18][25].

The domain should include an inflow and outflow surface, placed sufficiently far away from the
ship. The inlet boundary should be placed 1 − 2LP P upstream of the ship and the outlet boundary
at 3 − 5LP P downstream of the ship, where LP P is the length between perpendiculars of the hull
[26]. Other boundaries of the domain should be placed at least 1LP P away from the hull. [23]
Figure 6 shows the computational domain.

Figure 6: The size of the computational domain.

12
4.1.1 Boundary conditions
In addition to the symmetry boundary condition on the ship’s center plane, boundary conditions
for the inflow and outflow should be defined [23]. The inflow boundary is often imposed as a
velocity field, and defined as a velocity inlet since the flow is initiated from this boundary [23]
and it is suitable for incompressible flows [7]. For the outflow boundary, a pressure outlet with a
Neumann condition is recommended [23]. On the hull, a no-slip wall boundary condition is usually
applied [7], see figure 7.

Figure 7: The boundary conditions for the computational domain.

4.2 Mesh
In order to use numerical equations, the domain is divided into smaller cells, called a mesh. The
mesh can be built out of structured or unstructured grids. Structured grids for three dimensional
domains are build out of hexahedral elements while the unstructured grid cells can have hexahedral,
tetrahedral, polyhedral or several other shapes, see figure 8. Unstructured grids are in general
slower and have less accuracy than equivalent structured grids. [23] Converting the mesh to
polyhedral cells can decrease the overall cell count [27].

Figure 8: Different shapes of cell elements that can be used to build a mesh.

In order to capture the flow on the free surface with high accuracy, orthogonal grids should be used
[23]. Since the waves created on the surface are much longer than they are high, high resolution on
the grid in vertical direction is required [28]. To capture this, a cell size in the vertical direction,
i.e. along the z-axis should be
LW L
z= (25)
1000

where LW L is the length of the waterline on the hull [28].

13
4.2.1 Boundary layer
When a fluid and a structure interacts, shear stresses arise on the surface of the structure which
creates a boundary layer. When the free stream reaches the structure with a uniform velocity a
laminar boundary layer starts to grow at the structure surface. The boundary layer, which affects
the velocity profile, grows along the surface and after some distance it goes into a transition region
after which the boundary layer becomes turbulent. Close to the surface, the boundary layer will
remain laminar and the turbulence will increase further away from the structure surface.

To capture this behavior when simulating a fluid-structure interaction, ANSYS FLUENT uses
a dimensionless wall distance to characterize the flow near the structure surface. The dimension-
less wall distance is defined as
u∗ y
y+ = (26)
ν

where y is the distance to the structure surface, ν is the kinematic viscosity and u∗ is a friction
velocity,
r
τw
u∗ = (27)
ρ

in which ρ is the water density and τw is the wall shear stress defined as [10]

∂U
τw = ρν (28)
∂y y=0

Experiments have shown that the near-wall region can be divided into sublayers. The innermost
layer, the viscous sublayer, consists of a flow that is almost laminar and where the viscosity domi-
nates and y + ≤ 5. The outermost layer, the log-law layer, is fully turbulent and thus, turbulence
dominates. In between these two layers there is a layer called the buffer layer where viscosity and
turbulence are equally important and where 5 ≤ y + ≤ 60. [27]

To properly model the near-wall region, in this case the boundary layer close to the hull, prism
layer mesh can be used. This can be done by either using near-wall turbulence models or by using
wall functions. Depending on the level of accuracy required, chosen turbulence model and whether
wall functions are used or not, the number of grid points can be determined in terms of y + . Near-
wall turbulence models resolve the flow in the laminar sub-layer all the way down to the wall,
requiring at least 3 points in the boundary layer. Therefore, it is recommended to use a y + ≤ 1
with an expansion ratio of 1.2 resulting in 20 points within the boundary layer. Wall functions are
semi-empirical formulas that without resolving the viscous sub-layer, bridge the solution variables
in the turbulent log-law layer and the corresponding quantities on the wall. The first point from
the wall is recommended to be within the log-law layer. Thus, the dimensionless wall distance
value is recommended to be 30 ≤ y + ≤ 100 with an expansion ratio of 1.2, resulting in 15 points
within the boundary layer. [23]

Choosing y + value, i.e. near-wall models or wall functions, is a trade-off between computational
effort and accuracy. Near-wall models is a more rational approach, but the highly refined grids
can lead to numerical instability making the computations heavy. On the other hand, wall func-
tions are based on two-dimensional flows at zero pressure gradients and the analytical expressions
become less valid with increasing pressure gradient. If possible, wall functions should be avoided.
[23]

14
However, once the y + is chosen, the distance y of the first point in the boundary layer can be
determined as

y + LP P
y= q (29)
C
ReL 2f

where Cf is the friction coefficient and ReL is the Reynolds number based on the length between
perpendiculars, defined as
U LP P
ReL = (30)
ν

In regions where high resolution is required, i.e. near the free surface and in the boundary layers,
tetrahedral cells should be avoided. Hexahedral or prismatic cells result in better convergence rates
and higher accuracy and should therefore be used in regions requiring high resolution. [23]

4.3 Turbulence
The flow around a hull traveling with high speed is a turbulent flow [10]. Turbulent flow, in con-
trast to laminar flow, is characterised by random and chaotic variations in the flow’s velocity and
speed, resulting in a three-dimensional flow with vortices [29].

The Navier-Stokes equations are the governing equations for fluid flows consisting of the conti-
nuity and the momentum equations. The equation of continuity implies a material balance over a
stationary fluid element. The momentum equations, also called the equation of motion, describes
the momentum balance which according to Newton’s second law requires that the rate of change
in momentum on the fluid particle is equal to the force acting on it. [30] Analytical solutions for
solving the Navier-Stokes equations for turbulent flows do not exist, so they need to be treated nu-
merically [10]. To solve the equations directly by direct numerical simulation (DNS) which resolve
the flow completely, is extremely computationally expensive and time consuming. Therefore, the
most common way to simulate turbulent flows is based on the Reynolds Averaged Navier-Stokes
(RANS) equations. These are simplified Navier-Stokes equations which are not as accurate as
DNS, but requires less computational effort. The idea of RANS equations is to describe the flow
as the turbulent viscosity fluctuations separated from the mean flow velocity. The main differences
between the original Navier-Stokes equations and the RANS equations is the Reynolds stress term,
which introduces the coupling between the turbulent fluctuations and the mean velocity. To model
this term is the sole purpose of RANS turbulence modelling and can be modelled by determine a
eddy viscosity. To determine this viscosity, and solve the RANS equations, a turbulence model is
needed. [30]

The three most commonly used turbulence models are the standard k- model, the k-ω model
and the SST k-ω model. The SST k-ω model is a combination of the standard k- and k-ω models,
which has shown good performance for complex flows. In the boundary layer the k-ω model is
used because of its validity in regions of low turbulence, i.e. close to walls. For the free flow,
the standard k- model is used, formulated on k-ω form, since it is a robust model with good
predictions for fully turbulent flows. [10] For simulating ship hydrodynamics the SST k-ω model is
the most commonly used turbulence model [8][23][25][31]. When using a two-equation turbulence
model as the SST k-ω, the time step ∆t must be
L
∆t ≤ 0.01 (31)
U

where L is the length and U is the speed of the boat [23].

15
4.4 Two-phased flows
The intended simulations contain two fluids, water and air, which need to be simulated as two
different phases. This can be done by using the multiphase method in ANSYS FLUENT. The two
phases are separated by a free surface which is resolved with the volume of fluid (VOF) formulation
and open channel boundary conditions.

The VOF formulation enables modelling of two immiscible fluids and is in general used to compute
time-dependent solutions. However, it can also be used for steady-state calculations if the solution
is independent of the initial conditions and there are distinct inflow boundaries for the two different
phases. [27]

The multiphase method divides the computational domain into two different phases based on
where the free surface is placed, see figure 9, and the fluid’s characteristics need to be defined.
The open channel flow model enables to define the location of the free surface. However the model
requires that the open channel boundary condition for the inlet is defined as either pressure inlet
or mass flow rate and the outlet as either pressure outlet or outflow boundary. [27] Necessary for
simulations with a free surface is also to define the direction and magnitude of the gravitational
acceleration g [7].

Figure 9: The multiphase method divide the computational domain into two different phases.

4.4.1 Numerical ventilation

A common problem when applying the VOF formulation on planing hulls is a phenomenon called
numerical ventilation. The simulated resistance is reduced due to a thin layer of air between the
hull and the water, decreasing the friction. Furthermore, the air under the hull might have an
impact on the pressure distribution and the trim of the hull, which also affects the resistance [32].
This phenomenon is completely artificial, meaning that it only appears in simulations and not in
real situations. Therefore, it is necessary to remove the effect of the numerical ventilation. One
way to do this is to use a user defined function (UDF) and loop through every cell in the boundary
layer close to hull, and for cells where the volume fraction of air is below a certain limit, it is set to
0 [10]. It is shown that low values of the dimensionless wall distance y + , first order modelling of the
convection terms and large time steps increase the effects of numerical ventilation. Consequently,
it is recommended to have y + ≈ 30 and to model the convection term using a second order
scheme. [33] Moreover, a modified HRIC scheme decreases the numerical ventilation significantly
in comparison with a regular HRIC scheme. [32]

4.5 Fluid-structure interaction

When simulating a hull, it is assumed that the hull will reach an equilibrium, i.e. a steady position
and orientation with respect to the free surface, when moving with constant speed. Before reaching
this equilibrium, the hull will have a dynamic behaviour where the interaction between the fluid,
in this case water, and the structure, which in this case is the hull, needs to be considered. This is

16
done by solving the equations of motion and rotation of the boat as it is influenced by the forces
and moments from the surrounding fluids and gravitational acceleration. [10] The boat can be
considered as a rigid body which in general is allowed to move in six degrees of freedom (DOF);
translation in three directions and rotation in three directions, along and around the x, y and z
axes, respectively. The translation motions are called surge η1 , sway η2 and heave η3 along the
x, y and z axes, respectively, and around the same axes are the rotational motions called roll η4 ,
pitch η5 and yaw η6 , see figure 10. [34]

Figure 10: Translation motions surge η1 , sway η2 and heave η3 and the rotational motions roll η4 ,
pitch η5 and yaw η6 along and around the x, y and z axes, respectively.

To solve the equations of motion and rotation, ANSYS FLUENT uses a six DOF solver. The
solver follows an iterative procedure until the equilibrium is reached and the running attitude is
identified. A six DOF solver requires that the mass and the moments of inertia are specified. The
six DOF solver is often limited to two DOF, heave and pitch, through a UDF. The UDF is then
written to only allow for translation along the z-axis, heave η3 , and rotation around the y-axis,
pitch η5 , and is defining the required properties. [10] Heave and pitch are earlier mentioned as
changes in draught and trim, which define the running attitude of the ship.

4.6 Dynamic mesh

In order to capture motions in several degrees of freedom when modelling, a dynamic mesh is
required. ANSYS FLUENT has three different methods for dynamic meshing: smoothing methods,
dynamic layering and remeshing methods. For complex dynamic problems, these methods can be
combined. [35] The smoothing methods change the shape of the nodes, while there are no changes
in the number of nodes and their connectivity. The remeshing methods collect cells in the mesh
that are invalid, e.g. cells with negative volume, and updates the mesh with new cells. If using
dynamic layering, a cell layer close to the moving surface can be added or removed, based on the
height of the cells next to the moving surface. [36]

4.7 Schemes
The numerical methods in ANSYS FLUENT can be density-based or pressure-based. The density-
based solver can either use implicit or explicit numerical methods, where the difference is that the
implicit method uses both unknown and known values from neighbouring cells while the explicit
only uses known values. [27] When applying the pressure-based solver, two different algorithms
are available: a segregated and a coupled algorithm. The coupled algorithm results in a reduced
convergence time, but requires a higher computational cost than the segregated algorithm. [36]
The pressure-based solver is required when appling the VOF model [27].

Continuous equations need to be discretized in order to be solved numerically. For steady-state

flows, spatial discretization is needed. For cases with transient flow, both spatial and temporal

17
discretizations are required. The spatial discretization in ANSYS FLUENT is done with an upwind
scheme, meaning that values in a cell are derived from the cell in the upstream direction, compared
to the normal velocity. This can be done with a number of different schemes: first-order upwind,
second-order upwind, power-law, and QUICK. The temporal discretization can be done with either
implicit or explicit time integration. The implicit time integration is stable regardless of the size
of the time steps, but has a higher computational cost than the explicit integration. However,
the explicit time integration cannot be used to compute incompressible flows and requires that a
density-based explicit solver is used. [27]

According to ITTC [23], the second-order upwind scheme is used in the majority of industrial
CFD codes. It is recommended for all convection-diffusion transport equations since it is reason-
ably accurate and robust. [23]

4.8 Quality
Convergence of the solution, a grid independence study and the dimensionless wall distance y + are
commonly used to demonstrate the quality of the solution.

One way to check the convergence of the solution is to define a convergence criterion. The conver-
gence criterion can be that the residual has decreased to a sufficient degree. Simply, residuals are
the change in the equation over an iteration [37] and the default criterion is that the residual will be
reduced to 10−3 [35]. When the convergence criterion has been reached, the solution has converged
[35][37]. However, sometimes the residual will not fall below the criterion and the solution can be
considered as converged if the solution no longer changes with respect to the number of iterations
[35].

Grid independence should be checked through a mesh convergence study, meaning that the mesh is
checked to be good enough to not affect the solution. A mesh convergence study can be performed
by starting with a coarse mesh, calculating a fluid property, e.g. drag force, followed by refining the
mesh and calculating the same property again. By repeating this procedure until the fluid property
has converged to a certain value, it can be assured that the mesh is good enough to not affect the
solution. In order to save computational cost, a coarser mesh can be used, under the assumption
that the deviations are known and that the coarser mesh provides results with acceptable accuracy.

If the quality is demonstrated with the dimensionless wall distance, the value of y + is checked.
Depending on if a near-wall turbulence model or a wall functions is used, the value of y + should
correspond to the limits presented in section 4.2.1.

4.9 Visualisation
The result of the simulations relevant for the resistance prediction is firstly the drag force, which
in ANSYS FLUENT represents the total resistance force containing both the pressure and viscous
forces. It is also of interest to check the contribution and magnitude of the pressure and viscous
forces which can be done through a force report. Note that if the port-starboard symmetry is used,
only half of the hull is simulated and thus, only half of the total forces are calculated.

In order to verify the quality, the residuals will be plotted and should be checked. A contour
plot of the water volume fraction on the hull is also of interest in order to see if numerical ventila-
tion has occurred.

18
5. Evaluation of methods
To evaluate the different methods, a set of criteria is required. The project triangle presented in
figure 11 is often used to illustrate time, cost and performance as competing constraints as well as
criteria to measure whether or not the project goal is met. [38]

Figure 11: The project triangle.

However, there might be more competing constraints than these three when deciding which method
to implement in the ship design process. For a small company like X Shore, one limitation is the
accessibility of each method. A clarification of what each criterion means in this study is presented
below.

Time refers to the total time consumption, including determining input values, performing calcu-
lations, and post-processing and interpreting results.

Cost is the total economical cost of applying a method. This includes the cost of software li-
censes, necessary tools such as high-performance computers, and the cost of work hours required
to apply the method.

Performance can be scope, technology or quality [38], and is sometimes substituted by one
of these in the project triangle. In this study, performance is substituted by quality and consists
of the accuracy and precision of the results of each method. Accuracy refers to the deviations
between results and the true value, and precision refers to the consistency of the results, i.e. that
the error is consistent.

Accessibility includes the tools, facilities and competence required to apply the method as well
as interpret the results. Low accessibility indicates that more tools, facilities and competence is
required, compared with a method with higher accessibility.

Each method in this study will be evaluated with regard to the four criteria in figure 12. Most
likely, one method will not fulfill all criteria, and choosing the most suitable method for X Shore
will be a trade-off between them. It is therefore necessary to perform a systematic assessment of
each method.

Figure 12: Evaluation criteria for this study.

19
5.1 Evaluation tool
Multi-criteria decision analysis (MCDA) is a tool used to assess and order a finite number of op-
tions with regard to several criteria. The tool includes a wide range of approaches with varying
complexity, where most of the approaches have the performance matrix in common. In the most
basic performance matrix, each row represents an alternative and each column a criterion. The
criteria are weighted to indicate importance, and the options are assigned a score for each crite-
rion, that represents how well the option meets that particular criterion. However, MCDA lacks
a reference option, and consequently, the tool does not indicate if the assessed options are better
than the option of doing nothing. [39]

The Pugh Matrix is a system engineering tool used to evaluate and compare design concepts
[40]. It is very similar to MCDA, but has a reference that the alternatives are compared with.
Since model tests already have been performed and can serve as a reference method, the Pugh
Matrix is used for evaluation and the design concepts are replaced with the resistance prediction
methods. The Pugh Matrix is a tool that is easy to use whenever a decision between a number
of alternatives is needed and provides a simple way of taking multiple factors into account in the
decision making [40].

The basic concept of the matrix can be seen in table 2, in which three methods are evaluated
against four criteria. The idea of the matrix is to select one method as the baseline, which the
other methods are compared against for each criteria. The baseline method will be valued as 0
for each criteria and the other methods will be compared and evaluated on a scale from −2 to
+2 depending on if it is worse or better than the baseline method for each criteria. To get an
even better differentiation and to gain more a robust assessment, the criteria are weighted from
1 − 5 depending on the importance of the criteria. [40] Finally, the total score is calculated for
each method by multiplying the weight with the evaluated score for each criteria respectively and
summing up the calculated score for each method [40].

Table 2: Pugh Matrix.

Criteria Weight Baseline Method 1 Method 2 Method 3

Criteria 1 4 0 -1 -1 +1
Criteria 2 1 0 +1 +2 -1
Criteria 3 3 0 +1 -1 +2
Criteria 4 5 0 +2 +2 -2
Score 0 7 5 -1

In a Pugh Matrix, the method ending up with the highest score is the winner. However, there is
quite often not a clear winner but a clear loser when performing a Pugh Matrix and interpreting
the results often includes performing a sanity check. The quality of the decision is also clearly
depending on the selection of criteria and how well-defined the criteria are. [40]

20
6. Results
The purpose of this chapter is to present how to apply Holtrop & Mennen, the Savitsky method
and a CFD-simulation. The results from applying each method on Eelex 2020 will be compared
and evaluated against the criteria from chapter 5. To enable a comparison, the characteristics for
Eelex 2020 identified in the conducted model tests are used as input values in the three applied
methods. For the same reason, the water density ρ and kinematic viscosity ν are assumed to be
the same as in the model tests, ρ = 1026.06 kg/m3 and ν = 1.1892 · 10−6 m2 /s. Furthermore, the
results from the model tests are re-calculated and does not include resistance from appendages.
The sections below describe how to find the input values if no model test results are available or
if any of the input values are missing from the tests. The latter is the case for some of the input
values used in the applied methods.

6.1 Results from previous studies

Several studies comparing resistance from CFD with experimental values have been conducted.
A study from MIT that evaluated hull resistance for speeds corresponding to Froude numbers
Fn ≤ 0.41, reports that the maximum error was an over-prediction by 8.2% [7]. Another report
studying two displacing hulls at Froude numbers 0.22 < Fn < 0.44, presents a maximum error of
1.3% and 4.1% for the hulls respectively [33]. In contrast to the findings from MIT, another study
shows that for planing hulls and Fn > 1.3, CFD under-estimates the total resistance by 8.2% [10].
Brizzolara and Serra [41] conclude that an under-estimation of 10% can be expected when CFD
is applied. Furthermore, CFD is shown to be more accurate than the Savitsky method [41] and
Çakıcı et al. [25] found a difference of 7.8% when comparing CFD and the Savitsky method at
Fn = 1. Nikolopoulos and Boulougouris [17] have applied Holtrop & Mennen on 11 different hulls
operating in speeds corresponding to Fn < 0.2. In some cases, the method over-estimates the
resistance, while it under-estimates it in other cases. The maximum error is an over-prediction of
16%. However, they point out that in cases where the hull geometry is close to the applicability
limits, the accuracy might decrease. In conclusion, it is common that the CFD results are within
10% of the experimental values, while the errors for the Savitsky method and Holtrop & Mennen
generally are larger.

The resistance is difficult to predict for planing hulls. According to Brizzolara and Serra [41],
one reason for this is that both the viscous and pressure components are related to the trim mo-
ment and the dynamic lift force in a non-linear way. Consequently, it is crucial to accurately predict
trim and sinkage in order to obtain an accurate result of the total resistance. Frisk and Tegehall
[10] identified difficulties when simulating free sinkage and trim in ANSYS FLUENT. Instead of
finding an equilibrium position, the free surface moved with the dynamic mesh, which caused the
hull to oscillate with increasing amplitude.

6.2 Holtrop & Mennen

This section presents the suggested approach to Holtrop & Mennen, followed by the results from
applying the method on Eelex 2020. The suggested approach is a Python 3 program developed
by the authors to calculate resistance according to Holtrop & Mennen. The program is based
on equations in Appendix A and the graphical user interface is based on an open source library,
graphics.py [42].

21
6.2.1 How to apply Holtrop & Mennen
The program starts with showing an input window and the inputs required are presented below in
table 3 in the same order as they appear in the program’s input window. All input values should
be given in SI-units. The interface of the input window can be seen in Appendix C in figure C.1.

Table 3: Input parameters for Holtrop & Mennen.

Input Description Symbol Figure reference

A value reflecting the afterbody
form of the hull, i.e. if it has
Afterbody form a V-shaped, normal shaped or - -
U-shaped afterbody the input
value is 1, 2 or 3 respectively.

The immersed part of the

The transverse area
transverse area of the AT Figure 13b
of the transom
transom at zero speed

Area of immersed The immersed midship section

AM Figure 13c
midship section at LW L /2

The horizontal area

The waterplane area AW P Figure 13d
of the hull at the waterline

Width of the boat

Beam B -
at the widest point

Weight The total weight of the boat m -

Length at the level where

Length on waterline LW L Figure 14
it sits in the water

Length between Distance between aft and

LP P Figure 14
perpendiculars forward perpendiculars

Draught at forward Measured in the perpendicular

DF P Figure 14
perpendicular of the bow

Draught at aft Measured in the perpendicular

DAP Figure 14
perpendicular of the stern

Longitudinal
Measured from the aft LCB Figure 14
center of buoyancy

22
(b) The immersed part of the transverse area of
(a) The hull on the free surface. the transom at zero speed.

Figure 13: Clarification of AT , AM and AW P .

Figure 14: Figure showing LW L , LP P , DF P , DAP and LCB.

The results are then presented in a new output window, the interface can be seen in Appendix C
in figure C.2. The first row presents the different speeds in knots used in the calculations. The
second row shows the Froude number Fn for the different speeds, calculated according to equation
24. The Froude number indicates whether or not the vessel is displacing. The next seven rows in
the table presents the resistance components in newton in the total resistance equation 17 for the
Holtrop & Mennen method. The last three rows present the total resistance, first the resistance
force in newton, followed by the resistance power in horsepower and watts.

6.2.2 Holtrop & Mennen applied on Eelex 2020

Table 4 presents the input values used when applying Holtrop & Mennen on Eelex 2020. The
total time required to perform the method is approximately 30 minutes, where the main part is to
determine the input values. When the input values are determined, the calculation time is a few
seconds, and no post-processing is required.

23
Table 4: The input values used when applying Holtrop & Mennen on Eelex 2020.

Symbol Value Unit

Afterbody form 1
Transverse area of transom AT 0.447 m2
Area of immersed midship section AM 0.4381 m2
Waterplane area AW P 12.73 m2
Beam B 2.550 m
Length on waterline LW L 7.414 m
Length between perpendicular LP P 7.489 m
Weight m 2600 kg
Draught at forward perpendicular DF P 0.417 m
Draught at aft perpendicular DAP 0.382 m
Longitudinal position of the center of buoyancy LCB 2.664 m

The results are shown in table 5.

Table 5: The resulting values when applying Holtrop & Mennen on Eelex 2020.

4 knots 5 knots 6 knots 7 knots 8 knots

Froude number Fn 0.24 0.3 0.36 0.42 0.48

Frictional resistance RF 71.4 N 107.5 N 150.2 N 199.3 N 254.8 N

Form factor 1 + k1 2.2 2.2 2.2 2.2 2.2

Resistance of appendages RAP P 0N 0N 0N 0N 0N

Wave-making and
RW 41.2 N 265.6 N 1002.1 N 1509.0 N 1984.3 N
wave-breaking resistance
Resistance due to bulbous
RB 0N 0N 0N 0N 0N
bow near the surface
Resistance of immersed
RT R 138.5 N 194.6 N 248.8 N 296.0 N 330.8 N
transom stern

Model-ship resistance RA 19.6 N 30.7 N 44.2 N 60.1 N 78.6 N

Total resistance Rtotal 358 N 729 N 1628 N 2308 N 2960 N

Total resistance power RW 736 W 1877 W 5027 W 8311 W 12181 W

Total resistance power Rhp 1 hp 2 hp 6 hp 11 hp 16 hp

The contribution to the total resistance from the resistance components in the Holtrop & Mennen
method are illustrated in figure 15. The method is applied for the speed range 2−8 knots and it can
be seen that the greatest impact on the total resistance as the speed increases is the wave-making
and wave-breaking resistance.

24
Figure 15: The contribution of the resistance components in Holtrop & Mennen.

The applicability criteria for Holtrop & Mennen, presented in section 3.1, are checked for each
speed. As can be seen in table 6, the length-beam ratio criteria is not fulfilled for any of the speeds
and the Froude number criteria is not fulfilled for the speed of 8 knots. However, the prismatic
coefficient is met for all the displacement speeds.

Table 6: Criteria for Holtrop & Mennen applicability.

Froude number Prismatic coefficient Length-beam ratio

Fn < 0.45 0.55 ≤ CP ≤ 0.85 3.9 ≤ LW L /B ≤ 9.5
4 knots 0.24 0.77 2.9
5 knots 0.30 0.77 2.9
6 knots 0.36 0.77 2.9
7 knots 0.42 0.77 2.9
8 knots 0.48 0.77 2.9

If any of the criteria are not met, the results for that speed will be printed in red in the output
window of the developed program.

6.3 Savitsky
This section presents the suggested approach to the Savitsky method, followed by the results from
applying the method on Eelex 2020. The suggested approach is a Python 3 program developed by
the authors to calculate resistance according to Savitsky. The program is based on equations in
Appendix B and the graphical user interface is based on an open source library, graphics.py [42].
The specific weight of water ϕ is assumed to be 9.789 kN/m3 .

25
6.3.1 How to apply the Savitsky method
The program starts with showing an input window and the inputs required are presented below in
table 7 in the same order as they appear in the program’s input window. All input values should
be given in meter, kilogram or degrees. The interface of the input window can be seen in Appendix
D in figure D.1.

Table 7: Input parameters in the Savitsky method.

Input Description Symbol Figure reference

Displacement Total weight of the boat ∆ -

Length between Distance between aft and

LP P Figure 14
perpendiculars forward perpendiculars

Longitudinal Distance from stern to the centre

LCG Figure 16a
centre of gravity of gravity, measured along the keel

Vertical Distance from keel to the centre of

V CG Figure 16a
centre of gravity gravity, measured normal to the keel

Width of the boat

Beam B Figure 16b
at the widest point

Angle of the v-shape at the stern,

Deadrise angle measured from horizontal plane β Figure 16b
to bottom of the boat

Angle between keel Inclination of the thrust

Figure 16a
and propeller shaft line relative to the keel

Distance between
Measured normal to the
propeller shaft and f Figure 16a
extension of the shaft line
centre of gravity

(a) Figure showing LCG, V CG, and f . (b) Figure showing B and β.

Figure 16: Clarification on input values for the Savitsky method.

The results are then presented in a new window, see figure D.2 in Appendix D. The first row
presents the different speeds in knots used in the calculations. The second row shows the Froude

26
number Fn for the different speeds, calculated according to equation 24. The Froude number
indicates whether or not the vessel is fully planing. The third row in the table presents the
equilibrium trim angle, i.e. the angle the boat will have when going in constant speed, and the
fourth row presents the draught at the transom. The last three rows present the total resistance,
first the resistance force in newton, followed by the resistance power in watts and horsepower.

6.3.2 The Savitsky method applied on Eelex 2020

Table 8 presents the input values when applying the Savitsky method on Eelex 2020. The total
time required to perform the method is approximately 30 minutes, where the main part is to
determine the input values. When the input values are determined, the calculation time is a few
seconds, and no post-processing is required.

Table 8: The input values when applying the Savitsky method on Eelex 2020.

Symbol Value Unit

Displacement ∆ 2600 kg
Length between perpendiculars LP P 7.498 m
Longitudinal centre of gravity LCG 3.081 m
Vertical centre of gravity V CG 0.718 m
Beam B 2.550 m
Deadrise angle β 15 deg
Angle between keel and propeller shaft 6 deg
Distance between propeller shaft and COG f 0.6 m

The input in table 8 resulted in the values presented in table 9.

Table 9: The resulting values when applying the Savitsky method on Eelex 2020.

12 knots 16 knots 20 knots 24 knots 32 knots

Froude number Fn 0.72 0.96 1.20 1.44 1.92
◦ ◦ ◦ ◦
Equilibrium trim τ 3.97 4.27 3.81 3.23 2.37◦
Resistance force Rtotal 2604 N 3114 N 3379 N 3690 N 4641 N
Resistance power RW 16.1 kW 25.6 kW 34.8 kW 45.5 kW 76.4 kW
Resistance power Rhp 21.5 hp 34.4 hp 46.6 hp 61.1 hp 102.4 hp

The applicability criteria for the Savitsky method, presented in section 3.2, are checked for each
planing speed, table 10.

27
Table 10: Criteria for Savitsky applicability.

Speed coefficient Trim angle Wetted length-beam ratio

0.6 ≤ CV ≤ 13 2◦ ≤ τ ≤ 15◦ Λ≤4
12 knots 1.23 3.97◦ 2.5
16 knots 1.65 4.27◦ 2.07
20 knots 2.06 3.81◦ 1.87
24 knots 2.47 3.23◦ 1.77
◦
32 knots 3.29 2.37 1.63

As can be seen in the table 10, all the criteria are fulfilled for each speed which indicates that the
method should be applicable on Eelex 2020. If any of the criteria are not met, the results for that
speed will be printed in red in the output window of the developed program.

After further investigation, it was found that the applicability criteria for the Savitsky method
is met even for the speeds 7 and 8 knots even though they are not planing speeds. The results
from applying the Savitsky method for 7 and 8 knots can be seen in table 11.

Table 11: The resulting values when applying the Savitsky method on Eelex 2020 for 7 and 8
knots.

7 knots 8 knots
Froude number Fn 0.42 0.48
◦
Equilibrium trim τ 2.88 3.04◦
Resistance force Rtotal 1657 N 1823 N
Resistance power RW 5966 kW 7500 kW
Resistance power Rhp 8.0 hp 10.1 hp

6.4 CFD
This section presents how the CFD simulation on Eelex 2020 was conducted. The presented setup is
the suggested approach for how to do the simulations and they are performed in ANSYS FLUENT
19.0. A step-by-step tutorial on how to perform the simulations has been made and was given to
X Shore.

6.4.1 Assumptions and limitations

To start with, the fluids air and water are assumed to be incompressible resulting in that velocity
and pressure fields are independent from the temperature field. The approach is based on the
theory presented in chapter 4 and it should be noted that the ITTC guidelines [23] are intended
for applications with high Reynolds numbers and Froude numbers of the order of 0.1 and rarely
above 1. These conditions are fulfilled for Eelex 2020 in displacement speeds, but when reaching
top speeds, the guidelines have not been proven valid.

Several attempts of dynamic meshing were made, but problems with negative cell volume oc-
curred. The problem can probably be fixed by reducing the time step and/or refining the mesh.
However, due to time and hardware limitations, it was decided to focus on static simulations.
Therefore, the following results are obtained from static simulations and no dynamic mesh have
been used. Consequently, the simulations do not capture hull motion, such as heave and pitch,

28
and the running attitude is hence not identified by the simulations themselves. To capture the
impact of the running attitude, the draught and trim were needed to be manually implemented
in the static setup. The discovered hardware limitations pointed out the importance of sufficient
hardware performance in this type of simulations. Initially, an attempt to perform both dynamic
and static simulations was made on a computer with 16 GB RAM and a 1.33 GHz processor with
8 cores. The attempts were unsuccessful due to lack of RAM, which resulted in both mesh limi-
tations and extremely high computational time. Therefore, the computer used in this project has
64 GB RAM and a 3.6 GHz processor with 8 cores, which was sufficient for the static simulations.

6.4.2 Geometry and computational domain

The geometry of Eelex 2020 was imported as a CAD-file in STEP format, since it is a compatible
file format with ANSYS FLUENT. The port-starboard symmetry was used, and only half of the
hull was simulated. The computational domain was built as a rectangular block around the hull,
with the smallest dimensions possible according to the ITTC guidelines in order to reduce the
number of cells. The coordinate system was located at the symmetry plane, 38 mm below the free
surface at zero speed and 465 mm forward the aft of the the hull. Ideally, the coordinate system
should be placed at the draught at zero speed in order to simplify the setup. Measured from
the coordinate system, the inlet and outlet were placed approximately 2LP P and 3LP P forward
and backward, respectively, and the other boundaries were placed approximately 1LP P away. In
order to enable for refinements around the free surface, a smaller rectangular block was built
around the free surface area. This rectangular block had the same length and width as the whole
computational domain, but the height was set to 0.25 m upwards and 0.5 m downwards measured
from the coordinate system. Before proceeding, the imported hull surface was subtracted from the
domain. The computational domain and the associated dimensions can be seen in figure 17 and
table 12.

Figure 17: The computational domain.

Table 12: The dimensions of the computational domain measured from origo.

Inlet Outlet Top Bottom Wall

Computational domain 16 m −24 m 8m −8 m 8m
Refinement block 16 m −24 m 0.25 m −0.5 m 8m

29
6.4.3 Mesh
The domain was meshed using the function body of influence in order to achieve the refinements
around the surface. To capture the boundary layer, an automatic prism layer with 15 layers was
added to the hull with a first layer thickness of 2.7 · 10−4 m and a growth rate of 1.2. This
corresponds to a dimensionless wall distance y + = 30 for 7 knots and was set in order to prevent
numerical ventilation. The prism layer in the aft of the hull can be seen in figure 18. For the hull
surface, the function face sizing was used and the element size was set to 0.1 m. The final mesh
was chosen based on the grid dependence study, see section 6.4.5.

Figure 18: The prism layer in the aft of the hull.

6.4.4 Model setup

In the model setup, the mesh was converted to polyhedral cells. The two models applied to the
simulations was the viscous model and multiphase model. For the viscous model, the SST k-ω
model was chosen and for the multiphase, the VOF was used as model, open channel flow as VOF
sub-model, implicit as volume fraction formulation and the body force formulation was set as an
implicit body force. The material for the two phases in the multiphase was defined as air and
water. For water, the density and viscosity were changed to 1026.06 kg/m3 and 0.0012 kg/ms
respectively. The gravity force was defined as 9.81 m/s2 in negative z-direction. To satisfy the
time step criterion for the SST k-ω turbulence model, the time step was calculated according to
equation 31 and resulting values used in the simulations are presented in table 13. The number of
iterations was set to 1000 for all the performed simulations.

Table 13: The time steps used in the simulations.

Time step Time step

4 knots 0.03 12 knots 0.011
5 knots 0.02 16 knots 0.009
6 knots 0.02 20 knots 0.007
7 knots 0.02 24 knots 0.006
8 knots 0.015 32 knots 0.004

The boundary condition for the inlet was set as an pressure inlet, due to the fact that the open
channel flow model is used. The outlet was set to a pressure outlet, the hull, top and bottom were
set to wall and the symmetry plane to symmetry. For the in- and outlet the turbulence intensity
and the turbulent viscosity ratio was set to default, i.e. 5 % and 10, respectively. The free surface
was varying with the speed and the bottom level was set to −8 m.

Since the VOF model is applied, the pressure-based solver is used and the solution is set to be
steady state. The solution method used is a coupled scheme and the second-order upwind scheme

30
is chosen for all but the gradient, the pressure and volume fraction discretizations were least square
cell based, PRESTO! and compressive is used, respectively. The quality of the simulations was
ensured by a residual criteria of 10−3 and a mesh convergence study.

6.4.5 Mesh convergence

A mesh convergence study was made in order to ensure the quality of the results. The study was
performed on four different meshes for the Froude number Fn = 0.42, corresponding to 7 knots.
The number of cells, cell size at the free surface, if the solution converged or not and the resulting
total resistance coefficient CT can be seen in table 14, for each mesh.

Table 14: Grid dependence study of Eelex 2020 for Fn = 0.42.

Mesh number Number of cells Cell size at free surface Converged CT

1 4218791 0.095 m No -
2 5563201 0.085 m Yes 10.5 · 10−3
3 7750496 0.075 m Yes 10.6 · 10−3
4 88799451 0.03 m Yes 10.5 · 10−3

As can be seen in table 14, a refinement of the mesh from 5.56 million to 88.8 million cells does
not affect the total resistance coefficient. However, an even coarser mesh, mesh number 1 with
4.2 million cells, results in a solution that does not converge. The mesh used in the simulations is
mesh number 2, since a finer mesh will contribute to higher computational cost without increasing
the accuracy. With mesh number 2, simulating one speed, including setting up the computational
domain, running the calculations and interpreting the results, can be done within one day. The
entire speed range can be simulated in approximately one week, excluding the mesh convergence
study. The height of the cells at the free surface should ideally be 0.01% of the hull’s length of
the waterline. However, that resulted in a mesh size beyond the computer’s capacity, so the cell
size needed to be adjusted. For mesh number 2, the cell size at the free surface is 0.085 m, which
corresponds to 1.1% of the waterline length. However, the mesh convergence study shows that
refining the mesh at the free surface to 0.04% of the waterline length, as in mesh number 4, does
not affect the accuracy. In summary, mesh number 2 is assumed to give sufficient accuracy, and
can be seen in figure 19.

(a) The mesh from the symmetry plane. (b) Overview of the mesh.

Figure 19: The 5.56 million mesh.

Note that the mesh is not hollow, the mesh in figure 19b is only showing mesh on the outer edges
of the computational domain.

6.4.6 Results from the CFD simulations

The CFD simulations on Eelex 2020 has been performed for the whole speed range, i.e. for 4 − 32
knots, with two different setups regarding the location of the free surface. Both setups are simu-

31
lated without trim, because the identified method to change the trim angle requires modifications
in the CAD-model and remeshing of the computational domain. This procedure is time consum-
ing and due to time limitations the simulations were initially performed without any trim angle.
Consequently, this will affect the accuracy of the results for the higher speeds.

The location of the free surface is referred to as ∆W L, which is the change of waterline length,
where ∆W L = 1 is the waterline length at zero speed. It is assumed that the change in waterline
length is proportional to the change in draught, e.g. ∆W L = 0.9 corresponds to 90% of the wa-
terline, and 90% of the draught at zero speed. The first setup is presented in table 15 where the
free surface in located at the draught at zero speed for all speeds up to 12 knots, ∆W L = 1. For
the higher speeds, a lift force is assumed to be acting on the hull and therefore the free surface is
assumed to be lower, ∆W L = 0.9. The differences in total resistance compared to the model tests
results are presented as Diff. in res. in table 15.

Table 15: The resulting resistance in when applying CFD on Eelex 2020 with setup 1.

Total resistance ∆W L Diff. in res.

4 knots 390 N 1 11.43%
5 knots 542 N 1 −11.58%
6 knots 822 N 1 −13.84%
7 knots 1014 N 1 −32.26%
8 knots 1284 N 1 −43.39%
12 knots 2596 N 1 −21.09%
16 knots 3268 N 0.9 −14.7%
20 knots 4666 N 0.9 19.15%
24 knots 6254 N 0.9 45.87%
32 knots 10156 N 0.9 77%

In order to investigate the importance of the location of the free surface level, the second setup
uses the ∆W L from the model tests to place the free surface level. The used values of ∆W L, as
well as the total resistance compared to the model tests results for the second setup can be seen
in table 16.

32
Table 16: The resulting resistance in when applying CFD on Eelex 2020 with setup 2.

Total resistance ∆W L Diff. in res.

4 knots 390 N 1 11.43%
5 knots 542 N 1 −11.58%
6 knots 822 N 1 −13.84%
7 knots 1008 N 0.99 −32.67%
8 knots 1134 N 0.96 −49.21%
12 knots 1832 N 0.86 −44.32%
16 knots 2192 N 0.74 −42.78%
20 knots 3144 N 0.72 −19.71%
24 knots 4362 N 0.72 1.77%
32 knots 8046 N 0.76 40.84%

It can be seen in table 15 and 16 that the location of the free surface have a great impact on the
results, especially for the planing speeds, which was expected due to the impact of the running
attitude. For semi-planing speeds, a more accurate location of the free surface results in less accu-
rate resistance predictions, which might be explained by the lack of trim angle. In order to get the
most accurate resistance predictions, it has been decided to proceed with the results from setup 1
for speeds up to 16 knots and the results from setup 2 for higher speeds.

The pressure resistance, viscous resistance and total resistance forces are presented in table 17

Table 17: The resulting resistance when applying CFD on Eelex 2020.

4 knots 5 knots 6 knots 7 knots 8 knots

Pressure resistance 320 N 436 N 690 N 840 N 1018 N
Viscous resistance 70 N 106 N 132 N 174 N 266 N
Total resistance 390 N 542 N 822 N 1014 N 1284 N
12 knots 16 knots 20 knots 24 knots 32 knots
Pressure resistance 1950 N 2222 N 1980 N 2680 N 4860 N
Viscous resistance 646 N 1044 N 1164 N 1682 N 3186 N
Total resistance 2596 N 3268 N 3144 N 4362 N 8046 N

According to the theory in section 2.1, the viscous resistance should be dominating in the displace-
ment speeds. However, it can be seen in table 17 that the pressure resistance is larger than the
viscous resistance for all speeds in the CFD simulations. Figure 20 shows the volume fraction on
the bottom of the hull, i.e. the distribution of the two phases air and water occupying the hull
surface for all different speeds. Water is displayed in red and air in blue.

33
(a) 4 knots. (b) 5 knots.

(c) 6 knots. (d) 7 knots.

(e) 8 knots. (f) 12 knots.

(g) 16 knots. (h) 20 knots.

(i) 24 knots. (j) 32 knots.

Figure 20: Volume fraction on the bottom of the hull for all speeds.
34
6.5 Comparison
This section will present a comparison between the results from applied methods and the values
from the conducted model tests. The model test results will serve as a reference value to determine
the accuracy of the other methods. For displacement, the speeds 4 − 8 knots are regarded and the
model tests are compared with results from Holtrop & Mennen and CFD simulations, as well as
the Savitsky method for the higher speeds in the range. For planing, the speeds 12 − 32 knots are
regarded and model tests are compared with the Savitsky method and CFD simulation results.
More detailed information about the results can be found in Appendix E.

6.5.1 Resistance predictions

The resulting resistance force from all the applied methods can be seen in figure 21. Since Holtrop &
Mennen is a method for displacement hulls and Savitsky is a method for planing hulls, theoretically,
the semi-empirical methods should leave a gap where the resistance can not be predicted. This
gap refers to the Froude numbers which characterize a semi-planing hull, which neither of the
semi-empirical methods are valid for. However, since the Savitsky method is applicable from 7
knots for Eelex 2020, the semi-planing speed range is covered by the semi-empirical methods.

Figure 21: The resulting resistance force from all the applied methods on Eelex 2020.

35
For displacement speeds, i.e. the speeds 4 − 8 knots the resulting resistance force for Eelex 2020
can be seen in figure 22.

Figure 22: Resulting resistance force for Eelex 2020 in displacement speeds.

It can be seen in figure 22 that Holtrop & Mennen over-estimates the resistance, while CFD under-
estimates the resistance for all speeds except for 4 knots. The difference in resistance force in
percentage between Holtrop & Mennen and model test and CFD and model tests, respectively,
indicates how well the methods predicts resistance assuming that the model tests provide an
accurate prediction, see table 18. The difference in resistance force between the Savitsky method
and model tests for 7 and 8 knots can also be seen in the table.

Table 18: Deviation in resistance force for Holtrop & Mennen, CFD and the Savitsky method from
model test results.

4 knots 5 knots 6 knots 7 knots 8 knots

Froude number 0.24 0.30 0.36 0.42 0.48
Holtrop & Mennen 2.2% 18.9% 70.6% 54.2% 30.5%
CFD 11.43% −11.58% −13.84% −32.6% −43.39%
Savitsky - - - 10.7% −19.6%

The greatest deviation for Holtrop & Mennen, 70.6%, is found Froude number Fn = 0.36. It is
a much larger deviation than the errors found in previous studies, where a maximum of 16% was
found, however for Fn < 0.2. For the same Froude number, Fn = 0.36, the deviation for CFD is
−13.84% and significantly lower than the deviation for Holtrop & Mennen. The greatest deviation
for CFD is found for Fn = 0.48 and is −43.39%. For the lower speeds in the range, the devia-
tions are close to the 10% that are found in previous studies. Although the error increases with
the speed, CFD seems to result in more accurate predictions in general. However, the Savitsky
method provides results with better accuracy than both Holtrop & Mennen and CFD for 7 and 8
knots.

36
For planing speeds, i.e. 12 − 32 knots, the resulting resistance force for Eelex 2020 can be seen in
figure 23.

Figure 23: Resulting resistance force for Eelex 2020 in planing speeds.

In figure 23 it can be seen that CFD under-estimates the results for the lower speeds in the
range, while it for the higher speeds over-estimates the resistance force. The Savitsky method is
consistently under-estimating the resistance and the curves for the Savitsky method and model
tests have similar shapes. The difference in resistance force in percentage between the Savitsky
method and model tests, and CFD and model tests respectively can be seen in table 19.

Table 19: Difference in resistance force in percentage between model tests and Savitsky.

12 knots 16 knots 20 knots 24 knots 32 knots

Froude number 0.72 0.96 1.20 1.44 1.92
Savitsky −20.8% −18.7% −13.7% −13.9% −18.8%
CFD −21.09% −14.7% −19.71% 1.77% 40.84%

For the Savitsky method, the largest deviation of −20.8% is found for Fn = 0.72. This is a
significant smaller deviation than the ones found for Holtrop & Mennen at displacement speeds.
The CFD results are less consistent than the results from the Savitsky method, which has an
average error of 17.18%.

37
The Savitsky method shows good agreement on the resulting trim angles for the model tests, see
figure 24, for the speeds 12 − 32 knots. The greatest deviation of −15.3% are found for 16 knots
and the smallest deviation of −4.8% for 12 knots.

Figure 24: The resulting trim angle from model tests and the Savitsky method.

It can also be noted that Savitsky seems to under-predict the trim for the speeds 12 − 32 knots.

6.5.2 Power predictions and efficiency

The resulting resistance power for displacement and planing speeds can be seen in table 20 and 21
respectively.

Table 20: Resulting resistance power from applying model tests, Holtrop & Mennen, CFD and the
Savitsky method on Eelex 2020 in displacement speeds.

4 knots 5 knots 6 knots 7 knots 8 knots

Froude number 0.24 0.30 0.36 0.42 0.48
Model tests 0.7 kW 1.6 kW 2.9 kW 5.4 kW 9.3 kW
Holtrop & Mennen 0.74 kW 1.88 kW 5.03 kW 8.31 kW 12.18 kW
CFD 0.80 kW 1.39 kW 2.54 kW 3.65 kW 5.28 kW
Savitsky - - - 5.56 kW 7.5 kW

Table 21: Resulting resistance power from applying model tests, the Savitsky method, and CFD
on Eelex 2020 in semi-planing and planing speeds.

12 knots 16 knots 20 knots 24 knots 32 knots

Froude number 0.72 0.96 1.20 1.44 1.92
Model tests 20.3 kW 31.5 kW 40.3 kW 52.9 kW 94.0 kW
Savitsky 16.1 kW 25.6 kW 34.8 kW 45.5 kW 76.4 kW
CFD 16.03 kW 26.90 kW 32.35 kW 53.86 kW 132.46 kW

38
Table 22 presents the different components of the total efficiency. The first four rows represent
the input values required to calculate the propulsion efficiency ηD , where the open water propeller
efficiency is an estimation for the single propeller used for Eelex 2020. The last three rows present
the gear box efficiency, motor efficiency and battery capacity for Eelex 2020.

Table 22: Efficiency for Eelex 2020.

Symbol Value Unit

Number of propellers 1 -
Open water propeller efficiency ηO 60 %
Relative rotary efficiency ηR 100 %
Hull efficiency ηH 106 %

Propulsion efficiency ηD 63 %
Gear box efficiency ηG 96 %
Motor efficiency ηM 95 %
Battery capacity EB 120 kWh

The resulting range for Eelex 2020 from the different methods, assuming constant speed and
including a sea margin of 15% to capture the influence of waves and wind, can be seen in figure
25.

Figure 25: Resulting range for the different methods.

39
It can be seen in figure 25 that the range decreases significantly from approximately 300 − 50
nautical miles (NM) for increasing speeds in the range of 4 − 12 knots, and both Holtrop &
Mennen and CFD show this behaviour. After 12 knots, the curve flattens out, and the decrease
in range is not as steep as for the lower speeds, according to all methods applied in the planing
speed range. The corresponding time for each range is presented in table 23 and 24.

Table 23: Resulting time and range for displacement speeds.

Holtrop &
Model test CFD Savitsky
Mennen
Time Range Time Range Time Range Time Range
4 knots 86h 344 NM 81h 327 NM 75h 300 NM - -
5 knots 37h 188 NM 32h 160 NM 43h 216 NM - -
6 knots 20h 124 NM 12h 72 NM 23h 142 NM - -
7 knots 11h 78 NM 7h 50 NM 16h 115 NM 10 h 70 NM
8 knots 6h 51 NM 5h 39 NM 11h 91 NM 8h 64 NM

Table 24: Resulting time and range for planing speeds.

Model test Savitsky CFD

Time Range Time Range Time Range
12 knots 3h 35 NM 3h 45 min 45 NM 3h 45min 45 NM
16 knots 1h 50min 30 NM 2h 20min 37 NM 2h 15min 35 NM
20 knots 1h 30min 29 NM 1h 45min 34 NM 1h 50min 37 NM
24 knots 1h 27 NM 1h 20min 31 NM 1h 26 NM
32 knots 40min 20 NM 45min 25 NM 30min 14 NM

It can be seen in table 24 that all of the three methods for resistance predictions in planing speeds,
and especially in the cruising speed of 24 knots, give similar predictions of the range. However, for
the displacement speeds the results of the range predictions are more varying.

6.6 Evaluation
Table 25 presents the evaluation of the methods in a Pugh Matrix. Model tests are set as a base-
line, and each method is rated from -2 to +2 on each criterion, where -2 means that the method
performs worse than the baseline on that particular criterion, and +2 better than the baseline.

The weights are relative values, where the most important criterion is assigned the highest value
of the weighting scale, i.e. 5, and the least important criterion is assigned the lowest value, i.e. 1.
Since the weights are not absolute values, the weighting must be re-evaluated if new criteria are
added and the ranking changes. A criterion with the weight 1, is therefore not irrelevant but of
less importance than criteria with a higher value.

40
Table 25: Pugh Matrix with X Shore’s criteria.

Holtrop &
Criteria Weight Model tests Savitsky CFD
Mennen
consumption
Low time

Preparing calculations 4 0 +2 +2 +1
Performing calculations 3 0 +2 +2 -1
Interpreting results 5 0 +2 +2 +2
Low cost

Equipment* 2 0 +2 +2 +1
quality

Accuracy 1 0 -2 -1 -1
High

Precision 3 0 -2 0 -1
accessibility

Equipment* 5 0 +2 +2 +1
High

Competence 4 0 +2 +2 0

Score 0 38 45 14

* Equipment refers to software licenses, hardware, tools and facilities.

The method with the highest total score of 45 is the Savitsky method. Holtrop & Mennen has a
slightly lower score, 38, due to its low scores on the quality criteria. CFD has the lowest total score,
14, since it is much more time consuming and has a lower accessibility than the semi-empirical
methods. It can be seen in the Pugh Matrix, table 25, that low time consumption and cost, and
high accessibility have higher priority than high accuracy, when choosing a method to predict resis-
tance in the design phase. Accuracy is therefore assigned the weight 1, not because it is irrelevant,
but because it has lower priority. However, precision which is the other quality criteria is weighted
higher since the consistency of the results determines if the method can be used at all.

A low time consumption is of high importance, due to the fast-paced design processes at the
company. It can be seen in the matrix that it is more important that determining input values and
interpreting results are less time consuming than the importance of the time spent on perform-
ing the calculations. This means that the most important aspect of low time consumption is the
active work hours spent on the task. If running a CFD-simulation takes hours, or days, without
demanding active work hours and deliver results within time plan, it can still be a suitable method.
The cost of equipment is ranked lower than both time consumption and accessibility. While cost
is an important aspect, equipment cost is assumed to be spread out over a long period of time.
It might therefore be less expensive to invest in equipment to enable in-house approaches than
to constantly hire external consultants. Cost will consequently have less importance and not be
a deciding factor. The accessibility of the equipment is however of bigger importance in order to
enable in-house approaches. The same goes for competence, since competence deficiencies in the
company require external consultants. Note that the weighting is set based on the purpose of this
study, i.e. to find a suitable approach to predict resistance for X Shore as a start-up company.

41
7. Discussion and Conclusion
All resistance predictions are made on a bare hull and in calm water conditions. When converting
the resistance to range, a sea margin has been included in order to increase the accuracy of the
range estimations. However, the estimated ranges for different speeds are, despite the sea mar-
gin, higher than the ranges that can be expected from a real life scenario, where resistance from
appendages are present. Adding resistance from appendages will give more accurate results, but
requires modifications of the methods investigated in this study. Several aspects considering the
efficiency might influence the outcomes of the methods, e.g. the hull efficiency which is calcu-
lated using simplified equations and the open water propeller efficiency which is assumed to be
constant over the entire speed range. Moreover, energy losses from the batteries to the motor are
not accounted for. Since the study is limited to one hull, no conclusions can be drawn about the
performance of the methods in general.

The quality of the outcome of the Pugh Matrix depends on the selection of criteria. In order
for the Pugh Matrix to guide the decision making process, all relevant criteria must be included,
and no irrelevant criteria should be present in the matrix. The criteria in this study was selected
after discussions between the authors and X Shore, and are assumed to be sufficient for the pur-
pose of this study. However, to use this evaluation method in the future, X Shore needs to ensure
that all relevant aspects are covered in the selected criteria. The company might prioritize the
criteria differently in future design processes, and the recommendations in this report might not
be suitable. However, the same evaluation process may still be useful in order to choose the most
suitable method. Finally, it should be clarified that even though the selected criteria are relevant
and the weighting corresponds to the company’s priorities, the method with the highest score is
not necessarily the most suitable in all cases. While a Pugh Matrix serves as a tool in the decision
making, decisions should not be based solely on the matrix scores.

The methods evaluated in this study is supposed to fill a gap in the early design phase. It is
of importance that the methods have a sufficient precision to serve as a guide when making design
changes. The methods should give predictions of the right magnitude, as well as capture differ-
ences between different designs, but the exact value of the resistance is not important in the early
design phase. In later design phases, when the design is set and more accurate predictions may be
valuable, the recommendations in this study may not be suitable.

The results from Holtrop & Mennen on Eelex 2020 deviate between 2.2% and 70.6% from the
model test results, which is a significantly larger error than 16% shown in a previous study. It can
also be concluded that the method has a low precision, and the predictions do not follow a clear
pattern. The method is previously shown to be less accurate in cases where the hull geometry is
close to the applicability limits, and the fact that the length-beam ratio of Eelex 2020 is below
the applicability limit is a possible explanation to the large deviations and low precision. Because
Holtrop & Mennen is developed from model tests performed on a number of hulls, the method is
only applicable for hull geometries similar to the ones the equations are based on. This is clearly
a disadvantage of the method, since it only can be applied when designing hulls with traditional
shapes and dimensions. To summarize, Holtrop & Mennen does not provide accurate resistance
predictions for Eelex 2020. Despite the low accuracy and precision, Holtrop & Mennen is a quick
and user friendly method that does not require neither high competence nor advanced equipment.

The results from Holtrop & Mennen show that the wave resistance is the dominant resistance
component as the speed increases. This corresponds well with the theory presented in section 2.1,
which states that the pressure resistance, i.e. the wave resistance, increases with increasing speed.
This indicates that although Holtrop & Mennen over-estimates the resistance in an inconsistent
way, the method seems to capture the theoretical behaviour of the resistance forces. It may there-
fore be able to apply the method to compare different hull designs, because even though the value
of the resistance predictions are inaccurate, it may accurately indicate differences between hull
designs. This is however just a speculation, and needs to be confirmed by further investigations.

42
For low speeds, corresponding to Fn ≤ 0.36, the CFD results deviate maximum 13.84% from the
model test results. The deviations are within acceptable limits, but slightly higher than the 10%
error found in previous studies. For 7 and 8 knots, the error is larger, 32.6% and 43.39% respec-
tively. Compared to Holtrop & Mennen, the CFD results are more consistent, especially for the
lower speeds in the displacement range. CFD resulted in a total score of 14 in the Pugh Matrix,
which is less than half of the total score for Holtrop & Mennen. The drawback with CFD is the
time consumption, which is significantly higher than the time for Holtrop & Mennen. However,
the time required to perform a CFD simulation covering all speeds simulated in this study takes
approximately one week in total time, and three days in active time, excluding the mesh conver-
gence study. The time consumption is considered acceptable and although it is higher than the 30
minutes Holtrop & Mennen takes, one week can be reasonable to spend on resistance predictions.
Another aspect influencing the Pugh Matrix outcome is the highly prioritized accessibility, where
CFD requires both high competence and suitable hardware and licence, which Holtrop & Mennen
does not. If CFD will be applied in-house, X Shore would need to invest in suitable equipment and
competence, which might not be an obstacle due to that low cost is not highly prioritized. Once
X Shore have equipment and competence, the accessibility will increase. Based on the arguments
mentioned above, in combination with the very low precision and accuracy from Holtrop & Men-
nen, CFD is a more suitable method for X Shore than Holtrop & Mennen, despite the contradictory
Pugh Matrix scores. However, the Savitsky method applied for 7 and 8 knots predicts results with
better accuracy than both Holtrop & Mennen and CFD.

Compared with the model test results, the Savitsky method under-predicts the results by 20.8% −
13.7%, with an average deviation of 17.18% for the speeds 12 − 32 knots. The maximum error
is for F n = 0.72, which means that although the applicability criteria are fulfilled, the boat has
not reached planing and a large error can therefore be expected. It can be seen in figure 23, that
although the Savitsky method under-predicts the resistance for the speed range 12 − 32 knots, the
shape of the curve is similar to the curve for model tests. For 7 knots, the Savitsky method over-
predicts the resistance with 10.7%, which deviates from the behaviour of the method for the higher
speeds. Nevertheless, the Savitsky method provides the most precise results of all applied methods
for the speeds that fulfills the applicability criteria of the Savitsky method. The average deviation
of the absolute values for the Savitsky method applied to the speeds 7 − 32 knots is 17.58%. By
modifying the method and include a correction factor of 1.1758 the accuracy can be improved, see
Appendix F. This correction factor has been added in the developed Python program.

The predicted trim angles from the Savitsky method correspond well with the angles obtained
in the model tests, and the error is between 4.8% and 15.3%. The maximum deviation is 0.77◦ and
occurs at F n = 0.96. Like Holtrop & Mennen, the Savitsky method is a semi-empirical method
based on experiments, and its applicability is limited to hull shapes similar to the ones that the
equations are derived from. However, Eelex 2020 fulfills the applicability criteria, which indicates
that the method should give acceptable results. Holtrop & Mennen and the Savitsky method have
in common that the input values are easily changed, which enables quick adjustments of the hull
characteristics and indications of their impact on the resistance. According to section 2.1, LCG
should be positioned at 40%−45% of the chine length from the transom for displacement hulls, and
25% − 35% of the chine length for planing hulls. At Eelex 2020, LCG is positioned approximately
40% forward of the transom, which is a good position for displacement, but not optimal for planing
speeds. With the flexibility that the Savitsky method offers, LCG can easily be adjusted in order
to determine how sensitive the total resistance is to the position. To make a similar study with
CFD would take significantly more time and effort.

The CFD simulations for planing speeds vary more than the results from the Savitsky method.
While the smallest error is very close to the model test result, only 1.77% deviation at Fn = 1.44,
the largest error is as much as 40.84% at Fn = 1.92. Previous studies show that CFD should
yield results with less than 10% error, and that they generally are smaller than the errors from the
Savitsky method. The results from this study show the opposite, and one possible explanation is
that the ITTC guidelines used are developed for Froude numbers of the order 0.1 and rarely above
1. Thus, above 16 knots, Fn > 1 for Eelex 2020, the ITTC guidelines may not be appropriate.

43
Furthermore, the fact that trim angle is neglected and the running attitude is not accounted for
properly, affects the accuracy negatively. According to the Pugh Matrix, the Savitsky method
is the most suitable method for resistance predictions in planing speeds, with a total score of 45
compared to 14 for CFD. Due to that CFD both shows lower precision, and requires more time
to perform, the recommendation is to apply the Savitsky method for resistance predictions in the
planing speed range.

For semi-planing, i.e. 0.4 < Fn < 1 and speeds 7 − 16 knots, the running attitude of the hull
varies significantly compared to displacement speeds, and consequently the resistance is harder to
predict. Since the trim angle has been neglected from the simulations, this might be the explana-
tion to the large deviations of the CFD results, especially for 7 and 8 knots. Possibly, a dynamic
mesh which captures the hull motions, and therefore identifies the running attitude, can improve
the CFD results in this speed range.

In contrast to the semi-empirical methods, the applicability of CFD is not limited by hull ge-
ometry. The limitations for CFD are rather competence and computer capacity. The differences
between the results from setup 1 and setup 2 show that the resistance predictions are sensitive to
the location of the free surface. Estimating the free surface level accurately requires knowledge of
the behaviour of planing hulls, alternatively data from model tests or other methods. For planing
speeds, the largest deviation is 40.84% for 32 knots when the free surface location is changed ac-
cording to model test data. It should be noted that ∆W L is larger for 32 knots than for 20 and 24
knots which could be a source of error. Adjusting the free surface to the same level as for 20 and
24 knots would decrease the error, but the result would probably still be over-estimated. Qualified
estimations of the free surface has been proven to be of importance. Another interesting aspect
is the effect of a potential implemented trim in the simulations. All the performed simulations in
this study are done without any trim, but a trim angle would probably improve the results since it
would increase how accurate the running attitude is modelled. Estimating the trim would in the
same way as the free surface estimations require a qualified guess, and since the Savitsky method
has shown good agreement of the trim angles, a suggestion is to combine CFD and the results of
trim angle from the Savitsky method. Combining CFD and the Savitsky method has been tested
for 12 and 32 knots, see Appendix F. None of the simulations did converge, and it can be concluded
that in order to implement trim in the CFD simulations, at least a new mesh convergence study
needs to be conducted.

The selection of methods and models for the CFD simulations is based on theory in chapter
4. However, other methods and models are available and no further investigation regarding the
impact of them has been performed. In this study, a mesh convergence study was performed for
one Froude number. Consequently, the dimensionless wall distance y + is the same in all simula-
tions, which is an inaccurate assumption. To increase the accuracy of the simulations and ensure
the quality of the mesh, mesh convergence studies should be made over the whole range of Froude
numbers and the value of the dimensionless wall distance should be verified. The blue areas on the
hull in figure 20 symbolize air, which may indicate that numerical ventilation occurs. This would
decrease the total resistance prediction, but looking at figure 20i for 24 knots, a large area of air
in the middle of the hull can be seen. However, the result for 24 knots has the smallest deviation
of all simulated speeds, which may indicate that the blue areas have another explanation than
numerical ventilation. By checking and adjusting the dimensionless wall distance, the effects from
the potential numerical ventilation can be reduced and the overall quality can be improved. Since
the y + value is supposed to capture the flow close to the hull surface, poor meshing of the boundary
layer will affect the total resistance. The fact that the y + value has not been checked in this study
might explain why the viscous resistance is lower than the pressure resistance, even for low speeds.
Another way of increasing the accuracy and quality of the simulation is to change the convergence
criteria, i.e. lower the residuals. However, the simulations have shown good convergence and stable
solutions with the residuals used in this study, and the quality is assumed to be sufficient in order
to determine the suitability of CFD as a method for X Shore.

44
Besides the possibility to easily evaluate different hulls by simply changing the CAD-file, CFD
has a number of advantages. Although the purpose of applying CFD in this study was to predict
resistance, the simulations can also predict e.g. pressure distribution on the hull, wave pattern,
and turbulence. For that reason, it might be useful in the later design phases and even for higher
speeds, if a more complex setup is applied. Additionally, CFD has a lower economical cost than
model tests. The major advantage of model tests is that it provides accurate results. Even though
it is both costly and time consuming, there are several reasons why it may be useful to perform
occasionally, e.g. when predictions with high accuracy are needed. Additionally, model tests can
be used to verify and adjust other methods. However, it is not necessary to include model tests in
every design process, and if used, it should not be applied in early design phases. As X Shore has a
quick manufacturing process and access to the full-scale hulls, another way to validate and adjust
the methods is to perform full-scale tests where total resistance and trim angle are measured.

7.1 Conclusion
From the results, it can be concluded that the recommended approach for X Shore to implement
in their design process to predict resistance is the Savitsky method for all speeds that fulfils the
applicability criteria of the method. If predictions for lower speeds are of interest, CFD is the
recommended method. When applying the Savitsky method, it is recommended to follow the
procedure in section 6.3 and use the program developed by the authors, where the correction
factor has been included. The CFD simulations should be applied according to section 6.4.

45
8. Future work
One focus for future work should be the mesh used in the CFD simulations. The mesh convergence
study have only been conducted for one Froude number, and to improve the accuracy mesh conver-
gence should be checked for all Froude numbers corresponding to speeds between 4 and 32 knots.
A suggestion is to perform the CFD simulations with a fixed trim on the hull, in order to determine
the sensitivity of the results to changes of the trim angle. It would also be of interest to compare
the results from this study with results obtained from simulations performed with a dynamic mesh.

Another improvement that would result in more accurate estimations of the range would be to
include appendages in the methods. This requires a re-evaluation of the recommended systematic
approach, since it may influence the accuracy of the different methods. To add resistance from
appendages in Holtrop & Mennen is straightforward, since the method include it by default. How-
ever, for the Savitsky method and CFD, it is more complicated. The Savitsky method does not
handle appendages by default and need to be modified. In order to capture the influence from
appendages in CFD simulations, appendages need to be properly meshed. Furthermore, the CFD
methods and schemes would need to be re-evaluated to be suitable for simulations with appendages.

Even though the semi-empirical methods, and especially Holtrop & Mennen, deviate significantly
from model test results, it could be of interest to apply the methods on different hulls and inves-
tigate whether it can be used for comparisons of hulls.

46
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48
A. Holtrop & Mennen

Nomenclature
ABT = transverse bulb area at the position of the bow
where the still-water surface intersects the stem RA = model-ship resistance
AM = midships cross-sectional area RAP P = resistance of appendages
AT = immersed part of the transverse area of RB = additional pressure resistance of bulbous
the transom at zero speed bow near the water surface
AW P = waterplane area Re = Reynolds number
B = maximum breadth RF = frictional resistance according to ITTC
CA = correlation allowance coefficient formulation
CB = block coefficient Rtotal = total resistance of a ship
Cf = friction coefficient RT R = additional pressure resistance of im-
CM = midship section coefficient mersed transom stern
CP = prismatic coefficient RW = wave-making and wave-breaking resis-
CW P = waterplane area coefficient tance
D = average draught RW −A = wave-making and wave-breaking resis-
DA P = draught at aft perpendicular tance for Fn < 0.4
DF P = draught at forward perpendicular RW −B = wave-making and wave-breaking resis-
Fn = Froude number tance for Fn > 0.55
Fni = Froude number based on the immersion RW −C = wave-making and wave-breaking resis-
FnT = Froude number based on the transom im- tance for 0.4 < Fn < 0.55
mersion S = wetted surface of the hull
g = gravitational acceleration SAP P = wetted surface of the appendages
hB = position of the center of the transverse U = ship velocity
area ABT above the keel line (1 + k1 ) = form factor describing the viscous re-
lcb = longitudinal position of the center of buoy- sistance of the hull form in relation to RF
ancy forward of 0.5L as percentage of L (1 + k2 ) = appendage resistance factor
LW L = length on waterline ∇ = volume displacement
LR = length of the run ν = kinematic viscosity
LP P = length between perpendiculars ρ = density of the fluid
PB = coefficient as a measure for the emergence

If nothing else is specified, the equations in this appendix come from Holtrop & Mennen [19].

The total resistance is subdivided into

Rtotal = RF (1 + k1 ) + RAP P + RW + RB + RT R + RA (A.1)

and is determined by the following steps.

Step 1:
The first term, RF is the frictional resistance according to the ITTC formulation [18]
1 2
RF = ρU Cf S (A.2)
2

where ρ is the density of the water, U is the speed of the boat and S is the wetted surface of the
hull. Cf is the friction coefficient [23] defined as
0.075
Cf = (A.3)
[log10 (Re) − 2]2

I
where Re is the Reynolds number defined as
U LP P
Re = (A.4)
ν
where LP P is the length between perpendiculars and ν is the kinematic viscosity [22].

The wetted surface of the hull can be approximated as

p
S =LW L (2D + B) (CM )(0.453 + 0.4425CB − 0.2862CM
(A.5)

B ABT
− 0.003467 + 0.3696CW P ) + 2.38
D CB

where LW L is the waterline length, B is the maximum breadth and ABT is the transverse sectional
area of the bulb at the position where the still-water intersects the stem. The average draught D
is defined as
DF P + DAP
D= (A.6)
2

where DF P is the draught at the forward perpendicular and DAP is the draught at the aft per-
pendicular of the hull. CM is the midship section coefficient, CB is the block coefficient and CW P
is the waterplane area coefficient defined as
AM
CM = (A.7)
BD
∇
CB = (A.8)
LP P BD
AW P
CW P = (A.9)
LB

where AM is the immersed midship sectional area at LW L /2, ∇ is the displacement volume and
AW P is the waterplane area [9].

Step 2:
The frictional resistance is multiplied with a form factor (1 + k1 ) describing the viscous resistance
of the hull form in relation to RF . The form factor is defined as

0.92497
B
(1 + k1 ) = c13 (0.93 + c12 (0.95 − CP )−0.521448 (1 − CP + 0.0225lcb)0.6906 ) (A.10)
LR

where lcb is the longitudinal position of the center of buoyancy forward of 0.5LW L as a percentage
of LW L . The prismatic coefficient CP is based on LP P and defined as [9]
∇
CP = (A.11)
LP P AM

In the form factor formula, LR is a parameter reflecting the length of the run and is defined as

1 − CP + 0.06CP lcb
LR = LW L (A.12)
4CP − 1

The form factor does also contain two coefficients, c12 and c13 . The first one is defined as

II
 0.2228446
D D

 LW L
 when LW L > 0.05





 2.078
c12 = 48.20 D − 0.02 + 0.479948 when 0.02 < D
< 0.05 (A.13)
 LWL LW L







0.479948 D
when LW L < 0.02

The other coefficient c13 is defined as

c13 = 1 + 0.003Cstern (A.14)

and accounts for the specific shape of the afterbody related to the coefficient Cstern which have
the tentative guidelines presented in table A.1.

Table A.1: Tentative guidelines for the specific shape of the afterbody of the ship.

Afterbody form Cstern

V-shaped sections -10
Normal section shape 0
U-shaped sections with Hogner stern +10

Step 3:
The appendage resistance is the next term in the total resistance formulation and it is defined as

RAP P = 0.5ρU 2 SAP P (1 + k2 )eq Cf (A.15)

where SAP P is the wetted surface of the appendages and (1 + k2 ) is the appendage resistance
factor. Tentative values for the factor (1 + k2 ) are presented in table A.2, obtained from resistance
tests with bare and appended ship models.

Table A.2: Tentative values for the appendage resistance factor (1 + k2 ).

Approximate (1 + k2 ) values
Rudder behind skeg 1.5 - 2.0
Rudder behind stern 1.3 - 1.5
Twin-screw balance rudders 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 keels 1.4

III
A combination of appendages is determined as
P
(1 + k2 )SAP P
(1 + k2 )eq = P (A.16)
SAP P

Step 4:
Following, the wave resistance RW is depending on the Froude number Fn , based on the waterline
length LW L and defined as
U
Fn = √ (A.17)
gLW L

For the speed range Fn < 0.4, the wave resistance is defined as
d −2
RW −A = c1 c2 c5 ∇ρge(m1 Fn +m4 cos(λFn ))
(A.18)

and for the speed range Fn > 0.55 the wave resistance is defined as
d −2
RW −B = c17 c2 c5 ∇ρge(m3 Fn +m4 cos(λFn ))
(A.19)

where c1 , c2 , c5 , c17 , m1 , m3 , m4 and λ are coefficients, g is the gravitational acceleration and

d = −0.9. [43] [19] For the speed range 0.4 < Fn < 0.55 the wave resistance is defined as

(10Fn − 4)(RW −B0.55 − RW −A0.4 )

RW −C = RW −A0.4 + (A.20)
1.5

which is an interpolation formula where RW −A0.4 is the wave resistance for Fn = 0.4 and RW −B0.55
is the wave resistance for Fn = 0.55 according to equation A.18 and A.19 respectively [43].

The first coefficient c1 is determined as

1.07961
D
c1 = 2223105c3.78613
7 (90 − iE )−1.37565 (A.21)
B
and  0.33333
B B



 0.229577 LWL
when LW L < 0.11





c7 = B B (A.22)
when 0.11 < < 0.25
 LW L

LW L




 LW L
B
0.5 − 0.0625

when > 0.25
B LW L

The angle of the waterline at the bow in degrees with reference to the center plane is called the
half angle of entrance, iE , neglecting the local shape at the stem. If this angle is unknown, it can
be determined as

0.34574 0.16302
0.80856 LR
−( B
L
) (1−CW P )0.30484 (1−CP −0.0225lcb)0.6367 100∇
B L3
iE = 1 + 89e WL (A.23)

The second coefficient c2 accounts for the reduction of the wave resistance due to the action of a
bulbous bow and is defined as
√
c2 = e(−1.89 c3 )
(A.24)

IV
where the coefficient determining the influence of the bulbous bow on the wave resistance is defined
as

0.56A1.5
BT
c3 = √ (A.25)
BD(0.31 ABT + DF P − hB )

where hB is the position of the center of the transverse area ABT above the keel line. Next
coefficient express the influence of a transom stern on the wave resistance and is defined as
1 − 0.8AT
c5 = (A.26)
BDCM

where AT is the immersed part of the transverse area of the transom at zero speed. Following, c17
is defined as [43]
2.00977 1.40692
−1.3346 ∇ LW L
c17 = 6919.3CM −2 (A.27)
L3W L B
The remaining coefficients m1 , m3 , m4 and λ are determined as follows [43] [19]
1/3
LW L ∇ B
m1 = 0.0140407 − 1.75254 − 4.79323 − c16 (A.28)
D LW L LW L

where 
2 3
8.07981CP − 13.8673CP + 6.984388CP
 when CP < 0.80
c16 = (A.29)

1.73014 − 0.7067CP when CP > 0.80


0.326869 0.605375
B D
m3 = −7.2035 (A.30)
LW L B

−3.29
m4 = c15 0.4e(−0.034Fn )
(A.31)

where
L3W L



 −1.69385 when ∇ < 512




LW L

−8.0 L3W L
c15 = −1.69385 + ∇1/3
when 512 < < 1727 (A.32)

 2.36 ∇




 L3W L
0.0 when > 1727

∇

and 
LW L LW L

1.446CP − 0.03
 B when B < 12
λ= (A.33)
LW L

1.446CP − 0.36 when > 12

B

Step 5:
The additional resistance RB due to the presence of a bulbous bow near the surface is defined as
−2
0.11e(−3PB ) Fni
3 1.5
ABT ρg
RB = 2 (A.34)
1 + Fni

where PB is the coefficient of the measure for the emergence of the bow defined as

V
√
0.56 ABT
PB = (A.35)
DF P − 1.5hB

and Fni is the froude number based on the immersion determined as

U
Fni = p √ (A.36)
g(DF P − hB − 0.25 ABT ) + 0.15U 2

Step 6:
The additional pressure resistance RT R due to the immersed transom is defined as

RT R = 0.5ρU 2 AT c6 (A.37)

where 
0.2(1 − 0.2FnT )
 when FnT < 5
c6 = (A.38)

0 when FnT ≥ 5


related to the Froude number based on the transom immersion

U
FnT = q (A.39)
2gAT
B+BCW P

Step 7:
The last contribution to the total resistance is the model-ship correlation resistance RA is supposed
to describe primarily the effect of the hull roughness and the still-air resistance. It is defined as
1 2
RA = ρU SCA (A.40)
2

where CA is the correlation allowance coefficient based on results of speed trials

r
−0.16 LW L 4
CA = 0.006(LW L + 100) − 0.00205 + 0.003 C c2 (0.04 − c4 ) (A.41)
7.5 B

where 
DF P DF P
 LW L
 when LW L ≤ 0.04
c4 = (A.42)
DF P

0.04 when > 0.04

LW L

VI
B. Savitsky method

Nomenclature

a = distance between the viscous drag compo- LCG = longitudinal center of gravity measured
nent Df and the center of gravity, measured along the keel from the transom
normal to Df Re = Reynolds number
B = beam of the boat U = ship velocity
c = the distance between the center of pressure U1 = speed at the hull bottom
and the center of gravity, measured parallel to V CG = vertical center of gravity measured from
the keel the keel line
Cf = friction coefficient β = deadrise angle
CL0 = lift coefficient γ = specific weight of water
CLβ = planing coefficient ∆ = displacement mass of the boat
Cp = center of pressure = the inclination of the propeller axis relative
CV = planing coefficient to the keel
D = total drag Λ = mean wetted length-beam ratio
Df = viscous drag component ν = kinematic viscosity
f = distance between the propeller axis and the ρ = density of water
center of gravity, measured normal to the pro- τ = trim angle
peller axis ϕ = γ/g
g = gravitational acceleration

If nothing else is specified, the equations in this appendix come from Savitsky [20].

Note that all input values must be expressed in imperial units, and not SI-units.

Step 1:
Calculate the planing coefficients CV and CLβ :
U
CV = p (B.1)
(gB)
∆
CL β = 1 2 2
(B.2)
2 ϕU B

where g is the gravitational acceleration, B is the beam of the boat, ∆ is the displacement, U is
the speed of the boat and ϕ is defined as
γ
ϕ= (B.3)
g

where γ is the specific weight of water.

Step 2:
Making the following steps for a number of different values of τ will enable solving a moment
equation and through linear interpolation finding the equilibrium trim angle.

Solve for CL0 :

0.60
CLβ = CL0 − 0.0065βCL0 (B.4)

VII
where β is the deadrise angle. The mean wetted length-beam ratio Λ is determined by solving
following equation for Λ:

0.0120Λ1/2 + 0.0055Λ5/2

CL0 = τ 1.1 (B.5)
CV 2

The speed at the hull bottom U1 differs from the speed of the boat U and is approximated to

U1 = 0.95U (B.6)

Reynold’s number is calculated by

U1 ΛB
Re = (B.7)
ν

where ν is the kinematic viscosity of water. The frictional coefficient is calculated according to
ITTC [23] and defined as
0.0075
Cf = (B.8)
[log10 (Re) − 2]2

The viscous component of drag Df is defined as

ϕU1 2 ΛB 2 (Cf + ∆Cf )

Df = (B.9)
2cos(β)

where ∆Cf = 0.0004 according to ATTC standard. The total drag D can then be calculated by

ϕU1 2 ΛB 2 (Cf + ∆Cf )

D = ∆tan(τ ) + (B.10)
2cos(β)cosτ

The center of pressure Cp can be calculated as

1
Cp = 0.75 − 5.21CV2 (B.11)
Λ2 + 2.39

and is needed to determine the distance between the center of pressure and the center of gravity,
measured parallel to the keel:
c = LCG − Cp ΛB (B.12)

where LCG is the longitudinal center of gravity measured along the keel from the transom. The
distance between the viscous component of drag Df and the center of gravity, measured normal
to Df is defined as
b
a = V CG − tanβ (B.13)
4

where VCG is the vertical center of gravity measured from the keel line. Finally, the momentum
equation that should be fulfilled in order to obtain the equilibrium trim angle is

(1 − sin(τ )sin(τ + ))c
∆ − f sinτ + Df (a − f ) = 0 (B.14)
cosτ

where is the inclination of the propeller axis relative to the keel and f is the distance between
the propeller axis and the center of gravity, measured normal to the propeller axis.

VIII
Step 3:
When equation B.14 has been calculated for different values of τ , i.e f (τ ), linear interpolation can
be used to find the equilibrium trim angle τeq that fulfills the equation.

τ2 − τ1
τeq = τ1 − f (τ1 ) (B.15)
f (τ2 ) − f (τ1 )

Step 4:
When the equilibrium trim angle τeq is identified, equations B.5-B.9 need to be calculated again,
with τeq . Finally, by using the obtained values in

ϕU1 2 Λeq B 2 (Cf eq + ∆Cf )

D = ∆tan(τeq ) + (B.16)
2cos(β)cosτeq

the total resistance is evaluated. The index eq means that the value is calculated with the equilib-
rium trim angle τeq , and τ1 and τ2 should be chosen so that f (τ1 ) < 0 and f (τ2 ) > 0. The unit of
D is pound-force, lbf, and can be converted to newton:
D
DN = 9.81 (B.17)
2.205

IX
C. Program interface for Holtrop &
Mennen
Figure C.1 shows the input window in the Python program developed for Holtrop & Mennen. The
output window is presented in figure C.2.

Figure C.1: User interface of Holtrop & Mennen program.

Figure C.2: The results from applying Holtrop & Mennen on Eelex 2020.

X
D. Program interface for the Savitsky
method
Figure D.1 shows the input window in the Python program developed for the Savitsky method.
The output window is presented in figure D.2.

Figure D.1: User interface of the Savitsky program.

Figure D.2: The results from applying the Savitsky method on Eelex 2020.

XI
E. Results
Table E.1 presents the resulting resistance force from applying model tests, Holtrop & Mennen,
CFD and the Savitsky method on Eelex 2020 in displacement speeds 4 − 8 knots.

Table E.1: Resulting resistance force from applying model tests, Holtrop & Mennen, CFD and the
Savitsky method on Eelex 2020 in displacement speeds.

4 knots 5 knots 6 knots 7 knots 8 knots

Froude number 0.24 0.30 0.36 0.42 0.48
Model tests 350 N 613 N 954 N 1497 N 2268 N
Holtrop & Mennen 358 N 729 N 1628 N 2308 N 2960 N
CFD 390 N 542 N 822 N 1014 N 1284 N
Savitsky - - - 1657 N 1823 N

Table E.2 presents the resulting resistance force from applying model tests, the Savitsky method
and CFD on Eelex 2020 in semi-planing and planing speeds 12 − 32 knots. Table E.3 and E.4
present the resulting trim angle and the difference in trim angle from model tests and the Savitsky
method.

Table E.2: Resulting resistance from applying model tests, the Savitsky method, and CFD on
Eelex 2020 in planing speeds.

12 knots 16 knots 20 knots 24 knots 32 knots

Froude number 0.72 0.96 1.20 1.44 1.92
Model tests 3290 N 3831 N 3916 N 4286 N 5713 N
Savitsky 2604 N 3114 N 3379 N 3690 N 4641 N
CFD 2596 N 3268 N 3144 N 4362 N 8046 N

Table E.3: Resulting trim angles from applying model tests and the Savitsky method on Eelex
2020 in planing speeds.

7 knots 8 knots 12 knots 16 knots 20 knots 24 knots 32 knots

Froude number 0.42 0.48 0.72 0.96 1.20 1.44 1.92
Model tests 1.43◦ 2.80◦ 4.17◦ 5.04◦ 4.28◦ 3.52◦ 2.56◦
Savitsky 2.88◦ 3.04◦ 3.97◦ 4.27◦ 3.81◦ 3.23◦ 2.37◦

Table E.4: Difference in trim angle in percentage between model tests and Savitsky.

7 knots 8 knots 12 knots 16 knots 20 knots 24 knots 32 knots

Froude number 0.42 0.48 0.72 0.96 1.20 1.44 1.92
Savitsky 101.4% 8.57% −4.8% −15.3% −11.0% −8.2% −7.4%

XII
F. Modifying and combining methods
F.1 Modifying the Savitsky method with a correction factor
By adding a correction factor of 1.1758 to the total resistance from the Savitsky method, the
accuracy of the method is improved, see figure F.1.

Figure F.1: Resulting resistance force from the modified Savitsky method compared to the model
tests for Eelex 2020.

F.2 Combining CFD and the Savitsky method

The estimated values for trim angles and draught from the Savitsky method were used in the CFD
simulations in order to investigate if the accuracy of the resistance predictions could be improved.
The trim angle was implemented in the CAD file, where the hull was rotated around the coordinate
system. The same computational domain, mesh and setup as in the simulations without trim angles
were used. The free surface was located to generate a draught at transom corresponding to the
values presented in table F.1. The simulations were made for two different speeds, 12 knots and
32 knots.

Table F.1: Values used in the CFD simulations with trim angles.

Draught at
Speed Trim angle Converged
transom
12 knots 3.97◦ 0.55 m No
32 knots 2.37◦ 0.28 m No

None of the simulations with implemented trim converged, and for 32 knots, an error indicating
a too coarse mesh appeared. To combine CFD and the Savitsky method, another mesh and/or
setup is required.

XIII
TRITA TRITA-SCI-GRU 2020:185

www.kth.se

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