CFD ANALYSIS AND SHAPE MODIFICATION IN ORDER TO...
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i CFD ANALYSIS AND SHAPE MODIFICATION IN ORDER TO DEVELOP AN EFFICIENT FISHING VESSEL HULL André Georges Monteiro de Carvalho Projeto de Graduação apresentado ao Curso de Engenharia Naval e Oceânica da Escola Politécnica, Universidade Federal do Rio de Janeiro, como parte dos requisitos necessários à obtenção do título de Engenheiro. Orientador: Paulo Tarso Themistocles Esperança, D.Sc. Rio de Janeiro Outubro de 2018
Transcript of CFD ANALYSIS AND SHAPE MODIFICATION IN ORDER TO...
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CFD ANALYSIS AND SHAPE MODIFICATION IN ORDER TO
DEVELOP AN EFFICIENT FISHING VESSEL HULL
André Georges Monteiro de Carvalho
Projeto de Graduação apresentado ao Curso de Engenharia Naval e
Oceânica da Escola Politécnica, Universidade Federal do Rio de
Janeiro, como parte dos requisitos necessários à obtenção do título
de Engenheiro.
Orientador: Paulo Tarso Themistocles Esperança, D.Sc.
Rio de Janeiro
Outubro de 2018
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CFD ANALYSIS AND SHAPE MODIFICATION IN ORDER TO
DEVELOP AN EFFICIENT FISHING VESSEL HULL
André Georges Monteiro de Carvalho
PROJETO DE GRADUAÇÃO SUBMETIDO AO CORPO DOCENTE DO CURSO
DE ENGENHARIA NAVAL DA ESCOLA POLITÉCNICA DA UNIVERSIDADE
FEDERAL DO RIO DE JANEIRO COMO PARTE DOS REQUISITOS
NECESSÁRIOS PARA A OBTENÇÃO DO GRAU DE ENGENHEIRO NAVAL E
OCEÂNICO.
Examinado por:
Orientador: Prof. Paulo Tarso Themistocles Esperança , D.Sc.
Profa. Marta Cecilia Tapia Reyes, D.Sc.
Prof. Carl Horst Albrecht, D.Sc.
RIO DE JANEIRO – RJ – BRASIL
SETEMBRO DE 2018
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Georges Monteiro de Carvalho, André
CFD analysis and shape modification in order to develop
an efficient fishing vessel hull / André Georges Monteiro
de Carvalho. – Rio de Janeiro: UFRJ/Escola Politécnica,
2018.
XIX, 87 p.: il.; 29,7 cm
Orientador: Paulo Tarso Themistocles Esperança
Projeto de Graduação – UFRJ/POLI/Engenharia Naval e
Oceânica, 2018.
Referências Bibliográficas: p. 86-88
1. CFD. 2. Modificação de geometria de casco. 3. Análise
hidrodinâmica. I. Tarso, Paulo Themistocles Esperança. II.
Universidade Federal do Rio de Janeiro, Escola Politécnica, Curso de
Engenharia Naval e Oceânica. III. CFD analysis and shape
modification in order to develop an efficient fishing vessel hull
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AGRADECIMENTOS
Eu gostaria de agradecer a Adrian por me dar a oportunidade de fazer meu estágio em
Vicusdt e por confiar em mim este projeto muito interessante e importante como parte do
meu treinamento para me tornar um engenheiro completo.
Sou profundamente grato ao meu colega de trabalho Marcos Meis, por usar todas as suas
habilidades didáticas de professor comigo, explicando a teoria CFD e a parte prática de
usar StarCCM + e Aitor também, por me dar conselhos muito importantes sobre a
construção de um casco em Rhinoceros e usar sua experiência para modelar a nova
geometria final do casco utilizado nesse estudo.
Agradeço ao professor Jean-Marc Laurens por ter me encorajado a fazer minha tese de
mestrado à Vicusdt e me orientado ao longo dos anos de intercâmbio e ao professor Paulo
de Tarso por ter me orientado ao longo desse trabalho de conclusão de curso.
Agradeço a Capes por ter me dado a oportunidade de fazer o Duplo Diploma na ENSTA
Bretagne, que foi sem dúvida a experiência de vida mais enriquecedora que poderia ter
tido no campo acadêmico, profissional e pessoal. Além disso, é importante agradecer as
professoras Anna Carla Araújo e Marta Cecilia Tapia Reyes, que sempre se mostraram
disponíveis para oferecer suporte e nos ajudar a resolver os mais diversos problemas que
surgiram ao longo do intercâmbio.
Eu também gostaria de agradecer a todas funcionárias da secretaria de curso, aos meus
colegas da Equipe Solar Brasil e da turma de 2012.2 que dividiram muitos momentos de
risadas, noites sem dormir e formaram uma grande família para mim. Vocês estarão
sempre presentes em minha memória.
Por fim, agradeço a minha família que tanto se esforçou para me proporcionar a melhor
educação e que muitas vezes abdicou de seu próprio sonho para que seus filhos pudessem
alcançar os deles.
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Resumo do Projeto de Graduação apresentado à Escola Politécnica/UFRJ como
parte dos requisitos necessários para a obtenção do grau de Engenheiro Naval e Oceânico.
Análise CFD e modificação da geometria de casco para o desenvolvimento de um navio de
pesca mais eficiente
André Georges Monteiro de Carvalho
September/2018
Orientador: Paulo Tarso Themistocles Esperança, D.Sc.
Curso: Engenharia Naval e Oceânica
Mais barato e mais rápido do que os testes experimentais, os estudos numéricos são muito
importantes na fase de projeto de uma nova embarcação por diversos motivos e o que será
abordado neste estudo é a redução do consumo de combustível. Esses estudos não devem
ser vistos como custo, mas sim como um investimento, porque se for bem feito, pode ajudar
o armador a economizar milhões de dólares durante a fase operacional e definir se o projeto
é lucrativo ou não. Este projeto baseia-se num pedido feito por um armador para renovar a
sua frota de navios de pesca. Antes de passar para a fase de construção, o armador solicitou
um estudo de um novo formato do casco e sistema de hélice para reduzir o consumo de
combustível. Para isso, foi utilizada uma série de softwares indispensáveis no campo de
engenharia naval. O projeto foi iniciado usando rhinoceros para modelar a geometria do
casco, seguido de Starccm + para fazer as análises hidrodinâmicas numéricas e finalmente
concluir com o uso de NavCad para prever a potência necessária que deve ser instalada na
embarcação, bem como a primeira estimativa de um projeto de hélice. Este relatório cobrirá
a explicação de todos esses passos importantes, abordando os principais pontos teóricos que
existem por trás desses códigos e expondo os resultados obtidos com este estudo.
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Abstract of the Course Conclusion Project presented to the Department of Naval and
Oceanic Engineering of the Polytechnic School as a partial fulfillment of the requirements
for the degree of Bachelor in Naval and Oceanic Engineering (B.Sc.)
CFD analysis and shape modification in order to develop an efficient fishing vessel hull
André Georges Monteiro de Carvalho
September/2018
Advisor: Paulo Tarso Themistocles Esperança, D.Sc.
Course: Naval Engineering
Cheaper and faster than experimental tests, numerical studies are very important in the
project phase of a new vessel for many reasons and the one that will be approached in this
study is the reduction of fuel consumption. It shouldn’t be seen as cost, but rather as an
investment, because if it is well done, it can helps the shipowner save thousands euros during
the operational phase and define if the project is profitable or not. This project is based on a
request made by a shipowner to renew his fleet of fishing vessels. Before move into the
construction phase, the shipowner asked for a study of a new hull shape and propeller system
in order to reduce fuel consumption. For this, a number of software application
indispensable in the Naval engineering field was used. The project was started by using
Rhinoceros to model the hull geometry followed by using Starccm+ to make the numerical
hydrodynamic analyses and finally concluding with the use of NavCad to predict the
required power that should be installed in the vessel as well as the first estimation of a
propeller design. This report will cover the explanation of all these important steps,
approaching the main theoretical points that exists behind these codes and exposing the
results obtained with this study.
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Table of Contents
List of figures...................................................................................................................... xii
List of Graphics ................................................................................................................. xiv
List of tables ........................................................................................................................xv
Acronyms........................................................................................................................... xvi
Dimensionless number ...................................................................................................... xvi
Nomenclature .................................................................................................................... xvii
1. Introduction ....................................................................................................................1
1.1. The context of the study .............................................................................................1
1.2. The purpose of the study ............................................................................................1
1.3. Computational and software resources .......................................................................2
2. Organization ...................................................................................................................3
3. Theory ............................................................................................................................5
3.1. Development of Navier Stokes Equations ..................................................................5
Conservation of mass and momentum ....................................................................5
The transport Theorem ............................................................................................6
The Continuity Equation .........................................................................................9
Euler’s Equations ....................................................................................................9
Stress relation in a Newtonian Fluid .....................................................................10
The Navier-Stokes Equations ...............................................................................12
3.2. Turbulence ................................................................................................................13
The physics behind turbulent flows ......................................................................14
Turbulence modelling ...........................................................................................14
Boundary Layer ....................................................................................................18
3.3. The volume of fluid (VOF method) .........................................................................22
3.4. Fluid Structure interaction ........................................................................................23
Rigid Body Motion ...............................................................................................23
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Dynamic hull simulation .......................................................................................25
Estimative .............................................................................................................26
3.5. Extrapolations results ...............................................................................................27
Resistance decomposition .....................................................................................27
Coefficients calculation ........................................................................................29
3.6. Power ........................................................................................................................32
Effective Power (𝑷𝒆) ............................................................................................32
Thrust Power (𝑷𝑻) ................................................................................................33
Wake fraction coefficient (w) ...............................................................................33
Thrust deduction coefficient (t) ............................................................................34
Delivered Power (𝑷𝑫) ..........................................................................................34
Shaft Power (𝑷𝑺) ..................................................................................................35
Break Power (𝑷𝒃) .................................................................................................35
3.7. Efficiency ..................................................................................................................35
Gear efficiency ( η G) ...........................................................................................35
Shaft efficiency (η s) .............................................................................................36
Hull efficiency (η H) ............................................................................................36
Propeller efficiency-behind hull (η b) ...................................................................36
Propulsive efficiency (η p) ....................................................................................37
4. Numerical Methods ......................................................................................................38
4.1. The finite volume method (FVM) ............................................................................38
4.2. Spatial discretization schemes ..................................................................................39
4.3. Temporal discretization schemes ..............................................................................40
5. Database .......................................................................................................................41
6. Hull modeling ...............................................................................................................42
6.1. 3D Construction ........................................................................................................43
6.2. Geometry validation .................................................................................................43
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7. CFD Simulations for marine resistance prediction ......................................................45
7.1. Computational domain definition .............................................................................47
ITTC Recommendations .......................................................................................47
Kelvin angle ..........................................................................................................48
Damping wave reflection ......................................................................................48
7.2. Mesh generation .......................................................................................................49
Regular mesh ........................................................................................................49
Overset mesh .........................................................................................................51
7.3. Model definitions and properties ..............................................................................54
Turbulence model .................................................................................................54
Hull motion ...........................................................................................................55
Boundary layer thickness ......................................................................................55
7.4. Boundary and initial conditions ................................................................................56
Boundary condition ...............................................................................................57
Initial condition .....................................................................................................59
7.5. Choice of Mesh .........................................................................................................59
7.6. Post-processing .........................................................................................................61
Pressure around the hull ........................................................................................62
Free surface wave pattern .....................................................................................62
Body force in X .....................................................................................................63
Body orientation around Y....................................................................................63
Translation along Z ...............................................................................................64
Body moment around Y and body force along Z .................................................64
CFD wave cut .......................................................................................................65
7.7. Results ......................................................................................................................65
8. Extrapolation of the model results ...............................................................................66
8.1. Calculation of the form factor ..................................................................................66
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8.2. Results of the extrapolation ......................................................................................67
9. - Hull modifications .....................................................................................................68
9.1. Motivation for the hull modifications .......................................................................68
9.2. Validation of the Form Factor value .........................................................................71
9.3. Main hull modifications............................................................................................72
Modifications at the bow ......................................................................................72
Modifications at the aft part ..................................................................................73
9.4. Results ......................................................................................................................75
Coefficient changes ...............................................................................................75
Free surface elevation ...........................................................................................76
10. NavCad analysis .......................................................................................................77
10.1. Propeller design ....................................................................................................77
10.2. Power break (Pb)...................................................................................................78
11. Efficiency ..................................................................................................................79
11.1. EIVs calculation ....................................................................................................79
11.2. Efficiency results ..................................................................................................81
12. Conclusion ................................................................................................................83
12.1. Conclusion from the point of view of the client ...................................................83
12.2. Conclusion from the point of acquired knowledge ...............................................84
12.3. Future works .........................................................................................................84
Bibliography ........................................................................................................................85
Attachments ...................................................................................................................... - 1 -
1) Organization of the project ........................................................................................ - 1 -
2) Gantt Diagram ........................................................................................................... - 2 -
3) Propulsion system efficiency .................................................................................... - 5 -
4) Numerical Scheme .................................................................................................... - 6 -
4.1.1 First order upwind scheme ......................................................................... - 6 -
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4.1.2 Central differencing scheme ....................................................................... - 6 -
4.1.3 Second order upwind scheme ..................................................................... - 7 -
5) Vessels main characteristics ...................................................................................... - 8 -
6) Geometry Validation ................................................................................................. - 9 -
7) Main characteristics of Mar de Maria full scale ...................................................... - 10 -
8) CFD results .............................................................................................................. - 11 -
9) Turbulence models in Starccm+ .............................................................................. - 13 -
10) Overset mesh construction ................................................................................... - 19 -
11) Propeller design ................................................................................................... - 21 -
12) Effective power .................................................................................................... - 21 -
Comparison between effective power required for each vessel ..................................... - 21 -
13) Propeller data ....................................................................................................... - 22 -
14) NavCad and EIV results ...................................................................................... - 23 -
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List of figures
Figure 1: Schematic illustration of a boundary layer at a flat plate. ....................................18
Figure 2: Schematic illustration of the functions used to approximate the relation between
u+ and y+ in each of the sub-layers. ....................................................................................21
Figure 3: Nomenclature of the 6 DOF of a vessel ...............................................................23
Figure 4: Interactive procedure of a hull simulation ...........................................................25
Figure 5: Illustration of the variables that influence on the calculation of the power
required ................................................................................................................................32
Figure 6: the red points represent the chosen vessels to create the 3D hull geometry. The
blue points represent all the fishing vessels that Vicus already worked before ..................42
Figure 7: Transforming the 2D plans in a 3D geometry using Rhinoceros .........................43
Figure 8: ITTC recommendation to define the computational domain ...............................47
Figure 9: Kelvin wake pattern behind a moving object. ......................................................48
Figure 10: Schematic illustration of the regular mesh structure ..........................................50
Figure 11: Schematic illustration of the overset mesh structure.. .......................................52
Figure 12: Illustration of the use of the “mesh alignment” ...............................................53
Figure 13: Illustration of the mesh done in the simulation of vessel called Ana Barral.. ...54
Figure 14: Illustration of the domain boundary conditions .................................................57
Figure 15: Bottom and side view of the pressure around the hull. ......................................62
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Figure 16: Top and perspective view of the surface wave pattern. .....................................62
Figure 17: Cross section used to capture the elevation of the free surface .........................69
Figure 18: Original bow geometry of Mar de Maria ...........................................................72
Figure 19: Modified bow geometry of Mar de Maria .........................................................72
Figure 21: Details of the modification in the aft part of Mar de Maria ...............................73
Figure 20: Comparison between the original (green) and modified aft part of Mar de Maria
(grey) ...................................................................................................................................73
Figure 22: Twisted rudder (front view) ...............................................................................74
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List of Graphics
Graphic 1: Bare-hull resistance ...........................................................................................63
Graphic 2: Pitch average ......................................................................................................63
Graphic 3: Sinkage average .................................................................................................64
Graphic 4:Moment around Y average .................................................................................64
Graphic 5:Force along Z average ........................................................................................64
Graphic 6: CFD wave cut ....................................................................................................65
Graphic 7: form factor calculation for the Mar de Maria ....................................................66
Graphic 8: Ct, Cw and Cv curves for the model simulation of the original Mar de Maria
geometry ..............................................................................................................................68
Graphic 9: Illustration of the area that will be calculated to quantify the dissipated energy
in form of wave. This is a graphic of a free surface elevation taken from the Siempre Juan
Luis simulation at 8.5 knot ..................................................................................................69
Graphic 10:Comparison between the area below the graphic taken from the free surface
elevation for the operational speed of each vessel model simulation ..................................70
Graphic 11: Tendency of the integral of surface elevation of the free surface and Cw/Ct .71
Graphic 12: Comparison between the total resistance coefficient (Ct) from the modified
and original Mar de Maria hull geometry ............................................................................75
Graphic 13: reduction of the wave creation between the different vessels geometries .......76
Graphic 14: Power break prediction on NavCad .................................................................78
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Graphic 15: Graphic of EIV X Fn for each vessel ..............................................................81
Graphic 16: EIV tendency line ............................................................................................82
List of tables
Table 1: Geometry validation ..............................................................................................44
Table 2: Main features for the real and model scale of Mar de Maria ................................45
Table 3: Relative difference between the results from the towing tank test and the results
found using CFD simulations with Overset mesh and with Regular mesh. ........................60
Table4:Final comparisons between the Original and the new geometry of Mar de Maria.83
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Acronyms
DOF: degree of freedom
FVM: finite volume method
PDE: partial differential equation
EIV: estimate index value
EEDI: Energy Efficiency Design Index
RANS: Reynold Average Navier-Stokes
VOF: Volume of fluid
FSI: Fluid-structure interation
Dimensionless number
𝑅𝑒: Reynold number = 𝑽𝑳
𝑣
𝑦+ =𝑢∗𝑦
𝑣
𝑢+ =𝑢
𝑢∗
𝐶𝑖 =𝑅𝑖
0,5𝜌𝑈𝑠2𝑆
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Nomenclature
J: volume of fluid
𝜌: fluid density
𝑆: surface that involves the volume of fluid
𝑄: general vector that is continuous and differentiable in the volume J
𝑛: exterior normal vector pointing out of Jon the surface S
𝜏𝑖𝑗: stress tensor
𝜇: viscous shear coefficient or coefficient of viscosity
𝛿𝑖𝑗: Kroenecker delta function
𝑣: Kinematic viscosity coefficient
𝑈𝑖: velocity time average
𝑃: pressure time average
ui: fluctuating component of the velocity
p: fluctuating component of the pressure
𝑘: Turbulent kinetic energy
𝑣𝑡: Turbulent viscosity
𝜗𝑘, 𝜗𝜀 , 𝐶𝜇, 𝐶𝜀1, 𝐶𝜀2 are constants used in the 𝑘 − 휀 𝑚𝑜𝑑𝑒𝑙
𝑃𝑘 is the production of turbulent kinectic energy
𝜗𝑤, 𝛽∗, 𝜗𝑘𝑤 𝐶𝜇, 𝐶𝑤1, 𝐶𝑤2 are constants used in the 𝑘 − 휀 𝑚𝑜𝑑𝑒𝑙
휀: Turbulent kinetic energy dissipation
𝜔: Turbulent kinetic energy specific dissipation
𝜏𝑤: wall shear stress
𝑢∗: friction velocity
𝑃∞: undisturbed free stream pressure
𝑈ℎ𝑢𝑙𝑙: hull velocity
𝜌𝑖: density in the phase i
𝑣𝑖: viscosity in the phase i
𝑆𝑖,𝑠: source term responsible to account the momentum exchange across the interface
between the different phases, due to surface tension forces.
𝑅𝑦: Gyration radius around axis y
𝐼𝑦: Moment of inertia around axis y
𝑋𝑐𝑚, 𝑌𝑐𝑚, 𝑍𝑐𝑚: Coordinates of the center of mass of a ship
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𝐿𝐶𝐵: Longitudinal center of buoyancy
𝐿𝑂𝐴: Length overall
𝐿𝑤𝑙: water line length
∆: Displacement
𝑅𝑇 ∗: total resistance
𝑅𝑣 ∗: viscous resistance
𝑅𝑤 ∗: wave resistance
𝐶𝑇 ∗: total resistance coefficient
𝐶𝑣 ∗: viscous resistance coefficient
𝑪𝑭 ∗ : frictional resistance coefficient
*If there is an index “s” is for values relative to the real scale. If there is an index “m” is
for values relative to the model scale.
𝐶𝑤: wave resistance coefficient (is the same for the full-scale and for the model)
1 + 𝑘: form factor
𝑢∗: friction velocity
𝑦+: dimensionless wall distance
𝑢+: dimensionless velocity
𝐶𝑎𝑎: is the coefficient that considers the air resistance, and in this case of study will be
neglected.
𝐶𝑎: is the correlation coefficient (roughness allowance).
𝐾𝑠: constant that represents the roughness
𝑃𝑒: effective power
𝑃𝑇: Total power
𝑉𝑎: velocity of the arriving water at the propeller
𝑤: wake fraction
𝑡: thrust deduction coefficient
𝑃𝐷: delivered power
𝑃𝑆: shaft power
𝑃𝐵: break power
hG: gear efficiency
hs: shaft efficiency
hH: hull efficiency
hb: propeller efficiency-behind hull
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hR: rotative efficiency
h0: open water propeller efficiency
hp: propulsive efficiency
𝛿: Boundary layer thickness
𝜑: flow general variable
Γ: diffusive source
S(𝜑): source source of a general variable
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1. Introduction
1.1. The context of the study
As the need for hydrodynamics studies increases, the CFD approach is more and
more appreciated. On the other hand, the experimental tests in towing tanks and other basins
are very expensive and must anticipate long period before to schedule a tank test. Numerical
simulations are completely adequate for classical studies, because numerical codes have
already been validated for similar cases and thus reliable. Therefore, CFD is well recognized
by R&D companies for economical reasons, as it allows companies the likes of Vicus dt,
allowing them to increase the price competitiveness and accuracy during client negotiation.
1.2. The purpose of the study
Vigo is a very well-known city for its fishing industry. There are many shipyards
and ship-owners specialized in this field. Vicus dt is one of the few companies able to
provide a complete consulting service, solving not only hydrodynamics problems, but also
structural problems.
Specifically, in this case of study, the client is a ship-owner that wants to renew
around 20 ships of its fleet. These ships are divided in two different groups by value of GT:
vessels with around 600 GT that, are deployed in the Pacific Ocean, and vessels with around
200GT that, are deployed in the Atlantic Ocean.
Before it constructing the news vessels, the client contracted Vicus dt to design two
new hulls and propellers, one for each group, in order to reduce the total resistance and,
consequently, the operational cost. This case of study will focus on the vessels that are
deployed in the Pacific Ocean, and will explain the method used to develop a new hull
geometry and propeller.
2
1.3. Computational and software resources
To develop this project, Vicus dt provided three softwares licenses:
Rhinoceros: generating the hull geometries that were used in CFD
simulations;
Starccm+: preparing the mesh and launch the CFD simulations.
NavCad: predicting the motor power that should be installed in the vessel
and making a preliminary propeller geometry.
The computer resources included 2 screens and one laptop with processor Intel i3-
6006U (2.0GHz; 3MB), memory 4GB (1x4GB) DDR4 2133MHz, which was used to
prepare the hull geometry and simulations on Starccm+. However, due to the high
computational requirements involved into running the simulation, they were normally done
using another computer, Intel® Xeon® CPU E-2640 v3 @ 2.60GHz (2processors), 32GB,
64bits, which was also used to launch others CFD simulations from other projects that ran
in parallel. Consequently, this computational limit was a strong constraint, which increase
the concern of reducing the quantity of cells in each simulation.
3
2. Organization
This section will give an overview about how the project was organized. The project
was divided in several steps which are briefly explained in this section. The following
section delves into each step in more depth.
Before starting to work on the project, it was decided to make a revision of
the most important theoretical concepts that were to be covered in this case
study.
In order to create an important database to use as reference to create the new
optimal hull, the beginning of the project was dedicated to do a static analysis
of all the fishing vessels that Vicus had already worked with before.
To analyze the range of vessels that were of interest to the client, it was
decided to group the boats by GT, and focus on the vessels that were
approximately in the range of the study, that is, around 600GT.
Based on the information that was available for each vessel, it was elected
one as reference to represent the group of vessels, which has around 600GT.
The chosen vessel is called Mar de Maria.
Creating 3D geometry from 2D planes of Mar de Maria
Creating and validating the CFD simulation using the 3D geometry generated
in the previous step and taking as reference a towing tank test done using the
Mar de Maria scale model.
Choosing 10 vessels among all the fleet that could give a general view of the
fleet hull geometry.
Modelling the 3D geometry of these 10 vessels.
Creating a hull resistance curve for each 3D hull geometry model done in the
previous phase. In order to do that, it was necessary to launch marine
resistance simulations at 3 different speeds for each one of these vessels,
including for the proposed hull geometry dimension.
Comparing the values taken from these curves and elect the vessels that have
the best results in order to use its geometries to inspire the construction of
the new geometry.
4
Take the vessel Mar de Maria as a start point to begin the hull modifications
in order to reduce the bare-hull resistance.
Use NavCad to create the first propeller design for the new hull geometry
and consider the propulsion system to create a break power curve for each
vessel.
Create a curve of efficiency, using EIV criteria for all vessels and analyze
how much more efficient the new geometry is comparing to the other boats.
All these steps are described in detail in the following sections. To better
understanding the organization of the project, please see the diagrams in annex 1 and 2.
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3. Theory
The objective of this chapter is to provide the theoretical basis to understand all the
subject that will be approached in this report. It starts with an introduction explaining the
theory that exist behind the CFD calculations with the deduction of Navier Stokes equations
and explaining the turbulence and boundary layer theory. Besides that, it will be also
explained how the vessel motion is calculated, followed by result extrapolations and the
factors that can influence the vessel efficiency.
3.1. Development of Navier Stokes Equations
Before starting the simulations using no matter what CFD code, it is mandatory to
first know what the theory behind these kinds of code is. In this section, it will be approached
important concepts that will enable us to deduce the Navier Stokes equations, which are the
fundamental base for CFD codes and responsible to describe the motion of a viscous fluid
[3].
Conservation of mass and momentum
It will be defined a volume of fluid v(t) in order to focus our attention on a group of
particles. If the fluid density is denoted by 𝜌, the total mass of fluid in this volume is given
by the integral ∭𝜌 𝑑J. Conservation of mass requires that this integral must be constant or:
𝑑
𝑑𝑡∭𝜌 𝑑v = 0
v
Similarly, the momentum density of a fluid particle is equal to the vector 𝜌𝑉, with
components 𝜌𝑢𝑖. Respecting the Newton second law:
𝑑
𝑑𝑡∭𝜌𝑢𝑖 𝑑J = ∬𝜏𝑖𝑗𝑛𝑗
𝑆
𝑑𝑆
v
+ ∭𝐹𝑖 𝑑J
v
(3.1)
(3.2)
6
Using the divergence theorem [1]:
∭∇.𝑄 𝑑v = ∬𝑄. 𝑛
𝑆
𝑑𝑆
v
Rewriting this expression in the indicial notation, we have:
∭∂𝑄𝑖
∂𝑋𝑖 𝑑v = ∬𝑄𝑖. 𝑛𝑖1
𝑆
𝑑𝑆
v
Q is representing a general vector that is continuous and differentiable in the volume
v, and the unit normal n is the exterior normal vector pointing out of v on the surface S.
Using the equation 3.3b to transform the surface integral into 3.2:
𝑑
𝑑𝑡∭𝜌𝑢𝑖 𝑑v =
v
∭(𝜕𝜏𝑖𝑗
𝜕𝑋𝑗+ 𝐹𝑖 ) 𝑑v
v
Equations (3.1) and (3.4) express the conservation laws of mass and momentum for
the fluid, in terms of an arbitrary prescribed material volume v(t). The need to consider this
volume integral and specially its time derivative is inconvenient. To overcome this problem,
we first consider the evaluation of the time derivative, bearing in mind that the volume of
integration is itself a function of time.
The transport Theorem
Let us consider a general volume integral of the form
𝐼(𝑡) = ∭𝑓(𝑥, 𝑡)𝑑v
v(t)
Here f is an arbitrary differentiable scalar function of position x and time t that will
be integrated over a prescribed volume v(t), which may also vary with time. Therefore, the
(3.3 a)
(3.3 b)
(3.4)
(3.5)
7
boundary surface S of this volume will also change with time, and its normal velocity is
denoted by 𝑈𝑛.
In the usual manner of elementary calculus, we consider the difference
∆𝐼 = 𝐼(𝑡+∆𝑡) − 𝐼(𝑡)
∆𝐼 = ∭ 𝑓(𝑥, 𝑡 + ∆𝑡)𝑑𝑣
v(t+∆t)
− ∭𝑓(𝑥, 𝑡)𝑑v
v(t)
By the definition of derivation:
𝜕𝑓(𝑥, 𝑡)
𝜕𝑡=
𝑓(𝑥, 𝑡 + ∆𝑡) − 𝑓(𝑥, 𝑡)
∆𝑡
𝑓(𝑥, 𝑡 + ∆𝑡) = 𝑓(𝑥, 𝑡) + ∆𝑡 ∗𝜕𝑓(𝑥, 𝑡)
𝜕𝑡
Replacing the equation 3.7 in the 3.6 and doing a similar decomposition for the
volume v(t) and simplifying the nomenclature, passing from f (x,t) to just f, we obtain:
∆𝐼 = ∭(𝑓 + ∆𝑡𝜕𝑓
𝜕𝑡) 𝑑v
v+∆v
− ∭𝑓 𝑑v
v
∆𝐼 = ∭𝑓 𝑑v
v+∆v
+ ∭(∆𝑡𝜕𝑓
𝜕𝑡)𝑑v
v+∆v
− ∭𝑓 𝑑v
v
∆𝐼 = ∭𝑓 𝑑v
v
+ ∭𝑓 𝑑v
∆v
+ ∆𝑡 ∭(𝜕𝑓
𝜕𝑡) 𝑑v
v
+ ∆𝑡 ∭(𝜕𝑓
𝜕𝑡) 𝑑v
∆v
− ∭𝑓 𝑑v
v
∆𝐼 = ∭𝑓 𝑑v
∆v
+ 𝑂[(∆𝑡)2] + ∆𝑡 ∭(𝜕𝑓
𝜕𝑡)𝑑v
v+∆v
(3.6)
(3.7)
(3.8)
8
Where 𝑂[(∆𝑡)2]denotes a second-error proportional to (∆𝑡)2.
To evaluate the integral over the volume ∆v, we note that this thin region has a
thickness equal to the distance between S(t) and S(t+∆𝑡). This thickness is the normal
component of the distance traveled by S(t) in the time ∆𝑡, wich is equal to 𝑈𝑛∆𝑡. So, we can
infer:
𝑑v = 𝑈𝑛∆𝑡 ∗ 𝑑𝑆
Replacing the equation (3.9) in (3.8):
∆𝐼 = ∬(𝑈𝑛 ∆𝑡 𝑓) 𝑑𝑆
𝑆
+ ∆𝑡 ∭(𝜕𝑓
𝜕𝑡)𝑑v
v
+ 𝑂[(∆𝑡)2]
Dividing both sides by ∆𝑡 and taking the limit as this tends to zero:
𝑑𝐼
𝑑𝑡= ∭(
𝜕𝑓
𝜕𝑡)𝑑J
v
+ ∬(𝑓 𝑈𝑛) 𝑑𝑆
𝑆
The equation (3.10) is known as the transport theorem. Physically, the surface
integral in this equation represents the transport of the quantity of f out of the volume v.
Rewriting the value of 𝐼 as in the equation (3.5) and knowing 𝑈𝑛 = 𝑉. 𝑛 = 𝑈𝑖𝑛𝑖 we
can obtain from the equation (3.10):
𝑑𝐼
𝑑𝑡= ∭(
𝜕𝑓
𝜕𝑡) 𝑑v
v
+ ∬(𝑓 𝑈𝑛) 𝑑𝑆
𝑆
𝑑
𝑑𝑡∭𝑓𝑑v
v(t)
= ∭(𝜕𝑓
𝜕𝑡)𝑑v
v(t)
+ ∬(𝑓𝑈𝑖𝑛𝑖)𝑑𝑆
𝑆
(3.9)
(3.10)
9
Applying the divergence theorem (3.3b)
𝑑
𝑑𝑡∭𝑓𝑑J
v(t)
= ∭(𝜕𝑓
𝜕𝑡+
𝜕(𝑓𝑢𝑖)
𝜕𝑥𝑖)𝑑v
v(t)
The Continuity Equation
Returning to equation (3.1), expressing conservation of mass, we immediately have
from equation (3.11):
𝑑
𝑑𝑡∭𝜌 𝑑J
v
= ∭(𝜕𝜌
𝜕𝑡+
𝜕(𝜌𝑢𝑖)
𝜕𝑥𝑖)𝑑v
v
= 0
𝜕𝜌
𝜕𝑡+
𝜕(𝜌𝑢𝑖)
𝜕𝑥𝑖= 0
𝜕(𝜌𝑢𝑖)
𝜕𝑥𝑖= 0
∇ . 𝑉 = 0
Care must be taken to not confuse V that represents the vector velocity of the flow,
with v that represents the volume of fluid that is being observed.
Euler’s Equations
Applying the transport theorem (3.10) to the conservation of momentum (3.4), it
follows that:
∭(𝜕(𝜌𝑢𝑖)
𝜕𝑡+
𝜕(𝜌𝑢𝑖𝑢𝑗)
𝜕𝑥𝑗)𝑑v
𝑣
= ∭(𝜕(𝜏𝑖𝑗)
𝜕𝑥𝑗+ 𝐹𝑖)𝑑v
v
0
(3.11)
(3.12)
(3.13)
(3.14)
10
The volume in question is arbitrary, so we can write the integrands alone, in the
form:
𝜕(𝜌𝑢𝑖)
𝜕𝑡+
𝜕(𝜌𝑢𝑖𝑢𝑗)
𝜕𝑥𝑗=
𝜕(𝜏𝑖𝑗)
𝜕𝑥𝑗+ 𝐹𝑖
𝜌 [𝜕(𝑢𝑖)
𝜕𝑡+
𝜕(𝑢𝑖𝑢𝑗)
𝜕𝑥𝑗] =
𝜕(𝜏𝑖𝑗)
𝜕𝑥𝑗+ 𝐹𝑖
Expanding the left side using the chain rule:
𝜕(𝑢𝑖)
𝜕𝑡+ 𝑢𝑗
𝜕(𝑢𝑖)
𝜕𝑥𝑗+ 𝑢𝑖
𝜕(𝑢𝑗)
𝜕𝑥𝑗=
1
𝜌
𝜕(𝜏𝑖𝑗)
𝜕𝑥𝑗+
𝐹𝑖
𝜌
𝜕(𝑢𝑖)
𝜕𝑡+ 𝑢𝑗
𝜕(𝑢𝑖)
𝜕𝑥𝑗=
1
𝜌
𝜕(𝜏𝑖𝑗)
𝜕𝑥𝑗+
𝐹𝑖
𝜌
Stress relation in a Newtonian Fluid
A Newtonian fluid is a fluid in which the viscous stresses are proportional to the
local strain rate [2, Ch.1, Sc.2]. That is equivalent to saying that those forces are proportional
to the rates of change of the fluid’s velocity vector. Most common fluids, including water
and air are consider as Newtonian for all practical purposes.
Finally, we must relate the stress tensor 𝜏𝑖𝑗 to the kinematic properties of the fluid.
The task here is analogous to the relations in solid mechanics [3, Ch.3, Sc.3.6 ].
0
(3.15)
11
As result, the stress tensor 𝜏𝑖𝑗 can be expressed as:
𝜏𝑖𝑗 = −𝑝𝛿𝑖𝑗 + 𝜇 (𝜕𝑢𝑖
𝜕𝑥𝑗+
𝜕𝑢𝑗
𝜕𝑥𝑖)
Where 𝜇 is viscous shear coefficient or coefficient of viscosity and 𝛿𝑖𝑗 is the
Kroenecker delta function, define as: 𝛿𝑖𝑗 = 1; 𝑖 = 𝑗0; 𝑖 ≠ 𝑗
To understand better what the equation (3.16) physically represents, we will explain
the physically meaning of the two matrices. The first represents the normal pressure stress.
The second is the viscous stress tensor, proportional to the viscosity coefficient 𝜇. The
diagonal elements of the viscous stress are associated with elongations of fluid elements,
and the off-diagonal elements are due to shearing deformations.
[𝜏𝑖𝑗] = [
−𝑝 0 00 −𝑝 00 0 −𝑝
] + 𝜇
[ 2
𝜕𝑢
𝜕𝑥
𝜕𝑢
𝜕𝑦+
𝜕𝑣
𝜕𝑥
𝜕𝑢
𝜕𝑧+
𝜕𝑤
𝜕𝑥𝜕𝑣
𝜕𝑥+
𝜕𝑢
𝜕𝑦2
𝜕𝑣
𝜕𝑦
𝜕𝑣
𝜕𝑧+
𝜕𝑤
𝜕𝑦𝜕𝑤
𝜕𝑥+
𝜕𝑢
𝜕𝑧
𝜕𝑤
𝜕𝑦+
𝜕𝑣
𝜕𝑧2
𝜕𝑤
𝜕𝑧 ]
(3.17)
(3.16)
12
The Navier-Stokes Equations
The Navier-Stokes equations express the conservation of momentum for a
Newtonian fluid. To obtain them, the stress-strain relations (3.16) are substituted in Euler’s
equations (3.15).
𝜏𝑖𝑗 = −𝑝𝛿𝑖𝑗 + 𝜇 (𝜕𝑢𝑖
𝜕𝑥𝑗+
𝜕𝑢𝑗
𝜕𝑥𝑖)
𝜕𝜏𝑖𝑗
𝜕𝑥𝑗= −
𝜕𝑝
𝜕𝑥𝑖+ 𝜇
𝜕
𝜕𝑥𝑗(𝜕𝑢𝑖
𝜕𝑥𝑗+
𝜕𝑢𝑗
𝜕𝑥𝑖)
𝜕𝜏𝑖𝑗
𝜕𝑥𝑗= −
𝜕𝑝
𝜕𝑥𝑖+ 𝜇
𝜕2𝑢𝑖
𝜕𝑥𝑗𝜕𝑥𝑗+ 𝜇
𝜕2𝑢𝑗
𝜕𝑥𝑗𝜕𝑥𝑖
from the continuity equation (3.13):
𝜕2𝑢𝑗
𝜕𝑥𝑗𝜕𝑥𝑖=
𝜕
𝜕𝑥𝑖(𝜕𝑢𝑗
𝜕𝑥𝑗) = 0
Thus:
𝜕𝜏𝑖𝑗
𝜕𝑥𝑗= −
𝜕𝑝
𝜕𝑥𝑖+ 𝜇
𝜕2𝑢𝑖
𝜕𝑥𝑗𝜕𝑥𝑗
Applying the equation (3.17) to the Euler’s equation, we derive the Navier-Stokes
equations:
𝜕𝑢𝑖
𝜕𝑡+ 𝑢𝑗
𝜕𝑢𝑖
𝜕𝑥𝑗= −
1
𝜌 𝜕𝑝
𝜕𝑥𝑖+ 𝑣
𝜕2𝑢𝑖
𝜕𝑥𝑗𝜕𝑥𝑗+
𝐹𝑖
𝜌
Where 𝑣 =𝜇
𝜌 is the kinematic viscosity coefficient.
(3.17)
(3.18 a)
13
Written in the vector form:
𝜕𝑉
𝜕𝑡+ (𝑉. ∇𝑉) = −
1
𝜌 ∇𝑝 + 𝑣 ∇2𝑉 +
𝐹𝑖
𝜌
In Cartesian coordinates:
𝜕𝑢
𝜕𝑡+ 𝑢
𝜕𝑢
𝜕𝑥+ v
𝜕𝑢
𝜕𝑦+ 𝑤
𝜕𝑢
𝜕𝑧= −
1
𝜌 𝜕𝑝
𝜕𝑥+ 𝑣 ∇2𝑢 +
𝐹𝑥
𝜌
𝜕v
𝜕𝑡+ 𝑢
𝜕v
𝜕𝑥+ v
𝜕v
𝜕𝑦+ 𝑤
𝜕v
𝜕𝑧= −
1
𝜌 𝜕𝑝
𝜕𝑦+ 𝑣 ∇2v +
𝐹𝑦
𝜌
𝜕𝑤
𝜕𝑡+ 𝑢
𝜕𝑤
𝜕𝑥+ v
𝜕𝑤
𝜕𝑦+ 𝑤
𝜕𝑤
𝜕𝑧= −
1
𝜌 𝜕𝑝
𝜕𝑧+ 𝑣 ∇2𝑤 +
𝐹𝑧
𝜌
3.2. Turbulence
The vorticity generation constitutes an effective mechanism of mixing layers that is
totally absent in the flow state commonly referred as laminar. The transition from laminar
state to the turbulent state affects the resistance and is directly related to the Reynolds
number. For a flat plate, the transition to a turbulence flow takes place at Reynolds number
around 106. [4, Ch3, sc. 3.2]
In the case of this project we are working with the value of Re around 108 and
because of that we must consider the effects of turbulence.
(3.18 c)
(3.18 b)
14
The physics behind turbulent flows
Turbulence is a flow state characterized by an apparently chaotic three-dimensional
vorticity [5, Ch.1, Sc.1.2]. It is characterized as a three-dimensional because it has the ability
to generate new vortices from old one [6, Ch. 25, Sc.1.5] where turbulent kinetic energy is
dissipated from the largest to the smallest turbulent scales. On the smallest turbulent scales,
known as Komolgorov scales [5, Ch.2, Sc.2.2], the energy is dissipated into heat due to
viscous forces.
Turbulence modelling
One of the most used approach for studying turbulent flows is the Reynold-average
Navier-Stokes (RANS) model. RANS system is derived by means of the Reynolds
decomposition [7]. The instantaneous velocity and pressure can be decomposed as:
𝑈𝑖 = 𝑈𝑖 + 𝑢𝑖
𝑃 = 𝑃 + 𝑝
Where 𝑈𝑖 and 𝑃 denote the time average quantities while 𝑢𝑖 and p are the fluctuating
components of the velocities and the pressure. By inserting the Reynold decomposition into
Navier-Stokes equation(3.18a) and in the continuity equation (3.13) the RANS equation is
obtained. These are written as:
𝜕𝑈𝑖
𝜕𝑥𝑖= 0
𝜕𝑈𝑖
𝜕𝑡+ 𝑈𝑗
𝜕𝑈𝑖
𝜕𝑥𝑗= −
1
𝜌 𝜕𝑝
𝜕𝑥𝑖+ 𝑣
𝜕2𝑢𝑖
𝜕𝑥𝑗𝜕𝑥𝑗−
𝜕𝑢𝑖𝑢𝑗
𝜕𝑥𝑗+ 𝑔𝑖
It can be noticed that the RANS equations are very similar to the Navier-Stokes
equations, except for the additional term including 𝑢𝑖𝑢𝑗, referred to as the Reynolds stress
tensor.
(3.19)
(3.20
a)
(3.20 b)
15
A common approach for closing the RANS system is based on the turbulent viscosity
model: using the Boussinesq hypothesis [8] that allows to relate the Reynolds stresses to the
mean velocity gradients. In this assumption, the Reynolds stress tensor is modeled as a
diffusion term by introducing a turbulent viscosity,𝑣𝑡 ,according to:
−𝑢𝑖𝑢𝑗 = 𝑣𝑡 (𝜕𝑈𝑖
𝜕𝑥𝑗+
𝜕𝑈𝑗
𝜕𝑥𝑖) −
2
3𝑘𝛿𝑖𝑗
In this equation, 𝑘 is the turbulent kinetic energy defined as:
𝑘 =1
2𝑢𝑖𝑢𝑖
However, it was created one unknown, 𝑣𝑡, by using a model to describe how the
turbulent viscosity depends on the flow, RANS equations can be solved. The so called two-
equation turbulence models, such as the 𝑘 − 휀 𝑚𝑜𝑑𝑒𝑙 and 𝑘 − 𝑤 𝑚𝑜𝑑𝑒𝑙, use two additional
transport equations to describe the turbulent viscosity. In order to explain better these
models, they will be briefly described here based on the theory applied in the reference [15].
If needed to go deeper in the concept, is advised look the references [9], [11] and also the
annex 9, which brings the main information about the models of turbulence used in
Starccm+.
(3.21)
(3.22)
16
3.2.2.1 The standard 𝒌 − 𝜺 𝒎𝒐𝒅𝒆𝒍
In this model, the transport equations for the turbulent kinetic energy and dissipation,
휀, are used to obtain the turbulent viscosity. It has been described by Launder [12]. The
model equations for 𝑘 and 휀 are:
𝜕𝑘
𝜕𝑡+
𝜕
𝜕𝑥𝑗(𝑘𝑈𝑖) =
𝜕
𝜕𝑥𝑗[(𝑣 +
𝑣𝑡
𝜗𝑘)
𝜕𝑘
𝜕𝑥𝑗] + 𝑃𝑘 − 휀
𝜕휀
𝜕𝑡+
𝜕
𝜕𝑥𝑗(휀𝑈𝑖) =
𝜕
𝜕𝑥𝑗[(𝑣 +
𝑣𝑡
𝜗𝜀)
𝜕휀
𝜕𝑥𝑗] +
휀
𝑘(𝐶𝜀1𝑃𝑘 − 휀𝐶𝜀2)
𝑣𝑡 = 𝐶𝜇
𝑘2
휀
Where 𝜗𝑘 , 𝜗𝜀 , 𝐶𝜇, 𝐶𝜀1, 𝐶𝜀2 are constants and 𝑃𝑘 is the production of turbulent kinectic
energy. The latter is define using the Boussinesq approximation, described in the equation
(3.21)
𝑃𝑘 = −𝑢𝑖𝑢𝑗 𝜕𝑈𝑖
𝜕𝑥𝑗
The standard 𝑘 − 휀 𝑚𝑜𝑑𝑒𝑙 gives good predictions for free flows with small pressure
gradients. It assumes that the flow is fully turbulent which limits its applicability to high
Reynolds number flows [10], which is most of cases studied by Naval engineers. However,
over time, it has been observed that this model cannot be used to describe the wake behind
a moving hull in a satisfactory manner [13].
(3.23)
(3.24)
17
3.2.2.2 The standard 𝒌 − 𝝎 𝒎𝒐𝒅𝒆𝒍
In this model, described by Wilcox [10], the transport equations for the turbulent
kinetic energy and its specific dissipation, 𝜔, are used in a similar way as for the standard
𝑘 − 휀 𝑚𝑜𝑑𝑒𝑙. The specific dissipation is related to the dissipation according to:
𝜔 ∝휀
𝑘
The model equations for 𝑘 and 𝜔 are:
𝜕𝑘
𝜕𝑡+
𝜕
𝜕𝑥𝑗(𝑘𝑈𝑖) =
𝜕
𝜕𝑥𝑗[(𝑣 +
𝑣𝑡
𝜗𝑘𝑤)
𝜕𝑘
𝜕𝑥𝑗] + 𝑃𝑘 − 𝛽∗𝑘 𝜔
𝜕𝜔
𝜕𝑡+
𝜕
𝜕𝑥𝑗(𝜔𝑈𝑖) =
𝜕
𝜕𝑥𝑗[(𝑣 +
𝑣𝑡
𝜗𝑤)
𝜕𝜔
𝜕𝑥𝑗] +
𝜔
𝑘(𝐶𝑤1𝑃𝑘 − 𝜔𝐶𝑤2)
𝑣𝑡 = 𝑘
𝜔
Where 𝜗𝑤 , 𝛽∗, 𝜗𝑘𝑤 𝐶𝜇, 𝐶𝑤1, 𝐶𝑤2 are model constant.
The 𝑘 − 𝜔 𝑚𝑜𝑑𝑒𝑙 has the advantage that it is also valid close to walls and in regions
of low turbulence. Thus, it is valid in the low turbulent Reynold number region close to
walls, meaning that this model can be used in the whole flow domain. However, using this
model, the results are very sensitive to the choice of boundary and initial conditions.
(3.25)
(3.26)
18
Figure 1: Schematic illustration of a boundary layer at a flat plate. [16, Ch.7]
3.2.2.3 The SST 𝒌 − 𝝎 𝒎𝒐𝒅𝒆𝒍
In order to make use of the collective advantages of the k-ε and k-ω models, was
developed the shear stress transport (SST) model by combining the two models into one
using blending functions [13]. In this hybrid model, the k-ω model is used in the boundary
layer while the k-ε model, is used in the free flow. It has been recognized for its good overall
performance [14] and it is the most commonly used turbulence model for simulations of
ship hydrodynamics.
Boundary Layer
When a fluid flows along a surface, shear stresses give rise to a boundary layer in the
vicinity of the surface. The structure of a boundary layer near the edge of a flat plate is
illustrated in figure 1, where the incident flow has a uniform velocity profile with
velocity 𝑈0. When the flow reaches the plate, a laminar boundary layer starts to grow at the
surface. After some distance, the boundary layer goes into a transition region after which a
turbulent boundary layer is developed, and the turbulence increases further away from the
wall [16].
U0 U (y)
Fully turbulent
sub-layer
Buffer sub-layer y
Viscous sub-layer
Laminar boundary
layer
Re<5 ∗ 105
Transition
Region
Turbulent boundary
layer
Re>3*106
19
To characterize the flow near a wall, a dimensionless wall distance and velocity
are introduced, define as:
𝑦+ =𝑢∗𝑦
𝑣
𝑢+ =𝑢
𝑢∗
Where 𝑦 could be defined as a normal distance from a wall you should place your
first grid-line [17], 𝑢 the free stream velocity and 𝑢∗is a friction velocity, defined as:
𝑢∗ = √𝜏𝑤
𝜌
Where 𝜏𝑤is the wall shear stress, defined as:
𝜏𝑤 = 𝜌𝑣𝜕𝑈
𝜕𝑦|
In the boundary layer, there is a high velocity gradient in the wall normal
directions, influencing directly the results of pressure field and consequently the forces
applied on the hull vessel. This implies that a high spatial resolution is required in order
to capture the effects near the wall. A common method used to work around this problem
is refine the geometry mesh near to the wall and also apply the wall functions, which are
empirical models used to estimate the flow variables near to the wall.
Standard wall functions assume that the boundary layer can be described as a flat
plate boundary layer and describe the time-averaged velocity as function as the
dimensionless wall distance.
Inside the turbulent boundary layer, there are three different sublayers, which are
regulated by a different wall function. These regions are called: viscous sub-layer, buffer
sub-layer and fully turbulent sub-layer [35].
𝑦=0
(3.27 a)
(3.28)
(3.29)
(3.27 b)
20
3.2.2.4 Viscous sub-layer
In the viscous sub-layer, it can be shown that the 𝑢+has the same value as 𝑦+.
This relation is valid for 𝑦+ < 5.
𝑢+ = 𝑦+
3.2.2.5 Fully turbulent sub-layer
In the fully turbulent-layer, the velocity follows the logarithmic law of the wall,
meaning that 𝑢+is proportional to the natural logarithm of 𝑦+. This relation gives a good
approximation for 𝑦+ > 30.
𝑢+ =1
𝐾𝑙𝑛 𝑦+ + 𝐶+
Where K and 𝐶+ represent constant, and from experiences, 𝐾 ≈ 0.41 and 𝐶+ ≈ 5.
3.2.2.6 Buffer sub-layer
Between these sublayers, in the buffer sub-layer, is not possible to use either the
linear approximation or the logarithmic law. The buffer sub-layer is a transition from
linear to logarithmic 𝑦+dependence and it is situated between the range of 5 < 𝑦+ < 30
.
Like is possible to observe in the figure 2, the largest variation from either law
occurs approximately where the two curves intercept, a value of 𝑦+ ≈ 11 . That means,
before 𝑦+ ≈ 11 is more accurate use the linear approximation. After this value, the
logarithmic approximation should be used.
(3.30)
(3.31)
21
Figure 2: Schematic illustration of the functions used to approximate the relation between u+ and y+ in each of the
sub-layers.
To circumvent this problem, nowadays the CFD codes propose a function 𝑓 that
try to make the transition from linear to logarithmic 𝑦+dependences, in order to decrease
the numerical error when 𝑦+ is in the buffer sub-layer.
22
3.3. The volume of fluid (VOF method)
In order to simulate a hull moving in water, models are needed to resolve the
interface between water and air. The most frequently used method to capture the free
surface in ship hydrodynamics is the Volume of Fluid (VOF) method.
In the VOF method, each phase is marked with a function that represents the
volume fraction of the phases [18]. If 0 < 𝛾 < 1, there is an interface present and the
properties of the phases are averaged in order to get a single set of equations. The average
density and viscosity are:
𝜌 = 𝛾𝜌1 + (1 − 𝛾)𝜌2
𝑣 = 𝛾𝑣1 + (1 − 𝛾)𝑣2
Where 𝜌𝑖 , 𝑣𝑖 are respectively the density and the viscosity of the phase i and 𝛾
represents the volume of fraction in the phase 1.
Then, a modified set of the Navier-Stokes equations can be used for the averaged
fluid properties:
𝜕𝑢𝑖
𝜕𝑥𝑖= 0
𝜕𝑢𝑖
𝜕𝑡+ 𝑢𝑗
𝜕𝑢𝑖
𝜕𝑥𝑗= −
1
𝜌 𝜕𝑝
𝜕𝑥𝑖+ 𝑣
𝜕2𝑢𝑖
𝜕𝑥𝑗𝜕𝑥𝑗+ 𝑔𝑖 + 𝑆𝑖,𝑠
(3.30)
(3.31)
23
This formulation of the Navier-Stokes equations contains an additional source
term, 𝑆𝑖,𝑠, accounting for the momentum exchange across the interface due to surface
tension forces. This surface tension force must be correctly modelled which can be an
issue. The surface is captured by solving a transport equation:
𝜕𝛾
𝜕𝑡+ 𝑢𝑖
𝜕𝛾
𝜕𝑥𝑖= 0
The free surface waves affect the forces on the hull, so is very important to get
an accurate and stable solution of equation.
3.4. Fluid Structure interaction
In order to simulate the dynamic behavior of a hull before reaching the
equilibrium, the fluid-structure interaction (FSI) between the hull and the fluids has to be
taken into account. This is done by solving the equations of motion and rotation of the
vessel under the influence of the forces and moments from the surrounding fluids and
gravity. [15, pp14]
Rigid Body Motion
A vessel can be approximated as a rigid body which can move in three dimensions
and rotate around the three axes, totalizing 6 DOF. For the case that is being studied in
this thesis, will be considered just two DOF: pitch (rotation around axis Y) and heave
(translation along axis Z).
Figure 3: Nomenclature of the 6 DOF of a vessel [26]
(3.32)
24
Another important fact is that will be considered that the vessel has no velocity.
The flow around the vessel hull is generated by an imposed fluid velocity, 𝑈𝑏 , relative
speed between the vessel and the fluid, calculated in the section 7.3.1.1.
For a rigid body, the translation motion of the center of gravity is described by Newton’s
second law:
𝑚𝑑𝑈𝑏
𝑑𝑡= 𝐹
Where m is the mass, 𝑈𝑏 is the velocity of the boat and 𝐹 is the sum of forces
acting on the body.
The rotation of the body expressed in the body coordinates, is described by Euler’s
equations:
𝑀𝑑Ω
𝑑𝑡+ Ω x (m . Ω ) = 𝜏
Where Ω is the angular velocity of the body, 𝜏 is the resultant torque acting on the
body and M is the tensor of the moments of inertia and it represents:
𝑀 = [
𝑀𝑥𝑥 𝑀𝑥𝑦 𝑀𝑧𝑥
𝑀𝑦𝑥 𝑀𝑦𝑦 𝑀𝑧𝑦
𝑀𝑧𝑥 𝑀𝑧𝑦 𝑀𝑧𝑧
]
(3.35)
(3.36)
(3.37)
25
Dynamic hull simulation
Under most circumstances, a hull moving with a constant speed will reach a
steady position and orientation with respect to the free surface. In order to discover this
equilibrium position, a DOF solver can be implemented in the solution process as
shown in the figure 4. When the motions and rotations have ended and the final position
is reached, the net forces and moments acting on the hull are considered zero, respecting
a tolerance established by the user.
Figure 4: Iterative procedure of a hull simulation used to describe the fluid-structure interaction [15, pp16]
26
Estimative
In order to simulate the hull dynamics, it is needed calculate some important input values,
as the moment of inertia and the center of mass. For the purpose of calculating these
values, some estimates will be done that will be explained below.
3.4.3.1 Center of mass
Due to the lack of information about the loading conditions, the center of mass
will be approximated as follows:
𝑋𝑐𝑚 = 𝐿𝐶𝐵
𝑌𝑐𝑚 = 0𝑍𝑐𝑚 = 𝑇
3.4.3.2 Moment of Inertia
As explained in the section 3.4.1, in this case of study it will be considered just 2
DOF: pitch and heave. In order to capture these movements, is not necessary to calculate
all the elements of M, the tensor of the moments of inertia (3.37) It is sufficient to
calculate the value of 𝑀𝑦𝑦. To calculate this exact value, we must have all the load
distribution on board, but often it is a data that we don’t have, so we must estimate this
value as follow:
It is possible to estimate the value of the gyration radius as [19]:
𝑅𝑦 = 0.225 ∗ 𝐿𝑂𝐴
Replacing the definition of gyration radius in the expression 3.38, will
appear an estimate for 𝐼𝑦:
𝑅𝑦 = √𝐼𝑦
∆
𝐼𝑦 = (0.225)2 ∗ 𝐿𝑂𝐴2 ∗ ∆
(3.38)
(3.39)
(3.40)
27
3.5. Extrapolations results
All the simulations were made considering the model scale, so after obtained the
results, it is necessary to extrapolate this values to the real scale and the calculus that
enable to do this are going to be described in this section.
The simulations are made considering the model scale mainly for two reasons:
The simulations using model scales can save a lot computational effort,
because it is needed less cells to have an accurate result.
The simulation using the model scale is going to be very useful in this
study because it will be possible to compare directly with the towing tank
results made with the “base ship” and use this comparison to validate the
mesh used in the simulations.
However, the results that are going to be useful for us are the hydrodynamics
characteristics for the full-scale vessel and to find them, it is needed extrapolate the values
from the model scale. To make this extrapolation possible, it will be used some theoretical
concepts about the resistance decomposition and about the method ITTC 1978
performance prediction method [27, Ch3, Sc.3.2.3].
Resistance decomposition
The measured calm-water resistance is usually decomposed into various
components, although all these components usually interact and most of them cannot be
measured individually. The concepts of resistance decomposition help in designing the
hull form as the designer can focus on how to influence individual resistance components.
28
Is necessary know how the resistance is composed, so is possible discover what
are the variables that we need to extrapolate to find the value of the total resistance for
the vessel in real scale. In this case of study, it will be applied the method ITTC 1978.
This method simplifies the decomposition of the total resistance in just two components:
viscous and wave resistance.
𝑅𝑇 = 𝑅𝑣 + 𝑅𝑤
With,
𝑅𝑇: total resistance
𝑅𝑣: viscous resistance
𝑅𝑤: wave resistance (is the same for the full-scale and for the model for the same
Fr, Froude number)
These resistances forces could be also expressed as non-dimensional coefficient
of the form:
𝐶𝑖 =𝑅𝑖
0,5𝜌𝑈𝑠2𝑆
Thus, is easily noticed that the following is also valid:
𝐶𝑇 = 𝐶𝑣 + 𝐶𝑤
𝐶𝑇: total resistance coefficient
𝐶𝑣: viscous resistance coefficient
𝐶𝑤: wave resistance coefficient (is the same for the full-scale and for the model)
(3.41)
(3.42)
(3.43)
29
Coefficients calculation
The method ITTC1978 has become a widely-accepted procedure to calculate the
coefficients on the equation 3.41 and evaluate models test. This method will be
decomposed in several steps, that will be presented below, to be better understood.
3.5.2.1 Determine 𝑪𝑻𝒎
One of the simulation output results, is the model total resistance (𝑅𝑇𝑚). In order
to extrapolate this value, the first step is to determine the total resistance coefficient in the
model test (𝐶𝑇𝑚):
𝐶𝑇𝑚 =𝑅𝑇𝑚
0,5𝜌𝑚𝑈𝑚2 𝑆𝑚
With,
𝜌𝑚: water density in the model test
𝑆𝑚: wetted surface in the model test
𝑅𝑇𝑚:Total resistance in the model test
𝑈𝑚: model velocity
Frequently, the value of 𝑅𝑇𝑚 is given by the model simulation, 𝑆𝑚 is given by
Rhinoceros and 𝑈𝑚 is given by the expression 7.2.
(3.44)
30
3.5.2.2 Determine 𝑪𝑭𝒎
Need to determine the frictional resistance coefficient for the model (CFm),
following ITTC 1957 [30].
𝐶𝐹𝑚 =0,075
(𝑙𝑜𝑔10𝑅𝑛𝑚 − 2)2
With,
𝑅𝑛𝑚: Reynold number of the model = 𝑉𝑚𝐿𝑤𝑙
𝑣𝑚
𝑉𝑚: model velocity
𝐿𝑤𝑙: water line length
𝑣𝑚: kinematic viscosity
3.5.2.3 Determine 𝑪𝑽𝒎
The viscous resistance coefficient for the model could be determined as:
𝐶𝑉𝑚 = (1 + 𝑘) 𝐶𝐹𝑚
3.5.2.4 Determine 𝑪𝒘
Need to determine the wave resistance coefficient (𝐶𝑤). As saw in the section
3.5.1, 𝐶𝑤 is the same for model and ship. Applying the equation 3.46 on equation 3.43:
𝐶𝑤 = 𝐶𝑇𝑚 − (1 + 𝑘) 𝐶𝐹𝑚
The factor (1+k) is called form factor and will be determined in the follow section.
(3.45)
(3.46)
(3.47)
31
3.5.2.5 Determine the form factor (1+k)
Using the Prohaska method, the form factor could be determined in a least square
fit of α in the follow function [29].
𝐶𝑇𝑚
𝐶𝐹𝑚= (1 + 𝑘) + 𝛼
𝐹𝑛4
𝐶𝐹𝑚
The form factor depends basically on the boat geometry. Due to the fact that in
this study all the boats that are being studied belong to the same vessel family and have a
very similar hull shape, it is possible to consider that the form factor will be the same for
all of them. To calculate this value, it is going to be used a vessel called Mar de Maria,
and use this same value for the other vessels. The numerical details about this calculation
can be seen in the section 8.
3.5.2.6 Determine 𝑪𝑻𝒔
As a final step, it is needed to calculate the value of the total resistance of the ship
(𝐶𝑇𝑠),
𝐶𝑇𝑠 = 𝐶𝑤 + (1 + 𝑘)𝐶𝐹𝑠 + 𝐶𝑎 + 𝐶𝑎𝑎
𝐶𝐹𝑠: is the frictional resistance coefficient calculated as explained in the section
8.2.2, but for the full-scale ship.
𝐶𝑎𝑎: is the coefficient that considers the air resistance, and in this case of study
will be neglected.
𝐶𝑎: is the correlation coefficient (roughness allowance). The expression for 𝐶𝑎 is
given below:
𝐶𝑎 = 0,105 ∗ (√𝐾𝑠
𝐿𝑤𝑙
3
− 0,64)
𝐾𝑠: represents the roughness = 1,5*10−4 m.
(3.49)
(3.50)
(3.48)
32
3.5.2.7 Total resistance for the ship 𝑹𝑻𝒔
To finally find the value of the total resistance for a ship, is just apply the value
of 𝐶𝑇𝑠, on the equation 3.42 and extract directly the value of 𝑅𝑇𝑠.
3.6. Power
In this section it will be presented some theoretical concepts that allows to predict
the power required by the vessel to achieve certain speed. The figure 5 illustrates all the
variables that influences on the power prediction and where they act, turning it easier to
understand the physical meaning for each of them.
Figure 5: Illustration of the variables that influence on the calculation of the power required [33]
Effective Power (𝑷𝒆)
It corresponds to the effective (towing) power, 𝑃𝑒, necessary to move the ship
through the calm water [34].
𝑷𝒆 = 𝑉 ∗ 𝑹𝑻
With, V representing the vessel velocity and 𝑅𝑇 the total resistance calculates,
which in this case of study will be calculated by the CFD simulations.
(3.51)
33
Thrust Power (𝑷𝑻)
The thrust power represents the power required by the propeller to move the ship
at speed V.
𝑷𝑻 = 𝑇 ∗ 𝑽𝒂
With, T representing the thrust required to move the ship at speed V and 𝑉𝑎 the
velocity of arriving water at the propeller.
Care must be taken because 𝑉 ≠ 𝑉𝑎 and 𝑅𝑇 ≠ 𝑇. These differences will be
explained in the sections 3.6.3 and 3.6.4 respectively.
Wake fraction coefficient (w)
Due to the phenomena of boundary layer, discussed on the section 3.2.3, there is
a velocity difference between the flow along the sides of the hull and the velocity of the
vessel. Additionally, the ship’s displacement of water will also cause wake waves. All
this implies that the propeller behind will be working in a wake field. This means that the
velocity of arriving water 𝑉𝑎 at the propeller is different from the ship speed V.
The used wake fraction coefficient w is defined as:
𝑤 =𝑉 − 𝑉𝑎
𝑉
𝑉𝑎 = (1 − 𝑤) ∗ 𝑉
The value of the wake fraction coefficient depends largely on the shape of the
hull, but also on the propeller’s location and size, and has great influence on the
propeller’s efficiency.
(3.52)
(3.54)
(3.53)
34
For ships with one propeller, which is the case that will be studied, the wake
fraction, 𝑤 , is normally in the region of 0,20 to 0,45, corresponding to a flow velocity to
the propeller of 80% to 55% of the ship’s speed V [34,Ch2].
Thrust deduction coefficient (t)
The rotation of the propeller causes the water in front of it to be “sucked” back
towards the propeller. This results in an extra resistance on the hull generally called
“thrust deduction fraction”, F. This means that the thrust force T on the propeller must
overcome both, the ship’s resistance 𝑅𝑇, and this “loss of thrust“, F.
As in the previous section, the thrust deduction coefficient is defined as:
𝑡 =𝐹
𝑇=
𝑇 − 𝑅𝑇
𝑇
𝑅𝑇 = (1 − 𝑡) ∗ 𝑇
The shape of the hull may have also a significant influence in the value of t. For a
ship with one propeller, t is normally in the range of 0,12 to 0,30. A ship with a large
block coefficient has a large thrust deduction coefficient [34, Ch2].
Delivered Power (𝑷𝑫)
Represents the delivered power to the propeller. Transforms a rotational power
into a thrust power. It is defined as:
𝑃𝐷 = 2𝜋𝑛𝑄𝐷
With, n representing rotations per second and 𝑄𝐷 is the moment delivered to the
propeller.
(3.55)
(3.56)
(3.57)
35
Shaft Power (𝑷𝑺)
Represents the power delivered by the engine to the shaft. Is the engine power
installed (𝑷𝒃) , discounting the gear efficiency.
Break Power (𝑷𝒃)
Represents the engine power installed. This power is measured at the crankshaft
with the brake dynamometer, it is the highest power that can be measured in the vessel.
3.7. Efficiency
The efficiency of a propulsion system is one of the most important points in the
project, because is directly related to the fuel consumption, thus is a decisive factor to
define if a vessel will be profitable or not. The total propulsive efficiency, hp, is composed
by a set of other efficiencies coming different parts of the propulsive system and an
illustration of this can be seen in the annex 3.
Gear efficiency ( η G)
It represents the efficiency due to the mechanic transmission inside the gear box.
This value is normally taken between 0,97 and 0,98. The gear efficiency can be
calculated as:
η G = 𝑃𝑆
𝑃𝐵
(3.58)
36
Shaft efficiency (η s)
It represents the efficiency due to the mechanic transmission done by the shaft.
Frequently equal to 0,97 or 0,98. The gear efficiency can be calculated as:
η S = 𝑃𝐷
𝑃𝑆
Hull efficiency (η H)
The hull efficiency is defined as the ratio between the effective power (𝑃𝑒) and
the thrust that the propeller gives to the water. It can be expressed as:
ηH = 𝑃𝐸
𝑃𝑇=
𝑅𝑇∗𝑉
𝑇∗𝑽𝒂=
𝑅𝑇
𝑇∗
𝑉
𝑽𝒂=
1−𝑡
1−𝑤
For a ship with one propeller, the hull efficiency is usually in the range of 1.1 to
1.4 with the high values for ships with high block coefficient.
Propeller efficiency-behind hull (η b)
The propeller efficiency behind the hull is composed by others two types of
efficiency: the rotative efficiency (hR) and the open water propeller efficiency (h0) [34].
It can be also calculated doing the ratio between the thrust power 𝑃𝑇, which the propeller
delivers to the water, and the power 𝑃𝐷, which is delivered to the propeller. Thus, the
propeller efficiency for a propeller behind the ship is defined as:
η b = 𝑃𝑇
𝑃𝐷= η R * η 0
(3.59)
(3.61)
(3.60)
37
3.7.4.1 Open water propeller efficiency (η 0)
The propeller efficiency h0 is related to working in open water, the propeller works
in a homogeneous wake field with no hull in front of it. This efficiency depends especially
on the speed of advance 𝑉𝑎, thrust force T, rate of revolution n and the propeller geometry.
Frequently the value of h0 can vary between 0,35 and 0.75, with the high value
being valid for propellers with a high speed of advance 𝑉𝑎.
3.7.4.2 Open water propeller efficiency (η R)
The actual velocity of the water flowing to the propeller behind the hull is neither
constant not at right angles to the propeller’s disk area, but has a kind of rotational flow.
Therefore, compared with the propeller working in open water, the propeller’s efficiency
is affected by the factor hR.
On ships with a single propeller, the rotative efficiency is frequently around 1.0
to 1.07, in other words, the rotation of the water has a beneficial effect.
Propulsive efficiency (η p)
The propulsive efficiency must not be confused with the open water propeller
efficiency h0. It is equal to the ratio between the effective power 𝑃𝐸 and the necessary
power delivered to the propeller 𝑃𝐷 .
η p = 𝑃𝐸
𝑃𝐷=
𝑃𝐸
𝑃𝑇∗
𝑃𝑇
𝑃𝐷 = η H * η B = η H * η 0 * η R
(3.62)
38
4. Numerical Methods
In this chapter, the numerical methods used for treating the mathematical model shown
in the precedent chapter.
4.1. The finite volume method (FVM)
The finite volume method (FVM) is a numerical method of discretizing a
continuous partial differential equation (PDE), into a set of algebraic equations. The first
step of the discretization is to divide the computational domain into a finite number of
volumes, forming what is known as mesh or grid. Next, the PDE is integrated in each
volume by using the divergence theorem, yielding an algebraic equation for each cell. In
the centers of the cell, cell-average values of the flow variables are stored in so called
nodes. This implies that the spatial resolution of the solution is limited by the cell size
since the flow variable do not vary inside a cell.
The FVM is conservative, meaning that the flux leaving a cell through one of its
boundaries is equal to the flux entering the adjacent cell through the same boundary.
A stationary transport equation involving diffusion and convection of a general
flow variable, 𝜑, can be written as:
𝜌𝑈𝑖𝜕𝜑
𝜕𝑋𝑖=
𝜕
𝜕𝑋𝑖(Γ
𝜕𝜑
𝜕𝑋𝑖)+S(𝜑)
Where Γ is the diffusivity and S is the source term which may depend on 𝜑. It can
be noted that the equations in Chapter 3 governing the transport of 𝑢, k, ε, ω and 𝛾 are
all written on these forms. By using the FVM, this equation can be written on discrete
form as
𝑎𝑝𝜑𝑝 = ∑𝑎𝑛𝑏𝜑𝑛𝑏
𝑛𝑏
+ 𝑆𝑖𝑗
(4.1)
(4.2)
39
Where,
𝑎𝑝 = ∑𝑎𝑛𝑏
𝑛𝑏
+ 𝑆𝑝
In these equations, where the summations run over all the nearest neighbors of
each cell, 𝜑𝑝 is the value of the flow variable in the present cell and 𝜑𝑛𝑏 are the values of
the flow variable in the neighboring cells. 𝑆𝑖𝑗 and 𝑆𝑝 are constants and flow variable
depending parts of the source term, respectively. Furthermore, 𝑎𝑝 is the discretization
coefficient associated to the present cell, 𝑎𝑛𝑏 are discretization coefficient describing the
interaction with its neighboring cells. The discretization coefficients depend on the
discretization schemes used to approximate the values of the flow variables on the cell
faces. By using appropriate discretization schemes to determine the coefficients of
equation 4.2 and 4.3, a set of algebraic equations for the cell values is obtained [15].
4.2. Spatial discretization schemes
The convection and diffusion terms in the equation 4.1 are usually discretized
using different numerical schemes that estimate the faces values of the flow variables
[15]. Frequently, it is recommended to discretize the diffusion terms by using a central
differencing scheme and for the convection term is often used the upwind scheme [20].
It is important to have in mind that it is usually recommended to start a numerical
solution process with lower order schemes. However, the low accuracy of these schemes
can lead to a numerical diffusion higher order schemes should therefore be used to obtain
a more physically correct result. The second order upwind scheme is often considered as
a suitable discretization scheme since it exhibits a good balance between numerical
accuracy and stability [21].
To give a general idea about these methods, they are rapidly described in the
annex4.
(4.3)
40
4.3. Temporal discretization schemes
For the transient problems, the transport equation must also be discretized in
time. This is done by integrating the PDE over a time step, Δ𝑡, in addition to the spatial
discretization. In order to solve this integrated equation, the cell values of the flow
variables must be evaluated at a certain time.
Implicit time integration means the flow variables are evaluated at the feature
time, 𝑡 + Δ𝑡. Since these are not known in the current time step, implicit time
integration requires interaction. In comparison to explicit time integration, where the
flow variables are evaluated at the current time so that interaction is avoided, the
implicit time integration is more computationally expensive. On the other hand, implicit
time integration is unconditionally stable, meaning that is stable for all time step size
[15].
41
5. Database
At the beginning of the project, was dedicated to doing a static analysis of all the
fishing vessels that Vicus had already worked with before in order to create an important
data base to use as reference in designing the new optimal hull.
In order to analyze the range of vessels that were of interest to the client, it was
decided to group the boats by GT, and focus on the vessels that were around 600GT and
chose one of them to be considered as a “base ship”. Thus, it was possible to start doing
the geometry modification, as will be explained in the section 9.
The choice of the “base ship” was made based on two factors:
Vicusdt had to have the 2D plans of the ”base ship” as a minimum in order
to reconstruct the 3D geometry.
The “base ship” had to have been submitted to a towing tank test and the
results had to be available in order to validate the mesh used in the CFD
simulation.
It was decided to take as “base ship “a vessel called Mar de Maria, because it
complied with the above mentioned criteria and which has the most similar capacity
(607GT) comparing to the capacity required by the client(600GT).
42
6. Hull modeling
This study consists of the development of a more efficient hull geometry and this
is only possible if we analyze the general hydrodynamic comportment of the fleet and
select the vessels that have the best efficiency in order to start the modifications based on
their geometry.
To do this, it was decided to select 10 vessels that reliably represented the entire
range of the fleet, as shown in the figure below.
Figure 6: the red points represent the chosen vessels to create the 3D hull geometry. The blue points represent all the
fishing vessels that Vicus had already worked before
In order to run the CFD simulation for marine resistance prediction for each one
of these vessels, it was necessary to have their 3D hull geometry. Since some hulls 3D
geometries are not available, it was needed to construct them from the 2D plans before
starting the CFD simulations.
This step was done using Rhinoceros and will be not considered in the vessels
appendages. More information about these vessels main characteristics can be seen in the
annex 5.
43
6.1. 3D Construction
This is a very laborious part of the project and some considerations will be done
in order to simplify this process without compromising the quality of results. In this
example, it is possible to notice that the geometry of upper works were simplified in order
to make easier the 3D geometry construction. This is a normal procedure and is safe to
assume that this simplification is not going to influence the final results significantly
because the force exerted by air is insignificant when compared to the hydrodynamic
forces.
6.2. Geometry validation
In order to accomplish a simulation that represents the reality as closely as
possible, the 3D geometry model generated must be checked to evaluate if accurately
represents the real vessel or not. For this step, the hydrostatics values taken from the
vessel project and from 3D geometry models generated by Rhinoceros are compared.
Rhinoceros
Figure 7: Transforming the 2D plans in a 3D geometry using Rhinoceros
44
The data compared were: displacement, wetted surface and center of buoyancy
(Xb and Zb). At the table below is shown the percentage of discrepancy between the
Hydrostatics values taken from Rhinoceros and from the vessel project.
Table 1: Geometry validation
Based on the table above, it is possible to verify that the discrepancy average
between the values given by Rhino are all lower than 2.0%, confirming that the 3D
geometries generated accurately represent the real vessel. The exact values for each of
these hydrostatic values can be verified in the annex 6.
1 The hydrostatics information about Tronio were not available for the company.
Discrepancy
Vessels
Displacement
(t) S.wetted(m²) Xb Zb
Ana Barral 0,08% 1,19% 0,02% 0,05%
Bonito Dos 0,41% 0,46% 0,03% 0,31%
Loucenzas 0,86% 1,09% 1,78% 1,95%
Mar de Maria 3,11% 6,94% 0,62% 2,20%
Novo Airiño 0,22% 0,31% 3,18% 0,19%
O Taba 0,87% 1,90% 0,19% 0,71%
Siempre Juan Luis 0,67% 1,98% 3,85% 1,28%
Talasa 1,06% 1,83% 3,34% 0,89%
Tronio1 - - - -
Xuxo 0,04% 2,73% 0,04% 0,00%
AVERAGE 0,73% 1,84% 1,30% 0,76%
45
7. CFD Simulations for marine resistance prediction
Before starting the simulations with many different hull geometries in order to
find an optimum, it was necessary to validate the mesh that was to be used in these
simulations. For this, it will be used the results of a towing tank test that was done using
a ship model of the vessel Mar de Maria. The model had a scale factor of 10.8 and its
main characteristics are expressed in the following table. There are more details of the
main characteristics of the full-scale model of Mar de Maria in the annex7.
Table 2: Main features for the real and model scale of Mar de Maria
An important common characteristic in all numerical simulations made is the fact
that the vessel has no velocity and just 2 DOF (sinkage and trim), the flow around the
vessel hull is generated by an imposed fluid velocity, calculated in section 7.3.1.1 and the
hull motions are calculated as discussed in section 3.4.
Mar de Maria
Real Model
Lpp(m) 36 Lpp(m) 3,33
B(m) 9 B(m) 0,83
T(m) 3,5 T(m) 0,32
V(knots)
10
V(m/s)
1,56
13 2,03
13,5 2,11
Scale
Factor 10,8
46
In the process of mesh validation, it will be used two different strategies to
construct the mesh:
Regular mesh, which the domain remains fixed on a vessel referential.
What moves is the flow around the hull surface, which consequently
makes the vessel move, along with the “regular mesh” too, keeping the
same distance from one cell to another.
Overset mesh, which the domain is divided into one stationary background
region and one moving overset region close to the hull.
Then, it will be chosen the method that gives the best cost-benefits analyses,
comparing the simulation results with the towing tank test and taking into consideration
the computational effort required. Once the choice is made, it will be applied the same
mesh to run all the other simulations with different vessels. Thus, the scale of each model
should change in order to fit different hull geometries inside the same domain.
The method to construct these two types of mesh are very similar, but the
particular points of each one will be described on the section 7.2.
47
7.1. Computational domain definition
The computational domain definition is a very important step, because the effects
from the boundaries can affect directly the flow around the hull. To avoid that, we adopted
three important criteria:
ITTC Recommendations
For the height and length of the domain, it was used the values suggested by ITTC
[25, sc.2.5, pp 5]
Figure 8: ITTC recommendation to define the computational domain
48
Kelvin angle
It was used the Kelvin angle to calculate the minimum domain width (𝑌𝑑) in order
to capture all wave pattern, trying to reduce the influence of waves reflections.
Figure 9: Kelvin wake pattern behind a moving object. [27, pp. 67]
Doing a very simple trigonometric calculation using Kelvin angle (19,470),
considering 𝑋𝑑 the distance from the forward perpendicular until the end of the domain,
the value of 𝑌𝑑 is given as:
𝑌𝑑 = 𝑋𝑑 ∗ 𝑡𝑔(19.47)
𝑌𝑑 = (5𝐿𝑝𝑝) ∗ 𝑡𝑔(19.47)
𝑌𝑑 = (5 ∗ 3.33) ∗ 0.35
𝑌𝑑 = 5.8 ≅ 1,76𝐿𝑝𝑝
To have a merge, will be adopted 𝒀𝒅 = 𝟐𝑳𝒑𝒑.
Damping wave reflection
In Starccm+, it is possible to activate the “damping wave reflection”, that is a tool
that decrease the wave reflection from the boundaries [23].
(7.1)
49
7.2. Mesh generation
Like it was explained at the beginning of the section 7, it were used two
procedures to generate the mesh: “regular mesh and overset mesh”.
In both cases, to capture more precisely the flow behavior, a higher mesh density
was focused on certain regions of the domain: the bow, the aft and the free surface.
Furthermore, was constructed a boundary layer with prism layer mesh and set the height
of the first cell layer to obtain a proper value of 𝑦+ that respects the range of values
allowed to be used by the turbulent model adopted.
The particularities of each of these methods will be presented in the sections 7.2.1
and 7.2.2.
Regular mesh
This method is the most natural way to construct a mesh for CFD simulation. The
domain and all the others refined regions are completely fixed relative to the vessel,
moving together with it.
A study of the size refinement was done to analyze the arrangement that best
represents the flow and has less computational effort. Once that was created a first mesh
with all the volume meshes needed, it was created a macro on Starccm+ to generate
automatically these regions, turning faster the mesh generation and consequently the
analyses their best arrangement. It is important to highlight that the mesh density has an
anisotropic and a slowly graduation in order to get a better flow representation and also
to avoid wave reflections due to abrupt mesh transitions [24]. This effect can be even
more reduced by changing the template Default Growth Rate from “slow” to “very slow”
during the mesh generation [23].
50
The final arrangement of the refinements blocks contains at total 27 volumes
mesh:
Volumes meshes to capture the wake
Volumes meshes to capture the flow around the hull
Volumes meshes to capture the free surface
Volumes meshes to capture the flow around the bow
6 Volumes meshes to capture the flow around the aft part
The total number of mesh cells using this method was around 1 750 000 cells.
In figure 10, the structure of the regular mesh is illustrated.
(a) Overview (b) Symmetry plan
(c) Prism layer at the bow (d) Surface mesh at the bow
Figure 10: Schematic illustration of the regular mesh structure. (a) shows an overview of the mesh. (b) shows the
mesh symmetry plane with all the refined volume meshes. (c) shows the prism layer at the bow. (d) shows the
surface mesh at the the bow
51
Overset mesh
In this method, the mesh is divided into two overlapping meshes, one stationary
background region and one moving overset region close to the hull. All cells maintain
their shape and the mesh motion is determined by solving the equations of motion and
rotation of the vessel.
The background mesh exchange information with the moving mesh in the
following way. First, the cells around the interface of the overset mesh are identified and
labelled as donor cells. Then the cells in the background closest to the donor cells are
identified as acceptor cells. The background cells that are completed cover by the overset
region are inactivated. The donor and acceptor cells transfer information between the
meshes. Each acceptor cell has one or more donors cells. Choosing the donors cells can
be done differently, the method used in this study is the linear interpolation.
In this study, both parts were meshed with trimmed, with local and wake
refinements. Since the whole overset region sinks and trims, for it, was needed to increase
the refinement zones around the free surface at the overlap were needed in order to
maintain a uniform cell height around the hull, as is shown in figure 11d.
Care had to be taken so that the cells in the overlapping region have the same size
and form a continuous layer around the overset region. The size continuity is a crucial
point, because if the cells in the overlapping region have not the same size, it will be not
possible to construct a continuous layer around the free surface, and so interpolation
errors will appear, changing significantly the simulation results. In order to avoid this
problem, it was used two strategies: to place the “meshed alignment” at the free surface
and also to generate a mesh with an integer number of cells. These strategies will be
covered in detail in the section 7.2.2.1 and 7.2.2.2.
The big advantage with the overset method is that only a part of the mesh is
moving without requirement for altering the grid topology. A drawback is that the
interpolation between the cells can cause numerical errors [23].
52
The total number of cells meshing using this method was around 850 000 cells.
In figure 11, the structure of the overset mesh is illustrated.
(a) Overview (b) Symmetry plan
(d) Height continuity on free surface (c) Prism layer at the bow
overset
mesh
Cells height
continuity
Figure 11: Schematic illustration of the overset mesh structure. (a) illustration of an overview of the mesh. (b) illustration
of the mesh symmetry plane with the overset region and the background mesh. (c) illustration of the prism layer at the
bow. (d) illustration of the interface between the overset and background region.
53
7.2.2.1 Integer number of cells
The “mesh alignment location” is a tool on Starccm+ that allows to choose a point
to start the mesh construction. It is possible to impose this point at the free surface,
creating a clear distinction between water (represented by the red color) and air
(represented by the blue color), capturing the free surface with higher quality. The figure
12a represents the mesh without placing the “mesh alignment” on the free surface,
showing that in just one cell there are air and water (this mixture is represented by the
yellow strip), even if the surface is not disturbed by the flow interaction with the boat,
showing that the free surface is not being captured properly. The figure 12b represents
the mesh when the “mesh alignment” is placed on the free surface, allowing to create a
clear distinction between water and air, each one in different cells. Because of these
factors, it is possible to better model the free surface, thus having more precise results.
(a) (b)
Figure 12: Illustration of the use of the “mesh alignment” (a) represents the mesh without placing the
“mesh alignment” on the free surface (b) represents the mesh when the “mesh alignment” is placed
on the free surface.
54
7.2.2.2 Integer number of cells
With the intention of reducing numerical errors due to interpolation, it is advisable
to construct an overset mesh region which its limits coincide exactly with the adjacent
cell from the background region, like is shown in the figure 13b. Consequently, the
distance between the centroids of the adjacent cells will be smaller and consequently the
interpolation error will also be smaller.
Figure 13: Illustration of the mesh done in the simulation of vessel called Ana Barral. (a) represents one of the
source of interpolation error since the overset region is ending in the middle of one cell from background region. (b)
represents the correct way to construct an overset mesh to avoid interpolation errors. Notice that the end of the
overset mesh is exactly at the beginning of one cell from the background region.
In annex 10 there is a scheme that explains better how avoid this problem.
7.3. Model definitions and properties
Based on the theoretical background presented in the chapter 3, this section will
describe the appropriate models used in the CFD simulations.
Turbulence model
The turbulence was modelled using the RANS equations with the SST 𝑘 − 휀 two
layer approach. In this approach, the computation is divided into two layers. In the layer
next to the wall, the turbulent dissipation rate ε and the turbulent viscosity 𝜇𝑡 are specified
as functions of wall distance. The values of ε specified in the near-wall layer are blended
smoothly with the values computed from solving the transport equation far from the wall.
55
The equation for the turbulent kinetic energy is solved across the entire flow domain. This
explicit specification of ε and 𝜇𝑡 is arguably no less empirical than the damping function
approach, and the results are often as good or better.
The advantage of using this model is that it is considered the model that offer the
most mesh flexibility. It can give good results on fine meshes (that is, low 𝑦+meshes),
and also produces the least inaccuracies for intermediate meshes (that is, 1<𝑦+ <30) [23].
Hull motion
The hull motion was adopted with just 2 DOF: vertical translation, and pitching
rotation. This was done by allowing only the translational motion along the z-axis and
rotational motion around the y-axis. This setup allows us to use symmetry conditions and
include only half the geometry, thus reducing the computational cost of the simulation.
To solve the equations of motion, the mass, the center of mass and the moment of
the inertia were estimated, like is specified in the section 3.4.3. However, care had to be
taken at the time of entering these values in Starccm+, because the simulation runs with
just half of the vessel, so the mass and the inertia has also been divided by 2.
Boundary layer thickness
As explained in the section 3.2.3, it is very important to well define a boundary
layer because this region is characterized by high velocity gradients in the wall-normal
direction, influencing directly the results of pressure field and consequently the forces
applied on the hull vessel. To capture these variations, it is needed to refine this region
and to do this, two parameters are usually used: boundary layer thickness and number of
prism layer.
To calculate the thickness of the boundary layer, is possible to use the
approximation made by Blasius and Prandtl, that are used normally to describe laminar
and turbulent flow respectively on a flat plate. In this case of study, even for the
simulations in model scale the value of Reynolds number are higher than 3. 106,
56
characterizing the flow as completely turbulent. Thus, to calculate the boundary layer
thickness, will be used the Prandtl formula, that is expressed by:
𝛿 ≅0.37𝑥
(𝑅𝑒𝑥)1
7⁄
In this case of study was decided to fix the vessel model length as 4m, in order to
fit different hull geometries inside the same domain, like was explained in the beginning
of the section 7. Considering also that the model flow velocity doesn’t exceed 2.5m/s, it
is possible calculate the maximum value of 𝛿. Overestimating this value, is guaranteed
that the boundary layer will be captured. Replacing these considerations in the expression
7.1:
𝛿 ≅0.37 ∗ (4)
(1.12 ∗ 107)1
5⁄
𝜹 ≅ 𝟎. 𝟎𝟓𝟕𝒎
So, it will be considered that the boundary layer thickness is not going exceed
6cm. This was the first estimate that was made, but it generates large number of cells.
Based on the company’s preview experiences this thickness was set to 4cm without
compromising the results.
To evaluate if the boundary layer thickness is appropriate, it should be checked if
the values of Y+ are among the range of values allowed to be used for the model of
turbulence adopted. If it is not, the number of prism layer could be also modified. In this
study, it was set as 6.
7.4. Boundary and initial conditions
Before attempting to solve the Navier-Stokes equations, it is necessary to impose
appropriate physical conditions of the fluid domain. In fact, it is precisely these conditions
that distinguish different flow problems [3, pp 63].
(7.1)
57
Boundary condition
The boundary conditions are inputs values defined on the computational boundary
domain which establish restrictions to the numerical method, which are essential for the
representation of the physical model. The boundary conditions adopted in this case of
study were:
Figure 14: Illustration of the domain boundary conditions
7.4.1.1 Velocity Inlet
In the inlet of the computational domain, is imposed as a boundary condition a
constant velocity along the normal direction of the Inlet surface. The velocities were
chosen according to the velocity frequently adopted by the studied vessel. Is important to
remember that the CFD analysis has to account for model scale. Thus, to discover the
velocity that will be used as input, it must be used the similitude law of Froude, allowing
to discover the velocity of the model depending just on the velocity of the real vessel and
the scale model, like is shown in the follow expression [28,Ch.1, pp10]:
𝑈𝑚 =𝑈𝑅
√𝜆
With,
𝑈𝑚: model scale vessel velocity
𝑈𝑅: real scale vessel velocity
𝜆: scale model
(7.2)
58
7.4.1.2 Pressure Outlet
As we are working with a subsonic flow, it is needed just to specify the pressure
on the surface that is receiving this boundary conditions [23]. In this case of study, the
pressure will be define by the function Hydrostatic Pressure of FlatVofWave, which is
used to calculate the pressure taking into consideration the free surface elevation.
7.4.1.3 Wall function
On the vessel surface, it is imposed a wall function as a boundary condition which
determines:
Impenetrability conditions: the velocity along the normal velocity of the
surface is equal to zero.
No slip condition: the velocity is equal to zero on the wall, allowing to
capture the boundary layer.
7.4.1.4 Symmetry Plane
Since the vessel and the motions that are being studied are symmetric in relation
to the plan XZ, we can use a boundary condition known as Symmetry plane to save
computational efforts. This condition represents an imaginary simulation plan.
Mathematically, the symmetry plan is created in order to represent the simulations results
along the total domain, but just having to construct half of this.
It is worth to highlight that the shear forces on the symmetry plan are zero. The
pressure and velocity on the symmetry plan are calculated by extrapolating the values of
adjacent cells.
59
Initial condition
In addition to the boundary conditions, the user must establish the initial
conditions of the system at the beginning of the flow simulation.
In order to create an initial condition, it is utilized the VOF Waves Model. This
tool is provided by StarCCM+ and it simulates the free surface water level changes over
time during the simulation. The initial values of pressure, velocity and volume fraction
are calculated by this model [24].
The parameters required to use this model are: the pressure, velocity and volume
fraction. These values are calculated using respectively the functions: Hydrostatic
Pressure of FlatVofWave, Velocity of FlatVofWave, Volume Fraction of Heavy Fluid of
FlatVofWave (to calculate the volume fraction of water) and Volume Fraction of light
Fluid of FlatVofWave (to calculate the volume fraction of air).
7.5. Choice of Mesh
Following the information given along this section, it was used the Mar de Maria
geometry to create one simulation using the method of overset mesh and other simulation
using the method of regular mesh.
The results of these two simulations were compared with the towing tank results
done with the Mar de Maria model. Considering the difference between these results and
the computational effort, it will be decided which method of mesh construction should be
adopted.
The following table shows the difference in percentage between the values
obtained doing CFD simulations with overset mesh and with regular mesh, comparing
them with the towing tank test.
60
Table 3: Relative difference between the results from the towing tank test and the results found using CFD
simulations with Overset mesh and with Regular mesh.
According to the above table, is possible to notice regardless of the proximity of
the results between the two methods of mesh with the towing tank results, that there is a
huge difference between the computational efforts. The overset mesh has almost 50% less
cells than the regular mesh. The reason of this huge difference is mainly because of the
reduction of the volume mesh that need to be created and also because the wake was
captured using a triangle geometry, and not a rectangle as was used on the regular mesh,
which can also contribute to reduce the number of cells.
CFD Simulation with Overset mesh CFD Simulation with Regular mesh
V(knot) Rt(Kn) Cf Ct 1+K Rt(Kn) Cf Ct 1+K
7 0,50% 0,10% 3,99% 2,60% 0,94%
0,13%
3,84%
2,95%
10 4,60% 0,21% 1,20% 2,60% 4,25%
0,18%
1,36%
2,95%
13 0,87% 0,22% 2,61% 2,60% 1,13%
0.24%
2,48%
2,95%
61
7.6. Post-processing
After choosing the most appropriate mesh, this same mesh will be used to run all
the simulations. As the main intention is to analyze the hull-bare resistance, it is necessary
to create a resistance-curve for each vessel. In this case of study, it was decided to create
this curve through the analysis of three different speeds for each ship. Totalizing at the
end 33 CFD simulations for the 10 vessel existing geometries plus 3 simulations for the
new geometry.
For each of these simulations were analyzed some important visual information
as the pressure around the hull and the wave pattern. Also, it was analyzed important
graphics which represent the body force in X, body force along Z, body moment around
Y, translation along Z and body orientation around Y. For these graphics it is very
important to quantify what is being analyzed. For this, it will be taken an average among
the points plotted in the graphic. To extract this value, the beginning of the simulation
will be discarded, because as we can see in the graphics, the software still in the iterative
process to find the vessel’s equilibrium position as explained in the section 3.4.2.
The following sections will better explain the analysis. For this, was chosen use
as example the simulation of the vessel called Siempre Juan Luis at 8.5knot, Fn = 0.294.
62
Pressure around the hull
This information is very important to identify the points in the hull where the
biggest efforts are applied.
Figure 15: Bottom and side view of the pressure around the hull.
Free surface wave pattern
This information is important to check if the constructed mesh is capturing the
wave pattern.
Figure 16: Top and perspective view of the surface wave pattern. The numbers in this figure will be explained in the
section 7.5.7
1
2 3
63
Body force in X
This information represents the hull bare resistance, which is essential to predict
the power efficiency as explained in the section 3.6.
Graphic 1: Bare-hull resistance = -23.65N
Body orientation around Y
This information allows to evaluate the trim angle that the vessel has when it is in
cruise situation.
Graphic 2: Trim average = 3.94 degrees
-50
-40
-30
-20
-10
0
10
0 5 10 15 20 25 30 35
Res
itan
ce (
N)
Physical Time (s)
6-DOF Body Force X: Rigid Body Force (N)
-6
-5
-4
-3
-2
-1
0
1
0 5 10 15 20 25 30 35
Pit
ch (
deg
)
Physical Time(s)
6-DOF Body Orientation Y: Rigid Body Angle (deg)
64
Translation along Z
This information allows to evaluate the sinkage that the vessel has when it is in
cruise situation. This value is normally captured to compare with the values from the
towing tank test, but in this case of study, this value is not available to do the comparison.
Graphic 3: Sinkage average = 0.038m
Body moment around Y and body force along Z
This information allows to evaluate if the iterative process presented in the section
3.4.2 is successfully reaching the equilibrium position of the vessel. As is possible to
notice in the following graphics, the residual moment around Y and the residual force
along Z are very small, meaning that the procedure explained in the section 3.4.2 was
converged to a hull in equilibrium position.
-0,06
-0,05
-0,04
-0,03
-0,02
-0,01
0
0,01
0 5 10 15 20 25 30 35
Sin
kage
(m)
Physical Time(s)
6-DOF Body Translation Z: Rigid Body Translation (m)
-100
-50
0
50
100
0 10 20 30 40
Mo
men
t a
rou
nd
Y (
N.m
)
Physical Time(s)
6-DOF Body Moment Y: Rigid Body Moment (N-m)
-60
-40
-20
0
20
40
60
80
0 10 20 30 40
Forc
e al
on
g Z
(N
)
Physical Time(s)
6-DOF Body Force Z: Rigid Body Force (N)
Graphic 4: Moment around Y average = 0.0003m Graphic 5: Force along Z average = 0.00048N
65
CFD wave cut
Graphic 6: CFD wave cut
This graphic is generated by intersecting a perpendicular plan, distant 10% of the
breadth from the vessel, with the free surface, as explained in the section 9.
Comparing this graphic with the figure 16, is possible to perceive that the highest
values for the surface elevation are concentrated in the bow and in the aft part of the vessel
(identified by the numbers 1 and 3 respectively) and the lowest values are concentrate
around the main section of the vessel (identified by the number 2).
Thus, it is possible to notice that this graphic is very important because it allows
to quantify the information given in the section 7.5.2 and will be very useful in the phase
of modifications of the hull, in section 9, and to compare the results.
7.7. Results
The same post process was made for all simulations. The results of the post
processing for each boat are attached in a table on the annex 8.
1
2
3
66
8. Extrapolation of the model results
After extracting the results from the CFD simulations, it is needed to extrapolate
the results from the simulations using the model scale to the real scale. To make this, will
be applied the theory described in the section 3.5.
8.1. Calculation of the form factor
One factor that is very important to calculate is the form factor. Like it was
explained in the section 3.5.2.5, the form factor depends basically on the boat geometry.
Due to the fact that in this study all the boats that are being studied belong to the same
family of vessel and have a very similar hull shape, it is possible to consider that the form
factor will be the same for all of them.
Thus, it was chosen the “base ship”, Mar de Maria, to be used to calculate its form
factor and use this same value for the other vessels. Three simulations, each one sets with
a difference velocity, will be run, in order to get three different values for the ratios 𝑪𝑻𝒎
𝑪𝑭𝒎
and 𝐹𝑛4
𝐶𝐹𝑚 to construct the linear tendency line equation, as in the equation 3.48 and get the
value of the factor form, which is represented by the constant term.
Graphic 7: form factor calculation for the Mar de Maria
Thus, is possible to noticed at the graphic above that 1+K= 1.276.
y = 0,2341x + 1,2763
0,00
0,50
1,00
1,50
2,00
2,50
0,00 1,00 2,00 3,00 4,00
𝑪𝑻𝒎/𝑪𝑭𝒎
𝐹𝑛^4/𝐶𝐹𝑚
Form factor
67
With this estimation for the value of the form factor, it will be possible to calculate
the value of the vessel total resistance (𝑅𝑇𝑠), as shown in the section 3.5. This value will
be very important in the section 10, where it will be used to dimension the engine using
NavCad.
8.2. Results of the extrapolation
After doing the extrapolations in order to discover the value of the vessel total
resistance (𝑅𝑇𝑠), it is possible to use the expression 3.51 to calculate the effective power
required for each hull geometry. A graphic comparing the effective power of each vessel
is attached in the annex 12.
68
9. - Hull modifications
9.1. Motivation for the hull modifications
The hull modifications were done in order to reduce the bare hull resistance. As it
is possible to see in the graphic 8, usually the wave coefficient represents a very important
part of the total coefficient when the vessel is working at its operational speed or greater
speeds. In the graphic, it is possible to see how Cw, Cv and Ct evolves with respect to
Froude number.
Graphic 8: Ct, Cw and Cv curves for the model simulation of the original Mar de Maria geometry at Fn equal to
0,191;0,274 and 0,356
In this case of study, it is possible to notice that the participation of Cw on Ct
became more important than the participation of Cv after Fn ≅ 0.295.
Taking into account that the design speed is 13 Knots, that is Fn ≅ 0.33, beyond
the point where Cv = Cw, it was decided to focus the hull modifications on the reduction
of Cw, and to achieve this objective, we will try to reduce the waves creation induced by
the boat. To make this analysis possible, it was created in Starccm+ a plane that intersects
orthogonally the free surface. This plan is placed distant 10% of breadth from the vessel
as shown in the figure 17.
0
0,002
0,004
0,006
0,008
0,01
0,150 0,200 0,250 0,300 0,350 0,400
Ct,
Cv,
Cw
Fn
Original Mar de Maria
Ct
Cv
Cw
69
Figure 17: Cross section used to capture the elevation of the free surface, taking as origin of the cross section the
point: (0;0.1*B; T). With B and T representing respectively the vessel breadth and the draft.
It allows to plot the elevation of the free surface, like it is shown in the graphic 9.
Thanks to this plot, it will be possible to compare the amount of water that is being
displaced not only visually, but also quantitatively, because the integration of this curve
will give a value that represents amount of water displaced as wave in the model scale,
which could be physically seen also as the energy is dissipated in form of wave. Thus,
the vessel modifications will be inspired considering the geometries that gives less energy
dissipation in form of wave.
As explained in section 5, in this case of study, it will be selected one vessel
among the vessels which has the most similar cargo capacity to with was demanded by
the client. The main dimensions will be maintained, but the bow and aft geometry can be
modified.
WL
Graphic 9: Illustration of the area that will be calculated to quantify the dissipated energy in form of wave. This is
a graphic of a free surface elevation taken from the Siempre Juan Luis simulation at 8.5 knot
70
To give an idea about how start the modifications, will be considered that the
waves present behind the aft perpendicular, 𝑋 < 0, are resulted by the aft part of the ship
and the waves present in 𝑋 > 0 are resulted by the bow part of the ship, considering the
zero as the longitudinal position of the aft perpendicular. Integrating the free surface
elevation curve in these two parts separately, will be possible to quantify which vessels
have the most efficient aft and bow geometries (as shown in the graphic 10) and so, use
their geometries as inspiration in the construction of the new hull geometry.
It is important to highlight the fact that joining the best bow and aft geometries
will not necessarily give an optimal overall result, but it still a valid start the optimization
doing this. It will give the directions for a first hull new geometry which can be continuous
modified in order to achieve an optimal result.
Graphic 10:Comparison between the area below the graphic taken from the free surface elevation for the operational
speed of each vessel model simulation
In this graphic is possible to notice that there is a vessel that generates an extreme
low value of elevation of free surface. It is the vessel called Tronio, the biggest vessel
among the analyzed vessels with 55m of LOA, much bigger than the others vessels, and
consequently, for the velocities that were studied, it has a Froude number not big enough
to create significant perturbations in the free surface. So, its geometry will be not used as
an example to inspire the hull modifications.
0,0000,0250,0500,0750,1000,1250,1500,1750,2000,225
Integration of the free surface elevation curve(m^2)
Bow part
Aft part
71
9.2. Validation of the Form Factor value
Furthermore, the value of the integration of the all surface elevation curve, as
shown in graphic 9, can give an idea if the consideration made on the section 8.1, which
considered that the form factor (1+k) could be taken as the same for all vessels without
significant errors led us to these results, is valid or not.
As explained before, the graphic 10 is a way of quantifying the dissipation of
energy as wave. A mathematical expression that gives the same physical meaning is 𝐶𝑤
𝐶𝑡,
which using the theory from the section 3.5.1 and 3.5.2 could be also written as:
𝐶𝑤
𝐶𝑡= 1 −
𝐶𝑣
𝐶𝑡= 1 −
𝐶𝑓 ∗ (1 + 𝑘)
𝐶𝑡
Even if the integral of the free surface elevation, represented in graphic 9 and the
expression 𝐶𝑤
𝐶𝑡 don’t have the same units and order of magnitude, they have the same
physical meaning and if the approximation made for the value of (1+k) is valid, the both
curves must represent the same tendency, as shown in the follow graphic.
Graphic 11: tendency of the integral of surface elevation of the free surface and Cw/Ct
In this graphic, is possible to see clearly that both graphics respect the same
tendency, allowing to infer that the approximation made for the form factor is acceptable.
0%
10%
20%
30%
40%
50%
60%
70%
80%
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
Cw
/Ct
Surf
ace
Ele
vati
on
(1+K) Validation
SurfaceElevation
Cw/Ct
(9.1)
72
9.3. Main hull modifications
As it was explained in the section 5, the hull geometry used as a start point was
from the vessel called Mar de Maria, which is a vessel that have 607GT, very close from
the 600GT required by the client.
Maintaining the main vessel dimensions of Mar de Maria, the bow and aft
geometry were modified by analyzing the geometry from the vessels which have best
results taking into account the graphic 10, aiming to decrease the energy dissipation
created by the waves. The main modifications that were done are going to be listed in this
section.
Modifications at the bow
In the graphic 10, is possible to see that the Mar de Maria has already an efficient
bow geometry comparing with the others, so, subtle modifications were done in this
region. Considering the geometry of Novo Airiño, which also has a good bow efficiency,
the bulb height and length were slightly increased, as can be seen in the figure 20.
Figure 18: original bow geometry of Mar de
Maria
Figure 19: modified bow geometry of Mar de
Maria
73
Modifications at the aft part
Comparing the figures 21 and 22 is possible to perceive that was made significant
modifications in the aft part of the vessel, which will be listed above with their respective
qualitative effects.
9.3.2.1 Creation of a bulb at the aft part
As we can notice, in the original design, there is not a bulb at the aft part of the
vessel. Represented by the number 1, the creation of a bulb in this region, allows to
improve the direction of the flow toward the propeller, creating a more homogeneous
velocity and pressure field in this region. Consequently, the mechanical wear of the
propeller due to the transition between regions having different pressures will decrease
and its efficiency will be increased.
9.3.2.2 Geometry continuity (rudder bulb)
Represented by the number 2, it is possible to perceive that there was a concern
to create a geometry continuity between the rudder geometry and the propeller shaft. The
main goal in doing this was to reduce the wear on the rudder caused by the flow impact
which is accelerated by the propeller and brutally stopped by the rudder.
Figure 21: Comparison between the original (green)
and modified aft part of Mar de Maria (grey)
Figure 20: details of the modification in the aft part
of Mar de Maria
1
2
3
74
9.3.2.3 Twisted rudder
Represented by the number 3, and more in detail in the figure 22, it shows that
was implanted a twisted rudder, which has asymmetric trailing edge in order to assure a
better course keeping and minimizes rudder “hunting”, decreasing the drag.
Figure 22: Twisted rudder (front view)
75
9.4. Results
After the modifications were made on the hull geometry, simulations using the
new hull geometry of Mar de Maria were launched with the same conditions as the
previous ones using the old geometry. Thus, will be possible to analyze the changes
brought on the coefficients values and also in the elevation of the free surface.
Coefficient changes
In the graphic below is possible to notice the difference between the total
coefficient values from the simulation using the original geometry and from the
simulation using the modified geometry.
Velocity
(Knot) Fn Ct
7 0.191 +0,2%
10 0.274 -1,77%
13 0.356
-
24,44%
Graphic 12: Comparison between the total resistance coefficient (Ct) from the modified and original Mar de Maria
hull geometry
It is possible to infer through these data that there is not a huge difference between
the values of the total coefficient when the vessel is working at low speed, but when it is
at the design speed (𝐹𝑛 ≅ 0,33), there is a reduction of 24,44% in the total resistance
coefficient.
0,004
0,005
0,006
0,007
0,008
0,009
0,01
0,150 0,250 0,350 0,450
Ct,
Cv,
Cw
Fn
Ct from Original Hull Vs Ct from New Hull
Ct originalhull
Ct new hull
76
Free surface elevation
In order to confirm the results exposed in the previous section, is also valid to
check how less energy is being dissipating as wave.
Graphic 13: reduction of the wave creation between the different vessels geometries
The integral of the free surface curve for the original hull geometry of Mar de
Maria is 0,200𝑚2 and for the new geometry is 0.137, it is means a reduction of 31.5%,
confirming the positive result taken from the section 9.4.1.
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
Ana Barral BonitoDos
Loucenzas OriginalMar deMaria
New Marde Maria
NovoAiriño
O Taba SiempreJuan Luis
Talasa Tronio Xuxo
Integration of the free surface elevation curve (m^2)
77
10. NavCad analysis
NavCad is a software tool for the prediction and analysis of vessel speed and
power performance and according to these analysis, it can also provide a first estimative
of a propeller design.
To make the power and speed prediction, the bare-hull resistance has to be
calculated and to do this, NavCad uses statistical methods that are implemented in the
software, but this is a methodology much less accurate than the CFD simulations. That is
the reason why in this study it was used CFD simulations to estimate the bare-hull
resistance and NavCad to estimate the RPM and motor power required, propulsion overall
efficiency and percentage of cavitation for each CFD simulation, that is, for each tested
velocity for the 10 vessels.
To start NavCad analysis, it is needed to consider the results from CFD simulation
and input the velocities and the bare hull resistance for each one of them. Then, it is
possible to choose the option “sizing by thrust”, which means that the calculations will
be done in order to respect the thrust that was calculated from the input values of bare-
hull resistance from CFD analysis. To estimate the thrust and the break power, it was
decided to use the method Holtrop and also add 15% of a design margin.
10.1. Propeller design
As explained in the beginning of this section, NavCad can propose a first
estimative of a propeller design. A more detailed study on the design of the propeller will
be done in the next steps of this project, but in this study it will be approached only the
first estimation made by NavCad.
For the design of this first propeller estimative, it is necessary to enter the
information of the maximum diameter of the propeller. To calculated this, must be
considered a safety distance ℎ1 ≅ 0,1𝑚, between the propeller and the lowest point of
hull to avoid collision with the bottom of the sea or docking area and also a distance ℎ2 ≅
78
0,2𝐷 (20% of the propeller diameter) from the highest part of the propeller and the hull,
to avoid a sudden deceleration of the flow caused by the proximity of the hull and the
propeller and consequently overloading the propeller. For better understanding, check the
scheme made in the annex 11. Besides that, the design and the main data of the propeller
is also attached in the annex 13.
10.2. Power break (Pb)
Applying the theory approached in the section 3.6, NavCad calculates the value
of Pb, which are represented in the follow graphic.
Graphic 14: Power break prediction on NavCad
According to this graphic, is possible to notice that the power break efficiency
required by the vessel Mar de Maria at 13knots (its cruising speed) passed from 1513Kw
to 1000 Kw, that means a reduction of 33,78%. Considering that the engine will work
with 85% of its capacity, it is necessary to look for an engine that works in the range
around 1200Kw.
The main results taken from NavCad for each vessel are attached in the annex14.
0
300
600
900
1200
1500
1800
2100
2400
2700
3000
3300
3600
0,00 0,10 0,20 0,30 0,40 0,50
Pb
(K
w)
Fn
Power Break (Pb) New Mar deMaria
Novo Airiño
Siempre JuanLuis
Bonito Dos
Loucenzas
O Taba
Talasa
Xuxo
Original Mar deMaria
Tronio
Ana Barral
79
11. Efficiency
The efficiency of a vessel is a measure that goes beyond simply analyzing how
much power is required for a given speed to be achieved. It is a complex factor that
depends on a variety of variables as the volume of cargo transported, vessel velocity,
specific fuel consumption and the required engine power.
There are many definitions and metrics to evaluate the efficiency of ship. In this
study, it was chosen the Estimated Index Value (EIV) because it formed the basis for the
current regulation of the design efficiency by requiring ships to have a maximum Energy
Efficiency Design Index (EEDI). The EIV is a simplified form of the EEDI [31].
In order to assess how much better the new hull’s design is, it will be created an
EIV reference line from the analysis of all the 10 vessels that had been studied. The
reference line is the best fit of a power function through the EIVs of these vessels. A value
above the reference line means that a ship emits more CO2 per ton-mile under standard
conditions. As will be seen in the follow section, the parameters relative to the CO2
emission will be considered as fixed values, so the values that are above the EIV reference
line could be also considered less efficient than the average comparable ship [32], which
is normally an information that draw more attention for the ship-owner than the
CO2 emition.
11.1. EIVs calculation
The EIVs have been calculated in conformity with the resolution of the Marine
Environment Protection Committee, MEPC.212(63), and it could be expressed as:
𝐸𝐼𝑉 = 𝐶𝐹 ∗𝑆𝐹𝐶𝑀𝐸𝑖 ∗ ∑ 𝑃𝑀𝐸𝑖 + 𝑆𝐹𝐶𝐴𝐸𝑖 ∗ 𝑃𝐴𝐸
𝑁𝑀𝐸𝑖=1
𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑦 ∗ 𝑉𝑟𝑒𝑓
(11.1)
(9.1)
80
Where,
The subscripts 𝑀𝐸 and 𝐴𝐸 refer to the main and auxiliary engine(s), respectively.
𝐶𝐹: carbon emission factor is constant for all engines and equal to 3.1144g CO2/g
fuel.
𝑆𝐹𝐶𝑀𝐸𝑖: specific fuel consumption for all ship types is constant for all main
engines and equal to 190g/kWh.
𝑆𝐹𝐶𝐴𝐸𝑖: specific fuel consumption for all ship types is constant for all auxiliaries
engines and equal to 215g/kWh.
𝑃𝑀𝐸𝑖: represents 75% of the total installed main power (𝑀𝐶𝑅𝑀𝐸𝑖) expressed in
Kw.
𝑃𝐴𝐸: is the auxiliary power expressed in Kw. For the ships with a main engine
power below 10 000Kw, 𝑃𝐴𝐸 is define as [32]:
𝑃𝐴𝐸(𝑀𝐶𝑅𝑀𝐸𝑖<10000𝑘𝑤) = 0.05 ∗ (∑ 𝑀𝐶𝑅𝑀𝐸𝑖
𝑛𝑀𝐸
𝑖=1
+∑ 𝑃𝑃𝑇𝐼(𝑖)
𝑛𝑃𝑇𝐼𝑖=1
0.75)
𝑃𝑃𝑇𝐼(𝑖): represents the innovative mechanical energy efficiency technology. It is
excluded from the calculation. 𝑃𝑃𝑇𝐼(𝑖) = 0.
𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑦: defined as 100% of dead weight tonnage. In this case of study, it will
be used directly the cargo volume that can be transported.
𝑀𝐶𝑅𝑀𝐸𝑖: Main machine power, this value is the same as the “power break” given
by NavCad
𝑉𝑟𝑒𝑓: refers to the design speed.
(11.2)
(9.2)
81
Applying all these considerations in the formula 9.1, results in the following
expression:
𝐸𝐼𝑉 = 481.81 ∗𝑀𝐶𝑅𝑀𝐸𝑖
𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑦 ∗ 𝑉𝑟𝑒𝑓
It is possible to see in the equation 11.3, the measurement of efficiency using the
EIV method does not considers just the main engine power, but the ratio between this
value and the capacity multiplied by the vessel velocity. This is very useful because we
are also taking into account that a vessel that has a higher velocity or capacity of transport
can be more efficient than the other even if its main engine power is higher.
11.2. Efficiency results
In this section, it will be generated a similar graphic to the graphic 14, but to
create the curve it will be not considered just the break power as made in the section 10,
but will be considered the value of EIV.
Graphic 15: Graphic of EIV X Fn for each vessel
0
20
40
60
80
100
120
140
160
180
200
220
0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45
EIV
Fn
New Mar de Maria
Novo Airiño
Siempre Juan Luis
Bonito Dos
Loucenzas
O Taba
Talasa
Xuxo
Original Mar deMariaTronio
Ana Barral
(11.3)
(10.3)
82
In this graphic, it is possible to see clearly that the boats that required less break
power are not necessarily the most efficient. Comparing the graphic 14 and 15 is possible
to affirm that in this case of study it is almost the opposite. For example, the vessel Bonito
Dos which in the graphic 14 is the vessel that requires less power, has at the same time
the highest curve of EIV.
Furthermore, is possible to notice that the EIV curve that represents the modified
hull geometry of Mar de Maria is below than the EIV curve taken from the old geometry,
meaning that the new geometry is more efficient than the old one. In order to make it
clearer, the graphic 16 shows a reference line getting the values of EIV calculated from
the design velocity for each vessel. The red point on the graphic represents the EIV value
from the original hull of Mar de Maria and the green point is for the modified one.
Graphic 16: EIV tendency line
It is possible to notice that the value of EIV for the original hull geometry was
above the tendency line, and as explained in the beginning of this section, it means that
the vessel is less efficient than the average of the studied vessels. However, after the
modification, the efficiency was increased, and as the graphics shows, the EIV value for
the modified vessel is below the curve, showing that the modifications were successfully
done and reached the objective.
0
50
100
150
200
250
0,25 0,27 0,29 0,31 0,33 0,35 0,37 0,39 0,41
EIV
Fn
EIV tendecy line
Original Marde Maria
Modified Marde Maria
83
12. Conclusion
As it is possible to perceive, this work was very rich in terms of results presented
to the client and also in terms of knowledge acquired by the student. Thus, in this chapter
it will be presented an overview of the main result for each one of this field and also a
prediction for the future works.
12.1. Conclusion from the point of view of the client
From the point of view of the client, it is important to highlight the reduction
provoked by these changes in the main coefficient related to the vessel consumption and
which calculation was explained in the previous section. The follow table resumes these
results.
Table 4: Final comparisons between the Original and the new geometry of Mar de Maria
This result shows clearly that the study achieved the main objective and
consequently made the customer satisfied with our work.
Original Mar de
Maria
New Mar de
Maria % reduction
Ct 0,00896 0,00677 24,44%
EIV 81,635 58,12 28,81%
Peff(Kw) 636,91 460,16 27,75%
Pb(Kw) 1513 1001,9 33,78%
84
12.2. Conclusion from the point of acquired knowledge
This project approached the main points that concern the career of a naval
engineer. It allowed me to be in touch with the problems concerning hull geometry
modelling, CFD studies, propulsion efficiency, which made me familiar with softwares
that are essential for a technical engineer formation as Rhinoceros, Starccm+ and
NavCad. Furthermore, the experience of working in a real project for a company that
really depends on the work of each one that is part of it was very enriching for the
construction of a sense of responsibility. Undoubtedly, this project was an essential step
to prepare me to face the labor market nowadays.
12.3. Future works
The next steps for advancing this project are:
To continue working on this first new geometry, varying the bow and aft
geometry and doing the simulations in parallel in order to find a geometry
with a better performance.
Improve the study of the propeller, using Propcad.
Do the self propulsion simulation, to extract better estimation for the value
of wake fraction (w) and thrust deduction (t), which are currently estimate
by using NavCad.
Repeat the NavCad analysis using the coefficient values taken from the
self-propulsion simulation.
Choose an engine that satisfies these new necessities.
Besides that, as explained in the section 1, the client wants renew all his
fleet of vessel. On this project was approached the study for the one part
of the fleet, that is means, for the vessels that works in the Pacific, which
have normally 600GT. As the next step of this project, it is needed to do
the same study but for the vessels that works in the Atlantic, which
normally are smaller, with around 200GT.
85
Bibliography
[1] http://mathworld.wolfram.com/DivergenceTheorem.html
[2] Applied Hydrodinamics ; Hubert Chanson
[3] Marine Hydrodynamics; J.N Newman
[4] An Introduction to computational fluid dynamics. The finite volume method; H.K.
Versteeg and W.Malalasekera
[5] Introductory lectures on turbulence; J.M McDonough
[6] Compendium of ship hydrodynamics ; Jean-Claude Dern, Jean-Marc Quenez and
Philip Wilson
[7] REYNOLDS,O., On the Dynamical Theory of Incompressible Viscous Fluids and
the Determination of the Criterion. Philosophical Transaction of the Royal Society of
London. A, 186: 123-164, 1895. ISSN 02643820. URL
htttp://www.jstor.org/stable/90643
[8] François G Schmitt. About Boussinesq’s turbulent viscosity hypothesis: historical
remarks and a direct evaluation of its validity. Comptes Rendus Mécanique, Elsevier
Masson, 2007, 335 (9-10), pp.617-627.
[9] WILCOX, d.c., Turbulence Modeling for CFD. DCW Industries, Inc. La Canada,
CA, 1998.
[10] WILCOX, d.c., Turbulence Modeling for CFD. DCW Industries, Inc. La Canada,
CA, 2006.
[11] Davidson, L. An Introduction to Turbulence Models. Gothenburg: Chalmers
University of Technology; 2003.
[12] Launder, B. E., Spalding, D. B. Lectures in Mathematical Models of Turbulence.
London: Academic Press; 1972.
[13] Larsson, L., Raven, H. C. Ship Resistance and Flow. Ed. by Paulling, J. R.
Principles of Naval Architecture Series. Jersey City: Society of Naval Architects and
Marine Engineers; 2010.
[13] Menter, F. R. Two-Equation Eddy-Viscosity Turbulence Models for Engineering
Applications. The American Institute of Aeronautics and Astronautics Journal.
1994;32(8):1598–1605. doi: 10.2514/3.12149
86
[14] Bardina, J., Huang, P., Coakley, T. Turbulence Modeling Validation, Testing, and
Development. Technical report 110446. National Aeronautics and Space Administration
(NASA), 1997.
[15] David Frisk, Linda Tegehall. Prediction of High-Speed Planning Hull Resistance
and Running Attitude, Department of Shipping and Marine Technology, Chalmers
University of Technology, Gothenburg, Sweden 2015.
[16] White, F. M. Fluid mechanics. 7th ed. Boston: McGraw-Hill; 2003.
[17] https://www.cfd-online.com/Wiki/Dimensionless_wall_distance_(y_plus)
[18] D. Lörstad, L.Fuchs. A volume of Fluid (VOF)method for handling solid objects
using Cartesian grids, Departament of Heat ans Power Engineering, Lund Institute of
Technology, Sweden
[19] Seakeeping Ship Behaviour in Rough Weather, by ARJM Lloyd, pp 73
[20] Fluent ANSYS guide:
https://www.sharcnet.ca/Software/Fluent6/html/ug/node992.htm
[21] 26thITTCSpecialistCommitteeonCFDinMarineHydrodynamics.Practical
Guidelines for Ship CFD Simulations. Technical report 7.5-03-02-03. Revision 01.
International Towing Tank Conference (ITTC), 2011
[22] André Bakker. Lecture5-Solution Methods, Applied Computational Fluid
Dynamics, Dartmouth University
[23] Starccm+ Theory Guide 11.02.009
[24] Resistance ship analysis presentation made by Cd-Adapco
[25] ITTC-Recommended Procedures and Guidelines. Practical Guidelines for ship
CFD application, 7.5-03-02-03, 2011
[26] K.Roncin, Lecture of Monoeuvrabilité-Simulations-Ensta Bretagne
[27] Volker Berttram. Practical Ship Hydrodynamics,1st ed,2000.
[28] Y. Doutreleau, J.Marc Laurens, L. Jodet. Resistance et Propulsion du Navire. Ensta
Bretagne
[29] Sverre Steen. Making speed-power prediction from model tests. Institut for marin
teknikk. NTNU.
[30] ITTC-Recommended Procedures and Guidelines. Practical Guidelines. Testing and
Extrapolation Methods Resistance Uncertainty Analysis, Example for Resistance Test,
7.5-02-02-02, 2002
87
[31] Jasper Faber, Maarten`t Hoen. Historical trends in ship design efficiency. Delft, CE
Delft, March 2015.
[32] Resolution MEPC.212(63). 2012 Guidelines on the method of calculation of the
attained energy efficiency design index (EEDI) for new ships. Annex8. March 2012.
[33] Lectures of Jean-Marc Laurens: « hydrodynamique Naval – rendement et
propulsion »
[34] Basic Principles of Ship Propulsion. MAN Diesel & Turbo.
[35] M.Chmielewski, M.Gleras. Three-zonal Wall Function for k-휀 Turbulence Models.
Warsow University of Technology, Institute of Heat Engineering. 18/02/2016
- 1 -
Attachments
1) Organization of the project
Theory Review
Statistical Analysis
Elect a ship to be considered as start point to the
generate the mesh and hull geometric modifications
Construct the 3D from 2D plans for this vessel
Construct and validate the mesh that will be used in
the simulations
Construct the 3D geometry of 10 vessels that gives a
general view about all the client fleet
Create a resistance curve for
each of these vessels using
CFD simulations
Start modifying the 3D
geometry from the reference
ship to optimize its resistance
Generate the first propeller design for the
new hull geometry using NavCad
Calculate the Break Power required for each vessel
Compare the vessel efficiency using EIV method
- 2 -
2) Gantt Diagram
Activities Description Start Duration End
1 Hull optimization of fishing boats 04/04/2017 122 04/08/2017
1.1 Theory Review 04/04/2017 6 10/04/2017
1.2 Statistical analysis of all previous fishing
vessels that Vicus worked before 04/04/2017 6 10/04/2017
1.3
Analysis of the characteristics of each
vessel and chose the "base ship" to be
considered as a start point for the hull
geometry
04/04/2017 10 14/04/2017
1.4 Construction of a 3D geometry form the
2D plan of Mar de Maria 10/04/2017 5 15/04/2017
1.5 Start doing the Starccm+ tutorials for
Marine resistance prediction 15/04/2017 7 22/04/2017
1.6 Create a CFD simulationn using the 3D
geometry generated in the previous step. 22/04/2017 10 02/05/2017
1.7 Validate the mesh used in the simulation,
comparing with the towing tank results 02/05/2017 2 04/05/2017
1.8 Chose 10 vessels in order to give a general
idea about all the client fleet geometry. 02/05/2017 2 04/05/2017
1.9 Create the 3D geometry for all the vessels
chosen in the previous step 04/05/2017 21 25/05/2017
1.10 Run the simulation for 3 different
velocities for each vessel 25/05/2017 43 07/07/2017
1.11
Modify the original 3D from Mar de Maria
and run CFD simulations using this
geometry
10/06/2017 22 02/07/2017
- 3 -
1.12 Create and compare the resistance curves 27/05/2017 44 10/07/2017
1.13 Preliminary analysis of the new hulls
resistances 10/07/2017 3 13/07/2017
1.14 Use NavCad to calculate the value of Pb
and create the first propeller design 13/07/2017 3 16/07/2017
1.15
choose a motor for the new hull geometry
and estimate the fuel consumption using
NavCad
16/07/2017 1 17/07/2017
1.16 Create a efficient curve, using the EIV
criteria 17/07/2017 3 20/07/2017
1.17 Compare the results from the modified
geometry with the original geometry 20/07/2017 3 23/07/2017
1.18 Write the report 25/04/2017 101 04/08/2017
- 4 -
- 5 -
3) Propulsion system efficiency
This illustration is very useful to clarify the local where each type of power acts and the
origin of each type of efficiency. This schema was taken from the Jean Marc’s lecture
[33].
Position where each power acts and the origin of each type of efficiency [33]
- 6 -
4) Numerical Scheme
4.1.1 First order upwind scheme
This method assumes that the values of 𝜑 at the face is the same as the cell centered
value in the cell upstream of the face. The main advantages are that it is easy to
implement and that it results in very stable calculation, but also diffusive [22]. This is
often the recommended scheme to start with.
Illustration of the First order upwind scheme
4.1.2 Central differencing scheme
The value of 𝜑 at the face is determined by linear interpolation between the cell
centered values [22]
Illustration of the Central differencing scheme
- 7 -
4.1.3 Second order upwind scheme
The value of 𝜑 at the face is determined from the cell values in two cells upstream of
the face. This is more accurate than the first upwind scheme, but regions with strong
gradients it can result in face values that are outside of the range of cell values. It is the
necessary to apply limiters to the predicted face values and it is one of the most popular
numerical schemes because its combination of accuracy and stability [22].
Illustration of the Second order upwind schem
- 8 -
5) Vessels main characteristics
Vessel
Geometry Characteristics
LOA(m) B(m) T(m) Despl (T) GT
Cargo
Capacity(m^3)
Ana Barral 47,75 9,60 3,40 759,22 665 447,00
Bonito Dos 21,00 6,00 2,68 206,36 126 65,27
Loucenzas 28,45 7,00 3,00 289,90 235 143,00
Mar de Maria 42,15 9,00 3,50 660,20 607 289,16
Novo Airiño 50,00 10,20 4,41 1179,81 742 550,00
O Taba 32,25 7,50 3,00 344,74 300 147,16
Siempre Juan
Luis 27,30 7,00 3,00 294,70 236 71,16
Talasa 39,00 8,10 3,76 694,00 427 325,00
Tronio 55,00 10,20 4,00 1640,88 1058 515,00
Xuxo 29,79 7,50 3,00 312,17 238 210,00
- 9 -
6) Geometry Validation
hydrostatiques values Rhino values Discrepancy
Ves
sel
s
Dis
pla
cem
en
t (t
)
Swtt
e
d (
m²)
Xb
Zb
Dis
pla
cem
en
t (t
) Sw
tte
d(m
²)
Xb
Zb
Dis
pla
cem
en
t (t
) Sw
tte
d (
m²)
Xb
Zb
An
a
Bar
ral
75
9,2
2
46
9,6
2
19
,25
2,0
3
75
8,6
0
47
5,2
1
19
,24
2,0
3
0,0
8%
1,1
9%
0,0
2%
0,0
5%
Bo
nit
o D
os
20
6,3 6
1
72
,3 2
7,6
6
1,6
0
20
5,5 1
1
71
,5 2
7,6
6
1,6
0
0,4
1%
0,4
6%
0,0
3%
0,3
1%
Lou
cen
z
as 2
89
,90
25
2,7
4
10
,74
1,8
0
29
2.3
9
255.5
0
10.9
3
1.8
4
0.8
6%
1.0
9%
1.7
8%
1.9
5%
Mar
de
Mar
ia
66
0,2
0
47
4,3
0
17
,79
2,0
5
63
9,6
6
44
1,3
6
17
,68
2,1
0
3,1
1%
6,9
4%
0,6
2%
2,2
0%
No
vo
Air
iño
11
79
,8 1
61
2,9
4
18
,00
2,6
7
11
77
,2 4
61
4,8
2
18
,57
2,6
7
0,2
2%
0,3
1%
3,1
8%
0,1
9%
O
Tab
a
34
4,7 4
3
01
,2 0
12
,35
1,8
3
34
1,7 6
2
66
,3 6
12
,33
1,8
4
0,8
7 %
11
,57 %
0
,19 %
0
,71 %
Sie
mp
re
Juan
Luis
29
4,7
0
22
8,9
5
9,3
0
3,2
0
29
6.6
7
233.1
4
9.6
6
3.2
5
0.6
7%
1.9
8%
3.8
5%
1.2
8%
Tala
sa
69
4,0
0
40
4,2
5
14
,72
2,2
4
68
6,6
7
41
1,6
4
15
,21
2,2
2
1,0
6%
1,8
3%
3,3
4%
0,8
9%
Tro
nio
16
40
,8 8
78
1,8
8
21
,22
2,0
4 - - - - - - - -
Xu
xo
31
2,7
0
23
5,8
0
11
,42
1,8
1
31
2,8
4
24
2,2
5
11
,41
1,8
1
0,0
4%
2,7
3%
0,0
4%
0,0
0%
- 10 -
7) Main characteristics of Mar de Maria full scale
Mar de Maria
LOA(m) 42,14
Lpp(m) 36
T(m) 3,5
B(m) 9
Dsp max(t) 660
Pot(Kw) 1500
Fuel consumption( g/kw/h) 220
Fishing days per year 310
- 11 -
8) CFD results
Vessel Velocity
(m/s) Fn
Resistance
(N) Sinkage (m) Pitch(deg)
Moment Y
(N.m) Force Z(N)
Ana Barral
1,041 0,186 -9,880 0,0350 -1,1030 0,0055 0,0010
1,487 0,266 -21,420 -0,0120 0,4600 0,0184 -0,0117
1,934 0,346 -68,360 -0,0220 0,7500 -0,0025 0,0260
Bonito Dos
1,346 0,230 -22,988 -0,0269 0,4350 0,0087 -0,0351
1,906 0,326 -63,615 -0,0434 0,5960 -0,0050 0,0700
2,467 0,422 -173,960 -0,0650 0,8600 -0,0039 -0,1000
Loucenzas
1,156 0,207 -19,960 -0,1150 -0,5600 0,2500 0,0800
1,638 0,294 -36,500 -0,1230 -0,4500 -0,2200 0,0880
2,120 0,380 -107,880 -0,1323 -0,4800 -0,0750 -0,2000
Original
Mar de Maria
1,095 0,191 -12,060 -0,0224 -0,7350 0,1810 -0,0045
1,564 0,274 27,780 -0,0308 -0,5490 -0,0170 -0,0240
2,033 0,356 75,120 -0,0444 -0,4480 0,0044 -0,0179
New
Mar de Maria
1,095 0,191 -12,082 -0,0113 -0,4320 0,0181 -0,0050
1,564 0,274 -22,660 -0,0190 -0,1659 -0,0070 0,0070
2,033 0,356 -56,748 -0,0308 0,0805 0,0074 0,0023
Novo Airiño
1,017 0,182 -10,546 -0,0041 -0,7050 -0,0094 0,0364
1,454 0,259 -23,134 -0,0111 -0,5190 0,0077 -0,0029
1,889 0,337 -65,902 -0,0218 -0,4459 0,0136 -0,0280
O Taba
1,086 0,193 -13,080 -0,0221 0,4080 -0,0151 -0,0015
1,538 0,273 -29,388 -0,0308 0,6425 0,0039 -0,0224
1,991 0,354 -80,994 -0,0444 0,7945 -0,0124 -0,0178
Siempre Juan Luis
1,180 0,208 -23,634 -0,0120 -1,5900 0,0129 -0,0720
1,672 0,294 -45,960 -0,0385 -1,3816 -0,0980 0,0001
2,164 0,381 -127,360 -0,0490 -1,3580 -0,0040 -0,0320
Talasa 0,988 0,172 -11,610 -0,0254 1,7744 0,0088 0,0277
- 12 -
1,400 0,244 -24,214 -0,0322 1,9105 -0,0097 -0,0075
1,810 0,315 -78,353 -0,0433 1,8852 0,0649 0,0194
Tronio
0,970 0,167 -9,120 -0,0134 0,6539 0,0217 0,0204
1,386 0,238 20,160 -0,0130 0,6540 0,0210 0,0200
1,802 0,309 -63,630 -0,0281 0,8028 0,0437 0,0204
Xuxo
1,318 0,240 -33,233 -0,1095 0,8287 0,0252 0,1117
1,883 0,342 -83,416 -0,1245 0,8524 -0,0231 -0,0736
2,448 0,444 -364,653 -0,1540 0,7356 0,0423 0,0106
- 13 -
9) Turbulence models in Starccm+
This section was taken directly from the Starccm+ user guide and explains the four RANS
turbulence models that are used in Starccm, and go deeper in the explication of the the
main ones, which are K-Epsilon and K-Omega models.
Deciding on a RANS Turbulence Model
There are four major classes of RANS turbulence models currently in STAR-CCM+. This
section presents broad guidelines as to the applicability of each of these.
Spalart-Allmaras models are a good choice for applications in which the boundary
layers are largely attached and separation is mild if it occurs. Typical examples
would be flow over a wing, fuselage or other aerospace external-flow
applications. The Spalart-Allmaras models for RANS equations are not suited to
flows that are dominated by free-shear layers, flows where complex recirculation
occurs.
K-Epsilon models provide a good compromise between robustness,
computational cost and accuracy. They are generally well suited to industrial-type
applications that contain complex recirculation, with or without heat transfer.
K-Omega models are similar to K-Epsilon models in that two transport equations
are solved, but differ in the choice of the second transported turbulence variable.
The performance differences are likely to be a result of the subtle differences in
the models, rather than a higher degree of complexity in the physics being
captured.
Reynolds stress transport models are the most complex and computationally
expensive models offered in STAR-CCM+. They are recommended for situations
in which the turbulence is strongly anisotropic, such as the swirling flow in a
cyclone separator.
- 14 -
Deciding on a K-Epsilon Model
The Standard K-Epsilon model and the Realizable K-Epsilon model are suitable
for coarse meshes, where the wall-cell y+ values are typically 30 and above. The
Realizable model generally gives results at least as good as the Standard model,
but typically better.
The Standard Two-Layer K-Epsilon model and the Realizable Two-Layer K-
Epsilon model offer the most mesh flexibility. They can be used with the same
meshes as the high-Reynolds number versions. They give good results on fine
meshes (that is, low-Reynolds number type or low-y+ meshes), and also produce
the least inaccuracies for intermediate meshes (that is, 1<y+<30).
The Standard low-Reynolds number model, Abe-Kondoh-Nagano low-Reynolds
number model, and V2F low-Reynolds number model are recommended for truly
low-Reynolds number applications. The standard low-Reynolds number model is
suitable for natural convection flows, for situations when it is desirable to have a
low-Reynolds number version of the Standard K-Epsilon model, or when a Non-
linear Constitutive Model is needed with a low-Reynolds number treatment.
The Elliptic Blending model accurately models near wall anisotropy and is valid
both in the low Reynolds number boundary layer and high Reynolds number bulk
flow. The model is well suited for internal flows, heat transfer modeling, and other
cases where accurate near wall modeling is important. The Lag Elliptic Blending
model improves the sensitivity of the Elliptic Blending model to anisotropy of
turbulence and streamline curvature and rotation. The Lag Elliptic Blending
model is better suited for external aerodynamics simulations.
If there is uncertainty as to which turbulence model to use in a given situation, the
Realizable Two-Layer K-Epsilon model would be a reasonable choice. If the mesh
is coarse, it provides results that are quite close to the version without the two-
- 15 -
layer formulation. If the mesh is fine enough to resolve the viscous sublayer, it
produces results similar to a low-Reynolds number model.
The K-Epsilon models and wall treatments available in STAR-CCM+ are:
K-Epsilon Model Wall Treatment
Standard high-y+
Standard Two-Layer all-y+
Realizable high-y+
Realizable Two-Layer all-y+
Standard Low-Reynolds Number low-y+, all-y+
Elliptic Blending low-y+, all-y+
Lag Elliptic Blending low-y+, all-y+
Abe-Kondoh-Nagano Low-Reynolds Number low-y+, all-y+
V2F Low-Reynolds Number low-y+, all-y+
- 16 -
K-Omega Turbulence
A K-Omega turbulence model is a two-equation model that solves transport equations for
the turbulent kinetic energy k and the specific dissipation rate w, that is, the dissipation
rate per unit turbulent kinetic energy (w ~ 휀/𝑘), to determine the turbulent viscosity.
One reported advantage of the K-Omega model over the K-Epsilon model is its improved
performance for boundary layers under adverse pressure gradients. Perhaps the most
significant advantage, however, is that it may be applied throughout the boundary layer,
including the viscous-dominated region, without further modification. Furthermore, the
standard K-Omega model can be used in this mode without requiring the computation of
wall distance.
The biggest disadvantage of the K-Omega model, in its original form, is that boundary
layer computations are sensitive to the values of w in the free stream. This translates into
extreme sensitivity to inlet boundary conditions for internal flows, a problem that does
not exist for the K-Epsilon models. The versions of the model included in STAR-CCM+
have been modified in an attempt to address this shortcoming.
There are three versions of the K-Omega model in STAR-CCM+:
Standard K-Omega Model
SST K-Omega Model
SST K-Omega Detached Eddy Model
Standard K-Omega Model
Wilcox revised his original model in 1998, and then in 2006, to account for several
perceived deficiencies in the original version (1988).
However, the validation results published in Wilcox’s book are typically for two-
dimensional, primarily parabolic, flows. Until further validations for complex flows are
widely published, the corrections should be used with caution. Therefore, each correction
has been included as an option in STAR-CCM+.
- 17 -
SST K-Omega Model
The problem of sensitivity to free-stream/inlet conditions was addressed by Menter, who
recognized that the 휀 transport equation from the standard K-Epsilon model could be
transformed into an w transport equation by variable substitution.
The transformed equation looks similar to the one in the standard K-Omega model, but
adds an additional non-conservative cross-diffusion term containing the dot product
∆𝑘 . ∆𝑤. Inclusion of this term in the w transport equation potentially makes the K-Omega
model give identical results to the K-Epsilon model. Menter suggested using a blending
function (which includes functions of wall distance) that would include the cross-
diffusion term far from walls, but not near the wall. This approach effectively blends a
K-Epsilon model in the far-field with a K-Omega model near the wall. Purists may object
that the blending function crossover location is arbitrary, and could obscure some critical
feature of the turbulence. Nevertheless, the fact remains that this approach cures the
biggest drawback to applying the K-Omega model to practical flow simulations.
In addition, Menter also introduced a modification to the linear constitutive equation and
dubbed the model containing this modification the SST (shear-stress transport) K-Omega
model. The SST model has seen fairly wide application in the aerospace industry, where
viscous flows are typically resolved and turbulence models are applied throughout the
boundary layer.
In complex flow—for example in strong swirl, streamline curvature, shear layer, or
boundary layer flow—turbulence is anisotropic. The anisotropy of the Reynolds stresses
not only affects the flow field but also the turbulent transport of scalars (temperature,
concentration, passive scalar). As eddy viscosity models tend to strongly under predict
the anisotropy of the Reynolds stresses, Spalart suggested a redefinition of the Reynolds-
stress tensor to add quadratic relations. Differential Reynolds stress models perform much
better than eddy viscosity models but often are unstable when used in complex flows.
Explicit algebraic Reynolds stress models (EARSM) are a good compromise. These
models are derived from Reynolds stress models and extend the classical Boussinesq
- 18 -
approximation by adding non-linear functions of the strain and vorticity tensors. A cubic
model was proposed by Wallin and Johansson, that was further improved by Hellsten.
SST (Menter) K-Omega Detached Eddy
The SST (Menter) K-Omega Detached Eddy model allows you to combines features of
the SST (Menter) K-Omega RANS model in the boundary layers with a large eddy
simulation (LES) in unsteady separated regions.
The delay factor introduced by Menter and Kuntz has been adopted for the DES Version
of the SST K-Omega Model formulation of the SST K-Omega model. This modification
enhances the ability of the model to distinguish between LES and RANS regions on
computational meshes where spatial refinement could give rise to ambiguous behavior.
In addition, the improved delayed detached eddy simulation (IDDES) formulation is
available as an option. This combines DDES with an improved RANS-LES hybrid model
aimed at wall modeling in LES when the grid resolution supports it.
- 19 -
10) Overset mesh construction
Illustration of the overset region, showing the parameters used to calculate the dimensions of this region.
Due to the fact that the “mesh alignment” is placed on the free surface, the mesh start
growing on free surface and grows in two directions: above and below the free surface.
Because of that, the calculus to make a properly overset mesh region will be also split in
two parts: one that calculate the upper limit of the box and other to calculate the lower
limit of it.
a) Upper limit (𝑯𝟏)
The minimum height value of the upper overset box surface must respect the
follow relation:
𝐻1 = 2ℎ1 ≥ 𝑇 + 𝑛 ∗ 𝑏
Lower limit (𝑯𝟐)
The minimum height value of the upper overset box surface must respect the
follow relation:
𝐻2 = 2ℎ2 ≥ 𝑛 ∗ 𝑏 − 𝑇
With,
𝐻1: the minimum distance from the free surface until the upper limit of the overset
mesh.
𝐻2: the minimum distance from the free surface until the lower limit of the overset
mesh.
T: Draft
- 20 -
b: cell base size on the overset region
n: minimum integer number that satisfy the relation
WL: water line
Care must be taken because T is not coincident with 𝐻2. T is measured since the center
of the keel line until the water line and 𝐻2 is measured since the lowest point of the vessel
until the water line.
The coefficient 2 before the ℎ1 and ℎ2 is required just to give a minimum distance
between the vessel hull and the overset borders.
- 21 -
11) Propeller design
Illustration of the factors that influence in the dimensioning of the maximum diameter of the propeller
12) Effective power
Comparison between effective power required for each vessel
0
100
200
300
400
500
600
700
800
900
1000
1100
0,10 0,20 0,30 0,40 0,50
Pef
f
Fn
New Mar deMaria
Novo Airiño
Siempre JuanLuis
Bonito Dos
Loucenzas
O Taba
Talasa
Xuxo
Original Mar deMaria
Tronio
Ana Barral
- 22 -
13) Propeller data
Propeller Data
D 3 m
Ae/Ao 0,463 -
P/D 0,862 -
P 2,583 m
Hub immersion 2,884 m
N 4 -
Rpm 205,41 rpm
Propeller design taken from Rhinoceros Propeller data taken from NavCad
- 23 -
14) NavCad and EIV results
Vessel Ana
Barral
Bonito
Dos Loucenzas
Original Mar
de Maria
New Mar de
Maria
Novo
Airiño O Taba
Siempre
Juan Luis Talasa Tronio Xuxo
Geometry
Characteristics
LOA(m) 47,75 21,00 28,45 42,15 42,15 50,00 32,25 27,30 39,00 55,00 29,79
B(m) 9,60 6,00 7,00 9,00 9,00 10,20 7,50 7,00 8,10 10,20 7,50
T(m) 3,40 2,68 3,00 3,50 3,50 4,41 3,00 3,00 3,76 4,00 3,00
Despl (T) 759,22 206,36 289,90 660,20 660,20 1179,81 344,74 294,70 694,00 1640,88 312,17
GT 665 126 235 607 607 742 300 236 427 1058 238
Cargo
Capacity(m^3) 447,00 65,27 143,00 289,16 289,16 550,00 147,16 71,16 325,00 515,00 210,00
Model Scale 11,93 5,25 7,11 10,80 10,80 12,50 8,06 6,83 9,75 13,75 7,45
Curve of
Resistance
V1 (knot) 7,00 6,00 6,00 7,00 7,00 7,00 6,00 6,00 6,00 7,00 7,00
V2 (knot) 10,00 8,50 8,50 10,00 10,00 10,00 8,50 8,50 8,50 10,00 10,00
V3 (knot) 13,00 11,00 11,00 13,00 13,00 13,00 11,00 11,00 11,00 13,00 13,00
Peff1(Kw) 51,96 21,58 14,07 51,91 47,60 62,05 20,18 22,78 12,80 23,63 49,28
Peff2(Kw) 167,83 84,86 55,59 176,32 130,43 204,28 65,74 69,68 40,76 156,06 179,78
Peff3(Kw) 751,37 298,51 199,42 636,91 453,45 818,75 241,4900 255,44 193,93 471,42 1032,86
Froude
Fn1 0,1662 0,2149 0,1846 0,1769 0,1769 0,1625 0,1734 0,1885 0,1577 0,1549 0,2105
Fn2 0,2375 0,3044 0,2615 0,2528 0,2528 0,2321 0,2456 0,2670 0,2234 0,2213 0,3007
Fn3 0,3087 0,3939 0,3384 0,3286 0,3286 0,3017 0,3179 0,3455 0,2891 0,2877 0,3909
EIV
EIV1 8,001 26,548 7,901 12,356 11,331 7,765 11,012 25,706 3,163 3,158 16,152
EIV2 18,090 73,692 22,035 29,379 21,733 17,895 25,322 55,505 7,109 14,600 41,248
EIV3 62,299 200,310 61,082 81,635 58,120 55,172 71,877 157,230 26,136 33,926 182,287
Engine
RPM1 559 758 794 668 713 601 446 684 525 455 780
RPM2 826 1200 1220 1000 1000 884 657 990 760 782 1200
RPM3 1390 1860 1826 1535 1507 1375 1000 1518 1200 1113 2236
PB1 90,2 55,6 33,9 111,5 101,2 159,8 40,1 48,9 31,9 43,9 112,7
PB2 299,1 223,4 126,4 378,7 267,3 525,5 129 151,6 94,2 317,2 433,8
PB3 1642,6 888,8 475,3 1513 1001,9 2279,5 525 630,3 449,1 1020,7 3360
Overall
Propulsion
Efficiency
EFFOA1 68,33% 50,60% 57,41% 57,70% 58,66% 50,54% 59,67% 55,34% 61,38% 57,99% 51,90%
EFFOA2 66,57% 46,61% 55,55% 56,14% 59,29% 49,77% 58,62% 54,55% 60,95% 56,25% 49,19%
EFFOA3 54,28% 60,52% 50,83% 50,11% 54,05% 44,36% 53,18% 48,09% 54,77% 53,43% 36,47%
Propeller
Cavitation1 2,00% 2,00% 2,00% 2,00% 2,60% 2,00% 2,00% 2,00% 2,00% 2,00% 2,00%
Cavitation2 2,10% 4,50% 2,00% 4,90% 5,50% 2,00% 2,00% 2,00% 2,00% 2,00% 4,40%
Cavitation3 17,60% 29,20% 2,40% 26,70% 25,00% 7,60% 4,70% 7,10% 4,60% 2,00% 89,00%
