Bilge keels

"''"Verification of CFD calculations with experiments on a rolling cylinder

with bilge keel in a free surface"

W'tmiim:«.

Master Thesis

By

Maris Paap

Delft University of Technology &

Bluewater Energy Services b.v.
 Bilge keels
^^\l erification of CFD calculations with experiments on a rolling
cylinder with bilge keel in a free surface"

Date: 15-09-2005

Author: M.W.Paap
Address: Molenstraat 5
261 U Z Delft
The Netherlands
Tel: +31 614 395 097
E-mail: marispaap@hotmail.com

Company: Bluewater Energy Services B.V.

Department: Technology & Development
Address: Marsstraat 33
2132 HR Hoofddorp
The Netherlands
Tel: +31 23 568 2800
Web: www.bluewater.com

University: Delft University of Technology

Department: Mechanical, Maritime and Materials Engineering
Specialization: Ship Hydromechanics
Address: Mekelweg 2
2628 CD Delft
The Netherlands
Tel: +31 15 278 6873
Web: www.ocp.tudelft.nl
 Preface

Preface

The work presented in this report is executed as a graduation project for the specialization Ship
Hydromechanics. This specialization is a master variant of the faculty of Mechanical, Maritime and Materials
Engineering, University of Technology Delft. The project is initiated by and performed in co-operation with
Bluewater Energy Services b.v. Hoofddorp as a part of an extensive research program to reduce the roll
motion of moored FPSO vessels. The project is supervised by Ir J.J. van der Cammen (Bluewater) and Prof,
dr. ir. J.A. Pinkster (TU-Delft).

A word of thanks for the people of Bluewater for giving the opportunity to perform this project. Special
thanks to Mathieu, Jeroen, Robert and Christien for their time, support and valuable advises.

Delft, 15-09-2005
Maris Paap

I
Verification of CFD calculations witli expenments on a rolling circular cylinder with bilge keels in a free surface.
University of Technoiogy Delft & Biuewater Energy Services b.v.

Table of contents

PREFACE I

TABLE OF CONTENTS 11

ABSTRACT VI

1 INTRODUCTION 8

2 RESEARCH APPROACH 10

2.1 C O M P O N E N T S O F ROLL DAMPING 10

2.2 CONTRIBUTION OF THE BILGE KEEL TO THE TOTAL ROLL DAMPING 11

2.3 DIMENSIONLESS COEFFICIENTS 12

2.4 GLOBAL SETUP INVESTIGATED MODEL 14

2.5 SELECTED CFD PACKAGE: FLUENT 15

3 THEORY: NUMERICAL TURBULENCE MODELING 16

3.1 T H E PHENOMENA OF TURBULENCE 16

3.2 GOVERNING EQUATION: N A V I E R - S T O K E S 17

3.3 D I R E C T N U M E R I C A L SIMULATION (DNS) 18

3.4 L A R G E E D D Y SIMULATION (LES) 18

3.5 R E Y N O L D S - A V E R A G E D N A V I E R S T O K E S EQUATIONS (RANS) 19

3.5.1 K-e model 20

3.5.2 Reynolds Stress Model (RSM) 21

4 SOLUTION ALGORITHM OF FLUENT 23

4.1 DISCRETIZATION SCHEME: QUICK 24

4.2 F R E E SURFACE MODELING: V O F 24

4.3 ITERATIVE SOLVING THE PRESSURE-VELOCITY COUPLING: PISO 25

4.3.1 Under Relaxation factor 26

4.3.2 Pressure interpolation scheme: PRESTO!. 27
4.4 SOLVING DISCRETIZED EQUATIONS: MULTI-GRID METHOD 27

4.4.1 Convergence Criteria 28

4.5 U N S T E A D Y CALCULATIONS 29

4.5.1 Time step 29

4.5.2 Total flow time 30

5 COMPUTATIONAL MODEL 31

5.1 COMPUTATIONAL GRID 31

5.1.1 Boundary layer near the body 33

5.1.2 Keel zone with unstructured grid 34
5.1.3 Grid refinement for free surface modeling 34
5.1.4 Far field cells for wave damping 35
5.2 BOUNDARY CONDITIONS 36

5.3 INITIAL CONDITIONS 37

5.4 B O D Y AND ITS MOTION 37

5.5 OUTPUT 40

5.5.1 Forces and moments on the hull. 40

5.5.2 Registration wave height. 40
 Table of Contents

5.5.3 Visual location of large eddies. 40

5.6 REPETITIVE COMPUTATIONS 40

5.7 O V E R V I E W COMPUTATIONAL .SETTINGS 41

6 FLUENT CALCULATIONS: INFLUENCE COMPUTATIONAL SETTINGS 42

6.1 G E N E R A L SETUP 43

6.1.1 Computational domain 43

6.1.2 Input parameters 44
6.1.3 Hull motion 44
6.1.4 Output 45
6.2 INVESTIGATED SETTINGS 45

6.2.1 Reference run, coarse grid 46

6.2.2 Reference run, fine grid 46
6.2.3 Altered turbulence model: RSM. 47
6.2.4 Increased rotational inner grid. 47
6.2.5 Opposite initial start angle 48
6.2.6 Including free surface 49
6.3 POSTPROCESSING 50

6.4 RESULTS 51

7 FLUENT CALCULATIONS: INVESTIGATION KEEL SHAPES 53

7.1 SETUP 53

7.2 INVESTIGATED KEEL SHAPES 53

7.3 POSTPROCESSING 55

7.4 RESULTS 56

7.4.1 Equivalent linear damping 56

7.4.2 Velocity field. 57
7.4.3 Pressure field. 58
7.4.4 Turbulence field. 59
7.3 CONCLUSIONS AND RECOMMENDATIONS 59

8 EXPERIMENTS 60

8.1 SETUP 60

8.1.1 Sign conventions 60

8.1.2 Layout 61
8.1.3 Tested keels 62
8.1.4 Test program 63
8.1.5 Measured parameters 64
8.1.6 Determination of mass properties experimental model 66
8.2 POSTPROCESSING 67

8.2.1 Governing equations 67

8.2.2 Hydrodynamic coefficients 70
8.2.3 Keel forces 72
8.2.4 Energy balance 73
8.3 RESULTS 73

8.3.1 Sensitivity mass properties experimental model 75

8.3.2 Frequency content measured signals. 77
8.3.3 Repeatability of the experiments 78
8.3.4 Time domain 79
8.3.5 Hydrodynamic coefficients 81
8.3.6 Drag coefficients keel 84
Verification ofCFD calculations with experiments on a rolling circular cylinder with bilge keels In a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

8.3.7 Viscous fraction of damping 85

8.3.8 Visual registration 86

9 COMPARISON FLUENT <-> EXPERIMENTS 87

9.1 T I M E DOMAIN 89

9.2 F O R C E S ON THE KEEL 91

9.2.1 Force amplitude. 91

9.2.2 Phase shift 92
9.3 F O R C E S ON THE MODEL 93

9.4 R A D I A T E D WAVES 95

9.5 FREQUENCY DOMAIN 96

9.6 LOCATION LARGE EDDY 98

10 CONCLUSIONS AND RECOMMENDATIONS 102

10.1 CONCLUSIONS 102

10.2 RECOMMENDATIONS 103

LIST OF SYMBOLS AND ABBREVIATIONS 104

INDEX OF FIGURES 106

INDEX OF TABLES 109

REFERENCES 110

APPENDICES Ill

A. EXAMPLE USER DEFINED FUNCTION (UDF) 111

B. EXAMPLE STANDARD JOURNAL FOR REPETITIVE COMPUTATIONS 112

C. RESULTS KEEL SHAPE SELECTION 114

D. DETAILED LAY OUT EXPERIMENTAL SETUP 117

E. RESULTS STATIC INCLINATION TESTS 122

F. PROPERTIES MODEL 124

G. HYDRODYNAMIC COEFFICIENTS 125

H. RESULTS COMPARISON CFD-EXPERIMENTS 128

I. VISUAL COMPARISON 134

iv
Table of Contents

V
Verification ofCFD calculations with experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.vt

Abstract
Recent investigations reveal that the motions of moored Floating Production Storage and Offloading (FPSO)
vessels are in some weather conditions considerably larger than preferable. Reducing the motions is
demanded for the oil-handling processes on board and safety of the crew. Especially the roll motion leads to
problems. Due to the cost benefit, the oil-industry searches for entirely passive devices without the demand
of operational effort or maintenance to mitigate this roll motion.

One option is to apply non-conventional bilge keel configurations like multiple keels, extraordinary large
keels or "exotic" keel shapes. Traditional calculation methods for roll damping can not incorporate such
configurations. This leads to the first goal of this thesis; investigate if CFD can be used for calculations on
these kinds of non-conventional keel configurations. A basic verification of a CFD package is performed with
several different bilge keel shapes on as many flow-aspects as possible. This will give a reliable indication of
the value of CFD in future bilge keel design.

To reach this first goal, experiments are performed with a basic model to gather verification data and
investigate the flow phenomena. A CFD package is chosen and its details investigated to achieve a model
that describes the real flow around the keel as accurately as possible. In the end of this thesis the results of
the experiments are compared with the CFD calculations.

At the University of Technology Delft, the CFD package Fluent is available that, based on the specifications,
should be able to perform 2D computations on a rolling cross section of an FPSO. It can include any kind of
prescribed roll motion, radiation of waves, far field wave damping, grid refinement around the keel,
turbulence modeling and it can record the forces and moments on the body.

When the default settings of Fluent are used the solution diverges and no realistic results are found. Altered
descretization schemes (Quick), pressure interpolation scheme (PRESTO!) and solution algorithm (PISO) are
demanded. To model the free surface, the Volume Of Fluid (VOF) method is used, which has to include
"Implicit Body Force Treatment" for a stable solution. To model the turbulence around the keel, the standard
two equation k-£ model is used with additional Renormalization Group (RNG) theory for better performance.

The reliability of the executed experiments is investigated by checking the repeatability, frequency content
and the influence of the mass of the experimental model on the results. Repeating test runs under the same
conditions gave the same results. The frequency content of the measured signals revealed that most signals
contain one dominating first harmonic, which makes the use of this harmonic in the comparison sufficient.
Considerable deviations of the calculated mass properties have an average influence on resulting force
amplitudes and phase shifts of 2-3%.

To make a thorough investigation of the capabilities of Fluent on these kinds of computations with non-
conventional keels, the following aspects are included in the comparison:
• Amplitude trends of the forces and moment on the whole model and the separate forces on the
keel.
• Phase shift trends between the roll motion and those measured forces and moments
• The height of the radiated waves
• The frequency content to investigate if Fluent predicts the higher harmonics as well
• Time domain comparison shows if the detailed force behavior during oscillation is similar
• Location of the prime vortex at the keel tip is visual investigated

VI
 Abstract

The conclusion of this thesis is that the agreement found between the CFD calculations and experiments is
good; CFD can be used in the future to perform computations on cross sections of an FPSO in a pure roll
motion. As many as possible 2D phenomena present at a real cross section of an FPSO section were
included in the comparison and the resemblance gives confidence that reliable results will be achieved when
future keel optimizations are performed with this Fluent CFD model.

To reach the main goal of this project, the verification of CFD, several bilge keel shapes are investigated.
Studying the effects of those altered shapes gives insight in why a certain keel shape performs better than
another. The will lead to basic advises for further bilge keel optimization, which is a secondary goal of this
thesis. Within the boundaries of the simplified model used in this thesis the following conclusion can be
drawn: a normal thin flat plate performs best. This keel generates more damping than the examined
different shaped keels due to stronger and more profound vortex development and as a result generates
more turbulence.

As a result of the research performed in this thesis it is recommended to do future optimization of bilge keel
configurations with the CFD model described in this thesis. A cross section of a real FPSO has to be used to
achieve a more realistic insight in the effects of the altered keel configurations. The results of these
calculations are expected to be reliable, both qualitatively and quantitatively. It is expected that
improvement of damping will not be found in altered keel shapes; a thin flat plate seems to perform best. It
is recommended to investigate the gain in damping performance with multiple keels and/or extraordinarily
large keels.

VII
Verification of CFD calculations with experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

1 Introduction

The concept of a Floating, Production, Storage and Offloading (FPSO) vessel in the chain of oil production is
used all over the world, especially at locations of great water depth or at small oil fields. These vessels
combine several tasks; production of crude oil, first treatment of this oil, temporary storage in her cargo
hold and finally offloading to a shuttle tanker. For all these tasks, an FPSO has a vast amount of deck
equipment with processes running all the time. Safe operation of these systems and the crew demands a
low level of motions of the vessel.

Despite all kind of arrangements like bilge keels, weathervane capacity etc. recent investigations reveal that
the motions of FPSOs are still considerably larger than preferable. Especially the roll motion leads to
problems due to the large rotation angles of this motion. Rolling of an FPSO occurs at certain combinations
of wave, wind and current.

Several options are possible to mitigate the roll motion of moored vessels at sea. Systems like active use of
thrusters, anti-roll tanks etc. all demand continuous operational effort and/or maintenance. The prime
purpose of FPSO's is to produce as much oil as possible with as little resources as possible. Therefore it is
still of interest to find a device that damps the roll motion entirely passively. A bilge keel is in compliance
with this demand.

The shape of those bilge keels is still the same as at the time they were introduced; a flat plate
perpendicular to the bilge plating. These keels are kept relatively small and straight forward to minimize the
drag of the ship at forward speed. FPSO's are moored at the same location for a long time, which makes
resistance at forward speed not a design issue. This makes room for the application of non-conventional
bilge keel, which could mitigate the roll damping better than the commonly used flat plate. Non conventional
keels include extremely large keels, multiple keels and "exotically" shaped keels.

To investigate the effects of altered bilge keel configurations, traditional calculation methods are not
suitable. Empirical methods are based on normal keels and do not include small changes in the keel shapes.
Diffraction calculations do not incorporate the viscosity of the water, which makes these computations not
suitable; the local flow around a bilge keel in strongly influenced by viscosity.

The use of a Computational Fluid Dynamics (CFD) package is a way to avoid the shortcomings of traditional
calculation methods. CFD includes all characteristics of the water itself and can handle any kind of bilge keel
shape. The main disadvantage of CFD packages at this moment is that the results of these computations can
not all be taken for granted. Many flow-types are investigated and verified but time-domain calculations of
flows around oscillating keels still demand verification with measured data.

This verification of the performance of a modern CFD package to compute the flow around oscillating keels
is the goal ofTihis thesis. The performance of CFD will be investigated both qualitatively and quantitatively at
different aspects of the flow behaviour. This leads to the central research question of this thesis: "To what
extent is it possible to establish similarity between experiments with an oscillating plate and numerical
calculations with CFD?" The verification of the CFD package will be performed with several different keel
shapes; this gives the opportunity to study the consequences of those shapes on the flow behavior. This will
result in some basic design advises on performance of certain keels, these advises are a secondary goal of
this thesis.

8
 Introduction

To find an answer to this central question this thesis will execute experiments to gather measurement data
and to study the basic flow phenomena. This measured data and a visual registration of the flow
phenomena is compared with the results of CFD computations. To optimize these computations, the
phenomena of turbulence and the background of CFD packages Is investigated in this thesis as well. It is
aimed to establish such a resemblance between experiments and the results of the CFD calculations that
CFD can be used in the future for bilge keel optimization without the demand of new experiments.

To be sure that these future CFD calculations give reliable results, many phenomena of the flow around a
real FPSO are included in this verification. This means that the influence of wave propagation, the motion of
the vessel itself etc. have to be Included. This thesis will use a simplified representation of a bilge keel under
an FPSO; only the bilge and the keel will be modeled. The effects of the detail flow behavior of different keel
shapes will be studied which will lead to basic recommendations for an optimal performing bilge keel shape.

The following structure can be found in this report. It starts in chapter 2 with describing the background of
roll damping and the basic setup of this thesis' model. Chapter 3 describes the theory of turbulence
modelling used in many modern CFD packages. Chapter 4 describes the solution algorithm the selected CFD
package (Fluent) uses to solve turbulent flows. This solution algorithm demands a computational model as
input, which is described in chapter 5. The effects of several settings of the computational model on the
results of the calculations are uncertain. Chapter 6 investigates the effects of five of those settings. Chapter
7 compares several keel shapes with each other in order to select the shapes that will be tested during the
experiments; this comparison is entirely based on CFD calculations. This chapter also investigates the effect
of the altered shapes on the flow behaviour to get insight why certain keel shapes perform better than
others. The executed experiments are described in chapter 8, where also the results of those experiments
can be found. The actual comparison between the experiments and CFD can be found in chapter 9. The
overall conclusions of this thesis are described in chapter 10 where also recommendations for further
research are given.

9
Verification of CFD calculations with experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

2 Research approach

This chapter describes the background of roll damping and the contribution of a bilge keel to the total
damping in order to lay the foundations for the chosen basic model setup of this thesis. It starts with
describing the components of roll damping in the first paragraph, the second paragraph will show that a
bilge keel significantly influences this damping process. Paragraph 2.3 investigates the dimensionless
coefficients that describe the flow around a full-size bilge keel to indicate what coefficients are of importance
when experiments are executed at model scale. These coefficients lead to the basic experimental setup
described in paragraph 2.4. The last paragraph gives the consequences of this setup for the demands on the
CFD package and describes why the package Fluent is chosen.

2.1 Components of roll damping

To be able to investigate the influence of the bilge keel on the total damping, first the components of roll
damping will be discussed. It based on the work of Ikeda, Tanaka, and Himeno ([6], [7], [8], [9], [1]).
Considering a vessel in pure roll motion, the following components of roll damping (B44) can be found:

(2-1) B^=B,+ B^ + 5„ ,„„ + B^,, + 5,,

Where the damping components are defined as:

• Bf = Friction at the hull
• B„ = Radiation of waves
• Be,huii - Eddy damping due to the bare hull
• Be,bk = Increased eddy damping due to the presence of a bilge keel
• Bbk = Drag of the bilge keel

The friction damping is caused by the tangential stresses at the hull and is relatively small compared with
the other components. It can be predicted with Kato's formula for ships at zero advance speed ([6]).

Radiation of waves is caused by the rectangular shape of the vessels' cross section. This rectangle pushes
water upwards causing a wave crest and a wave through when rolling in the opposite direction. This wave
propagates away from the vessel. This is a potential phenomenon for low to moderate roll angles and can
thus be described with potential theory. The wave damping increases linear with the roll amplitude.

Eddy damping of the bare hull is caused by flow separation at the corners of the cross section. At these
relatively sharp corners the flow separates from the hull causing the development of a vortex. This vortex
gives a low pressure just behind the corners of the cross section. The achieved pressure distribution along
the hull produces a hydrodynamic moment that works in the opposite direction of the roll motion which
damps the system. The process of vortex development is by the viscosity of the water and depends on the
sharpness of the corners. In case of round bilges this phenomenon occurs as well, but the location of the
vortices is then uncertain.

The process of the increased eddy damping due to the presents of a bilge keel is similar to eddy damping
caused by the bare hull. The keel gives an additional sharp corner when the flow passes the bilge, which
gives an additional development of a vortex and increases the effect of the eddy damping.

10
 Research approach

The drag of the keel occurs when the flow passes the keel itself; viscous effects lead to an increased
pressure on the front side and a decreased pressure on the backside of the keel. This pressure distribution
gives a normal force on the keel in opposite direction of the roll motion, which causes damping. Due to the
dynamic behavior of roll motion it is difficult to predict these damping values; the numerous investigations to
stationary flows over fences are not applicable.

2.2 Contribution of tlie bilge l(eei to the total roll damping

To be able to asses the influence of a bilge keel on the total damping, the magnitude of the components
given in the previous paragraph will be investigated. The numbers presented in this paragraph will be based
on experiments performed by MARIN in 2004 ([1]). During these experiments various roll tests were
executed with a model (scale 1:40) of Bluewater's Glas Dowr. This vessel is a typical FPSO and will give a
general impression of the relative contribution of the components to the total roll damping.

The model was equipped with separate force receptors that measured the force on the bilge keels. This
yields directly to an estimation of the damping due to bilge keel drag (Bbk). The radiated waves were
calculated with the potential theory based program ITH-SHIPMO. The tests were performed with and
without bilge keels, which gives the opportunity to compute the eddy damping due to the bare hull and the
increased eddy damping due to the presence of the keel. A global impression of the results can be found in
figure 2-1; the values in this graph are based on forced oscillation tests in calm water.

2.5E+06 _
□ Increased eddy by keel
□ Drag force on keel

O.OE+00 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
5 7 9 11 13 15

Roll amplitude [deg]

Figure 2-1 Components of damping as a function of roll amplitude

On the vertical axis the Linear Equivalent Damping (Bei) is given, explanation about the calculation of Bei can
be found in paragraph 6.3. From the graph it can be concluded that the additional damping due to the
presence of the keel (drag and increased eddy) is around 73% of the total damping. The height of the keel
(0.7 m) is very small in comparison to the breadth of the ship (42 m), but apparently the influence of the
keel is significant.

Based on the information presented in this paragraph it appears to be legitimate to investigate if a certain
bilge keel shape can improve the damping capacity of a vessel. When a better performing keel is found, it
can be expected that it has considerable effect on the total damping of the vessel.

11
Verification ofCFD calculations with experiments on a rolling circular cylinder with b ilge keels in a hee surface.
University of Technology Delft & Bluewater Energy Services b.v.

2.3 Dimensionless coefficients

To start with the detailed investigations of flow behavior around a keel, the dimensionless parameters that
describe this flow are discussed in this paragraph. The force on a keel under a vessel in a pure roll motion is
determined by four parameters: roll period (T), roll amplitude (cpa), height of the keel (Hbk) and the distance
from the centre of roll to the keel (RQ). TO establish the dimensionless parameters, it is convenient to
incorporate a velocity. Here it is chosen to use the maximum velocity of the bilge keel tip during oscillation:
(Vo). With the motion approximately a sine function, the roll amplitude is not necessary anymore; RQ, T , VQ
and a sine function fix the roll amplitude. The dimensionless force on the keel (Cd) can now be described as:

(2-2) C,=f(T,V„H,,,R,)

With the aid of the kinematic viscosity (v) and these parameters the following dimensionless expression can
be derived:

K • f^hk ^hk ^hk

(2-3) Q-/

In this expression the first term is referred to as the Reynolds number, the second is the Strouhal number or
inverse Keulegan-Carpenter number and the third term is a shape factor. Leading to the following
dependence of the dimensionless keel force:

(2-4) C,=/(/?n,5r,/„„^J

To get an indication of these values on a rolling, full scale FPSO, table 2-1 is given. Here the sizes are taken
of B luewater's Glas Dowr and a roll period of 15 seconds, approximately the vessel's natural frequency.

Roll amplitude Strouhal Rn Fshape

[deg] [-] [-] [■]

5 0.050 6.63E+05 0.027

10 0.025 1.33E+06 0.027

15 0.017 1.99E+06 0.027

Table 2 - 1 Dimensionless coefficients based on full scale data

When the model tests of this thesis are performed with the same values of these dimensionless coefficients,
the flow around the keel is considered to be identical and the dimensionless forces on the keel will be the
same. For the Reynolds number is not demanded to reach these high values, it is commonly agreed that
above 10.000 [-] the influence of Reynolds becomes negligible [17].

12
 Research approach

A Stouhal number below 0.1 has influence on the flow behavior and it is advised to reach the same values
with the experiments. Rewriting the Strouhal number as function of the radius (RQ), roll amplitude and
height bilge keel, gives:

H hk
(2-5) St=-
^■K(Pa

This shows that the Strouhal number does not depend on the frequency of the roll motion. It is therefore
recommended perform a test sequence based on an amplitude range rather than frequency range [Ikeda: 7,
8].

The shape factor will be hard to reach at model scale; for example, scale 1:40 would mean that the model is
still reasonable large (breadth = 1 [m]) but the keel is too small to be investigated in detail (Hbk = 1.75
[cm]). In this thesis only a local shape factor, Rbnge/ Hbk is maintained instead of a global factor. Here Rbnge is
the radius of the bilge. This local ratio is prescribed by the demand that the keel should not stick out of the
outer rectangle of the cross section, see figure 2-2. It can be found that this ratio is always around 2.4.
Leaving the full scale shape factor values of table 2-1 will undoubtedly result in different global flow behavior
at model scale but the detailed behavior around the keel is expected to be similar. With this simplification it
is not possible to draw conclusions for bilge keels under full-scale FPSO's but it will give insight in the
detailed flow behavior.

Free surface

Outer rectangle

Figure 2-2 Typical cross section of an FPSO

13
Verification of CFD calcuiations witft experiments on a roliing circular cylinder with bilge keels in a free surface.
University of Tecinnology Delft & Bluewater Energy Services b.v.

2.4 Global setup Investigated model

The experiments in this thesis are performed to gather data for verification of CFD computations. They will
be zoomed on the bilge keel itself and its direct surroundings. This means that only the keel and the bilge
will be modeled, the influence of the straight parts of the cross section will be left outside this thesis. This
lead to the basic setup given in figure 2-3, both CFD calculations and experiments will have the same
appearance.

The flow around the keels of a moored FPSO in roll motion can be regarded as 2 dimensional, see previous
paragraph. It is chosen to use a 2D mode setup in this thesis. The cross section of the model will be taken
circular to diminish the influence of the radiated waves; the circle will be up to its central axis in the water;
see figure 2-3. The model is oscillated around this central axis; its motion will be a pure sine function.

Figure 2-3 Basic 2D setup model

Although the cross section of the model differs considerably from the cross section of an FPSO, the values of
the dimensionless parameters given in the previous paragraph have to be matched as closely as possible.
This means the Reynolds number has to be above 10.000 [-], the Strouhal number under 0.1 [-] and the
shape factor will be 2.4 [-].

14
 Research approach

2.5 Selected CFD package: Fluent

This thesis has to asses the capabilities of commercial CFD software on 2D computations with a rolling cross
section of an FPSO, extensive information can be found in [15] and [16]. Although the cross section will be
simplified, target is to take into account as many 2D phenomena that occur at a real FPSO as possible. To
reach this goal, the selected CFD package has to meet at least the following requirements:

• Friction at the hull

• Turbulent flow around the keel
• Radiation of waves
• Any prescribed motion of the body
• Grid refinement around the keel tip
• Recording forces and moments

The CFD packages that are currently available can be divided in three categories; experimental, semi-
commercial and commercial. SEPIMN and Dolfyn are examples of experimental packages; both do not give
the option of turbulent flows in combination with free surface waves. Experimental means that there are
numerous options to alter the code to meet the demands of the problem on hand. Altering is a specialized
job that takes considerable time and is therefore outside this thesis, which makes the experimental packages
not suitable. There is one semi-commercial package COMFLOW; this package is still not suitable because
there is no sophisticated turbulence model available and no option for rotational mesh to keep grid
refinement around the keel-tip.

The specification of the commercial packages FEIMLAB, CFX and STAR-CD show that these packages should
in theory be able to model all phenomena stated above. FEMLAB has little experience with marine related
flows and CFX is currently rewritten to be connected with the Finite Element package Ansys, which makes
both packages less suitable. All three are not available at the TU-Delft which means that technical support is
not available, in case educational licenses can be arranged.

Two commercial packages are available at the TU-Delft: FL0W3D and Fluent. The specification of both
packages also indicates that all demanded phenomena are possible. In this thesis Fluent is selected because
it has more sophisticated turbulence models and because there is more technical support available. At the
TU-Delft there is a Fluent-workgroup in which several users participate with various research backgrounds,
this will be beneficial in case of unforeseen problems. Fluent runs on the computer systems of the laboratory
of Aero and Hydro dynamics of mechanical engineering, where most of this thesis is executed.

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University of Technology Delft & Bluewater Energy Services b.v.

3 Theory: numerical turbulence modeling

This chapter describes the numerical background of turbulence modeling with a modern CFD package. The
text is a summary based on [3], [11], [13] and [2], readers interested in the details of the presented models
and formulations are directed to this literature. Although the calculations performed in this thesis are all 2D,
this chapter will present 3D equations for completeness.

The first paragraph of this chapter describes the phenomena of turbulence; the second paragraph gives a
method to describe viscous flows: Navier-Stokes (NS) equations. Paragraph 3.3 and paragraph 3.4 describe
the Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES), two methods to solve the NS-
equations. Unfortunately the available computer power does not allow computations with these models,
they are added to this report to give a complete overview of turbulence modeling and explain the downsides
of these methods.

Paragraph 3.5 gives the basics of the CFD method used in this thesis; Reynolds-Averages Navier-Stokes
(RANS) equations. This set of equations contains more unknown than equations, leading to a closure
problem. These closure problems are overcome with so-called turbulence models; two different models are
used in this thesis, they will be described in paragraph 3.5.1 and paragraph 3.5.2.

3.1 The phenomena of turbulence

Turbulent flows are determined by chaotic, random velocity structure. Identification of such flow is based on
a typical length and velocity scale and the kinematic viscosity; the Reynolds number (= UL/v).

At low Reynolds numbers a flow is called laminar and characterized by smooth and adjacent layers that slide
past each other in an orderly fashion. When the Reynolds number increases the flow becomes turbulent with
an unstable and chaotic character. In these flows, vortices can be detected with a wide variety of length-
and velocity-scales. Generally, these vortices are called eddies. The largest eddies have a typical length and
time scale comparable to the boundary conditions. These large eddies are strongly influenced by the mean
flow and therefore anisotropic. At this level, the so-called macro scale, the viscosity has no influence on the
flow behavior and the Reynolds number is high.

When a turbulent flow develops, the rotation rate of the large eddies increases and their cross-section
decreases. This is caused by the conservation of momentum during the process of vortex stretching see
figure 3-1. It creates transverse eddies at with small length and time scales, which makes a turbulent flow
inherently three dimensional. The process of initiating small eddies is repeated until the smallest eddies
reach micro scale or Kolmogorov-scale. With this process the kinetic energy of the mean flow is transported
to the small eddies, which is called the energy-cascade process.
Transverse eddies

. >■/
•4 1 '■ \— -I —► Vortex stretching
\ , \ ■ \ , \

▲ ▲

Decrease cross section

Figure 3-1 Vortex stretching

16
 Theory: numerical turbulence modeling

These smallest eddies at micro scale have a length scale of 0.1-0.01 [mm] and frequencies of 10 [kHz] in
typical engineering flows (source: [13]). The Reynolds number at this scale is low; around 1. The flow is
dominated by viscosity and has no influence of the mean flow, it is considered universal and isotropic. Here
the energy of the mean flow is dissipated by converting the kinetic energy of the small vortices into thermal
internal energy.

3.2 Governing equation: Navier-Stoltes

A fully developed isothermal turbulent flow with a constant density (p) can be described by a set of non-
linear coupled partial differential equations. These equations are deducted from the basic rules of
conservation of mass and momentum. These are known as the three Navier-Stokes equations and the
conservation of mass:

du. 3M, dp 3~M,

(3-1) —'- + u^—- = P8--::r + ^ ^ ^ fori,j = 1,2,3
y dt dxj J dx ax~;

du. .
(3-2) ^ = 0 for i,j = 1,2,3
dx.

Variable (u) refers to velocities and the scalar (x) to the directional derivatives of these velocities. Subscripts
i,j = 1,2,3 refer to the x,y,z direction respectively. This gives a set of four equations for four unknowns
(three velocities, u, and pressure, p). Thus for a flow with a constant density this set of equations can be
solved and should give the exact motion of the flow. Unfortunately, there is no analytical solution found for
these equations for general turbulent flows. It can only be solved analytically for simple laminar flows at a
low Reynolds numbers. Solving problems of normal engineering interest is not yet possible. A solution is
found in the Finite Volume Method (FVM) which is commonly incorporated in Computational Fluid Dynamics
(CFD) packages. The FVM will be explained in the next chapters by considering the following general model
equation:

d(p(p)
(3-3) —^^—- + div{p(pu.) = div{Tgmd(p) for i,j = 1,2,3
at

These equations are called the transport equation of variable cp, which can be any variable. When for
example cp = Ui these equations become identical the Navier Stokes equations. The first term on the left
hand side is the rate of increase of cp, the second is the convective term. On the right hand side the diffusive
(r=diffusion coefficient) and the source term of variable cp can be found. The key step of CFD using FVM is
to integrate this transport equation on a finitely sized Control Volume (CV) in order to get a discretized
version of these equations at every node. There are several techniques to perform this method; the most
common will be described in the next paragraphs.
When other scalar quantities like temperature, pollution concentration, turbulent viscosity etc. are to be
included in the problem, the same appearance of the transport equation is used.

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Verification of CFD calcuiadons witli experimen ts on a rolling circular cylin der with bilge keels in a free surface.
University of Technoiogy Delft & Bluewater Energy Services b.v.

3.3 Direct Numerical Simulation ( D N S )

DNS is a numerical technique that uses the exact time dependent transport equations as given above. This
technique incorporates computation of the velocities at all length and time scales; even the smallest eddies
at micro scale are calculated explicitly. This means there is no additional turbulence model needed to
describe the influence of the smallest eddies.
It makes it possible to study the instable and chaotic character of a turbulent flow in detail by giving the
entire 3D velocity field of the flow. Parameters like pressure can be calculated, which can not be measured
in a laboratory. This makes DNS a very powerful tool for many research areas.
The down side of this method is related to the time and length scale of those smallest eddies. To make a
useful numerical simulation of these eddies, the grid has to be very dense and the time step has to be small.
Additionally the grid has to be 3D, due to the inherent 3D character of turbulence. This limits the maximum
Reynolds number that can be simulated with the current computer power. Nevertheless, successful
simulations with turbulent flows at low Reynolds numbers are executed. The current computer power does
not allow the use of DNS for common engineering problems.

3.4 Large Eddy Simulation (LES)

Large Eddy Simulation demands less C PU-time than DNS and therefore more applicable. It filters the small
eddies from the velocity field and calculates only the large eddies explicitly. The small eddies are taken into
account with the aid of a subgrid-model. This is possible because of the universal character of the micro-
scale eddies; only the macro scale eddies are flow-dependent. The large advantage is that the grid and time
step can be based on the macro scale length and velocity. The grid has still to be 3D but there is
considerable reduction in CPU-time compared to DNS.
After the filtering the Navier-Stokes equation spatially with filter length If, it looks as follows:

3M, du dp d^u- dr.

(3-4) PSi-^ +M ^ +^ fori,j = 1,2,3
OX, ax, ax■

Here the tilde represents the scalars after filtering. The added terms, Ty are called the subgrid stresses that
are comparable to the Reynolds stresses described in the next paragraph. The key aspect of LES is to find
the best suitable description of these subgrid stresses. Several methods are already developed and research
is still executed to develop better models. Detailed description of these models will be outside this thesis, the
most common can be found in [2]. The power of this LES method is that the subgrid stresses depend only
on the micro scale behavior. This behavior is flow independent and isotropic which makes it reasonable easy
to describe in theory.

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 Theoty: numerical turbulence modeling

3.5 Reynolds-Averaged Navier Stokes equations (RANS)

The last described the technique of numerical simulation of turbulence flows will be based on the Reynolds-
Averaged Navier Stokes equations (RANS). The RANS equations are deducted from the general Navier
Stokes equation. This is based on the characteristic that variables of a turbulent flow can be decomposed in
an ensemble averaged and a fluctuating part:

M = M -I- u'
(3-5) _ , for i,j = 1,2,3
Pi = Pi + P,

These expressions are called the Reynolds-decomposition; the over-bar represents the ensemble average
and the accent the fluctuation. To be able to use these expressions they have to comply with the following
Reynolds conditions:

f+8^f+8

(3-6) £ ^ £
ds ds
fg = fg

The last part needed are new conservation laws for the ensemble average and fluctuating part of the
velocities. They can be found by substituting equation (3-5) in the conservation laws (3-1), giving the
following new conservation laws:

5", - n du
(3-7) —^
—- = 0
U and --^ = U for i,j = 1,2,3
ax, ÓX

Now the Reynolds-Averaged Navier Stokes (RANS) equation can be found by substituting the Reynolds
decompositions (3-5) in the Navier Stokes equations (3-1). With the aid of the (3-5) and the conservation
laws (3-7) the RANS equations read as follows:

' 3M, 3M,M, dp d-u, du'u'.

(3-8) p PS-^ + U-^-^-^ fori,j = 1,2,3
3r 9jr^ J dx, dx'j dXj

There is one new term in this equation; the fourth term on the right hand side. This new term leads to a
closure problem (more unknowns than equations) and makes it impossible to solve this set of equations
directly. The three momentum equations together have six new unknown values:

(3-9) U-Uj for i,j = 1,2,3

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Verification of CFD calculations with experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

These terms (3-9) are called Reynolds stresses. To be able to solve the transport equations, a RANS-based
CFD package demands a so-called turbulence model to calculate these Reynolds stresses. Turbulence
models used in this thesis are described below.
The advantage of using RANS equations is that there is no demand of a dense grid to capture the micro
scale eddies. It is also possible to perform 2D calculations because the 3D effects are looked after by the
turbulence models. This leads to a strong decrease in demanded CPU time.
The disadvantage of the RANS method is that the Reynolds stresses depend on the entire flow and therefore
makes them flow dependent. There is no universal RANS-based turbulence model that covers all turbulent
flows. This is the advantage of the subgrid-model of the LES method described in the previous paragraph;
this model depends only on the small eddy behavior that is isotropic and universal.

3.5.1 K-£ model

Besides the three transport equation for momentum, the K-£ model uses two additional transport equations
to describe the Reynolds stresses. One for the turbulent kinetic energy, K, one for the turbulent dissipation
rate, £. First there has to be an expression that connects known variables of the flow to the Reynolds
stresses. Most two-equation turbulence models are based on the Boussinesq closure hypothesis that reads
as follows:

(3-10) p(-^"'"y+T"I'<5;^) = m a7 +-
for i,j = 1,2,3

Two new scalars appear in this formula: 5ij the Kronecker delta (5|j =1 if i=j and 5^ = 0 if i # j) and the
turbulent viscosity: Pf This hypothesis relates the Reynolds stresses with the turbulent viscosity, other
values are known from the momentum transports equations. Now the K-£ model needs an expression to
describe this coefficient:

ir
(3-11) A=C„

In this expression C|j is an empirical constant with value 0.09, which is based on the condition that the K-E
model has to comply with a number of standard flows. Scalars K and e can be derived with the following two
transport equations:

die ~ die
—-+u. ax. (7.. dx.I J
dt ' dxj
(3-12) for i,j = 1,2,3
de — de ^de_
— + 11 — + -(c„/'-C,,£)
dt ' 3A-. dx.

20
 Theory: numerical turbulence modeling

This set of equations contains the following empirical constants: Ok = 1, Oe = 1.3, 02^ 1.3, Ou = 1.44 and
are known as the turbulent Prandtl numbers. P is the production of kinetic energy:

du, du 3M;
(3-13) P = -öu-u- —- ■M, —'- + — i for i,j = 1,2,3
dx- 8JC; dx:■' J a7

This relation is based on the assumption that the dissipation rate e is proportional with the production of
turbulent viscosity. At this point transport equations (3-12), the B oussinesq closure hypothesis (3-10) and
the momentum transport equations (3-8) together form a closed set of equations.
The K-£ model is considered as the work horse of numerical turbulent calculations; it gives satisfying results
for large range of engineering problems. A large drawback of this model is when complex strain fields or
significant body forces are involved. Under these circumstances the B oussinesq hypothesis (3-10) gives a
poor representation of the Reynolds stresses.

Additional turbulence components

Here several additive components are given that are added to the solution algorithm to improve the
accuracy of the K-£ model.
The accuracy of the turbulence computations are significant improved by adding an extra term in the e-
transport equation. This model is known as the Renormalization Group (RNG) method. The method uses the
so-called renormalization group theory to determine the coefficients and scalars in the transport equation.
This model performs better with swirling and rapidly changing flow velocities. It provides analytical formulas
for the Prandtl numbers instead of the fixed values used in the standard K-e model. It also provides an
analytically derived differential formula for the effect of viscosity at low Reynolds numbers. The latter
phenomenon will occur everywhere outside the keel-region in the computation on hand.
To specify both K and £ near the body wall, an additive Enhanced Wall Treatment (EWT) is also
incorporated. Near a wall the flow changes from molecular viscosity dominated to fully turbulence dominated
in a very thin region. To prevent the need of a very fine mesh near the wall, EWT is applied.

3.5.2 Reynolds Stress Model (RSM)

To overcome the problem with the B oussinesq hypothesis (see paragraph 3.5.1 ) the Reynolds Stress Model
(RSM) is introduced. RSM calculates the Reynolds stresses directly with six additional transport equations,
one for every Reynolds stress.
The general representation of the transport equation for the Reynolds stresses is as follows:

dii'-u
(3-14) for i,j = 1,2,3
dt

Pij is considered as the Rate of production of the Reynolds stresses. !,{. Transport of the Reynolds stresses
by diffusion, 0^: Transport of the Reynolds stresses due to the turbulent pressure-velocity interaction. The
last term (EIJ) is known as the dissipation of the Reynolds stresses.

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The right hand side terms of equation (3-14) demands several empirical constants that are derived from
standard flows. The transport equations for the Reynolds stresses also introduce new unknown scalars,
giving again a closure problem. The main advantage with RSM is that the closure terms have to provide
additional information of the equations of the Reynolds stresses. When these Reynolds stresses are found,
they are used in the momentum transport equations without approximation. The K-£ and K-CO models on the
other hand demand a closure hypothesis that gives a correlation between the Reynolds stresses and the flow
velocities. In other words, the already approximated Reynolds stresses are taken into account in the
momentum equations by another approximation (the Boussinesq hypothesis). This makes the K-e and K-W
more susceptible for errors in the method of turbulence modeling
More complex flows with anisotropic rapid changing strain rates are considered to be better modeled with
the RSM model. Globally it can be said that RSM covers more general engineering problems. The down side
is the increased CPU time as a result of the extra transport equations. Also the lack of verification of the
closure models and empirical constants can make the results of this model less reliable for certain flow
types.

22
 Solution algorithm of FLUENT

4 Solution algorithm of FLUENT

This chapter describes the algorithm that Fluent uses

to solve flow problems. An overview of this algorithm is
given in figure 4-1. A couple of decisions in this
algorithm are based on the created computational
model; this model is described in the next chapter.

Solving an unsteady numerical problem means three

loops have to be used. The outer loop is involved in
the advance in time. Details of this loop can be found
in paragraph 4.5. The inner loop is an iteration to
compute variables like velocity, pressure and
turbulence based on an initial guess of these values.
Every iteration the solution should become closer to
the exact solution; it will be explained in paragraph
4.3. The last loop, not given in the figure is
incorporated inside step 2 and 3; this loop is necessary
to solve the set of linear algebraic equations. This is
performed iteratively due to the size of the matrices
involved, it is described in paragraph 4.4.

Figure 4 - 1 Overview solution algorithm

Before FLUENT starts with solving the equations, several preparations have to be done. First the transport
equations (momentum and turbulence) have to be discretized. The applied discretization scheme is given in
paragraph 4.1. Secondly the computational model has to be defined, this is described in chapter 5. And the
last preparation is to manually set the initial values of all incorporated variables (see paragraph 5.3).

With these preparations done, the solution algorithm can start; briefly it works as follows: step 1 determines
the location of the free surface; the Volume of Fluid (VOF) method is used for this step and is described in
paragraph 4.2. Step 2 solves the discretized momentum equation (3-8), this solving method will be
explained in paragraph 4.3. After step 2 the velocity and pressure field are known and step 3 can start to
solve the discretized turbulent transport equations to determine the value of K and £.

The solution reached after these three steps can be regarded as a correction on the initial guessed values.
After each iteration the accuracy of the reached solution is checked and it is decided if another iteration is
necessary. The reached values of this iteration will be used as initial values at the next iteration. The
iterations stop when the convergence criteria are met, which will be explained in paragraph 4.4.1.

When the solution is converged, the whole computation is repeated on the next time level until the
maximum flow time is reached. The consequences of this time dependent computation are given in
paragraph 4.5.

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Verification of CFD calculations with experiments on a rolling circular cylinder with bilge keels in a &ee surface.
University of Technology Delft & Bluewater Energy Services b.v.

4.1 Discretization scheme: QUICK

As can be found in figure 4-1, step 2 and 3 solve the sets of momentum and turbulence equations. In the
appearance given in the previous chapter (equations (3-8) and (3-12)), they can be regarded as continuous
differential equations. They have to be discretized before the systems can be solved. Discretization means
approximating the transport equations by a system of algebraic equations for flow variables at some set of
discrete locations in space and time.

The discrete locations in space are the grid cells of the computational model. These cells divide the flow
domain in a finite number of control volumes. All cells together are called the mesh. The mesh of this
particular problem can be found in paragraph 5.1. A discretization scheme gives for flow variables like
pressure or velocity a relation between a particular cell and its neighbours. The discrete location in time
means that an unsteady flow will be divided in small time steps. A discretization scheme also relates
information of the previous time step with the current time step. The relations for space and time form an
equation for every cell, which yields to a system of equations with a size equal the number of cells. Solving
these systems will be explained in paragraph 4.4.

Fluent offers several first and second order schemes for this purpose. With the problem on hand it is decided
to use a second order scheme for a couple of reasons. A second order scheme performs better when the
mesh contains triangular cells, which is the case for the keel region, see paragraph 5.1.2. A second order is
also advised whenever the flow is not inline with the grid, which is almost always the case for triangular
cells. Higher order schemes involve the use of more neighbor cells and thus an improved accuracy in
comparison with first order schemes.

Within the second order schemes, the Quadratic Upstream Interpolation for Convective Kinetics (QUICK)
scheme is chosen. This scheme is considered to improve the results with rotating and swirling flows (see
[3]). The scheme is assumed to be less sensitive to numerical diffusion errors and therefore also has an
overall higher accuracy. Detailed description of this scheme can be found in [2] and [13].

4.2 Free surface modeling: VOF

Fluent offers several models to incorporate multiphase flows; every model is developed for its own specific
flow-type. The "geo-reconstructed VOF" method of Fluent is chosen for the following reasons. The Volume
of Fluid (VOF) method is made for flows with completely separated phases; the phases do not diffuse into
each other. The problem on hand uses only two phases; water and air, at normal temperatures evaporation
of water is negligible. Geo-reconstruction is added to the VOF scheme to define the free surface more
accurately.

All VOF methods use a scalar to determine the amount of a particular phase in a cell in comparison with the
total volume of that cell. This scalar is called Volume Fraction (VF). In the problem at hand there is only one
VF demanded (water or air); for example when the fraction of water is known in a particular cell, it is
automatically known that the remaining part of that cell is filled with air.

For the Volume Fraction a transport equation is established, comparable to the general transport equation
(3-3). Before geo-reconstruction starts, the VOF method solves the VF-transport equation to determine the
volume fractions in each cell of the mesh.

24
 Solution algorithm of FLUENT

This geo-reconstruction method contains the following three steps: step 1 determines the location of the
free surface in each cell based on the known volume fractions. It uses piece-wise linearization between the
cells. Step 2 calculates the mass balances at the faces of the cells. The third and last step determines the
new VF's based on the fluxes calculated in step 2. These steps are performed cell by cell by tracking all cells
at the free surface.

Fluid properties like density, viscosity etc. of a cell at the interface are influenced by the VF of that particular
cell. That is why the last part of the VOF method determines these new properties with the aid of the
following formula:

(4-1) (P^oc.fp.+i'^-aj^a

Here cp represents the new (average) value of a particular fluid property. The scalars ((p„) and (cpa) represent
the value of that property of respectively water and air. Scalar (a„) is the volume fraction of water. With the
new VF's and fluid properties known, the momentum equations can be solved; this will be described in the
next paragraph.

4.3 I t e r a t i v e solving the pressure-velocity coupling: PISO

Initiate: p*, u*.

2,1: Solve set of discretized momentum equations

u««,
r
2.2: Solve first pressure-correction equation

P'
r
2.3: Update pressure and velocities

r
2.4: Solve second pressure-correction equation: p"

1r ""
2.5: Update pressure and velocities

▼ U|, p

Figure 4-2 Detail steps of the PISO algorithm

FLUENT and other CFD packages incorporate an iterative solution algorithm to solve the discretized
momentum transport equations. It is mathematically very difficult to solve this coupled set of differential
equations due to the non-linear terms and complicated way in which the three equations are coupled. All
velocity components appear in all equations and in an unsteady version of the transport equation, they also
appear at different time levels. Especially for large numbers of grid cells (order 10^) solving these systems
with direct solvers becomes virtual impossible. Therefore iterative solution algorithms are introduced to
overcome this problem.

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Verification ofCFD calculations with experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

All iterative algorithms have the same general structure: they start with a guessed pressure (p*) field. With
a known pressure field it is possible to solve the momentum equations and determine a temporal velocity
field. Then a pressure correction (p') is calculated based on these temporal velocities. With these pressure
corrections the final velocities and pressures are determined.

Several solution algorithms have been developed in the past, each best suitable for a certain flow problem.
FLUENT has three available: SIMPLE, SIMPLEC and PISO. The main difference between the first two and
PISO is that PISO uses two pressure-correction equations instead of one. This means more calculations per
iteration, but the velocity and pressure fields at the end of the iteration are generally more accurate. This
means in most cases less iteration steps.

For the flow problem on hand it is chosen to use the PISO algorithm; this is generally recommended when
unsteady flows are involved. Due to the second pressure correction it is possible to enlarge the time step,
which means a considerable decrease in demanded CPU-time. It is also possible to increase the Under-
Relaxation-Factors (URF, explained in paragraph 4.3.1) of the velocities, which means faster convergence
and thus also decreased CPU-time.

The main steps of the PISO algorithm are given in figure 4-2, it will be treated briefly below and details can
be found in [13].

Step 2.1 uses the guessed pressure (p*) field to compute the temporal velocities (u,**). The set of
momentum transport equations needs guessed velocities (Uj*) to determine several coefficients that are
present in the discretized versions of those equations. Both guessed (p*) and (u;*) are taken from the
previous iteration or initial values. With the velocity and pressure field computed, step 2.2 solves the first set
of pressure-correction equations, giving the fist pressure correction p'. Now step 2.3 uses the pressure
correction (p') to update the pressure (p**) and velocities (Uj***). With the new pressure and velocity field
the second pressure-correction term (p") will be calculated (step 2.4). Finally step 2.5 updates the velocity
and pressure for the last time.
The final values of pressure and velocities computed by the PISO process are used to solve the turbulence
transport equations, step 3 of figure 4-1 and will be explained in paragraph 4.4.

4.3.1 Under Relaxation factor

In the iteration loop described in this chapter, Fluent uses sets of correction equations to compute the
variables (velocity, pressure, volume fraction, turbulent viscosity etc.) at a new iteration step. This process is
susceptible to divergence and it is therefore necessary to control the rate of change of these variables.
Fluent uses so called Under Relaxation Factors (URF) to reduce the rate of change during each iteration.
This can be described as follows:

(4-2) (p = (p„„+/3-A(p

In this expression, q> is one of the fluid variables, cpoid is the value of that variable at the previous iteration,
Acp is the computed change at the current iteration and scalar (P) is the under-relaxation factor.

26
 Solution algorithm of FLUENT

The URF is user defined and has a value between 0 and 1. A value of 1 means that the rate of change of
that variable is maximal at every iteration. This will lead to fast convergence but will in most cases also leads
to oscillation or in the worst case to divergence of the solution. Zero means that there is no correction at all
and the solution will not converge. An optimal URF is flow type, iteration scheme, and discretization scheme
dependent. In the problem on hand the URF is determined by investigation of the convergence behavior
during preliminary calculations. The URF's that are used in the end can be found in paragraph 5.7.

4.3.2 Pressure interpolation scheme: PRESTO!

A pressure interpolation scheme is needed just before step 2.1 of figure 4-2because Fluent stores the
pressure values at the cell centers and solving the transport equation of step 2.1 needs the pressure at the
cell faces. Fluent offers several schemes to solve this pressure interpolation; PREssure STaggering Option
(PRESTO!) is chosen.

PRESTO! is recommended when significant body forces are expected in the solution; this is the case because
the gravity is taken into account and the difference in density between water and air is large. When the VOF
method is applied, the PRESTO! scheme is recommended as well. The last reason is that this scheme is
mainly built for handling large pressure gradients and this phenomenon can be expected near the keel tip in
the problem on hand.
In preliminary tests to investigate if Fluent can do the demanded computations it was discovered that Fluent
gives unrealistic velocity and pressure fields when the PRESTO! scheme is not used.

4.4 Solving discretized equations: Multi-grid method

In Step 2 and 3 of figure 4-1 the momentum and turbulence transport equations have to be solved. The
discretized versions of these transport equations form a very large set of linear equations. The matrix of
these equations has the same number of rows and columns as the number of grid cells, which means for the
problem on hand a matrix of approximately 80.000 x 80.000 cells. It is impossible to solve this set of
equations with direct methods like Gauss elimination or LU-decomposition. Fluent uses an iterative method
called Multi-Grid (MG) that can handle these large matrices.

This paragraph will treat the MG method briefly; readers interested in the details are directed to [2]. Based
on the initial guess of the solution, MG starts with calculating the residuals from the solution as described in
paragraph 4.4.1. The information on these residuals is brought to a coarser grid level. At this grid level
correction calculations are carried out to diminish the residuals at this coarse level. When a satisfying
decrease in residuals is found, the corrections are brought back to a finer grid level, where it is used to
improve the solution.

Two remarks on the MG method can be made; first is the number of grid levels. When the inability to reduce
the residuals on certain grid level is detected, an additive coarser level is incorporated. This process repeats
until the largest grid level is reached that can handle the global errors in the solution. In other words, the
more global an error in the approximated solution, the more grid levels are demanded. The second remark is
on the iterative character of the MG method. As mentioned in the introduction of this chapter, a third
iterative loop is used to solve the system of equations. This third iterative loop is performed at all grid levels
to solve the set of correction equations. Fluent solves this system by the iterative point-implicit Gauss-Seidel
method. Details can be found in [2].

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Verification ofCFD calcuiations witti experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

There are mainly two reasons to use a MG method; first is the size of the system that has to be solved. On a
coarser level, several fine grid cells are handled as one large grid cell this means that to describe the total
grid domain less equations are needed. Solving a system of linear equations demands a quadratic number of
computer operations, this means that when the grid is twice as coarse the demanded CPU time diminishes 4
times. The second advantage is the ability to reduce global errors much faster. When global errors (initiated
by boundary conditions or poor initial guesses) are present on a fine grid, it takes many iterations before this
global error is "transported" to distant cells. Every iteration the error can at most be given to its neighbour
when the standard Gauss-Seidel solving method is used. On a coarser grid this global error is "felt" much
faster by distant grid cells. This latter advantage improves the convergence speed significantly.

4.4.1 Convergence Criteria

The Convergence Criteria determines when the inner iteration loop of figure 4-1 stops. Deciding when to
stop an iteration process is important from both accuracy and efficiency point of view. Every flow variable
(velocities, pressure, K and e) has its own convergence criterion. The concept of a Convergence Criterion
works as follows: When the set of linear equations of an arbitrary flow variable (cp) is exactly solved, it can
be written as:

(4-3) A(p^-b = 0

Where (q)e) is the exact solution, (A) the matrix containing the set of equations and vector (b) contains the
boundary conditions. Before the exact solution is found there will be a residual (d) associated with the
approximate solution (cp):

(4-4) A^-& = | j |

After each iteration these residuals are calculated for every variable. When all residuals are below a certain
user defined value the solution is considered converged. This user defined value is called the Convergence
Criterion (CC). When the solution algorithm is converged, the solution algorithm moves to the next time
level, see figure 4-1

Very low CC values (order 10"^) lead to accurate solutions but it will take a considerable number of iterations
to reach these low values. On the other hands high values (order 10'^) can lead to instability or divergence
of the solution. It is recommended to choose the CC value such that during every iteration-loop all residuals
drop around 3 orders of magnitude. For the problem on hand this is investigated by preliminary calculations
and the CC values that are chosen in the end can be found in paragraph 5.7.

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 Solution algorithm of FLUENT

4.5 Unsteady calculations

The outer loop of figure 4-1, describes the advance in time for unsteady calculations. The problem on hand
is clearly unsteady due to the oscillating motion of the body.

The Fluent algorithm uses an implicit method to incorporate unsteady calculations. Implicit means that the
values of fluid variables, (velocities, pressure, viscosity etc.) at a new time-step are computed with values of
the surrounding cells at that new time step and values of the cell itself of the previous time step. The other
option is to use an explicit method that uses only information of the previous time step to compute the
variables on the new time step.

The consequences of unsteady calculations are found in the discretization schemes; they have to incorporate
the information of the previous time step and the rate of change in time, see paragraph 4.1.

4.5.1 Time step

The time step is has a large influence on the stability of the solution algorithm. Generally the maximum time
step is dictated by the velocity of the flow and cell size; the time step has to be chosen small enough that a
fluid particle does not pass more than one grid cell between time steps. In this case the hull motion and free
surface determined the size of the time step. The rotating inner mesh (see paragraph 5.1) slides along the
stationary outer mesh. The time step has to be chosen such that a node at the sliding mesh does not pass
more than one cell of the outer stationary mesh at a time step. The same principle applies for the grid cell at
the free surface; the water level should not change every time step such that it passes more than one cell.
When the water level does, the time step has to be decreased. It can be concluded that the finer the grid at
the interface and at the free surface, the smaller the time step has to be.

For the problem on hand, it was hard to use the rules given above to establish the maximum time step. It
was easier and to determine the maximum time step by an investigation of the solution behavior of
preliminary calculations. In the end the time step was set at 0.001 [s]. It was possible to take a slightly
higher value (for example 0.005 [s]); the solution was still stable, but the convergence behavior became
poor. This price of a slow convergence did not overcome the five times faster advance in time; the lowest
CPU-demands are found with a time step of 0.001 [s].

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Verification of CFD calcufations witfi experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

4.5.2 Total flow time

The outer loop, enabling the advance in time ends when the total flow time is crossed. This is a user defined
value and depends on the type of problem. Time independent flow can be stopped when the result becomes
constant. Time dependent flows are stopped when the results become periodic.

To determine the periodicity of this particular problem, the momentum registration on the keel was taken.
An example graph of this registration can be found in figure 4-3, in this figure the roll angle and keel force
are given dimensionless. It can be seen that the momentum graph becomes periodic after a number of
periods. The time it takes to become periodic depends on the frequency, roll amplitude and keel type. Every
computational run was therefore investigated separately and generally stopped after three subsequent
identical periods. Long term computations revealed that after three identical periods the solution continued
to be periodic.

Flow time t/T [-]

Figure 4-3 Typical registration of the force on the l<eel

The non-periodic start of each computation is mainly caused by the development of the turbulent viscosity.
This parameter was initially set for the water at rest, which is clearly not suitable for the water around the
keel. Iteratively solving transport equations for turbulent kinetic viscosity and turbulent dissipation rate have
to lead to a certain level of turbulent viscosity for the water around the keel. This process takes time. When
the amplitude of the hull motion is not increased gradually in time, the viscosity becomes initially
extraordinary high due to the sudden start; it takes considerable number periods to dissipate this excess of
viscosity. To avoid this viscosity peak a start function is used; figure 4-3 shows the result of a start function
which increases the roll amplitude during the first four periods. There is still an overshoot, but it is
considerable less than without a start function.

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 Computational model

5 Computational model

This chapter describes the computational model that is solved by Fluent with the solution algorithm
described in chapter 4. Throughout this thesis several slightly altered models are used for several calculation
types. All models have the same basic settings and are in compliance with the thesis setup described in
paragraph 2.4. This chapter gives the details of the basic settings and explains why certain settings and
models are selected. In the following chapters these basic settings are altered or additional models are used;
these changes are described in the involving chapters.

At the end of this thesis the results of Fluent calculations are compared with experimental data. The
computational model used in this comparison is based on experience gained during preliminary calculations
and literature research. To support the text written in this chapter, figures and other examples will be taken
from this final model. Altered computational models of other the chapters are similar.

5.1 Computational Grid

The whole computational domain has to be divided into small control volumes, called grid cells in order to
solve the discretized transport equations. Constructing a computational grid is a constant tradeoff between
accuracy and CPU-time; when a grid is coarse the systems that have to be solved are small which implies
short-CPU times. The downside is that a coarse grid is unable to represent small velocity or pressure
gradients in the flow field. A very fine grid will be more accurate but can take undesirably long CPU-times.
An additional disadvantage of a fine grid is that discretization gives a small round off error for every grid cell;
more grid cells imply more round off errors

The computational grid used in this thesis can be found in figure 5-1 and figure 5-2. It is divided in two main
regions: a rotational inner and a stationary outer region. The inner region oscillates with the body to keep
the grid refinement around the keel tip. The outer part damps the fluid motions and radiated waves.

Two different types of grid cells can be found in this model: unstructured grid in the keel zone (see figure
5-2) and structured grid everywhere else (see for example outer region in figure 5-2). Structured grid
consists of ordered quadrilateral cells, which gives considerable computational advantages. The
disadvantage is that opposite zone-walls must have the same number of nodes. (Nodes are defined as the
location where cell faces are connected with zone edges.) This constraint limits the modeling freedom.
Especially complex shapes are difficult to model with a structured grid. Therefore an unstructured grid with
triangular cells is used at the keel zone.

, Figure 5-1 Overview computational body

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Verification of CFD calculations with experiments on a rolling circular cylinder wit/t bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

The following four zones can be detected in figure 5-1 and figure 5-2 will be treated separately in this
paragraph:
1: Boundary layer near the body
2: Keel zone with unstructured grid
3: Grid refinement for free surface modeling
4: Far field cells for wave damping

Figure 5-2 Computational domain around body

The grid is created with the program "Gambit". This is a standard mesh-generator compatible with several
CFD packages, including Fluent. The main advantages of this program are the automatic mesh generator
and the extended options to adapt the model to user preferences. Gambit defines the model, grid and all
boundary types. The rotational part and stationary part of the grid were created separately in Gambit and
merged with the program "Tmerge". This merged grid is imported in FLUENT, where all other initial
conditions are added to the model, see further in this chapter.

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 Computational model

5.1.1 Boundary layer near the body

Near the wall of the body the tangential velocities in the flow rapidly changes. This demands a careful
description of the computational grid near the wall with a very fine grid normal to the wall. This refinement
strongly influences the computed wall friction. The boundary layer near a body wall can be divided in 3
regions:
1. Laminar sub layer (y"^ < 5)
2. Buffer region (5 < y^ < 30)
3. Turbulent region (y"^ > 30)
Where the parameter y"^ represents a dimensionless distance from the wall to an arbitrary point in the flow.
There are two options to model these layers near the wall. Firstly the grid can be very fine which can
describe even the laminar sub layer accurately. This could lead to undesirable large number of cells,
especially at high Reynolds numbers. Another option is to use a coarse grid near the wall with an additional
wall-function that takes care of the detail flow behaviour in the boundary layer.

-,—
, 1—

l\
A
/
1 Keel wall

Figure 5-3 Details boundary layer near body

For a first estimation if the grid is fine enough to describe the layers sufficiently, more or less arbitrary grid
sizes were chosen:
• Thickness first layer: 0.001 [mm]
• Thickness growth factor: 1.3 [-]
• Number of layers: 7
With these boundary layer cells, preliminary computations were carried out and the y"^ values of the adjacent
cells to the body wall were plotted. Plots at different point during the oscillation were investigated and it
could be concluded that all y^ values were within a range of 1 - 20 [-] for the cell both at the keel and
cylinder. This means that at some locations (when y* = 1) the sub layer will be described accurately and at
some locations (when y"^ > 5) an additional wall function is demanded or a grid refinement. In this
computational model it is chosen to keep the grid sizes given above and incorporate Enhanced Wall
Treatment (EWT). This will add a wall function to all boundary layers, also where the grid is fine enough.

The lengths of the cells have no specific demands; it is taken as large as possible to limit the number of
cells. The length is only bounded by the condition that away from a body wall the maximum thickness-length
ratio should preferably not be above 1:20. The resulting grid near the body-wall can be found in figure 5-3.

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Verification ofCFD calculations with experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

5.1.2 Keel zone with unstructured grid

In this zone it is chosen to use an unstructured grid with triangular cells, see figure 5-2. This makes it
possible to have a grid refinement around the keel tip and have a grid coarsening at the connection edges
(red circle in figure 5-2) with the other zones. This choice is beneficial when several keel shapes have to be
modeled. Unstructured grid is generated automatically when the nodes along the edges are defined; there is
no demand to make sub-zones to get an identical number of grid nodes at opposite edges.

The size of the cells in this zone is determined by the size of the cells adjacent at this zone, the boundary
layer cells and refinement at the keel tip. At the tip it is generally recommended to have at least 3 cells.
Away from the tip, adjacent cells should preferably have an area growth factor of maximum 1.3. This growth
factor is also preferred with cells of adjacent zones.

Due to the rotation of the inner grid cells it is not possible to keep the area growth factor below 1.3 at the
zone edges (see for example transition refined zone -> outer region). At these locations numerical problems
can be expected. With a small time step and low convergence criteria, it is possible to limit the influence of
this phenomenon on the results. When the recommended growth factor has to be maintained at every
location at all times, the number of demanded grid cell would become undesirably large.

5.1.3 Grid refinement for free surface modeling

The solution algorithm uses the Volume of Fluid (VOF) method to describe the radiated waves at the free
surface, as described in paragraph 4.2. This method demands a minimum number of cells to be able to
describe waves, which are fairly small due to the circular shape of the body. Therefore the grid at the
expected free surface has to be refined.

The maximum height of the cell at the free surface is determined by the expected maximum wave heights.
Preliminary estimations expected wave heights around 3-5 [mm] for these kinds of oscillations with this
circular model. VOF method uses three cells for interpolation to define the exact location of the interface.
This means that there are at least 10 cells demanded in height to describe a sinusoidal wave. This should be
sufficient, further refinement is preferable but that will lead to extraordinary large CPU-times. It is chosen to
have a maximum ceil height at the free surface of 0.4 [mm]. This implies that waves under 4 [mm] are not
described with the demanded 10 cells, which could lead to poorly predicted wave heights.

Figure 5-4 Grid refinement for free surface

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 Computational model

When a stationary grid is involved, these small cell heights do not have to lead to large numbers of grid
cells; small waves can be described by a thin refined zone (see right side of figure 5-4). The problem is the
rotating part of the grid: the refined zone has to be at the free surface during oscillation. This means a
demanded refined zone twice the oscillation amplitude, see left side of figure 5-4. The radiated waves were
recorded at the location of the red line, further handling of radiated waves can be found in paragraph 8.2.4.

The length of the cells at the free surface is limited by the common assumption that cells away from wall
boundaries should have a maximum aspect ratio of 20. This target value is not reached everywhere; it would
lead to too many cells. The down-side of aspect ratios above 20 is poor convergence behavior and thus
larger CPU-times.

5.1.4 Far field cells for wave damping

Fluent has no standard model to treat far-field behavior of waves; they should be transported out of the
control volume without reflection. This solved by increasing the cell sizes away from the oscillating body.
Gradually the cell sizes at the free surface become too large to describe the radiated waves, the waves are
numerical absorbed and the waves disappear. This numerical trick is a common method and can be found in
[18] for example

Fbw-trne [s]

Figure 5-5 Typical wave registration

The danger of reflection due to the absorption is not really investigated. It can be visually seen that the
waves disappear totally and the wave height registration does not give an indication of reflection, see figure
5-5. This figure gives the registration of the computational run with the small keel, a frequency of 0.6 [Hz]
and amplitude of 15 [deg].

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Verification ofCFD calculations with experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

5.2 Boundary conditions

In figure 5-6 it can be seen that boundary condition have to be defined at four locations. The locations will
be treated separately.

__, -^9?5L
^"^ jn
Figure 5-6 Overview boundary conditions

Body wall (1)

At the body wall, a no-slip condition is incorporated. The
velocities normal and tangential to these faces are set at zero.

Outer wall (2)

The outer wall has the same no-slip condition as the body wall:
zero velocities normal and tangential to the wall surface. The wall
functions to compute viscous stress at the wall are also used at
these walls although the fluid here is completely at rest. Once
these functions are turned on in Fluent, they apply to all wall
boundary types.

__^.,_ .^^_=_ Figure 5-7 Grid interface

Interface between rotating and stationary grid (3)

Exchange of flow information across the interface demands special attention as the cells adjacent to the
interface are not lined up with each other. The fluxes across the interface are computed using intersecting
edges of the adjacent cells, rather than the edges of the cells itself. This can be explained with the aid of
figure 5-7. In this figure, 5 cells adjacent to the interface are given, three (I, I I and III) belonging to the
rotational inner mesh and two (IV and V) belonging to the stationary outer mesh. When for example the flux
across the interface into cell IV is demanded, edge E-F is ignored and edges ejb and b-f are used to bring
the information into cell IV from cells I and I I , respectively.

Free surface (4')

At the free surface boundary conditions are also demanded, but Fluent incorporates these conditions
automatically when the fluid types (water and air) are defined.

36
 Computational model

5.3 Initial conditions

The variables included in the problem on hand (velocities, pressure, density, K, E. Volume fraction) have to
be initiated before the solving algorithm can start. These initial guesses have to be as accurate as possible; a
guess close to the exact solution will lead to fast convergence. An overview of the initial values is given in
table 5-1.

Water | Air unit

x-velocitles 0 [m/s]

y-velocities 0 [m/s]

pressure Reference pressure: 101325 [Pa]

Density 998.2 1 1.225 [kg/m3]

Turbulent kinetic energy l.OE-06 [mVs^]

Turbulent dissipation rate l.OE-4 [mVs^]

Volume Fraction Free surface halfway the cylinder [-]

Table 5-1 Initial conditions

As both phases (water and air) are initial at rest and incompressible, the initial settings of velocity and
density speak for themselves. For the density the default values of Fluent are taken. The volume fractions in
all cells are known as soon as the location of the free surface is defined. This surface is set halfway the
cylinder in agreement with the experiments.

A fixed reference pressure is set at 101325 [Pa], equal to the average atmospheric pressure. This pressure is
preferably defined at a location in the computational domain where there are as little pressure fluctuations
as possible. In this case the reference pressure is defined in the air phase and away from the rotating
model. With the pressure known at one location, the initial pressures in all other cell can be defined.

The initial values of the turbulence parameters are somewhat more complicated; they have a considerable
influence on the hull forces. When these levels are poorly guessed, it will take a long time before the excess
or shortage of turbulence is removed. There are all kind of rules how to get the best estimation of the
turbulence levels, but these are mainly developed for fluids in motion. In this case a very long preliminary
computation showed the final levels of turbulent kinetic energy and turbulent dissipation rate for the water
at rest. These levels were taken as initial values for all other computations afterwards.

5.4 Body and its motion

The hull is created in the program Gambit that also generates the mesh. Figure 5-1 gives the cylinder with
the reference keel, the other keels can be found in chapter 7, which handles the keel shape investigation.
The hull was constructed in two parts: cylinder and keel. This made it possible to measure the forces on
these parts separately.

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Verification ofCFD calculations with experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

Fluent makes a motion of the hull possible by prescribing the velocity of the surrounding grid. Any wall
attached to this grid gets the same velocity, which means in this case prescribing the velocity of the inner
mesh, the keel and cylinder are taken along automatically. Fluent can handle any kind of prescribed rotation
and even rotations depending on current forces on the hull. The latter makes it possible to handle free
floating objects. This thesis only incorporates prescribed rotation around one axis. All other motions are
constrained.

The function that prescribes the demanded motion has to be defined in an external file, based on C**
programming language. Fluent calls this function a "user-defined-function" (UDF). A function is defined that
gives the rotational velocity as function of time. At a particular time step. Fluent reads from the UDF what
rotational speed corresponds with the current flow-time and uses this value to move the body. The file
containing the UDF can be found in appendix A, the used function for the rotational speed is explained
below.

As many numerical simulations that incorporate a body motion, a gradual start of the motion is needed to
avoid large initial peaks in variables as viscosity or pressure. It can take considerable number iterations to
get rid of those initial peaks. The final motion of the hull will be a pure sine:

(5-1) (p{t) = ^^, • sin cot

A start function is defined that slowly increases the amplitude from zero to the final value for the first 4
periods, the frequency will be constant during the whole computation. A function f(t) is demanded that
starts at t = 0 with the value zero and after 4 periods the function must have a value of 1. It is chosen to
use a part of a sine function, see figure 5-8.

2 3
Oscfetbn period: T

Figure 5-8 Start function

This start function can be written as follows:

f n ^
(5-2) / ( 0 = 4sin \---t-^n

Here T is the period of the ultimate motion. A sine is chosen because now the hull accelerates very slowly in
the beginning to let the turbulent viscosity adjust to the motion. When the flow time reaches 4 periods, the
start function has become 1.

38
 Computational model

Fluent demands a prescribed rotational velocity instead of a prescribed roll angle, this leads to the following
formula for rotational speed of the first 4 periods:

(5-3) (j>it) = f (t)-CO-(p^-sin ox- f \t)-0^-cos cot forO<t<4T

Here f'(t) is the derivative of the start function f(t), co and q)a is the frequency and amplitude of the ultimate
motion. After 4 periods, function f(t) has reached a value of 1 and from this point the value of f(t) stays 1.
This gives the following formula for the rotational speed:

(5-4) ^(t) = CO-(p^- cos ox for t > 4T

These functions can be found in appendix A where the file of the UDF is given. With these functions for the
rotational velocity, the hull will make a motion as given in figure 5-9. The roll angle is given arbitrary; every
computational run has its own roll angle, but the motion has the same appearance.

Oscillation period: T

Figure 5-9 Roll motion

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5.5 Output

5.5.1 Forces and moments on the hull

Fluent calculates the forces and moments by integrating the pressure at the surface of a particular part of
the body. Forces in z and y direction and moment around the x-axis are registered for the keel and cylinder
separately. The values of these parameters at every time step are written in a text file by Fluent. Post
processing these files to compare with the experiments is performed with I^ATLAB and is described in
paragraph 8.2.

5.5.2 Registration wave height

Fluent has no standard model to register the radiated waves, this is solved as follows: Fluent records every
25 time steps the density inside every cell along a vertical line. This line is at a location with a sufficiently
fine grid at the free surface, see paragraph 5.1.3. The records are stored in a text file. MATLAB reads this
file and starts with looking for the location of the free surface. This location is where the density of water
changes to the density of air. A linear interpolation between the neighboring control volumes under and
above makes the localization of the surface more accurate. The wave height found is known at every time
step and can be compared with the experiments see paragraph 9.4.

5.5.3 Visual location of large eddies

To make an assessment of the capabilities of Fluent to determine the location of the eddies occurring at the
tip of the keel, plots are made of the velocity fields. Fluent plots a vector in every cell, where the vector is in
the same direction as the flow and its length gives the local flow velocity. These plots are used in paragraph
9.6 for comparison with the visual registration during the experiments. In these figures not all vectors are
plotted; some have been skipped because the fine grid would make it impossible to detect individual vectors.

5.6 Repetitive computations

The input parameters given in this chapter and the chosen solution algorithm described in the previous
chapter stayed the same for every computational run. Computations were madej/vith several amplitudes,
frequencies and keel shapes, but all other parameters are not changed. In Fluent it is possible to make a
"journal" which contains all standard inputs. This journal is an externally written file; Fluent reads this file
and automatically initiates the solution. Such a journal gives considerable advantages with repetitive
computations in which very similar initial settings are needed. In appendix B a typical journal can be foundr^^^

40
 Computational model

5.7 Overview computational settings

Table 5-2 below contains the computational settings as far as the default settings are altered. The given
settings are used to compare the Fluent computations with the experiments. All these settings can be found
in the journal of appendix B, and are thus used for all computational runs. Explanation of the settings can be
found in this chapter and the previous; the second columns of table 5-2 indicate the concerning paragraph.
Chapter 6 describes the results of preliminary computations with altered settings; the relevant settings for
those computational models will be given in that chapter.

Category Details in paragrapli Description Input

Numerical solver Segregated implicit unsteady first order

Solving algorithm scheme PISO

Solution algorithm §4.3
Convergence Criteria 0.001

RNG K- £ model
Enhanced wall treatment
Turbulence modeling §3.5.1
Pressure gradient effect
RNG-differential viscosity

VOF, Geo-reconstructed
Multiphase model §4.2 Include Implicit Body Force treatment
Specified operational density

Momentum QUICK
Discretization scheme §4.1 Turbulent viscosity QUICK
Turbulent dissipation rate QUICK

Interpolation scheme §4.3.2 For pressure PRESTO!

Pressure 0.6
Density 0.6
Body-forces 0.6
Momentum 0.6
Under Relaxation Factors §4.3.1
Volume Fraction 0.6
Turbulent kinetic energy 0.6
Turbulent dissipation rate 0.6
Turbulent viscosity 0.6

Reference pressure 101325 [Pa]

Initial Conditions §5.3 Turbulent kinetic energy l.OE-06 [-]
Turbulent dissipation rate l.OE-04 [-]

Amplitude range 4, 8, 12, 16 [deg]

Hull motion §5.4
Frequency range 0,4 and 0,6 [Hz]

Rotating <-> Stationary grid Interface

Boundary condition §5.2 Outer walls No-slip wall
Body No-slip wall

Table 5-2 Overvievi/ computational settings

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University of Technology Delft & Bluewater Energy Services b.v.

6 FLUENT calculations: Influence computational settings

This chapter describes several computations with altered settings of Fluent to validate assumptions made in
the previous chapter. Most choices made to reach the computational model of the previous chapter were
based on literature research. This chapter investigates 5 altered input parameters. These are selected
because literature left uncertainties about the influence of these parameters on this particular problem.

The calculations described in this chapter are performed with a similar computational model as described in
the previous chapter, of course besides the altered settings. The body (keel and cylinder) is also oscillated
around one axis and the moment needed for this oscillation is recorded. From this moment-time graph, the
linear equivalent damping is calculated as will be explained in paragraph 6.3. The investigation of this
chapter is based on comparing these damping parameters. Unfortunately it is not possible to compare the
results with any kind of experiments; they are not available at this point. Also literature gives little indication
about forces on oscillating fences; they are mainly about stationary flows.

The altered computational settings are chosen to answer the following 5 questions:

1. What is the influence of the cell size

2. What is the influence of an altered turbulence model?
3. What is the influence of the interface between the stationary and rotating grid?
4. What is the influence of the initial start angle?
5. What is the influence of the free surface?

To make this investigation, 6 groups of computational runs were performed. The resulting damping and
CPU-time are compared to investigate the influence of the altered settings, see paragraph 6.4. An overview
of the 6 executed runs can be found in figure 6-1. The runs can be divided in two groups, based on the grid
size around the keel; three runs are performed on a coarse grid and three on a fine grid. The arrows given
in the figure are the only legitimate comparisons, comparing other results is not recommended because the
computational models differ at more than one setting. The numbers of this figure refer to the questions
given above.

Coarse grid
4 5
Opposite initial start angle Reference run, coarse grid Include free surface

ih_
1

Fitie grid
^'
3 2
Increased rotational domain Reference run, fine gird RSM turbulence model

Figure 6-1 Comparison diagram computational settings

42
 FLUENT calculations: influence computational settings

6.1 General setup

6.1.1 Computational domain

The basic lay-out of the computational model will be the same as described in the previous chapter. This
means rather academic representation of a bilge keel under a vessel; only the keel and the bilge are
modeled. All models of this chapter have the same general setup of the computational domain as given in
figure 6-2. This paragraph describes briefly what this model looks like; detail can be found in the paragraphs
of the previous chapter

The computational grid is divided in two main sections; a stationary outer part and rotational inner part. The
grid refinement around the keel is kept at the keel during the oscillation by rotating the inner grid with the
same rotational velocity as the keel. Cylinder and keel are modeled as a no-slip wall, see paragraph 5.2. The
outer domain is circular and is enclosed by an outer no-slip wall. The interface exchanges the flow
information (velocities, pressures, turbulence levels, etc.) between the two domains.

Figure 6-2 General setup of the computational domain

In this chapter the cylinder has a diameter of 30 [cm] and the keel has a length of 7.5 [cm]. These
dimensions are in consensus with models used during previous investigations at Bluewater. After the
computations of this chapter were performed, the setup of the experiments of this thesis was determined
and it was decided to use a larger body. Despite this smaller body, the results of this chapter give an
estimation of the consequences of the altered settings.

43
Verification ofCFD calculations witfi experiments on a rolling circular cylinder wiOi bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

6.1.2 Input parameters

To draw a conclusion on the effects of a certain altered setting, every setting is computed with 1 roll
amplitude and 4 frequencies. Decision on the values of these variables was made based on the expected
maximum capacities of the experimental set-up described in paragraph 8.1. All computational runs were
made with an amplitude of 10 degrees and a frequency range of co = 2, 4, 6, 8 [rad/s].

The time step in this chapter depended on the frequency involved. As can be read in the previous chapter
the time step depended on the rate of change of the rotational mesh between two succeeding time steps.
This means that a computational run with a lower frequency can be performed with a larger time step.
Preliminary computations revealed that the following time steps lead to the smallest CPU-demands: time
step = 0.01, 0.005, 0.003, and 0.002 [s] for the frequencies co = 2, 4, 6, and 8 [rad/s] respectively.
Additional time step considerations can be found in paragraph 4.5.1.

Other parameters like Under Relaxation factor, solution algorithm, discretization scheme etc. are chosen
based on literature research. Explanations of these parameters can be found in the chapter 5 and the initial
values can be found in table 5-1.

6.1.3 Hull motion

The motion of the body at the computational runs of this chapter is prescribed with the same kind of user-
defined function as described in paragraph 5.4. Based on the current flow time, this function gives the
rotational velocity of the body. This rotational velocity imposed on to the inner circular mesh, which gives
the same velocity to the keel and cylinder. This gives the advantage that possible grid refinements around
the keel tip stay around the keel tip during oscillation.

The motions described in this chapter have no start function; a cosine is applied to have a start at a zero
speed. The consequences of this rather fast start are detected during the computations of this chapter. The
use of a smooth start function as described in paragraph 5.4 is a recommendation based on this chapter's
experience. The frequencies and amplitude involved can be found in the previous paragraph. The general
motion with dimensionless roll amplitude can be found in figure 6-3.

Oscillation period: T

Figure 6-3 Hull motion

44
 FLUENT calculations: influence computational settings

6.1.4 Output

Fluent calculates the moment on the hull at every time step and writes these values in an external file. The
moment is calculated by integrating the pressure distribution along the hull. The external file is further
handled by MATLAB to process the data, see paragraph 6.3.

6.2 Investigated settings

An overview of the investigated setting can be found in table 6-1. Details can be found in the subsections of
this paragraph.

Fine grid Coarse grid

Case number 1 2 3 4 5 6
Increased rotational inner

Reference run, coarse

fine

Opposite initial start

Altered turbulence
model: RSM

=3
Vl
Reference run,
domain

angle
grid

grid
CI

1
Detail in paragraph: § 6.2.4 § 6.2.3 § 6.2.2 §6.2.1 §6.2.5 § 6.2.6

Motion Frequency: 2, 4, 6, 8 [rad/ s] at amplitude of 10 [deg]

Turbulence model K-£ RSM K-£ K-£ K-E K-E

No of grid cells 80.000 40.000 40.000 16.300 16.300 65.500

Initial roll angle 10 [deg] 10 [deg] 10 [deg] 10 [deg] - 10 [deg] 10 [deg]

Diameter inner mesh 2.1 [m] 0.7 [m] 0.7 [m] 0.7 [m] 0.7 [m] 0.7 [m]

Free Surface No No No No No Yes, VOF

Table 6-1 Overview investigation computational setting

45
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University of Technology Delft & Bluewater Energy Services b.v.

6.2.1 Reference run, coarse grid

Figure 6-4 gives the computational domain of the reference run with a coarse grid. The left figure gives an
impression of the hull, keel and the rotational inner mesh; the right is a detail of the grid cells around the
keel. All input parameters are identical to the standard model given in paragraph 5.7.

Figure 6-4 Computational domain reference run with coarse grid

All flow phenomena take place around the keel and this is therefore the only region that has to be modeled
with care. Transitions to adjacent regions are as smooth as possible; relative difference of cell area of two
adjacent cells should be small, see paragraph 5.1. Especially on the edge between the structured and
unstructured grid of the inner rotational part, this factor is exceeded. Investigation at this location show that
the pressure and velocity gradients at these locations are small; no numerical problems are expected.

6.2.2 Reference run, fine grid

The computations of this run are used as a reference are for the altered turbulence model and altered
rotational domain. These three models have in common that the refinement around the keel is identical. The
run with the RSM model has an identical mesh; the run with the altered rotational domain has a larger inner
mesh than presented in this paragraph. An impression of this fine grid around the body can be found in
figure 6-5.

Figure 6-5 Computational domain reference run with fine grid

46
 FLUENT calculations: influence computational settings

These grid refinements investigate if cell size has influence on the results. Ideally the mesh should have no
influence because it has nothing to do with the flow behavior itself. A common recommendation with CFD
computations is that grid should be fine enough to describe the flow gradients sufficiently. This implies that
it is an iterative process to find the optimum grid size; a computational run is demanded to compute the
velocity and pressure gradients but for this run a computational grid is already needed. It is also uncertain
when a gradient is "sufficiently" taken into account.

To overcome these uncertainties, the fine grid described in this paragraph is compared with the coarse grid
of the previous paragraph. When a large difference between these two runs is found, the computational grid
is of influence and thorough investigation is demanded to establish what mesh describes the true flow
behavior.

6.2.3 Altered turbulence model: RSM

The computational model of these runs is identical to the reference run with a fine grid of paragraph 6.2.2,
only the turbulence model is changed. Instead to the standard K- e model the more sophisticated Reynolds
Stress Model is used. Details of this model and the difference with K- e are described in paragraph 3.5.2.
Chapter 3 describes also the K-CÜ model; this model is not tested in this thesis. Due to the demanded CPU-
time per computation, it is decided to investigate only one altered turbulence model. RSM is the most
sophisticated model available in Fluent and is therefore chosen for this comparison.

The RSM model is expected to be more sensitive to initial conditions than the K- e model. Poor initial guesses
will lead to poor convergence behavior or even instability. Therefore the computations are started with a K- £
model and when periodicity is reached the computations are continued with RSM.

RSM turbulence models are expected to have better performance when the flow contains regions with
anisotropic, high velocity gradients. This behavior can be expected around the keel. When there is a large
difference found between the RSM and K- E model it is not said that the RSM gives the best approximation of
the true flow behavior. The RSM model has its own specific shortcomings and is not as thoroughly validated
as the K- £ model. When large differences are found in the results of this comparison, at least one of the
models is not suitable for these kinds of flows around keels and verification experiments are demanded to
detect what model performs best.

6.2.4 Increased rotational inner grid

These computations are added to the investigation because the influence of the interface on the results was
uncertain. There was not much experience with a combined rotating and stationary grid. Investigations at
the interface edge of the behavior of variables like velocities and pressure showed that the transition across
the interface was not entirely smooth.

47
Verification ofCFD calcu lations with experiments on a rolling circu lar cylinder with bilge keels in a free su rface.
University of Technology Delft & B luewater Energy Services b.v.

The size of the inner mesh used in this subsection is based

on the premise that all relevant fluid motions should stay
inside this mesh, see figure 6-6. This would mean that
possible errors due to information exchange across the
interface will have no influence on the results. On both
sides of the interface the fluid will be at rest, leading
inevitably to a smooth transition to the outer stationary
domain. Although the fluid in the outer domain are
everywhere at rest, this domain is demanded to prevent
computational instabilities.

Figure 6-6 Computational domain with enlarged inner grid

When a significant difference between these runs and the reference runs is found, the interface has
influence on the solution. This would mean that it is advised not to use an interface and it thus becomes
impossible to have an oscillating keel with the aid of those 2 domains and an interface. This would mean
that alternatives have to be found to simulate oscillation of the body.

6.2.5 Opposite initial start angle

During the preliminary computations an asymmetric behavior of the force on the keel was detected. Figure
6-7 gives the recorded moment on the body, 20 periods after the computation was started. B oth trend and
maximum at the negative roll angles are not identical to those at the positive roll angles. The applied motion
is a pure cosine and thus a symmetric behavior of the force on the keel is expected.

2. S T
]
1 Moment
^.-••" • - . / ^ " \ i J Roll motbn
,.• ^'::::.. N.
' .'■■ /
"■-. ' \

1
_/ 1
. . 0 / 0.?5 0.5"-, \ 0>5 .--'l

-1 y *•.
' " • ■
"^~-\.-'''
- ■ " ■ ^ ^ -
!
^

2.S
i

Tme t/T [-]

Figure 6-7 Asymmetric behavior of keel force

The only asymmetric part during these computational runs is the initial start angle. The motion is from the
start a cosine thus a start with zero velocity at a maximum roll angle. It is expected that the start at one side
is "remembered" by subsequent oscillation periods. The computations of this subsection are performed to
investigate if the same phenomenon occurs when the calculations are started with an opposite initial roll
angle.

48
 FLUENT calculations: influence computational settings

When the results of these runs are the identical to the reference runs, it can be concluded that the
asymmetric behavior is not caused by the initial angle and further investigation or comparison with the
experiments is demanded. When the trends of the results are similar but mirrored in the horizontal x-axis of
figure 6-7, the initial angle is the cause and a start function is demanded. A start function will gradually
increase the roll amplitude during the first periods to diminish the effects of start errors on the subsequent
periods.

6.2.6 Including free surface

A free surface demands considerably more grid cell to describe the radiated waves sufficiently accurately.
There will be large CPU advantages when it is not needed to include a free surface in the computational
domain.

Figure 6-8 Computational domain including free surface

The cross section of the body is circular, which should lead to little interference with the free surface;
radiated waves are expected to be small. Radiation of waves is a way of dissipating energy and will
therefore increase the damping capacity of the system. When the waves are small, it can be expected that
the influence of wave radiation has no significant influence on the total damping and there is no point in
incorporating a free surface in the computational model.

Besides to the omission of radiated waves, not incorporating a free surface possibly leads to a change of
flow behavior around the keel. In other words it is possible that the flow around the keel "feels" the
presence of the free surface nearby. When the flow behavior changes due to the presence of a free surface
it will be hard to compare the computations with the experiments where the free surface is inevitably
present.

The large number of grid cells is needed because of the motion of the body; as can be seen in figure 6-8, for
an area twice the roll amplitude, grid refinement is required to keep a fine grid at the free surface during
oscillation. Outside the rotational mesh, the grid is still fine at the free surface but this area is thin and
consequently does not need many cells.

A last remark on the height of the cells at the free surface; in paragraph 4.2 it is stated that a wave should
at least be described with 10 cells in vertical direction. This would mean that with an expected wave height
of 3-5 [mm] the maximum cell height should be around 0.4 [mm]. Unfortunately when the computational
runs of this chapter were performed, the wave height was expected to be around 10-20 [mm] and therefore
a cell height of 1.05 [mm] is used with these runs. The final model used for the verification computations of
chapter 9 needs considerable more cells at the free surface to describe the small waves accurately.

49
Verification of CFD calculations with experimen ts on a rolling circular cylin der with bilge keels in a free surfyce.
University of Technology Delft & Bluewater Energy Services b.v.

6.3 Post processing

A typical example of a graph showing the recorded moment on the body can be found in figure 6-9. A non
periodic part can be found during the first 2-3 periods of the computations. This number depends on the
frequency used and sometimes also on the computational settings.

Flow time [s]

Figure 6-9 Moment - time graph

To make an overview of the influences of the altered input parameters, the damping this system generates
will be compared. Damping is a combination of the generated force on keel and the phase shift between this
force and the roll motion. A method to compute this damping is proposed as can be found in [1]. The
method is based on the assumption that the net work done by the oscillation can be related to a so-called
equivalent linear damping (Bei). Computations uses the recorded moment on the hull (M(t)), the roll velocity
(cp'(t)), the period (T), frequency (co) and the roll amplitude (cpa). This equivalent damping reads as:

2^{M{t)*0{t))dt
(6-1) fiw = [Nms/rad]
T*o/*0'^

With the aid of MATLAB this damping is computed from the output files of Fluent. The damping can be
computed for every period, which yields figure 6-10. This figure gives the linear equivalent damping of the
reference run with a coarse grid; all other runs show similar graphs.
5
4.5
4 O O □ O -B—a- D o D D -B—e —B—frequency = 8
[rad/s]

'^"o-(,.,,-'^"^"^'^-^-oo-o-o-(^-^o o o ■0-- frequency = 6

[rad/s]

^—X—X—X—><—X—X—X—^—X—X—K—X—X - X — frequency = 4
[rad/s]

4--H !-.■ + ■-H y-~\ h--+-- + - - - ( - - + - - + - - H 1 h-- + -4- - - frequency = 2

[rad/s]

10
No. periods

Figure 6-10 Linear Equivalent Damping (Bei) of reference runs with a coarse grid

50
 FLUENT calculations: influence computational settings

To make an overview of the computations the average value of calculated Bei values is compared. Not all
computational runs had the same length, leading to different numbers of periods to base the average on.
This could lead to small differences. The first recorded period is left out of the average because it differs
substantially from the eventually constant value. This deviation would have a large influence if a small
number of periods are included and no influence if a long term computations was performed. Results of the
MATLAB computations can be found in the next paragraph.

6.4 Results

The overview of table 6-2 is the result all involved computational runs. All runs are processed as described in
the previous paragraph. The results are presented as percentage of the reference run with a coarse grid.

One remark has to be made on the CPU-times given in this table; those values are added to the table just to
give an impression of the price that has to be paid for a certain altered setting. They are average values for
the computer systems available at the laboratory for Aero and Hydro Dynamics, which are standard Pentium
2000 MHz processors with 2 GB ram memory. The absolute values will be different when other systems are
used.

Fine grid Coarse grid

Case number 1 2 3 4 5 6

Reference run, coarse grid

Opposite initial start angle

Altered turbulence model:
Increased rotational inner

fine grid

Include Free surface

RSM
grid

Reference run,

No periods included average 7 12 7 66 30 23

2 100.0 95.9 100.0 100 99.0 106.2

Linear equivalent damping:
4 100.5 95.3 99.0 100 100.0 103.1

[Percentage of reference 6 100.3 95.2 99.7 100 99.3 84.0

computation witti coarse grid]
8 99.0 95.5 102.6 100 100.0 80.6

CPU time:
average 7.1 6.0 3.8 1.1 1.1 3.93
[hour / period]

Table 6-2 Results computational settings comparison

Influence grid size

The first research question of this chapter was if the grid size has a significant influence on the results, in
this case the damping. When comparing case number 3 and 4 of table 6-2, it appears that the differences
are small and the offset is not constant or linear. The runs on a fine grid do not contain many periods, which
could be a reason for the small fluctuations in the offset. Especially when the 3.5 times larger CPU demand
is taken into account, it appears not useful to incorporate a fine grid.

51
Verification ofCFD calculations with experiments on a rolling circular cyliixier with bilge keels in a free surface.
University of Teclinology Delft & Bluewater Energy Services b.v.

Influence altered turbulence model: RSM

The second question was to look if another turbulence model would give other results. Turbulence models
aim to incorporate the behavior of the small eddies and should therefore give the same results for a certain
flow computation. To answer this question, case numbers 2 and 3 have to be compared; a constant offset of
around 5 percent is found. This difference is significant and makes the use of the standard K-£ model
doubtful. It should not be concluded that the RSM performs better than the K-E model. Both K-E and RSM
model have their specific flow types where they perform poorly. In this thesis it is chosen to use the K-E
model based on the CPU advances, but it is left as a strong recommendation to investigate the RSM model
with this kind of flows.

Enlarged rotational inner grid

The third question is about the influence of the interface between the inner stationary mesh and the outer
rotating mesh. Comparing case 1 and 3 with each other, it can be concluded that the influence is negligible.
This means that the interface can be used safely. This means that a motion of cylinder and keel can be
incorporated in this thesis without further investigation.

Influence initial start angle

Asymmetric behavior of the moment-time graph leads to the fourth question. When starting with an initial
angle of -10 [deg] in stead of +10 [deg] the moment-time graph is identical but mirrored in the horizontal
axis. This means that .the asymmetric behavior is purely caused by the start angle. The asymmetric start
initiates an error that is "remembered" by the subsequent periods. Comparing case 4 and 5 of table 6-2 with
each other shows no significant influence on the damping. Based on this comparison it is concluded that a
start function is demanded that gradually increases the roll amplitude. The use of a start function will not
lead to other damping values but it will decrease the initial error and therefore give more realistic moment
behavior. ^ .:~:

Influence free surface

The fifth and last question of this chapter involved the influence of a free surface. It was expected that the
radiated waves were small due to the circular cross section and thus the influence of a free surface would be
negligible. When case 4 and 6 of table 6-2 are compared, it can be concluded that the presence of a free
surface has a significant influence on the damping. The offset between these two cases is not constant
which makes it impossible to compare the computations without free surface with the experiments; both
trend and absolute values will be different. When the body can radiate waves, it is expected that the
damping increases; the system has an additional way of dissipating energy. Table 6-2 shows the opposite
for the higher frequencies. This can be explained by the changed flow behavior due to a free surface;
apparently the flow around the keel "feels" the presence of a free surface nearby. The influence of the free
surface can increase the damping (at low frequencies) or decrease the damping capacities of the system (at
high frequencies). It has to be concluded that the free surface has to be incorporated in the computational
model, despite the circular shape of the model.

52
 FLUENT calculations: Investigation keel shapes

7 Fluent calculations: investigation keel shapes

This chapter describes Fluent calculations performed to investigate the effects of unusual keel shapes.
Unusual is defined as different from the normal flat plates (or profiles) that are currently used as bilge keels
under vessels. This chapter compares the damping capacities of those shapes and explains the flow behavior
due to the oscillating motion of the keel. The results are used to decide what keel shapes will be tested
during the experiments, which can be found in the next chapter.

This entire investigation is based on Fluent calculations; no reference to literature or experiments will be
made. The used computational model will be identical to the one described in the previous chapter as
"reference keel with coarse grid", see paragraph 6.2.1; this time with different keels attached to the
cylinder. Every keel in this chapter is oscillated at one frequency and one amplitude.

The conclusions from this chapter will be based the damping capacity of the system and a visual
presentation of the velocity, pressure and turbulence fields. Comparison of the damping will show if damping
can be improved when unusual keel shapes are applied. The visual presentations will give more insight in
the reason why a particular shape has better damping capacities than others.

7.1 Set up

The computational domain used in this chapter is identical to the one described in paragraph 0. This means
two separate meshes: a rotational inner mesh and a stationary outer mesh, both circular. The mesh will
have the same coarse grid as described in paragraph 6.2.1 and a small inner rotational part. No free surface
will be included and a K-E turbulence model is used in order to decrease the CPU-time. The motion of the
body is identical to the one given in paragraph 6.1.3.

All computational runs of this chapter will be at one frequency of 8 [rad/s] and one amplitude of 10 [deg].
Paragraph 6.1.2 gives a recommended time step for runs at this frequency: 0.002 [s]. All other input
parameters and initial conditions can be found in table 5-2.

7.2 Investigated keel shapes

There are virtually unlimited options to shape a keel that could perform better than a flat plate. CPU-times
with this kind of flow computations dictate that these options have to be limited. In this thesis it is chosen to
select shapes based on their expectations to move as much water forward as possible; when a large amount
of water is pushed forward, the pressure on the suction side of the keel will be low and the generated vortex
will be strong. Strong vortices will dissipate more energy and will thus give more damping to the system.

The thickness and sharpness of the keel is not investigated in this thesis. With a length/thickness ratio of 20,
the keels are considered thin and sharp. Further decrease in thickness of the keel tip might improve the
damping on model scale but the effect on full scale is not expected to be significant. Also variances in the
third dimension like holes in the keel or waved keel edges won't be investigated. The CPU demand of the
current 2D flow problem is already large; 3D computations are simply impossible within the available time.

53
Verification ofCFD calculations with experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

The height of current bilge keels under vessels is limited by the premise that they have to stay inside the
outer rectangle of the cross section, see paragraph 2.3. Leaving this premise in real life would lead to
several problems when the vessel is docked or moored to a quay. With the computational runs of this
chapter the reference keel is the only one that is in line with this premise; the other keels have the same
total height as this reference keel. This gives the altered shapes a fair chance; a smaller frontal area will
inevitably lead to less damping capacities, regardless the keel shape.

Selected for the investigation of this chapter are T, Y and Circle shaped keels. The T and Y shapes are
expected to have an optimal damping performance at a certain size or angle of the top flange; therefore
they will be tested with several flange options. The Circle shaped keel is selected because it is the shape
with the highest drag coefficient in stationary flows and might therefore also perform well in these oscillating
conditions. These considerations lead to three basic keels, given in figure 7-1.

B * 0.075

r4-i 1
J j

i
0.008 0.075

^^---^ 1 f

Figure 7-1 Investigated keel shapes, a: y-shape, b: t-shape, c: circle shape

The following 7 variances of the basic keels are selected for the investigation of this chapter:

1. Y-shape, 8 = 30 [deg]
2. Y-shape, 9 = 60 [deg]
3. Y-shape, 0 = 90 [deg]
4. T-shape, p = 1
5. T-shape, 3 = 0.75
6. T-shape, p = 0.5
7. Circle shaped

These seven keel shape options will be compared with a reference keel. This keel is identical to the one used
in the previous chapter, see paragraph 6.1.1 . It consists of a normal flat plate with a thickness of 8 [mm]
and a length of 75 [cm] and will further be referred to as "reference keel".

54
 FLUENT calculations; investigation keel shapes

7.3 Post processing

The output of all computations is a file containing the moment on the cylinder and keel at every time step.
MATLAB is used to compute the linear equivalent damping from these file, this procedure is described in
paragraph 6.3.

Figure 7-2 is an example of an output file from this keel shape comparison, it gives the moment on the body
and the roll angle. This figure gives the graphs of the reference keel and the circle shaped keel; graphs of
the other keels are similar. First it can be noticed that the circle shaped keel generates a considerably higher
moment in comparison with the reference keel. It can also be seen that the phase shift of the reference keel
is larger than the shift of the circle shaped keel. The phase shift can be regarded as the time between the
maximum value of the roll motion and the maximum moment. The phase shift will be given in degrees
instead of seconds, where one period corresponds to 360 degrees; this makes it possible to compare runs at
different frequencies.

Time [t/T]

Figure 7-2 l^oment behavior altered keel shapes

A large damping capacity of a keel is reached when it generates a large moment in combination with a large
phase shift. This makes it not straight forward to detect the best performing keel shape directly from these
kinds of graphs. Therefore the equivalent linear damping is computed to find the best keel shape.

55
Verification ofCFD calculations witii experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

7.4 Results

7.4.1 Equivalent linear damping

With the method described in the paragraph 6.3 the linear equivalent damping is calculated. In figure 7-3 an
overview of the comparison is given. In this figure the results are given as a percentage of the damping of
the reference keel. This gives a quick overview of the performance of the keel shapes; the absolute values
of the damping are not given in this comparison.

Equivalent linear clamping

100 _

Ref. Keel l.Y-shape, 2. Y-shape, 3.Y-shape, 4. T-shape, 5. T-shape, 6. T-shape, 7. Two circles
e = 30 [deg] e = 60 [deg] 6 = 90 [deg] f=i. p = 0.75 f = 0.5

Figure 7-3 Result keel shape comparison

The first thing to notice is that there is no keel shape in this investigation that performs better than the
reference keel. Under these circumstances and with this model lay-out, the normal flat plate has the best
capacities to reduce the roll motion.

This investigation shows the importance of the phase shift between the roll motion and the generated keel
force. In figure 7-2 it can be seen that the circle shaped keel generates a higher keel force that the
reference keel, but figure 7-3 shows that the damping of the circle shaped keel is 21.7% lower than the
reference keel.

The Y-shapes have a minimum damping when the top-bars have an angle of 60 degrees. It was expected
that a certain angle has better capabilities to push the water outward and thereby producing more damping.
With a maximum damping of 68.6% of the reference keel, it can be concluded that Y-shapes do not improve
the damping. The T-shapes give an even lower damping, the maximum damping is half the damping of the
reference keel. In the following subparagraphs, this decrease will be explained.

A last remark should be made on the height of the tested keels, all altered keel shapes stick outside the
rectangle around the cross section of the vessel. The reference keel is the only keel that stays inside this
rectangle. When the height of the altered keels is decreased to be able to stay inside this rectangle, their
damping performances will decrease, making the reference keel even more superior.

56
 FLUENT calculations: investigation keel shapes

7.4.2 Velocity field

This paragraph investigates the influence of a particular keel shape on the velocity field around the keel.
This will bring more understanding to the question why a certain keel shape performs better than another.

Figure 7-4 and figure 7-5 give four velocity fields; these are selected to explain the trend in the resulting
flow velocities. The conclusions drawn in this paragraph also apply to the plots of the other keel shapes. The
arrows in the plots give the direction of the flow and the length the local velocity. In all plots the keel moves
from the left to the right. These plots are given to show the difference between the keel shapes; the
absolute values are of less concern. For an indication: the small blue arrows visible represent a velocity of
0.05 [m/s] and the largest yellow arrow represent a velocity around 0.7 [m/s].

Every plot is zoomed in on the flow around the keel; the underside of the cylinder (1) and the interface (2)
can be seen. The interface is the line between the stationary and rotational mesh; it is the black circle at the
underside of every plot. One has to be careful with comparing the plots; the scales of the plots can be a little
different.

Figure 7-4 Velocity field of reference keel and option 1: y-shape 30 [deg]

Figure 7-5 Velocity field of option 3: y-sliape 90 [deg] and option 7: circle shaped

Looking at the graphs of figure 7-4 and figure 7-4 the first thing noticed is the stronger and more profound
vortex at the reference keel compared to the other keels. This can be seen by the higher, fully rotating
velocities. At all other keels the vortices are more spread out and considerably weaker. Also the influence of
the vortex of the reference keel on the surrounding water seems to be more spread out. When the velocities
around the interface are compared, it is clear that the velocities are larger in case of the reference keel.

57
Verification of CFD calculations with experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

When option 1 and 3 are compared it looks as if the vortices become weaker when the angle between the
flange sides increases. Or more general when the keel consists of two or more tips with a certain distance in
between, the vortex will be spread out and consequently be weaker than when the keel consists of one
single tip. This statement is confirmed by the circle shaped keel.

7.4.3 Pressure field

This paragraph gives an indication of the consequences of a certain keel shape on the pressure distribution.
Figure 7-6 gives the keels at a motion from the left to the right with a pressure in Pascal. An indication of
the pressures in the figures; green represents a pressure of zero Pascal, the darkest blue represents -350
Pascal and light red +300 Pascal. These numbers are taken relative to the operation pressure of the
computational model, which was set at zero Pascal and explains the negative values. Away from the keel the
pressure is zero everywhere due to the absence of gravity in these computations.

The left plot of the reference keel shows a gap in the pressure field. The pressure in this gab is outside the
indicator limits of the plots; it is estimated around -750 Pascal. The limits are not altered to keep the same
outer limits for both plots and thus the same colors for the same pressures. With an indicator limit of -750
Pascal, the pressure differences of the right plot won't be visible anymore.

Figure 7-6 Pressure field of the reference keel and option 3: y-shape 90 [deg]

The two plots of figure show similar trends as the velocity fields of the previous section; the reference keel
induces a stronger vortex. This stronger vortex means in this case a lower pressure at the location of the
vortex. It is also clear from the plots that the pressure on the whole suction side of the keel is considerably
lower in case of the reference keel. This will lead to larger forces on the keel.

58
 FLUENT calculations: Investigation keel shapes

7.4.4 Turbulence field

This last comparison is to show the consequences of the vortices on the amount of generated turbulence.
Turbulence is a way to dissipate energy; more generated turbulence means more energy dissipation en thus
a better damping of the system.

Figure 7-7 shows similar trends as the other keel shapes. The turbulence is presented with the aid of a
turbulent kinetic energy, which is given in [mVs^]. In the figures, blue represents a value of 0 and yellow
represents 0.04 [m^/s^].

Figure 7-7 Turbulence field of the reference keel and the circle shaped keel

It can be noticed that stronger vortices have no significant influence on the maximum intensity of the
generated turbulence; both plots give more or less the same maximum value. But energy dissipation is all
about the amount of turbulence a keel can generate. Then the superiority of the reference keel becomes
clear; the amount of turbulence is much larger. In case of the reference keel most of turbulence can be
found outside the interface, while the turbulence of the circle shaped keel does not even reach the interface.

7.5 Conclusions and recommendations

Although these computations are rather academic and based on a computer model, it can be concluded that
there is no "exotic" keel shape that is expected to generate more damping than a normal flat plate. The
relation between these calculations and a full size vessel should be treated with care. The cross sections
used in this comparison are significantly different from those of a normal vessel, which could lead to
different performances of a certain keel shape. Also scale effects between the tested model and full scale
vessels could lead to other optimum keel shapes.

The superior performance of the flat plate is due to its ability to generate stronger and more profound
vortices. These strong vortices lead to a lower pressure at the suction side of the keel, which results in more
drag force on the keel. Strong vortices lead to more turbulence; more turbulence means more energy
dissipation and thus more damping.

It is now necessary to select a keel shape to be tested during the experiments described in chapter 8. The
experiments of this thesis are meant to verify the computer model rather than to find the best keel shape. In
this light it seems appropriate to select besides the reference keel also a keel with different flow phenomena
and damping performances compared to the reference keel. This would make the verification more reliable
for other non-verified keel options. Therefore the circle shaped keel is selected, because the moment graph
and velocity field are rather different from the reference keel. It is expected that those differences can be
seen in the experiments as well, which would make a solid premise that the computer model is reliable for
these kind of computations.

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Verification of CFD calculations with experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

8 Experiments

The experiments described in this chapter are performed to verify the CFD calculations performed with the
computational model of chapter 5. The selected experimental set up will give a first insight of the current
CFD performances with these kinds of computations on a rolling, moored vessel.

The model used in these experiments is a cylindrical tube with one keel attached in the centre on the
underside of that tube, which is in line with the basic setup given in paragraph 2.4. The model is oscillated
around the central x-axis of the tube leading to a pure roll motion. The tube is placed half-submerged and
crosswise in the test basin of the laboratory of ship hydromechanics of the TU-Delft. The circular tube
represents the bilge of an FPSO. The round shape of the tube gives a minimum interference with the free
surface. The cross section of the tube and keel will be constant along the model, which makes these
experiments nearly 2D. Prime interest in these experiments are the forces on the keel, therefore these
forces are measured separately. The radiated waves are measured and the flow behavior is visually recorded
by a camera.

This chapter starts with describing the lay-out of the experiments and the executed test program, then
paragraph 8.2 will describe the post-processing of the measured data. The achieved results will be
presented in paragraph 8.3.

8.1 Setup

8.1.1 Sign conventions

The sign conventions given in this paragraph are valid throughout the remainder of this report. All sign and
motion conventions are taken in line with to the normal conventions in marine technology.

Figure 8-1 Sign conventions

The axis is earth bound with the origin in the centre of rotation at one side of the model. The axis definition
can be found in figure 8-1, a clear indication on which side of the basin the origin is located can be found in
figure 8-2. The model is hinged in such way that it can perform pure roll only, all other motions are
constrained. The forces in heave and sway direction are measured, for that reason those two directions are
mentioned in figure 8-1. As it is a pure 2D experiment, no motion or forces are expected in the x-direction
and are left out this sign convention.

60
 Experiments

8.1.2 Lay out

The experiments were performed in towing tank No 2 of the laboratory of ship hydromechanics at TU-Delft.
This tank's main dimension are 70 * 2.73 [m] with a water depth of 1.19 [m] (see figure 8-2). On one side
there is a "beach" to reduce reflection of waves, on the other side there is a wave maker, this last device
won't be used during these experiments. Every run is stopped before reflecting waves from the wave maker
side reach the model, this prevents interference of those waves with the measurements.

wave maker

*^«ter depth: i . i g
m

Figure 8-2 Location of tlie model in the tank

The model was placed in the middle of the tank, just in front of the windows to make it possible to record
the flow behavior visually, figure 8-3 gives an overview of the model. The outer wall of the cylinder is
constructed from a PVC tube, closed by two wooden endplates. In the middle of the tube, on the underside,
three keel parts are attached; two static dummy parts to make the flow 2D and one section to measure the
perpendicular keel force. All equipment like hinges and force receptors inside the model are supported by
PVC half-circles (see figure 8-3: "PVC-support"). The lead ballast is distributed evenly through the model and
is supported by yellow foam (see figure 8-3: "Foam support"). To be able to make a visual recording of the
vortices around the keel an ink injection is applied with its release point at the tip of one of the dummy
parts. The radiated waves are recorded with a standard wave height gauge, located at the beach side of the
model.
Foam support,
PVC support

\ Lead ballast

Hinge and force

receptors
Dummy part
PVC tube /
Measured section Dummy part
End plates
Keel

Figure 8-3 Overview model

All details of the construction of the hull and keels, force receptors, ballast, wave height registration,
oscillator, data acquisition and hinging of the model can be found in appendix D.

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University of Teclinology Delft & Bluewater Energy Services b.v.

8.1.3 Tested keels

The following four keel shapes are selected to be tested in during these experiments:
1. Reference keel, this keel will be a normal flat thin plate, (size: Hbk == 95 [mm], t = 3 [mm])
2. Small keel, this keel will give information on the ability of Fluent to compute small differences in keel
size and it will be used to investigate the influence of keel height and the damping characteristics,
(size: Hbk = 78 [mm], t = 2.5 [mm])
3. "Saw-blade" shaped keel, this keel is expected to produce more vortices due to the changing keel
height, (size: Hbk = 104 [mm], t = 2.5 [mm])
4. "Academic" keel: Selection of this keel will be based on distinct force and flow behavior compared to
the reference keel. Based on the conclusions of chapter 7, the circle shaped keel is chosen for this
purpose, (sizes: Hbk = 91 [mm], t = 4.5 [mm])

The height of the reference keel is based on the Hbk/Rbiige ratio at a full size FPSO, or in other words the
whole keel is inside the outer rectangle of the cross section, see paragraph 2.3. The height of saw blade keel
is not based on the same premises as the reference keel. To make a comparison between the reference keel
and this saw-blade keel it was necessary to keep the frontal area of the keels equal and thus a larger
maximum Hbk. The height of the circle shaped keel is based on an imaginary circle trough the tip of the
reference keel and a midpoint at the centre of rotation. The circle shaped keel stays inside this imaginary
circle. This way this circle shaped keel will not stay inside the outer rectangle of vessels' cross section, like
the reference keel does. The computations of chapter 7 indicate that shrinking this keel until it stays inside
this rectangle will make its performance even worse compared to the reference keel.

PVC part

Figure 8-4 Overview tested keels

The thickness of the keels depends on the height of these keels; the height/thickness ratio must be above
20. This ratio will ensure that the keels are regarded as flat plates. As long as this ratio is preserved, the
thickness of the keels depended on the available stainless steel plates in the laboratory. The circle shaped
keel is made of PVC tubes and the height/thickness ratio is of less concern. The PVC part has a thickness of
4.5 [mm], which was also determined by the available material.

62
 Experiments

8.1.4 Test program

The test program of the experiments can be divided in two main parts: one part consists of the actual test
runs that will be compared with the CFD computations; the other part is executed to support this
comparison. The latter runs are demanded to compute values like mass of the model, check if the
experiments give the same results when repeated or check the calibration of the sensors. All executed tests
runs can be found in table 8-1.

Verification test runs Support test runs

0.6 0.4 0.2 0.8 0.5 Static Clieck Static Visual

Test group Total
[Hz] [Hz] [Hz] [Hz] [Hz] zero reFieatability inclination registration

Dry oscillation - 10 10 - - - - 20

Reference Iceel I l l - - 47 7 14 - 89

Small keel 1 1 1 - - 32 3 12 9 77

Saw blade 1 1 1 - - 24 - 17 4 66

Circle sliaped
1 1 1 - 28 13 12 8 82
keel

Bare hull 1 1 1 - 5 - 10 - 36

Total no of runs 35 35 35 10 35 136 23 65 21 370

Table 8-1 Overview test runs

The 7 runs per frequency of table 8-1, refer to 7 tested roll amplitudes, being: 4, 6, 8, 10, 12, 14 and 16
[deg]. The maximum amplitude was based on the physical maximum roll angle of the model. The minimum
is also based on the minimum Reynolds number. The number of amplitudes was limited by the total
available tank time.
The bare hull runs were used to determine the solid mass moment of inertia of the model, as described in
paragraph 8.1.6. All "static zero" runs were performed prior to every oscillation to determine the orientation
of the keel under the model and check the calibration of all sensors. The latter check was demanded
because of wringing of the model, this made the force receptors deviate from their target value of zero. The
"static inclination" runs were performed to determine the location of the COG of the model and the weight of
the keel also described in paragraph 8.1.6 .

The selected frequencies depend on both the limits of the model and a minimum Reynolds number. The
maximum frequency depends on the strength of the force sensors that registered the applied moment. The
minimum frequency depends on the premise to keep the Reynolds number above 10.000 [-], see paragraph
2.3.

Several runs were performed twice to determine the repeatability of the experiments. The number of visual
registrations is limited due to the time-consuming post-analyses. These visually recorded runs provide a
number of additive test runs to check the repeatability of the experiments.

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Verification ofCFD calcuiations with experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

8.1.5 Measured parameters

During the experiments, 8 forces, 1 roll motion and the radiated waves are measured. The wave heights will
be treated further in paragraph 8.2.4. This leaves the following 9 measured signals that will discussed in this
paragraph:

Fkeei,r(t) Force on keel, right strain gauge.

Fkeei,i(t) Force on keel, left strain gauge.
FYRCI) Force in y-direction on total model, right strain gauge.
FyiCt) Force in y-direction on total model, left strain gauge.
FzR(t) Force in z-direction on total model, right strain gauge.
FzL(t) Force in z-direction on total model, left strain gauge.
Fmi(t) Force in pure normal stress gauge for applied moment, aft.
Fm2(t) Force in pure normal stress gauge for applied moment, front.
cD(t) Roll motion, measured by an inclinometer and a pot-meter

The roll motion was recorded by an inclinometer and a pot-meter, the latter is part of the control system of
the oscillator. Due to technical characteristics of the inclinometer and the hydraulic system, it is decided to
use the roll angles registered by the inclinometer and the phase shift of the pot-meter. The technical details
where this choice is based on can be found in appendix D. This combination of the two signals is taken when
following computations refer to the roll motion (<t»(t)).

An overview of the forces on the model can be found in figure 8-5. Every force is measured on two
locations; the indicators left and right are defined as standing on the beach and looking to the wave maker
of the test basin, see figure 8-2. Indicators Aft and Front are defined standing on the beach as well. One
remark on the registered forces keel forces (Fkeei): the direction of those forces rotates with a model-bound
axis, see paragraph 8.2.3.

Figure 8-5 Measured forces and motion

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 Experiments

As the model is a prismatic cylinder all

phenomena are expected to be 2D and the
problem will be treated as 2D further in this
report. There are no forces expected in the x-
direction and no other motions than the roll
Surface elevation
motion around the x-axis. Waves are expected A
Waterline
to be constant along the width of the basin.
Any effects of the sides of the basin are
expected to be negligible. The demanded 2D
forces and main dimensions can be found in
figure 8-6. The roll angle is positive in the
given direction. Definition of the surface
elevation is also given in this figure. Change
of the centre of buoyancy (due to the
presence of the keel) during to the roll motion
is neglected.

Figure 8-6 Force equilibrium 2D

The measurements of the experiments have to be translated to 2D values for comparison with 2D CFD
calculations. The following three forces and one moment will be compared with CFD:
1. Force in y-direction
2. Force in z-direction
3. Applied moment
4. Perpendicular force on the keel

These forces and moment can be computed from the 8 measured forces as follows (conventions and
parameters, see figure 8-6):

Force in y-direction:
(8-1) F^ (0 = F^ it) + F^, (0 - {F,„, (0 + F,„2 (0} ■ sin(^(f))
Force in z-direction:

(8-2) F, it) = Fy, (t) + Fy, (t) + {F,„, (0 + F„,2 (0} • cos(0(O)
Applied moment:
(8-3) F,it) = r^-{F„,(t)-F„„{t)}

Perpendicular keel force:

65
Verification of CFD calculations with experiments on a rolling circular cylinder with bilge keels In a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

8.1.6 Determination of mass properties experimental model

This paragraph describes what method is used to gather the demanded mass properties of the model. These
properties are needed to subtract the mass of the experimental model from the measured signals in order to
achieve the hydrodynamic coefficients, see paragraph 8.2.1.

The weight of the total model including ballast could not be measured directly due to the used strain gauges
in z-direction; these were too fragile to support the total weight. The mass of the empty model and the lead
ballast blocks are weighed separately; summation gives an accurate estimation of the total mass.

To determine the distance rg, from COG to the roll centre, static inclination test were performed, see table
8-1. Together with the known total mass and applied moment, the value of rg can be determined. These
calculations were performed for the total model and the keel-part separately. A full review of the static
inclination tests can be found in appendix E.

The whole model is drawn in the 3D CAD program Rhinoceros, small adjustments were made to meet the
measured values of mass and rg. With a realistic distribution of the components in the model. Rhinoceros
was able to calculate the solid mass moment of inertia of the model. In appendix F a table can be found
with all properties of the model parts. Depending on the performed test run, parts of this list were taken to
determine the model parameters of that particular test run. A summary of these main model parameters can
be found in table 8-2.

Mass Rg Mass moment of Inertia

[kg] [m] [kg*m2]
Keel type model keel etc. model keel etc. model keel etc.
Flat plate, Hbk = 95 [mm] 273.4 4.64 0.011 0.231 4.953 0.285
Flat plate, Hbk = 78 [mm] 271.5 4.07 0.010 0.217 4.763 0.214
Saw blade 273.0 4.41 0.011 0.225 4.917 0.249
Circle 274.4 5.30 0.010 0.232 4.998 0.318

Table 8-2 Mass properties model

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 Experiments

8.2 Post processing

8.2.1 Governing equations

The deduction of the governing equations presented in this paragraph is based on [5, 10]. The general
equations of motion are given by Newton's second law:

6
(8-5 ) ^ m- J X Xj - F, for i = 1,...6

With:
mij = 6 x 6 matrix with masses and solid mass moments of inertia of the body
Fi = Force or moment in the direction i
Xj - Acceleration of the body in j direction

When a body free floating in water is involved, added mass (ay), damping (bij) and spring (QJ) coefficient are
introduced. The equations of motion become, see also [12]:

(8-6 ) X {{™'.y + «w )^y + ^'. / ' "^y + ^'7 ' •*; 1 = ^^ *°'' = ^'-^

In equation (8-6) the indices i and j refer to the direction of the motion: Surge, Sway, Heave, Roll, Pitch and
Yaw, in this order. The indices (i,j) of the added mass, damping and spring terms (a, b, c respectively) refer
to the couple coefficients; force in i-direction as a reaction on a motion in the j-direction, all other motions
constrained. F; is a vector with the applied forces and moments in the considered direction. In the
experiments executed in this project the roll motion is prescribed, all other motions are constrained. In the
experimental arrangement the forces in y and z (sway and heave) direction are measured as well (see
paragraph 8.1.5). That gives the opportunity to calculate the couple coefficients roll-sway and roll-heave.
This gives the following equation for the considered problem:

(8-7) (m,4 + a-4 )^' + bf^ ■(p + c.^-(p = F. for i = 2,3,4

With the roll motion, speed and accelerations defined as follows:

(p = 0„ sin(6X)

(8-8) ^ = 0a (ÜCOS iüX)

(p = -0^ ■ CO' ■ sin(fyf)

In (8-7) the mass moment of inertia of the rigid body and hydrodynamic added mass moment of inertia are
defined as the in-phase part of the equation. The damping coefficients are considered as the out-of-phase
part. The spring term is in-phase with the motion. F2 and F3 are the measured forces in y and z direction
respectively, F4 is the applied moment by the hydraulic motor.

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Verification of CFD calculations with experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

Computation of the restoring forces

The left hand side of equation (8-7) contains 3 restoring forces that are in this case known when the mass
distibution of the model is known. The mass properties of the model are described in paragraph 8.1.6.

Equation (8-7) contains two couple terms (C24 and

C34) and one spring term (C44). Treatment of these
terms differs from calculations with a normal ship;
a rotation around the roll centre does not change
the body under the waterline, which makes the
spring terms only dependent on the mass
properties. Couple term C24 is equal to zero; a
static rotation around the x-axis, has no influence
on the force in the y-direction. The same holds \t
» F„
for C34; a static inclination does not influence the
force in z direction. During a rotation around the
X-axis, the mass of the model is compensated by
the buoyancy force at all times. Spring term C44 <P ^ A
would be equal to zero when the centre of gravity
is located in the centre of rotation. During the
preparation of the experiments it was a target to
get Rg as small as possible.

Figure 8-7 Moment equilibrium

The results of the static inclination given in table 8-2, indicate that Rg is too large to neglect the C44 spring
term. This term can be computed with the mass of the model (M), the gravitational acceleration (g) and the
distance from the centre of roll to the COG (rg) see figure 8-7. The spring term is not linearized due to the
considerably large roll angles. The three spring terms now read as:

C24 • 0 = 0
(8-9) C34-^ = 0
C44 • ^ = F • r • sin 0 = M • g • r^ • sin (p

Computation of the known mass terms

Equation (8-7) contains three mass terms: m24, m34 and m44. The first two are related to the force in y and z
direction respectively, the last related to the moment around the z-axis. The influence of the distribution of
the mass will be taken into account; the solid mass moment of inertia is not equal to zero. Unlike a normal
ship in waves, these experiments were performed with the COG outside the center of roll; therefore the
acceleration of the mass center on force equilibrium must be included. There will be no linearization of the
motion. The model will be treated as a rigid body, any influence of model flexibility will be neglected.

Computation of these mass acceleration terms are all based on the general equations of motions. With the
aid of the motion equilibrium in y and z direction (see figure 8-7), the m23 and m34 terms can be found. The
expressions incorporate the mass (1^) of the model, the distance from the centre of roll to the COG (rg) and
the roll angle (cp). The term m44 is equal to the mass moment of inertia around the x-axis, found in
paragraph 8.1.6. This gives the following expressions for the mass terms:

68
 Experiments

m,^ = -M ■ r^ ■ cos (/>

(8-10) m34 = - M ■ r^ ■ sin ^

The m24 term reads as the force in y-direction as a result of acceleration around the x-axis, all other motions
constrained. Similar statements can be made for the m34 and m44 terms The minus sign are a result of the
chosen direction of the positive rotation, see figure 8-7.

Equations of motion
The spring and mass terms of equations ((8-9) and (8-10)) can substituted in equation (8-7), giving the
following equations of motion:

{-M -r^ -008^ + 024)^ + ^24 ■è = Fiit)

(8-11) (- M • r^ • sin <z> -I- a^^ ) ^ + b^^0 = F, ( 0
{-1,, + a^)^ + b^ ■ ^ + M ■ g ■ r^ -sin^ = F^{t)

Bringing all known values to the right, keeping the hydrodynamic terms on the left, gives the following 3
basic equation of motion:

(8-12) a,4^' + b,^<p = F-it) for i = 2,3,4

Where F '2(t), F '3(t) and F '4(t) represent the signals measured from the experiments without the influence of
the mass distribution of the model. These signals are divided by the length of the model (2.7m) to get all
forces per meter model. This is consistent with the F luent calculations; the forces generated by this program
are per meter model and do not include the mass of the model.

69
Verification of CFD calculations with experiments on a rolling circular cylinder witli b ilge keels in a free surface.
University of Technology Delft & B luewater Energy Services b.v.

8.2.2 Hydrodynamic coefficients

Before the damping and added mass coefficients can be computed from the signals given in equations
(8-11), Fourier Analyses (FA) have to be performed. After FA are performed the signal can be written as a
sine with a certain frequency (co), time (t) and
■
phase shift (£i<p). Definition of the phase shift is
' ^ --^ ■ ■ . ,
Roll rrotwn [-]
0.75 Keel force [-] clarified by figure 8-8, which gives a
0.5
dimensionless roll motion and keel force. This
0.25 / ■ '

'••. figure is an example of a phase shift of -60

\ ,
0.1 .• 0.2 0.3 0.4 o.\ 0.6 ''•■!
0.8 0.9 / l
degrees; the keel force is 60 [deg] behind the
0.25

-0.5 ,•* / roll motion. The force signals of equation (8-12)

_J>C...
0.75
\ ^ can now be written as:
TiDe[t/T]

Figure 8-8 Definition of the phase shift

(8-13) ^ ' ( 0 , = F,a ■ S^nikOX + £,. ) for i = 2,3,4

In this equation i=2 is associated with the roll-sway components, i=3 with the roll-heave components and
i=4 gives the damping and added mass coefficients itself. Scalar (k) in this equation is 1 when i = 2 or 4 and
2 when i = 3. This means that the second harmonic will be used in case of the roll-heave coefficients.
Physical this can be explained as follows; when the model makes half a period the component force on the
keel in z-direction becomes first positive and then negative or visa versa. This means that in one period the
force becomes twice positive and twice negative. Or in other words the signal will behave with twice the
applied oscillation frequency. Further comparison of the forces in this z-direction will therefore based on the
second harmonic

Substituting this equation (8-13) in equations (8-12), gives:

(8-14) a J + b,J = F,„ ■ siniküX + f,,) for i = 2,3,4

With the motion, velocity and acceleration of equation (8-8), gives the equations of motion:

-a^^(f>^-TJO^ ■sm{üx) +bi.^-</)^,-CD-cos{QX)-

(8-15) for k = 2,3,4
Fu ■-cM£f^}-micot) +^F^^ ■ mi£p^^cosiüJa
The added mass and damping terms can be found by filling cot = 0 and cot = n/2 in equation (8-15):
cot = 0 gives the three damping terms that are considered out-of-phase:

^2. sinCff.)
(8-16) ^24 = [kgm/s]

(8-17) ^..4 = [kgm/s]

70
 Experiments

(8-18) b^ = — [kgmVs]

And Cut = n/2 gives the three added mass terms that are considered in-phase:

(8-19) «02,44 = ■ :r^— [kgm]

<t>a -co'

F,. •cos(ff^^)
(8-20) «34 -
a,4 = ■ ^^— [kgm]
t -co'

F, ,-cosCf^^^)
(8-21) «04444 = '. r-^ [kgm^]
t (O'

Dimensionless representation:
To be able to compare these results with other research projects all hydrodynamic terms will be presented
dimensionless (in compliance with [14]). The added mass and damping terms will be made dimensionless
with parameters of the keel itself. Dimension analyses give the following dimensionless representations of
the hydrodynamic added mass terms and two couple coefficients:

«94 «34 «44

(8-22) ^ ^ , '-^—r, ^ [-]

pAB' pAB' pAB^L

And the couple coefficients and damping term:

,o,,> ^24 ^34 ^44

(8-23) 1 = , T=, j =B^ [-]
p-A-B'-j^ p.A-B'.M pAB'-L-^^^
With:
p Water density [kg/m^]
B B readth of the model [m]
L Length of the model [m]
g Gravitational acceleration [m/s^]

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University of Technology Delft & B luewater Energy Services b.v.

8.2.3 Keel forces

Measurements during the experiments included a direct

measurement of the force perpendicular to the bilge keel. The
hydrodynamic force can be extracted from the signal with the aid of
a free body diagram of the keel. The mass of the keel and distance
(rg) are computed from the static tests and can be found in table 8-2.
The free body diagram of figure 8-9 holds for the whole oscillation Fkeel(t)

period. The following calculations are based on a keel-bound

coordinate system, with t-direction perpendicular to the keel and n- CP
direction in line with the keel.

Force equilibrium in t-direction reads:

(8-24) Y.^,=M,,,r
a ,
FWi(t)
With at the acceleration in t-direction, gives:

Figure 8-9 free body diagram keel

(8-25) F^ ■ sin(^(0) + F,,,,(0 - F\^^, it) = M,,,„ • a, (t)

With a , (t) = r ■ (pit) and F^ = M^^^, ■ g , gives:

(8-26) M,,.,., ■ g ■ sinm)) + F,,,(t) - F'^, (t) = M,,,„ • r ■ 0(r)

Now the hydrodynamic force on the bilge keel can be calculated:

(8-27) F\.e, (0 = -F,.,,(r) + M,^, ■ r ■ 0(t) - M,,„, ■ g ■ sinim)

Equation (8-27) is performed at every time step, yielding a time trace with exclusively hydrodynamic forces
and no influence of the mass of the keel. Fourier Analysis (FA) is performed on these signals to be able to
write the signal as a pure sine with a certain oscillation frequency (co), time (t) and phase shift (£keei(p):

(8-28) C / ( 0 = F„.,.„-sin(ö;f + £,,,,^)

Next to the keel force amplitude and phase shift, the drag coefficient (Q) of the keel will be compared with
CFD computations. The Cd -value uses density of water (p), maximum velocity of the keel tip (VQ) and frontal
area of the keel (A) and is defined as:

(8-29) C,=
\-p-Vo'-A

72
 Experiments

8.2.4 Energy balance

The energy balance is discussed in this paragraph to investigate the ratio between the viscous and non-
viscous damping. The results of the computations presented in this paragraph can be found in paragraph
8.3.7, where the results of the experiments are given.

1 Control volume 1
1
1
1
Eoscillator

E™ve Ewave

'S!"- h — '
Basin bottom EvisoxJS
1
1
1

Figure 8-10 Energy balance

The energy balance is given by figure 8-10; the energy applied by the oscillator is dissipated by radiated
waves and generation of turbulence. The way turbulence dissipates energy is described in paragraph 3.1.
The wave energy is considered as the non-viscous energy dissipation, the turbulence is considered as the
viscous part of the dissipated energy. The latter will therefore be referred to as Evicous, which gives the
following energy balance:

(8-30) ^oscillator ^'t^.wave + Evicous [Nm]

During the experiments the wave height is measured with a wave-height-gauge giving the registration of the
surface elevation: ^(t). With Fourier Analyses (FA) this signal can be rewritten as a pure sine wave amplitude
(^), oscillation frequency (co), time (t) and phase shift (z^^):

(8-31) ^(0 = C,sin(6;r + f . J [m]

Typical examples of a wave registration and the sine approximation are shown in figure 8-11. From the
recorded signal only the fist couple of wave periods is taken; large standing waves in the basin might
otherwise deteriorate the FA. These large waves might be initiated by neighboring experiments or the
opening one of the doors of the laboratory.

Measured signal
First Harmonic

Time [s]

Figure 8-11 Typical surface elevation registrations and extracted first harmonic

73
Verification ofCFD caiculations with experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

The radiated waves are shallow waves (A/h>20), which makes the group velocity (rg) equal to the phase
velocity and the radiated wave energy is defined as:

(8-32) ^.«v. = T P g - ^ ' c , [Nm]

Equation (8-32) gives the radiated energy during one period at one side of the model.

The input energy of the oscillator is calculated with the applied moment at a time step (M(ndt)) and the
change of roll angle at that time step (A(p(ndt)). Giving the following formula for one period, with N time
steps:
N

(8-33) £,„a7/«,«r = Z ^ ( " ^ ^ ) " M>(.ndt) [Nm]

n=\

All terms on the right hand side of equation (8-32) and equation (8-33) are now known, which make is
possible to compute the dissipated viscous energy.

74
 Experiments

8.3 Results

8.3.1 Sensitivity mass properties experimental model

This paragraph investigates to what extend the results presented in this chapter are sensitive to the
computed mass properties. When a small deviation of the mass properties leads to large changes of the
results, the value of these experiments will be questionable.

Two typical signals will be investigated. Firstly the moment on the model, which is influenced by the mass
properties related to the whole model. And secondly the force on the keel, that is influenced by the mass
properties of the keel itself. From these signals the influence on the amplitude and phase shift of the first
harmonic will be investigated. The coefficients presented in this chapter are based on these two values.

The calculations of this chapter start with determination of the expected maximum difference between the
calculated properties and the true mass properties. These expectations are based on the results of the
experiments or the calculations, see paragraph 8.1.6. Based on the expected deviations, the altered mass
properties are determined. With the altered properties, the hydrodynamic forces are recalculated and
compared with the original values. The original values are set on 100%; table 8-2 and table 8-3 present the
deviations from this 100%. The values presented in the tables are an average of the 7 amplitudes per
frequency and the red numbers indicate large differences above 5%.

Moment on the whole model

The hydrodynamic moment on the model is influenced by the mass moment of Inertia (l^), mass (M) and
distance (Rg) from the centre of roll to the Centre of Gravity (COG). The values of l^ and M are changed by
plus and minus 3%. This percentage is slightly larger than the expected accuracy of the determination of
these values. The Rg is changed by 10%; because this parameter is more sensitive: 10% percent means 1.1
[mm] difference in the location of COG. Table in appendix E shows that 1 [mm] is the difference found
between the results of the static inclinations.

frequency Ixx + 3 % Ixx - 3 % M +3% M -3% Rg +10% Rg -10%

0.2 102.0 98.1 92.9 107.7 78.8 127.6

0.4 102.0 98.0 98.1 102.0 93.9 106.8

0.6 101.9 98.2 99.2 100.8 97.5 102.6

Moment amplitude [%]

frequency Ixx + 3 % Ixx - 3 % M +3% M -3% Rg +10% Rg -10%

0.2 99.1 101.0 104.1 96.7 116.2 92.7

0.4 98.6 101.4 101.3 98.7 104.6 95.8

0.6 98.8 101.2 100.5 99.5 101.8 98.3

Table 8-3 Influence mass properties on hydrodynamic moment of the model

From table 8-3 it can globally be said that the phase shift is more sensitive to the mass properties than the
moment amplitudes. This was to be expected as the mass properties are in-phase with the roll motion and
the hydrodynamic forces are almost out of phase (50-70 [deg], see figure 9-4 ). The frequencies, 0.4 and
0.6 [Hz] will be compared with the CFD calculations. At these frequencies the average accuracy is around 2-
3%, which can be regarded as accuracy of these experiments when it comes to the forces on the whole
model.

75
Verification ofCFD calculations with experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

Forces on the keel

These forces are influenced by the mass of the keel (Mkeei) and distance (Rg.keei) from the centre of roll to the
COG of the keel, see paragraph 8.1.5. Both values are altered by 10%. The considerably small property
values of the keel are easily influenced by defects like glue, bolts and the sealing hose (see appendix D).
The selected deviations are the maximum expected from these kinds of experimental setups.

frequency Mk«i -10% Mkeel + 1 0 % R<,.k«i -10% R, keel + 1 0 %

0.2 135.3 78.6 99.0 101.0

0.4 107.0 93.7 98.9 101.1

0.6 102.2 97.9 98.9 101.1

frequency Mkeel -10% Mkeel + 1 0 % R, keel -10% R, keel +10%

0.2 79.3 123.8 100.9 99.1

0.4 95.1 105.2 100.9 99.1

0.6 98.6 101.5 100.7 99.3

Table 8-4 Influence mass properties on hydrodynamic keel force

From table 8-4 it can be concluded that the force amplitudes are less influenced than the phase shifts, for
the same reason as stated above. The influence of Rg is small or even negligible. At 0.4 and 0.6 [Hz] the
accuracy is again around 2-3 percent, except for the influence of the mass at 0.4 [Hz]. It can be concluded
that the results from the measurements on the keel have also an accuracy around 2-3%.

76
 Experiments

8.3.2 Frequency content measured signals

Most results given in this chapter are based on the first harmonic (or second harmonic in case of z-direction)
of the measured signal. This paragraph will investigate the frequency contents of each signal to see if this
harmonic indeed dominates the signal. This content will also reveal if other "pollution" is present in the
signal that can make the results unreliable.

MATLAB is used to perform a Fast Fourier Transformation and to plot a frequency spectrum of every test
run. This paragraph will give three particular graphs, discuss the frequency spectrum and will conclude what
test runs are less reliable.

Ffaquancy content Frequency content Fpoquoncy content

' ' ' ' ' I 16, r- ^ ' ^ p > . 1 ^i ' ' ■ ' '

.1 il. . , - I ,[ Ik , . S— i L A ii I ol—<^ ' ' • •—' '■'"■ '-—

0 05 ' I S 2 25 3 35 « D 05 1 1 5 2 3Ï 3 35 » 0 05 t l 5 2 25 3 35 4
Frequency [Hz] Frequency [Hz] Frequency [Hz]

(a) (b) (c)

Figure 8-12 a) one dominating frequency, b) Influence higher harmonics, c) Influence hydraulic system

Figure 8-12 gives three typical contents of the signal found during the experiments. These graphs are taken
from of the recorded forces on the keel at different frequencies. Plot (a) has one dominating frequency, any
higher harmonics or pollutions is not visible. Plot (b) clearly shows a significant influence of higher order
terms, but still a distinct peak at the oscillation frequency. Plot (c) is particular for oscillations at a lower
frequency (0.4 [Hz] in this example); low damping made that the control system of the oscillator was too
stiff, causing the model to vibrate; these vibrations are visible in the signals.

The consequences of these frequency contents will be handled per plot type. Type (a) gives only a small
error when using the first harmonic; this harmonic dominates the signal and covers all work performed by
the model. The higher harmonics visible at (b) are expected to be caused by the flow behavior itself and as
long as the first (or second) harmonic dominates the content, no problems are expected using this signal.
The pollution from type (c) can be extracted from the signal by a proper Fourier Analyses (FA). The
frequency of the pollution (3.2 [Hz]) is far away from the oscillating frequency (0.2-0.6 [Hz]), which will give
reliable results. Care should be taken when using the phase shift of this signal; this shift is easily influenced
by the pollution. The pollution (vibration of the model in this case) can also cause additive forces at the
oscillation frequency or at higher harmonics. This effect is expected to be small.

As stated above, all signals without a dominating frequency peak at the first or second harmonic should be
treated with care. All signals are investigated on their frequency content and the results are given in table
8-5. This table gives an overview of all test runs; Y(es) means that there is one distinct frequency in the
signal. N (o) means that the pollution is more dominant than the first harmonic. Y and N refer to all roll
amplitudes at that particular frequency; in a couple of cases the larger roll amplitudes contain no pollution.
This is indicated by the numbers with + sign; 8+ means that all runs with roll amplitude of 8 and higher
contain no pollution.
Verification ofCFD calculations with experiments on a rolling circular cylinder with b ilge keels in a free surface.
University of Technology Delft & B luewater Energy Services b.v.

From table 8-5 it can be concluded that all runs at 0.2 [Hz] have to be treated with care. At 0.4 and 0.6 [Hz]
the applied moment and keel force are reasonable reliable; no significant influence of the pollution is
expected on both the phase shift and force amplitude. The small keel has less damping than the reference
keel, resulting more influence of the vibration of the model. The last remark is on the bare hull; table 8-5
shows better results despite the very small damping. For these runs the stiffness of the system was lowered
to avoid excessive vibrations. The downside is that the system will deviate more from the prescribed motion.
Correction on this deviation can cause vibration as well; this can be seen at the 0.6 [Hz] runs.

Saw Blade
Reference

shaped
3
1

Circle
keel

keel
re 5 £
re
s E CD
If)

Frequency 0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6
Force y-directlon N 8+ Y N 16+ Y N 10+ Y Y Y 6+ Y Y Y
Force z-direction N N 10+ N 16+ 10+ N 14+ 12+ N 10+ 6+ 6+ 8+ N
Applied Moment IV 10+ 6+ N 16+ Y N 10+ Y 10+ 10+ 6+ N N N
Keel force N 12+ Y N 12+ Y 14+ 8+ Y 6+ Y 6+ Y Y N

Table 8-5 Overview results investigation frequency content

8.3.3 Repeatability of the experiments

The value of executed experiments depends on the level of repeatability of the test runs; when the
experiments are repeated under the same conditions, the results should be the same. Figure 8-13 gives the
phase shift and keel force amplitude. A check of on the other signals and test runs gives the same
impression of the repeatability.

-10 ---&--■ Smal keel, 0.6 [Hz]

— * — Repeat test r\jns small keel
■■-□-■■ Crete diaped keel, 0.6[H;]
-?o
«• -30

i ^

Roll amplitude [deq] Roll amplitude [deg]

Figure 8-13 Checl< repeatability; keel force amplitude & pliase sliift

From the two graphs above it can be concluded that both parameters the show little difference when the
same test is repeated. This gives confidence for the reliability of the test results when it comes to
repeatability.

A remark on the values of the circle shaped keel; those repetition runs performed with a short rest period
between the runs, due to time pressure. The water level of the basin was not totally at rest before the next
run was started, this explains the behavior of those lines. The results presented in paragraph 8.3.4 - 8.3.8
will be based on the first series of runs, where enough rest time was taken into account.

78
 Experiments

8.3.4 Time domain

This paragraph investigates several typical time-domain results to study the detail effects of a changed
frequency, amplitude and keel shape. Most of the phenomena found in these figures will be confirmed by
other results presented in this chapter; the phenomena are typical for most experimental runs. To be able to
compare the time traces, all variables (time, force, roll amplitude) of the graphs on the next page are given
dimensionless. The dimensionless representation of the force is in line with the one given by equation
(8-29). The presented graphs are taken from the recorded force on the keel.

Figure 8-14 gives the influence of frequency on the keel force, in paragraph 2.3 it was stated that frequency
has no influence on the flow behavior and thus not on the keel force. The left graph of figure confirms that
the frequency has no influence in case of the reference keel. The right graphs shows that for the circle
shaped keel this statement does not hold. Apparently the lower speeds at 0.2 [Hz] improves the generation
of drag force considerably.

Figure 8-15 shows the influence of roll amplitude on the keel force. Both reference keel and circle shaped
keel show a dependence on roll amplitude, this is in compliance with the statement given in paragraph 2.3.
It can be found that the difference between 0.4 and 0.6 [Hz] is much smaller than the difference between
0.2 and 0.4 [Hz] this will be explained later on.

In figure 8-16 the deference between keels at a high frequency is investigated; as expected, no difference is
found between the small keel and the reference keel. The right graph of this figure shows that the circle
shaped keel has both a different phase shift and a different force behavior. ~

The difference between the keels at a low frequency is given in figure 8-17; still no difference between the
small keel and the reference keel. The right graph also presents a similar trend: phase shift and force
behavior are different. It can even be seen that the circle shaped keel generates a higher drag force
compared to the reference keel, but at a smaller phase shift. Apparently the circle shaped keel generates a
higher force when the velocities of the keel are low.

All presented graphs in this paragraph show a small vibration when the force on the keel changes sign. This
could be caused by hydrodynamic phenomena, but it could also be a small margin in the construction of the
experimental model.

79
Verification of CFD caicuiations witli experiments on a rolling circular cylinder witit bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

Figure 8-14 Frequency dependence: Reference keel (left) & Circle shaped keel (right)

Figure 8-15 Amplitude dependence: Reference keel (left) & Circle shaped keel (right)

Figure 8-16 Keel comparison, frequency: 0.6 [Hz], roll amplitude: 16 [deg]

Figure 8-17 Keel comparison, frequency: 0.2 [Hz], roll amplitude: 16 [deg]

80
 Experiments

8.3.5 Hydrodynamic coefficients

This paragraph will give the resulting 6 hydrodynamic coefficients extracted from the measured signals to
investigate the consequences of the altered keel shapes. To illustrate the trends found in the graphs, this
paragraph will present the results of the runs at a frequency of 0.6 [Hz]. The graphs of the other
frequencies (0.2 and 0.4 [Hz]) show similar trends and can be found in appendix G.

Roll coefficients
Added mass Damping
009

-B-
'Reference 1
0.0
O Small
0.07 - ^ Saw Blade
X Circle shaped
— h Bare Hull
0.06

Q. 005
^ j 0 06

f
004
O O -B- Reference
0.03
O Small
^1^ Saw Blade
X Circle shaped 0.02
-1- Bare Hull |
0.01

I. I. I, I

-^ <|)a [deg]

Figure 8-18 Added mass and damping coefficient

The left graph of figure 8-18 shows a linear dependence of the added mass on the Toll amplitude, a
theoretical explanation for this behavior is not found at this point. The right graph gives a linear dependence
of the flat keels (reference, small and saw-blade keel) and a less than linear dependence of the circle shaped
keel. The behavior of the flat keels is also found by the measurements on the keel of the Glas Dowr, see
paragraph 2.2. The behavior of the circle shaped keel indicates that this shape becomes less efficient when
the roll amplitude increases compared to the flat keels.

Both graphs of figure 8-18 show that the coefficients of the bare hull runs are close to zero. This should
theoretical be the case since the only forces at the bare hull are due to friction at the cylinder wall. These
forces are several orders smaller than the forces on the keel. That the practical results are also close to zero
indicates that possible errors due to the hinging and lining of the model have no influence on the roll-roll
coefficients.

The left graph shows a considerable higher added mass coefficient for the circle shaped keel, this can be
explained by looking at the shape of this keel. This shape is among the flat plates best capable to take along
water during the oscillation. Unfortunately the resulting force is in-phase with the roll motion and therefore
the advantage of this shape is not visible in the damping terms. The damping of the circle shaped keel is
smaller compared to the reference keel, as predicted by the CFD calculations of paragraph 7.5.

In the damping coefficient it can be found that the keels consisting of flat plates (reference, small and Saw-
blade keels) are linear with the roll amplitude. The circle shaped keel is less than linear with the roll
amplitude; apparently the circle shaped keel gets less efficient compared to the flat plates, when the
amplitude increases.

Two last remarks on the flat plate keels; there is little difference between the reference keel and the saw-
blade keel. The changing keel height along the length of the keel does not induce more turbulence. When

81
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University of Technology Delft & Bluewater Energy Seivices b.v.

the reference keel is compared with the small keel, a small decrease in added mass and damping can be
found, this will be explained in paragraph 8.3.6.

Roll-sway couple coefficients

Added mass Damping

'
-B- Reference
06 0 Small
-5|(- Saw Blade
■9 05 ■ X C ircle shaped oj
—I- Bare Hull 0
- 1 " 04
X ■
■X-

X-

r-
01 \
n 4, 1^ 1-, 1-^—— 1 - . — 1 1 -
6 8 10 12 14 16
- > i>!. [deg]

Figure 8-19 C ouple coefficients roll-sway

Discussion of results
Looking at the left graph of figure 8-19, it can be noticed that the added mass couple coefficient of the bare
hull runs is not equal to zero. After subtraction of the influence of the mass distribution of the model this
should be the case because the only damping is the very small friction damping and this damping has no
component in the y-direction. The mass properties of the model do not have enough influence on these
results to account for this deviation. The suggestion is that the lining of the model was not accurate enough
and a wringing force between the axes and hydraulic engine was present. This wringing force is probably in-
phase with the roll motion and has therefore a significant influence on the added mass term that is in-phase
as well.

The damping term is out-of-phase, which explains that the bare hull damping terms reaches almost zero.
This in-phase wringing force will have a significant influence on the phase shifts of the hydraulic forces,
especially because these forces are almost out-of-phase (-50/-70 [deg], see paragraph 9.2.2). This makes
the absolute values of these roll-sway coefficients of the other keels less reliable. The trends and relative
differences between the keels can be trusted as these wringing forces were present in all experimental runs.

The added mass roll-sway couple coefficient show a declining value up to a roll amplitude of 12 [deg]. An
explanation for this trend is not found at this moment.

Comparing the graphs of figure 8-18 and figure 8-19, it can be concluded that the trends are similar.
Therefore the same conclusions can be made on the roll-sway trends as made on the roll-roll terms above.
Due to the small roll angles it could be expected those terms are similar.

82
 Experiments

Roll-heave couple coefficients

Added mass Damping

' X
-B- Reference
O Small X
- ^ Saw Blade
■X Circle shaped
—1- B ate Hull .X

"?- x

.X
x
■m^ ^
X

® —
-8^
1 , 4i=— 11— 11 — h -
6 8 10
- > <tla [*9]

Figure 8-20 Couple coefficients roll-heave

The added mass terms in the left graph of figure 8-20 are all negative, which is a result of the selected
phase shift. The computation of the roll-heave couple terms are based on the second harmonic (see
paragraph 8.2.1), which gives two options for the phase shift between the roll motion and the force signal.
The phase shift is chosen in a way that it gives positive damping terms.

Both graphs of figure 8-20 show that the coefficients of the bare hull reaches almost zero. The wringing
force suggested in the previous section influences these coefficients as well but the influence will be
considerably smaller because of the small roll angles. The small roll angles generate only a small force in the
z-direction and considerably large ones in the y-direction. This suggestion is confirmed by the fact that the
coefficients of the bare hull increase with an increasing roll amplitude.

The damping couple coefficient of the circle shaped keel is almost three times larger than the coefficients of
the flat plate keels (reference, small and saw-blade). This can be the result of a larger force in this direction
or a larger phase shift. Comparison of this keel with the reference keel reveals that the forces are
comparable but the phase shift is twice as large for the circle shaped keel. Thus the forces in the z-direction
are generated more around the maximum roll angles for this keel. The velocities around the roll amplitude
are small, apparently the circle shaped keel performs better at small velocities; this statement is confirmed
by the graphs of figure 8-21.

The same remarks can be made about the flat plate keels as with the roll-roll terms stated above; the saw
blade does not differ from the reference keel and coefficients of the small keel are slightly below the ones of
the reference keel.

83
Verification ofCFD calculations with experiments on a rolling circular cylinder with b ilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

8.3.6 Drag coefficients Iceei

Calculation of the drag coefficients presented in this paragraph can be found in paragraph 8.2.3. From the
graphs of figure 8-21 the first thing to be noticed is the large Cd-values of the circle shaped keel, especially
at low frequencies. Apparently this keel shape is beneficial to generate a large force on the keel, which was
already expected from paragraph 7.3. Unfortunately this force is more in-phase with the roll-motion
compared to the reference keel, which is a disadvantage when it comes to damping capacity. Computation
of the Cd-values does not include the phase shift.

Frequency == 0.2 [Hz] Frequency == 0.4 [Hz]

1
X
-B- Reference -B-
' Reference
40 40
O Small O Small
- ^ Saw B lade - ^ Saw Blade
35
X
■x
—1-
Circle shaped
B are Hull
3S
■X
—1-
Circle shaped
B are Hull
j
30 30

'x.. w=" X

'x - .
X . X,
15 X-
X
16
g-
^■^—-©^ ii. ^ .]
10
n
10
i
5 5
-1
• ^ — ) — h H-T H f 0 — Il p—.^..^ 1 ' 1 '——M
[deg]
-^ 0a [''^s]

Frequency = 0.6 [Hz]

45
- &
' Reference
40
O Small
35 . -*-
■X
Saw B lade
Circle shaped
\
—1- B are Hull
30

25

20
f"

15 X
■ X-
10

0 —1' 1 ' ——+■' h' H

10 12 14 16
-^ <i>a [deg]

Figure 8-21 Drag coefficients

Another interesting result from this Cd-comparison is the constant values above 10-12 degrees roll amplitude.
At larger roll amplitudes the Cj-value appears to become constantly around 8 [-]. A keel of a full-size FPSO
operates in this area of large roll angles, which means that when it comes to designing a full size bilge keel,
taking Cd = 8 [-] is a good first estimation, based on the results of these experiments.

The Cd-values given in the graphs above are well above the values found at fully developed flows over a flat
plate. [17] gives Cd = 1.4 [-] for 2D developed flows over a stationary flat plate placed on the ground. The
flow around an oscillating keel does not get the chance to become fully developed due to the accelerations.
The so-called start-vortices in such undeveloped flow produce a significantly higher Cd-value on the keel.
[17] confirms these higher values with measurements and observations during starts of stationary flows.

84
 Experiments

Looking at the results of the small keel it can be noticed that the difference with the reference plat is almost
negligible. This contradicts the results of the hydrodynamic terms discussed in the previous paragraph.
Explanation is found in the velocity of the keel tip; calculations of the dimensionless hydrodynamic terms do
not include this velocity. The Cj-value on the other hand incorporates this velocity, leading to a correction for
the smaller keel, giving a relativeJy higher Cd-value. It should be concluded that the smaller keel has the
same damping performance, but a larger keel will generate higher forces and more damping.

Comparing the graphs of the flat plate keels (reference, small and saw-blade keels) it can be found that their
performance is frequency independent. This was to be expected from paragraph 2.3, where it is stated that
force on the keel depends on the roll amplitude and not on the frequency. The performance of the circle
shaped keel on the other hand depends on the frequency. This is probably caused by the larger efficiency of
this keel to drag water along with the motion at low frequencies; this is also stated in paragraph 2.3.

8.3.7 Viscous fraction of damping

Computation of these viscous fractions can be found in paragraph 8.2.4. The graphs of figure 8-22 show
that the amount of viscous damping is large compared to the wave damping. The fist explanation can be
found in the circular cross section of the hull; this shape gives little interference with the free surface. The
second reason for the large viscous damping is the location of the keel under the model and the small roll
angles; this will not generate high waves at the free surface.

Frequency = 0.4 [Hz] Frequency = 0,6 [Hz]

=1=

—> iPa [deg]

Figure 8-22 Comparison l<eel shapes on viscous fraction of damping

Comparing the graphs at the different frequencies it becomes clear that the viscous fraction increases when
the frequency decreases. This trend is continued at a frequency of 0.2 [Hz]; at this frequency there are
almost no radiated waves, giving a viscous fraction around 99%. Common feeling contradicts this result; a
slower motion leads to a less viscous effects. The explanation lays in the efficiency of wave making; when
the keel travels on relative small velocity through the water, the water has time to pass the keel without
being pushed forward. Also the velocity the pushed fluid receives from the keel is not enough to reach the
surface. When the velocity increases (0.6[Hz]) more fluid will be pushed forward by the keel at a higher
velocity. Consequently higher waves will be generated at the surface.

This phenomenon of decreasing viscous damping with increasing frequency is not expected to continue,
there is probably an optimum frequency. At very high frequencies the keel will not be able to push any water
in the direction of the free surfaces and the waves will disappear. The physical maximum of the test setup
does not allow investigations to this phenomenon.

85
Verification ofCfD calculations with experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

The graphs show a relatively small viscous percentage at the circle shaped keel; which means that this keel
generates less turbulence and/or more wave in comparison with the other keels. This can be explained by
the figures of paragraph 7.4.2; due to the shape of this keel there are no strong vortices but the amount of
water taken along with the motion is large. The combination of these phenomena leads to smaller viscous
damping.

The decrease of viscous damping with increasing roll amplitude is found at all keel shapes. In line with the
reasoning of the frequency dependence, an increase was expected when the keel velocity and thus roll
amplitude increases. Apparently the smaller distance from the free surface to the keel at larger roll angles
give larger radiated waves. The effect of those larger waves makes the viscous fraction to decrease with
increasing roll amplitude instead of vice versa.

8.3.8 Visual registration

With the aid of an ink injection the location of the large eddy behind the keel was found. The resulting
images indicate that with a strong light and a mirror on the bottom of the tank, the location of the vortex
can be recorded clearly (see figure 8-23).

Figure 8-23 Typical result visual registration

Bright stripes were attached to the end of the keels to have an indication of the length scale of at the
recorded images and thus be able to locate the large eddy quantitative. Unfortunately these stripes were not
visible at the resulting images, leaving the estimation of the eddy location only qualitative.

Interesting to see at these locations of the large eddies is the effect of increased roll amplitude and the
behavior of the eddy during a roll motion. These effects are described Paragraph 9.6, where the results of
the experiments are compared with the Fluent output.

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 Comparison FLUENT <-> Experiments

9 Comparison FLUENT <-> Experiments

This chapter compares the CFD computations with the results of the experiments; the computational CFD
model can be found in chapter 5, the experiments are described in chapter 8. The computational model is
not adapted to get a better agreement with the results of the experiments. Based on literature and
experience of preliminary computations it is expected that this model should perform best on calculations
with these kinds of flows around oscillating keels.

The goal of CFD in this thesis is to compute the hydrodynamic coefficients, given in paragraph 8.2.2. These
coefficients are based on the amplitude and phase shift of the first harmonic, see (8-16) to (8-21). The
comparison in the paragraphs of this chapter will therefore compare these amplitudes and phase shifts.
When a satisfactory similarity is found with these variables, it can be expected that the coefficients agree as
well. The same holds for the viscous fraction of paragraph 8.2.4; this fraction is based on the wave height,
which makes a comparison of the heights sufficient.

To get an impression of the detailed performance of CFD, this chapter will start with a comparison of time-
domain graphs. Paragraph 9.2 and paragraph 9.3 will discuss the amplitudes and phase shift trends of the
forces and moments on the keel and model. The radiated waves are compared in paragraph 9.4. In
paragraph 9.5 it is investigated if Fluent predicts the higher harmonics in the signal. The last paragraph
gives an impression of capabilities of Fluent to predict the location of the large eddies. A visual registration
of the experiments and plots generated by Fluent will be compared in this paragraph.

Due to a limited available computational time it was not possible to perform CFD computations of all
experimental runs. An overview of the test runs that are compared can be found in table 9-1. The reference
and small keel are compared at 0.4 and 0.6 [Hz]. The runs at 0.2 [Hz] were less reliable due to the large
amount of pollution in the experimental output, this is described in paragraph 8.3.2. The circle shaped keel
is only verified at 0.6 Hz, just to investigate the capabilities of CFD with this kind of unusual shape. The saw
blade keel is not verified for mainly two reasons; firstly the experimental data showed that there was no
difference with the reference keel. Secondly this shape would demand a 3D calculation, which is not possible
to perform within the available time.

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University of Technology Delft & Bluewater Energy Services b.v.

Amplitude 4 6 8 10 12 14 16

Experiments
Experiments

Experiments

Experiments

Experiments

Experiments

Experiments
CFD
CFD

CFD

CFD

CFD

CFD

CFD
U-

0.2 X X X X X X X

Reference keel 0.4 X X X X X X X X X X X

0.6 X X X X X X X X X X X

0.2 X X X X X X X

Small keel 0.4 X X X X X X X X X X X

0.6 X X X X X X X X X X X

0.2 X X X X X X X

Circle shape keel 0.4 X X X X X X X

0.6 X X X X X X X X X X X

0.2 X X X X X X X

Saw blade keel 0.4 X X X X X X X

0.6 X X X X X X X

Table 9-1 Overview compared test runs

The results of the experiments, presented in the various graphs of this chapter are computed with methods
given in paragraph 8.2, the output of the Fluent computations received a similar treatment.

In case percentages of deviation between CFD and experiments are mentioned in this chapter, it refers to a
difference between the trends of both graphs. The values are achieved by calculating the difference between
the second order trend lines through each graph.

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 Comparison FLUENT <-> Experiments

9.1 Time domain

This paragraph investigates the time-domain graphs and looks at the effect of filtering on both CFD and
experimental output. Figure 9-1 and figure 9-2 (next page) give four time-domain graphs of 2 typical runs;
one taken from the reference keel to see the effect of higher harmonics (third harmonic in this case). And
the other run is taken from the small keel, where a strong influence of the model's vibration can be found in
the signal. The phenomena found in these signals are typical; they are found in many other signals as well.

These graphs are achieved by a Fourier filtering; first the signal is transformed to frequency domain, the
frequency content above a certain frequency is set to zero, and the signal is transformed back to the time-
domain giving the graphs in figure 9-1 and figure 9-2.

The information is filtered above four different frequencies to be able to see the effect of certain harmonics;
an overview is given in table 9-2. This table gives in the first and third column the filtering frequency and in
the second and fourth column the effect of this filtering; it gives the part that is left in the signal. For
example the first row of the oscillation at 0.6 [Hz]; it states that above 0.7 [Hz] all information is filtered
from the signal and as a consequence only the first harmonic is left in the signal.

Oscillation frequency: 0.6 [Hz] Oscillation frequency: 0.4 [Hz]

Filter: Signal includes: Filter: Signal includes:

>0.7 [Hz] 1^ Harmonic >0.5 [Hz] 1^ Harmonic

>1.3 [Hz] 1=* - 2"" Harmonic >1.7 [Hz] 1^ - 4'" Harmonic

>1.9 [Hz] 1"' - 3'" Harmonic >2.5 [Hz] 1 * - 6* Harmonic

!=•- 6* Harmonic and 1="-9* Harmonic and

>3.7 [Hz] >3.7 [Hz]
possible vibration model vibration model

Table 9-2 Overview filtering effect

From figure 9-1, the small keel at 0.6 [Hz], it can be concluded that both first and second harmonic are well
predicted by the CFD computations. There is not much difference between the two graph, which indicates
that the influence of the second harmonic is not very strong. The influence of the third harmonic can be
found at both CFD and experiments; the experiments give a stronger influence of this harmonic compared to
CFD. The capabilities of CFD could be the reason for this difference, but it could also be a small margin in
the experimental setup. This small vibration around the sign change of the forces is found in most signals.
The difference in peak-height is caused by the larger third harmonic of the experimental output.

Figure 9-2, taken from the reference keel at 0.4 [Hz], shows that the lowest four harmonics are well
predicted by CFD. Also the deviation from a pure sine shape, visible at the second graph is found in the CFD
output. The third graph gives the first signs of the vibration in the model; the fourth graph is highly
influenced by this vibration. This sequence of these graphs show that the vibration can be filtered from the
signal and leads to a close comparison with the CFD output.

89
Verification ofCFD calculations wiOi experiments on a rolling circular cylinder with b ilge keels in a free surface.
University of Technoiogy Delft & B luewater Energy Services b.v.

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90
 Comparison FLUENT <-> Experiments

9.2 Forces on the keel

9.2.1 Force amplitu de

To investigate if F luent can predict the flow behavior around an oscillating keel, this paragraph starts with a
comparison of the predicted amplitude of the keel force. The results presented in figure 9-3 are the results
of the measurement directly on the keel during the experiments and the F luent calculations on the separated
keel part. F ourier analyses on the output signals give the presented amplitudes of the first harmonic, see
paragraph 8.2.1. Here the reference keel and the circle shaped keel are given; the small keel is similar to the
reference keel and can be found in appendix H.

Reference l<eel Circle shaped keel

60 1 45
■■a--- Experinents, 0,6 [Hz]
-*—CFD, 0,6 [Hz] 40 '3*
g so J Z
■ -Q- • - Expe*ients, 0,4 [Hz] ^ 35
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■•■&■■■ Experiments, 0.6 [Hz]
10 ü'"
5 —*—CFD, 0.6 [Hz]
0 —

Roll amplitude [deg] Roll amplitude [deg]

Figure 9-3 Comparison l<eel force; reference & circle sliaped l<eel

The similarity between the CF D computations and the experiments is good for the reference keel and small
keel. It can be concluded that F luent predicts the forces on the keel both quantitatively and qualitatively very
well. The difference found between CFD and experiments for these force amplitudes is typical 3-6% which is
considered accurate in the light of the maximum accuracy of the experiments.

The similarity with the circle shaped keel is less accurate, the trend is more or less predicted but the
absolute deviation is around 20%, which is too large to have confidence in the CF D results. This difference
could also be caused by stiffness problems of the experimental model; the small PVC tubes that represent
the keel, showed small deflections during the oscillations.

From these graphs the same behavior of the keel force can be found as presented in paragraph 8.3.6, where
the drag coefficients are computed. The less than quadratic dependence on the roll amplitude of the
reference keel suggests a decreasing drag coefficient with increasing roll amplitude. Also the linear behavior
of the circle shaped keel suggests a stronger decrease of the drag coefficient. Both these trends can be
found in paragraph 8.3.6.

91
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9.2.2 Phase shift

A Second parameter that determines the flow around an oscillating keel is the phase shift between the roll
motion and the keel force. This paragraph investigates if Fluent can predict the phase shifts found during
the experiments.

Paragraph 2.3 states that the flow around the keel depends on the roll amplitude and not on the frequency;
the graphs of the small keel and the reference keel conform this statement. B oth experiments and CFD show
no significant difference between the runs at 0.6 [Hz] and 0.4 [Hz].

The graphs of the small and reference keel in figure 9-4 show that the phase shift is going to -90 degrees
when the roll amplitude increases (or the Strouhal number decreases). This means that the damping starts
to dominate the motion. At large Strouhal numbers (or small amplitudes) the phase shift reaches zero and
the added mass dominates the motion. In other words, small roll amplitudes with a large keel give a high
the added mass; large roll amplitudes with a small keel give high damping.

Reference keel Circle shaped keel

0
8 10 12 14 16 2 4 6 8 10 12 14 16 la

■ ■-A-- - Expenments, 0.6 [Hzl ■■-*---Ejtperifients, 0.6 [ m ]

■to
— * — C F D , 0.6[Hz]
-, -20 — * — C F D , O.6IH1]
■--0--Experiments, 0,4 [Hz]
— • — CFD, 0.4 [Hz]
Ï -15-

•20
*^^
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1 ■30
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-35
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-40

-45

Roll amplitude [deg] Roll amplitude [deg]

Small keel
14 16

-10
-■ft--- Expertments, 0.6 [Hi]

- - * — c r o , 0.6 [ H I )
^ -20
. -a-. ■ Experiments, 0.4 [Hz]

'T: -30 -■ CFD, 0.4 [Hï]

Roll amplitude [deg]

Figure 9-4 Comparison phase shift, roll motion <-> keel force

From figure 9-4 it can be found the phase shift of the reference and small keel is well predicted by CFD; the
deviations are around 3-6%.

Figure 9-4 gives also the phase shift of the circle shaped keel; it can be found that the predicted trend by
Fluent is similar to the experimental results. When comparing the phase shift of this keel with the other
keels, it can be found that the phase shift is considerable smaller. This means that this circle shape relatively
generates more added mass and less damping. This statement is confirmed by the graphs of paragraph

92
 Comparison FLUENT <-> Experiments

8.3.5. The absolute difference (around 30%) between the calculations and the experiments is probably
caused by the experimental setup; the geometry of the circle shaped keel of the experiments did not
completely match the one of the computational model. Also the stiffness of the circle shaped keel was not
sufficient during the experiments.

9.3 Forces on the model

The forces on the whole model are used to compute the hydrodynamic coefficients, as described in
paragraph 8.2.2. Every hydrodynamic term is calculated from the force amplitude and phase shift of the first
harmonic of the output signal. To investigate these parameters of every run would mean that 18 graphs
have to be compared (3 keels, 6 parameters). These graphs have all similar trends and it is therefore more
convenient to present a table with an overview of the comparison results.

To be able to give an overview of the verification of the parameters that determine the hydrodynamic
coefficients, a so called accuracy level is introduced. In table 9-3, 3 different levels can be found with the
following explanation.

Accuracy level 1:
The similarity between the CFD computations and ■■■a--- Experiments, 0,6 [Hz]
— * — C F D , 0.6 [Hz]
the results of the experiments is accurate; both qualitative ■■■D---Experiments, 0.4 [Hz] AA
— • — C F D , 0.4 [Hz]
and quantitative the values are well predicted. The / ^
difference between CFD and experiments is at most 10%
at this level. This level of accuracy means that the _.fl ./ ^ '
performance of CFD is considered enough to rely on future ^,>^ '"""'^
calculations on these kinds of flows around oscillating t^:::^^^^
keels. An example of this accuracy level is given in figure i 8 10 1
Roll amplitude [deg]
9-5.

Figure 9-5 Comparison amplitude force y-direction, reference Iceel

Accuracy level 2:
At this accuracy level the similarity between the ''1
Expenments, 0.6 [Mi]

CF=D, 0.6 [Mz]

calculations and the experiments is reasonable; the trends f 60 j

Experiments. 0.4 [Hz]

are more or less predicted and the absolute difference are

c
1
50 - CFD, 0.4 [Hz]

not too large. The discrepancy is expected to have no ■a 40 \

significant influence on the computed hydrodynamic ^

Ü

coefficients. Deviations between CFD and experiments can ■S

be around 10-20% at this level; an example is given in Q.

0 -^
figure 9-6.
Roll amplitude [deg]

Figure 9-6 Comparison phase shift force z-direction, reference keel

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University of Technology Delft & Bluewater Energy Services b.v.

Accuracy level 3: le

At this level, the difference between the Experinents, 0.6 [Hz]

CFD, 0.6 [Hzj

experiments and the calculations is large; deviations Expenments, 0.4 [Hi]

above 20%. The trends are predicted but the absolute CFD, 0,4 [Hz]

difference is too large to achieve reliable hydrodynamic

coefficients. An accuracy level of 3 is not necessarily the
result of poor Fluent computations, also the experimental
setup gives reason for inaccuracies as will be discussed Roll amplitude [deg]
below. An example of this level can be found in figure 9-7.

Figure 9-7 Comparison phase shift force y-direction, reference keel

Discussion of results
Table 9-3 shows that all force amplitudes are well predicted by the CFD calculations. A reason for this
similarity could be that the mass distribution of the experimental model has little influence on the force
amplitudes; the forces due to the mass distribution are in-phase with the motion, the hydrodynamic forces
are almost out-of-phase (see figure 9-4).

Total force y-direction Total force z-direction Applied moment around x-axis

Hydrodynamic
a24 + b24 334 + b34 344 + b44
coefficient

Force amplitude Phase shift Force amplitude Phase shift Force amplitude Phase shift

Reference keel 1 3 1 2 1 1

Small keel 1 3 1 2 1 1

Circle shaped keel 1 3 1 2 1 3

Table 9-3 Overview accuracy levels between CFD and experiments

All the phase shifts of the circle shaped keel are not accurate, at best the trend can be found but in some
cases the discrepancy is too large to get reliable hydrodynamic coefficients. A possible explanation is given
in the previous paragraph; the keel of the experimental setup differs from the theoretical shape and the
PVC-tubes were not stiff enough

The phase shift of the small and reference keel are well predicted in case of the applied moment. This
means that the hydrodynamic added mass and damping will be accurately computed with CFD software.

The agreement at the forces in y- and z- direction is less satisfying, which means that the couple coefficients
are not accurately predicted by Fluent. A possible explanation is found in paragraph 8.3.5. Here it is
suggested that an in-phase torsion moment in the experimental setup makes the forces in y-direction
unreliable. The influence on the forces in z-direction is less because of the small roll angles. Both
phenomena are confirmed in table 9-3.

94
 Comparison FLUENT <-> Experiments

9.4 Radiated waves

Paragraph 8.2.4 uses the wave height and applied moment to compute the amount of viscous damping. The
performance of Fluent to predict the applied moment is discussed in the previous paragraph, this paragraph
will compare the height of the radiated waves. The phase shift between the radiated waves and the applied
moment is not considered as it is not used in the energy balance for the viscous part of the damping. Figure
9-8 gives the results of the reference keel and the circle shaped keel; the small keel is similar to the
reference keel and can be found in appendix H.

Reference keel Circle shaped keel

■ Experiments, 0.6 [Hz] ■-Ö-■■ Experiments, 0.6 [Hz]

-CFD, 0.6 [Hz] — * — C F D , 0.6 [Hz]

■ Expenments, 0.4 [Hz]

-CFO, 0.4[Hz]

lU 10

Roll amplitude [deg]

Roll amplitude [deg]

Figure 9-8 Comparison wave height: reference & circle shaped keel

Looking at the graphs it can be noticed that the trends are well predicted in all cases; quadratic behavior of
the flat keels, linear behavior of the circle shaped keel. This linear behavior is in compliance with the keel
forces presented in paragraph 9.2.1. The CFD calculations seem to over predict the radiated wave height
constantly, an explanation for this difference is not found.

The absolute values of the wave heights are less accurately predicted when the heights decrease. The
largest waves are found at the reference keel at 0.6 [Hz], these waves are predicted within 5% accuracy by
CFD. The smaller waves found at the circle shaped keel and small keel at 0.6 [Hz], have an average
deviation of 10%. The predictions of the smallest waves are very poor: deviations above 35% are found at
the frequencies of 0.4 [Hz]. This leads to the assumption that the CFD model did not have sufficiently fine
grid cells at the free surface. In paragraph 5.1.3 it is stated that 10 cells in vertical direction should be able
to model a wave; this number is possibly not enough.

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9.5 Frequency Domain

The comparisons of the previous paragraphs are based on the first harmonic derived from the signals.
Paragraph 8.3.2 shows that there are higher harmonics present in most signals. This paragraph investigates
if these higher harmonics are found by Fluent as well.

The higher harmonics are defined by the peaks in the frequency content which is found by a Fourier
Transformation (FT) on the output signals. The height of those peaks depends heavily on the number of
periods incorporated in the FT as illustrated in figure 9-9. Due
to the large CPU time, most CFD runs are stopped as soon as — ExperimenCs

three identical periods were found during the calculations; this CFD w t f i 1 penod
CFD with 4 penods
-CFD wth 8 penods
is not enough to perform a solid FT. Therefore this paragraph
will present the results of one long-term Fluent calculation, all ^ 20000 -

other runs are not investigated. The computational runs with

the reference keel and an oscillating frequency of 0.4 [-]
contain at least 8 identical periods, the results of these runs
can be found in figure 9-10. The investigated signal is the frequency [HZJ
recorded force o n t h e keel.
Figure 9-9 Influence incorporated number of periods

Oa = 4 [deg] Oa = 8 [deg]
Experiments Expenmenls
- CFD calculations - CFD calculations

0 05 1
-^^AA
15 2 35 3 35 4 0 06 1 15 2
frequency [Hz]
25 3
k. 3,5
. ^<
frequency [Hz]

Oa = 12 [deg] Oa = 16 [deg]

__
.,.. ..., ' Experiments
" CFD calculations

400 \
2000
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to
1 1500

200 \ 1000

100
\ 500

0 06
M i 1 15 2 25
JV 3 36
-J
4 0
V-^-A-
05 1 15 3
frequency [Hz]
26 3 3.5 4
frequency [Hz]

Figure 9-10 Frequency content higher harmonics

96
 Comparison FLUENT <-> Experiments

Looking at the graphs of figure 9-10, it becomes clear that in case of the 8,12,16 [deg] roll angle the second
and third harmonic are predicted well by CFD. In all cases the fourth harmonic, the highest visible in these
graphs is not found in the CFD output.

All frequency peaks above 2 [Hz] in the experimental output are caused by the hydraulic system that
oscillated the model, also discussed in paragraph 8.3.2. Especially at small roll amplitudes the small forces
cause not enough damping to obtain a smooth operating control-system; at low damping levels, the system
is considered too "stiff" causing the system to vibrate.

There is one remarkable omission of the second harmonic in the experimental output at 4 [deg] roll
amplitude. This is the only harmonic found by CFD and not by the experiments. Looking at the trend of
increasing second harmonic with decreasing roll amplitude, it can be expected that this second harmonic
should be in the signal at 4 [deg]. The large influence of the hydraulic system on the frequency content
could damp the second harmonic at this roll amplitude.

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9.6 Location large eddy

This paragraph gives a visual comparison between the experiments and CFD on the location of the large
eddy. The smaller eddies can not be recorded with a normal camera nor are they explicitly computed by the
2D Fluent computations. The ink is released at the tip of the keel at some point during the oscillation, this
means that flow behavior in front of the keel is not visible. In a few cased CFD gave different flow field when
clockwise rotation was compared with counterclockwise rotation. Those differences were small, but a reason
for this phenomenon is not found at this point. The direction given in the figures below are arbitrarily picked
from the results.
To illustrate the CFD performance on this comparison, two series will be investigated with the small keel.
The small keel was selected because the output of these experimental runs was best visible. The first series
investigates how the eddy progresses during one oscillation. The second series investigates the effect of
increasing roll amplitude. At the end of this paragraph an impression is given of the effect of an altered keel
shape.
Comparison during oscillation

Figure 9-11 Small keel at t/T = 2/11 pi

Figure 9-12 Small keel at t/T = 1/2 pi

Figure 9-13 Small keel at t/T = 19/22 pi

98
 Comparison FLUENT <-> Experiments

Discussion of results
First the horizontal position of the eddy behind the keel is investigated; at the beginning of the motion the
eddy is very close to the keel, this becomes clear from both experimental and CFD output. When the motion
progresses, the eddy stays more or less behind in the flow, in other words the distance between the eddy
and the keel increases, this is also predicted by CFD.

During the oscillation the vertical position of the eddy appears to maintain the same vertical position, both
CFD and experiments show this phenomenon. On the other hand the CFD appears to predict the vertical
position constantly lower than found with the experiments.

Comparison effect increased roll amplitude

Figure 9-14 Small keel at t/T = 1/2 pi, phlA = 4 [deg]

Figure 9-15 Small keel at t/T = 1/2 pi, phiA = 8 [deg]

Figure 9-16 Small keel at t/T = 1/2 pi, phiA = 14 [deg]

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The experiments show a vertical position of the eddy in line with the keel tip at al roll angles. CFD predicts
this position at a roll angle of 4 and 8 degrees amplitude, at 14 degrees CFD gives a slightly lower position.
The horizontal distance between the keel and the eddy increases with an increasing roll amplitude; this is
predicted well by the CFD computations.

Overall it can be concluded that the location of the largest eddy in the flow is well predicted by the CFD
calculations.

Comparison effect altered keel shape

Figure 9-17 Circle shaped keel

In chapter 7 it is stated that the circle shaped keel shows less profound and strong vortices at the keel tips.
This statement is confirmed by the experiments, see left image of figure 9-17. The ink in this image is clearly
more diffused when compared with the images above. More diffusion means that the velocities are less
defined and thus the vortex less profound.

The location of the vortex is hard to establish in these graphs, due to the diffusion of the ink. It appears that
the vertical position of vortex is slightly lower at the experimental output.

100
Comparison FLUENT <-> Experiments

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10 Conclusions and Recommendations

10.1 Conclusions

On the verification of CFD

The conclusion of this thesis is that the agreement found between the CFD calculations and experiments is
good; CFD can be used in the future to perform computations on cross sections of an FPSO in a pure roll
motion. All major phenomena present at a 2D cross section of an FPSO section were included in the
comparison and the resemblance was found to be accurate. Accurate in this thesis means that the similarity
between CFD and experiments is good seen in the light of the maximum accuracy of the experimental data
and it is accurate enough to be confident in using CFD in future calculations without verification by new
experiments. The foundations of this main conclusion will be discussed below per aspect of the comparison:

Force and moment ampiitude

All amplitudes of forces and moments found during the experiments are accurately predicted by the CFD
calculations, both quantitatively and qualitatively. The deviation found between CFD and experiments on the
force amplitudes is around 3-7%

Phase shift
The phase shift between the recorded forces and the roll motion are well predicted when it concerns the
phase shifts of the applied moment by the oscillator; deviations of 1-5% are found. At the other forces
(measured on the keel and in z- and y-direction) larger differences between CFD and the experiments were
found; up to 20%. These differences are attributed to the experimental setup rather than to the accuracy of
the CFD calculations; the phase shifts of these signals were sensitive to the mass distribution of the model
and to small misalignments of the experimental setup.

Radiated waves
The trends of the radiated wave heights are in all cases well predicted by CFD. The absolute values of the
radiated waves at high oscillation frequencies are predicted within an accuracy of 5-10%. This percentage
does not hold for the results at lower frequencies where the prediction was poor: deviations around 30-40%
are found. The absolute heights of the waves at these low frequencies are small: 1-3 [mm]. Damping due to
radiation of these waves has a small influence on the total damping; <5%, which makes these deviations of
limited importance when it comes to predicting the total damping of the model.

Higher harmonics
Most signals recorded during the experiments contain higher harmonics; generally these harmonics are
predicted by CFD as well. In some cases the strength of the harmonics found in the CFD output deviates
from the experimental results. The differences are small; they could be the result of errors in the
experimental setup as well.

Location prime vortex

A visual investigation of the results shows that the location of the prime vortex is predicted reasonably well
by CFD. The behavior of the vortex is generally similar, the vertical position predicted by CFD is in some
cases slightly lower. Due to the use of a ink injection is was not possible to achieve absolute values for the
found differences.

102
 Conclusions and Recommendations

On the effect of altered keel shape

Based on the experiments and CFD calculations performed in this thesis, it can be concluded that of the
investigated shapes the normal thin flat plate keel performs best. Larger damping is found for this keel due
to stronger and more profound vortex development and as a consequence generation of more turbulence.
Using these results at full scale has to be done with care; the investigated model deviates considerably from
a full size FPSO.

One keel shape with a distinct flow behavior is a keel that looks in 2D like two half circles with the backsides
attached to each other, called the "circle shaped keel". This keel generates significantly more added mass
and a larger drag force on the keel compared to a normal flat plate keel. However, the phase shift of the
force generated by this keel is smaller, which makes the damping capacities of this keel lower than a flat
plate keel.

1 0 . 2 Recommendations

To continue the research to find a bilge keel configuration that improves the damping capacity of an FPSO
the following recommendations are given based on the investigation performed in this thesis

• A cross section of an FPSO has to be used in the computational model described in this thesis to
investigate the effects of altered bilge keel configurations more accurately.
• Based on investigated model in this thesis, a flat plate seems to perform better than any other
investigated keel shape. It is recommended to focus further research to improve the damping on
multiple keels or extraordinarily large keels rather than altered keel shapes.
• These large or multiple keels bring another challenge to the damping issue; these keels will stick
outside the outer rectangle of the FPSO's cross section. This phenomenon has consequences for the
FPSO when it is docked or moored at a quay; the consequences of this need to be investigated
parallel to the development of those large keels.
• In the next version of Fluent it will be possible to take motions in the heave and sway direction into
account as well. It can be interesting to see how CFD performs on these coupled motions
• Prescribing the response of an FPSO based on a certain pressure distribution along the hull makes it
possible to investigate the damping behavior of a cross section in waves. This is possible within
Fluent, but the results of 2D calculations are uncertain.

103
Verification ofCFD calculations with experiments on a tvlling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

List of symbols and abbreviations

A Frontal area bilge keel [ml

aij Added mass terms
B Breadth of the model [m]
Bbk Damping component: Drag of the bilge keel [Nms/rad]
Be,bk Damping component: Increased eddy damping due to bilge keel [Nms/rad]
Be,hull Damping component: Eddy damping due to the bare hull [Nms/rad]
Bel Equivalent Linear Damping [Nms/rad]
Bf Damping component: friction at the hull [Nms/rad]
bij Damping terms
bij Restoring terms
Bw Damping component: Radiation of waves [Nms/rad]
c Empirical constant [-]
CC Convergence Criteria
Cd Drag force on the keel [-]
CFD Computational Fluid Dynamics
COG Center of Gravity
DNS Direct Numerical Simulation
E Energy [Nm]
EWT Enhanced Wall Treatment
F(t) Recorded force on the model [N]
Fkeel Perpendicular force on the keel [N]
FPSO Floating Production, Storage and Offloading vessel
tshape Shape factor [-]
FVM Finite Volume Method
g Gravitational acceleration [m/s^]
h Water depth [m]
Hbk Height of the keel [m]
Ixx Solid mass moment of Inertia [kg*m^]
L Length of the model [m]
L Typical length scale [m]
LES _. Large Eddy Simulation _ .
M(t) Recorded applied moment [Nm]
MG Multi-grid method
mij _ mass terms
Number of time steps [-]
Navier Stokes

104
 List of symbols and abbreviations

P Pressure [N/m^]
P Rate of turbulence production [-]
PISO Solving method pressure-velocity coupling
PRESTO! PREssure STaggering Option
QUICK Quadratic Upstream Interpolation for Convective Kinetics
Ro Distance from the center of roll to the keel [m]
RANS Reynolds Averaged Navier Stokes
R-bilge Radius of the bilge [m]
Rg Distance from center of roll to COG [m]
Rn Reynolds number t-]
RSM Reynolds Stress Model
St Strouhal number [-]
t Flow time [s]
T Roll period [s]
t Time [s]
U Velocity undisturbed flow [m/s]
UDF User Defined Function
Ui Velocity in certain direction i - 1,2,3 for x, y, z direction respectively [m/s]
Vo Maximum velocity of the bilge keel tip [m/s]
VF Volume Fraction
VOF Volume of Fluid Method
Dimensionless distance to wall [-]
Cw Volume fraction of water [-]
p Under Relaxation factor [-]
r Diffusion coefficient [-]
Kronecker delta [-]
Turbulent dissipation rate [mVs^]
Phase shift [rad]
£
Surface elevation [m]
Wave amplitude [m]
Turbulent kinetic energy [m^/s^]
Wave length [m]
K
Dynamic viscosity [kg/m s]
A Turbulent viscosity [kg/m s]
IJ Kinematic viscosity [mVs]
Mt
V Density [kg/m^]
p Empirical constant [-]
o
Arbitrary variable [depends]
(p
Amplitude roll angle [rad]
Roll angle [rad]
Roll velocity [rad/s]
Roll acceleration [rad/s^]
CO Roll frequency [rad/s]

105
Verification of CFD calculations with experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

Index of figures

FIGURE 2-1 COMPONENTS OF DAMPING AS A FUNCTION OF ROLL AMPLITUDE 11

FIGURE 2-2 TYPICAL CROSS SECTION OF AN FPSO 13
FIGURE 2-3 BASIC 2D SETUP MODEL 14
FIGURE 3-1 VORTEX STRETCHING 16
FIGURE 4-1 OVERVIEW SOLUTION ALGORITHM 23
FIGURE 4-2 DETAIL STEPS OF THE PISO ALGORITHM 25
FIGURE 4-3 TYPICAL REGISTRATION OF THE FORCE ON THE KEEL 30
FIGURE 5-1 OVERVIEW COMPUTATIONAL BODY 31
FIGURE 5-2 COMPUTATIONAL DOMAIN AROUND BODY 32
FIGURE 5-3 DETAILS BOUNDARY LAYER NEAR BODY 33
FIGURE 5-4 GRID REFINEMENT FOR FREE SURFACE 34
FIGURE 5-5 TYPICAL WAVE REGISTRATION 35
FIGURE 5-6 OVERVIEW BOUNDARY CONDITIONS 36
FIGURE 5-7 GRID INTERFACE 36
FIGURE 5-8 START FUNCTION 38
FIGURE 5-9 ROLL MOTION 39
FIGURE 6-1 COMPARISON DIAGRAM COMPUTATIONAL SETHNGS 42
FIGURE 6-2 GENERAL SETUP OF THE COMPUTATIONAL DOMAIN 43
FIGURE 6-3 HULLMOnON 44
FIGURE 6-4 COMPUTATIONAL DOMAIN REFERENCE RUN WITH COARSE GRID 46
FIGURE 6-5 COMPUTATIONAL DOMAIN REFERENCE RUN WITH FINE GRID 46
FIGURE 6-6 COMPUTATIONAL DOMAIN WITH ENLARGED INNER GRID 48
FIGURE 6-7 ASYMMETRIC BEHAVIOR OF KEEL FORCE 48
FIGURE 6-8 COMPUTATIONAL DOMAIN INCLUDING FREE SURFACE 49
FIGURE 6-9 MOMENT-TIME GRAPH 50
FIGURE 6-10 LINEAR EQUIVALENT DAMPING (BEL) OF REFERENCE RUNS WITH A COARSE GRID 50
FIGURE 7-1 INVESTIGATED KEEL SHAPES, A: Y-SHAPE, B: T-SHAPE, C: CIRCLE SHAPE 54
FIGURE 7-2 MOMENT BEHAVIOR ALTERED KEEL SHAPES 55
FIGURE 7-3 RESULT KEEL SHAPE COMPARISON 56
FIGURE 7-4 VELOCITY FIELD OF REFERENCE KEEL AND OPTION 1: Y-SHAPE 30 [DEG] 57
FIGURE 7-5 VELOCITY FIELD OF OPTION 3: Y-SHAPE 90 [DEG] AND OPTION 7: CIRCLE SHAPED 57
FIGURE 7-6 PRESSURE FIELD OF THE REFERENCE KEEL AND OPTION 3: Y-SHAPE 90 [DEG] 58
FIGURE 7-7 TURBULENCE FIELD OF THE REFERENCE KEEL AND THE CIRCLE SHAPED KEEL 59
FIGURE 8-1 SIGN CONVENTIONS 60
FIGURE 8-2 LOCATION OF THE MODEL IN THE TANK 61
FIGURE 8-3 OVERVIEW MODEL 61
FIGURE 8-4 OVERVIEW TESTED KEELS 62
FIGURE 8-5 MEASURED FORCES AND MOTION 64
FIGURE 8-6 FORCE EQUILIBRIUM 2D 65
FIGURE 8-7 MOMENT EQUILIBRIUM 68
FIGURE 8-8 DEFINITION OF THE PHASE SHIFT 70
FIGURE 8-9 FREE BODY DIAGRAM KEEL 72
FIGURE 8-10 ENERGY BALANCE 73
FIGURE 8-11 TYPICAL SURFACE ELEVATION REGISTRATIONS AND E>0"RACTED FIRST HARMONIC 73
FIGURE 8-12 A) ONE DOMINATING FREQUENCY, B) INFLUENCE HIGHER HARMONICS, C) INFLUENCE HYDRAULIC SYSTEM 77
FIGURE 8-13 CHECK REPEATABILITY; KEEL FORCE AMPLITUDE & PHASE SHIFT 78
RGURE 8-14 FREQUENCY DEPENDENCE: REFERENCE KEEL (LEFT) & CIRCLE SHAPED KEEL (RIGHT) 80
FIGURE 8-15 AMPLFTUDE DEPENDENCE: REFERENCE KEEL (LEFT) & CIRCLE SHAPED KEEL (RIGHT) 80
FIGURE 8-16 KEEL COMPARISON, FREQUENCY: 0.6 [HZ], ROLL AMPLFTUDE: 16 [DEG] 80
FIGURE 8-17 KEEL COMPARISON, FREQUENCY: 0.2 [HZ], ROLL AMPLFTUDE: 16 [DEG] 80
FIGURE 8-18 ADDED MASS AND DAMPING COEFFICIENT 81
FIGURE 8-19 COUPLE COEFFICIENTS ROLL-SWAY 82
FIGURE 8-20 COUPLE COEFFICIENTS ROLL-HEAVE 83
FIGURE 8-21 DRAG COEFFICIENTS 84
FIGURE 8-22 COMPARISON KEEL SHAPES ON VISCOUS FRACTION OF DAMPING 85
FIGURE 8-23 TYPICAL RESULT VISUAL REGISTRATION 86
FIGURE 9-1 TIME TRACES, SMALL KEEL, F=0.6 [HZ], OA = 16 [DEG] 90
FIGURE 9-2 TIME TRACES, REFERENCE KEEL, F=0.4 [HZ], OA = 4 [DEG] 90
FIGURE 9-3 COMPARISON KEEL FORCE; REFERENCE & CIRCLE SHAPED KEEL 91
FIGURE 9-4 COMPARISON PHASE SHIFT, ROLL MOTION <-> KEEL FORCE 92
FIGURE 9-5 COMPARISON AMPLITUDE FORCE Y-DIRECTION, REFERENCE KEEL 93
FIGURE 9-6 COMPARISON PHASE SHIFT FORCE Z-DIRECTION, REFERENCE KEEL 93
FIGURE 9-7 COMPARISON PHASE SHIFT FORCE Y-DIRECTION, REFERENCE KEEL 94
FIGURE 9-8 COMPARISON WAVE HEIGHT: REFERENCE & CIRCLE SHAPED KEEL 95
FIGURE 9-9 INFLUENCE INCORPORATED NUMBER OF PERIODS 96
FIGURE 9-10 FREQUENCY CONTENT HIGHER HARMONICS 96

106
 Index of figures

FIGURE 9-11 SMALL KEEL AT T/T = 2/11 PI 98

FIGURE 9-12 SMALL KEEL AT T/T = 1/2 PI 98
FIGURE 9-13 SMALL KEEL AT T/T = 19/22 PI 98
FIGURE 9-14 SMALL KEEL AT T/T = 1/2 PI, PHIA = 4 [DEG] 99
FIGURE 9-15 SMALL KEEL AT T/T = 1/2 PI, PHIA = 8 [DEG] 99
FIGURE 9-16 SMALL KEEL AT T/T = 1/2 PI, PHIA = 14 [DEG] 99
FIGURE 9-17 CIRCLE SHAPED KEEL 100
FIGURE C-1 REFERENCE KEEL & OPTION 1:Y-SHAPE 30 [DEG] 114
FIGURE C-2 OPTION 3: Y-SHAPE 90 [DEG] & OFHON 4: T-SHAPE 1*H 114
FIGURE C-3 OPTION 6: T-SHAPE 0.5*H & OPTION 7: CIRCLE SHAPED KEEL 114
FIGURE C-4 TURBULENCE INTENSITY INDICATOR [%] 114
FIGURE C-5 REFERENCE KEEL & OPTION 1: Y-SHAPE 30 [DEG] 115
FIGURE C-6 OPTION 3: Y-SHAPE 90 [DEG] & OPTION 4: T-SHAPE 1*H 115
FIGURE C-7 OPTION 6: T-SHAPE 0.5*H & OPTION 7: CIRCLE SHAPED KEEL 115
FIGURE C-8 PRESSURE INDICATOR [%] 115
FIGURE C-9 REFERENCE KEEL & OPTION 1: Y-SHAPE 30 [DEG] 116
FIGURE C-10 OPnON 3: Y-SHAPE 90 [DEG] & OPTION 4: T-SHAPE 1*H 116
FIGURE C-11 OPTION 6: T-SHAPE 0.5*H & OPHON 7: CIRCLE SHAPED KEEL 116
FIGURE C-12 VELOCITY INDICATOR [M/S] 116
FIGURE D-1 HINGE AND STRAIN GAUGES 117
FIGURE D-2 STRAIN GAUGE KEEL 117
FIGURE D-3 INK INJECTION 118
FIGURE D-4 DISTRIBUTION BALLAST LEAD 119
FIGURE D-5 BLOCK DIAGRAM HYDRAULIC SYSTEM 120
FIGURE G-1 ADDED MASS ROLL-ROLL COEFFICIENT 125
FIGURE G-2 DAMPING ROLL-ROLL COEFFICIENT 125
FIGURE G-3 COUPLE ADDED MASS ROLL-SWAY COEFFICIENT 126
FIGURE G-4 COUPLE DAMPING ROLL-SWAY COEFFICIENT 126
FIGURE G-5 COUPLE ADDED MASS ROLL-HEAVE COEFFICIENT 127
FIGURE G-6 COUPLE DAMPING ROLL-HEAVE COEFFICIENT 127
FIGURE H-1 COMPARISON AMPLITUDE KEEL FORCE & PHASE SHIFT 128
FIGURE H-2 COMPARISON FORCE AMPLITUDE Y- & Z-DIRECTION 128
FIGURE H-3 COMPARISON CD-VALUE & CENTRE OF PRESSURE 128
FIGURE H-4 COMPARISON ADDED MASS & DAMPING COEFFICIENT 128
FIGURE H-5 COMPARISON COUPLE COEFFICIENT ROLL-SWAY 129
FIGURE H-6 COMPARISON COUPLE COEFFICIENT ROLL-HEAVE 129
FIGURE H-7 COMPARISON WAVE HEIGHT & PERCENTAGE VISCOUS DAMPING 129
FIGURE H-8 COMPARISON AMPLITUDE KEEL FORCE & PHASE SHIFT 130
FIGURE H-9 COMPARISON FORCE AMPLITUDE Y- & Z-DIRECTION 130
FIGURE H-10 COMPARISON CD-VALUE & CENTRE OF PRESSURE 130
FIGURE H-11 COMPARISON ADDED MASS & DAMPING COEFFICIENT 130
FIGURE H-12 COMPARISON COUPLE COEFFICIENT ROLL-SWAY 131
FIGURE H-13 COMPARISON COUPLE COEFFICIENT ROLL-HEAVE 131
FIGURE H-14 COMPARISON WAVE HEIGHT & PERCENTAGE VISCOUS DAMPING 131
FIGURE H-15 COMPARISON AMPLTTUDE KEEL FORCE & PHASE SHIFT 132
FIGURE H-16 COMPARISON FORCE AMPLITUDE Y- & Z-DIRECTlON 132
FIGURE H-17 COMPARISON CD-VALUE & CENTRE OF PRESSURE 132
FIGURE H-18 COMPARISON ADDED MASS & DAMPING COEFFICIENT 132
FIGURE H-19 COMPARISON COUPLE COEFFICIENT ROLL-SWAY 133
FIGURE H-20 COMPARISON COUPLE COEFFICIENT ROLL-HEAVE 133
FIGURE H-21 COMPARISON WAVE HEIGHT & PERCENTAGE VISCOUS DAMPING 133
FIGURE I-l REFERENCE KEEL AT T/T = 0, FREQUENCY = 0.6[HZ], ROLL AMPLITUDE = 14 [DEG] 134
FIGURE 1-2 REFERENCE KEEL AT T/T = 2/11 PI, FREQUENCY = 0.6[HZ], ROLL AMPLITUDE = 14 [DEG] 134
FIGURE 1-3 REFERENCE KEEL AT T/T = 4/11 PI, FREQUENCY = 0.6[HZ], ROLL AMPLITUDE = 14 [DEG] 134
FIGURE 1-4 REFERENCE KEEL AT T/T = 1/2 PI, FREQUENCY = 0.6[H2], ROLL AMPLITUDE = 14 [DEG] 135
FIGURE 1-5 REFERENCE KEEL AT T/T = 15/22 PI, FREQUENCY = 0.6[HZ], ROLL AMPLITUDE = 14 [DEG] 135
FIGURE 1-6 REFERENCE KEEL ATT/T = PI, FREQUENCY = 0.6[HZ], ROLL AMPLITUDE = 14 [DEG] 135
FIGURE 1-7 SMALL KEEL ATT/T = 0, FREQUENCY = 0.6[HZ], ROLL AMPLITUDE = 14 [DEG] 135
FIGURE 1-8 SMALL KEEL AT T/T = 2/11 PI, FREQUENCY = 0.6[HZ], ROLL AMPLITUDE = 14 [DEG] 136
FIGURE 1-9 SMALL KEEL ATT/T = 4/11 PI, FREQUENCY = 0.6[HZ], ROLL AMPLITUDE = 14 [DEG] 136
FIGURE M O SMALL KEEL AT T/T = 1/2 PI, FREQUENCY = 0.6[HZ], ROLL AMPLITUDE = 14 [DEG] 136
FIGURE M l SMALL KEEL ATT/T = 15/22 PI, FREQUENCY = 0.6[HZ], ROLL AMPLITUDE = 14 [DEG] 136
FIGURE 1-12 SMALL KEEL ATT/T = 19/22 PI, FREQUENCY = 0.6[HZ], ROLL AMPLITUDE = 14 [DEG] 137
FIGURE 1-13 SMALL KEEL ATT/T = PI, FREQUENCY = 0.6[HZ], ROLL AMPLITUDE = 14 [DEG] 137
FIGURE 1-14 CIRCLE SHAPED KEEL AT T/T = 0, FREQUENCY = 0.6 [HZ], ROLL AMPLITUDE = 16 [DEG] 137
FIGURE 1-15 CIRCLE SHAPED KEEL AT T/T = 2/11 PI, FREQUENa = 0.6 [HZ], ROLL AMPLITUDE = 16 [DEG] 137
FIGURE 1-16 CIRCLE SHAPED KEEL AT T/T = 4/11 PI, FREQUENCY = 0.6 [HZ], ROLL AMPLITUDE = 16 [DEG] 138
FIGURE 1-17 CIRCLE SHAPED KEEL AT T/T = 1/2 PI, FREQUENCY = 0.6 [HZ], ROLL AMPLITUDE = 16 [DEG] 138
FIGURE 1-18 CIRCLE SHAPED KEEL AT T/T = 15/22 PI, FREQUENCY = 0.6 [HZ], ROLL AMPLITUDE = 16 [DEG] 138
FIGURE 1-19 CIRCLE SHAPED KEEL AT T/T = 19/22 PI, FREQUENCY = 0.6 [HZ], ROLL AMPLITUDE = 16 [DEG] 138
FIGURE 1-20 CIRCLE SHAPED KEEL AT T/T = PI, FREQUENCY = 0.6 [HZ], ROLL AMPLITUDE = 16 [DEG] 139
FIGURE 1-21 SMALL KEEL AT T/T = 1/2 PI, PHIA = 4 [DEG], FREQUENCY = 0.6 [HZ] 139

107
Verification ofCFD calculations with experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

FIGURE I-22SMALL KEEL AT T/T = 1/2 PI, PHIA = 8 [DEG] , FREQUENCY = 0.6 [HZ] 139
FIGURE 1-23 SMALL KEEL AT T/T = 1/2 PI, PHIA = 14 [DEG] , FREQUENCY = 0.6 [HZ] 139
FIGURE 1-24 CIRCLE SHAPED KEEL AT T/T = 1/2 PI, PHIA = 8 [DEG] , FREQUENCY = 0.6 [HZ] 140
FIGURE 1-25 CIRCLE SHAPED KEEL AT T/T = 1/2 PI, PHIA = 12 [DEG] , FREQUENCY = 0.6 [HZ] 140
FIGURE 1-26 CIRCLE SHAPED KEEL AT T/T = 1/2 PI, PHIA = 16 [DEG] , FREQUENCY = 0.6 [HZ] 140

108
 Index of

Index of tables

TABLE 2-1 DIMENSIONLESS COEFFICIENTS BASED ON FULL SCALE DATA 12

TABLE 5-1 INFTIAL CONDITIONS 37
TABLE 5-2 OVERVIEW COMPUTATIONAL SETTINGS 41
TABLE 6-1 OVERVIEW INVESTIGATION COMPUTATIONAL SETTING 45
TABLE 6-2 RESULTS COMPUTATIONAL SETTINGS COMPARISON 51
TABLE 8-1 OVERVIEW TEST RUNS 63
TABLE 8-2 MASS PROPERTIES MODEL 66
TABLE 8-3 INFLUENCE MASS PROPERTIES ON HYDRODYNAMIC MOMENT OF THE MODEL 75
TABLE 8-4 INFLUENCE MASS PROPERTIES ON HYDRODYNAMIC KEEL FORCE 76
TABLE 8-5 OVERVIEW RESULTS INVESTIGATION FREQUENCY CONTENT 78
TABLE 9-1 OVERVIEW COMPARED TEST RUNS 88
TABLE 9-2 OVERVIEW FILTERING EFFECT 89
TABLE 9-3 OVERVIEW ACCURACY LEVELS BETWEEN CFD AND EXPERIMENTS 94
TABLE E-1 DATA STATIC INCLINATION TEST, REFERENCE KEEL, HBK = 95 [MM] 122
TABLE E-2 DATA STATIC INCLINATION TEST, SMALL KEEL, HBK = 78 [MM] 122
TABLE E-3 DATA STATIC INCLINATION TEST, CIRCLE SHAPED KEEL 123
TABLE E-4 DATA STATIC INCLINATION TEST, SAW BLADE 123
Veriffcation ofCFD calculations with experiments on a rolling circular cylinder with bilge keels in a free surfyce.
University of Technology Delft & Bluewater Energy Services b.v.

References
[I] Dallinga, R.P. and Engelenburg, R van, FPSO ROLL Damping in Calm water and waves. MARIN,
Wageningen, 2004

[2] Ferziger, J.H. and Peric M., Computational Methods for Fluid Dynamics, New York, 2002

[3] Fluent inc., FLUENT 6.1 user's guide, 2003.

[4] Gerritsma J., Bewegen en Sturen I, Golven, University of Technology Delft, September 1997.

[5] Gerritsma J., Golven scheepsbewegingen, sturen en manoeuvreren. University of Technology Delft,
June 1994.

[6] Ikeda Y., Components of Roll Damping of Ship at Forward Speed, Published in the Journal of the
Society of Naval Architects of Japan, 1978.

[7] Ikeda Y., On Eddy Making Component of Roll Damping Force on Naked Hull, Published in the Journal
of the Society of Naval Architects of Japan, 1977.

[8] Ikeda Y., On Roll Damping Force of Ship -Effect of Hull Surface Pressure Created by Bilge Keels-,
Published in the Journal of The Kansai Society of Naval Architects, Japan 1979.

[9] Ikeda Y., Velocity Field around Ship Hull in Roll Motion, Published in the Journal of The Kansai
Society of Naval Architects, Japan 1978.

[10] Journee, J.M.J., Massie, W.W. Offshore Hydromechanics, First Edition, Delft University of
Technology, 2001

[II] Nieuwstadt, F.T.M., Turbulentie, Utrecht: Epsilon Uitgaven, 1998

[12] Standing, R.G., Prediction of viscous roll damping and response of transportation barges in waves,
BMT Fluid Mechanics Ltd. Teddington U.K. 1991

[13] Versteeg, H.K., Malalasekera, W., An introduction to Computational Fluid Dynamics, London, 1995

[14] Vugts, J.H., Cylinder motions in beam waves. Shipbuilding laboratory. Delft University of
Technology, 1968

[15] WS Atkins Consultants and members of the NSC, Best Practical Guide for Marine Applications of
CFD, 2003

[16] WS Atkins Consultants and members of the NSC, Marnet CFD Final Report and Review of the state-
of-the-art in the application of CFD in the Marine and Offshore Industries, 2003

[17] White F.M., Fluid Mechanics, Singapore, 1999.

[18] Yeung R.W., Roddier D., Alessandrini B., Gemntaz L. and Liao S.W. On Roll Hydrodynamics of
Cylinders Fitted with Bilge Keels, University of California at Berkeley.

110
Appendices

A. Example User Defined Function (UDF)

#include "udf.h"
#include "threads.h"
#define FLUIDJD 1

DEFINE_ADJUST(varsliding,domain)
{
Thread *thread = Lookup_Thread(domain,ll);
real current_time ;
real rot_speed;
real frequency;
real T ;
real omega;
real p i ;
real amplitude;
real amplitude_rad ;
real f ;
real f f ;
pi = 3.141592654 ;

frequency = 0.6 ; / * [Hz] */

amplitude = 1 6 ; / * [deg] */

current_time = RP_Get_Real("flow-time");
T = 1 /frequency ;
omega = frequency * 2 * p i ;
amplitude_rad = amplitude * pi /180 ;

If (current_time < (4*T)) {

f = 0.5*(sin((0.25*pi/T)*current_time-0.5*pi))+0.5 ;
ff = 0.5*(cos((0.25*pi/T)*current_time-0.5*pi))*(0.25*pi/T);
rot_speed = f*amplitude_rad*omega*sln(omega*current_time)-ff*amplitude_rad*cos(omega*current_time);
}

else{
rot_speed = amplitude_rad*omega*sin(omega*current_time);
}

/ * MessageO("\n rot_speed =%e \n",rot_speed);

MessageO("\n current_time =%e \n",current_time); */

/ * Rotational origin */
THREAD_VAR(thread).fluid.origin[0]=0.;
THREAD_VAR(thread).fluid.origin[l]=0.;

/ * Translational speed: */
THREAD_VAR(thread).fluid.velocity[0]=0.;
THREAD_VAR(thread).fluld.velocity[l]=0.;

/ * Rotational speed */
THREAD_VAR(thread).fluid.omega=rot_speed;
}
Verification of CFD calculations with experimen ts on a rolling circular cylin der with bilge keels in a fiee sur^ce.
University of Technology Delft & Bluewater Energy Services b.v.

B. Example standard journal for repetitive computations

; Read mesti file

re TUD002.msli
; C hecl< grid
gc
; Reorder domain
g r reorder-domain
; Display grid
dg

; start unsteady calculation, first order

define models unsteady-lst-order? y

; Enable multi-phase
define models multiphase vof 2 geo-reconstruct 0.25 y y

; Enable k-epsilon turbulence model with near wall treatment

define models viscous ke-rng? y
define models viscous near-wall-treatment enhanced-wall-treatment? y
define models viscous near-wall-treatment wf-pressure-gradient-effects? y
define models viscous rng-differential-visc? y

; C opy water-liquid from data-base

define materials copy fluid water-liquid

; Define phases
define phases phase-domain 2 air no
define phases phase-domain 3 water y water-liquid

; Define operating conditions

define operating-conditions operating-pressure 0
define operating-conditions gravity y 0 -9.81
define operating-conditions reference-pressure-location 25 1
define operating-conditions operating-density? y 1.225

; Define boundary conditions (activate moving inner mesh)

define boundary-conditions fluid 11 mixture n n n n y O O O O O n n n

; Set grid interface

define grid-interfaces create interface innerjnterface outerjnterface n n

; Hook up UDF
define user-defined compiled-functions compile libudfy varsliding.c "
define user-defined compiled-functions load libudf
define user-defined function-hooks "none" "varsliding" "none" "none" "none" "none"

; Set solution controls of solver

solve set discretization-scheme pressure 14
solve set discretization-scheme flow 22
solve set discretization-scheme mom 4
solve set discretization-scheme k 4
solve set discretization-scheme epsilon 4
solve set under-relaxation pressure 0.6 ^ ■ '—
solve set under-relaxation density 0.6
solve set under-relaxation body-force 0.6
solve set under-relaxation mom 0.6
solve set under-relaxation mp 0.6
solve set under-relaxation k 0.6
solve set under-relaxation epsilon 0.6
solve set under-relaxation turb-viscosity 0.6

; Set initial turbulence values and Initialize the solution

solve initialize set-defaults mixture k le-06
solve initialize set-defaults mixture epsilon 0.0001
solve initialize initialize-flow

112
 Appendices

; Make line for wave height measurement

surface line-surface wave 0.53 -1.19 0 0.53 1.2 0

; Patch Vol-Frac to water-level of y=0

adapt mark-inout-rectangle y n -35 35 -1.5 0
solve patch water () hexahedron-rO () mp 1
/
; plot the volume fraction contours
display set contours filled-contours? y
display contour water vof 0 1

; Set and plot the monitors

solve monitors residual plot? y
solve monitors residual convergence-criteria 0.001 0.001 0.001 0.001 0.001
solve monitors force moment-coefficient y 9 8 () y y "moment_time_graph_keel_and_cyl.txt" y 1 y 0 0
solve monitors force drag-coefficient y 9 8 () y y "drag_time_graph_keeLand_cyl.txt" y 2 y 1 0
solve monitors force lift-coefficient y 9 8 () y y "lift_time_graph_keel_and_cyl.txt" y 3 y 0 1

113
Verification of CFD calculations with experiments on a rolling circular cylinder with b ilge keels in a free surfyce.
University of Technology Delft & B luewater Energy Services b.v.

C. Results keel shape selection

Results turbulence viscosity registration

Figure C-1 Reference keel & Option 1: Y-shape 30 [deg]

Figure C-2 Option 3: Y-shape 90 [deg] & Option 4: T-shape 1*H

Figure C-3 Option 6: T-shape 0.5*H & Option 7: Circle shaped keel

-I— -F--I--I— 1— 1— -I— -I— 1— -I— -I- O O O O O O O O C D O

o o o o o o o o o o o O O O O O O O O O O
+ + + + + + + + + + + + + + + + + -i- + + +
U(Uii:ituiUii:>iucuiUiU(U(U(U(D(U(Dii:>(UiU(U(U
o o c n o i c o c o r ^ r ^ c D C D L O L o o L o o L O t o L o o i - n c i
■^ocoh-a:iLO'«3-o7Cvi-'— ' ^ ' : | - ' t n c ^ c \ i c > j - ^ - ' — o o

Figure C-4 Turbulence intensity indicator [%]

114
 Appendices

Results Pressure registration

Figure C-5 Reference l<eel & Option 1: Y-shape 30 [deg]

Figure C-6 Option 3: Y-shape 90 [deg] & Option 4: T-shape 1*H

Figure C-7 Option 6: T-shape 0.5*H & Option 7: Circle shaped keel

CM CM C\
] CM en CD C i o C3
O O <=> + -I- -K + -h
+ + -H -I-
03
+ 03 03 03
CO
03
LO
03 03 03 03 03
03 03 03 03 03 CD LO CD LO CD
LO -I— -^ OO T— LO
LO ■•- CO
-I— • ' - ■ r - C M C M C M C ^ O O
r o C O C M C M C N ] - ^ - ^ - ^ I

'*

Figure C-8 Pressure indicator [%]

Verification ofCFD calcu lations with experiments on a rolling circu lar cylinder with bilge keels in a free su rface.
University of Technology Delft & B luewater Energy Services b.v.

Results Pressure registration

.-.■•ri; i '■

Figure C-9 Reference keel & Option 1: Y-shape 30 [deg]

J ■■
rv^ /''■■

■--■"Ci"

Figure C-10 Option 3: Y-shape 90 [deg] & Option 4: T-shape 1*H

'■:"'4'''^ y^/-^.'\ ■'■■■. ■

'

Figure C-11 Option 6: T-shape 0.5*H & Option 7: Circle shaped keel

I J l — 1— 1— -1— -I— -I— X--I— 1— ■■— 1— -1— T - - I — -I— 1— 1— 1— t.V"-"

O O O O C 3 O C 3 O O O O O O C 3 C 3 C 3 O I 0 O O O
4 I I I I I I I I I I I I I I I I I I I -i-
O O O O O O O O O O O O O O O O O O O O O
O l f l O l O c p l q O l O O l O O L O O l O O l O O l O O O O
■ ■ " ^ o i a i o d a d f ^ K t b u s L o i o - ^ - ^ c d c o o J o i - i - ^ - i - i o o

Figure C-12 Velocity indicator [m/s]

116
 Appendices

D. Detailed lay out experimental setup

Force receptors
A total of 8 force receptors were installed in the model. Two stain gauges were attached to each hinge to
measure the forces in y and z direction respectively, these were standard "blocks": 80 [kg] for the z
direction and 40 [kg] for the y direction. Two pure normal stress gauges were used to measure the applied
moment from the hydraulic motor on the model. These receptors can theoretically only withstand normal
forces; forces in the other direction will snap the slim
connection bolts. In these experiments the possibility
that the force is not totally inline with these force
receptors will be neglected. The maximum load of strain gauge Strain gauge
those normal stress gauges is 500 [N], it turned out y-direction z-direction
that these receptors determined the maximum
frequency that could be reached during the
experiments. The last two force receptors were used
to measure the force on the middle part of the keel.
These were also normal blocks and were mounted to a Steel plate
black PVC support in the model, see figure 8-3. The
maximum load on these receptors is 100 [N] each.

Figure D-1 Hinge and Strain gauges

Bilge keels
There are two different keel parts: "static keels" and a "measured keel". These two have a different
construction method.

strain gauge
keel force

PVC support
steel Filling

steel keel holder

Figure D-2 Strain gauge keel

The static keels are connected to the PVC tube by fixed steel "keel holders". These keel holders are glued in
a PVC pocket, which on its turn is glued in the large PVC tube, making it a watertight construction. The keel
is connected to the steel holders by means of a M5 bolt and RVS fillings. These fillings are used to get rid of
the small tolerances between the keel and the keel holders and they make the lining of the three keel parts
more easily.
The measured keel is mounted to a steel keel holder as well (see Figure D-2). This keel holder is connected
to a steel filling; this filling is connected to the force receptor. This force receptor is bolted to the side of the
black PVC supports by means of some aluminum parts. To get a watertight connection one groove are

117
Verification ofCFD calculations with experiments on a rolling circular cylinder with bilge keels in a free surface.
Univereity of Technology Delft & Bluewater Energy Services b.v.

made in the keel holder and one in the PVC pocket. This PVC pocket is glued in the large PVC-tube. A silicon
tube and two rubber rings in the two grooves made it all watertight and at the same time keeping the keel
detached from the model. ___

The keels had to be changed when there was still a buoyancy force on the model; the force receptors in the
z-direction were not strong enough to withstand the weight of the model and the ballast blocks. To change
the keels, the water level was lowered around 13 [cm] (this took around 45 [minutes]), after lowering it is
possible to walk in the tank with a dry-suit. With the model at its maximum roll angle it was possible to
change keel keels. This meant unscrewing the all six M5 bolts, fetching the fillings, change keels, replace
fillings and screw the bolts back in.
All oscillations had to be performed with the keel totally vertical keel at the mid-point of the motion. This
was ensured by measuring ones the deviation between an exact vertical keel and the given signal by the
inclinometer. After this measurement the calibration of the inclinometer is never changed.
To prevent a water flow through the small cap between the keel and the hull, a filling of sealant is used.
Special care is demanded on the dynamic keel, because the sealant would connect the keel to the hull,
which could disturb the keel measurements. Keels that were prepared outside the test section could be
sealed with the aid of a dummy hull; after the sealant was dry it is ensured that it does not stick to the hull.
The first keel had to be treated on the test site; here two layers of paper prevented the sealant from sticking
to the hull. One layer of paper was removed when the sealant was dry. A discovered disadvantage of this
paper was that it curled after a couple of days in the water. The seal was not as smooth as it should be but
it is considered satisfying. In both cased the small remaining gap between the sealant and the hull was
closed by the pressure difference of the upstream and downstream side of the keel.

Figure D-3 Ink injection

Ink injection
To get a visual impression of the flow around the keel a device was made to inject ink, see figure d-3. The
ink was "methylene blue" and is injected by a needle. First it went trough a flexible small hose to reduce the
interference with the motion of the model. The second and last part it went trough a brass tube to get the
ink right at the tip of the bilge keel. The ink was diverging in the water very quick, therefore it was
demanded to release the ink at the very tip of the keel. The flow was recorded by a normal steady cam,
which was located in front of the underwater window. The midpoint of the camera was located exactly along
the edge of the keel.

In x-direction the release point was located around 30 cm from the window, here it is expected that the flow
is without interference with the wall and close enough to record on camera. To be able to determine the
scale on the screen, bright yellow stripes were stick to the edge of the keel with intervals of 10 [mm].

118
 Appendices

The model blocks the light from above, which made it too dark around the keel to see the ink injection. To
overcome this problem a mirror was placed at the bottom of the tank. A strong light from above aimed on
this mirror illuminated the bottom of the model and the keel. To improve the recording of the flow even
more, all light behind the camera was dimmed and the experiments were performed after sunset to prevent
reflection from the sunlight.

Wave height gauge

All tests are performed in calm water, to get an impression of the capabilities of FLUENT to predict the wave
heights radiated from the oscillation, a wave height gauge is placed at the beach side of the model. A
standard wave height gauge from the laboratory was used. This device is based on the resistance between
two parallel strings.

Ballast
Around 190 kilos of lead ballast is put in the model to compensate the buoyancy force. A model in floating
condition has advantage that the strain gauges in vertical direction can be kept small and sensitive. On the
other hand the small gauges could not support the model including ballast when the water level is lowered.
This made bilge keel changes not an easy task and made dry oscillation to determine the solid mass moment
of inertia of the total model impossible.

Figure D-4 Distribution ballast lead

The ballast was positioned to get the vertical centre of gravity exactly on the centre of rotation; this would
make the forces during oscillation smaller and make it less difficult to subtract the influence of the mass
distribution. In horizontal direction the mass was located such that the model floated in the water without
trim. Figure D-4 gives a drawing of the ballast blocks, an exact location, centre of gravity and a mass
moment of inertia can be found in the appendix F.

Hinging and lining of the model

The model was connected to the lorry by two hinges and the axis of the hydraulic motor. The challenge is to
get those three axes inline with each other. The two hinges can be lined during construction with the aid of
a steel bar that keeps the hinges inline when they were bolted into the model. The lining of the hydraulic
motor had to be done when the model was already hanging under the towing lorry. The weight of the motor
and the impossibility to do accurate measurement, made it hard to get the lining accurate enough. The
lining can be tested with a static inclination; static under a certain angle should only add a momentum
compared to the initial situation. After a first series of inclination tests, it was decided to do the lining all
over. After the second lining, it was improved, but not accurate enough to consider it as a pure momentum.
In the appendix E the results of the static inclination can be found. Force equilibrium in z and y direction
should be zero during those inclination tests. It can be found this is the case within 2-3 percent of the model

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University of Technology Delft & Bluewater Energy Services b.v.

weight. It is not expected that those small deviations from zero lead to inaccuracies during comparison with
CFD. "^ -—

Oscillator
The model is oscillated by means of a hydraulic motor, powered by a hydraulic pump system. The motor is a
house-made model of mechanical engineering. The pump is taken from the laboratory of ship
hydromechanics; the one of the six-lag oscillator.
The hydraulic pump only generated 100 bar work pressure to the system of the motor. All options to control
the motion are in the PID-system of the motor. The motor is controlled by a normal feedback system, with
the following block diagram:

he reference signal is produced by a standard signal generator of the laboratory; this signal was registered
as described in paragraph 0. Care is demanded when using this signal because it can have a large
discrepancy with the actual motion because it is tapped in front of the summing junction. At the summing
junction the reference signal and the feedback signal are subtracted and produces the error signal e. This
error signal is amplified by the gain, K in the controller. This factor K determines the actual stiffness of the
whole system; a large K means a large reaction on a certain difference between the reference signal and
feedback signal. A stiff system can cause overshoot of the motion or even instability due to a lack of
damping. On the other hand a weak system leads to a large discrepancy between the given motion and the
actual motion. An optimum value of the gain is determined experimentally and was set during the tests at 7
on a scale of 0 to 10.
Other control options as calibration points were determined prior to the experiments, but won't be handled
any further.
Registration
ref. signal -- -
Motion

On
Input reference
Controller Actuator Hydraulic valve
Signal

Pot meter *

Figure D-5 Block diagram hydraulic system

The signal from the controller is used by the actuator to change the signal to a hydraulic flow that steers the
valves. The position of the valves determines the direction of the flow through the motor and therefore the
oscillation motion. This motion is measured by a standard pot-meter, which gives the feedback signal to the
summing junction.

Registration of roll motion

The motion is registered by an inclinometer, which gave the possibility to put this device at a random
location in the model. The used inclinometer is a "Jewell LSO 14,5". This meter has a bandwidth or -3db-
point of 15 [Hz]. This means that at the tested maximum frequency of 0.8 [Hz] this meter is not expected to
have an overshoot. At the -3db-point this inclinometer has a phase shift of -90 degrees, which is an
indication that at the tested frequency a small phase can be expected.
To overcome this problem, the phase angle of the reference signal of the hydraulic system will be taken. As
stated above the amplitude of this reference signal can not be used but the phase angle at 0.8 [Hz] is
expected to be reliable. To the opinion of the builders of the hydraulic system, a phase shift is only expected
above 1000 Hz.

120
 Appendices

During the following computations in this report the roll amplitude is taken from the inclinometer and the
phase angle is taken from the reference signal.

Data acquisition
Amplifier
All signals from the strain gauges were put through the standard amplifiers of the hydrodynamics laboratory.
These analog amplifiers are type MCA 100, build by Peekel.

Filter
All signals were put through a filter with a cut off frequency above 10 Hz. This should be high enough to
keep all hydrodynamic effects within the signal; it is above the 16'*^ harmonic of the roll motion. A filter
normally introduces a small phase shift in the signal, that's why all signals were put through. The home-build
"variable stat" filter was used

AD - converter
To be able to read the signals with the computer, the standard AD-converter was used.

DAStank
DAStank is the standard software on the lorry (written by ir.J Ooms) and is used to do the actual data
acquisition. The runs are started and stopped with this program and with the calibration factors properly
given; it gives the real forces as output. All signals were stored in binary files with the raw voltage output of
the AD-converter. This made it possible to make adjustments to the calibration factors afterwards.

Datamanager
Datamanager was used to do the first signal investigation; it displays graphs of the signals and produces the
first three harmonic of every signal. This made it possible to make a quick estimation of the reliability of the
test run.

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Verification of CFD calculations with experiments on a rolling circular cylinder with b ilge keels in a h-ee surface.
University of Technology Delft & B luewater Energy Services b.v.

E. Results static inclination tests

Keel Type Flat Plate

Height B ilge keel 95 [mm]
Mass total model 2 7 3 . 4 [kg]

Run name run086 run090 run094 run098 run096 run092 runOSS run084
Static angle tdeg] 16.0 14.0 11.9 9.7 -10.0 -12.0 -13.9 -15.6

D e t e r m i n a t i o n mass k e e l , including connection c o m p o n e n t s , excluding bouyancy effect

Force perpendicular to keel, right [N] 5.84 5.13 4.39 3.61 -3.55 -4.31 -4.99 -5.64
Force perpendicular to keel, left :N] 5.56 4.83 4.08 3.30 -3.81 -4.53 ■5.21 ■5.83
Total force [N] 11.41 9.96 8.47 6.91 -7.36 ■8.85 -10.20 -11.47 Average
Weight keel etc. [kg] 4.60 4.59 4.56 4.55 4.71 4.71 4.71 4.72 4.64

D e t e r m i n a t i o n Rg (distance COG < - > c e n t r e of roil)

Force on aft moment receptor [N] 17.82 13.24 7.00 2.25 10.10 14.73 19.50 23.58
Force on front moment receptor [N] ■22.66 -22.32 -22.69 -21.10 38.75 48.03 57.58 66.88
Total applied moment cum] 8.09 7.11 5.94 4.67 -5.73 -6.66 -7.62 -8.66 Average
Rg [m] 0.011 0.011 0.011 0.010 0.012 0.012 0.012 0.012 0.011

D e t e r m i n a t i o n force equilibrium in y-direction

Force y-direction by blocks in hinges [N] ■29.77 -26.43 ■23.91 -19.78 7.08 6.58 4.92 2.66
Force y-directlon by moment receptors [N] 1.33 2.19 3.24 3.18 8.47 13.06 18.49 24.36
Total force y-direction [kg] -2.90 ■2.47 -2.11 -1.69 1.59 2.00 2.39 2.75 Average
Percent deviation from zero, based on model «eight [%] 1.06 0.90 0.77 0.62 0.58 0.73 0.87 1.01 0.8

D e t e r m i n a t i o n force equilibrium in z-direction

Force z-direction by blocks in hinges [N] ■3.32 1.94 14.00 15.35 ■39.09 -55.75 -64.81 ■75.52
Force z-direction by moment receptors [N] -4.65 -8.81 -15.35 -18.58 48.11 61.39 74.83 87.11
Total force z-directton [kg] -7.97 -6.87 ■1.36 -3.23 9.02 5.63 10.03 11.59 Average
Percent deviatton from zero, based on model /veight [%] 2.91 2.51 0.50 1.18 3.30 2.06 3.67 4.24 2.5

Table E-1 Data static inclination test. Reference keel, Hbk = 95 [mm]

Keel Type Flat Plate

Height B ilge keel 78 [mm]
Mass total model 271.5 [kg]

Run name run328 run330 run332 run334 run346 run344 run342 run340
Static angle [deg] 23.0 19.9 16.2 12.3 -10.1 ■13.9 -17.7 -19.7

D e t e r m i n a t i o n mass k e e l , including connection c o m p o n e n t s , excluding bouyancy effect

Force perpendicular to keel, right [N] 7.25 6.38 5.25 4.08 -3.19 ■4.41 -5.62 -6.22
Force perpendicular to keel, left [N] 7.28 6.26 5.07 3.84 -3.27 -4.47 -5.69 -6.29
Total force [N] 14.53 12.63 10.32 7.92 ■6.46 ■8.89 -11.31 -12.50 Average
Weight keel etc. [kg] 4.09 4.08 4.07 4.08 4.03 4.05 4.08 4.08 4.07

D e t e r m i n a t i o n Rg (distance COG < - > c e n t r e o f roll)

Force on aft moment receptor [N] 39.00 29.00 19.26 10.23 19.25 29.34 40.18 45.67
Force on front moment receptor [N] -14.17 -16.34 ■18.40 -18.61 43.07 62.08 81.14 90.79
Total applied moment [Nm] 10.64 9.07 7.53 5.77 -4.76 -6.55 ■8.19 ■9.03 Average
Rg [m] 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010

D e t e r m i n a t i o n force equilibrium in y-direction

Force y-direction by blocks in hinges [N] -33.20 -31.73 -28.27 -23.12 7.38 3.70 -3.43 ■8.31
Force y-direction by moment receptors [N] -9.69 -4.30 -0.24 1.79 10.98 22.01 36.94 45.89
Total force y-directton [kg] -4.37 -3.67 •2.91 ■2.17 1.87 2.62 3.42 3.83 Average
Percent deviatkxi from zero, based on model weight [%] 1.61 1.35 1.07 0.80 0.69 0.97 1.26 1.41 1.1

D e t e r m i n a t i o n force equilibrium in z-direction

Force z-direction by blocks in hinges [N] -27.48 -15.40 ■3.56 5.82 -57.31 -82.85 -107.98 ■119.60
Force z-direcdon by moment receptors [N] 22.86 11.91 0.82 -8.19 61.35 88.72 115.55 128.51
Total force z-direction [kg] -4.62 -3.49 -2.74 -2.38 4.04 5.87 7.58 8.91 Average
Percent deviation from zero, based on model weight [%] 1.70 1.29 1.01 0.88 1.49 2.16 2.79 3.28 1.8

Table E-2 Data static Inclination test, Small keel, Hbk = 78 [mm]

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 Appendices

Keel Type Circle

hieight Bllge keel 88 [mm]
Mass total model 274.4 [kg]

Run name run426 run428 run430 run432 run444 run442 run440 run438
Static angle [deg] 23.5 19.7 15.8 12.2 -10.1 -14.0 -17.9 -19.5

Determination mass Iceel, including connection components, excluding bouyancy effect

Force perpendicular to keel, right [N] 7.83 6.66 5.40 4.24 -3.24 -4.54 ■S.86 -6.40
Force perpendicular to keel, left [N] 7.45 6.20 4.90 3.72 ■3.47 -4.74 -5.99 -6.53
Total fofce [N] 15.28 12.85 10.30 7.96 -6.71 -9.28 -11.85 -12.94 Average
Weight keel etc. [kg] 5.30 5.29 5.27 5.25 5.31 5.32 5.33 5.34 5.30

Determination Rg (distance COG <-> centre of roll)

Force on aft moment receptor [N] 38.99 29.07 17.07 9.67 20.54 31.60 43.33 48.63
Force on front moment receptor [N] -16.53 -17.74 -19.45 -19.54 44.33 64.19 85.18 94.67
Total applied moment [Nm] 11.11 9.36 7.30 5.84 -4.76 -6.52 -8.37 -9.21 Average
Rg [m] 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010

Determination force equilibrium in y-direction

Force y-diredion by blocks in hinges [N] -37.90 -34.54 ■30.98 ■25.44 7.33 3.46 -4.48 -9.42
Force y-direction by moment receptors [N] -8.96 -3.82 0.65 2.08 11.35 23.12 39.45 47.90
Total force y-direction [kg] ■4.78 -3.91 -3.09 -2.38 1.90 2.71 3.57 3.92 Average
Percent deviation from zero, based on model weight [%] 1.74 1.42 1.13 0.87 0.69 0.99 1.30 1.43 1.2

Determination force equilibrium in z-direction

Force z-direction by blocks in hinges [N] -24.54 -13.74 -0.12 8.16 -59.86 -87.29 -115.12 -127.01
Force z-direction by moment receptors [N] 20.60 10.67 -2.29 -9.65 63.87 92.96 122.30 135.06
Total force z-direction [kg] -3.94 -3.07 -2.41 -1.48 4.01 5.67 7.18 8.05 Average
Percent deviatton from zero, based on model weight [%] 1.44 1.12 0.88 0.54 1.46 2.07 2.62 2.93 1.6

Table E-3 Data static inclination test, Circle shaped keel

Keel Type Saw Blade

Height Bilge keel 104 [mm]
Mass total model 273 [kg]

Run name run294 run296 run298 run300 run310 run312 run314 run316
Static angle [deg] 22.6 19.3 15.8 12.3 -10.3 ■14.1 ■17.8 -19.5

Determination mass keel, including connection components, excluding bouyancy effect

Force perpendicular to keel, right [N] 7.86 6.83 5.67 4.53 -3.24 ■4.57 ■5.82 -6.40
Force perpendicular to keel, left [N] 7.49 6.37 5.15 3.97 -3.66 -4.96 -6.20 ■6.77
Total force [N] 15.35 13.20 10.82 8.50 -6.90 -9.53 -12.02 -13.17 Average
Weight keel etc. [kg] 4.45 4.44 4.44 4.44 4.33 4.36 4.39 4.41 4.41

Determination Rg (distance COG <-> centre of roll)

Force on aft moment receptor [N] 30.96 19.53 8.87 0.21 12.43 23.54 34.18 38.98
Force on front moment receptor [N] -29.31 -31.72 -32.77 -32.11 40.25 60.99 80.94 89.69
Total applied moment [Nm] 12.05 10.25 8.33 6.46 -5.56 -7.49 ■9.35 -10.14 Average
Rg [m] 0.012 0.012 0.011 0.011 0.012 0.011 0.011 0.011 0.011

Determination force equilibrium in y-direction

Force y-directon by blocks in hinges [N] -31.03 -29.18 ■24.96 ■19.50 12.56 8.70 1.99 -1.84
Force y-direction by moment receptors [N] -0.63 4.04 6.49 6.81 9.38 20.61 35.18 42.89
Total force y-direction [kg] -3.23 -2.56 -1.88 -1.29 2.24 2.99 3.79 4.18 Average
Percent devBtton from zero, based on model waght [%] 1.18 0.94 0.69 0.47 0.82 1.09 1.39 1.53 1.0

Determination force equilibrium in z-direction

Force z-direction by blocks in hinges [N] 0.45 14.02 25.87 34.30 -42.45 -70.92 -96.59 -107.49
Force z-direction by moment receptors [N] 1.52 -11.50 -23.00 -31.17 51.84 81.97 109.61 121.32
Total force z-direction [kg] 1.97 2.52 2.86 3.13 9.39 11.06 13.02 13.83 Average
Percent deviatkxi from zero, based on model weight l%] 0.72 0.92 1.05 1.15 3.44 4.05 4.77 5.07 2.6

Table E-4 Data static Inclination test, Saw blade

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Verification ofCFD calculations with experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

F. Properties model

Group description Material Average density Mass COG all part! Ixx
X z
rkg/m3] [kg] [m] [m] tm] [kgm2]
Model Tube PVC gray 1200.0 37.64 0.000 -0.031 1.350 2.3184
Keel holders PVC black 1400.0 0.44 0.000 -0.235 1.352 0.0244
Support PVC PVC black 1400.0 11.27 0.000 -0.124 1.365 0.3542
Support foam Foam 34.5 1.50 0.000 •0.127 1.372 0.0453
Static keel holder Steel 7830.0 0.35 0.000 -0.255 1.351 0.0230
End plates Wood 600.0 3.94 0.000 0.000 1.350 0.1145

Keel force receptor Dynamic keel holder Steel 7635.5 1.24 0.000 -0.214 1.351 0.0576
Keel force receptors Steel 9326.7 0.96 0.000 -0.109 1.351 0.0119
Filling keel receptor Steel 7594.3 1.04 0.000 -0.159 1.351 0.0256
Corner piece Aluminum 3088.1 0.49 0.000 -0.096 1.351 0.0049
Filling keel receptor I I Aluminum 2592.6 0.56 0.000 -0.107 1.351 0.0067
Plate Aluminum 2619.0 0.88 0.000 -0.107 1.351 0.0122
Half force receptor Steel 9326.7 0.48 0.000 -0.128 1.351 0.0080

Axis Plate Steel 7604.2 2.92 0.000 -0.035 1.351 0.0045

Axis house Steel 9942.6 1.86 0.000 -0.010 1.353 0.0012
Axis Steel 13626.6 2.02 0.000 -0.001 1.353 0.0002
M8 connection Steel 7830.0 0.03 0.000 0.018 1.353 0.0000
Small filling Aluminum 2690.0 0.09 0.000 0.041 1.353 0.0002
Corner piece Aluminum 2690.0 0.49 0.000 0.066 1.364 0.0022
Half force receptor Steel 7830.0 0.44 0.000 -0.128 1.351 0.0067

keel FP92 static part Steel 7709.8 3.62 0.000 -0.299 1.351 0.3262
FP92 measured part Steel 7744.3 2.14 0.000 -0.299 1.351 0.1928
SB104, measured part Steel 7603.2 1.74 0.000 -0.299 1.351 0.1570
FP75 static part Steel 7663.9 2.44 0.000 -0.291 1.351 0.2071
FP7S measured part Steel 7677.7 1.44 0.000 -0.291 1.351 0.1222
TC88, Static part, PVC I PVC 1512.7 0.94 0.000 -0.294 0.423 0.0826
TC88, Static part, PVC I I PVC 1461.7 1.46 0.000 -0.294 2.277 0.1279
TC88, Static part, Steel Steel 7429.6 1.64 0.000 -0.279 1.351 0.1280
TC88, measured part, PVC PVC 1452.3 1.70 0.000 -0.294 1.352 0.1490
TC88, measured part. Steel Steel 7546.0 0.98 0.000 -0.279 1.351 0.0765

Lead Nol Lead 12503.5 22.52 0.000 0.005 2.238 0.0222

No2 Lead 13069.8 23.54 -0.081 0.011 2.148 0.1788
No3 Lead 11759.5 21.18 0.080 0.011 0.567 0.1589
No4 Lead 12781.1 23.02 0.080 0.011 2.146 0.1727
No5 Lead 13125.3 23.64 -0.081 0.011 0.563 0.1796
No6 Lead 12492.4 22.50 0.000 0.006 0.465 0.0222
No7 Lead 10359.5 25.36 0.000 -0.020 1.496 0.3069
No8 Lead 10596.4 25.94 0.000 0.050 1.496 0.2973
No9 Lead 10643.2 4.78 0.000 0.072 0.469 0.0410
Nolo Lead 11890.1 5.34 0.000 0.072 0.534 0.0458

Miscellaneous Bolts, glue, kit, fillings, etc [-] [-] 1.00 0.000 -0.250 1.350 0.0000
Inclinometer metal 1625.0 0.65 0.000 -0.064 2.021 0.0033

124
 Appendices

G. Hydrodynamic coefficients

Frequency = 0.2 [Hz] Frequency = 0.4 [Hz]

■X X

< 0 06
O.
(13
0 04 -

0 02 ■

O H 1-

-0 02
4 6 10 12 14 16
- > * a [deg]

Frequency = 0.6 [Hz]

-B- Reference 1
<) Small
- ^ Saw Blade
X Circle shaped
- f - Bare Hull |

Figure G - 1 Added mass roll-roll coefficient

Frequency = 0.2 [Hz] Frequency = 0.4 [Hz]

0 05 ^^.
Jfc;S=
o
0 04 <
~?- ^^#=*^o
0 03 . X' - ■ x-

"-» 0.02 '^'^ • Ö X

X ■
X
0.01 ■

1
n ii 1, 1 ,■ i-i i-T r-i
12 14
- > (tig [ * g ] - > <j)a [ * g ]

Frequency = 0.5 [Hz]

-B- Reference
() Small
^ 1 ^ Saw Blade
X Circle shaped
- 1 - Bare Hull |

Figure G-2 Damping roll-roll coefficient

125
Verification of CFD calculations with experiments on a rolling circular cylinder with b ilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

Frequency = 0.2 [Hz] Frequency = 0.4 [Hz]

19
X .
18 xj
.X- xl 0.9
x X

i.r -X-.
X O.B \
16 X
X ■
O
O
o1 X

07

1 5 O
O- o — ^
O
06 ^-'^ \
O ^ ^K"
1 4
-r^ fj ^:^=:==^=z=^==^
^

J
05
13
Q - i ^ — - ^ ^
^ K^ 04
1.2
:>tl*~~-
u——1,- +-I 1 ,
12 14 6 8 ID 12 14 16
^a [deg] ■H> (|)a [deg]

Frequency = 0.6 [Hz]

- B - Reference 1
■ O Small
0.7 -^ Saw Blade
X Circle shaped
06 -h Bare Hull |

< 06

04

H 1 h H 1 h

10 12 14 16

->

Figure G-3 Couple added mass roll-sway coefficient

Frequency = 0.2 [Hz] Frequency == 0.4 [Hz]

0.18

016
^ 03

0,14 „^*-' oJ ^ ^
3
f^ 012
J)r .o S 0.25
^ i : ^
J

© j ^ f ^ O
© '^^
- ^ 02 J
^ «1
.1.5!*=***0 at
i ^ -^o
<
- 006 < ^-^è^
,rfi:^-e ■X ■
X- X
■■X ■ X I X

^■^.oX
S2 ^ g - " - " ' ^ x X JD ■X
. 006 H
^^^-^ "^ t »i X \
O04 S
002
j\ 0,05

n 1, 1, 1, 1 , 1 , 1 , M -h— IT Ii —1-, h ^ — H
8 10 12 14 16 10 12 14 16
-^ <l>a [*9]

Frequency = 0.6 [Hz]

- B - Reference 1
<) Small
^1^ Saw Blade
■X Circle shaped
- H Bare Hull |

Figure G-4 Couple damping roll-sway coefficient

126
 Appendices

Frequency = 0.2 [Hz] Frequency = 0.4 [Hz]

Frequency = 0.6 [Hz]

o -h h -B- Reference
8mall
-0.02 (> Saw Blade
-*-
X Circle shaped
■00<
-h Bare Hull |
-006
CD

O. 008 ■

r "■■'
-0.12

-0.14

-0.16 -

-0.18

Figure G-5 Couple added mass roll-heave coefficient

Frequency = 0.2 [Hz] Frequency = 0.4 [Hz]

0 07
X1
006
x
3 005 \
..X '
ë
~?- 004
■\

X"
<:
CD

.X'
^ , 003
ih

y 002
x' ^
-^Bj
X

0.01
r: h
6
b
8
h
10
H i —— H — —-f
12 14
1
16
- > * . [deg]

Frequency = 0.6 [Hz]

-B- Reference 1
X O Small
Saw Blade
01 ..x" ] ^^
X Circle shaped
-1- Bare Hull |
.■X''
d 0o.oe
e _x'
0 06

.x'
0.04
...x'
^=a=*^ \
X
0 02

-*-
n : M- li 1 1 1 1 hr— F1
14 16
~> $3 W^l

Figure G-6 Couple damping roll-heave coefficient

127
Verification ofCFD calculations witii experiments on a rolling circular cylinder witii b ilge keels in a free surface.
University of Teclinoiogy Delft & B luewater Energy Services b.v.

H. Results comparison CFD-experiments

Keel: Flat plate. Hbk: 95 rmml.
60
-A-■- Expertnents, 0,6 [Hz]
2 4 6 8 10 12 14 16 1£
— A — C F D , 0,6 [Hz]
g 50- -10
■■□■ Expeitnents, 0,4 [Hz) ■■-A--- Expenments, 0,6 [Hz]

— ■ — C F D , 0.4 [Hz] —*—CF=D, 0.6[Hz]

£ 40 -20
- - -o- - ■ Experiments, 0.4 [ Hz]
1 30 — • — C F D , 0.4 [Hz]
f. 30 -
3
■g. 20 <::::
<E -50
^"^^^
-60 "^ ^^^^:z
10
^^ÖÏS,
0 8 10 12
Roll amplitude [deg]
Roll amplitude [deg]

Figure H-1 Comparison Amplitude keel force & Phase Shift

■■&■■- Experiments, 0.6 [Hz] - Experiments, 0.6 [Hz]

-*—CFD, 0.6 [Hz] -CFD, 0.6 [Hz]
-a--- Experiments, 0.4 [Hz] - Experinents, 0.4 [Hz]

80 - - • — C F D , 0.4 [Hz] -CFD, 0.4 [Hz]

60 j

-ë 40 I
3

Roll amplitude [deg]

Figure H-2 Comparison force amplitude y- & z-direction

U-i 120

16
Q, ^
n
X
H
r^ l ^ .
1? ■ ^
an
rfi ^i&..
10

f( ) 8
^fe;,;^!;^;^^;^^^^^J A i ^ - r r . ^ 3
60

-a- ■ Experiments, 0.6 [Hz]

» 6
- < W 1 0.6 [Hz]
O
11
40

4 -
Expertnents, 0.4 [Hz]
V.
■D-
70
2
- L I - Ü , 0.4 [Hz] u
0
~"*"
) 2 4 6 8 10 12 14 16 18 -10 12 14 16 « -
Roll amplitude [deg] Roll amplitude [deg]

Figure H-3 Comparison CD-value & Centre of Pressure

0,09 ^ Experinents, 0.6 [Hz]

r^ 0.09 - 0.08 i CFD, 0.6 [Hi}
Tl- 0,08 ExperiTients, 0,4 [Hz]
0,07 1
% 0.06 !
CFD, 0.4 [Hz]

*- 0 07 -
0.05 !
g 0.06
0,04 -
S 0,05
■A-- Expertnents, 0.6 [Hz] 0.03 i
a 0.04
- * — CFD, 0.6 [Hz] 0,02 ^
t 0.03 -
n- - - Expertnents, 0.4 [Hz] 001 ■
1 °« -m—CFD, 0.4 [HZ]
•* 0,01 -
0.5 1 1,5
Strouhal number [-] Strouhal number [-]

Figure H-4 Comparison Added Mass & Damping coefficient

128
 Appendices

■ a- - - Experitients, 0,6 [Hi]

-•—CFD, 0.6[H2] Experiments, 0.6 [Hl]

■Q-- Experiments, 0.4 [Hi] CFD, 0.6 [Hi]

-•—CfD.OAfH!] Experiments, 0,4 [Hz]

CFD, 0.4 [Hz]
5
« 0.5
£ 0.1

° 0.3
V
Q. 0.2

0.1

o
S 2 2.5
Strouhai number [-] Strouhai number [■]

Figure H-5 Comparison couple coefficient roll-sway

0.045 &••• Expertnents, 0.6 [Hz]

0 0.5 1 1.5 2 2.5 3 4,
0.02 O04 ^—CFD, 0.6 [Hz]
°--- Expertnents, 0.4 [Hz]
-0.04 0.035
*—CFD, 0.4 [Hz]
-0,06 0.03
■0.08 J 1^ 0.025
^).l
i 0.02
■O.U -■-br- Expertnents, 0.6 [Hz]

■OM *— -CFD, 0.6 [Hz]

0.015

-.-0-- Experinents, 0.4 [Hz] 0.01

•0.16
-CFD, 0.4 [Hz] 0.005
0.18 2

0
•0.2
strouhai number [-]
Strouhai number [-]

Figure H-6 Comparison couple coefficient roll-heave

• Expertnents, 0.6 [Hz]

□ 0.--- ■ ■■□■... ...□--.
^
-CFD, 0 6 [Hz] s
B
90

80
a Ö

—i^::^ '^^^U^
• Expertnents, 0.4 [Hz]
■s 70
* ■---li
■ -A

I s -CFD, 0.4 [Hz]

^ 60

1 50

40
■-■fr- Experiments, 0.6 [Hz]
w
■A — A — CFD 0.6 [Hz]
20
s .. -a-. Experments. 0.4 [Hz]

CFD 0.4 [Hz]

n

8 10 12 Roll amplitude [deg]

Roll amplitude [deg]

Figure H-7 Comparison Wave height & Percentage viscous damping

129
Verification of CFD calculations with experiments on a rolling circular cylinder with b ilge keels In a free surface.
University of Technology Delft & B iuewater Energy Services b.v.

Keel: Flat plate, Hbk: 78 [mml

■ a ■ Expertnents, 0,6 [Hz] 14 16 1

-4—CFD, 0.6 [Hz]
■ -fi- ■ ■ Experiments, 0.5 [Hz]
■a- ■ Expertnents, 0.4 [Hz]
- * — C r o , 0.5[Hz]
- • — CFD, 0.4 [Hz] ,-, -20
■ -D- ■ ■ Expeiiments, 0.4 [Hz]
-•—CH3, 0,4[Hz]

-50

■60-

8 10 12 -70

Roll amplitude [deg] Roll amplitude [deg]

Figure H-8 Comparison Amplitude keel force & Phase Shift

- Experiments, 0.6 [Hz] ■Lr- Experinents, 0.6 [Hz]

- CFD, 0. -»—CFD, 0.6[Hz]
■ Experiment! -D ■ - Experinents, 0.4 [Hz]
12
- CFD, 0.4 - • — CFD, 0.4 [HzJ
10

40 e

30 e

10 ^ 2

oi 8 10 12
O
6 8 10 12 14 16
Roll amplitude [deg] Roll amplitude [deg]

Figure H-9 Comparison force amplitude y- & z-direction

Expertnents, 0.6 [Hz]

140 OTJ, 0.6 [Hz]

120 Expenments, 0.4 [Hz]

CH), 0.4 [Hz]
£ 10

■A-- Expertnents, 0.6 [Hz]

Q - • — CFD, 0.6 [Hz]
■□-■- Experiments, 0 4 [Hz]
- • — CFD, 0.4 [Hz]

8 10 12 14 16 6 8 10 12

Roll amplitude [deg] Roll amplitude [deg]

Figure H-10 Comparison CD-value & Centre of Pressure

0.1 0.08 1 Expertnents, 0.6 [Hz]

r-, 0.09 i[Hz]
0.07
•^ 0.08 Expertnents, 0.4 [Hz]

0.07 ^ l[H2]
% 0.06 % 0-05
? 0.05
T.
0.04 -
r^
u
irt 0-04
0.03
■ Expertnents, 0,6 [Hz]
i 0.03
-CFD, 0.6 [Hz]
■o 0.02
F
- Expertnents, 0.4 [Hz]
< 0.01 -CFD, 0.4 [Hz]

0.5 1,5 2 2.5 1.5 2 2.5

Strouhal number [-] strouhal number [-]

Figure H-11 Comparison Added Mass & Damping coefficient

130
 Appendices

■ Experrnents, 0.6 [Hz]

-CFD, 0.6 [Hz] fi--- Experinents, 0.6 [Hz]

- Experiments, 0.4 [Hz] * — CFD, 0.6 [Hz]

-CFD, 0.4 [Hz] Q- ■ ■ Expcrinents, 0.4 [Hz]

CFD, 0 4 [Hz]

S 0.

1.5 2 2.5 3
Strouhal number [-] Strouhal number [-]

Figure H-12 Comparison couple coefficient roll-sw/ay

0.05 Experhients, 0.6 [Hz]

-0.02 - 0.045 CFD, 0.6 [Hz]

V 0.04 Experinents, 0,4 [Hz]
r^ ^).04
CFD, 0.4 [Hz]
0.035
s; -O IK 2
Ï ■»■<» ? 0.03

0.025

i -i fi- ■ Experinents, 0.6 [Hz]

■*—CFD, 0.6 [Hz]
1
a»
0.02

0.015
8 -0.12
o- - - ExperiTients, 0.4 (Hz]
0.01
Q- 0.14 •—CFD, 0.4 [Hz]
0.005
Ö -0.16 J
0
-0.18 J

-0.2 Strouhal number [ Strouhal number [-]

Figure H-13 Comparison couple coefficient roll-heave

100
-&-'- Experiments, 0.6 [Hz]
^ 90
-*—CFD, 0.6 [Hz]
■□■ ■ - Experinents, 0.4 [Hz] TO
O 80

O 70

£ 60

Iro»-
«"
g
30
■ Expeiiments, 0.6 [Hz]
—»—CFD, 0.6[Hz]
40
^ 20 ...a... Experiments, 0.4 [Hz]
10 ] —«—CFD, 0.4 [Hz]
ot- ,
6 8 10 12 0 2 4 (
Roll amplitude [deg] Roll amplitude [deg]

Figure H-14 Comparison Wave height & Percentage viscous damping

131
Verification of CFD calculations with experiments on a rolling circular cylinder with b ilge keels in a free surface.
University of Technology Delft & B luewater Energy Services b.v.

Keel: Circle. Hbk: 91 rmml

10 12 14 16 18

■ ■ -fl- - ■ Expertnents, 0.6 [Hz]

—*—CFD, 0.6 [Hz]

ff -15
a 30
1. 7S e -20
iS -7*1
^
é
?n
»
Ti 15
F -35
< 10 ---L-- Expertnents, 0.6 [Hz]
-40
5 — * — 0=0, 0.6 [H2]
•45
0
Roll amplitude [deg]
Roll amplitude [deg]

Figure H-15 Comparison Amplitude keel force & Phase Shift

■fi--- Expertnents, 0.6 [Hz]

-*—CFD, 0.6 [Hz]

§ 60 I

F
<
6 8 10 12 14 16 18 10 12 H 16 18
Roll amplitude [deg] Roll amplitude [deg]

Figure H-16 Comparison force amplitude y- & z-direction

35 1
■ Experiments, 0.6 [Hi]
30
-CFD, 0.6 [HI]

IL 25
c
a 20

1.5-

■ Experinents, 0 6 [Hz]

-CFD, 0.6 [Hz]

6 8 10 12 14 16 I S 10 12 14 16 18

Roll amplitude [deg] Roll amplitude [deg]

Figure H-17 Comparison CD-value & Centre of Pressure

0.16

O.H

0.12

■a.
0.1
-A- ^
0.08 Ü

0.06

0.04 E ■ Expertnents, 0.6 [Hz]

■•■£.--■ Experiments, 0,6 [Hz]
a 0.01
0.02 - CFD, 0.6 [Hz]
—*—CFD, 0.6[Hz]
0
s 2 2.5 3
Strouhal number [-] Strouhal number [-]

Figure H-18 Comparison Added Mass & Damping coefficient

132
 Appendices

o.s 0.4

0.7 0.35
"■'■■-i....

r-'- ^ ^ - ^
0.3 -

0.25

0.2 -

8 0.3- 0.15

u --■a--- Experinents, 0,6 [Hz] ■ Experinents, 0.6 [Hz]

0.1 -CfD, 0.61Hz]
—*—CFD, 0.6 [Hz]
0
> 2 2.5
Strouhal number [ strouhal number [

Figure H-19 Comparison couple coefficient roll-sway

0.12 ■&-■■ Experiments, 0.6 [Hz]

-•—CFD, 0.6 [Hz]

^ 0.1

m
XI 0,08
.5
B 0.06

V 0.04
a
3
O
" 0.02

5 2 2.5
strouhal number [-] Strouhal number [

Figure H-20 Comparison couple coefficient roll-heave

100
■.fi..- Experiments, 0,6 [Hz]
90
-*—CFD, 0.6 [Hz] 80
,-, 6
E 70
Ê 5

30

2 0 ] ■ ■&■■■ Experiments, 0.6 [Hz]

10 J . -*—CFD, 0.6[Hz)

ol
6 B 10
Roll amplitude [deg] Roll amplitude [deg]

Figure H-21 Comparison Wave height & Percentage viscous damping

133
Verification ofCFD calculation s witit experimen ts on a rolling circular cylin der with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

I. Visual comparison

Figure I - l Reference keel at t/T = 0, frequency = 0.6[Hz], roll amplitude = 14 [deg]

<:r^ *=^ ■

^/y/^^;^^'-^^
>-
-^-.^.e^^ y:^
^"^^
' '^^/^/././ / ^-S
Figure 1-2 Reference keel at t/T = 2/11 pi, frequency = 0.6[Hz], roll amplitude = 14 [deg]

Figure 1-3 Reference keel at t/T = 4/11 pi, frequency = 0.6[Hz], roll amplitude = 14 [deg]

134
 Appendices

Figure 1-4 Reference keel at t/T = 1/2 pi, frequency = 0.6[Hz], roll amplitude = 14 [deg]

-'. \ L I

^1 '

Figure 1-5 Reference keel at t/T = 15/22 pi, frequency = 0.6[Hz], roll amplitude = 14 [deg]

Figure 1-6 Reference keel at t/T = pi, frequency = 0.6[Hz], roll amplitude = 14 [deg]

:- \ymms^
-:/

Figure 1-7 Small keel at t/T = O, frequency = 0.6[Hz], roll amplitude = 14 [deg]

135
Verification of CFD calculations with experiments on a rolling circular cylinder wiOi bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

." /

'"V\.\ -^ .^ r
Figure 1-8 Small keel at t/T = 2/11 pi, frequency = 0.6[Hz], roll amplitude = 14 [deg]

Figure 1-9 Small keel at t/T = 4/11 pi, frequency = 0.6[Hz], roll amplitude = 14 [deg]

Figure I-IO Small keel at t/T = 1/2 pi, frequency = 0.6[Hz], roll amplitude = 14 [deg]

Figure I - l l Small keel at t/T = 15/22 pi, frequency = 0.6[Hz], roll amplitude = 14 [deg]

136
 Appendices

Figure 1-12 Small keel at t/T = 19/22 pi, frequency = 0.6[Hz], roll amplitude = 14 [deg]

Figure 1-13 Small keel at t/T = pi, frequency = 0.6[Hz], roll amplitude = 14 [deg]

Figure 1-14 Circle shaped keel at t/T = 0, frequency = 0.6 [Hz], roll amplitude = 16 [deg]

Figure 1-15 Circle shaped keel at t/T = 2/11 pi, frequency = 0.6 [Hz], roll amplitude = 16 [deg]

137
Verification otCFD calculations with experiments on a rolling circular cylinder with bilge keels in a free surface.
University of Technology Delft & Bluewater Energy Services b.v.

. ^-.----^
\\ > < ^ ^/-""-- \ '

^T:;

Figure 1-16 Circle shaped keel at t/T = 4/11 pi, frequency = 0.6 [Hz], roll amplitude = 16 [deg]

'rtf
''HI,

Figure 1-17 Circle shaped keel at t/T = 1/2 pi, frequency = 0.6 [Hz], roll amplitude = 16 [deg]

'r,'

Figure 1-18 Circle shaped keel at t/T = 15/22 pi, frequency = 0.6 [Hz], roll amplitude = 16 [deg]

Figure 1-19 Circle shaped keel at t/T = 19/22 pi, frequency = 0.6 [Hz], roll amplitude = 16 [deg]

138
 Appendices

Figure 1-20 Circle shaped keel at t/T = pi, frequency = 0.6 [Hz], roll amplitude = 16 [deg]

; ' « m
////.
V^^/V,
Figure 1-21 Small keel at t/T = 1/2 pi, phiA = 4 [deg], frequency = 0.6 [Hz]

Figure I-22Small keel at t/T = 1/2 pi, phiA = 8 [deg], frequency = 0.6 [Hz]

\^
U#//; ;..

Figure 1-23 Small keel at t/T = 1/2 pi, phiA = 14 [deg], frequency = 0.6 [Hz]

139
Verification ofCFD calculations with experiments on a rolling circular cylinder with bilge keels in a free surface
University of Technoiogy Deift & Biuewater Energy Services b.v.

v,-.;;.r.
Figure 1-24 Circle shaped keel at t/T = 1/2 pi, phiA = 8 [deg], frequency = 0.6 [Hz]

V '-

Figure 1-25 Circle shaped keel at t/T = 1/2 pi, phiA = 12 [deg], frequency = 0.6 [Hz]

'V ^> ,:-- ^ ^

,' %

Figure 1-26 Circle shaped keel at t/T = 1/2 pi, phiA = 16 [deg], frequency = 0.6 [Hz]

140