Free Decay Testing of a Semisubmersible Offshore Floating

Master’s thesis

carried out at the Department of Mathematical Sciences and Technology of the Norwegian University of Life Sciences and submitted in partial fulﬁlment of the requirements for the M.Sc. degree at the Department of Mechanical Engineering of the University of Kassel

Free decay testing of a semisubmersible oﬀshore ﬂoating platform for wind turbines in model scale

Examiners: Prof. Dr.-Ing. Martin Lawerenz Author: Felix Kelberlau Prof. Dr.Ing. Tor Anders Nygaard Abstract

Wind turbines can be installed on floating platforms in order to use wind resources, which blow above the deep sea. The presented research investigates the hydrodynamic behaviour of the Olav Olsen Concrete Star floater in free decay motion. The floater is a new design of a semisubmersible offshore floating platform for wind turbines. The thesis describes the physical free decay testing of a 1/40 plastic scale model in a water basin as well as numerical simulations of the same test cases with the aero- hydro-servo-elastic code 3Dfloat. Tests are performed for the heave and pitch degree of freedom and without mooring lines. The experimental results are compared with the simulation results in order to find adequate added mass and damping coefficients for the numerical model and to validate 3Dfloat for the prediction of heave motions.

The overall added mass coefficient for the vertical direction is found to be CM = 2.71 and the damping behaviour is simulated by using a Morison coefficient of CD = 8.0 and a considerable amount of additional linear damping. Subsequently, the results are discussed in order to find possible causes for discrepancies between the results.

I “We are like tenant farmers chopping down the fence around our house for fuel when we should be using Nature’s inexhaustible sources of energy – sun, wind and tide. [...] I hope we don’t have to wait until oil and coal run out before we tackle that.”

Thomas Edison (1931) (Newton, 1989, p. 31)

II Preface and acknowledgement

This thesis is submitted in partial fulfilment of the requirements for the Master of Science degree in “Erneuerbare Energien und Energieeffizienz” (Renewable Ener- gies and Energy Efficiency) at the Department of Mechanical Engineering of the University of Kassel. It was carried out between April and September 2013 at the Department of Mathematical Sciences and Technology of the Norwegian University of Life Sciences in As,˚ Norway. I would like to express my very great appreciation to Professor Dr.Ing. Tor An- ders Nygaard for supervising this thesis project and being helpful in all questions concerning my stay in Norway. His support was more than one could expect. I wish to thank Professor Dr.-Ing. Martin Lawerenz for giving me the freedom that was needed to work on this Master’s thesis abroad. Trond Landbø and JoséAzcona Armendárizdeserve my gratitude for giving a good introduction to the project and maintaining my work progress by providing all needed information. In the phase of physical model building, Bjørn Brenna was supporting me with tools and useful advice, what I would like to acknowledge. My special thanks go to the colleagues in our office for making the work not only educational but also convivial.

III Contents

List of tables VII

List of ﬁgures VIII

Nomenclature X

1 Introduction 1

2 Thesis background 4 2.1 Development of wind energy usage ...... 4 2.1.1 Early mechanical use of wind energy ...... 4 2.1.2 Wind turbines for producing electricity ...... 7 2.1.3 Oﬀshore technologies ...... 10 2.2 Olav Olsen Concrete Star Wind Floater ...... 14 2.3 Project description ...... 16

3 Theoretical bases 17 3.1 Coordinates system and degrees of freedom ...... 17 3.2 Oscillations and damping ...... 18 3.3 Scaled model testing ...... 20 3.3.1 Scaling methodology ...... 20 3.3.2 Froude similitude ...... 22 3.3.3 Reynolds invariance ...... 23 3.4 Keulegan-Carpenter number ...... 24 3.5 Hydrodynamic forces ...... 25 3.5.1 Morison equation ...... 25 3.5.2 Linear damping ...... 29

IV 4 Methodology of model testing 30 4.1 Test case description ...... 30 4.1.1 General information about the performed tests ...... 30 4.1.2 Heave drop test case ...... 31 4.1.3 Pitch drop test case ...... 31 4.2 Testing environment ...... 31 4.2.1 Physical testing in water basin ...... 31 4.2.2 Numerical simulation with 3Dﬂoat ...... 33 4.3 Model description ...... 34 4.3.1 General assumptions ...... 34 4.3.2 Physical scale model ...... 35 4.3.3 Numerical scale model ...... 39

5 Physical and numerical test results 45 5.1 General observations ...... 45 5.1.1 Heave drop test case ...... 45 5.1.2 Pitch drop test case ...... 46 5.2 Period length ...... 48 5.2.1 Heave drop test case ...... 48 5.2.2 Pitch drop test case ...... 50 5.3 Damping behaviour ...... 50 5.3.1 Heave drop test case ...... 50 5.3.2 Pitch drop test case ...... 53 5.4 Keulegan-Carpenter number ...... 53 5.4.1 Heave drop test case ...... 53 5.4.2 Pitch drop test case ...... 55

6 Discussion of results 57 6.1 Period length and added mass ...... 57 6.1.1 Heave motion ...... 57 6.1.2 Pitch motion ...... 58 6.2 Damping coeﬃcients ...... 59 6.2.1 Heave motion ...... 59

V 6.2.2 Pitch motion ...... 61

7 Conclusion and outlook 63

Bibliography i

Declaration iv

A Appendix v Data sheets of sensors ...... v Drop test results, single test runs ...... viii 3Dﬂoat input ﬁle ...... x Manual calculation of added mass ...... xiii

VI List of tables

3-1 Conversion of units between translational and rotational movements . 20 3-2 Scale factors commonly used for Froude scaling ...... 23 3-3 Inﬂuence of skin friction and form drag on the drag term ...... 25

4-1 Key figures of the OO Star floater in full scale and model scale . . . . 35 4-2 Masses of the different parts of the scale model and their ballasts . . 39 4-3 Volume of water displaced by the scale model ...... 39 4-4 Cross section areas of one leg of the physical and numerical model . . 44

5-1 Key ﬁgures of the results of 14 cm heave drop test ...... 47 5-2 Key ﬁgures of the results of pitch drop test ...... 49

VII List of ﬁgures

2-1 Model of a Persian vertical axis windmill ...... 5 2-2 Overview of the historical development of European horizontal axis windmills ...... 6 2-3 Wind turbine in Denmark, designed by Poul La Cour ...... 8 2-4 Common types of bottom fixed foundations for offshore wind turbines 11 2-5 Types of floating platforms for offshore wind turbines ...... 12 2-6 Concrete Star Wind Floater by Dr. techn. Olav Olsen AS ...... 15

3-1 Coordinates system and degrees of freedom ...... 18 3-2 Polygons of forces for prototype and model ...... 21 3-3 Influence of Reynolds number on drag coefficient ...... 24 3-4 Potential flow around a circular cylinder ...... 26 3-5 Relative importance of drag vs. inertia forces within Morison equation and its applicability for small vs. large structures ...... 28

4-1 Breadboard construction for moving the model in heave direction . . 32 4-2 Breadboard construction for turning the model in pitch direction . . . 33 4-3 Scale model of the OO Star platform with tower, without ballast . . . 36 4-4 Sketch of the OO Star platform in model scale ...... 38 4-5 Visualization of the numerical scale model in three views ...... 41 4-6 Cross section of one of three legs, divided into bucket, pontoon and tower area ...... 42 4-7 Cross section of one of three legs, divided into bucket, pontoon and tower areas: grey and black for physical model, coloured for numerical model ...... 44

5-1 Heave drop test results, experimental mean data ...... 46

VIII 5-2 Pitch drop test results, experimental mean data ...... 48 5-3 Heave drop test damping behaviour, experimental and numerical data 51 5-4 Heave drop test results, experimental and numerical data ...... 52 5-5 Pitch drop test damping behaviour, experimental and numerical data 54 5-6 Pitch drop test results, experimental and numerical data ...... 55

6-1 Picture of the DeepCWind 1/8 scale model ...... 61

A-1 Data sheet of Honeywell 945-L4Y-2D-1C0 ultrasonic distance sensor . vi A-2 Data sheet of SBG IG 500-N inertial navigation system ...... vii A-3 Heave drop test results, single test runs ...... viii A-4 Pitch drop test results, single test runs ...... ix A-5 3Dfloat input file 1/4 ...... x A-6 3Dfloat input file 2/4 ...... xi A-7 3Dfloat input file 3/4 ...... xii A-8 3Dfloat input file 4/4 ...... xiii A-9 Manual calculation of added mass ...... xiii

IX Nomenclature

Symbols Symbol Description A Area C Coeﬃcient D Characteristic Diameter/Spring rate F Force F r Froude number g Gravitational constant K Constant KC Keulegan-Carpenter number L Characteristic length λ Scale factor m Mass π Number Pi Re Reynolds number ρ Density S Submerged length T Period u Velocity u˙ Acceleration V Volume x Distance in x-direction y Distance in y-direction z Distance in z-direction ζ Damping ratio

X Indices Symbol Description 0 Undamped A Amplitude B Buoyancy b Bucket crit Critical D Drag d Displaced FK Froude-Krylov G Gravitational

H2O Water I Inertial l Linear M Added mass m Model p Pontoon/Pressure/Prototype res Resulting S Spring s Structural t Tower V Viscous

XI 1 Introduction

In view of the limited availability of fossil fuels and the knowledge about anthro- pogenic climate change, it is a common goal of mankind to reduce the emission of greenhouse gases to the atmosphere. One of the major steps to reach this goal is to substitute the use of fossil fuels by extracting power from renewable sources. Wind energy is supposed to be an important component of this future energy system. Al- though in most regions there are lots of suitable locations for harvesting the wind resources onshore, eﬀorts in oﬀshore technologies should also be strengthened for the following reasons:

• The energy yield of a wind turbine, which is installed above the open seas, is in general considerably higher than onshore due to steadier and stronger winds.

• Many big cities are located close to a coastline and so transmission losses are low.

• The visual and audible impact of an oﬀshore wind park disappears with rising distance from the coastline.

Despite these arguments only 2% of the wind energy capacity installed today is located offshore (Sawyer, 2012, p. 15). Nearly all of these turbines are built in shallow waters of 30 m or less and use foundations that are rigidly connected to the seabed (Xiaojing et al., 2012, p. 304). However, in some regions no shallow waters are found close to the coastline and to be able to profit from wind resources that are available above deeper water, floating platforms need to be used, because the existing methods of bottom fixed support structures are not economically applicable. The development of those floating platforms for wind turbines is based on the expertise of offshore oil and gas industry, but a lot of research needs to be done to

1 1 Introduction adapt the knowledge to the smaller size of the turbine platforms and to handle the additional wind loads. One of the central topics in many projects is the platforms’ hydrodynamic behaviour and how it can be determined by numerical simulations. For this, free decay tests deliver information about the natural periods and the damping behaviour of newly developed structures and are often the first tests that are made. This thesis gives an answer to the question, how the free decay movement in heave and pitch of the Olav Olsen Concrete Star Wind Floater, which is a certain semisubmersible floating platform for wind turbines, can be described by the aero-hydro-servo-elastic code 3Dfloat. 3Dfloat uses the Morison equation for the calculation of hydrodynamic forces. This equation is often used for the determina- tion of wave forces on circular cylindrical members of fixed offshore structures, but little is known about its applicability on oscillating structures of other shape in a quiescent fluid. At first, a 1/40 plastic scale model of the floater is built and test runs are performed in a water basin to supply measurement data of the actual free decay motion. These data are then analysed in terms of periods and damping to obtain proper coefficients for added mass and quadratic as well as linear damping. Then a numerical model for 3Dfloat is generated, based on the physical properties of the plastic model and the coefficients calculated from the obtained experimental results. The plaus- ibility is discussed, based on previous research. As a final step the experimental and numerical datasets of the same test cases are compared in order to find possible reasons for discrepancies. Chapter 2 describes the historical background that shows how floating offshore wind turbines developed from the wind mills of ancient ages. It classifies the position of the thesis in the overall project and gives a description of the investigated Olav Olsen Concrete Star platform. The following chapter 3 presents all theoretical bases, which are applied in subsequent explanations. It describes the coordinates system and the degrees of freedom that are referred to in the description of the oscillating motions of the platform. The principle and the laws of scale model testing are presented afterwards. The Keulegan-Carpenter number is then introduced, because it has an influence on the hydrodynamic forces, which are presented last but not least.

2 1 Introduction

In chapter 4, the performance of the drop tests is demonstrated. The testing environment is presented for both, the physical experiments and the numerical simulations and also the relevant properties of both, the real and the virtual model, are explained. The results of the testing are described and visualized by plots in chapter 5, while their discussion follows in chapter 6. Chapter 7 sums up the main ﬁndings, gives an answer to the research question and oﬀers an outlook on the further testing campaign.

3 2 Thesis background

This chapter embeds the subject of the thesis in its surrounding. It gives a deeper insight into the topic but is not essential to understand the testing procedure and results. Section 2.1 gives the historical background of wind energy usage in subsections 2.1.1 and 2.1.2 and also presents the state of the art in offshore wind technology in subsection 2.1.3. Afterwards the investigated offshore floating platform for wind turbines is presented in section 2.2 and the overall research project is described in the last section 2.3.

2.1 Development of wind energy usage

This thesis is about improvements for the most modern wind power plants, which make it possible to produce electrical energy with low impact to human and nature and at competitive costs. While the research in the field of offshore floating platforms for wind turbines is still in an early stage, it has been a long way to get there. This development was always driven by the human needs of the respective time. The following chapter describes some milestones that show, how these demands were satisfied. It is just a glimpse into the development. For further information it is referred to Hau (2013), who delivers a comprehensive overview about the history of wind energy.

2.1.1 Early mechanical use of wind energy

Pumping water and milling corn are two examples of activities, which were done for centuries, before people were able to beneﬁt from electrical energy. Since these daily works were exhausting and ineﬃcient, when done by muscle power, humans were searching for ways to facilitate their physical work.

4 2 Thesis background

Figure 2-1: Model of a Persian vertical axis windmill (Deutsches Museum)

It is assumed by historians, that the first windmills were used in the orient for irrigation as early as 1.700 B.C. (Gasch and Twele, 2012, p. 15). The first certain information about a machine, that converts the energy of the wind into useable power, dates back to the year 644 AD. It is from the Persian region and describes a vertical axis windmill for the milling of grain. As shown on figure 2-1, the wind was led through an inlet gap, where it pressed against the blades on one side of the rotor. The resulting drag force led to a spinning movement of the vertical axis. The disadvantage of these windmills is that they are depending on wind blowing from a certain direction. Also the demand of the Chinese for draining rice fields may have led to the first remarkable usage of wind energy. To create the necessary asymmetry the Chinese windmills did not use walls for directing the air, but had flapping sails that turn out of the wind, while moving against the wind direction. Due to a lack of reliable evidence, there is some uncertainty about which concept was the first. One thing that both have in common is, that they use a direct driven vertical shaft for energy transfer. The occidental counterpart was developed during the outgoing 12th century in Western Europe. The main difference to the oriental origins was that they had a horizontal axis of rotation. This different concept made it necessary to invent a

5 2 Thesis background

Figure 2-2: Overview of the historical development of European horizontal axis windmills. (Gasch and Twele, 2012, p. 24) completely different way of absorbing the wind energy. Like an aircraft propeller the rotor turns in a plane perpendicular to the wind direction. The deflection of the airstream, which flows through this plane, causes a lift force, which accelerates the shaft in its horizontal rotation. Lantern gears were used to change the direction of the movement, so that it is available at the bottom of the mill. The windmills spread over the whole western world and were slightly improved during the following centuries. Figure 2-2 shows the development of the European types of mills. The last stage of classical windmills, which was developed in the mid-19th century, was the so called western mill. It had more than 20 wings made of metal sheets and was used mainly in North America for pumping water (Gasch and Twele, 2012, p. 25). For a long time, the only competitors for windmills were watermills, but their usage was limited to a few locations close to flowing waters. This situation started to change, when the first steam engine was put into operation in 1785. Its main advantages were the independency of the weather and the bigger power output. Though, it took the 19th century to outrun the classical windmills, which experienced a short revival during the Second World War, when fuel was scarce.

6 2 Thesis background

2.1.2 Wind turbines for producing electricity

Large-scale exploitation of electrical energy started in 1882 with the first power plants and justifies the idea of producing electricity from wind energy. It was not later than 1887, when the American inventor Charles Francis Brush used a western mill for driving a generator to produce electrical energy and stored it in a battery. One of the first, who systematically development the use of wind energy, was Poul La Cour from Denmark (Jamison, 2012, p. 14). He performed wind tunnel tests to improve traditional windmill technology to perfection and did pioneer work in the field of electricity generation out of wind power. Additionally he was faced with the problem of energy storage for delivering a constant energy supply to rural regions. His solution was hydrogen production by electrolysis for operating gas lamps, whenever the light is needed. His results were used for a small wind industry in the beginning of the 20th century, which was boosted by the rising fuel prices during the First World War. A typical La Cour wind power station is shown in figure 2-3. Its power output was approximately 30 kW with a rotor diameter of around 18 m. While La Cour’s turbine designs were closely related to the traditional and reliable windmills, other manufacturers entered the market after the war with more modern concepts. They benefited from the outcomes of research on airplanes, which was adaptable to the blades of wind turbines and helped to improve them. The new knowledge was further processed on a scientific basis by Albert Betz. In Betz (1926) he sums up, that not more than 16/27 of the energy transported by the wind can be converted into mechanical energy by a discoidal turbine. His findings and summaries of previous works are an important part of the fundamentals for modern wind energy converter design. With this theoretical background ideas for large wind turbines came up. Hau (2013, p. 29) gives the example of the German steel construction engineer Hermann Honnef, who developed the concept for a 20 MW wind power plant in the 1930s, which should be operated to support conventional power plants in a large- scale. His ideas showed the way forward, but could not be realized, as they were too ambitious with regard to feasibility. Instead, two turbines with a rotor diameter of 30 m and a rated power output of 100 kW were built in USSR. One of them, called WIME D-30 was operating from 1931 to 1942 with good results in terms of

7 2 Thesis background

Figure 2-3: Wind turbine in Denmark, designed by Poul La Cour. (Hau, 2013, p. 26)

8 2 Thesis background reliability. The plans for continuation with bigger multi-megawatt projects were destroyed by the war. Solely in the USA the development continued during the war and a 1.25 MW turbine with a rotor diameter of 53 m was connected to the grid in 1941. It was in operation until 1945 but revealed, that the costs were too high compared with energy from conventional power plants. After the war, there was a revival of interest in wind energy in Europe. It started with the upcoming awareness of the limited reserves of coal in the 1950s and ended around 10 years later with the availability of cheap oil from the Middle East. This phase was characterized by the inventions of the German company Hütter, which used fibreglass blades and an electrohydraulic pitch control for their prototype W34. Another name that should be mentioned when describing this phase is Johannes Juul, a student of Poul La Cour. For a site in Gedser, Denmark, he used an asynchronous generator and a robust stall-regulation to demonstrate a cheap, reliable and safe system, compared to other wind turbines. The last uprising of wind energy that still continues to the present day was caused by the oil price shocks in 1973 and 1979. They led to the question about alternatives to the dependency on oil. Two different approaches should give an answer: On the one hand the governments of several countries like the USA, Germany, Sweden and some others invested enormous amounts of money in huge projects for developing giant wind turbines. One of the most popular was the ’Growian’, a 3 MW turbine with a rotor diameter of 100 m that was erected in 1983. It was not possible to put it into a permanent operation, due to technical problems. As a result, it was dismantled only four years later and many wind energy critics used this defeat as an argument, that it would not be possible to serve the energy demand by anything else than conventional sources. On the other hand, small Danish manufacturers started to fabricate wind turbines with diameters between 12 and 15 m and a rated power output of 30 to 75 kW (Gasch and Twele, 2012, p. 31). A market could be established, because the Danish and later the German government granted a feed-in tariff for electricity from wind energy. The experiences from these small installations gave the knowledge for steady improvements and the development of larger systems. Today the wind industry is still on this track and is able to manage projects in a size that could not be handled before.

9 2 Thesis background

2.1.3 Oﬀshore technologies

Based on the idea that it should be possible to erect wind turbines at sea, the world’s first offshore wind farm was built in 1991. The Vindeby site consists of 11 turbines with 450 kW each. They were set up in the three to five meter shallow water of the North Sea close to the Danish coastline. Major challenges to be addressed were the resistance of the turbines against corrosion and lightning strikes, that are a bigger issue offshore than onshore (Prinds, 2011, p. 43). Now, more than twenty years later, this forerunner in offshore wind energy is still in use and due to its success, other countries mainly in northwest Europe followed with bigger projects. Outside of Europe, China and the United States are planning to move offshore. To show the motivation for these plans, Lynn (2012, p. 153f) mentions four arguments for using the wind that is blowing along the open sea:

• The bigger a wind farm is, the cheaper each turbine can be produced due to the economies of scale. This results in the need for large areas, which are available only oﬀshore, when looking at countries with a dense population.

• Development trends towards bigger turbine sizes with rotor diameters over 100 m. Their visual impact and space requirement can be handled by setting it up on the open sea.

• The wind conditions are better. In general, the average wind speed over the water is higher than onshore and the air turbulence is less. Therefore more full-load hours can be reached.

• While the oil under the north sea is running low, the huge industry for pumping it remains. A lot of the workforce and know-how can be adapted and used for harvesting wind instead of oil.

It is a business case to decide whether these advantages outweigh the higher costs of building a structure offshore. Taking into account the higher construction and also operating costs into account, Hau (2013, p. 677) mentions that the better specific energy yield does not necessarily lead to an improved economical balance. Anyway, a lot of projects initiated by the big energy suppliers tend to build huge wind farms offshore. Their order of magnitude around 1 GW is similar to the rated output of

10 2 Thesis background

Figure 2-4: Common types of bottom ﬁxed foundations for oﬀshore wind turbines: monopile, tripod, jacket and gravity based (STRABAG) conventional power stations and therefore large wind farms have the potential to be a real alternative.

Bottom based foundations and support structures

All existing offshore wind turbines are standing in water depths of less than 50 m and have bottom fixed foundations. Figure 2-4 shows a selection of the most common types of foundation structures that are currently in use. The first image shows a monopile, which is the most popular kind of structure and is used for more than 70% of all offshore wind turbines. It is a tube of 3.5 to 6 m in diameter that is rammed into the seabed at sites with rather shallow water (<30 m). Monopiles are cheap and simple in construction. The second image shows a so called tripod support structure, which is common for oil and gas platforms but quite new and still unusual for being used by wind industry. It consists of braces that connect a central column with three piles that found the structure to the seabed. High bending moments can be derived what gives tripods a good stability against turning and makes them suitable for installations in deeper water (30 to 50 m). The disadvantages are the complexity of the structure and the bigger cross section area leads to higher wave and current loads compared to monopiles. Beside of the tripods, jacket structures are also possible to set up in water that is deeper than 30 m. Three or four legs are stiffened by a supporting structure. This

11 2 Thesis background

Figure 2-5: Types of floating platforms for offshore wind turbines: spar buoy, TLP, semisubmersible (SFFE) concept is also inherited from oil and gas industry and results in low loads and good material efficiency. Unfortunately the high number of welding nodes demands for a lot of manual work, what raises the costs. When the water is very shallow (<10 m), gravity based foundations, like the one depicted in the last picture, can be taken into consideration as an independent type of structure. These are very heavy ballasts, put on the seabed. They are holding the turbine in place by their sheer weight. A big advantage compared to piling is, that no hammering is needed for installation. This protects fish and sea mammals from extremely loud noise. As gravity based structures are made out of concrete, they are very durable and can be used for more than a turbines lifetime.

Floating platforms

Most of the global wind is blowing above deep water. For example 61% of the US American offshore wind resources are found over deep water and in Japan nearly the complete offshore wind potential is located above the deep sea. The situation is similar for various European countries. (Main(e) International Consulting, 2013, p. 4) The use of bottom based structures in more than 50 m deep water for harvesting wind energy would be prohibited by too high cost. For these sites, floating platforms are seen as an alternative. They are designed to swim on the ocean’s surface and are connected to the ground solely by anchored mooring lines, that hold the installations in their position. So it is possible to assemble these platforms onshore and tow them

12 2 Thesis background to the sites after completion, what gives a significant cost advantage. A rule of thumb says, that any offshore work is five to ten times more expensive than the same work on land (Gasch and Twele, 2012, p. 521). Furthermore floating platforms can easily be decommissioned or moved to another site. Different concepts exist and it is not yet decided, which of them is superior. Fig- ure 2-5 gives an overview of the most promising concepts. Although all of them are already in use for other offshore applications, a lot of tasks in research and development have to be coped with in order to adapt the technology for supporting wind turbines. This mainly concerns the limited knowledge about the dynamic behaviour in wind and waves, and it has to be investigated, how the resulting movements influence the turbines efficiency and the fatigue of the components. The spar buoy is a long cylindrical tube reaching deep into the water. It is ballasted by a heavy weight in its lowest point and above it is filled with air. As a result, the center of gravity is lower than center of buoyancy. So slight deflection from the upright position leads to a restoring moment what makes the spar buoy stable due to its construction. The weight of the whole structure calms the movement in all degrees of freedom and the small diameter of the tube at water level makes it rather insensitive to the impact of waves. Catenary mooring lines are used to hold the buoy in position, so the installation is straightforward. One disadvantage is that spar buoys need deep water, as their draught is very high. As part of the Hywind project, the world’s first floating wind turbine was erected on a spar buoy in September 2009 in Norway. It has a rated power output of 2.3 MW and the draught is 100 m, while the diameter at sea level is just 6 m. It is operated commercially by StatoilHydro with nearly 3200 full load hours in 2010. Another concept is followed by tension leg platforms (TLP). In operating position, the buoyancy of these exceeds their weight force and that is why tensioned mooring lines, fixed to the seabed, are needed to hold them under water. These tension legs, which may be steel pipes, give the platform stability. When the platform tends to pitch, the tension in one of the legs increases and pulls it back in an upright position. The cross section area attacked by wave loads is also small. Both results in small motions. TLPs have a wide range of possible water depths and a low steel weight, but their installation is challenging and rather expensive. Semisubmersible and barge floaters are wide structures, which are only partly

13 2 Thesis background submerged. Their centre of gravity is above the centre of buoyancy, but when they are tilted out of the initial position, the centre of buoyancy moves sideways and a positive restoring moment arises, that pushes the platform upright again. Damping plates and the huge surface area in the water are used to stabilize the platform. Additionally the inertia coefficients are high. Anyway, the motions are relatively large, also since the contact surface at water level is higher than for spar buoy or TLP. A big advantage of these floaters is, that they can be fully assembled in a dockyard and mooring remains the single work that has to be done offshore. This makes the setup of semisubmersible and barge floaters cheap and quite independent from the weather. Floaters are very flexible concerning water depth. The WindFloat project has set up a 2 MW Vestas turbine on a semisubmersible platform off the coast of Portugal.

2.2 Olav Olsen Concrete Star Wind Floater

The platform investigated within this thesis is a patent pending design by the company Dr. techn. Olav Olsen AS (OO). The development of this semisubmersible floater was driven by the search for a cost effective alternative to fixed bottom structures. The floater shall support huge wind turbines under harsh environmental conditions. The aim of low costs is reached by a simple design that is shown on figure 2-6. The floater construction consists of three corner cylinders (buckets) that serve buoyancy, a star-shaped pontoon and a central tower with a transition unit to the tower of the wind turbine. The whole system including tower and turbine can be fully assembled on a floating barge, in a dry dock or on a quay. After testing it, it will be towed to the wind farm where it gets additionally water ballasted and connected to pre-set mooring lines that serve station keeping. Under operating conditions the pontoon will be fully submerged and only the upper thirds of the cylinders are visible from above the surface. For building the substructure, concrete is the chosen material. It offers good stability and the material fatigue is so low, that the design life is more than a hundred years. This makes sense, because unlike gas and oil reservoirs, wind resources are not limited in time. During the whole lifetime no maintenance is required. The platform

14 2 Thesis background

Figure 2-6: Concrete Star Wind Floater by Dr. techn. Olav Olsen AS

15 2 Thesis background is designed for mass production and the scale effects of concrete constructions are strong. The OO Concrete Star Wind Floater (OO Star) is intended for use in water deeper than 50 m. It needs to have stable motion characteristics while carrying 5 to 10 MW wind turbines. For the investigation a 6 MW Siemens wind turbine is chosen. In the numerical simulation it will be represented by a 5 MW reference wind turbine defined by Jonkman et al. (2009). The extension slab at the bottom of the floater is a water entrapment plate. It damps and slows down the motions of the platform and helps to reach the design aims concerning the allowed motion under the influence of waves, sea current and wind.

2.3 Project description

The work on this thesis is embedded in a two-year project granted by the Research Council of Norway. The title is “Concrete substructure for floating offshore wind turbines” and the overall dynamic behaviour of the OO Star shall be evaluated. Dr. techn. Olav Olsen AS is responsible for the construction of the tested platform (see subsection 2.2). The company is an independent structural and marine technical consultancy that was founded in 1962. The about 80 employees have their main focus concentrated on platforms for the oil and gas industry. OO designed many of the world’s biggest concrete offshore structures. The Norwegian Institute for Energy Technology (IFE), Norway’s national research company for nuclear and energy technology, is another project member. The IFE was founded in 1948 and has about 600 employees. They developed the software tool 3Dfloat, that is used for the numerical simulation of the test cases (see subsection 4.2.2). The Marine Renewables Infrastructure Network (MARINET) is an initiative, fun- ded by the European Commission, which shall contribute in the development of marine renewable energy systems by offering access to the test facilities of the part- ner institutions without charging for it. The Ecole´ Centrale de Nantes (ECN) in France offered its Hydrodynamic and Ocean Engineering Tank for MARINET. The project gains access for two weeks of scale model testing, which will take place after the completion of this thesis.

16 3 Theoretical bases

This chapter describes the theoretical background, which the explanations and calculations throughout the thesis are based on. It is divided into five sections. The first section 3.1 defines the used coordinates system and introduces the six degrees of freedom, in which floating platforms can move. The second section 3.2 explains the model, which is used for the description of a damped oscillating motion in one degree of freedom, like found in the free decay motion during the tests. In the third section 3.3 an insight is given into scaling methodology that is applied for scale model testing. The second last section 3.4 deals with the Keulegan-Carpenter number, which gives some information about the relative importance of drag and inertia forces. These and other hydrodynamic forces are the subject of interest in section 3.5.

3.1 Coordinates system and degrees of freedom

A Cartesian coordinates system is used for the description of the platform and its movements. Figure 3-1 shows the inertial system and the used terms for translation along and rotation around the axes. The origin of the three axes x, y and z is the intersection point of the central line through the tower and the still water level. The x-z plane is a plane of symmetry of the platform. Unlike bottom ﬁxed structures, ﬂoating platforms move in all 6 degrees of freedom. In the chosen right handed Cartesian coordinates system the translation in x-direction is called surge, in y- direction sway and in z-direction heave. Rotation around the x-axis is called roll, around the y-axis pitch and around the z-axis yaw. Without mooring lines, these movements are independent of each other, what means that e.g. a force in z-direction, applied to the center of gravity, leads to a pure heave motion without any movement in the other 5 degrees of freedom.

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Figure 3-1: Coordinates system and degrees of freedom

3.2 Oscillations and damping

When the tested model is put into water very slowly, it will start to ﬂoat, as soon as the gravitational force FG = msg is in equilibrium with the buoyancy force

FB = gρH2OVd, that is depending on the draught of the model. While FB may change, when the model is moved, FG remains constant. The diﬀerence between them is the spring force FS. It is also called hydrostatic restoring force and is caused by the change of buoyancy resulting from a changed amount of water displaced by the model. It is proportional to the displacement and is nonzero only for the heave, pitch and roll degrees of freedom. All the testing described in section 4 will investigate the movement of the model, when it is being released after forcing it into a certain position. Such a system can be described by a simpliﬁed equivalent model consisting of a mass m that is attached to a spring with the spring rate D. The mass m is the sum of the structural mass of the model and the mass of a certain amount of water that is moving with the model (see subsection about added mass in 3.5.1). For the acceleration of this summed mass, the inertial force FI = mx¨ needs to be applied. FS = Dx is the spring force that is necessary for the displacement of the model, for instance to further submerge

18 3 Theoretical bases or lift it. The sum of both forces is the resulting force

Fres = mx¨ + Dx. (3-1)

With no external force and an initial position in equilibrium, the resulting force is zero and the model is standing still. The very slow application of a force, leads to a displacement of the platform to a new position, in which FS equals Fres. When this external force Fres is then released abruptly, the mass starts to oscillate perpetually with no damping and its natural period

p T0 = 2π m/D. (3-2)

This undamped oscillation does not represent reality, because whenever water is forced to a motion, energy is dissipated. This dissipation decreases the inherent energy in the mass spring system and is therefore called damping. There are several forms of damping forces and the simplest one is the pure viscous damping. The damping force FD is in this case proportional to the velocity of the model in the quiescent ﬂuid. This damping term is added to equation 3-1 to get the equation of damped motion for the displacement of mass.

Fres = mx¨ + CDlx˙ + Dx (3-3)

When the value for the damping coeﬃcient is

p CDlcrit = 2 D/m, (3-4)

one says, that the movement is critically damped, what means, that the model returns to its initial equilibrium position as fast as possible without oscillation. The damping ratio ζ is deﬁned as C ζ = Dl (3-5) CDlcrit A system with 0 < ζ < 1 swings periodically around equilibrium with exponentially decreasing amplitude, whereas a higher ζ value causes a monotonous approach of the equilibrium position. It should be mentioned that the critical damping coeﬃcient

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Transl. dim. Symbol Unit Rotat. dim. Symbol Unit kgm kgm2 Force F s2 Torque M s2 Mass m kg Moment of inertia J kgm2 Distance x m Angle ϕ rad m rad Velocity u =x ˙ s Angular frequency ω =ϕ ˙ s m rad Acceleration a =x ¨ s2 Angular acceleration α =ϕ ¨ s2 kg kgm Damping Dtran s Damping Drot s Table 3-1: Conversion of units between translational and rotational movements is only valid for linear damping forces that are proportional to the velocity. The damping increases the period length T of the damped system as follows:

T T = 0 (3-6) p1 − ζ2

Some of the above mentioned terms are usually used only for the description of translational movements but all formulas may be used for rotations as well. Table 3-1 shows the corresponding terms.

3.3 Scaled model testing

Before an expensive structure like the OO Star platform is built in full scale, tests on scaled models are carried out, in order to predict the dynamic behaviour under the inﬂuence of the prevailing loads and to validate the results of numerical simulations. To determine the parameters of the model and testing environment, a closer look is taken to modeling laws.

3.3.1 Scaling methodology

When physical experiments with models are performed, it is tried to realize a similar behaviour between the model and the prototype of the investigated object. For reaching this similitude, several conditions need to be fulﬁlled. One is that the model has a similar geometry, what means that the ratio of lengths between model and prototype has to be a constant. This value is then called the scale factor λ

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J J J JFIp J J J J J FIm J^ J @I J @FV p JJ^ @ @IFV m @ ¡¡ ¡¡ FGp ¡ FGm ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡Fpm ¡F ¡ ¡ pp ¡ ¡ ¡ ¡ ¡ ¡ ?¡ ¡ ¡ ¡ ?¡ Figure 3-2: Polygons of forces for prototype and model

x y z m = m = m = λ (3-7) xp yp zp It follows from this deﬁnition, that areas and volumes of a scaled model compared to the prototype are also dependent on the scale factor:

A x y m = m m = λ2 (3-8) Ap xpyp

V x y z m = m m m = λ3 (3-9) Vp xpypzp Beside of the geometrical similitude, the distribution of masses of the prototype has to be reproduced in the model, so that the centre of gravity and buoyancy, the mass moment of inertia and the radii of gyration are identical. Additionally the similitude of forces has to be achieved. A ﬂuid element ﬂowing along a structure is subjected to several forces. Namely the gravity force FG, pressure force Fp and viscous force FV are considered in the following explanation, while the

21 3 Theoretical bases elastic force and the capillary force are neglected. All of them attack the ﬂuid element in a certain angle and their vector sum equals the resulting inertia force FI . The similitude of forces is achieved, if the polygon of forces is similar for model and prototype, like shown in ﬁg. 3-2. This would mean that

F F F F Gp = pp = V p = Ip . (3-10) FGm Fpm FV m FIm

Equation 3-10 is equivalent to a combination of Froude similitude

F F Ip = Im (3-11) FGp FGm and Reynolds similitude F F Ip = Im (3-12) FV p FV m By definition, the Froude number is the ratio of the inertia force to the gravitational force and the Reynolds number is the ratio of the inertia force to the viscous force. Maintaining these two dimensionless numbers would be sufficient to achieve the same dynamic behaviour between model and prototype in a fluid flow.

3.3.2 Froude similitude

The Froude number is the most important dimensionless number for the character- isation of objects, which are washed round by a ﬂuid with an open surface. This could be current and waves of water close to the surface. In these cases the gravitational forces have to be considered. For maintaining the Froude similitude, the following condition applies

up um F r = p = √ (3-13) gLp gLm

and all scale factors listed in table 3-2 are derived from this equation and can be used to obtain the prototype values from the model data and the other way round.

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Dimension Scale factor Length λ Area λ2 Volume λ3 Mass λ3 Mass per unit length λ2 Moment of Inertia Area λ4 Moment of Inertia Mass λ5 1 Time λ 2 1 Velocity λ 2 Acceleration 1 Angle 1 1 Angular velocity λ 2 Angular acceleration λ−1 Force λ3 Tension λ3

Table 3-2: Scale factors commonly used for Froude scaling

3.3.3 Reynolds invariance

It can easily be shown, that keeping on the Reynolds number would lead to very high resulting ﬂuid speeds in down scaled model tests. Chakrabarti (1994, p. 19) mentions that Reynolds similitude is quasi non-existent in scale model technology. Nevertheless, it is possible to get good scaling results, even if the Reynolds number is not constant between model and prototype, as long as it is high enough. Figure 3-3 shows, how the pressure coeﬃcient depends on the Reynolds number. While the

Cp value is a function of Re in the laminar and transitional flow state, it is about constant in turbulent flow. The constant pressure coefficient leads to a constant ratio of inertia force to dynamic pressure force

F F Ip = Im (3-14) Fpp Fpm and coupled with Froude similitude (eq. 3-11) the equation for the similarity of the polygons of forces 3-10 is then also fulﬁlled. This becomes obvious by looking at the fact, that very high Reynolds numbers equal very short FV vectors compared to FI in 3-2. The polygons become triangles, which are entirely deﬁned by the

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Figure 3-3: Influence of Reynolds number on drag coefficient (Jirka, 2007, p. 136), modified

Froude number and the geometrical similitude. Thus, it is acceptable to use a lower Reynolds number in the model than in the prototype, as long as it remains above the critical value (Jirka, 2007, p. 136). This is ensured for all presented testing throughout this thesis, because of the ﬂow separation that occurs generally at the sharp edges of the water entrapment plate (Newman, 1977, p. 20).

3.4 Keulegan-Carpenter number

The Keulegan-Carpenter number is a dimensionless number that is used to describe the relative influence of drag and inertia forces on objects with an oscillating relative velocity to its surrounding fluid. It is defined as:

u T KC = 0 (3-15) D

For the free decay testing described in this thesis, the characteristic diameter D is constant. Also the period T is nearly not changing during the tests. Hence, the amplitude of the velocity u0 is the only parameter that inﬂuences the changing KC value during the performance of the tests. Like visualized in ﬁgure 3-5 on page 28, a large KC number stands for a high relative importance of the drag forces, while for a small number the inertia forces are dominant.

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pure skin friction drag

mainly skin friction drag

mainly form drag

pure form drag

Table 3-3: Inﬂuence of skin friction and form drag on the drag term Images: (Wikipedia)

3.5 Hydrodynamic forces

3.5.1 Morison equation

The simulation tool 3Dfloat uses the Morison equation for calculating the hydrodynamic forces. This equation was published by Morison et al. (1950) and is a semi empirical formula describing the force exerted on a slender object in an unsteady fluid flow. It superposes the effects of inertia force and drag force and has the form

f = KD |u| u + KM u˙ . (3-16) | {z } | {z } drag inertia

The drag term is the sum of the skin friction drag, caused by the shear stress within the viscous fluid and the form drag, caused by the pressure gradient around the object. For a body that is not streamlined, the form drag dominates the drag term. This is visualized in table 3-3. In experimental estimation of the drag coefficient a combination of both effects is found as they cannot be separated without big effort. The drag coefficient is a function of the Reynolds number, the Keulegan-Carpenter number and the roughness of the member. The value of the drag force varies with

25 3 Theoretical bases

Figure 3-4: Potential ﬂow around a circular cylinder (Wikimedia Commons) the square of the velocity. For the force on a circular cylinder it applies:

SD F = C ρ |u| u (3-17) D D H2O 2

The inertia term is also representing two components. One is the Froude-Krylov force on the object in an unsteady flow. It can be illustrated by looking at the volume of water that is displaced by the object. If the object would be absent, the mass of this fluid volume would be accelerated by the force of the surrounding flow.

SD2π F = mu˙ = ρ u˙ (3-18) FK H2O 4

The other component is the effect of the added mass. It can be explained by the influence of the object on the fluid flow. Figure 3-4 shows a circular cylinder in an incompressible originally uniform flow field. The increased length of the lines in the surrounding of the object shows, that a member in the flow cross section leads to an increased fluid velocity. This is resulting in a higher pressure difference and increased force. The corresponding mass is called added mass. For instance, the velocity around a circular cylinder is twice the speed of the undisturbed flow. Thus the inertial force is also doubled. In this case the added mass is 100% of the mass of the object. For getting a rough idea about the quantity of added mass for a flat plate headed against the fluid flow, a rule of thumb, which is also used by Maniaci and Li (2011, p. 3) and many others, can be applied. It says that the added mass of a rectangular flat plate with a certain width and infinite length equals the mass

26 3 Theoretical bases of the water within a circular cylinder of the same diameter like the width of the flat plate. For a round flat plate, the added mass is assumed to be equivalent to the mass of the water in the smallest sphere around it. In general the inertial force can be defined by

FM = (m + mM )u ˙ = CM mu˙ (3-19) with m C = 1 + M (3-20) M m The value of the Keulegan-Carpenter number (see section 3.4) determines, which one of the two components, inertial or drag force, is dominating. The values on the y-axis in ﬁgure 3-5 are equivalent to the KC number. The diagram shows, that in states with a large KC value, the turbulent drag term in the Morison equation is dominant, while it can be neglected for low KC numbers on the y-axis. The resulting force on a slender object of the submerged length S determined with the Morison equation is given by:

πD2 D dF = C ρ u˙ + C ρ |u| u dS (3-21) M 4 D 2

The real added mass and drag coefficients CM and CD depend upon the state of the fluid motion, but in the equation, they are considered as constant. Little is known about the coefficients in accelerated systems and available numbers are mainly about cylinders with a circular cross section. It depends on the chosen wave model, which of them are the best to use. For linear wave theory a drag coefficient of 1.4 and an added mass coefficient of 2.0 is suggested for round cylinders in Agerschou and Edens (1965, p. 239). The Morison equation is only valid for members that are small compared to the wavelength. Looking at the x-axis of figure 3-5 shows, that for large structures, wave diffraction forces from potential flow theory have to be added to the overall hydrodynamic force.

27 3 Theoretical bases

Figure 3-5: Relative importance of drag vs. inertia forces within Morison equation and its applicability for small vs. large structures (Chakrabarti, 2005, p. 167)

28 3 Theoretical bases

3.5.2 Linear damping

Another approach to calculate the hydrodynamic forces on a submerged member is based on the potential flow theory. For the computation of these flow fields, numerical methods are used. When the submerged member is large in comparison to the wavelength, the fluid flow remains attached to the body and the water particle movement is affected in a large area around the object (Chakrabarti, 2005, p.160). Incident waves are diffracted due to the presence of the member. While the drag forces represented in the Morison equation are proportional to the square of the velocity, these diffraction forces are directly linear to the velocity.

29 4 Methodology of model testing

This chapter explains, how the physical testing and the numerical simulations throughout the thesis are performed. It is divided into three sections that describe the test cases, the testing environment and the used models. While the test cases, presented in section 4.1, are the same for the physical experiments and the numerical simulations, the description of the testing environment and of the used models in sections 4.2 and 4.3 is divided into subsections for both ways of testing.

4.1 Test case description

4.1.1 General information about the performed tests

The testing presented in this thesis, is only the first part of a comprehensive test program, which will give information about the coupled behaviour of the OO Star platform, carrying a wind turbine in wind and waves. Most of these tests take place in the hydrodynamics laboratory of Ecole´ Centrale de Nantes (ECN) in France. Only the first part of the test program, which is presented within this thesis, is performed in a basin at the Norwegian University of Life Sciences (UMB) in As,˚ Norway. It consists of free decay tests that do not require sophisticated equipment or huge testing facilities. The purpose of free oscillation tests is to determine the natural periods and damping coefficients of the model (Chakrabarti, 2005, p. 1034). The physical test results are used to find proper added mass and damping coefficients for the numerical model. Then, numerical simulations of the same test cases are carried out. The results of both, the physical experiments and numerical simulations are compared in order to evaluate the suitability of the 3Dfloat tool for the prediction of heave and pitch motions. The free decay tests are carried out for these two degrees of freedom, while the roll

30 4 Methodology of model testing motion is neglected, because it would deliver the same results like the pitch motion. Surge, sway and yaw motion cannot be considered, because the basin at UMB is not big enough for the installation of the mooring system, which delivers the restoring forces in these degrees of freedom.

4.1.2 Heave drop test case

For the heave drop tests, the floating platform is deflected from equilibrium position downwards in negative heave direction, by applying a force. After a while, when the surrounding water is quiescent again, the force is released and the model starts moving upwards. It will be found out, that the motion is subcritically damped and will therefore oscillate. This movement is a pure heave motion. The test is finished, when no more translation in z-direction can be detected. Two sets of heave drop tests are performed: One with an initial deflection of 14 cm and one with 8 cm.

4.1.3 Pitch drop test case

The pitch drop test starts with adding a torque around the y-axis of rotation, which cuts the midpoint between the centre of gravity and the centre of buoyancy. When the desired angular excursion is reached and the water is quiescent again, the torque is released and the model starts rotating freely. It will be found out, that the motion is subcritically damped and will therefore oscillate. This movement is a pure pitch motion around the centre of rotation. The test is ﬁnished, when no more rotation in z-direction can be detected. The chosen initial angular excursion is 8◦.

4.2 Testing environment

4.2.1 Physical testing in water basin

The physical experiments are executed in a round pool with a diameter of 5.8 m and a water depth of 1.05 m. The water temperature is 15 ◦C. For the heave drop testing the movement of the model in vertical direction is determined by distance measurements. A Honeywell 945-L4Y-2D-1C0 ultrasonic distance sensor is installed on the tower top and a non-moving plate is located above it. The vertical movement

31 4 Methodology of model testing

Figure 4-1: Breadboard construction for moving the model in heave direction of the tower top is measured in the time domain by the sensor output for the distance between sensor and plate. Attachment A-1 shows the specifications of the sensor. 150 kg of steel ballast is placed on the ground in the middle of the pool. A deflection roller is connected to this ballast and a second roller is positioned outside the pool, so that it is possible to pull down the platform with a rope, in order to perform the heave drop tests without the disturbance of a moving person inside the pool. Figure 4-1 gives an overview about the complete installation for the heave drop tests. The pitch movement is recorded with a SBG IG 500-N sensor, which uses gyroscopes to deliver data for the absolute angles of the platform in all rotational degrees of freedom. During the pitch drop tests, it is fixed to the tower top. See attachment A-2 for the specifications of the sensor. In order to rotate the model around its centre of rotation, two ropes are attached to the central tower: One above the centre of rotation and one below in the same distance. One of these ropes is led through a deflection roller, so that both lines can be pulled from one side outside of the pool. Figure 4-2 shows the installation for the pitch drop tests.

32 4 Methodology of model testing

Figure 4-2: Breadboard construction for turning the model in pitch direction

4.2.2 Numerical simulation with 3Dﬂoat

The numerical tests are performed with 3Dfloat, a software tool for the modelling of floating offshore wind turbines. Its code uses an aero-hydro-servo-elastic finite element model to estimate the movement of a structure under the simultaneous influence of wind, waves and current. It was developed by the Norwegian University of Life Sciences as well as the IFE and is used for analysis and comparison of con- ceptual designs (Cordle and Jonkman, 2011). The finite element structural model uses Euler-Bernoulli beams with 12 degrees of freedom. 6 on both end points of each beam. Each of these end points serves as a node, where forces can be transferred. Loads from gravity, buoyancy, water and wind attack distributed along the nodes of the model. Additional stiffness and damping can be applied to certain nodes. For calculation of hydrodynamics 3Dfloat uses the Morison equation, that is presented in subsection 3.5.1. Therefore not only strength and geometrical properties but also added mass and damping coefficients need to be given for all structural elements. The standard beam elements are round cylinders, but it is also possible to add rectangular shaped box members to the model. 3Dfloat calculates a water particle flow field for current and waves, but the structure and its movement do not interfere with

33 4 Methodology of model testing the flow field. Hence, the linear damping due to wave diffraction needs to be added manually. 3Dfloat is text based for all input and output. The best way to work with it, is to prepare a .txt file including all model definition data and simulation parameters. This Input file will then be processed by the program, which can write the desired simulation results in text based output files for post processing.

4.3 Model description

4.3.1 General assumptions

Figure 2-6 on page 15 shows the investigated floater with a 6 MW Siemens wind turbine in full scale. The mooring lines that are necessary for station keeping and that have considerable influence on the hydrodynamic behaviour in wind, waves and current, are not shown. For model testing a scale factor of 1/40 was chosen as a compromise between low cost and handling efforts on the one hand as well as good scaling results on the other hand. Several simplifications were made for modelling. In the first phase of the tests that is presented in this thesis, the rotor nacelle assembly on the top of the tower is represented by a clump mass. No aerodynamic effects are considered. Likewise, no mooring lines were used for the drop tests, due to the limited dimensions of the test basin. This can be accepted because the influence of the mooring system on added mass and damping is assumed to be low for semisubmersible floaters of this size. The influence of the mooring lines might be taken into account afterwards, when their effect is investigated in further tests that are not part of this thesis. The concrete central tower of the platform and the steel tower of the real wind turbine have a decreasing diameter with increasing height. The diameter of the tower in the model is constant along z-axis. The location of mechanical outfitting like working platforms, ladders, fairleads and so on is not taken into account in the models, only its overall mass is respected. Table 4-1 lists the key figures of the floating wind turbine in full scale and the representative values for the scale models. The stiffness matrix, shown in equation 4-1, is defined by the buoyancy and the geometry of the model. With g = 9.81 m/s

34 4 Methodology of model testing

Full scale Scaled Tower weight 350 t 5.469 kg RNA weight 310 t 4.844 kg Mechanical outﬁtting 200 t 3.125 kg Mooring tension1 2.354 ∗ 106 N 36.79 N Displacement 9870 t 154.2 kg Centre of buoyancy2 7.046 m 0.176 m Centre of gravity2 10.342 m 0.259 m Interface platform/tower2 30 m 0.75 m Tower height2 102 m 2.55 m Draught 20 m 0.5 m

1 in a downwards direction 2 from bottom of platform

Table 4-1: Key ﬁgures of the OO Star ﬂoater in full scale and model scale

3 and ρH2O = 1000 kg/m it accounts:   0 0 0 0 0 0   0 0 0 0 0 0   0 0 1743.75 N/m 0 0 0   C =   (4-1) 0 0 0 155.277 Nm/rad 0 0   0 0 0 0 155.238 Nm/rad 0   0 0 0 0 0 0

4.3.2 Physical scale model

For the tests in the basin, a physical model of the OO Star floater had to be built. For this 1/40 scale model, polycarbonate was the material of choice. This thermoplastic polymer is not brittle and nearly unbreakable. It can easily be cut by sawing and several different types of glue are available to bond polycarbonate. For this model, ACRIFIX R 1R 0192 was used. It is a light curing 1-component polymerization adhesive with some gap-filling capability. The application of this glue makes the resulting compound brittle. Figure 4-3 shows the plastic model of the OO Star platform with tower. A sketch

35 4 Methodology of model testing

Figure 4-3: Scale model of the OO Star platform with tower, without ballast

36 4 Methodology of model testing of the model of the platform is shown in figure 4-4. The three buckets and the central tower are manufactured from of pipes with a diameter of 250 mm and 200 mm respectively and a wall thickness of 3 mm. All other plastic elements are built out of 1.5 mm thick plane sheet. The three pontoon legs are completely flooded with water during the tests. Holes were drilled through the base and roof slab and the bulkheads in order to let water in and all air out, when the model is put into the water. There are three water inlets on the bottom of each pontoon ( 10 mm) and air outlets on the opposite ( 5 mm). The holes in the lower area of the bulkheads ( 5 mm) let the water flow between the compartments, in the case of different water levels. The bonds between the pipes and the base slab are the only connections, which need to be watertight, because the central tower and the buckets are only partly filled with water and additionally ballasted by sandbags in order to achieve the correct scaled values for masses, centre of gravity and buoyancy as well as moments of inertia. The manufacturing met the requirements for tolerances of DIN ISO 2768 T1 class c, but the real diameter of the buckets is 248 mm and therefore 2 mm less than intended. This results in a slightly decreased buoyancy of around 0.5% for the complete model and has minor influence on its hydrodynamic behaviour. Table 4-2 lists the masses of all parts of the platform and table 4-3 shows, how much water is displaced by the model, when it is submerged to the designed draught of 0.5 m. The difference of 3.754 kg between the sum of masses and the sum of the displaced water is caused by the missing mooring lines in the model. This mass equals a reduction of draught of 0.021 m, but the effect is partly eliminated by the before mentioned lower diameter of the outer columns. The real draught of the model is 0.483 m. While the pontoons and the pipes are quite stiff, the extensions of the base slab (water entrapment plates) can be deformed by hand. So some minor deformation will result also from the forces exerted by the fluid flow. The dimensions of the base slab are so large, that it was not possible to build the whole platform in one piece. Hence, the construction was divided into two components: Component 1 with two buckets and the central tower and component 2 with the third bucket. For testing, the two pieces are connected with overlapping joints and glue. The position of the interface between the two components is shown as a red line on the sketch of the base slab. This division makes transportation less complex.

37 4 Methodology of model testing

748.5

Figure 4-4: Sketch of the OO Star platform in model scale 38 4 Methodology of model testing

Plastic mass Water ballast Solid ballast Sum Pontoons1 3 X 1.313 3 X 20.663 - 65.928 Buckets 3 X 2.171 3 X 12.465 3 X 5.845 61.443 Central tower 1.668 7.560 3.695 12.923 Tower 4.221 - 1.270 5.491 Rotor nacelle assembly - - 5.000 5.000 Sum 16.341 106.944 27.500 150.785 1 including water entrapment plates

Table 4-2: Masses of the diﬀerent parts of the scale model and their ballasts in kg

Volume Sum Pontoons1 3 X 21.823 65.468 Buckets2 3 X 24.470 73.410 Central tower2 15.661 15.661 154.539 1 including water entrapment plates 2 to still water line at elevation 0.5 m

Table 4-3: Volume of water displaced by the scale model in litre (10−3 m3)

The tower is 1800 mm long and has the same outer diameter and wall thickness like the central tower of the platform. Both components are connected by an inner overlapping joint and tape. The tower top is equipped with a mounting device for the sensors.

4.3.3 Numerical scale model

The numerical 3Dfloat scale model is a simplified model, which represents the physical scale model during all simulations. The input file is printed in figures A-5 to A-8 in the appendix. Figure 4-5 shows a plot of the numerical scale model. Round objects are not shown. The input file defines four connected bodies. The first body represents the tower and the rotor nacelle assembly. It consists of two circular beams, which are composed of several stacked beam elements. These elements may be imagined as pieces of pipes, with a start and an end point, a diameter, wall thickness and the respective material parameters. All beam elements that are used in the model have a vertical

39 4 Methodology of model testing or horizontal orientation. The first beam, described in the input file, is representing the tower and the other is defined to correspond to the mass of the rotor nacelle assembly at the tower top. The second body contents beam elements to represent one of the outer buckets and a box member for one pontoon. This body is rotated twice by 120◦ to generate the third and fourth body for completing the model by adding a second and third bucket and pontoon. The beams for tower and buckets start in the same vertical depth. And their overall height, diameter and wall thickness equal the values of the respective elements in the physical model. The highest and lowest of these beam elements are closed by a massless lid, so that there will be no water flow between the inside of the pipes and the surrounding. In their lower area, the box member with a horizontal orientation is located between the beams. It has a cuboid shape with given length, width and height. While the length between the two beams and the height in z- direction equal the values from the physical model, the width is a constant value, which is equivalent to the changing width with length in the physical model. The beams and the box members are connected by massless beam elements, which have no influence on buoyancy and hydrodynamic forces and are so thick, that they are considered to be rigid. The densities, set in the geometry definition, are calculated from the real material values, so that the mass of the polycarbonate and its respective sand or water filling will be concentrated in the wall area. Unlike the physical model with its extended base and roof slab, the numerical model uses no explicit water entrapment plates. The influence of these plates on added mass and damping is taken over by the lowest beam elements of tower and buckets and the box member elements of the pontoons.

Added mass

The added mass for movement in z-direction, that is applied in the Morison equation

(see 3.5.1), is represented in the CM coeﬃcients of the pontoons and the lowest beam elements of buckets and tower. All other elements have only a physical mass due to their density and no added mass in z-direction.

40 4 Methodology of model testing

Figure 4-5: Visualization of the numerical scale model in three views. Objects, that should have a round shape in the respective view, are not shown

In order to estimate the distribution of the added mass to buckets, pontoons and tower, a closer look is taken at the wetted surfaces of the physical model. Figure 4-6 shows the cross section of one of the three legs of the platform in z-direction. The dark grey area in the middle between the red lines has a surface of 0.1864 m2. It is surrounded by water from two sides and therefore counts twice. The light grey area on the left is 0.0917 m2. This value is also multiplied by two but then reduced by the circular area of the bucket, whose top end is located above sea level and thus has only half of the impact of a fully submerged section. The black surface on the right has an area of 0.0173 m2 and is also only partly submerged because of the tower. This comparison of surfaces results in the conclusion that 70.2% of the added mass should be set to the pontoons, 25.3% to the lowest bucket elements and 4.5% to the lowest tower element. The absolute value of the model mass plus added mass will be derived from the results of the physical heave drop tests. The scale model behaves like a damped spring pendulum with low damping after some periods. When a damping ratio of ζ = 0.15 is assumed for late periods the inﬂuence of the remaining damping increases the period length by around 1%. This inﬂuence is neglected in the calculation of added mass. With the measured period T and the spring rate D, obtained from the

C33 element of the stiﬀness matrix (see eq. 4-1), the total mass equals:

T 2 m = D (4-2) 2π

This total mass is then divided into the three parts for the pontoons, buckets

41 4 Methodology of model testing

Figure 4-6: Cross section of one of three legs, divided into bucket, pontoon and tower area and the tower. With the masses of water, displaced by the considered elements mdp = 64.52 kg, mdb = 9.20 kg and mdt = 1.96 kg the added mass coeﬃcients are:

m − (mdp + mdb + mdt) CMp = 1 + 0.702 , (4-3) mdp

m − (mdp + mdb + mdt) CMb = 1 + 0.253 (4-4) mdb and m − (mdp + mdb + mdt) CMt = 1 + 0.045 . (4-5) mdt

In section 5.2 it is shown, that the period of the oscillation of the physical scale model in heave direction is T0 = 3.04 s. With this number, the total mass is calculated to 408.199 kg. With a structural mass of 150.785 kg, the overall added mass coeﬃcient in heave motion is CM = 2.71. Distributed on the three relevant elements, the coeﬃcients are:

42 4 Methodology of model testing

Pontoon: CMp = 4.61

Bucket: CMb = 10.18

Tower: CMt = 8.63 The CM values for movement in x and y-direction are set to 2 for all circular elements according to Agerschou and Edens (1965, p. 239). For the pontoon the added mass is set to 1.38 for y-direction (see A-9 in the appendix) and to 1 for x-direction.

Damping

In order to achieve the damping coefficients for the numerical model, the cross sections of the physical model with its damping plates are compared to the cross sections of the 3Dfloat model. Figure 4-7 reveals, that the real cross section is bigger than the numerical. Therefore the CD values need to be adjusted. The physical values are overtaken from the calculations for the added mass. The values for the numerical model are 0.1434 m2 for one pontoon, 0.0491 m2 for one bucket and 0.0105 m2 for a third of the tower. All of them count twice, because the elements are fully submerged. By dividing the real areas by the numerical, the following factors result: Pontoon: 1.30 Bucket: 1.37 Tower: 1.15 The assumed damping coefficients need to be multiplied by these factors to achieve the intended damping behaviour. It follows from the application of strip theory for heave motions and small angles in pitch motion, that the hydrodynamic damping effects are the same for all movement in z-direction, no matter whether they are caused by heave or pitch motions. Thus, the same damping coefficients can be used for the simulation of the free decay behaviour in the heave and pitch degree of freedom. This is done for the Morison damping values and would be possible for linear damping also, if the damping would not be added to one central point of the numerical model, but to the same elements in bucket, pontoon and tower that are used for Morison damping. For the simulations presented in this thesis, separate values for linear damping in heave and pitch are applied only to the point of origin, instead of distributing the linear damping forces

43 4 Methodology of model testing

Figure 4-7: Cross section of one of three legs, divided into bucket, pontoon and tower areas: grey and black for physical model, coloured for numerical model to diﬀerent nodes.

Physical Top Bottom Sum Bucket 0.0426 0.0917 0.1343 Pontoon 0.5591 0.5591 1.1182 Tower 0.0068 0.0173 0.0241

Numerical Top Bottom Sum Bucket 0.0491 0.0491 0.0982 Pontoon 0.1434 0.1434 0.2868 Tower 0.0105 0.0105 0.0209

Table 4-4: Cross section areas of one leg of the physical and numerical model in m2

44 5 Physical and numerical test results

The following chapter shows the test results obtained from both, the physical experiments and the numerical calculations of the same test cases. It is divided into the four sections 5.1 to 5.4 for general observations, period length, damping behaviour and the Keulegan-Carpenter number. Within each section two subsections for the heave and pitch drop test case are found. The chosen coeﬃcients for the 3Dﬂoat simulations and all other assumptions are only presented, while the arguments for choosing them are given in the next chapter 6. Likewise, the discussion of the observations follows in chapter 6.

5.1 General observations

5.1.1 Heave drop test case

For the physical heave drop tests several test runs with a drop height of -14 cm and a decreased initial position at -8 cm were performed. The measured signals of five of these runs are plotted in time domain in figure A-3 in the appendix. It is visible, that the signal is distorted by noise. This noise has a frequency of around 1.1 Hz and an amplitude of around 24 mV. Effective filtering with a low-pass has severe damping impact on the signal and cancellation by overlaying of a regular sine was not successful because of a slightly varying frequency in the noise. The best practice to cope with this issue, is to calculate the average over several test runs. This was done for the plot in figure 5-1. It shows the free decay of the movement for the heave drop tests with both heights. The graph is oscillating around equilibrium with damping. In table 5-1, the time and displacement values are listed for all extremes reached after each half period for the 14 cm drop. Only the first five periods are considered to deliver reliable values for the displacement because of the

45 5 Physical and numerical test results

Heave drop test 14cm vs. 8cm 8 14 cm 6 8 cm 4

m 0 −2 −2

−4

−6 Z−position in 10 −8

−10

−12

−14

−40 −30 −20 −10 0 10 20 30 Time in s

Figure 5-1: Heave drop test results, experimental mean data above mentioned distortion in the sensor’s signal. The 3Dfloat output graphs, plotted in figure 5-4, are undisturbed and show no conspicuousness in their shape. In the moment of release, the acceleration is higher in the 3Dfloat simulation than the acceleration of the physical model. This results in a phase shift between the numerical and experimental results.

5.1.2 Pitch drop test case

For the pitch drop test, several test runs with an initial rotation angle of around 8 degrees were performed. The measured signal of three of these runs is plotted in time domain in ﬁgure A-4 in the appendix. The signal is of a higher quality than the heave signal with no relevant noise. Thus, a longer time span can be used for determining the free decay behaviour. Anyway, an average is calculated over the

46 5 Physical and numerical test results

No. of t in s T in s extreme value abs. value period in cm in cm 0 0 -14.29 14.29 0.5 2.24 7.2 1 4.16 4.16 -2.78 2.78 1.5 5.81 1.73 2 7.32 3.16 -1.09 1.09 2.5 9.04 0.86 3 10.64 3.32 -0.62 0.62 3.5 12.12 0.62 4 13.58 2.94 -0.41 0.41 4.5 14.96 0.41 5 16.34 2.76 -0.28 0.28 5.5 18.38 0.28 6 19.68 3.34 -0.28 6.5 21.10 0.37 7 22.44 2.76 -0.23 7.5 24.56 0.34 8 25.90 3.46 -0.27 8.5 27.38 0.39 9 28.80 2.9 -0.18

Table 5-1: Key ﬁgures of the results of 14 cm heave drop test

47 5 Physical and numerical test results

Pitch drop test 8deg 8 Experimental mean data 6

−2 Angle in deg

−4

−6

−8

−10 0 20 40 60 80 100 120 140 Time in s

Figure 5-2: Pitch drop test results, experimental mean data single test runs for minimizing the small drift of the equilibrium position that is recognizable throughout the signal. The averaged displacement is plotted over time in ﬁgure 5-2. It shows the free decay of the movement from the moment of release and on. The graph is oscillating around equilibrium with damping. In table 5-2, the time and displacement values are listed for all extremes reached after each half period for the pitch drop.

5.2 Period length

5.2.1 Heave drop test case

The ﬁrst outcome that is regarded in the experimental heave drop test, is the oscillation period. Only full periods are considered. The ﬁrst full period, listed in

48 5 Physical and numerical test results

No. of t in s T in s extreme value abs. value period in deg in deg 0 0 -8.214 8.214 0.5 2.12 6.473 1 4.3 4.30 -4.974 4.974 1.5 6.48 3.666 2 8.62 4.32 -2.671 2.671 2.5 10.68 2.192 3 12.82 4.20 -1.789 1.789 3.5 14.88 1.567 4 16.94 4.12 -1.315 1.315 4.5 18.98 1.255 5 21.08 4.14 -0.966 0.965 5.5 23.18 1.004 6 25.3 4.22 -0.815 0.815 6.5 27.28 0.786 7 29.58 4.28 -0.730 0.729 7.5 31.54 0.649 8 33.64 4.06 -0.608 0.608 8.5 35.6 0.589 9 37.72 4.08 -0.575 0.575 9.5 39.8 0.428 10 41.88 4.16 -0.474 0.474 10.5 43.96 0.395 11 46.02 4.14 -0.411 0.411 11.5 48.08 0.343 12 50.12 4.10 -0.403 0.402 12.5 52.16 0.354 13 54.36 4.24 -0.337 0.337 13.5 56.32 0.318 14 58.42 4.06 -0.302 0.302 14.5 60.42 0.296 15 62.6 4.18 -0.256 0.255

Table 5-2: Key ﬁgures of the results of pitch drop test

49 5 Physical and numerical test results the third column of table 5-1, is longer than all others. The other values vary from 2.76 s to 3.46 s with no obvious systematics. A damped signal has a period T , which is longer than the period T0 of the undamped oscillation. In section 5.3 it will be shown, that the damping of the oscillation is low after a number of periods. Hence, the period of the 5th to 9th period is averaged, in order to get a value close to the undamped period. This mean value is 3.04 s, which equals a frequency of 0.33 Hz. The 8 cm drops deliver a similar mean value of 3.08 s. The period length in the numerical simulations is about constant after the ﬁrst two periods. Its mean value for 5th to 9th period is 3.17 s.

5.2.2 Pitch drop test case

The numbers in the third column in table 5-2 show full period lengths of between 4.06 s and 4.32 s for the physical pitch drop test. The standard deviation is lower than for the heave drop test results. The tendency to shorter periods in the later oscillations is the same. With period number 8 to 15 taken into account, the average period length is 4.13 s. If this period length is compared with the 3Dﬂoat simulations, a deviation is detected. It is visible in ﬁgure 5-6, that the oscillations in the numerical simulation have a longer period. Its value of 4.55 s is 10% higher.

5.3 Damping behaviour

5.3.1 Heave drop test case

The numbers for the extreme values in table 5-1 show, that the movement in the downwards direction is damped stronger than the movement in positive z-direction. In the periods 4 to 6 even no damping could be measured for the upwards movement. Thus, only full periods will be considered. The absolute values of the locations of the lower turning points are listed in the last column of the table. They correlate linearly to the remaining energy, that is still inherent in the oscillating system and not yet dissipated by any damping effects. The damping behaviour will be investigated for the first five full periods only, because the signal becomes unclear in later periods. In figure 5-3, the corresponding points are plotted as black circles on a logarithmic

50 5 Physical and numerical test results

Heave drop test damping behaviour 2 10 Experimental data Pure linear damping (445 kg/s) Pure Morison damping (CD=11.4) 1 10 3Dfloat data (CD=8, lin: 120 kg/s) m

−2 0 10

−1 10 Z−position in 10

−2 10

−3 10 0 1 2 3 4 5 No. of period

Figure 5-3: Heave drop test damping behaviour, experimental and numerical data between linear and quadratic damping y-scale. For getting a first impression about what effect dominates the damping characteristic, pure linear damping and pure Morison damping is also plotted. In both cases, the damping coefficients were adjusted to achieve the correct damping after the first full period. For linear damping, this needs 445 kg/s, which is about 53% of the value for critical damping. The Morison coefficient for achieving the same damping after one period is 11.4. A combination with a Morison coefficient of 8.0 and linear damping of 120 kg/s delivers the best approximation for the simulation of the real damping behaviour. The 3Dfloat results with these parameters are plotted as red asterisks. The maximum deviation between simulation and experiments is 13.8% after the 2nd period. The plot in figure 5-4 shows the experimental and numerical results for both drop heights in displacement over time.

51 5 Physical and numerical test results

Heave drop test experimental vs. 3Dfloat, 14cm 8 Experimental data 6 3Dfloat data (CD=8, lin: 120 kg/s) 4

m 0 −2 −2

−4

−6 Z−position in 10 −8

−10

−12

−14

−40 −30 −20 −10 0 10 20 30 Time in s

Heave drop test experimental vs. 3Dfloat, 8cm 8 Experimental data 6 3Dfloat data (CD=8, lin: 120 kg/s) 4

m 0 −2 −2

−4

−6 Z−position in 10 −8

−10

−12

−14

−40 −30 −20 −10 0 10 20 30 Time in s 52 Figure 5-4: Heave drop test results, experimental and numerical data for both drop heights 5 Physical and numerical test results

5.3.2 Pitch drop test case

In ﬁgure 5-6, the angular displacement over time is plotted for the experimental and numerical pitch drop test. The angular displacement can also be understood as the amount of potential energy that is still inherent in the system. The trigonometric error for this assumption is less than 0.7% for 8 deg and moves towards zero for smaller angles. For the assessment of the damping behaviour, the ﬁrst 15 oscillation periods are regarded. Subsequently, the graph shows much deviation from the expected sine wave form and too much changing of the equilibrium position to rely on the position of the extreme values. The experimental data is compared with the results of

3Dﬂoat simulations. For these the CD value of 8.0 is taken over from the heave drop simulations and a linear damping value of 6 kgm2/s was chosen to reach the best approximation between the experiment and the simulation. Figure 5-5 shows the angular displacement after each full period for experimental and numerical results. The highest deviation of 23,7% is found after the 15th period. Before, the deviation is always lower. The damping of pitch movement is way lower than the damping of heave movement. For example, after two full periods 7.6% of the energy is remaining for heave, but 32.5% for pitch. At last, it is observed carefully what might happen in the last oscillations, which were not considered for the approximation. One notices, that it could be, that the oscillations calculated by 3Dﬂoat come to a standstill, while the physical model might still be moving.

5.4 Keulegan-Carpenter number

5.4.1 Heave drop test case

The KC number in the heave drop test is decreasing with the amplitude of the oscillation. For calculating it, the characteristic diameter is set to the width of the base slab of the pontoons, which is around 0.38 m. The period is around 3 s. The vertical velocity reaches its maximum when crossing the equilibrium position. In the ﬁrst period it is 0.18 m/s and in the second 0.075 m/s. With these values, the KC

53 5 Physical and numerical test results

Pitch drop test damping behaviour

Experimental data 3Dfloat data (CD=8, lin: 6 kgm²/s)

0 10 Angle in deg

0 5 10 15 No. of period

Figure 5-5: Pitch drop test damping behaviour, experimental and numerical data

54 5 Physical and numerical test results

Pitch drop test 8deg 8 Experimental mean data 6 3Dfloat data (CD=8, lin: 6 kgm²/s)

−2 Angle in deg

−4

−6

−8

−10 −20 0 20 40 60 80 100 120 140 Time in s

Figure 5-6: Pitch drop test results, experimental and numerical data number becomes 1.4 for the ﬁrst period, 0.6 for the second and → 0 subsequently.

5.4.2 Pitch drop test case

The KC number in the pitch drop test is decreasing with the amplitude of the oscillation. Like in the heave drop, the characteristic diameter is set to the width of the base slab of the pontoons, which is around 0.38 m and the period is 4.13 s. For obtaining values for the vertical velocity, the angular velocity is multiplied by 0.688 m, the distance between the central tower and the bucket that is located in positive x-direction. The angular velocity, when crossing zero is 0.186 rad/s in the first period and 0.107 rad/s in the second. With these values, the KC number for the outer position is 1.0 for the first period, 0.6 for the second and → 0 subsequently. Looking at the flow velocity in x-positions between the bucket and the tower, the

55 5 Physical and numerical test results

KC number is decreasing linearly from the mentioned values to zero. The Keulegan- Carpenter number for the two other buckets in negative x-direction is 50% lower, since the shorter levers result in lower translational velocities.

56 6 Discussion of results

The results that were presented in the previous chapter 5 will be discussed in the following. All assumptions that were made are justified here and possible reasons are given for deviations from the expected results. In section 6.1 the results in terms of period length are used to find the correct added mass of the model and section 6.2 deduces the damping coefficients from the displacement data. Within the sections, the discussion is split into two subsections for the heave and the pitch degree of freedom.

6.1 Period length and added mass

6.1.1 Heave motion

The first full period measured in the heave drop tests, is longer than all others. A possible reason could be that the process of releasing the platform might have been not suddenly enough. Therefore this first value is ignored. The latter values are also not constant, what might be explained by the influence of remaining fluid movement from previous periods and also by the inaccuracies of the sensors signal. Theoretically, the undamped heave period is strictly depending on the total mass that is composed of the structural mass plus added mass. The 14 cm heave drop experiments deliver an averaged period of 3.04 s. This corresponds to a total mass of 408.2 kg and an added mass of 257.4 kg, when the structural mass of 150.8 kg is excluded. For getting a rough idea about whether this value is realistic, the rule of thumb, presented in 3.5.1 is used to calculate the added mass manually. With the assumptions shown in appendix A-9, this delivers a value of 244.8 kg, which is close to the experimental results, although the influence of the not submerged areas atop of the buckets and tower are neglected.

57 6 Discussion of results

A WAMIT1 analysis for the OO Star in full scale delivers a scaled value of 245.7 kg for added mass. Therefore, the experimental value is accepted and will be used for the added mass in the deﬁnition of the numerical model.

6.1.2 Pitch motion

For the period of heave motion, only the total quantity of added mass is important, but for the pitch motion it is crucial in order to achieve the correct moment of inertia, in what distance from the center of rotation the added mass comes into effect. The 3Dfloat simulations show a 10% longer period than the experiment. Hence, it can be assumed, that the chosen distribution of the added mass to the bucket and pontoon explained in subsection 4.3.3 is not exact. Less mass should be added to the buckets and more to the pontoons or tower in order to achieve a lower moment of inertia with the same amount of mass. The deviation can be partly explained by the simplifications that were made in the subsection about added mass in 4.3.3. The applied procedure for distributing the added mass is lacking in just looking at the wetted surface areas and not on the actual shape of the added water masses above and below them. It was assumed, that the upper areas of bucket and tower elements that end above still water level might be linearly excluded from the added mass calculation. But in reality this procedure underestimates the influence of these non-wetted surfaces on the added mass, because central areas move a higher amount of added mass, compared to areas close to the boundary of the elements. The bigger effect of subtracting areas, which are positioned in the centre of elements, could be considered by a more sophisticated approach, which uses spheres and cylinders instead of just areas for the circular beam and box member properties. Such a more complex approach could be justified for future investigations, when no experimental reference data (and no WAMIT analysis) is available. Tests were made with a

1“WAMIT is a computer program based on the linear and second-order potential theory for analyzing floating or submerged bodies [...]. The boundary integral equation method (BIEM) [...] is used to solve for the velocity potential and fluid pressure on the submerged surfaces of the bodies. Separate solutions are carried out simultaneously for the diffraction problem [...] and the radiation problems for each of the prescribed modes of motion of the bodies. These solutions are then used to obtain the relevant hydrodynamic parameters including added-mass and damping coefficients, exciting forces, response-amplitude operators (RAO’s), the pressure and fluid velocity, and the mean drift forces and moments.” (WAMIT Inc.)

58 6 Discussion of results reduced added mass of 20% less on the buckets, which was added to the pontoon elements. The simulation showed a decreased period of 4.45 s which is still 8% too much. Likewise, a reduction of the pontoon’s axial CM value has only minor influence on the period length. Thus, there must be another unknown effect. The WAMIT calculations for the full scale platform deliver a down scaled moment of inertia of 28.84 kgm2, what leads to a gyration radius of 0.343 m. This value cannot be reached, because the distance between the centre of rotation and the centre of the pontoon is already 0.396 m. So even adding all added mass to the pontoon that was previously attached to the bucket would result in a higher moment of inertia, but performed simulations for this scenario showed, that the period will be 4.00 s then. This is too low and adding no mass to the buckets would be unrealistic. Thus, for the presented testing, it is adhered to the original values and the longer period is accepted, although it might have a slight impact on the damping behaviour. Further testing needs to be done in order to find the cause of this deviation.

6.2 Damping coeﬃcients

6.2.1 Heave motion

During the experimental free decay in heave motion, the half periods in which the physical model is moving upwards are less damped than the downwards movement. This behaviour can partly be explained by the shape of the platform, because it can be assumed that the real drag coefficient is higher for a fluid streaming around the model in upwards than downwards direction. But another effect must be superim- posed to these different coefficients, because a differing drag does not explain why the damping in upwards direction comes to a complete standstill in several periods. A possible effect that could cause this might be the influence of the history of the motion. It could be for example that the first fast upwards movement after releasing the model, might have accelerated the water so vastly that its remaining motion in- fluences the damping effects in subsequent period. This cannot be validated without further investigation. Faltinsen (1987, p. 63) explains, that there is a significant influence of the sea bed on the damping characteristics, when the water depth to draft ratio is less than

59 6 Discussion of results two. In this experiment it is a little more than two, but possibly the basin bed still has a considerable but unknown influence. Also the limited stiffness of the plastic model, especially in the area of the water entrapment plates, due to the thin material thickness might have an influence that is not investigated here. It could influence the damping coefficients, especially for fast motions, compared to a completely stiff model. The heave damping coefficients, applied to the 3Dfloat model, were chosen so that it emulates the models movement in the experiment. Therefore the CD value of Morison equation was set to 8.0. Faltinsen (1990, p. 238f.) mentions that findings of Graham (1980) may be used for explaining experimental results, when the KC number is in a range of 0 to 10. His experiments show, that the CD value is then − 1 8∗KC 3 for a flat plate. This implies that for a KC number above 1, the CD value is less than 8.0 and the other way round. The Keulegan-Carpenter numbers in the experiment are in this order of magnitude for the first periods, but then they move towards zero. That means that the CD values rise. As this cannot be considered in the 3Dfloat model, the chosen values are supposed to be intermediate.

The 8.0 value is relatively high compared to the CD value of 4.8 that was chosen to match DeepCWind tank-test data with numerical simulation results (Robertson et al., 2012). Figure 6-1 shows the semisubmersible floating platform that was investigated in the DeepCWind project. Unlike the OO Star, it has no thin flat water entrapment plates attached. Thus, the DeepCWind floater is assumed to have a lower Morison damping than the OO Star. The above mentioned value of 4.8 is also suggested for the second phase of OC4 simulations, but no experimental reference data is available yet. From the visualization in figure 5-3, it becomes clear, that the real damping is characterized rather by quadratic Morison damping but does also need some amount of linear viscous damping from diffraction forces for getting enough damping, while moving with low velocity. Hence, in order to complete the damping forces, linear damping coefficients were added to the 3Dfloat model. To simulate the heave motion a linear damping coefficient of 120 kg/s was selected. This equals 14.2% of critical damping. IT shoudl be kept in mind that this value for critical damping is only for the linear damping. The sum of Morison and linear damping delivers higher amounts.

60 6 Discussion of results

Figure 6-1: Picture of the DeepCWind 1/8 scale model (Bangor Daily News)

The above mentioned damping coefficients were chosen to fit only the first five oscillation periods of the physical model. This was done, because the subsequent measurement signal does not deliver reliable numbers for the remaining energy in the system. But when the later sensors’ signal and the simulation results in figure 5-4 are compared qualitatively, it seems like the oscillation of the simulation results come to a standstill, before the physical model does. Therefore, it might be that the linear damping coefficient is too high to represent reality in low amplitudes. Another effect that could cause the continuous motion in the physical tests can be the influence of waves reflected from the basin walls. These returning waves excite small movements of the platform.

6.2.2 Pitch motion

The pitch motion has a lower damping in general, what can be explained by the shape of the ﬂoater. While pitching, the relative velocities between model and ﬂuid are higher for points with a big distance from the centre of rotation than for points

61 6 Discussion of results close to it. Therefore and because of the long corresponding levers the effect of a damping element is big, when it is located far from the central tower. By looking at the shape of the floater on figure 4-4, one can see that exactly in these outer areas, the damping is reduced by the buckets ending out of the water. The huge extensions of the base slab close to the central tower seem to have a big impact on damping in vertical direction. That cannot effect the pitching a lot. The simulations with a Morison damping of 8.0 and linear damping of 6 kgm/s, chosen for the pitch degree of freedom, deliver matching results with the experimental data. The value for linear damping equals only 2.9% of critical damping and is therefore less than for the heave degree of freedom. This nearly 80% lower value compared to heave damping makes sense, because the overall damping of pitch is also i.e. 77% lower, when the remaining energy after the first two oscillations is compared. A qualitative comparison of the last amplitudes shown in figure 5-6 reveals the possibility of over damping the small motions in 3Dfloat. This might be due to too much linear damping but needs to be further investigated.

62 7 Conclusion and outlook

It was the objective of this research to find out, if the 3Dfloat code is capable of simulating the movement of the OO Star semisubmersible floater in heave and pitch free decay motion and what parameters have to be used in the numerical model. The measurement data of a plastic model in 1/40 scale in a water basin of limited size therefore serve as a reference. Although reflections from the basin walls and bed might influence the free decay, it is presumed, that these experimental data represent the models behaviour in open waters. The measurements show oscillations with periods that correspond to added masses, that are close to the expectations. Some deviations from the expected behaviour, concerning the varying period lengths, suggest that the experimental results were influenced by the history of the fluid motion, especially while performing the heave drop tests. Further testing with more sophisticated measuring equipment will show, if this is the reason or if the poor sensors signal in heave tests causes the mentioned effect. The distribution of the added mass to the buckets, pontoons and tower of the model led to a deviation of about +10% in the period length of the pitch motion, when the numerical data is compared with the experimental results. This is for an unknown reason and can be corrected by measuring the moment of inertia for the physical model, what is part of the subsequent testing campaign at ECN. With regard to the damping characteristics, simulation results show good agree- ment with the experimental data, when a CD value of 8.0 is applied in the Morison equations and linear damping of 120 kg/s in heave and 6 kgm/s in pitch degree of freedom is added to the numerical model. The water entrapment plates at the bottom of the platform seem to contribute a high amount of the overall damping forces. The damping of heave motions is stronger than the pitch damping. Further testing should be done with additional plates in the bucket area to find out more about their effect.

63 7 Conclusion and outlook

It also needs to be investigated, if the numerical model represents the damping of very small motions properly. This can be done in the further testing campaign at ECN. There the model’s motion will be recorded with a laser tracking system, which will deliver more accurate measurement data. It can be summarised, that the 3Dfloat code is suitable for simulating the motion of the OO Star in heave and pitch, when the hydrodynamic forces achieved from Morison equations are supplemented by linear damping forces. The further progress of testing will show, if the results remain valid for coupled motions with other degrees of freedom and under the influence of wave and wind loads. There is also a need to further investigate, to what extent mooring lines influence the motion of the platform.

64 Bibliography

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iii Declaration

I herewith declare, that I have completed the present thesis independently making use only of the speciﬁed literature and aids. Sentences or parts of sentences quoted literally are marked as quotations; identiﬁcation of other references with regard to the statement and scope of the work is quoted. The thesis in this form or in any other form has not been submitted to an examination body and has not been published.

As,˚ 2nd October 2013 Felix Kelberlau

iv A Appendix

v A Appendix

Figure A-1: Data sheet of Honeywell 945-L4Y-2D-1C0 ultrasonic distance sensor

vi A Appendix

IG-500N - Specifications

PARAMETER SPECIFICATIONS REMARKS MECHANICAL DRAWING All dimensions are in millimeters Performance Aerospace Ground Marine / Subsea Roll/Pitch (Dynamic) 1.0° RMS 0.8° RMS 0.35° RMS Under good GPS availability Side view 36,40 Heading (Dynamic) 1.0° RMS 0.5° RMS 1.0° RMS Depends on heading aiding source Resolution < 0.05° < 0.05° < 0.05° Velocity (RMS) < 0.1 m/s < 0.1 m/s < 0.1 m/s Under good GPS availability 22 Position (SEP) < 2 m < 2 m < 2 m Under good GPS availability

Heave - - 10 cms or 10 % Whichever is greater 6 Sensing range 360° in all axes, no mounting limitation Solid state sensors

Inertial Sensors Accelerometers Gyroscopes Magnetometers Top view Measurement range ± 5 g ± 300 °/s ± 1.2 Gauss Refer to sensors options table Non-linearity < 0.2 % < 0.05 % < 0.2 % % of full scale Initial bias error ± 5 mg ± 0.5 °/s ± 0.5 mGauss Over temperature range Bias in-run stability ± 0.06 mg 20 °/hr - Allan variance - constant temperature M3 Scale factor stability < 0.1 % < 0.05 % < 0.5 % Over temperature range 35,80 Noise density 0.25 mg/√ Hz 0.05 °/s/√ Hz 0.01 mG/√ Hz Alignment error < 0.05° < 0.05° < 0.1° Bandwidth 250 Hz 240 Hz 500 Hz 1 k Hz gyroscopes coning integrals 49,40 Sampling rate 10,000 Hz 10,000 Hz 1,000 Hz Advanced anti-aliasing FIR ilters GPS Receiver Receiver type L1 frequency, C/A Code, 50 Channels, SBAS, 4 Hz Acquisition time < 1.0 s / 29 s Hot start / Cold start PRODUCT CODE .: standard product options Tracking sensitivity -160 dB Pressure Sensor IG-500N-G#A#P#-# Resolution 2.5 Pa / 20 cms / < 1 feet GYROSCOPES Pressure accuracy ± 50 Pa / ± 150 Pa Relative / Absolute 2: 75 °/s Sampling rate 50 Hz 3: 150 °/s 4: 300 °/s . Communication 5: 600 °/s 6: 1200 °/s Available data Euler angles, quaternion, rotation matrix, velocity, Each output can be enabled or position, heave, calibrated sensor data, delta angles, disabled by the user. Output rate is barometric data, device status, raw GPS data, UTC time user selectable ACCELEROMETERS reference, etc. 1: 2 g 2: 5 g . 3: 18 g Output rate 100 Hz for orientation, velocity and position 500 Hz in IMU mode only Serial interface RS-232, RS-422, TTL 3.3V or USB RS-422 only for S and O packages PROTOCOL MODE Binary proprietary protocol and NMEA/ASCII USB using an external adapter 1: RS-232 . CAN interface CAN 2.0A/B up to 1 Mbit/s Only available for S and O packages 2: Serial TTL 3: CAN 2.0A/B Physical 4: RS-422 Dimensions OEM 27 x 30 x 14 mm, 1.1 x 1.2 x 0.6” PACKAGING Dimensions box 36 x 49 x 22 mm, 1.4 x 1.9 x 0.9” B package B: Box small size . 36 x 49 x 25 mm, 1.4 x 1.9 x 1” S package S: Box with SyncOut Weight OEM 10 grams, 0.02 pounds O: OEM version Weight box 44 grams, 0.1 pounds B package 48 grams, 0.1 pounds S package Speciied temperature -40 to 85°C, -40 to 185°F Non-condensing environment Shock limit 1,000 g (Powered); 2,000 g (Unpowered) Shocks can afect performance OEM VERSION Operating vibration 3 g RMS (20 Hz to 2 k Hz per MIL-STD 810G) Valid for 18g accelerometers Electrical Operating voltage 3.3 V to 30 V Power consumption 800 mW @ 5.0 V High eiciency DC/DC converter SyncOut, Trigger Open drain pull-up voltage -0.3 to 25 V Open drain, use a pull-up resistor Start-up time < 1 s Valid data

SBG Systems EMEA (Headquarters) SBG Systems North America Phone: +33 1 80 88 45 00 Phone: +1 (773) 754 3272

Figure A-2: Data sheet of SBG IG 500-N inertial navigation system

vii A Appendix

Heave drop test 14cm 8 1st 6 2nd 3rd 4

m 0 −2 −2

−4

−6 Z−position in 10 −8

−10

−12

−14

−40 −30 −20 −10 0 10 20 30 Time in s

Heave drop test 8cm 8 1st 6 2nd 4

m 0 −2 −2

−4

−6 Z−position in 10 −8

−10

−12

−14

−40 −30 −20 −10 0 10 20 30 Time in s viii Figure A-3: Heave drop test results, single test runs A Appendix

Pitch drop test 8deg

1st 2nd 6 3rd

0 Angle in deg −2

−4

−6

−8

0 20 40 60 80 100 120 Time in s

Figure A-4: Pitch drop test results, single test runs

ix A Appendix

! 1:40 Scale model of OOStar concrete floater for test at ECN in September 2013 ! set_environment gravity: on buoyancy: on hydro_force: morison wind_force: ndrg waves: extrapolated_airy norder: -1 wind: no_mean_profile rho_water: 1000. rho_air: 1.29 nu_water: 1.e-5 nu_air: 1.e-6 wave_amplitude: 0. wave_direction: 0. current_speed: 0. current_direction: 0. wind_speed: 0. wind_ref_height: 90. wind_exponent: 0. wind_direction: 0. wave_ramps: 0. wave_rampe: 20. depth: 200. spectral_peak: 10. seed: .3 nfreq: 64. tshift_waves: 800. tshift_wind: 0. tsweeps: 0. tsweepe: -1. stretch: 0. wavelength: 156.07768226721353 pre_computed_waves_file: off random_mode: write_file random_file: irreg_waves.txt

!Global Morison coefficients are applied, if not specified elementwise wave_forces cm_morison: 2.0 cd_morison: 0.7

! Geometry definition define_material name: plast rho: 1225.0 e: 2.7e9 g: 0.97e9 ! real rho: 1154.4 define_material name: plast_sand_tower rho: 33065.94754 e: 2.7e9 g: 0.97e9 ! taking sand fill into account define_material name: plast_water_tower rho: 17145.473773 e: 2.7e9 g: 0.97e9 ! taking water fill into account define_material name: plast_sand_bucket rho: 41397.739541 e: 2.7e9 g: 0.97e9 ! taking sand fill into account define_material name: plast_water_bucket rho: 21311.3697705 e: 2.7e9 g: 0.97e9 ! taking water fill into account define_material name: plast_water_pontoon rho: 34141.84901 e: 2.7e9 g: 0.97e9 ! taking water fill into account define_material name: plast_rna rho: 10185.39066 e: 2.7e9 g: 0.97e9 ! adjusted to 5 kg RNA define_material name: plast_tower rho: 1636.37254 e: 2.7e9 g: 0.97e9 new_keypoints ! node # x y z [m] 1 0. 0. -.5 ! center column bottom 2 0. 0. -.4375 ! sand ballast upper level 2ton/m**3 3 0. 0. -.425 ! center column attachment 4 0. 0. -.2 ! water ballast upper level, also high-res area for elements -.2 to .2m 5 0. 0. 0. ! center still water node 6 0. 0. 0.2 ! high-resolution (wave forces) up to here 7 0. 0. .375 ! tower base 8 .1 0. -.425 ! center pontoon attachment 9 .5625 0. -.425 ! column 1 surface pontoon attachment 10 .6875 0. -.425 ! column 1 center pontoon attachment 11 .6875 0. -0.18625 ! water level 12 .6875 0. .0 ! column 1 swl 13 .6875 0. -.4375 ! sand level 14 .6875 0. -.5 ! column 1 bottom new_boxmember_properties ! Pontoon, see python script for calculations iprop: 1 ! Property set no material: plast_water_pontoon connect_mode: auto btype: beam_lid1_lid2 ea1: 3701700.0 ea2: 3701700.0 ! axial stiffness at keypoints 1 and 2 eiy1: 48440.7 eiy2: 48440.7 ! EIyy bending stiffness, STIFFNESS FOR BENDING ABOUT Z AXIS ! eiz1: 15988.05 eiz2: 15988.05 ! EIzz bending stiffness, STIFFNESS FOR BENDING ABOUT Y AXIS ! lx1: .4625 lx2: .4625 ! equiv section length for use in drag calculations ly1: .31 ly2: .31 ! section width for use in drag calculations lz1: .15 lz2: .15 ! section height for use in drag calculations m1: 46.808475 m2: 46.808475 ! mass per unit length at keypoint 1 and 2 ab1: 0.0465 ab2: 0.0465 ! area for buoyancy calculation pre_strain_long: .0 ! longitudal pre-strain. optional keywords from here pointmass1: 0. pointmass2: 0. cdx: 0. cdy: 1. cdz: 10.4 ! drag coefficients cmx: 1. cmy: 1. cmz: 4.61 ! inertia coefficients

Figure A-5: 3Dﬂoat input ﬁle 1/4 x A Appendix

new_body reftype: inertial name: center_column parent: inertial axes: normal

! Central tower ! new_elements_beam1 nelem 1 material plast_sand_tower connect_mode auto ! bottom to upper level sand x1 0. 0. -.5 dcyl1 .2 tcyl1 0.003 pre_strain_long 0. x2 0. 0. -.4375 dcyl2 .2 tcyl2 0.003 btype beam_lid1 cda 9.2 cma 8.63 new_elements_beam1 nelem 1 material plast_water_tower connect_mode auto ! upper level sand to pontoon attachment x1 0. 0. -.4375 dcyl1 .2 tcyl1 0.003 pre_strain_long 0. x2 0. 0. -.425 dcyl2 .2 tcyl2 0.003 btype beam cda 0. cma 0.01 new_elements_beam1 nelem 1 material plast_water_tower connect_mode auto ! pontoon attachment to upper level water x1 0. 0. -.425 dcyl1 .2 tcyl1 0.003 pre_strain_long 0. x2 0. 0. -.18174 dcyl2 .2 tcyl2 0.003 btype beam cda 0. cma 0.01 new_elements_beam1 nelem 1 material plast connect_mode auto ! upper level water to SWL x1 0. 0. -.18174 dcyl1 .2 tcyl1 0.003 pre_strain_long 0. x2 0. 0. .0 dcyl2 .2 tcyl2 0.003 btype beam cda 0. cma 0.01 new_elements_beam1 nelem 1 material plast connect_mode auto ! SWL to upper level high resolution x1 0. 0. .0 dcyl1 .2 tcyl1 0.003 pre_strain_long 0. x2 0. 0. .2 dcyl2 .2 tcyl2 0.003 btype beam cda 0. cma 0.01 new_elements_beam1 nelem 1 material plast connect_mode auto ! upper level high resolution to tower base x1 0. 0. .2 dcyl1 .2 tcyl1 0.003 pre_strain_long 0. x2 0. 0. .25 dcyl2 .2 tcyl2 0.003 btype beam cda 0. cma 0.01

! Tower ! new_elements_beam1 nelem 1 material plast_tower connect_mode auto ! SWL to tower base x1 0. 0. .25 dcyl1 .2 tcyl1 0.003 pre_strain_long 0. x2 0. 0. 2.05 dcyl2 .2 tcyl2 0.003 btype beam cda 0. cma 1.

! Rotor nacelle assembly ! new_elements_beam1 nelem 1 material plast_rna connect_mode auto ! SWL to tower base x1 -.125 0. 2.05 dcyl1 .05 tcyl1 0.025 pre_strain_long 0. x2 0. 0. 2.05 dcyl2 .05 tcyl2 0.025 btype beam new_elements_beam1 nelem 1 material plast_rna connect_mode auto ! SWL to tower base x1 0. 0. 2.05 dcyl1 .05 tcyl1 0.025 pre_strain_long 0. x2 .125 0. 2.05 dcyl2 .05 tcyl2 0.025 btype beam new_body name: pontoon_1 reftype: refnode xfind: 0. 0. 0. rotorder: 123 rotation: 0. 0. 0. axes: normal parent: center_column new_elements_beam1 nelem 1 material massless connect_mode auto ! attachment inside cylinder x1 0. 0. -.425 dcyl1 .2 tcyl1 0.03 pre_strain_long 0. x2 .1 0. -.425 dcyl2 .2 tcyl2 0.03 btype beamwb new_elements_beam1 nelem 1 material massless connect_mode auto ! attachment inside cylinder x1 .5625 0. -.425 dcyl1 .2 tcyl1 0.03 pre_strain_long 0. x2 .6875 0. -.425 dcyl2 .2 tcyl2 0.03 btype beamwb

! bucket_1 ! new_elements_beam1 nelem 1 material plast_sand_bucket connect_mode auto ! bottom to upper level sand x1 .6875 0. -.5 dcyl1 .25 tcyl1 0.003 pre_strain_long 0. x2 .6875 0. -.4375 dcyl2 .25 tcyl2 0.003 btype beam_lid1 cda 10.96 cma 10.18 new_elements_beam1 nelem 1 material plast_water_bucket connect_mode auto ! upper level sand to pontoon x1 .6875 0. -.4375 dcyl1 .25 tcyl1 0.003 pre_strain_long 0. x2 .6875 0. -.425 dcyl2 .25 tcyl2 0.003 btype beam cda 0. cma 0.01 new_elements_beam1 nelem 1 material plast_water_bucket connect_mode auto ! pontoon to upper level water x1 .6875 0. -.425 dcyl1 .25 tcyl1 0.003 pre_strain_long 0. x2 .6875 0. -0.17524 dcyl2 .25 tcyl2 0.003 btype beam cda 0. cma 0.01 new_elements_beam1 nelem 1 material plast connect_mode auto ! upper level water to SWL x1 .6875 0. -0.17524 dcyl1 .25 tcyl1 0.003 pre_strain_long 0. x2 .6875 0. 0. dcyl2 .25 tcyl2 0.003 btype beam cda 0. cma 0.01 new_elements_beam1 nelem 1 material plast connect_mode auto ! SWL to top of bucket. x1 .6875 0. 0. dcyl1 .25 tcyl1 0.003 pre_strain_long 0. ! lid2 will take water on deck into account x2 .6875 0. .25 dcyl2 .25 tcyl2 0.003 btype beam_lid2 cda 0. cma 0.01 new_boxmembers nelem_default: 2 ! member_no keypoint_1 keypoint_2 Property_Set optional nelem 1 8 9 1 nelem: 1 xi Figure A-6: 3Dﬂoat input ﬁle 2/4 A Appendix

new_body name: pontoon_2 reftype: refnode xfind: 0. 0. 0. rotorder: 123 rotation: 0. 0. 120. axes: normal parent: center_column copy_elements from_body: pontoon_1 to_body: pontoon_2 new_body name: pontoon_3 reftype: refnode xfind: 0. 0. 0. rotorder: 123 rotation: 0. 0. -120. axes: normal parent: center_column copy_elements from_body: pontoon_1 to_body: pontoon_3

! Just for station keeping, check of level add_node_stiffness find 0. 0. 0. f_lin 0. 0. -46.532754 0. 0. 0. 1 1 1. 2 2 1. 6 6 1.

!for linear damping add_node_damping find 0. 0. 0. 3 3 120 ! i.e. 100 eq. 71% of last periods Amplitude, 445 for pure linear damping, 844 for critical damping 5 5 29 ! i.e. 8, 20 for pure linear damping

! Initial conditions ! Heave drop apply_nodal_bc find: 0. 0. 0. apply_displ idofs: 1 idofe: 6 displ: 0. 0. -.1429 0. 0. 0. tstart: 0. tramp: 10. tend: 20.

! or Pitch drop !apply_nodal_bc find: 0. 0. 0. apply_displ idofs: 1 idofe: 6 displ: 0. 0. 0. 0. -0.14336 0. tstart: 0. tramp: 10. tend: 20.

! Sensorer inn her ! sensor 18 to 23 platform displacements swl ! monitor_element_b1 type orientation file turb1_swl.txt nmonitor 5 find 0. 0. 0. node 2 body_name inertial label PtfmSurge PtfmSway PtfmHeave Ptfm_xx Ptfm_xy Ptfm_xz Ptfm_yx

! plotting with Python script plot_geo.py monitor_element_b1 type: plot nmonitor: 100 file: turb1.plot

! Tecplot monitor_element_b1 type: tecplot nmonitor: 1000 file: turb1.dat

! adjustment of sensor position above monitor_element_b1 find 0. 0. -20 nmonitor 5 node 1 type gather file sensor_outputs.txt label oc4_gather toffset 0. sensors: nfact: 1 source: PtfmSurge scale: 1. unit: [m] ! rel. inertia nfact: 1 source: PtfmSway scale: 1. unit: [m] ! nfact: 1 source: PtfmHeave scale: 1. unit: [m] ! nfact: 1 source: Ptfm_xy scale: -57.295779513082323 unit: [deg] header: PtfmRoll nfact: 1 source: Ptfm_xx scale: 57.295779513082323 unit: [deg] header: PtfmPitch nfact: 1 source: Ptfm_yx scale: -57.295779513082323 unit: [deg] header: PtfmYaw print_nodes nodes.txt -1 -1 newmark dt: 0.01 nassemble: 1 nnewton: 10 nsubmin: 2 rwilson: 0.9 relax: 0.0 resid_newton: 1e-06 nmonitor: 100

!material damping damping type ratio2rayl ratio_1 .01 ratio_2 .01 omega_1 0.1 omega_2 1.

! Simulation length in ms ! 6000 for heave, 15000 for pitch step nstep 6000 method gen_alpha

Figure A-7: 3Dﬂoat input ﬁle 3/4

xii A Appendix

!printing and plotting to outputfile !print_nodes nodes.txt -1 -1

!print_b1_elements elements.txt -1 -1 3 2 3 2 print_solution solution.txt

END !______

Figure A-8: 3Dﬂoat input ﬁle 4/4

Manual calculation of added mass for z-direction according to areas shown in fig. 4.6 Bucket area considered as a rectangular plate Pontoon and tower areas considered as round plates Calculation based on rule of thumb presented in 3.5.1

Bucket: given area=0.1864m, length=0.4625m ->width=0.403m ->cylinder: 59.00kg

Pontoon: given area=0.0917m ->radius=0.171m ->sphere: 20.89kg

Tower: given area=0.0173m ->radius=0.074m ->sphere: 1.71kg total added mass: 81.6kg times 3: 244.8kg

This approach neglects the influence of the dry areas of buckets and tower

Manual calculation of added mass for pontoon y-direction Mass of displaced water from volume of pontoon (0.15m*0.31m*0.4625m) Mass of added mass from circular cylinder with radius and length mass of displaced water=21.506kg length=0.4625m radius=0.075m added mass=8.173kg

The real added mass is higher because of the influence of the water entrapment plate

Figure A-9: Manual calculation of added mass

xiii