DESIGN OF HYDRODYNAMIC TEST FACILITY AND SCALING PROCEDURE FOR OCEAN CURRENT RENEWABLE ENERGY DEVICES. William Valentine

DESIGN OF HYDRODYNAMIC TEST FACILITY AND SCALING PROCEDURE FOR OCEAN CURRENT RENEWABLE ENERGY DEVICES by William Valentine A Thesis Submitted to the Faculty of The College of Engineering and Computer Science in Partial Fulfillment of the Requirements for the Degree of Master of Science Florida Atlantic University Boca Raton, Florida August 2012

ABSTRACT Author: William Matthew Valentine Title: Institution: Design of Hydrodynamic Test Facility and Scaling Procedure for Ocean Current Renewable Energy Devices Florida Atlantic University Thesis Advisor: Dr. Karl D. von Ellenrieder Degree: Master of Science Year: 2012 Simulations have been carried out to validate a hydrokinetic energy system nondimensional scaling procedure. The requirements for a testing facility intended to test such devices will be determined from the results of the simulations. There are 6 simulations containing 3 prototype systems and 2 possible model facility depths to give a range of results. The first 4 tests are conducted using a varying current profile, while the last 2 tests use a constant current profile of 1.6 m/s. The 3 prototype systems include a: 6 m spherical buoy, a 12 m spherical buoy and a turbine component system. The mooring line used for the simulations is a 6x19 Wire Rope Wire Core of diameter 100 mm and length 1000 m. The simulations are implemented using Orcaflex to obtain the dynamic behavior of the prototype and scaled system. iii

DEDICATION I d like to dedicate this thesis to my loving family, friends and God for supporting me through my master s degree. Their faith and guidance helped me through all the obstacles in my path. I d like to thank my family and girlfriend for being patient and believing in me at all times.

DESIGN OF HYDRODYNAMIC TEST FACILITY AND SCALING PROCEDURE FOR OCEAN CURRENT RENEWABLE ENERGY DEVICES List of Tables...viii List of Figures...xi Nomenclature...xvi 1. Introduction......1 1.1. Problem Statement..2 1.2. Background...3 1.2.1. Hydrokinetic Systems and the Gulf Stream...3 1.2.2. Relevant testing facilities...5 1.2.2.1. Towing Tanks...6 1.2.2.1.1. Linear...7 1.2.2.1.2. Rotating Arm...7 1.2.2.1.3. Planar Motion Mechanisms...8 1.2.2.2. Water Tunnels...9 1.2.2.3. Cavitation Tunnels 11 1.2.2.4. Wave Making Basins 12 1.2.3. Flow Generation Methods.13 1.2.4. Wave Generation Methods 15 iv

1.2.5. Mooring Systems...16 1.2.5.1. Dynamics of Moored Systems...17 1.2.5.2. Scaling Mooring Lines...21 1.3. Proposed Contribution..23 2. Approach..24 2.1. Evaluation of Company Testing Needs 24 2.1.1. Testing Parameters of Existing Facilities..24 2.1.2. Information about Existing Hydrokinetic Energy Systems...26 2.1.3. Results of Evaluation of Testing Needs.27 2.2. Proposed Scaling Procedure.27 2.3. Determining Facility Size.30 2.4. Requirements and Constraints..31 2.4.1. Mission Requirements...31 2.4.2. Performance Requirements 32 2.4.3. Constraints.32 3. Evaluation of the Scaling Procedure...33 3.1. Environmental Parameters 34 3.2. Moored Prototype Systems...34 3.2.1. Spherical Buoy System 1...35 3.2.2. Spherical Buoy System 2...35 3.2.3. Turbine System..36 3.3. Prototype Mooring System...38 3.4. Testing Facility of Depth D m = 10 m....39 v

3.4.1. Scaled Quantities for Spherical Buoy System 1...39 3.4.2. Scaled Quantities for Spherical Buoy System 2...41 3.4.3. Scaled Quantities for Turbine System...42 3.4.4. Dynamic Behavior Results 43 3.5. Testing Facility of Depth D m = 40 m 51 3.5.1. Scaled Quantities for Spherical Buoy System 1...52 3.5.2. Scaled Quantities for Spherical Buoy System 2...53 3.5.3. Scaled Quantities for Turbine System...54 3.5.4. Dynamic Behavior Results 55 4. Conclusions......64 4.1. Recommendations for Future Work...65 A.0 Appendices.66 A.1 Prototype Environment 66 A.2 Prototype Mooring Line...66 A.3 Prototype System 1..67 A.4 Prototype System 2..67 A.5 Prototype System 3..68 A.6 Prototype System 1 Simulation 71 A.7 Prototype System 2 Simulation 72 A.8 Prototype System 3 Simulation 73 A.9 Test Description...75 A.10 Test 1, 3 and 5 Mooring Line...75 A.11 Test 2, 4 and 6 Mooring Line...76 vi

A.12 Model Environment.76 A.13 Model System 1...77 A.14 Model System 2...78 A.15 Model System 3...79 A.16 Model System 1 Simulation.83 A.17 Model System 2 Simulation.97 A.18 Model System 3 Simulation...111 A.19 Model System 1 Non-Dimensional Scaling Parameters 126 A.20 Model System 2 Non-Dimensional Scaling Parameters 127 A.21 Model System 3 Non-Dimensional Scaling Parameters 128 Bibliography...129 vii

LIST OF TABLES Table 1: Blade Geometry...37 Table 2: Prototype Environmental Parameters..66 Table 3: Prototype Mooring Line Parameters 66 Table 4: Prototype System 1 Parameters...67 Table 5: Prototype System 2 Parameters...67 Table 6: Prototype Lumped Turbine Parameters...68 Table 7: Prototype Turbine Components...68 Table 8: Prototype Turbine Blades 70 Table 9: Prototype System 1 Positions..71 Table 10: Prototype System 1 Static Results.71 Table 11: Prototype System 2 Positions 72 Table 12: Prototype System 2 Static Results.73 Table 13: Prototype System 3 Positions 73 Table 14: Prototype System 3 Static Results.74 Table 15: Test Specifications.75 Table 16: Mooring Line Parameters for sc = 40 75 Table 17: Mooring Line Parameters for sc = 10 76 Table 18: Test 1, 3 and 5 Model Environmental Parameters.76 Table 19: Test 2, 4 and 6 Model Environmental Parameters.77 viii

Table 20: Test 1 System 1 Parameters...77 Table 21: Test 2 System 1 Parameters...78 Table 22: Test 3 System 2 Parameters...78 Table 23: Test 4 System 2 Parameters...79 Table 24: Test 5 Model Lumped Turbine Parameters...79 Table 25: Test 5 Model Turbine Components...80 Table 26: Test 6 Model Lumped Turbine Parameters...81 Table 27: Test 6 Model Turbine Components...82 Table 28: Test 1 Model System 1 Positions...83 Table 29: Test 1 Model System 1 Static Results...84 Table 30: Test 2 Model System 1 Positions...85 Table 31: Test 2 Model System 1 Static Results...86 Table 32: Test 3 Model System 2 Positions...97 Table 33: Test 3 Model System 2 Static Results...98 Table 34: Test 4 Model System 2 Positions...99 Table 35: Test 4 Model System 2 Static Results.100 Table 36: Test 5 Model System 3 Positions.111 Table 37: Test 5 Model System 3 Static Results.112 Table 38: Test 6 Model System 3 Positions 113 Table 39: Test 6 Model System 3 Static Results.114 Table 40: Test 1 Non-Dimensional Parameters Comparison..126 Table 41: Test 2 Non-Dimensional Parameters Comparison..126 Table 42: Test 3 Non-Dimensional Parameters Comparison..127 ix

Table 43: Test 4 Non-Dimensional Parameters Comparison..127 Table 44: Test 5 Non-Dimensional Parameters Comparison..128 Table 45: Test 6 Non-Dimensional Parameters Comparison..128 x

LIST OF FIGURES Figure 1: Linear Towing Tank Test Facility....7 Figure 2: Rotating Arm Towing Tank Test Facility....8 Figure 3: PMM Towing Tank Test Facility.....9 Figure 4: Water Tunnel Test Facility.....10 Figure 5: Cavitation Tunnel Test Facility..11 Figure 6: Wave Making Basin Test Facility..12 Figure 7: Varying Current Profile with Depth...33 Figure 8: Turbine Performance Curves...37 Figure 9: Turbine Component Top View...37 Figure 10: Dynamic End A Tension (6m, sc = 40) 44 Figure 11: Dynamic End A Tension (12m, sc = 40)..44 Figure 12: Dynamic End A Tension (Turbine, sc = 40) 45 Figure 13: Dynamic End B Tension (6m, sc = 40) 45 Figure 14: Dynamic End B Tension (12m, sc = 40)..46 Figure 15: Dynamic End B Tension (Turbine, sc = 40) 46 Figure 16: Fluid Incidence Angle at End A (6m, sc = 40).47 Figure 17: Fluid Incidence Angle at End A (12m, sc = 40)...47 Figure 18: Fluid Incidence Angle at End A (Turbine, sc = 40).48 Figure 19: Dynamic X Position of End A (6m, sc = 40) 48 xi

Figure 20: Dynamic X Position of End A (12m, sc = 40)..49 Figure 21: Dynamic X Position of End A (Turbine, sc = 40).49 Figure 22: Dynamic Z Position of End A (6m, sc = 40).50 Figure 23: Dynamic Z Position of End A (12m, sc = 40)...50 Figure 24: Dynamic Z Position of End A (Turbine, sc = 40).51 Figure 25: Dynamic End A Tension (6m, sc = 10) 56 Figure 26: Dynamic End A Tension (12m, sc = 10)..57 Figure 27: Dynamic End A Tension (Turbine, sc = 10) 57 Figure 28: Dynamic End B Tension (6m, sc = 10) 58 Figure 29: Dynamic End B Tension (12m, sc = 10)..58 Figure 30: Dynamic End B Tension (Turbine, sc = 10) 59 Figure 31: Fluid Incidence Angle at End A (6m, sc = 10).59 Figure 32: Fluid Incidence Angle at End A (12m, sc = 10)...60 Figure 33: Fluid Incidence Angle at End A (Turbine, sc = 10).60 Figure 34: Dynamic X Position of End A (6m, sc = 10) 61 Figure 35: Dynamic X Position of End A (12m, sc = 10)..61 Figure 36: Dynamic X Position of End A (Turbine, sc = 10).62 Figure 37: Dynamic Z Position of End A (6m, sc = 10).62 Figure 38: Dynamic Z Position of End A (12m, sc = 10)...63 Figure 39: Dynamic Z Position of End A (Turbine, sc = 10).63 Figure 40: Test1 Current Profile 76 Figure 41: Test 3 Current Profile...76 Figure 42: Test 2 Current Profile...77 xii

Figure 43: Test 4 Current Profile...77 Figure 44: Test 1 End A Force...86 Figure 45: Test 1 End A Tension Separate 87 Figure 46: Test 1 End B Force...87 Figure 47: Test 1 End A Force Proof.88 Figure 48: Test 1 End A Tension Proof.88 Figure 49: Test 1 End A Tension Separate Proof..89 Figure 50: Test 1 End B Force Proof.89 Figure 51: Test 1 End B Tension Proof.90 Figure 52: Test 1 End A Fluid Incidence Angle Proof..90 Figure 53: Test 1 End A X Direction Motion Proof...91 Figure 54: Test 1 End A Z Direction Motion Proof 91 Figure 55: Test 2 End A Force...92 Figure 56: Test 2 End A Tension Separate 92 Figure 57: Test 2 End B Force...93 Figure 58: Test 2 End A Force Proof.93 Figure 59: Test 2 End A Tension Proof.94 Figure 60: Test 2 End A Tension Separate Proof..94 Figure 61: Test 2 End B Force Proof.95 Figure 62: Test 2 End B Tension Proof.95 Figure 63: Test 2 End A Fluid Incidence Angle Proof..96 Figure 64: Test 2 End A X Direction Motion Proof...96 Figure 65: Test 2 End A Z Direction Motion Proof 97 xiii

Figure 66: Test 3 End A Force.100 Figure 67: Test 3 End A Tension Separate..101 Figure 68: Test 3 End B Force.101 Figure 69: Test 3 End A Force Proof...102 Figure 70: Test 3 End A Tension Proof...102 Figure 71: Test 3 End A Tension Separate Proof 103 Figure 72: Test 3 End B Force Proof...103 Figure 73: Test 3 End B Tension Proof...104 Figure 74: Test 3 End A Fluid Incidence Angle Proof 104 Figure 75: Test 3 End A X Direction Motion Proof.105 Figure 76: Test 3 End A Z Direction Motion Proof.105 Figure 77: Test 4 End A Force.106 Figure 78: Test 4 End A Tension Separate..106 Figure 79: Test 4 End B Force.107 Figure 80: Test 4 End A Force Proof...107 Figure 81: Test 4 End A Tension Proof...108 Figure 82: Test 4 End A Tension Separate Proof 108 Figure 83: Test 4 End B Force Proof...109 Figure 84: Test 4 End B Tension Proof...109 Figure 85: Test 4 End A Fluid Incidence Angle Proof 110 Figure 86: Test 4 End A X Direction Motion Proof.110 Figure 87: Test 4 End A Z Direction Motion Proof.111 Figure 88: Test 5 End A Force.114 xiv

Figure 89: Test 5 End A Tension Separate..115 Figure 90: Test 5 End B Force.115 Figure 91: Test 5 End A Force Proof...116 Figure 92: Test 5 End A Tension Proof...116 Figure 93: Test 5 End A Tension Separate Proof 117 Figure 94: Test 5 End B Force Proof...117 Figure 95: Test 5 End B Tension Proof...118 Figure 96: Test 5 End A Fluid Incidence Angle Proof 118 Figure 97: Test 5 End A X Direction Motion Proof.119 Figure 98: Test 5 End A Z Direction Motion Proof..119 Figure 99: Test 6 End A Force.120 Figure 100: Test 6 End A Tension Separate 120 Figure 101: Test 6 End B Force...121 Figure 102: Test 6 End A Force Proof.121 Figure 103: Test 6 End A Tension Proof.122 Figure 104: Test 6 End A Tension Separate Proof..122 Figure 105: Test 6 End B Force Proof.123 Figure 106: Test 6 End B Tension Proof.123 Figure 107: Test 6 End A Fluid Incidence Angle Proof..124 Figure 108: Test 6 End A X Direction Motion Proof...124 Figure 109: Test 6 End A Z Direction Motion Proof 125 xv

NOMENCLATURE A Cross-sectional cable area a Imposed motion amplitude at the top of frequency w A b Buoy projected surface A c Cross-sectional area of bottom lying part of the cable B o Net buoyancy of buoy in water BCM Buoyancy Compensation Module C d Normal drag coefficient C db Buoy drag coefficient C f Frictional drag coefficient D Water depth d Stretched cable diameter D B Buoy diameter DDHT Dual Ducted Hydrokinetic Turbine D x Buoy drag force in x-direction D Static buoy drag force in x-direction ( x 0 ) xs b D z Buoy drag force in z-direction E Cable Young s Modulus e Local Strain xvi

E c Young s Modulus of bottom lying part of the cable FFF (, ) Distributed external force vector per unit stretched length t n F n Normal drag force F n0 Normal tangential drag force (v = 0) F n1 Dynamic normal drag force F t Tangential drag force F t 0 Static tangential drag force (u = 0) F t1 Dynamic tangential drag force GHT Gorlov Helical Turbine H Horizontal static force applied at the ends k eq Equivalent line stiffness k s Inserted spring constant L Suspended cable length l c Length of bottom lying part of the cable M Mass of buoy m Mass per unit un-stretched length Ma Added mass n local unit normal vector p Tangential displacement p 1 Lower end displacement p 2 Upper end displacement xvii

PMM Planar Motion Mechanism q Normal displacement RTT Rotech Tidal Turbine S Un-stretched Lagrangian co-ordinate measured from the lower end of the cable up to the material point of the cable SMD Soil Machine Dynamics T Tension vector along local tangential vector t Local unit tangential vector T 0 Static tension in cable T 1 Dynamic tension in cable t 0 Static tangential direction of the cable T a Static cable tension T i Total tension in cable segment i T Total tension in cable segment i+1 i 1 U Water/current velocity Vuv (, ) Material point velocity vector w ( ) ga - Immersed cable weight per unit length c w c ( xb, yb, z b) x, y, z co-ordinates of the buoy ( x, y, z ) Buoy velocity b b b ( x, y, z ) Buoy acceleration b b b Kinematic viscosity of water xviii

c Cable density w Water density B or b Buoy density Imposed motion frequency n Natural frequency two-dimensional cable added mass 0 1 Angle between horizontal and local tangential direction at cable material point 0 Static components of, means angle between the horizontal and the static tangential direction of the cable 1 Dynamic component of, means the dynamic deviation from the condition of static equilibrium i Angle in segment i Angle in segment i+1 i 1 0, i 0, i 1 angle relative to horizontal xix

1. INTRODUCTION Due to the energy crisis, there is an increasing demand on natural energy production. This demand includes natural resources such as wind, waves, thermal, biological, tidal and current. A large amount of interest is in ocean currents. The largest ocean currents that provide the best potential for energy production in the world are the Gulf Stream, Kuroshio current, and Agullas current. An advantage that South Florida has is that the Gulf Stream is close to its eastern shoreline. The Gulf Stream starts at the southern tip of Florida and flows northward along the coast of Florida. It then follows a north-eastern course along the rest of the United States and Newfoundland. This is a warm Atlantic Ocean current that is caused by western-intensification. After the Gulf Stream crosses the Atlantic Ocean, it splits into two parts. One part flows over Northern Europe, while the other part flows southward off the western coast of Africa. The Gulf Stream is an ocean current highly driven by the trade winds. The flow rate in the Gulf Stream varies depending upon location between 30 Sv 150 Sv (30 million 150 million cubic meters per second) [1]. An area of interest for energy production in this current is the portion of the Gulf Stream that passes between the coast of Florida and the Bahamas. This area is estimated to have an average kinetic energy flux of somewhere between 25 GW [2] and 20.6 GW [3]. This current starts at the surface and decreases in speed to a depth of 300 m as can be seen in figure 16 from [3]. The seafloor depth in this area can vary from 300 m to 800 m. 1

Systems that harvest this natural energy resource are called hydrokinetic energy systems. Due to the increase in demand for natural energy production, more companies are beginning to research and develop systems to harness this energy resource. Once these systems have a planned design, companies will want to test their systems at scale to make sure that their design works and is efficient before spending the money to build a full-sized system. These scaled systems will need to be tested in a facility capable of simulating the environmental conditions, the system may encounter during deployment, and to evaluate the performance of the system under those conditions. The testing parameters and conditions with which the tests are conducted for the model are crucial for an accurate system performance test. If the testing conditions do not accurately simulate the environmental conditions within which the full scale system will encounter, then the results obtained from the test can t be used as validation for the performance of the system under different conditions. Testing is of great importance when designing hydrokinetic energy systems. The most important part of a test is knowing what to test and simulate, otherwise time and money may be wasted performing meaningless tests. Thus, most testing facilities are designed to simulate environmental conditions that are unique/dominant in a specific deployment location. 1.1. Problem Statement Design a testing facility to test hydrokinetic energy devices. The facility should be designed with the intent of testing the hydrokinetic energy devices that will someday be implemented in the Gulf Stream. The design of this testing facility will have four parts: 2

1. Evaluate the testing needs of companies that produce hydrokinetic energy devices. 2. Provide a scaling procedure for hydrokinetic energy systems. 3. Evaluate this scaling procedure. 4. Provide a concept design of a testing facility that best meets these needs. 1.2. Background 1.2.1. Hydrokinetic Systems and the Gulf Stream There are many hydrokinetic energy systems in existence today, although mainly in the early deployment/evaluation stages. Some widely-recognized companies and their hydrokinetic energy systems are: Hydro Green Energy and their DDHT (array) [4]; Hydrovolts and their class II and class III turbines [5]; Sea Generation Ltd and their SeaGen system [6]; OpenHydro Ltd and their Open Center Turbine [7]; IHC Engineering Business and their Stingray [8]; GCK Technology Inc. and their GHT [9]; Lunar Energy Ltd and their RTT [10]; SMD Ltd and their TidEl [11]; Verdant Power and their Rite [12]; and Bourne Energy and their OCEANSTAR [13]. Not all of these systems are specifically designed for deep water applications; however the above systems provide only a minimum depth of operation and state their system can work in any depth greater than the minimum depth. None of these systems have been deployed and remained operational in deep water applications for more than a day. This is due to the challenges in which companies face when designing these systems. 3

Systems that will be placed in the Gulf Stream encounter problems such as deployment, mooring, maintenance and retrieval. These issues may be encountered due to the location of the Gulf Stream in the vertical water column above the seafloor and the velocity profile of the Gulf Stream. The location of the Gulf Stream offshore causes problems also for the deployment and retrieval process. Deploying a large system can require a complex system of cables and winches. Retrieving a large system that is moored to depths near 400m 800m requires detailed thought and planning. The systems that harvest the Gulf Streams energy will be designed to remain at sea for long periods of time before maintenance is required. In order to remain at sea, the system must be placed in the Gulf Stream and remain there on its own. This causes anchoring/mooring problems due to the velocity profile of the Gulf Stream with increasing depth. Systems will want to reside in the portion of the stream within which the optimum amount of energy can be produced [3]. In order to accurately decide on the depth of operation for the system, investigation of the Gulf Streams velocity profile is required. Velocity Measurements of the Gulf Stream (location Lat: 26 04.3 N, Long: 79 50.5 W) were taken over a 13 month time span, which started at February 2009 and ended at March 2010 [3]. These measurements show that the velocity of this current isn t always constant and can vary between 0.4 m/s to 2.5 m/s over time for a depth below the surface of 20 m. These data provide an average flow speed profile which can be seen to linearly decrease with increasing depth. From this datum, the optimum operational depth range for a system in this current would be 0 100 m [14]. 4

Mooring problems arise when designing a system to reside in this depth range. The system can t be directly anchored to the sea floor since the seafloor depths range from 300 m to 800 m in the Gulf Stream. Thought about the twisting of the mooring lines needs to be taken into account. A worst case environmental analysis needs to be done. The systems designed to harvest the Gulf Stream need to be extensively tested before placed into operation. These systems can be very costly to build at full-scale, therefore it is recommended that companies perform extensive testing so that they are confident that their design can operate efficiently in the Gulf Stream environment. To perform these tests, a testing facility that is capable of simulating the Gulf Stream environmental conditions is required. 1.2.2. Relevant Testing Facilities The most useful testing facilities are those which are well-suited to the particular requirements of the systems to be modeled/-tested. There are several facilities that have been well established for testing in specific areas of interest. Since the testing area of interest is broad for current/tidal extraction devices, this paper will consider several testing facilities that are most beneficial for fluid flow generation. The facility and tank types of interest in this paper are towing tanks, water tunnels/channels/flumes, cavitation tunnels and wave making basins. Each facility s average size, relative flow speeds, and flow generation ability/capability is described in detail in the subsequent sections. The testing ability/capability of each type of facility will be provided in the approach section of this report. 5

1.2.2.1. Towing Tanks These tanks are usually very long in comparison to their width and depth. A carriage is mounted on either a monorail or a dual-rail that extends the length of the tank. When testing in a towing tank, the test model is mounted and suspended from the carriage. The model can either be submerged or on the surface of the water during the test. The carriage will accelerate to the desired testing velocity in which the test and data collection will begin. The carriage will maintain this velocity for the entire length of the testing section of the tank. The carriage initiates the braking sequence once past the end of the test section, thus ending the test run. The towing carriages and their driving system vary for each tank. A towing carriage typically can reach speeds ranging from 3 15 m/s depending on the system driving the carriage. The driving system for these carriages can either be mechanically or electrically driven. Mechanically driven carriages are the slower of the two types, ranging from speeds of 1.5 5 m/s. Electric driven carriages have an electric drive system with drive wheels coupled to electric motors and horizontal guide wheels. The key issues that are encountered when designing any towing tank are: the connection of the model to the towing carriage, the instrumentation must be mounted on the towing carriage in order to continuously gather data, limited space on the towing carriage due to the previous issue, short test run times, need a very long test section in order to get a decent amount of data, difficulties in obtaining PIV or flow visualizations, the time taken and distance needed for the ramp up and ramp down speeds of the towing carriage. There are 3 main types of towing tanks that are used for more specific purposes, they are: Linear Towing Tanks, Rotating Arm Tanks, and Planar Motion Mechanism Tanks. 6

1.2.2.1.1. Linear Linear towing tanks are the most basic type of towing tank. These tanks roughly range from lengths of 20 m to 320 m, widths of 2 m to 15 m and depths of 2 m to 9 m. Most of these tanks have wave making devices on one side of the tank and a beach on the other side of the tank. The benefit of these wave making devices in a towing tank is to measure your systems overall performance in a wave environment. The draw back of this is that in the actual ocean environment, your system will encounter more than just waves in one direction. The information for this section was obtained from references [15] [22]. Figure 1: Linear Towing Tank Test Facility [53] 1.2.2.1.2. Rotating Arm Rotating Arm towing tanks are usually circular in shape as opposed to other towing tanks. These tanks roughly range from lengths of 30 to 50 m, widths of 20 to 50 m and depths of 5 to 12 m. These tanks have towing carriages that have an extended arm attachment system. The model is mounted to the end of this arm and the carriage begins to rotate the model about the center of the carriage. Wave making devices are used in these tanks to measure a system s performance as it encounters waves at many different 7

angles. These tanks allow full control of the yaw angle, surge speed and sway speed during testing. The information for this section was obtained from references [23], [24] and [56]. Figure 2: Rotating Arm Towing Tank Test Facility [56] 1.2.2.1.3. Planar Motion Mechanisms Planar Motion Mechanism towing tanks are similar to linear towing tanks in size. These tanks perform the same tests as the rotating arm tanks except in a narrower and longer tank such as the linear towing tank. The PMM can measure a models velocity derivatives, rotary derivatives and acceleration derivatives by producing a transverse oscillation at the bow and another transverse oscillation at the stern of a model as the model is towed with a constant velocity. The information for this section was obtained from references [25] [29]. 8

Figure 3: PMM Towing Tank Test Facility [59] 1.2.2.2. Water Tunnels A water tunnel is a recirculating fluid flow testing facility. These facilities are closed so the Reynolds Number is most favorably used for model scaling. These facilities are typically around 50 m long, 20 m wide and 10 m tall. The flow can be generated by a pump system or an impeller/propeller system. These facilities are typically capable of generating flow speeds of around 1 5 m/s. These tanks need several honeycombs, screens and corner vanes in order to have an accurate uniform flow velocity profile in the testing section of the tunnel. The downside to having such accurate flow profiles by using these facilities is that the tested model is not easily accessible to make changes or modifications to once placed in the test section. An example of water tunnel testing facility with a unique conveyer belt addition will be discussed. The flow is initially set into motion by turning on the motors which turn the impeller/propeller. The conveyer belt will also begin to move at the same time as the impeller/propeller. The conveyer belt forms the floor of the test section and moves at the flow speed to minimize the boundary 9

layer. The water flows through a diffuser then into a honeycomb system to straighten the flow and reduce lateral turbulence and velocity. The then enters the first turning vane system, which turns the flow 90 degrees while maintaining flow uniformity and avoiding boundary layer separation. Next, the flow enters the second turning vane system to turn the flow another 90 degrees. The flow then enters another honeycomb system to again straighten the flow before the test section. After flowing through the test section, the flow passes through another two sets of turning vanes and then into the impeller/propeller where the process is repeated. The key issues encountered when designing water tunnels are: accessibility to the test section, slow flow speeds, obtaining a proper flow characterization (by means of honeycombs, screens, turning veins, etc.), how and where the model will be mounted, difficulties in cleaning and maintaining, how will the data be collected and where will the instrumentation be mounted. The information for this section was obtained from references [30] [33]. Figure 4: Water Tunnel Test Facility [57] 10

1.2.2.3. Cavitation Tunnels Cavitation tunnels are pressurized water tunnels. Reynolds Number is most favorably used in these facilities. Cavitation tunnels are usually 15 m long, 10 m wide and 10 m tall. The flow in cavitation tunnels is generated in the same way as the flow generated in water tunnels. The flow generation requires pumps or impeller/propeller systems, honeycombs, screens and corner vanes for accurate fluid velocity profiles in the test section of the tunnel. These tunnels consist of a contracting and expanding piping system with some noise insulation systems and pressure control systems. The pressure control systems allows for pressure changes to be made in the test section of the tunnel. The key issues encountered when designing cavitation tunnels are: accessibility to test section, precision of machined components since this a pressure chamber, how and where will the model be mounted, difficulties in cleaning and maintaining, how will the data be collected and where will the instrumentation be mounted. The information for this section was obtained from references [34] [39]. Figure 5: Cavitation Tunnel Test Facility [58] 11

1.2.2.4. Wave Making Basins Wave Making Basins are large facilities having a wide range of testing capabilities. These basins are around 80 m long, 50 m wide and 10 15 m deep. Wave Making Basins have model towing capabilities, current generation capabilities and multidirectional wave generation capabilities. Models can be towed in the basin by means of an electric winch system at around speeds of 3 4 m/s. Current generation can be achieved by a nozzle manifold system at variable depths and speeds of around.5 m/s (possibly up to 1 m/s). Waves can be generated by different wave generating devices located at any or all sides of the basin. The wave height and period depends on the type of wave maker used, but typical wave heights of 280 mm and periods of 0.6 3.5 seconds. The key issues encountered when designing wave making basins are: slow flow speed generation capabilities, how will the model be pulled or move about within the basin, how will the data be collected and where will the instrumentation be mounted, difficult to obtain flow visualization and PIV measurements. The information for this section was obtained from references [40] [43]. Figure 6: Wave Making Basin Test Facility [42] 12

1.2.3. Flow Generation Methods Since the testing facility will be designed to simulate the Gulf Stream s environmental conditions, the facility should have the ability to produce a current stream that is a scaled representation of the Gulf Stream. A flow generation method is needed in order to produce a current stream. Several methods that have been used to generate flow in testing facilities will be examined: 1. Mechanically towed by a towing carriage, 2. Pumps, 3. Motor powered propeller, and 4. Multi-port jet nozzle/manifold. The characteristics of each method will be discussed below in more detail. The first method isn t technically flow generation, but rather apparent flow generation. In this method the fluid inside the tank remains still and the model being tested is dragged through the water by the towing carriage. For systems that need high speed tests, this is the most feasible method to use. The downside to this method is length of time in which this flow can be maintained. The apparent flow can only be maintained for very short intervals of time. A method that uses pumps is also used for flow generation. There are four main types of pumps: positive displacement pumps, hydraulic-ram impulse pumps, centrifugal pumps and axial flow pumps. Positive displacement pumps are commonly categorized as either rotary type or reciprocating type. Rotary positive displacement pumps create a vacuum from the rotation of the pump which captures and draws in the fluid. These pumps are very efficient but can only produce slow flow rates. Reciprocating positive displacement pumps use oscillating pistons to drive the fluid movement. These are used for systems that need to pump high viscous fluids. Hydraulic-ram impulse pumps take the 13

fluid in at a certain flow rate and pressure, then expels the fluid at a higher flow rate and pressure. These are used for pumping a fluid to a higher elevation. Centrifugal pumps use a rotating impeller to increase the flow rate and pressure of the fluid [44]. It is very easy to precisely control the flow rate at low speeds which is a big reason why these are commonly used in flow generation applications. Axial-flow pumps cause a pressure change between the upstream and downstream sections of the pump [44]. These pumps operate at low pressures and high flow rates. The performance curves for these pumps are less than the performance curves for the centrifugal pumps [44]. The thrust, head, and discharge coefficients for the above pumps are defined as:,, [44]. In the above formulas, the discharge rate is Q, the pressure head is H, the diameter of the impeller is D, and the rotational speed in revolutions per second is n. Another method for flow generation is the use of a motor powered propeller residing in the flow. The motor rotates the shaft thereby rotating the propeller attached to the end of the shaft. This rotation causes the fluid to be sucked through and accelerated by the propeller. This method is commonly used in re-circulating flow applications. The speeds in which could be produced vary widely based on the motor size, propeller size, propeller pitch and propeller RPM s. The non-dimensional advance ratio, thrust and torque coefficients for a propeller are:,, [45]. The last flow generation method is the multi-port jet nozzle/manifold. An array of jet nozzles can be oriented to generate the flow shape and conditions for the desired flow 14

stream. This stream would be easy to predict and control. A manifold system can be used whereby multiple arrays of these nozzles can be placed at different depths. This would allow for the production of any desired flow stream at any depth. 1.2.4. Wave Generation Methods Current flow isn t the only environmental flow condition that is needed to simulate the Gulf Stream environment. Since the velocity profile of the Gulf Stream starts at the surface of the ocean, then waves will be a factor on systems that are surfaced moored or in the deployment process. Waves in testing facilities are generated using wave makers. There are several types of wave makers: piston-type, flap-type and plunger-type [46]. The piston-type and flap-type are 2D wave makers whereas the plunger-type is a 1D wave maker. There also exists a 3D wave maker known as a snake [46]. These wave makers have equations that use their stroke length S, the tank depth h and the wave number k. The equations for the 1D and 2D wave makers are found by equating the volume of water displaced over a whole stroke and the volume of water in a wave crest. The piston-type wave maker uses a piston driven motion. The piston oscillates horizontally keeping its vertical position constant. The shallow water condition formula is [46]. The flap-type wave maker uses a flap driven motion. The flap is hinged at the bottom of the tank and oscillates about this hinged point. There is a reduction in the volume displacement for the stroke motion due to the limits in the 15

motion caused by the hinge. The shallow water condition formula is [46]. The plunger-type wave maker uses a cylinder driven motion. A cylinder oscillates vertically about the mean water line. If the cylinder has a radius R and the stroke length is R, then at full stroke the cylinder will range from fully emerged to half submerged. The shallow water condition formula is [46]. 1.2.5. Mooring Systems Mooring systems are needed for any system that desires to remain in one location in the ocean for a period of time. For systems that will be used to harvest energy from the Gulf Stream over long spans of time, the use of either a surface buoy system or a subsurface buoy system will be employed. Each of the buoy systems require an anchoring point on the sea floor that is connected to the mooring buoy through a cable system, single point or multi-leg. Surface buoys can be designed such that the environmental conditions on the oceans surface have either a great impact or little impact on the behavior of the buoy. These buoys can have a great range of shapes. A single point surface moored buoy system can have slack in the line allowing the buoy to move with the waves and current until the slack is diminished, or it can have a taut line that stretches some due to the motion of the buoy on the surface. A buoy is considered a surface buoy until the center of that buoy is 100m below the sea surface. This is due to the diminished effect that surface excitation has at 100m or more below the sea surface [47]. 16

A system in which the buoy is 100m or more below the sea surface are called subsurface buoys. The largest forces, on average, that these buoys experience are due to ocean currents. These buoys can be moored with slack in the line or with a taut line also. For a more detailed understanding of buoys and their purpose refer to [48]. 1.2.5.1. Dynamics of Moored Systems For a system moored in the ocean that is harvesting energy, it is crucial to know the dynamic behavior of that system. In order to have an accurate system behavior, the non-linear effects on the cable should be considered. The cable is assumed to have a negligible bending stiffness and is modeled as a slender rod. The Lagrangian coordinates are measured from the bottom most end of the cable up to a material point on the cable. In the equations below: m is the mass per unit length, s is the un-stretched Langrangian coordinate, v(u,v) is the velocity vector, T is the tension vector, t is the tangential unit vector, n is the normal unit vector, F(F t,f n ) is the external force per unit stretched length, e is the local strain, is the added mass, w is the submerged weight per unit length and is the angle between the horizontal and the local tangential direction of the material point. The equations of motion of a cable only expressed along the local tangential and normal directions of the moving dynamic reference are [49]: u T (1) m[ v ] wsin( ) Ft (1 e) t t s v (2) ( m ma) ( m) u T wcos( ) Fn(1 e) t t s 17

(3) e u v t s s ; (4) (1 ) v e u t s s ; (5) w ( ) c w gac; (6) T EAe A linear tension-strain relation (eq. (6)) is used. Using the separation principle, the external forces can be obtained [49]: (7) 1 Ft d wcf ( U cos( ) u) U cos( ) u 2 (8) 1 Fn d wcd( Usin( ) v) Usin( ) v 2 where ρ w is the water density, d is the stretched cable diameter, U is the water velocity, c f and c d are the frictional and drag coefficients for the cable. In eq. (7), F t is obtained using the separation principle by which a relation of the tangential force is made to the relative fluid-cable tangential velocity. In eq. (8), F n is obtained in a similar way using the normal force and the relative fluid-cable normal velocity. The out-of-plane motion is, to first order, decoupled from the in-plane motion of this cable configuration [49]. Projecting the non-linear eq. (1) (4) onto the static reference system, requires a relationship between the moving and static reference system using the velocity components and displacements [49]: (9) p q u cos( 1) sin( 1) t t ; (10) p q v sin( 1) cos( 1) t t where p and q represent the tangential and normal displacements and 1 is the dynamic angle. 18

Using equations (9) and (10), the unit vector relation for the 2 coordinate systems, subtracting the equilibrium equations, and assuming small displacements, the simplification equations, and assuming small displacements, the simplification procedure can be established [49]. The simplification procedure yields: (11) (12) p T d m T F t s ds 2 1 0 2 0 1 t1 2 q 0 1 T0 T1 1 ( m ma) T 2 1 T 0 1 Fn 1 1 T 1 t s s s s s (13) p d q 0 T 1 s ds EA (14) d ds q 0 p 1 e0 s (1 ) The external forces are obtained by the following: F ( F F )(1 e) F (1 e ) ; (16) Fn 1 ( Ft 1 Fn1)(1 e) Fn0(1 e0) (15) t1 t n1 1 t0 0 In these equations, Ft 0 and Fn 0 are found by setting u v 0. Neglecting the last 2 terms in eq. (12), assuming excitation frequencies are well below the first elastic frequency of the cable and using the hypothesis of large static tension relative to the total weight of the cable, then eq. (11) becomes trivial thus [49]: (17) q q q d ( ) t s s ds 2 2 2 0 m ma T 2 0 T 2 1 T 2 1 Fd (18) 1 q q F d c U U U 2 t t 2 d w d[( N ) N N] 19

L (19) UN Usin( ) d 0 ; (20) T1 keq[ p2 p1 q ds] ds 0 where (21) 1 1 1. k k ( EA/ L) eq s The above equations are for a cable-only analysis. Considering a buoy within 2 segments (i and i+1) of a line, the following equations of motion are obtained [49]: (22) ( M M ) x T 1cos( 1) T cos( ) D a b i i i i x (23) ( M M ) a zb Ti 1sin( i 1) Tisin( i) Dz B0 1 Dx Ap wcbdvr( U x b) (24) 2 ; (25) 1 Dz Ap wcbdvrz b 2 V ( U x ) z ; (26) 2 2 r b b Now applying a similar simplifications procedure, the following is obtained [49]: (27) ( M M ) p cos( ) ( M M ) q sin( ) T cos( ) T sin( ) a i a i 1, i 0, i 1, i T D cos( ) D sin( ) 1, i 1 xb 0, i 1 z 0, i 1 (28) ( M M ) p sin( ) ( M M ) q cos( ) T sin( ) T cos( ) a i a i 1, i 0, i 1, i T D sin( ) D cos( ) 0, i 1 1, i 1 xb 0, i 1 z 0, i 1 D D D (29) xb x xs 0, i 0, i 1 ; (30) x p cos( ) q sin( ) p cos( ) q sin( ) b i 0, i i 0, i i 1 0, i 1 i 1 0, i 1 (31) z p sin( ) q cos( ) p sin( ) q cos( ) b i 0, i i 0, i i 1 0, i 1 i 1 0, i 1 (32) 20

1.2.5.2. Scaling Mooring Lines The scaling of mooring lines for model testing has given many people trouble over the years. When scaling these lines, many factors need to be taken into account. These factors are discussed in Non-Linear Cable Response and Model Testing in Water [49] and Deep Water Mooring Dynamics [50]. The first paper describes the dynamic response of cables in water. The model testing of cables in water rather-then in air is highly important due to the vastly different response properties of cables in water as opposed to in air. The scaling of cables can be accomplished once the non-dimensional quantities are known. There are nine non-dimensional parameters for the analysis of the cables on [49]: L D ; d L ; wl T a ; Ud on which c f, c d and depend; c ; EA T ; a 2 2 a d ; L ;. ( T /( A)) w a c There are three more parameters in addition to the nine parameters above, when considering the analysis of a cable with a buoy attached [50]: a D ; B ; w B UDB on which c bd and M a depend. Papazolou and Mavrakos have shown that proper sealing can be achieved for the 2 cable only analysis [50]. They provide a parameter,, from which the response of cables primarily depends. 2 EA wl 2 ( )( ) where represents the ratio for the elastic H H stiffness to catenary stiffness of a mooring line. For small the response is dominated 21

by elastic stiffness where stretching occurs in the line [49]. For large the response is dominated by catenary stiffness where no stretching occurs. When modeling mooring lines, the scaling methods to be taken are not clear. To ensure the slenderness of the cable, d L should be matched for the model to the prototype. For proper static modeling, L D and wl T a must be kept the same [49]. For dynamic modeling, EA similitude is essential for the dynamic response, but is not needed for the T a static response. The problem occurs in finding the materials at these very small scales that will match the EA T a [49]. Papazoglou and Mavrakos show that by inserting buoys and springs in series with the cable at its lower end, they could artificially produce the required elastic stiffness (EA/L) needed for proper sealing [50]. Additional natural frequencies are introduced by these springs. With quasi-static stretching for small Ks compared to elastic stiffness the following is obtained for the natural frequency 3k approximation [50]: s n. If n comes out to be within the testing range of ml frequencies, then the test and results are invalid because the system excites a natural frequency. This occurs when the forcing frequency is near the natural frequency. 22

1.3. Proposed Contribution The goal of this project is to provide a system level design of a testing facility that meets the testing needs of companies. Companies will be allowed to pay in order to test their hydrokinetic energy systems in this facility. The contribution that will be made will consist of five parts: 1. Detailed Evaluation of the testing needs of companies 2. Trade Study to down-select from the possible tanks that are able to suit these needs 3. Scaling procedure for hydrokinetic energy systems 4. Evaluation and proof of the proposed scaling procedure 5. Design of a testing facility that best suits those need 23

2. APPROACH A test facility must meet the testing needs of companies, thus research about the needs and testing process of companies is required for a proper analysis. The size of the facility will be evaluated by the comparison of the dynamic tensions from several OrcaFlex simulations. Once the testing needs of industry and reasonable facility size are known, the facility requirements and constraints for the testing facility can be established. 2.1. Evaluation of Company Testing Needs 2.1.1. Testing Parameters of Existing Facilities Research on the testing parameters that existing facilities offer was carried out. Since there are many different types of facilities that can be used to test hydrokinetic energy systems, the testing parameters of each type of testing facility were collected and combined with facilities of the same type. This will provide information on what each type of facility most commonly tests. 1) Towing tanks a) Linear used for force and resistance measurements on systems or ship hulls, propulsion systems, propeller or turbine performance in open water, sea keeping 24

and maneuvering in head and following waves, mooring simulations and wake surveys. Information collectively obtained from references [15] [22]. b) Rotating Arm used for force and resistance measurements on submersible systems (submarines, torpedo s, etc.), accurate measurements of rudder performance, sea keeping and maneuvering simulations, all of these tests can be performed at high speeds. Information collectively obtained from references [23], [24] and [56]. c) Planar Motion Mechanisms used for force and resistance measurements on systems or ship hulls, measuring velocity derivatives, measuring rotary derivatives, measuring acceleration derivatives, propulsion systems, propeller or turbine performance in open water, sea keeping and maneuvering, experiments that require vertical and horizontal motions, rudder performance, mooring simulations and wake surveys. Information collectively obtained from references [25] [29]. 2) Water Tunnels used for cavitation tests of propellers or turbines, force and resistance measurements, measure noise produced by a system, measuring vibration produced by a system, flow visualization tests, deep water mooring and anchoring systems. Information collectively obtained from references [30] [33]. 3) Cavitation Tunnel used for cavitation tests of propellers or turbines, measuring the response and behavior of the system or the flow over a system to fluctuations in pressure, measurements of the noise produced by a cavitation propeller or turbine, performance measurements of hydrofoils and tidal energy turbine systems. Information collectively obtained from references [34] [39]. 25

4) Wave Making Basins used for extensive mooring system tests, sea keeping and maneuvering, deployment and retrieval process simulations, characterization of the dynamic behavior of the system, force and resistance measurements on systems or ship hulls and measurements of waves generated by ships. Information collectively obtained from references [40] [43]. 2.1.2. Information about Existing Hydrokinetic Energy Systems Research was carried out to gather information on as many hydrokinetic energy systems that were made available through the internet. The systems which provided the most available information were chosen and placed into a spread sheet. A total of 13 systems were used (references [4] [13] in addition to 3 other systems), since the most information was publically available for these systems. Out of the 13 total systems: 8 were horizontal axis turbine systems, 4 were vertical axis turbine systems and 1 was an oscillating system. The system rotor diameter for the 12 turbine systems ranged from 1 m 25 m with the average being 10 m. The designed flow speeds for all 13 systems ranged from 1.5 m/s 3.5 m/s with the average being 2.789 m/s. The information provided about the mooring systems is minimal and generalized for 9 of the 13 systems. The deployment wasn t given for 3 out of the 13 systems. The operational depth for most of these systems is greater than 30 m, which means they are deep water hydrokinetic energy systems. 26

2.1.3. Results of Evaluation of Testing Needs Due to the limited information available, some assumptions are made about the testing needs of companies. Researching the testing parameters offered at existing facilities and what information could be obtained about existing hydrokinetic energy system designs, allows some assumptions to be made about the testing needs of companies. Due to the limited facilities that offer detailed mooring tests along with the limited information companies provide online about their systems mooring plan, the performance of the system and the systems mooring were chosen as the most important testing needs of companies. 2.2 Proposed Scaling Procedure The scaling procedure proposed for hydrokinetic energy systems in this design facility will be described in detail below. Scaling systems as a whole is complex due to having to simultaneously scale the energy device and the mooring system. The proposed method uses a force scaling procedure. The most significant factors that need to be taken into account are depth of operation (D p ) for the prototype system and depth of testing facility (D m ). These two factors can be used to determine the required scale factor, which is (33) sc = D p /D m. The next scaling parameter of importance is the drag force (thrust force) on the moored system. The prototypes drag or thrust force will be used to estimate the End B tension, the lower end tension in the mooring line, later on in the steps thus the 27

formulas to find these are given below: (34) D xb = 0.5* w *C db *A b *(U bs ) 2, (35) TSR = d /U, (36) Thrust = w *C t *D 4 b *( d /2*pi*R b ) 2 and (37) Torque = w *C q *D 5 b *( d /2*pi*R b ) 2. factor: Step 1 of the scaling process involves the scaling of all quantities by the scale Lengths, Diameters ~ sc Mass, Force, Elasticity, Tension ~ sc 3 Cable mass per unit length ~ sc 2 Time ~ sc 1/2 Step 2 deals with the more detailed specifics regarding the moored device and the mooring line. For the moored device, the density of the device must scale using the formula (38) mb = pb *( pw / mw ). This must occur to ensure proper buoyancy and weight scaling. The scaled cables actual elasticity is found due to the scaled diameter of the cable. The cable must be of the same type as the prototype to ensure a reasonable density scale, however the scaling of the cable density is not of great importance. Step 3 is needed to find the appropriate spring to be used for testing. The needed scaled elasticity of the prototype line (EA eq ) found in Step 1 will be smaller than the model cables actual elasticity (E m A m ). It is impossible to find sufficient materials that will meet the needed scaled elasticity when the scaled diameter of the cable is used. Since 28

the elasticity is very important in dynamic response, the actual elasticity (E m A m ) needs to be scaled to the required elasticity. This is the basis for the method of artificially reducing the elasticity of a cable by the insertion of a weak spring in series with the lower part of the cable, proposed by Papazoglou and Mavrakos [49]. To solve for the necessary spring needed for the scaled system, this thesis s proposed method assumes that the drag (or thrust) on the moored device is equal to the tension (EndT pb ) at the anchored point (End B) of the system, assuming no waves present and in a current profile. This end tension gets scaled in Step 1 to give the needed model cable end point tension (EndT mb ). The model End B is connected to the spring that is anchored to the floor. Thus, the spring constant is set equal to the scaled End B tension divided by the un-stretched length of the spring. The mooring line is then treated as a spring system that has a spring constant equal to the actual elasticity of the cable (E m A m ) divided by the scaled length of the cable (L m ). The entire line (scaled cable + spring) is treated as a spring having a spring constant (k eq ) equal to the needed elasticity of the cable (EA eq ) divided by the sum of the scaled cable length and the un-stretched spring length (L eq = L m + L 0 ). Using the formula for springs connected in series, the un-stretched length of the spring can be found. The needed spring constant can then be found once the un-stretched spring length is known. The formulas below are used to solve for Step 3: (39) EndT pb = D xb or Thrust, (40) EA eq = E p A p /sc 3, (41) L m = L p /sc, (42) L eq = L m + L 0, (43) k m = E m A m /L m, (44) k s = EndT mb /L 0, (45) k eq = EA eq /L eq, (46) 1/k eq = 1/k s + 1/k m. Step 4 of the scaling procedure deals with finding the required water velocity profile needed for testing. As previously stated, this scaling method relies heavily on force scaling, thus the drag force (or thrust) on the model must equal the drag force (or 29

thrust) on the prototype divided by the cube of the scale factor. This force is found for the case when no waves are present. The basis for this step is to have a relative equivalence of the force coefficients. The user will have an idea of the static z position of their device in the vertical water column or rather where they would like it to be resting. This will allow for the velocity at the center of the moored device to be obtained for that z position. Knowing the velocity vs. depth profile will then allow the user to generate a fluid scale factor (sf). The fluid scale factor is obtained by taking the known velocities divided by the surface water velocity. When the user solves for the needed scaled velocity at the center of the scaled moored device, then the entire velocity profile needed for the model testing can be obtained after using an interpolation method and the fluid scale factor. This velocity is then compared to the Froude number scaling to be sure that the velocities match. The formulas needed for Step 4 are provided below: (47) U mbs = U pbs *[( pw / mw )*(D 2 pb /D 2 mb )*(1/sc 3 )] 1/2, (48) U m = U p *(R mb /R pb )*[ ( pw / mw )*(D 4 pb /D 4 mb )*(1/sc 3 )] 1/2. 2.3. Determining Facility Size and Testing Range Capabilities An OrcaFlex model of a 206W test turbine was created and tested by Allison Cribbs [51]. This test was done for Florida Atlantic University s Center for Ocean Energy Technology (COET), which is now called the Southeast National Marine Renewable Energy Center (SNMREC). The model includes an operational turbine and two surface buoys. A detailed discussion of how to use OrcaFlex as a modeling tool is provided in 30

[51] and [52]. Using the proposed scaling method, several sample systems are scaled so that the dynamic behavior of the line can be compared. The deep water mooring scaling method discussed in [49] and [50], where by the insertion of springs and additional buoys attached along a mooring line for accurate elastic stiffness scaling will be employed as a part of the scaling process. The dynamic tension in the prototype mooring line is compared to the dynamic tension in the model mooring line. The necessary facility size is estimated from the results of testing. In addition to these results, realistic cable sizes are taken into account when determining the size of the testing facility. Then a system level testing facility design can be created with the intent of testing deep water hydrokinetic energy devices that are designed for deployment into the Gulf Stream off the coast of Florida. 2.4. Requirements and Constraints Mandated = (M) Derived = (D) 2.4.1. Mission Requirements 1) Facility shall test hydrokinetic energy systems. (M) 2) Facility shall simulate the Gulf Stream environmental conditions. (M) a) Shall generate current flow stream, wind and waves. (D) i) Shall produce a velocity profile that linearly decreases with increasing depth starting at the surface. (D) 31

2.4.2. Performance Requirements 1) Facility shall meet the testing needs of companies. (M) a) Shall test the performance of a system. (D) i) Shall measure the free stream current velocity. (D) ii) Shall measure the systems six degree-of freedom motion. (D) iii) Shall measure the shaft performance. (D) (1) Shall measure the shaft RPM s. (D) (2) Shall measure the RPM s delivered to the generator. (D) iv) Shall measure the energy produced. (D) (1) Shall measure the power output from the generator. (D) v) Shall measure the thrust of the system. (D) vi) Shall measure the drag on the system. (D) b) Shall test the mooring configuration of the system. (D) i) Shall measure the total system drag. (D) ii) Shall measure the tension in the cables. (D) 2.4.3. Constraints 1) The metric system shall be used for the unit system. (D) 2) The fluid used for testing shall be fresh water. (D) a) Temperature shall be between 10 C - 15 C. (D) 3) The modeling program used shall be OrcaFlex 9.4. (D) 32

3. EVALUATION OF THE SCALING PROCEDURE The scaling procedure is tested to show the accuracy of the scaling procedure. Three prototype systems have been created for the purpose of this evaluation. Each system has 2 different scale factors allowing for a total of 6 tests. The environmental conditions and mooring system used are constant for the prototype systems throughout all 6 tests. For the first 4 tests (the spherical buoy s tests), the water profile varies with depth as shown in Figure 7. For the last 2 tests (the Turbine tests), the water profile is constant with depth. Information or symbols have a subscript p or m refer to prototype or model respectively. 0 Varying Current Profile -50-100 -150 Z (m) -200-250 -300-350 -400 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Current Velocity at 180 deg (m/s) Figure 7: Varying Current Profile with Depth 33

3.1. Environmental Parameters The following paragraph provides the prototype environmental parameters encountered during testing. The current profiles are 2 fold as listed above. These profiles can be seen to have a current velocity of 1.6 m/s at the surface of the water. The wave parameters chosen are of great importance if the moored object is near the surface due to the spring implementation in the scaling. Waves force the system and thus the period of excitation is very important since the testing results are invalid if resonance is reached. For this reason, a single wave train was chosen for simplicity having amplitude of 5 m with a period of 10 sec. The frequency of excitation is then p = 0.10 Hz. The water density and kinematic viscosity is taken to be sea water at 10 C which according to [55] gives pw = 1026.9 kg/m 3 and pw = 1.35x10-6 m 2 /s. The water depth is taken to be D p = 400 m and the gravitational constant used is g = 9.80665 m/s 2. The only model environmental parameters that are held constant are gravity, density and the kinematic viscosity. The water density and kinematic viscosity is taken to be fresh water at 10 C which according to [55] gives mw = 999.7 kg/m 3 and mw = 1.31x10-6 m 2 /s. 3.2. Moored Prototype Systems Three moored prototype systems were used for the six total tests. This allows for proof that the scaling procedure works not only for one type of moored of system. The first two systems are simple spherical buoys, whereas the turbine system is more complex. The first two systems are used to prove that the scaling of the mooring line 34

works and is accurate. The last system is used to show that the proposed scaling procedure is very helpful for hydrokinetic energy system testing at a scale level. 3.2.1. Spherical Buoy System 1 This buoy is a surface buoy that will follow the motion of the passing wave. The density was chosen such that the weight of the cable will not sink the buoy and is taken to be pb = 850 kg/m 3. The diameter of the buoy is d pb = 6 m and the volume of the buoy is V pb = 113.097 m 3. The drag coefficient of the buoy is taken to be C pdb = 0.47. The mass and mass moment of inertia of this buoy is: M p = 96132.735 kg, I xxpb = I yypb = I zzpb = 346077.847 kg*m 2. 3.2.2. Spherical Buoy System 2 This buoy is also a surface buoy that will follow the motion of the passing wave. The density was chosen to be enough such that the weight of the cable will not sink the buoy and is taken to be pb = 850 kg/m 3. The diameter of the buoy is d pb = 12 m and the volume of the buoy is V pb = 904.779 m 3. The drag coefficient of the buoy is taken to be C pdb = 0.47. The mass and mass moment of inertia of this buoy is: m pb = 769061.882 kg, I xxpb = I yypb = I zzpb = 11074491.095 kg*m 2. 35

3.2.3. Turbine System The turbine system consists of several components shown in Figure 9: 1.Aft Body (red), 2.Rotor (green), 3. Blades (purple), 4.Fwd Body (dark blue), 5.Left Rod (yellow), 6.Right Rod (yellow), 7.Left BCM (light blue) and 8.Right BCM (light blue). Since the topic of research of this paper is not to design a turbine system, a simplified model will be generated and then the components lumped together into 1 component. For even more simplification, a constant density of pb = 500 kg/m 3 is chosen for each part of the lumped system. The Aft Body is a cylinder of length 1 m. The outer diameter is 1 m and the inner diameter is 0.8 m. An origin coordinate system is needed so the center of mass of the lumped system can be found. The origin was chosen to be located at the far left side of the Aft Body. The center of mass of the Aft Body is at position (0.5, 0, 0). The Rotor is taken to be a cylinder of length 1 m and begins at the end of the Aft Body (position (1, 0, 0)). The outer and inner diameters are 1.2 m and 0.8 m respectively. The center of mass of the Rotor is taken to be at position (1.5, 0, 0). The turbine blade geometry is listed in table 1. The axis of the blades had to be aligned in order to get the proper center of mass and mass moment of inertias for the lumped system. Each blade element was separately analyzed to find the mass, center of mass, volume, weight and mass moment of inertia for each element after the proper axis alignments were taken into account. For more details about this alignment process refer to CorrectCenterOfMass.m code in the Appendix. The performance curves provided in 36

[2] were used as this turbines performance curves. These curves apply to a different turbine geometry, however the goal of this thesis is not to design a turbine but rather provide a scaling process for hydrokinetic energy systems. The scaling method assumes the user already knows information and details about the energy device planning to be scaled and thus deriving this information is not of great importance. The performance curves used are provided in Figure 8. Table 1: Blade Geometry span 1.202 1.202 1.202 1.202 1.202 1.202 1.202 1.202 1.202 chord 0.609 0.520 0.444 0.368 0.317 0.269 0.241 0.208 0.185 thickness 0.12 0.101 0.083 0.068 0.055 0.045 0.035 0.029 0.023 angle 24 18 17 16 15 13 11 10.50 10 0.8 0.7 0.6 Performance Curves of Turbine Cp Ct Cq 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 TSR Figure 8: Turbine Performance Curves Figure 9: Turbine Component Top View The Fwd Body is a cylinder of length 6 m. The outer and inner diameter is 1 m and 0.8 m respectively. The start of the cylinder begins at the end of the Rotor at position (2, 0, 0). The center of mass is found to be at (5, 0, 0). 37

The Left and Right Rod components are the exact same size and are both solid rods. The outer diameter and length of each is 0.8 m and 12 m respectively. The center of mass of the Left and Right Rod is located at positions (5, 6, 0) and (5, -6, 0) respectively. Each of these rods are connected to a BCM. Left and Right BCM have the same size and are of ellipsoid shape. The length in the x direction, width in the y direction and height in the z direction of each is 6 m, 6 m and 1.3 m respectively. The center of mass of the Left and Right BCM is located at positions (5, 15, 0) and (5, -15, 0) respectively. The center of gravity of the entire system is found by taking the weight of each element times its center of mass position from the origin and adding them all together then divide the result by the summation of all the weights. Finally, the mass moments of inertia are found about the center of gravity such that the product inertias of the system are close to zero. The lumped systems mass, mass moments of inertia and volume are found to be m pb = 32477.807 kg, I xxpb = 5.858214x10 6 kg*m 2, I yypb = 6.768467x10 4 kg*m 2, I zzpb = 5.912505x10 6 kg*m 2 and V pb = 64.955615 m 3. 3.3. Prototype Mooring System The prototype mooring system used in all 6 tests is detailed below. The mooring system was a simple single point system that connected the moored system (EndA) to the sea floor (EndB). The mooring line was 6x19 Wire Rope with Wire Core having a nominal diameter of d p = 100 mm = 0.100 m and a length of L p = 1000 m. The 38

submerged mass of the mooring line was m p = 34.735 kg/m with an elasticity of E p A p = 4.04x10 8 N. 3.4. Testing Facility Depth of D m = 10 m The above prototype systems are scaled using a model water depth of 10 m. This provides a scale factor sc = D p /D m = 40. The proposed scaling procedure is then carried out using the scale factor of sc = 40. The scaled environmental parameters then become a m = 0.125 m and = 3.977 Hz. The scaled mooring line parameters become d m = m 0.0025 m, L m = 25 m. The needed mooring elasticity found by following Step 1 of the proposed scaling procedure to be EA eq = 6.3125x10 3 N. The actual elasticity of a 6x19 Wire Rope with Wire Core having a nominal diameter of d m = 2.5 mm = 0.0025 m is found to be E m A m = 2.525x10 5 N. These scaled results are used with each of the systems below. 3.4.1. Scaled Quantities for Spherical Buoy System 1 The buoys density must be scaled so that the buoyancy to weight ratio matches that of the prototype. The density of the buoy must scale by using (38) to give = 827.486 kg/m 3. The diameter of the buoy becomes d mb = 0.15 m with a volume of V mb = 0.00177 m 3. The mass for the buoy becomes m mb = 1.462 kg. mb The assumption that states that the End B tension (EndT pa ) is equal to the drag force on the buoy is proven by placing the prototype parameters into OrcaFlex and then use the static state End B tension of the prototype system. Doing this provides 2 different 39

spring constants and un-stretched lengths for each test. Using the drag force assumption, the End B tension then becomes EndT pb_es = 4.4282x10 4 N. Using the OrcaFlex tension obtained, the End B tension is then EndT pb = 3.1787x10 4 N. After applying the scaling rules of Step 1 of the scaling procedure and using the formulas from Step 3 to obtain the required un-stretched spring length and spring constant and natural frequency, the following is obtained: L 0_es = 2.672 mm, L 0 = 1.915 mm, k s_es = 2.590x10 2 N/m, k s = 2.590x10 2 N/m, = 19.673 Hz and n = 19.673 Hz. The forcing frequency divided by each natural frequency was check for resonance m / n_es = m / n = 0.202 thus no resonance occurs. n_ es The buoys center position within the water column as the dynamic stage begins (also called static position) is z pbs = -2.836 m. Knowing the position of the buoy within the water column with no waves present allows for the drag force on the buoy at that state to be solved. Interpolation of the varying current profile allows for the solution of the current velocity at the center of the buoy, which is U pbs = 1.594 m/s. Using the drag force scaling described in the proposed scaling procedure, the required velocity at the center of the model buoy can be found to be U mbs = 0.256 m/s. Knowing the velocity at any depth for the model profile will allow for the solution of the entire model current profile. 40

3.4.2. Scaled Quantities for Spherical Buoy System 2 The scaled density for the buoy is found by using (38) to be mb = 827.486 kg/m 3. The diameter, volume and mass for the buoy are found by doing Step 1 and 2 of the scaling procedure to be: d mb = 0.3 m, V mb = 0.0141 m 3 and m mb = 11.698 kg. The assumption that states that the End B tension (EndT pa ) is equal to the drag force on the buoy will be proven by placing the prototype parameters into OrcaFlex and then use the static state End B tension of the prototype system. Doing this will provide 2 different spring constants and un-stretched lengths for each test. Using the drag force assumption, the End B tension then becomes EndT pb_es = 1.766x10 5 N. Using the OrcaFlex obtained tension, the End B tension is then EndT pb = 7.148x10 4 N. After applying the scaling rules of Step 1 of the scaling procedure and using the formulas from Step 3 to obtain the required un-stretched spring length and spring constant and natural frequency, the following is obtained: L 0_es = 10.655 mm, L 0 = 4.313 mm, k s_es = 2.589x10 2 N/m, k s = 2.589x10 2 N/m, _ = 7.964 Hz and n = 7.965 Hz. The forcing n es frequency divided by each natural frequency was check for resonance m / n_es = m / n = 0.499 thus no resonance occurs. The buoys center position within the water column as the dynamic stage begins (also called static position) is z pbs = -4.1903 m. At this position the velocity on the center of the buoy is found to be U pbs = 1.592 m/s. Using Step 4 of the scaling procedure, the needed model fluid velocity at the center of the buoy can be found to be U mbs = 0.255 m/s. 41

3.4.3. Scaled Quantities for Turbine System The scaled density for the lumped turbine system is found by using (38) to be = 486.7564 kg/m 3. The diameter, volume and mass for the buoy are found by doing Step 1 and 2 of the scaling procedure to be: d mb = 0.5759 m, V mb = 0.00102 m 3 and m mb = 0.494 kg. mb The assumption that states that the End B tension (EndT pa ) is equal to the Thrust force on the buoy will be proven by placing the prototype parameters into OrcaFlex and then use the static state End B tension of the prototype system. Doing this will provide 2 different spring constants and un-stretched lengths for each test. Using the thrust force assumption, the End B tension then becomes EndT pb_es = 2.938x10 5 N. Using the OrcaFlex obtained tension, the End B tension is then EndT pb = 3.106x10 5 N. After applying the scaling rules of Step 1 of the scaling procedure and using the formulas from Step 3 to obtain the required un-stretched spring length and spring constant and natural frequency, the following is obtained: L 0_es = 17.739 mm, L 0 = 18.756 mm, k s_es = 2.588x10 2 N/m, k s = 2.588x10 2 N/m, _ = 27.337 Hz and n = 27.336 Hz. The forcing n es frequency divided by each natural frequency was check for resonance m / n_es = m / n = 0.145 thus no resonance occurs. The turbines center position within the water column as the dynamic stage begins (also called static position) is z pbs = -22.6060 m. For the turbine system there was no varying current. A constant current of U pbs = 1.6 m/s was applied at all depths. Using 42

Step 4 of the scaling procedure, the needed model fluid velocity at the center of the turbine can be found to be U mbs = 0.256 m/s. 3.4.4. Dynamic Behavior Results The prototype system and environmental conditions were placed into OrcaFlex and evaluated so that the dynamic behavior of the prototype mooring line could be attained. The dynamic behavior of the mooring line includes End A and End B tensions, End A s x-z-position and fluid incidence angle of End A. In Figures 10 12 the dynamic End A tension is shown for the 6 m, 12 m and Turbine tests respectively. The graphs clearly show that the tension follows the behavior of the prototype tensions in each case. In Figures 13 15 the dynamic End B tension is shown for the 6 m, 12 m and Turbine tests respectively. It is clear that the larger the object with the higher scale factor, the closer the End B results match. This is intuitive since the smaller the object the smaller the drag force on the object. In Figures 16 18 the fluid incidence angle at End A is shown to match almost exactly for both the prototype and model system using the 6 m, 12 m and Turbine results. This means the moored system s dynamic motion matched the scaled moored system s dynamic motion. To only further aid this point, the dynamic X position and Z- position for End A is provided in Figures 19 21 and Figures 22 24. Comparing the fluid incidence angle, X position and Z position results of End A of the mooring line, it is clear that the dynamic behavior of the model system matches that of the prototype system. 43

12 10 Dynamic End A Tension for model and scaled prototype results Te dmasc =(Te dpa /(sc 3 )) Te dmaes Dynamic End A Tension (N) 8 6 4 2 0-2 0 2 4 6 8 10 12 14 16 Figure 10: Dynamic End A Tension (6m, sc = 40) 18 16 14 Dynamic End A Tension for model and scaled prototype results Te dmasc =(Te dpa /(sc 3 )) Te dmaes Dynamic End A Tension (N) 12 10 8 6 4 2 0-2 0 2 4 6 8 10 12 14 16 Figure 11: Dynamic End A Tension (12m, sc = 40) 44

14 12 Dynamic End A Tension for model and scaled prototype results Te dmasc =(Te dpa /(sc 3 )) Te dmaes Dynamic End A Tension (N) 10 8 6 4 2-2 0 2 4 6 8 10 12 14 16 Figure 12: Dynamic End A Tension (Turbine, sc = 40) 0.57 0.56 Dynamic End B Tension for model and scaled prototype results Te dmbsc =(Te dpb /(sc 3 )) Te dmbes Dynamic End B Tension (N) 0.55 0.54 0.53 0.52 0.51 0.5 0.49-2 0 2 4 6 8 10 12 14 16 Figure 13: Dynamic End B Tension (6m, sc = 40) 45

3 Dynamic End B Tension for model and scaled prototype results Te dmbsc =(Te dpb /(sc 3 )) 2.5 Te dmbes Dynamic End B Tension (N) 2 1.5 1 0.5-2 0 2 4 6 8 10 12 14 16 Figure 14: Dynamic End B Tension (12m, sc = 40) 10 9 8 Dynamic End B Tension for model and scaled prototype results Te dmbsc =(Te dpb /(sc 3 )) Te dmbes Dynamic End B Tension (N) 7 6 5 4 3 2 1 0-2 0 2 4 6 8 10 12 14 16 Figure 15: Dynamic End B Tension (Turbine, sc = 40) 46

90 Fluid Incidence Angle at End A 80 70 Fluid Incidence Angle (deg) 60 50 40 30 20 10 alpha dpa alpha dmaes 0-2 0 2 4 6 8 10 12 14 16 Figure 16: Fluid Incidence Angle at End A (6m, sc = 40) 90 Fluid Incidence Angle at End A 80 Fluid Incidence Angle (deg) 70 60 50 40 30 20 alpha dpa alpha dmaes 10-2 0 2 4 6 8 10 12 14 16 Figure 17: Fluid Incidence Angle at End A (12m, sc = 40) 47

90 Fluid Incidence Angle at End A 80 70 Fluid Incidence Angle (deg) 60 50 40 30 20 10 alpha dpa alpha dmaes 0-2 0 2 4 6 8 10 12 14 16 Figure 18: Fluid Incidence Angle at End A (Turbine, sc = 40) Dynamic X-Direction Motion of End A for model and scaled prototype results 0.1 X dmasc = X dpa /sc 0 X dmaes -0.1-0.2 X (m) -0.3-0.4-0.5-0.6-0.7-0.8-2 0 2 4 6 8 10 12 14 16 Figure 19: Dynamic X Position of End A (6m, sc = 40) 48

Dynamic X-Direction Motion of End A for model and scaled prototype results 0.1 0 X dmasc = X dpa /sc -0.1 X dmaes -0.2-0.3 X (m) -0.4-0.5-0.6-0.7-0.8-0.9-2 0 2 4 6 8 10 12 14 16 Figure 20: Dynamic X Position of End A (12m, sc = 40) Dynamic X-Direction Motion of End A for model and scaled prototype results 0.2 0.15 0.1 X dmasc = X dpa /sc X dmaes 0.05 X (m) 0-0.05-0.1-0.15-0.2-0.25-2 0 2 4 6 8 10 12 14 16 Figure 21: Dynamic X Position of End A (Turbine, sc = 40) 49

Dynamic Z-Direction Motion of End A for model and scaled prototype results -0.1-0.15 Z dmasc = Z dpa /sc -0.2 Z dmaes -0.25-0.3 Z (m) -0.35-0.4-0.45-0.5-0.55-0.6-2 0 2 4 6 8 10 12 14 16 Figure 22: Dynamic Z Position of End A (6m, sc = 40) Dynamic Z-Direction Motion of End A for model and scaled prototype results -0.2 Z dmasc = Z dpa /sc -0.22-0.24 Z dmaes -0.26-0.28 Z (m) -0.3-0.32-0.34-0.36-0.38-0.4-2 0 2 4 6 8 10 12 14 16 Figure 23: Dynamic Z Position of End A (12m, sc = 40) 50

Dynamic Z-Direction Motion of End A for model and scaled prototype results -0.5 Z dmasc = Z dpa /sc -0.55-0.6 Z dmaes -0.65-0.7 Z (m) -0.75-0.8-0.85-0.9-0.95-1 -2 0 2 4 6 8 10 12 14 16 Figure 24: Dynamic Z Position of End A (Turbine, sc = 40) 3.5. Testing Facility Depth of D m = 40 m The prototype systems will be scaled using a model water depth of 40 m. This provides a scale factor sc = D p /D m = 10. The proposed scaling procedure is then carried out using the scale factor of sc = 10. The scaled environmental parameters then become a m = 0.50 m and = 1.988 Hz. The scaled mooring line parameters become d m = 0.010 m m, L m = 100 m. The needed mooring elasticity is found by following Step 1 of the proposed scaling procedure to be EA eq = 4.04x10 5 N. The actual elasticity of a 6x19 Wire Rope with Wire Core having a nominal diameter of d m = 10.0 mm = 0.010 m is found to be E m A m = 4.04x10 6 N. These scaled results are used with each of the below systems. 51

3.5.1. Scaled Quantities for Spherical Buoy System 1 The buoys density must be scaled so that the buoyancy to weight ratio matches that of the prototype. The density of the buoy must scale by using (38) to give = 827.486 kg/m 3. The diameter of the buoy becomes d mb = 0.60 m with a volume of V mb = 0.1131 m 3. The mass for the buoy becomes m mb = 93.586 kg. mb The assumption that states that the End B tension (EndT pa ) is equal to the drag force on the buoy will be proven by placing the prototype parameters into OrcaFlex and then use the static state End B tension of the prototype system. Doing this will provide 2 different spring constants and un-stretched lengths for each test. Using the drag force assumption, the End B tension then becomes EndT pb_es = 4.428x10 4 N. Using the OrcaFlex obtained tension, the End B tension is then EndT pb = 3.179x10 4 N. After applying the scaling rules of Step 1 of the scaling procedure and using the formulas from Step 3 to obtain the required un-stretched spring length and spring constant and natural frequency, the following is obtained: L 0_es = 9.866 mm, L 0 = 7.072 mm, k s_es = 4.488x10 3 N/m, k s = 4.489x10 3 N/m, = 10.238 Hz and n = 10.238 Hz. The forcing frequency divided by each natural frequency was check for resonance m / n_es = m / n = 0.194 thus no resonance occurs. n_ es The buoys center position within the water column as the dynamic stage begins (also called static position) is z pbs = -2.836 m. Knowing the position of the buoy within the water column with no waves present allows for the drag force on the buoy at that state to be solved. Interpolation of the varying current profile allows for the solution of the 52

current velocity at the center of the buoy, which is U pbs = 1.594 m/s. Using the drag force scaling described in the proposed scaling procedure, the required velocity at the center of the model buoy can be found to be U mbs = 0.511 m/s. Knowing the velocity at any depth for the model profile will allow for the solution of the entire model current profile. 3.5.2. Scaled Quantities for Spherical Buoy System 2 The scaled density for the buoy is found by using (38) to be mb = 827.486 kg/m 3. The diameter, volume and mass for the buoy are found by doing Step 1 and 2 of the scaling procedure to be: d mb = 1.20 m, V mb = 0.9048 m 3 and m mb = 748.691 kg. The assumption that states that the End B tension (EndT pa ) is equal to the drag force on the buoy will be proven by placing the prototype parameters into OrcaFlex and then use the static state End B tension of the prototype system. Doing this will provide 2 different spring constants and un-stretched lengths for each test. Using the drag force assumption, the End B tension then becomes EndT pb_es = 1.765x10 5 N. Using the OrcaFlex obtained tension, the End B tension is then EndT pb = 7.148x10 4 N. After applying the scaling rules of Step 1 of the scaling procedure and using the formulas from Step 3 to obtain the required un-stretched spring length and spring constant and natural frequency, the following is obtained: L 0_es = 39.342 mm, L 0 = 15.926 mm, k s_es = 4.487x10 3 N/m, k s = 4.488x10 3 N/m, n_ es= 4.144 Hz and n = 4.145 Hz. The forcing frequency divided by each natural frequency was check for resonance m / n_es = m / n = 0.480 thus no resonance occurs. 53

The buoys center position within the water column as the dynamic stage begins (also called static position) is z pbs = -4.1903 m. At this position the velocity on the center of the buoy is found to be U pbs = 1.592 m/s. Using Step 4 of the scaling procedure, the needed model fluid velocity at the center of the buoy can be found to be U mbs = 0.51 m/s. 3.5.3. Scaled Quantities for Turbine System The scaled density for the lumped turbine system is found by using (38) to be = 486.7564 kg/m 3. The diameter, volume and mass for the buoy are found by doing Step 1 and 2 of the scaling procedure to be: d mb = 2.3036 m, V mb = 0.065 m 3 and m mb = 31.618 kg. mb The assumption that states that the End B tension (EndT pa ) is equal to the Thrust force on the buoy will be proven by placing the prototype parameters into OrcaFlex and then use the static state End B tension of the prototype system. Doing this will provide 2 different spring constants and un-stretched lengths for each test. Using the thrust force assumption, the End B tension then becomes EndT pb_es = 2.938x10 5 N. Using the OrcaFlex obtained tension, the End B tension is then EndT pb = 3.106x10 5 N. After applying the scaling rules of Step 1 of the scaling procedure and using the formulas from Step 3 to obtain the required un-stretched spring length and spring constant and natural frequency, the following is obtained: L 0_es = 65.499 mm, L 0 = 69.254 mm, k s_es = 4.486x10 3 N/m, k s = 4.485x10 3 N/m, n_ es= 14.226 Hz and n = 14.226 Hz. The forcing 54

frequency divided by each natural frequency was check for resonance m / n_es = m / n = 0.140 thus no resonance occurs. The turbines center position within the water column as the dynamic stage begins (also called static position) is z pbs = -22.606 m. For the turbine system there was no varying current. A constant current of U pbs = 1.6 m/s was applied at all depths. Using Step 4 of the scaling procedure, the needed model fluid velocity at the center of the turbine can be found to be U mbs = 0.513 m/s. 3.5.4. Dynamic Behavior Results The prototype system and environmental conditions were placed into OrcaFlex and evaluated so that the dynamic behavior of the prototype mooring line could be attained. The dynamic behavior of the mooring line includes End A and End B tensions, End A s x-z-position and fluid incidence angle of End A. In Figures 25 27 the dynamic End A tension is shown for the 6 m, 12 m and Turbine tests respectively. The graphs show that the tension follows the behavior of the prototype tensions in each case. In Figures 28 30 the dynamic End B tension is shown for the 6 m, 12 m and Turbine tests respectively. It is clear that the larger the object with the higher scale factor, the closer the End B results match. This is intuitive since the smaller the object the smaller the drag force on the object. In Figures 31 33 the fluid incidence angle at End A is shown to match almost exactly for both the prototype and model system using the 6 m, 12 m and Turbine results. This means the moored system s dynamic motion matched the scaled 55

moored system s dynamic motion. To only further aid this point, the dynamic X position and Z- position for End A is provided in Figures 34 36 and Figures 37 39. Comparing the fluid incidence angle, X position and Z position results of End A of the mooring line, it is clear that the dynamic behavior of the model system matches that of the prototype system. 600 500 Dynamic End A Tension for model and scaled prototype results Te dmasc =(Te dpa /(sc 3 )) Te dmaes Dynamic End A Tension (N) 400 300 200 100 0-5 0 5 10 15 20 25 30 35 Figure 25: Dynamic End A Tension (6m, sc = 10) 56

800 700 Dynamic End A Tension for model and scaled prototype results Te dmasc =(Te dpa /(sc 3 )) Te dmaes Dynamic End A Tension (N) 600 500 400 300 200 100 0-5 0 5 10 15 20 25 30 35 Figure 26: Dynamic End A Tension (12m, sc = 10) 800 700 Dynamic End A Tension for model and scaled prototype results Te dmasc =(Te dpa /(sc 3 )) Te dmaes Dynamic End A Tension (N) 600 500 400 300 200 100-5 0 5 10 15 20 25 30 35 Figure 27: Dynamic End A Tension (Turbine, sc = 10) 57

36.5 36 35.5 Dynamic End B Tension for model and scaled prototype results Te dmbsc =(Te dpb /(sc 3 )) Te dmbes Dynamic End B Tension (N) 35 34.5 34 33.5 33 32.5 32 31.5-5 0 5 10 15 20 25 30 35 Figure 28: Dynamic End B Tension (6m, sc = 10) 200 180 160 Dynamic End B Tension for model and scaled prototype results Te dmbsc =(Te dpb /(sc 3 )) Te dmbes Dynamic End B Tension (N) 140 120 100 80 60 40 20-5 0 5 10 15 20 25 30 35 Figure 29: Dynamic End B Tension (12m, sc = 10) 58

700 600 Dynamic End B Tension for model and scaled prototype results Te dmbsc =(Te dpb /(sc 3 )) Te dmbes Dynamic End B Tension (N) 500 400 300 200 100 0-5 0 5 10 15 20 25 30 35 Figure 30: Dynamic End B Tension (Turbine, sc = 10) 90 Fluid Incidence Angle at End A 80 70 Fluid Incidence Angle (deg) 60 50 40 30 20 10 alpha dpa alpha dmaes 0-5 0 5 10 15 20 25 30 35 Figure 31: Fluid Incidence Angle at End A (6m, sc = 10) 59

90 Fluid Incidence Angle at End A 80 Fluid Incidence Angle (deg) 70 60 50 40 30 20 alpha dpa alpha dmaes 10-5 0 5 10 15 20 25 30 35 Figure 32: Fluid Incidence Angle at End A (12m, sc = 10) 90 Fluid Incidence Angle at End A 80 70 Fluid Incidence Angle (deg) 60 50 40 30 20 10 alpha dpa alpha dmaes 0-5 0 5 10 15 20 25 30 35 Figure 33: Fluid Incidence Angle at End A (Turbine, sc = 10) 60

Dynamic X-Direction Motion of End A for model and scaled prototype results 0.5 X dmasc = X dpa /sc 0 X dmaes -0.5 X (m) -1-1.5-2 -2.5-3 -5 0 5 10 15 20 25 30 35 Figure 34: Dynamic X Position of End A (6m, sc = 10) Dynamic X-Direction Motion of End A for model and scaled prototype results 0.5 X dmasc = X dpa /sc 0 X dmaes -0.5-1 X (m) -1.5-2 -2.5-3 -3.5-4 -5 0 5 10 15 20 25 30 35 Figure 35: Dynamic X Position of End A (12m, sc = 10) 61

Dynamic X-Direction Motion of End A for model and scaled prototype results 0.8 0.6 0.4 X dmasc = X dpa /sc X dmaes 0.2 X (m) 0-0.2-0.4-0.6-0.8-1 -5 0 5 10 15 20 25 30 35 Figure 36: Dynamic X Position of End A (Turbine, sc = 10) Dynamic Z-Direction Motion of End A for model and scaled prototype results 0 Z dmasc = Z dpa /sc -0.5 Z dmaes -1 Z (m) -1.5-2 -2.5-5 0 5 10 15 20 25 30 35 Figure 37: Dynamic Z Position of End A (6m, sc = 10) 62

Dynamic Z-Direction Motion of End A for model and scaled prototype results -0.8 Z dmasc = Z dpa /sc -0.9-1 Z dmaes -1.1 Z (m) -1.2-1.3-1.4-1.5-1.6-5 0 5 10 15 20 25 30 35 Figure 38: Dynamic Z Position of End A (12m, sc = 10) Dynamic Z-Direction Motion of End A for model and scaled prototype results -2 Z dmasc = Z dpa /sc -2.2-2.4 Z dmaes -2.6-2.8 Z (m) -3-3.2-3.4-3.6-3.8-4 -5 0 5 10 15 20 25 30 35 Figure 39: Dynamic Z Position of End A (Turbine, sc = 10) 63

4. CONCLUSIONS From the results of the graphs, the dynamic behavior of the scaled model system closely resembles that of the prototype system. To further show the accuracy of scaling, the parameters that Papazoglou and Mavrakos provide for scaling mooring systems were then checked for each system. The results match almost perfectly for all the parameters of importance provided in [49] and [50]. The results from the smaller scale factor match the prototype results best. This would mean that if a facility wants to have the best accuracy of scale model testing, then the facility should be sufficiently large enough that the scale factor will become small. However, a facility of this magnitude would highly expensive to build and maintain. This is the reason for choosing the depths of 10 m and 40 m. Due to the results, the recommended target facility should be a minimum 8 m deep (suggested depth is 10 m). The larger the scale factor becomes, the more difficult the scaling due to the limitation of real world wire sizes. Owing to the length of the cable scaling to 25 m long with the scale factor of 40, the suggested length of the facility is at the minimum of 25 m. The large depth needed for scale testing mooring systems is something that cannot be avoided. Since companies do not have facilities of this size at their disposal, a facility similar to the proposed one could potentially bring in a large number of companies for testing. In addition, this facility would only further advance the research 64

and development of permanent hydrokinetic energy systems intended to be deployed in the Gulf Stream. 4.1. Recommendations for Future Work Additional testing could be done with different types of mooring lines and diameters that would help further prove the accuracy of this scaling method. The facility could be built and actual scaled tests could be run by following the scaling method proposed in this thesis. The actual scaled test results could then be compared to the results obtained from the Orcaflex simulations. 65

A.0 APPENDICES A.1 Prototype Environment Table 2: Prototype Environmental Parameters Parameter Value D p 400 m 1026.9 kg/m 3 pw 1.35x10-6 m 2 /s pw a p 5 m t p 10 s U p Varying or Constant 1.6 m/s U p0 A.2 Prototype Mooring Line Table 3: Prototype Mooring Line Parameters Parameter Type d p L p OD p m pair m p E p A p k p w p Value 6x19 Wire Rope Wire Core 0.100 m 1000 m 0.08 m 39.897 kg/m 34.735237606445828 kg/m 4.04x10 8 N 4.04x10 5 N/m 340.6363178732520 N/m 66

A.3 Prototype System 1 Table 4: Prototype System 1 Parameters Parameter Value d pb 6 m V pb 113.09733553 m 3 850 kg/m 3 pb m pb 96132.73519985 kg I xxpb 346077.84671945 kg*m 2 I yypb 346077.84671945 kg*m 2 I zzpb 346077.84671945 kg*m 2 A pbx = A pby = A pbz 28.274333882308138 m 2 A.4 Prototype System 2 Table 5: Prototype System 2 Parameters Parameter Value d pb 12 m V pb 904.77868423 m 3 850 kg/m 3 pb m pb 769061.88159878 kg I xxpb 11074491.09502245 kg*m 2 I yypb 11074491.09502245 kg*m 2 I zzpb 11074491.09502245 kg*m 2 A pbx = A pby = A pbz 113.0973355292326m 2 67

A.5 Prototype System 3 Table 6: Prototype Lumped Turbine Parameters Parameter Value d pb 23.036 m V pb 64.955614820371778 m 3 500 kg/m 3 pb m pb 32477.80741018589 kg I xxpb 5858213.699538750 kg*m 2 I yypb 67684.66617249245 kg*m 2 I zzpb 5912505.015000040 kg*m 2 Table 7: Prototype Turbine Components Aft Body Parameter Value OD 1 m ID 0.8 m L 1 m 500 kg/m 3 V 0.282743338823081 m 3 m 141.3716694115406 kg W 1.386382481834685 N B 2847.352341192076 N I xx 28.981192229365831 kg*m 2 I yy 26.271568565644635 kg*m 2 I zz 26.271568565644635 kg*m 2 Rotor Parameter Value OD 1.4 m ID 0.8 m L 1 m 500 kg/m 3 V 1.036725575684631 m 3 m 518.3627878423157 kg W 5083.402433393845 N B 10440.29191770428 N I xx 168.4679060487526 kg*m 2 I yy 127.4308520112359 kg*m 2 I zz 127.4308520112359 kg*m 2 Fwd Body 68

Parameter Value OD 1 m ID 0.8 m L 6 m 500 kg/m 3 V 1.696460032938488 m 3 m 848.2300164692438 kg W 831.8294891008109 N B 17084.11404715246 N I xx 173.8871533761950 kg*m 2 I yy 2631.633626095828 kg*m 2 I zz 2631.633626095828 kg*m 2 Left & Right Rods Parameter Value OD 0.8 m ID 0.0 m L 12 m 500 kg/m 3 V 6.031857894892403 m 3 m 301.5928947446202 kg W 29576.15961247329 N B 60743.51661209765 N I xx 36311.78452725226 kg*m 2 I yy 241.2743157956962 kg*m 2 I zz 36311.78452725226 kg*m 2 Left & Right BCM Parameter Value a 3 m b 3 m c 0.65 m 500 kg/m 3 V 24.504422698000386 m 3 m 12252.21134900019 kg W 120153.1484256727 N B 246770.5362366467 N I xx 23089.29228719087 kg*m 2 I yy 23089.29228719087 kg*m 2 I zz 44107.96085640070 kg*m 2 69

Table 8: Prototype Turbine Blades Blade span(m) chord(m) Gamma m blade V blade 1.202 0.6096 24 deg 45.6130152 kg 0.0912260304 m 3 1.202 0.5207 18 deg 31.79477512 kg 0.06358955024 m 3 1.202 0.4445 17 deg 22.38670909 kg 0.0447734182 m 3 1.202 0.3683 16 deg 15.1844933799 kg 0.03036898676 m 3 1.202 0.3175 15 deg 10.66669825 kg 0.0213333965 m 3 1.202 0.2692 13 deg 7.3937664399 kg 0.01478753288 m 3 1.202 0.2413 11 deg 5.1627582799 kg 0.01032551656 m 3 1.202 0.2083 10.5 deg 3.65549836 kg 0.00731099672 m 3 1.202 0.1854 10 deg 2.66306706 kg 0.00532613412 m 3 Top Blade span(m) chord(m) I x wtp (kg*m 2 ) I y wtp (kg*m 2 ) I z wtp (kg*m 2 ) 1.202 m 0.6096 6.90435149296713 5.491822234418 1.41252925854873 1.202 m 0.5207 4.54647397005430 3.8281015227063 0.71837244734793 1.202 m 0.4445 3.06396486959763 2.69536723787636 0.3685976317212 1.202 m 0.3683 1.99985964230277 1.82821806444979 0.17164157785298 1.202 m 0.3175 1.37387984573808 1.28427402486608 0.08960582087200 1.202 m 0.2692 0.93486329383018 0.89021194396481 0.0446513498653 1.202 m 0.2413 0.64664824809444 0.6215978178314 0.02505043026301 1.202 m 0.2083 0.45334055572988 0.44012322104345 0.01321733468643 1.202 m 0.1854 0.32826233089169 0.32063416171302 0.00762816917867 Bottom Blade 1 span(m) chord(m) I x wb1p (kg*m 2 ) I y wb1p (kg*m 2 ) I z wb1p (kg*m 2 ) 1.202 0.6096 6.90435149296713 2.43235250251615 4.47199899045098 1.202 0.5207 4.54647397005430 1.49580471618754 3.05066925386676 1.202 0.4445 3.06396486959763 0.95029003326004 2.11367483633759 1.202 0.3683 1.99985964230277 0.58578569950218 1.41407394280059 1.202 0.3175 1.37387984573808 0.38827287187052 0.98560697386756 1.202 0.2692 0.93486329383018 0.25604149839023 0.67882179543995 1.202 0.2413 0.64664824809444 0.17418727715512 0.47246097093932 1.202 0.2083 0.45334055572988 0.11994380627569 0.33339674945419 1.202 0.1854 0.32826233089169 0.08587966731226 0.24238266357943 Bottom Blade 2 span(m) chord(m) I x wb2p (kg*m 2 ) I y wb2p (kg*m 2 ) I z wb2p (kg*m 2 ) 1.202 0.6096 6.9043514929671 2.43235250251615 4.47199899045098 1.202 0.5207 4.5464739700543 1.49580471618754 3.05066925386676 1.202 0.4445 3.0639648695976 0.95029003326004 2.11367483633759 1.202 0.3683 1.9998596423027 0.58578569950218 1.41407394280059 1.202 0.3175 1.3738798457380 0.38827287187052 0.98560697386756 1.202 0.2692 0.9348632938301 0.25604149839023 0.67882179543995 1.202 0.2413 0.6466482480944 0.17418727715512 0.47246097093932 1.202 0.2083 0.4533405557298 0.11994380627569 0.33339674945419 70

1.202 0.1854 0.3282623308916 0.08587966731226 0.24238266357943 A.6 Prototype System 1 Simulation Table 9: Prototype System 1 Positions Buoy Mooring Line End A Mooring Line End B Buoy Buoy Mooring Line End A Mooring Line End B Initial Position Parameter Value X 0 Z -40 X 0 Z -3 X 44 Z Anchored Initial Static Position Parameter Value X -688.699747605058 Z -2.83615628977918 New Initial Position Parameter Value X 688.699747605058 Z -40 X 0 Z -3 X 732.699747605058 Z Anchored Table 10: Prototype System 1 Static Results Parameter Value Position and Water X pbs 0 Y pbs 0 Z pbs -2.83615628977918 71

Velocity for Buoy Center U pbs 1.594327687420442 m/s Dry Length 0.1638 m Portion Wet (PW) 0.9727 Buoy Buoyancy 1129690.41305 N Weight 942740.087648 N Drag 44281.97300019461 N Mooring Line EndF pa 165959.1374509470 N EndT pa 165957.2286285270 N End A alpha pa 83.890861000793805 deg Mooring Line EndF pb 31787.33613676160 N EndT pb (Orcaflex Tension) 31741.6746745531 N End B EndT pb es (Estimated 44281.97300019461 N alpha pb 0.00024393435175327 deg A.7 Prototype System 2 Simulation Table 11: Prototype System 2 Positions Buoy Mooring Line End A Mooring Line End B Buoy Buoy Mooring Line End A Initial Position Parameter Value X 0 Z 0 X 0 Z -6 X 600 Z Anchored Initial Static Position Parameter Value X -193.477248185559 Z -4.19031634090173 New Initial Position Parameter Value X 193.477248185559 Z 0 X 0 Z -6 72

Mooring Line End B X 793.477248185559 Z Anchored Table 12: Prototype System 2 Static Results Position and Water Velocity for Buoy Center Buoy Mooring Line End A Mooring Line End B Parameter Value X pbs 0 Y pbs 0 Z pbs -4.190316340901730 U pbs 1.591619367318197 m/s Dry Length 1.8097 m Portion Wet (PW) 0.84919167 Buoyancy 7737433.216620219 N Weight 7541920.701180676 N Drag 176526.6209074738 N EndF pa 204149.9338733550 N EndT pa 204147.178757109 N alpha pa 72.988198698464799 deg EndF pb 71498.91361312840 N EndT pb (Orcaflex Tension) 71478.62491914129 N EndT pb es (Estimated 176526.6209074738 N alpha pb 0 deg A.8 Prototype System 3 Simulation Table 13: Prototype System 3 Positions Buoy Mooring Line End A Mooring Line End B Initial Position Parameter Value X 0 Z -100 X 3.122172978679915 Z 0 X 500 Z Anchored Initial Static Position 73

Buoy Buoy Mooring Line End A Mooring Line End B Parameter Value X -391.333822431954 Z -22.605991465278 New Initial Position Parameter Value X 391.333822431954 Z -100 X 3.122172978679915 Z 0 X 891.333822431954 Z Anchored Table 14: Prototype System 3 Static Results Position and Water Velocity for Buoy Center Buoy Mooring Line End A Mooring Line End B Parameter Value X pbs 0 Y pbs 0 Z pbs -22.605991465277999 U pbs 1.6 m/s Dry Length 0.43 m Portion Wet (PW) 0.9813 Buoyancy 654132.198842 N Weight 318498.490039 N Thrust 293802.9900861568 N EndF pa 436947.0581818810 N EndT pa 436944.5073136070 N alpha pa 47.551688567896008 deg EndF pb 310638.9118257520 N EndT pb (Orcaflex Tension) 310634.2426596930 N EndT pb es (Estimated 293802.9900861568 N alpha pb 0 deg 74

A.9 Test Description Table 15: Test Specifications Test 1 Test 2 Test 3 Test 4 Test 5 Test 6 Parameter Value Prototype System 1 D m 10 m sc 40 Prototype System 1 D m 40 m sc 10 Prototype System 2 D m 10 m sc 40 Prototype System 2 D m 40 m sc 10 Prototype System 3 D m 10 m sc 40 Prototype System 3 D m 40 m sc 10 A.10 Test 1, 3 and 5 Mooring Line Table 16: Mooring Line Parameters for sc = 40 Parameter Type d m L m OD m m mair m m E m A m k m w m Value 6x19 Wire Rope Wire Core 0.0025 m 25 m 0.002 m 0.024935625 kg/m 0.021794974824206 kg/m 252500 N 10100 N/m 0.213735689859803 N/m 75

A.11 Test 2, 4 and 6 Mooring Line Table 17: Mooring Line Parameters for sc = 10 Parameter Type d m L m OD m m mair m m E m A m k m w m Value 6x19 Wire Rope Wire Core 0.010 m 100 m 0.008 m 0.39897 kg/m 0.348719597187301 kg/m 4040000 N 40400 N/m 3.419771037756841 N/m A.12 Model Environment Table 18: Test 1, 3 and 5 Model Environmental Parameters Parameter Value D m 10 m 999.7 kg/m 3 mw 1.31x10-6 m 2 /s mw a m 0.125 m t m 1.581138830084190 s (Test 1 and 3) U m varying profile (Test 5) U m const 0.256400706700896 m/s 0 Model Varying Current Profile 0 Model Varying Current Profile -1-1 -2-2 -3-3 -4-4 Z (m) -5 Z (m) -5-6 -6-7 -7-8 -8-9 -9-10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Current Velocity at 180 deg (m/s) -10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Current Velocity at 180 deg (m/s) Figure 40: Test 1 Current Profile Figure 41: Test 3 Current Profile 76

Table 19: Test 2, 4 and 6 Model Environmental Parameters 0 Parameter Value D m 40 m 999.7 kg/m 3 mw 1.31x10-6 m 2 /s mw a m 0.5 m t m 3.162277660168379 s (Test 1 and 3) U m varying profile (Test 5) U m const 0.512801413401792 m/s Model Varying Current Profile Model Varying Current Profile 0-5 -5-10 -10-15 -15 Z (m) -20 Z (m) -20-25 -25-30 -30-35 -35-40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Current Velocity at 180 deg (m/s) -40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Current Velocity at 180 deg (m/s) Figure 42: Test 2 Current Profile Figure 43: Test 4 Current Profile A.13 Model System 1 Table 20: Test 1 System 1 Parameters Parameter Value d mb 0.15 m V mb 0.001767150 m 3 827.48563638134 kg/m 3 mb m mb 1.46228782 kg I xxmb 0.00329015 kg*m 2 I yymb 0.00329015 kg*m 2 I zzmb 0.00329015 kg*m 2 77

Model System 1 A mbx = A mby = A mbz 0.017671458676443 m 2 Estimated Spring k s es 258.9459730942307 N/m L 0 es 0.002672008449717 m using estimated End B omega n es 19.673153222970132 Hz Spring k s 258.9540117470036 N/m L 0 0.001915257707127 m using Orcaflex End B omega n 19.673458584776839 Hz Table 21: Test 2 System 1 Parameters Model System 1 Estimated Spring using estimated End B Spring using Orcaflex End B Parameter Value d mb 0.6 m V mb 0.11309734 m 3 827.485636381341 kg/m 3 mb m mb 93.58642066 kg I xxmb 3.36911114 kg*m 2 I yymb 3.36911114 kg*m 2 I zzmb 3.36911114 kg*m 2 A mbx = A mby = A mbz 0.282743338823081 m 2 k s es 4488.396866966665 N/m L 0 es 0.009865877352802 m omega n es 10.238233534463051 Hz k s 4488.536203614728 N/m L 0 0.007071720764776 m omega n 10.238392449785580 Hz A.14 Model System 2 Table 22: Test 3 System 2 Parameters Parameter Value d mb 0.30 m V mb 0.01413717 m 3 827.485636381341 kg/m 3 mb m mb 11.69830258 kg I xxmb 0.10528472 kg*m 2 I yymb 0.10528472 kg*m 2 Model System 2 I zzmb 0.10528472 kg*m 2 A mbx = A mby = A mbz 0.070685834705770 m 2 Estimated Spring k s es 258.8612008840337 N/m 78

using estimated End B L 0 es 0.010655240886852 m omega n es 7.964289624885042 Hz Spring k s 258.9285393430006 N/m L 0 0.004313365831343 m using Orcaflex End B omega n 7.965325446621213 Hz Table 23: Test 4 System 2 Parameters Model System 2 Estimated Spring using estimated End B Spring using Orcaflex End B Parameter Value d mb 1.2 m V mb 0.90477868 m 3 827.485636381341 kg/m 3 mb m mb 748.69136531 kg I xxmb 107.81155660 kg*m 2 I yymb 107.81155660 kg*m 2 I zzmb 107.81155660 kg*m 2 A mbx = A mby = A mbz 1.130973355292326 m 2 k s es 4486.927481989917 N/m L 0 es 0.039342427889917 m omega n es 4.144747730003505 Hz k s 4488.094681945344 N/m L 0 0.015926273838804 m omega n 4.145286788725863 Hz A.15 Model System 3 Table 24: Test 5 Model Lumped Turbine Parameters Parameter Value d mb 0.5759 m V mb 0.001014931481568 m 3 486.7562566949069 kg/m 3 mb m mb 0.494024248770006 kg I xxmb 0.055693792408326 kg*m 2 I yymb 6.434752879251x10-4 kg*m 2 Model System 3 I zzmb 0.056209937671705 kg*m 2 Estimated Spring k s es 258.7860237243038 N/m L 0 es 0.017739256757493 m using estimated End B omega n es 27.336597077931003 Hz Spring k s 258.7752344598335 N/m 79

using Orcaflex End B L 0 0.017739256757493 m omega n 27.336597077931003 Hz Table 25: Test 5 Model Turbine Components Aft Body Parameter Value OD 0.025 m ID 0.02 m L 0.025 m 486.7562566949069 kg/m 3 V 4.417864669110649x10-6 m 3 m 0.002150423268921 kg W 0.021088448350164 N B 0.043311455221567 N I xx 2.755229813305010x10-7 kg*m 2 I yy 2.497627025882184 x10-7 kg*m 2 I zz 2.497627025882184 x10-7 kg*m 2 Rotor Parameter Value OD 0.035 m ID 0.02 m L 0.025 m 486.7562566949069 kg/m 3 V 1.619883712007237x10-5 m 3 m 0.007884885319377 kg W 0.077324310617268 N B 0.158808669145745 N I xx 1.601617330498439x10-6 kg*m 2 I yy 1.211479775633435x10-6 kg*m 2 I zz 1.211479775633435x10-6 kg*m 2 Fwd Body Parameter Value OD 0.025 m ID 0.02 m L 0.15 m 486.7562566949069 kg/m 3 V 2.650718801466389x10-5 m 3 m 0.012902539613526 kg W 0.126530690100984 N B 0.259868731329400 N I xx 1.653137887983006 x10-6 kg*m 2 I yy 2.501883071935255x10-5 kg*m 2 80

I zz 2.501883071935255x10-5 kg*m 2 Left & Right Rods Parameter Value OD 0.02 m ID 0 m L 0.3 m 486.7562566949069 kg/m 3 V 9.424777960769379x10-5 m 3 m 0.045875696403648 kg W 0.449886898136831 N B 0.923977711393423 N I xx 3.452146154374482x10-4 kg*m 2 I yy 2.293784820182381x10-6 kg*m 2 I zz 3.452146154374482x10-4 kg*m 2 Left & Right BCM Parameter Value a 0.075 m b 0.075 m c 0.01625 m 486.7562566949069 kg/m 3 V 3.828816046562560x10-4 m 3 m 0.186370016639818 kg W 1.827665523680875 N B 3.753659452535782 N I xx 2.195089352235861x10-4 kg*m 2 I yy 2.195089352235861x10-4 kg*m 2 I zz 4.193325374395913x10-4 kg*m 2 Table 26: Test 6 Model Lumped Turbine Parameters Parameter Value d mb 2.3036 m V mb 0.064955614820372 m 3 486.756256694906 kg/m 3 mb m mb 31.617551921280398 kg I xxmb 57.0304434261260 kg*m 2 I yymb 0.6589186948353 kg*m 2 Model System 3 I zzmb 57.5589761758256 kg*m 2 Estimated Spring k s es 4485.624411221266N/m L 0 es 0.065498794181514 m using estimated End B omega n es 14.226416169558551 Hz Spring k s 4485.437397303781 N/m L 0 0.069253946749188 m 81

using Orcaflex End B omega n 14.226119603762244 Hz Table 27: Test 6 Model Turbine Components Aft Body Parameter Value OD 0.1 m ID 0.08 m L 0.1 m 486.7562566949069 kg/m 3 V 2.827433388230815x10-4 m 3 m 0.137627089210943 kg W 1.349660694410493 N B 2.771933134180272 N I xx 2.821355328824330x10-4 kg*m 2 I yy 2.55757007450335x10-4 kg*m 2 I zz 2.557570074503356x10-4 kg*m 2 Rotor Parameter Value OD 0.14 m ID 0.08 m L 0.1 m 486.7562566949069 kg/m 3 V 0.001036725575685 m 3 m 0.504632660440124 kg W 4.948755879505138 N B 10.163754825327656 N I xx 0.001640056146430 kg*m 2 I yy 0.001240555290249 kg*m 2 I zz 0.001240555290249 kg*m 2 Fwd Body Parameter Value OD 0.1 m ID 0.08 m L 0.6 m 486.7562566949069 kg/m 3 V 0.001696460032938 m 3 m 0.825762535265657 kg W 8.097964166462958 N B 16.631598805081627 N I xx 0.001692813197295 kg*m 2 I yy 0.025619282656617 kg*m 2 I zz 0.025619282656617 kg*m 2 82

Left & Right Rods Parameter Value OD 0.08 m ID 0 m L 1.2 m 486.7562566949069 kg/m 3 V 0.006031857894892 m 3 m 2.936044569833447 kg W 28.792761480757171 N B 59.134573529179093 N I xx 0.353499766207947 kg*m 2 I yy 0.002348835655867 kg*m 2 I zz 0.353499766207947 kg*m 2 Left & Right BCM Parameter Value OD 0.3 m ID 0.3 m L 0.065 m 486.7562566949069 kg/m 3 V 0.024504422698 m 3 m 11.927681064948377 kg W 116.9705935155760 N B 240.2342049622901 N I xx 0.224777149668952 kg*m 2 I yy 0.224777149668952 kg*m 2 I zz 0.429396518338142 kg*m 2 A.16 Model System 1 Simulation Table 28: Test 1 Model System 1 Positions Buoy Mooring Line End A Mooring Line End B Initial Position Parameter Value X 0 Z -1 X 0 Z -0.075 X 21.1 Z -9.998 83

Spring End A Spring End B Buoy Buoy Mooring Line End A Mooring Line End B Spring End A Spring End B X 0 Z 0 X 21.101915257730477 Z Anchored Initial Static Position Parameter Value X 2.78556222165092 Z -0.0715799600600013 New Initial Position Parameter Value X -2.78556222165092 Z -1 X 0 Z -0.075 X 18.314437778349081 Z -9.998 X 0 Z 0 X 18.31635303607958 Z Anchored Table 29: Test 1 Model System 1 Static Results Parameter Value Position and Water X mbs 0 Y mbs 0 Velocity for Buoy Center Z mbs -0.0715799600600013 U mbs 0.255491716104629 m/s Dry Length 0.0034 m Portion Wet (PW) 0.9773333 Buoy Buoyancy 16.9319311554 N Weight 14.340144850003 N Drag 0.691905828128041 N Mooring Line EndF ma 2.60297575544038 N EndT ma 2.60294598546042 N 84

End A alpha ma 83.887196239009 deg Mooring Line EndF mb 0.497207942679174 N EndT mb 0.497207923578987 N End B alpha mb 0 deg Table 30: Test 2 Model System 1 Positions Buoy Mooring Line End A Mooring Line End B Spring End A Spring End B Buoy Buoy Mooring Line End A Mooring Line End B Spring End A Initial Position Parameter Value X 0 Z -4 X 0 Z -0.3 X 55 Z -39.992 X 0 Z 0 X 55.007071720764776 Z Anchored Initial Static Position Parameter Value X -18.2582464558261 Z -0.286319680053822 New Initial Position Parameter Value X 18.2582464558261 Z -4 X 0 Z -0.3 X 73.2582464558261 Z -39.992 X 0 Z 0 85

Spring End B X 73.265318176590878 Z Anchored Table 31: Test 2 Model System 1 Static Results Position and Water Velocity for Buoy Center Buoy Mooring Line End A Mooring Line End B Parameter Value X mbs 0 Y mbs 0 Z mbs -0.283615628977918 U mbs 0.510983432209259 m/s Dry Length 0.0137 m Portion Wet (PW) 0.977166667 Buoyancy 1083.45630720924 N Weight 917.769272165389 N Drag 44.281973000194618 N EndF ma 166.587673704267 N EndT ma 166.58576838051 N alpha ma 83.887094708783991 deg EndF mb 31.8209105716316 N EndT mb 31.8209091382278 N alpha mb 0 deg 12 10 Dynamic End A Force for model and scaled prototype results EndF dmasc =(EndF dpa /sc 3 ) EndF dmaes DynamicEndAForce(N) 8 6 4 2 0-2 0 2 4 6 8 10 12 14 16 Figure 44: Test 1 End A Force 86

Dynamic End A Tension (N) Dynamic End A Tension (N) 4 3 2 Dynamic End A Tension for scaled prototype results 1-2 0 2 4 6 8 10 12 14 16 Dynamic End A Tension for model results 15 10 5 0-2 0 2 4 6 8 10 12 14 16 Figure 45: Test 1 End A Tension Separate 0.57 0.56 Dynamic End B Force for model and scaled prototype results EndF dmbsc =(EndF dpb /(sc 3 )) EndF dmbes DynamicEndBForce(N) 0.55 0.54 0.53 0.52 0.51 0.5 0.49-2 0 2 4 6 8 10 12 14 16 Figure 46: Test 1 End B Force 87

12 Dynamic Model End A Force for the 2 different methods of spring solving EndF dma 10 EndF dmaes DynamicEndAForce(N) 8 6 4 2 0-2 0 2 4 6 8 10 12 14 16 Figure 47: Test 1 End A Force Proof Dynamic Model End A Tension for the 2 different methods of spring solving 12 Te dma 10 Te dmaes DynamicEndATension(N) 8 6 4 2 0-2 0 2 4 6 8 10 12 14 16 Figure 48: Test 1 End A Tension Proof 88

Dynamic End A Tension (N) Dynamic End A Tension (N) 15 10 5 Dynamic Model End A Tension using estimated tension method 0-2 0 2 4 6 8 10 12 14 16 Dynamic Model End A Tension using Orcaflex obtained tension method 8 6 4 2 0-2 0 2 4 6 8 10 12 14 16 Figure 49: Test 1 End A Tension Separate Proof 0.57 0.56 Dynamic Model End B Force for the 2 different methods of spring solving EndF dmb EndF dmbes DynamicEndBForce(N) 0.55 0.54 0.53 0.52 0.51 0.5 0.49-2 0 2 4 6 8 10 12 14 16 Figure 50: Test 1 End B Force Proof 89

Dynamic Model End B Tension for the 2 different methods of spring solving 0.57 0.56 Te dmb Te dmbes DynamicEndBTension(N) 0.55 0.54 0.53 0.52 0.51 0.5 0.49-2 0 2 4 6 8 10 12 14 16 Figure 51: Test 1 End B Tension Proof Fluid Incidence Angle at End A for the 2 different methods of spring solving 90 80 70 FluidIncidenceAngle(deg) 60 50 40 30 20 10 alpha dpa alpha dmaes 0-2 0 2 4 6 8 10 12 14 16 Figure 52: Test 1 End A Fluid Incidence Angle Proof 90

Dynamic Model X-Direction Motion of End A for the 2 different methods of spring solving 0.1 0 X dma X dmaes -0.1-0.2 X(m) -0.3-0.4-0.5-0.6-0.7-2 0 2 4 6 8 10 12 14 16 Figure 53: Test 1 End A X Direction Motion Proof Dynamic Model Z-Direction Motion of End A for the 2 different methods of spring solving -0.1-0.15-0.2 Z dma Z dmaes -0.25-0.3 Z(m) -0.35-0.4-0.45-0.5-0.55-0.6-2 0 2 4 6 8 10 12 14 16 Figure 54: Test 1 End A Z Direction Motion Proof 91

600 500 Dynamic End A Force for model and scaled prototype results EndF dmasc =(EndF dpa /sc 3 ) EndF dmaes DynamicEndAForce(N) 400 300 200 100 0-5 0 5 10 15 20 25 30 35 Figure 55: Test 2 End A Force Dynamic End A Tension (N) Dynamic End A Tension (N) 250 200 150 Dynamic End A Tension for scaled prototype results 100-5 0 5 10 15 20 25 30 35 Dynamic End A Tension for model results 600 400 200 0-5 0 5 10 15 20 25 30 35 Figure 56: Test 2 End A Tension Separate 92

36.5 36 35.5 Dynamic End B Force for model and scaled prototype results EndF dmbsc =(EndF dpb /(sc 3 )) EndF dmbes DynamicEndBForce(N) 35 34.5 34 33.5 33 32.5 32 31.5-5 0 5 10 15 20 25 30 35 Figure 57: Test 2 End B Force 600 Dynamic Model End A Force for the 2 different methods of spring solving EndF dma 500 EndF dmaes DynamicEndAForce(N) 400 300 200 100 0-5 0 5 10 15 20 25 30 35 Figure 58: Test 2 End A Force Proof 93

Dynamic Model End A Tension for the 2 different methods of spring solving 600 Te dma 500 Te dmaes DynamicEndATension(N) 400 300 200 100 0-5 0 5 10 15 20 25 30 35 Figure 59: Test 2 End A Tension Proof Dynamic End A Tension (N) Dynamic End A Tension (N) 600 400 200 Dynamic Model End A Tension using estimated tension method 0-5 0 5 10 15 20 25 30 35 Dynamic Model End A Tension using Orcaflex obtained tension method 600 400 200 0-5 0 5 10 15 20 25 30 35 Figure 60: Test 2 End A Tension Separate Proof 94

37 Dynamic Model End B Force for the 2 different methods of spring solving EndF dmb 36 EndF dmbes DynamicEndBForce(N) 35 34 33 32 31-5 0 5 10 15 20 25 30 35 Figure 61: Test 2 End B Force Proof Dynamic Model End B Tension for the 2 different methods of spring solving 37 Te dmb 36 Te dmbes DynamicEndBTension(N) 35 34 33 32 31-5 0 5 10 15 20 25 30 35 Figure 62: Test 2 End B Tension Proof 95

Fluid Incidence Angle at End A for the 2 different methods of spring solving 90 80 70 FluidIncidenceAngle(deg) 60 50 40 30 20 10 alpha dpa alpha dmaes 0-5 0 5 10 15 20 25 30 35 Figure 63: Test 2 End A Fluid Incidence Angle Proof Dynamic Model X-Direction Motion of End A for the 2 different methods of spring solving 0.5 X dma 0 X dmaes -0.5 X(m) -1-1.5-2 -2.5-3 -5 0 5 10 15 20 25 30 35 Figure 64: Test 2 End A X Direction Motion Proof 96

Dynamic Model Z-Direction Motion of End A for the 2 different methods of spring solving 0 Z dma -0.5 Z dmaes -1 Z(m) -1.5-2 -2.5-5 0 5 10 15 20 25 30 35 Figure 65: Test 2 End A Z Direction Motion Proof A.17 Model System 2 Simulation Table 32: Test 3 Model System 2 Positions Buoy Mooring Line End A Mooring Line End B Spring End A Spring End B Initial Position Parameter Value X 0 Z 0 X 0 Z -0.15 X 20 Z -9.998 X 0 Z 0 X 20.004313365831343 Z Anchored Initial Static Position Parameter Value X -0.170331982746872 97

Buoy Z -0.104954530452441 New Initial Position Buoy Mooring Line End A Mooring Line End B Spring End A Spring End B Parameter Value X 0.170331982746872 Z 0 X 0 Z -0.15 X 19.8296680172531 Z -9.998 X 0 Z 0 X 19.8339813830845 Z Anchored Table 33: Test 3 Model System 2 Static Results Position and Water Velocity for Buoy Center Buoy Mooring Line End A Mooring Line End B Parameter Value X mbs 0 Y mbs 0 Z mbs -0.104954530452441 U mbs 0.256400706700896 m/s Dry Length 0.045 m Portion Wet (PW) 0.85 Buoyancy 117.8071836924890 N Weight 114.721158996157 N Drag 2.758228451679279 N EndF ma 3.19956322280649 N EndT ma 3.19952021996218 N alpha ma 73.325234321773991 deg EndF mb 1.11774897462362 N EndT mb 1.117748966145 N alpha mb 0 deg 98

Table 34: Test 4 Model System 2 Positions Buoy Mooring Line End A Mooring Line End B Spring End A Spring End B Buoy Buoy Mooring Line End A Mooring Line End B Spring End A Spring End B Initial Position Parameter Value X 0 Z 0 X 0 Z -0.6 X 70 Z -39.992 X 0 Z 0 X 70.015926273838804 Z Anchored Initial Static Position Parameter Value X -9.31931719257251 Z -0.419817800886225 New Initial Position Parameter Value X 9.31931719257251 Z 0 X 0 Z -0.6 X 79.3193171925725 Z -39.992 X 0 Z 0 X 79.3352434664113 Z Anchored 99

Table 35: Test 4 Model System 2 Static Results Position and Water Velocity for Buoy Center Buoy Mooring Line End A Mooring Line End B Parameter Value X mbs 0 Y mbs 0 Z mbs -0.419817800886225 U mbs 0.510115413224024 m/s Dry Length 0.1802 m Portion Wet (PW) 0.849833 Buoyancy 7538.179725357622 N Weight 7342.154177617312 N Drag 176.5266209074739 N EndF ma 204.768170582766 N EndT ma 204.765418310353 N alpha ma 73.324914756910005 deg EndF mb 71.5354507701994 N EndT mb 71.5354501339077 N alpha mb 0 deg 18 16 14 Dynamic End A Force for model and scaled prototype results EndF dmasc =(EndF dpa /sc 3 ) EndF dmaes DynamicEndAForce(N) 12 10 8 6 4 2 0-2 0 2 4 6 8 10 12 14 16 Figure 66: Test 3 End A Force 100

Dynamic End A Tension (N) Dynamic End A Tension (N) 8 6 4 2 Dynamic End A Tension for scaled prototype results 0-2 0 2 4 6 8 10 12 14 16 Dynamic End A Tension for model results 20 15 10 5 0-2 0 2 4 6 8 10 12 14 16 Figure 67: Test 3 End A Tension Separate 3 Dynamic End B Force for model and scaled prototype results EndF dmbsc =(EndF dpb /(sc 3 )) 2.5 EndF dmbes DynamicEndBForce(N) 2 1.5 1 0.5-2 0 2 4 6 8 10 12 14 16 Figure 68: Test 3 End B Force 101

20 18 16 Dynamic Model End A Force for the 2 different methods of spring solving EndF dma EndF dmaes DynamicEndAForce(N) 14 12 10 8 6 4 2 0-2 0 2 4 6 8 10 12 14 16 Figure 69: Test 3 End A Force Proof Dynamic Model End A Tension for the 2 different methods of spring solving 20 18 16 Te dma Te dmaes DynamicEndATension(N) 14 12 10 8 6 4 2 0-2 0 2 4 6 8 10 12 14 16 Figure 70: Test 3 End A Tension Proof 102

Dynamic End A Tension (N) Dynamic End A Tension (N) 20 15 10 5 Dynamic Model End A Tension using estimated tension method 0-2 0 2 4 6 8 10 12 14 16 Dynamic Model End A Tension using Orcaflex obtained tension method 20 15 10 5 0-2 0 2 4 6 8 10 12 14 16 Figure 71: Test 3 End A Tension Separate Proof 2.2 Dynamic Model End B Force for the 2 different methods of spring solving EndF dmb 2 EndF dmbes DynamicEndBForce(N) 1.8 1.6 1.4 1.2 1 0.8-2 0 2 4 6 8 10 12 14 16 Figure 72: Test 3 End B Force Proof 103

Dynamic Model End B Tension for the 2 different methods of spring solving 2.2 Te dmb 2 Te dmbes DynamicEndBTension(N) 1.8 1.6 1.4 1.2 1 0.8-2 0 2 4 6 8 10 12 14 16 Figure 73: Test 3 End B Tension Proof Fluid Incidence Angle at End A for the 2 different methods of spring solving 90 80 FluidIncidenceAngle(deg) 70 60 50 40 30 20 alpha dpa alpha dmaes 10-2 0 2 4 6 8 10 12 14 16 Figure 74: Test 3 End A Fluid Incidence Angle Proof 104

Dynamic Model X-Direction Motion of End A for the 2 different methods of spring solving 0.1 0-0.1 X dma X dmaes -0.2-0.3 X(m) -0.4-0.5-0.6-0.7-0.8-0.9-2 0 2 4 6 8 10 12 14 16 Figure 75: Test 3 End A X Direction Motion Proof Dynamic Model Z-Direction Motion of End A for the 2 different methods of spring solving -0.2-0.22-0.24 Z dma Z dmaes -0.26-0.28 Z(m) -0.3-0.32-0.34-0.36-0.38-0.4-2 0 2 4 6 8 10 12 14 16 Figure 76: Test 3 End A Z Direction Motion Proof 105

800 700 Dynamic End A Force for model and scaled prototype results EndF dmasc =(EndF dpa /sc 3 ) EndF dmaes DynamicEndAForce(N) 600 500 400 300 200 100 0-5 0 5 10 15 20 25 30 35 Figure 77: Test 4 End A Force Dynamic End A Tension (N) Dynamic End A Tension (N) 400 300 200 100 Dynamic End A Tension for scaled prototype results 0-5 0 5 10 15 20 25 30 35 Dynamic End A Tension for model results 800 600 400 200 0-5 0 5 10 15 20 25 30 35 Figure 78: Test 4 End A Tension Separate 106

200 180 160 Dynamic End B Force for model and scaled prototype results EndF dmbsc =(EndF dpb /(sc 3 )) EndF dmbes DynamicEndBForce(N) 140 120 100 80 60 40 20-5 0 5 10 15 20 25 30 35 Figure 79: Test 4 End B Force 800 700 Dynamic Model End A Force for the 2 different methods of spring solving EndF dma EndF dmaes DynamicEndAForce(N) 600 500 400 300 200 100 0-5 0 5 10 15 20 25 30 35 Figure 80: Test 4 End A Force Proof 107

Dynamic Model End A Tension for the 2 different methods of spring solving 800 700 Te dma Te dmaes DynamicEndATension(N) 600 500 400 300 200 100 0-5 0 5 10 15 20 25 30 35 Figure 81: Test 4 End A Tension Proof Dynamic End A Tension (N) Dynamic End A Tension (N) 800 600 400 200 Dynamic Model End A Tension using estimated tension method 0-5 0 5 10 15 20 25 30 35 Dynamic Model End A Tension using Orcaflex obtained tension method 800 600 400 200 0-5 0 5 10 15 20 25 30 35 Figure 82: Test 4 End A Tension Separate Proof 108

140 130 Dynamic Model End B Force for the 2 different methods of spring solving EndF dmb EndF dmbes 120 DynamicEndBForce(N) 110 100 90 80 70 60 50-5 0 5 10 15 20 25 30 35 Figure 83: Test 4 End B Force Proof Dynamic Model End B Tension for the 2 different methods of spring solving 140 130 Te dmb Te dmbes 120 DynamicEndBTension(N) 110 100 90 80 70 60 50-5 0 5 10 15 20 25 30 35 Figure 84: Test 4 End B Tension Proof 109

Fluid Incidence Angle at End A for the 2 different methods of spring solving 90 80 FluidIncidenceAngle(deg) 70 60 50 40 30 20 alpha dpa alpha dmaes 10-5 0 5 10 15 20 25 30 35 Figure 85: Test 4 End A Fluid Incidence Angle Proof Dynamic Model X-Direction Motion of End A for the 2 different methods of spring solving 0.5 0 X dma X dmaes -0.5-1 X(m) -1.5-2 -2.5-3 -3.5-5 0 5 10 15 20 25 30 35 Figure 86: Test 4 End A X Direction Motion Proof 110

Dynamic Model Z-Direction Motion of End A for the 2 different methods of spring solving -0.8-0.9 Z dma Z dmaes -1-1.1 Z(m) -1.2-1.3-1.4-1.5-1.6-5 0 5 10 15 20 25 30 35 Figure 87: Test 4 End A Z Direction Motion Proof A.18 Model System 3 Simulation Table 36: Test 5 Model System 3 Positions Buoy Mooring Line End A Mooring Line End B Spring End A Spring End B Initial Position Parameter Value X 0 Z -2.5 X 0.078054324466998 Z 0 X 21 Z -9.998 X 0 Z 0 X 21.018756277244572 Z Anchored Initial Static Position Parameter Value X -1.27889462651261 111

Buoy Z -0.573561896038897 New Initial Position Buoy Mooring Line End A Mooring Line End B Spring End A Spring End B Parameter Value X 1.27889462651261 Z -2.5 X 0.078054324466998 Z 0 X 22.27889462651261 Z -9.998 X 0 Z 0 X 22.297650903757184 Z Anchored Table 37: Test 5 Model System 3 Static Results Position and Water Velocity for Buoy Center Buoy Mooring Line End A Mooring Line End B Parameter Value X mbs 0 Y mbs 0 Z mbs -0.573561896038897 U mbs 0.256400706700896 m/s Dry Length 0.0023 m Portion Wet (PW) 0.996 Buoyancy 10.1799323445 N Weight 4.8447228992 N Thrust 4.590671720096203 N EndF ma 6.83581472274654 N EndT ma 6.83577471160082 N alpha ma 47.616165995113988 deg EndF mb 4.85425163684623 N EndT mb 4.85425163489986 N alpha mb 0 deg 112

Table 38: Test 6 Model System 3 Positions Buoy Mooring Line End A Mooring Line End B Spring End A Spring End B Buoy Buoy Mooring Line End A Mooring Line End B Spring End A Spring End B Initial Position Parameter Value X 0 Z -10 X 0.312217297867992 Z 0 X 21 Z -39.992 X 0 Z 0 X 21.069253946749188 Z Anchored Initial Static Position Parameter Value X -68.1166017142767 Z -2.29419643931475 New Initial Position Parameter Value X 68.1166017142767 Z -10 X 0.312217297867992 Z 0 X 89.1166017142767 Z -39.992 X 0 Z 0 X 89.1858556610259 Z Anchored 113

Table 39: Test 6 Model System 3 Static Results Position and Water Velocity for Buoy Center Buoy Mooring Line End A Mooring Line End B Parameter Value X mbs 0 Y mbs 0 Z mbs -2.29419644098307 U mbs 0.512801413401792 m/s Dry Length 0.0094 m Portion Wet (PW) 0.9959 Buoyancy 651.4502568270513 N Weight 310.0622655488244 N Thrust 293.8029900861570 N EndF ma 437.480984417527 N EndT ma 437.478423591486 N alpha ma 47.614915284629006 deg EndF mb 310.670452796363 N EndT mb 310.670452650294 N alpha mb 0 deg 14 12 Dynamic End A Force for model and scaled prototype results EndF dmasc =(EndF dpa /sc 3 ) EndF dmaes DynamicEndAForce(N) 10 8 6 4 2-2 0 2 4 6 8 10 12 14 16 Figure 88: Test 5 End A Force 114

Dynamic End A Tension (N) Dynamic End A Tension (N) 10 8 6 4 Dynamic End A Tension for scaled prototype results 2-2 0 2 4 6 8 10 12 14 16 Dynamic End A Tension for model results 15 10 5 0-2 0 2 4 6 8 10 12 14 16 Figure 89: Test 5 End A Tension Separate 10 9 8 Dynamic End B Force for model and scaled prototype results EndF dmbsc =(EndF dpb /(sc 3 )) EndF dmbes DynamicEndBForce(N) 7 6 5 4 3 2 1 0-2 0 2 4 6 8 10 12 14 16 Figure 90: Test 5 End B Force 115

14 Dynamic Model End A Force for the 2 different methods of spring solving EndF dma 12 EndF dmaes DynamicEndAForce(N) 10 8 6 4 2-2 0 2 4 6 8 10 12 14 16 Figure 91: Test 5 End A Force Proof Dynamic Model End A Tension for the 2 different methods of spring solving 14 Te dma 12 Te dmaes DynamicEndATension(N) 10 8 6 4 2-2 0 2 4 6 8 10 12 14 16 Figure 92: Test 5 End A Tension Proof 116

Dynamic End A Tension (N) Dynamic End A Tension (N) 15 10 5 Dynamic Model End A Tension using estimated tension method 0-2 0 2 4 6 8 10 12 14 16 Dynamic Model End A Tension using Orcaflex obtained tension method 15 10 5 0-2 0 2 4 6 8 10 12 14 16 Figure 93: Test 5 End A Tension Separate Proof 6.5 Dynamic Model End B Force for the 2 different methods of spring solving EndF dmb 6 EndF dmbes DynamicEndBForce(N) 5.5 5 4.5 4-2 0 2 4 6 8 10 12 14 16 Figure 94: Test 5 End B Force Proof 117

Dynamic Model End B Tension for the 2 different methods of spring solving 6.5 Te dmb 6 Te dmbes DynamicEndBTension(N) 5.5 5 4.5 4-2 0 2 4 6 8 10 12 14 16 Figure 95: Test 5 End B Tension Proof Fluid Incidence Angle at End A for the 2 different methods of spring solving 90 80 70 FluidIncidenceAngle(deg) 60 50 40 30 20 10 alpha dpa alpha dmaes 0-2 0 2 4 6 8 10 12 14 16 Figure 96: Test 5 End A Fluid Incidence Angle Proof 118

Dynamic Model X-Direction Motion of End A for the 2 different methods of spring solving 0.2 0.15 X dma X dmaes 0.1 0.05 X(m) 0-0.05-0.1-0.15-0.2-0.25-2 0 2 4 6 8 10 12 14 16 Figure 97: Test 5 End A X Direction Motion Proof Dynamic Model Z-Direction Motion of End A for the 2 different methods of spring solving -0.5-0.55-0.6 Z dma Z dmaes -0.65-0.7 Z(m) -0.75-0.8-0.85-0.9-0.95-1 -2 0 2 4 6 8 10 12 14 16 Figure 98: Test 5 End A Z Direction Motion Proof 119

800 700 Dynamic End A Force for model and scaled prototype results EndF dmasc =(EndF dpa /sc 3 ) EndF dmaes DynamicEndAForce(N) 600 500 400 300 200 100-5 0 5 10 15 20 25 30 35 Figure 99: Test 6 End A Force Dynamic End A Tension (N) Dynamic End A Tension (N) 600 500 400 300 Dynamic End A Tension for scaled prototype results 200-5 0 5 10 15 20 25 30 35 Dynamic End A Tension for model results 800 600 400 200 0-5 0 5 10 15 20 25 30 35 Figure 100: Test 6 End A Tension Separate 120

700 600 Dynamic End B Force for model and scaled prototype results EndF dmbsc =(EndF dpb /(sc 3 )) EndF dmbes DynamicEndBForce(N) 500 400 300 200 100 0-5 0 5 10 15 20 25 30 35 Figure 101: Test 6 End B Force 800 Dynamic Model End A Force for the 2 different methods of spring solving EndF dma 700 EndF dmaes DynamicEndAForce(N) 600 500 400 300 200 100-5 0 5 10 15 20 25 30 35 Figure 102: Test 6 End A Force Proof 121

Dynamic Model End A Tension for the 2 different methods of spring solving 800 Te dma 700 Te dmaes DynamicEndATension(N) 600 500 400 300 200 100-5 0 5 10 15 20 25 30 35 Figure 103: Test 6 End A Tension Proof Dynamic End A Tension (N) Dynamic End A Tension (N) 800 600 400 200 Dynamic Model End A Tension using estimated tension method 0-5 0 5 10 15 20 25 30 35 Dynamic Model End A Tension using Orcaflex obtained tension method 800 600 400 200 0-5 0 5 10 15 20 25 30 35 Figure 104: Test 6 End A Tension Separate Proof 122

420 400 Dynamic Model End B Force for the 2 different methods of spring solving EndF dmb EndF dmbes 380 DynamicEndBForce(N) 360 340 320 300 280 260 240-5 0 5 10 15 20 25 30 35 Figure 105: Test 6 End B Force Proof Dynamic Model End B Tension for the 2 different methods of spring solving 420 400 Te dmb Te dmbes 380 DynamicEndBTension(N) 360 340 320 300 280 260 240-5 0 5 10 15 20 25 30 35 Figure 106: Test 6 End B Tension Proof 123

Fluid Incidence Angle at End A for the 2 different methods of spring solving 90 80 70 FluidIncidenceAngle(deg) 60 50 40 30 20 10 alpha dpa alpha dmaes 0-5 0 5 10 15 20 25 30 35 Figure 107: Test 6 End A Fluid Incidence Angle Proof Dynamic Model X-Direction Motion of End A for the 2 different methods of spring solving 0.8 0.6 X dma X dmaes 0.4 0.2 X(m) 0-0.2-0.4-0.6-0.8-1 -5 0 5 10 15 20 25 30 35 Figure 108: Test 6 End A X Direction Motion Proof 124

Dynamic Model Z-Direction Motion of End A for the 2 different methods of spring solving -2-2.2-2.4 Z dma Z dmaes -2.6-2.8 Z(m) -3-3.2-3.4-3.6-3.8-4 -5 0 5 10 15 20 25 30 35 Figure 109: Test 6 End A Z Direction Motion Proof 125

A.19 Model System 1 Non-Dimensional Scaling Parameters Table 40: Test 1 Non-Dimensional Parameters Comparison Papazoglou and Mavrakos Non-Dimensional Scaling Parameters Prototype Model Parameter Value Parameter Value L p /D p 2.5 L m /D m 2.5 d p /L p 0.0001 d m /L m 0.0001 w p L p /EndT pa 2.0525548702 w m L m /EndT ma 2.06063397540 E p A p / EndT pa 2434.3621747 EA eq / EndT ma 2434.36217475 a p /d p 50 a m /d m 50 L 2 p omega 2 p m p / 2.0930234792 L 2 m omega 2 m m m / 2.10126187373 pb / 0.8277339565 pw mb / 0.82773395656 mw a p /d pb 0.8333333333 a m /d mb 0.83333333333 Table 41: Test 2 Non-Dimensional Parameters Comparison Papazoglou and Mavrakos Non-Dimensional Scaling Parameters Prototype Model Parameter Value Parameter Value L p /D p 2.5 L m /D m 2.5 d p /L p 0.0001 d m /L m 0.0001 w p L p /EndT pa 2.0525548702 w m L m /EndT ma 2.06063397540 E p A p / EndT pa 2434.36217475 EA eq / EndT ma 2434.36217475 a p /d p 50 a m /d m 50 L 2 p omega 2 p m p / 2.09302347921 L 2 m omega 2 m m m / 2.10126187373 pb / 0.82773395656 pw mb / 0.82773395656 mw a p /d pb 0.83333333333 a m /d mb 0.83333333333 126

A.20 Model System 2 Non-Dimensional Scaling Parameters Table 42: Test 3 Non-Dimensional Parameters Comparison Papazoglou and Mavrakos Non-Dimensional Scaling Parameters Prototype Model Parameter Value Parameter Value L p /D p 2.5 L m /D m 2.5 d p /L p 0.0001 d m /L m 0.0001 w p L p /EndT pa 1.66858204922 w m L m /EndT ma 1.67514979074 E p A p / EndT pa 1978.96440430 EA eq / EndT ma 1978.96440430 a p /d p 50 a m /d m 50 L 2 p omega 2 p m p / 1.7014801682 L 2 m omega 2 m m m / 1.70817740078 pb / 0.82773395656 pw mb / 0.82773395656 mw a p /d pb 0.41666666666 a m /d mb 0.41666666666 Table 43: Test 4 Non-Dimensional Parameters Comparison Papazoglou and Mavrakos Non-Dimensional Scaling Parameters Prototype Model Parameter Value Parameter Value L p /D p 2.5 L m /D m 2.5 d p /L p 0.0001 d m /L m 0.0001 w p L p /EndT pa 1.66858204922 w m L m /EndT ma 1.67514979074 E p A p / EndT pa 1978.96440430 EA eq / EndT ma 1978.96440430 a p /d p 50 a m /d m 50 L 2 p omega 2 p m p / 1.70148016827 L 2 m omega 2 m m m / 1.70817740078 pb / 0.82773395656 pw mb / 0.82773395656 mw a p /d pb 0.41666666666 a m /d mb 0.41666666666 127

A.21 Model System 3 Non-Dimensional Scaling Parameters Table 44: Test 5 Non-Dimensional Parameters Comparison Papazoglou and Mavrakos Non-Dimensional Scaling Parameters Prototype Model Parameter Value Parameter Value L p /D p 2.5 L m /D m 2.5 d p /L p 0.0001 d m /L m 0.0001 w p L p /EndT pa 0.77958713788 w m L m /EndT ma 0.78265568751 E p A p / EndT pa 924.602537022 EA eq / EndT ma 924.602537022 a p /d p 50 a m /d m 50 L 2 p omega 2 p m p / 0.79495764393 L 2 m omega 2 m m m / 0.79808669373 pb / 0.48690232739 pw mb / 0.48690232739 mw a p /d pb 0.21705157145 a m /d mb 0.21705157145 Table 45: Test 6 Non-Dimensional Parameters Comparison Papazoglou and Mavrakos Non-Dimensional Scaling Parameters Prototype Model Parameter Value Parameter Value L p /D p 2.5 L m /D m 2.5 d p /L p 0.0001 d m /L m 0.0001 w p L p /EndT pa 0.77958713788 w m L m /EndT ma 0.78265568751 E p A p / EndT pa 924.602537022 EA eq / EndT ma 924.602537022 a p /d p 50 a m /d m 50 L 2 p omega 2 p m p / 0.79495764393 L 2 m omega 2 m m m / 0.79808669373 pb / 0.48690232739 pw mb / 0.48690232739 mw a p /d pb 0.21705157145 a m /d mb 0.21705157145 128

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