mainreport numerical study of the weptos single rotor · 2 numericalstudyoftheweptossinglerotor...

Main Report(Master Thesis, spring semester 2012)

Numerical study of the WEPTOS Single Rotor

Date of report: 13/08/2012

Author: Stephane RapucStudent-ID: s101701 E-Mail: [email protected]

Instructors: Harry B. Bingham, DTUJens Peter Kofoed

mailto:Stephane Rapuc<[email protected]>?subject=Main Report: Numerical study of the WEPTOS Single Rotor

Numerical study of the WEPTOS Single Rotor 1

Abstract

This master thesis took place from the February, 27th until August, 13th 2012. The mainsubject covered by this report is a numerical study of the wave energy converter WEPTOS(see http://www.weptos.com/). The full converter device is display in Figure 1. However, ithas been chosen to focus this study on only a single WEPTOS floater. Indeed, in front of thecomplexity of this energy converter and considering the lack of numerically relevant experimentalmeasurements made during the tests in Spain (i.e. no record of the floaters pitch motions) ithas been considered that going further into the numerical model by considering the full devicewas not reliable. Moreover, as one may see, while reading this report, the numerical study ofa single floater already appears as very problematic and required the use of a fully non linearhydrodynamic model. This floater, very similar to the “Salter Duck” can be seen from Figure 1(i.e. the orange part on the side of each arm) and Figure 2.

Figure 1: Picture from the experimental tests run on the full WEPTOS prototype in Spain duringthe Summer/Autumn 2011.

Figure 2: Picture from the experimental tests run on the single WEPTOS floater in Aalborg Flumeduring this thesis, June 2012.

The study of this rotor has then been divided in three parts. At first it has been tried to run anumerical model and to compare it with the experimental measurement made by Arthur Pecherin 2010 (see report [1]). However, it appears that the tests run at that time were not adaptedfor a comparison with a numerical model. It has therefore, been choose to run new experimentsin the AAU wave flume, in order to get accurate measurement of the pitch motions. During thisexperiment, it have been chosen to mainly focus on small regular waves. Then from these newexperimental data it has been possible to run and properly compare numerical models. Thesenumerical models are based on the linear theory. They has been made through the use of :

2 Numerical study of the WEPTOS Single Rotor

• Rhinoceros 3D [6] for meshing the shape of the WEPTOS Rotor.

• WAMIT [9] to obtain the hydrodynamic coefficient (added mass, damping, exciting forces)in the frequency domain for a given angle of the WEPTOS rotor.

• Matlab [5] to evaluate from the WAMIT coefficient, in the time domain, the pitch motionsof the WEPTOS rotor.

However, it appears that considering the rotor shape, dimensions and amplitude of motionsthe numerical model needed to be improved by adding non-linear hydrodynamic coefficients.Even with these improvements, the numerical model results are not necessarily perfectly fittingthe experimental measurements. It seems that to be perfect it would be necessary to go evenfurther into the discretization of the hydrodynamic coefficients in terms of water level and rotorangle.

Finally, one should understand that the main point of this report has been to evaluate how tosimulate the WEPTOS rotor motions through the use of different numerical models. Thus thefocus has been made on comparing the numerical calculations and the WEPTOS experimentalmeasurements. Considering this, no study on the possible power take off or efficiency of therotor have been done in this report.

If one required any of the Matlab files and/or WAMIT files used to make this study, feel freeto contact me by email at [email protected]


Preface and Acknowledgment

The present master thesis has been made and mainly written at Aalborg University (AAU),from end of February 2012 and August 2012. It comes as the conclusion of the Master of Sciencein Engineering Design and Applied Mechanics I undertook, at Copenhagen (DTU).

I would like to express my appreciation to my supervisors: Dr. Harry Bingham and Dr. JensPeter Kofoed for giving me the chance of doing this thesis and for helping all along with it.Special thanks to Dr. Harry Bingham for his time and availability, even though I was not basedat DTU during this thesis. Every time I asked for advice his tips appeared to be very helpful.I would also like to particularly thanks Dr. Jens Peter Kofoed for letting me use the Aalborguniversity’s wave flume as well as the prototype of the single WEPTOS floater.

Moreover, I would like to thank the numerous PhDs and all the employees from the AalborgWave Energy Research Group whom have been giving me very profitable advice and with whomI had interesting discussions during this 6 months of master thesis at AAU. In particular, Iwould like to thanks Andrew Zurkinden for his time and useful advices on the various issueI faced while I was trying to make running the numerical model. During this thesis Andrewalso gave me numerous very interesting documents that helped me to understand in details thedifferent theories necessary for this thesis. Furthermore, I thank Arthur Pecher for his helpduring the experimental tests and his availability all along these 6 months. Francesco Ferri hasalso been a very helpful person, always available when I needed some advice and whom helps meto understand more clearly some details on the methodology to pass from the frequency domainto the time domain.

Finally, my gratitude also goes to technicians from Aalborg wave laboratory whom have beenvery helpful and available when I was running the experimental tests.


Contents

List of Figures 9

List of Tables 15

1 Introduction 171.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2 Coordinate system definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Experimental study of the Weptos Rotor 192.1 Experimental Issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 Wave maker issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.2 Beach issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.3 Wave Maker specification . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.4 Flume specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.5 Rotor setup Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.5.1 Rotor Configuration . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.5.2 Still water level and Rotor rest angle . . . . . . . . . . . . . . . 222.2.5.3 Power Take Off (PTO) specification . . . . . . . . . . . . . . . . 22

2.2.6 Measurement setup specification . . . . . . . . . . . . . . . . . . . . . . . 252.2.6.1 Wave gauges setup . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.6.2 Potentiometer and Load cells measurement setup . . . . . . . . 252.2.6.3 Acquisition and amplification boxes . . . . . . . . . . . . . . . . 26

2.3 Mass and inertia measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.1 Theorical values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.2 Mass of the Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.3 Center of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.4 Inertia Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.5 Summary of the final values . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4 Wave tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4.1 List of the tests run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4.2 Evaluation of the wave measurement . . . . . . . . . . . . . . . . . . . . . 34

2.4.2.1 Regular wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4.2.2 Irregular wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4.3 Repeatability of the tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4.3.1 Regular wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4.3.2 Irregular wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5 Recorded pitch motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.5.1 Wave maker delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.5.2 Reflexion issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Linear study of the rotor 423.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1.1 Second Newton’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.1.2 Frequency domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.1.3 Time domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 WAMIT Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2.1 Geometric analysis of the system . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.1.1 Generalities on the geometric files . . . . . . . . . . . . . . . . . 44


3.2.1.2 Analysis of the main hole . . . . . . . . . . . . . . . . . . . . . . 443.2.2 Main WAMIT Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.2.1 Poten File.pot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2.2.2 Force File.frc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2.2.3 Geometric file File.gdf . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 WAMIT hydrodynamic coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4 Impulse response function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.5 Comparison with the experimental data . . . . . . . . . . . . . . . . . . . . . . . 48

3.5.1 Experimental and numerical Results . . . . . . . . . . . . . . . . . . . . . 483.5.2 Analysis of the results in the time domain . . . . . . . . . . . . . . . . . . 51

3.6 Limitation of the numerical model . . . . . . . . . . . . . . . . . . . . . . . . . . 523.6.1 Wave Amplitude and Non linearity . . . . . . . . . . . . . . . . . . . . . . 523.6.2 Large motion angle amplitude . . . . . . . . . . . . . . . . . . . . . . . . . 543.6.3 Slamming phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.6.4 Experimental wave reflection . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Solutions to improve and/or extend the numerical model 574.1 Addition of a friction moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.1.1 Details on the friction moment . . . . . . . . . . . . . . . . . . . . . . . . 574.1.2 Analysis of the RAO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2 Addition of a non linear hydrostatic restoring moment . . . . . . . . . . . . . . . 584.2.1 Details of the solutions used . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2.2 Analysis of the RAO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2.3 Analysis of the results in the time domain . . . . . . . . . . . . . . . . . . 61

4.3 Fully non linear numerical model . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.2 WAMIT hydrodynamic coefficients . . . . . . . . . . . . . . . . . . . . . . 634.3.3 Analysis of the RAO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.3.4 Analysis of the results in the time domain . . . . . . . . . . . . . . . . . . 67

4.4 Remaining limitation of the models . . . . . . . . . . . . . . . . . . . . . . . . . . 684.4.1 Discretization of the non-linear hydrodynamic coefficients . . . . . . . . . 684.4.2 Retardation function convolution . . . . . . . . . . . . . . . . . . . . . . . 694.4.3 Other phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5 Extension of the numerical model 715.1 Addition of a Power Take Off moment . . . . . . . . . . . . . . . . . . . . . . . . 71

5.1.1 Results for 6N pre-loading . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.1.1.1 Linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.1.1.2 Non Linear model . . . . . . . . . . . . . . . . . . . . . . . . . . 725.1.1.3 Analysis of the Results . . . . . . . . . . . . . . . . . . . . . . . 73

5.1.2 Results for 15N pre-loading . . . . . . . . . . . . . . . . . . . . . . . . . . 735.1.2.1 Linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.1.2.2 Non Linear model . . . . . . . . . . . . . . . . . . . . . . . . . . 745.1.2.3 Analysis of the Results . . . . . . . . . . . . . . . . . . . . . . . 75

5.2 Results for irregular waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.2.1 Without PTO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.2.2 With 6N pre-loaded PTO . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.2.3 Analysis of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6 Conclusion 816.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.2 Future possible study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81


References 82


List of Figures

1 Picture from the experimental tests run on the full WEPTOS prototype in Spainduring the Summer/Autumn 2011. . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Picture from the experimental tests run on the single WEPTOS floater in AalborgFlume during this thesis, June 2012. . . . . . . . . . . . . . . . . . . . . . . . . . 1

3 Scheme representing the different coordinate systems used during the calculations 18

4 Broken wave maker panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Fixed wave maker panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6 Rock beach without water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7 Rock beach with water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

8 Geomerty flume details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

9 Simple Scheme explaining the PTO principle. . . . . . . . . . . . . . . . . . . . . 23

10 Experimental measurement : Pitch, PTO 1 and PT0 2 representing the two loadsensor (on each side of the metallic wheel). This is done for the case of H=4cmand T=1.20s and PTO pre-tension of 6 N . . . . . . . . . . . . . . . . . . . . . . . 24

11 Simple scheme of the positioning of the wave gauges . . . . . . . . . . . . . . . . 26

12 Picture of the full setup used to measure the wave sequences . . . . . . . . . . . 26

13 Rotor setup use during all the wave states. . . . . . . . . . . . . . . . . . . . . . 27

14 Acquisition and amplification boxes used to acquire the measurement from thewave gauges, potentiometer and PTO load cells. . . . . . . . . . . . . . . . . . . 28

15 Inertia test with configuration W0. Initial angle 65◦. . . . . . . . . . . . . . . . . 30

16 Inertia test with configuration W5. Initial angle 90◦ . . . . . . . . . . . . . . . . 30

17 Numerical model results and experimental measurements of the inertia test forthe configuration configuration W0. The numerical model has been run for the 3couples of coefficients given in the Table 5 . . . . . . . . . . . . . . . . . . . . . . 31

18 Numerical model results and experimental measurements of the inertia test forthe configuration configuration W5. The numerical model has been run for the 3couples of coefficients given in the Table 5 . . . . . . . . . . . . . . . . . . . . . . 31

19 Scatter diagram of the error, for the configuration W0, of 1000 numerical modelsrun for 10x10 friction coefficients and for 10 inertias. . . . . . . . . . . . . . . . . 32

20 Scatter diagram of the error, for the configuration W5, of 1000 numerical modelsrun for 10x10 friction coefficients and for 10 inertias. . . . . . . . . . . . . . . . . 32

21 Scatter diagram of the mean error of both configurations run of 1000 numericalmodels run for 10x10 friction coefficients and for 10 inertias. . . . . . . . . . . . . 32

22 Time evolution of the friction moment for the case W0, with I22 =4.960E-02 andMfric = 1.800E − 03 ∗ Faxle + 0.01. . . . . . . . . . . . . . . . . . . . . . . . . . . 33

23 Time evolution of the friction moment for the case W5, with I22 =4.960E-02 andMfric = 1.800E − 03 ∗ Faxle + 0.01. . . . . . . . . . . . . . . . . . . . . . . . . . . 33

24 Non aligned wave elevation measurements for H=0.04m and T=1.00s. . . . . . . 35

25 Aligned wave elevation measurements, through the 1st crest after 10s, for H=0.04mand T=1.00s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

26 Non aligned wave elevation measurements for H=0.04m and T=1.00s. . . . . . . 36


27 Aligned wave elevation measurements, through the linear wave theory, for H=0.04mand T=1.00s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

28 Non aligned wave elevation measurements for WS10. . . . . . . . . . . . . . . . . 36

29 Manual aligned wave elevation measurements for WS10. . . . . . . . . . . . . . . 36

30 Manual averaged wave elevation measurements for WS10. . . . . . . . . . . . . . 37

31 AwaSys averaged wave elevation measurements for WS10. . . . . . . . . . . . . . 37

32 Full pitch motion recorded file from WaveLab. For H = 02 cm and T = 0.75 s. . 39










42 Shape of the rotor without the hole at 50.64 degrees, trimmed . . . . . . . . . . . 45

43 Shape of the rotor with the main hole at 50.64 degrees, trimmed . . . . . . . . . 45

44 Meshing of the rotor without the hole at 50.64 degrees, trimmed . . . . . . . . . 45

45 WAMIT added mass in function of the wave frequency and the number of panels.With the rotor at the rest angle for 107mm of water depth. . . . . . . . . . . . . 47

46 WAMIT and extended damping coefficient in function of the wave frequency andthe number of panels. With the rotor at the rest angle for 107mm of water depth. 47

47 WAMIT exciting force (absolute, real and imaginary part) in function of the wavefrequency and the number of panels. With the rotor at the rest angle for 107mmof water depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

48 RAO calculated by WAMIT in function of the wave frequency and the number ofpanels. With the rotor at the rest angle for 107mm of water depth. At this anglethe resonance frequency is about 0.87rad/s. . . . . . . . . . . . . . . . . . . . . . 47

49 Impulse response of the exciting force K55, as a function of the number of Panels 48

50 Comparison of the retardation function K55,rad, as a function of the number ofPanels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

51 Comparison of the experimental and numerical (in frequency and time domain)RAO. The results are given in terms of pitch amplitude degree for 1 cm waveamplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

52 Comparison between the numerical model and the experimental measurement.For H=0.02m and T=1.20s, considering a linear numerical model. . . . . . . . . . 51







58 Snapshot from video during the experiments at AAU. The condition are for reg-ular wave with a wave height of 8 cm . . . . . . . . . . . . . . . . . . . . . . . . . 53

59 Snapshot of the extreme down rotor position for H=0.05m and T=1.00 s . . . . . 54

60 Snapshot of the extreme up rotor position for H=0.05m and T=1.00 s . . . . . . 54

61 Evolution of the added mass at infinite frequency in function of the rotor pitchangle. Pitch equal zero degree is equivalent to the rest position. . . . . . . . . . . 55

62 Snapshot of the rotor slamming. Wave height 5cm, wave period 1s, regular waves 55

63 Comparison of the experimental and numerical (in frequency and time domain)RAO. The results are given in term of pitch amplitude [deg] for 1 cm wave height. 57

64 Linear and non-linear hydrostatic restoring moment in function of the rotor pitchangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

65 Comparison of the experimental and numerical (in frequency and time domain)RAO. The results are given in term of pitch amplitude [deg] for 1 cm wave height.The time domain calculation have been run with the linear model including a non-linear hydrostatic restoring moment and structural friction moment . . . . . . . . 60

66 Comparison between the numerical model and the experimental measurement.For H=0.03m and T=1.00s, considering a linear numerical model including non-linear hydrostatic restoring moment and the structural friction. . . . . . . . . . . 61








74 Meshing of the rotor at -10◦ (-60◦ compare to the rest position), trimmed . . . . 64

75 Meshing of the rotor at 10◦(-40◦ compare to the rest position), trimmed . . . . . 64

76 Meshing of the rotor at 160◦ (+120◦ compare to the rest position), trimmed . . . 64


77 Damping B55 in function of the wave frequency for the rotor angles from -120◦

to 0◦ compare to the rest position. . . . . . . . . . . . . . . . . . . . . . . . . . . 65

78 Damping B55 in function of the wave frequency for the rotor angles from 0◦ to+120◦ compare to the rest position. . . . . . . . . . . . . . . . . . . . . . . . . . 65

79 Exciting force X5 in function of the wave frequency for the rotor angles from -120◦

to 0◦ compare to the rest position. Note: the colors are identical to the one givenFigure 78 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

80 Exciting force X5 in function of the wave frequency for the rotor angles from 0◦

to +120◦ compare to the rest position. Note: the colors are identical to the onegiven Figure 78 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

81 K55,rad the radiation impulse-response function due to an impulsive velocity ofthe body, given for the rotor angles from -120◦ to 0◦ compare to the rest position. 65

82 K55,rad the radiation impulse-response function due to an impulsive velocity ofthe body, given for the rotor angles from 0◦ to +120◦ compare to the rest position. 65

83 K5,diff the diffraction impulse-response function, given for the rotor angles from-120◦ to 0◦ compare to the rest position. . . . . . . . . . . . . . . . . . . . . . . . 66

84 K5,diff the diffraction impulse-response function, given for the rotor angles from0◦ to +120◦ compare to the rest position. . . . . . . . . . . . . . . . . . . . . . . 66

85 Comparison of the experimental and numerical (in frequency and time domain)RAO. The results are given in term of pitch amplitude [deg] for 1 cm wave height.The time domain calculation have been run with full non-linear hydrodynamiccoefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

86 Comparison between the numerical model and the experimental measurement.For H=0.03m and T=1.00s, considering a fully non-linear numerical model in-cluding the structural friction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67






92 Evolution of the added mass at infinite frequency in function of the water levelfor the rotor at its rest position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

93 Comparison of the experimental and numerical (in frequency and time domain)RAO. The results are given in term of pitch amplitude [deg] for 1 cm wave height.For the linear numerical model considering a 15N pre-loaded PTO. . . . . . . . . 71


94 Comparison of the experimental and numerical (in frequency and time domain)RAO. The results are given in term of pitch amplitude [deg] for 1 cm wave height.For the fully non-linear numerical model considering a 6N pre-loaded PTO andthe structural friction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

95 Comparison of the experimental and numerical (in frequency and time domain)RAO. The results are given in term of pitch amplitude [deg] for 1 cm wave height.For the linear numerical model considering a 15N pre-loaded PTO. . . . . . . . . 74

96 Comparison of the experimental and numerical (in frequency and time domain)RAO. The results are given in term of pitch amplitude [deg] for 1 cm wave height.For the fully non-linear numerical model considering a 15N pre-loaded PTO andthe structural friction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

97 Wave State WS60 full calculation run for 150 seconds. Numerical Model : Linearhydrostatic - No PTO - No friction. . . . . . . . . . . . . . . . . . . . . . . . . . 76

98 Wave State WS60 zoom on result between 60s and 90s. Numerical Model : Linearhydrostatic - No PTO - No friction. . . . . . . . . . . . . . . . . . . . . . . . . . 76

99 Wave State WS60 full calculation run for 150 seconds. Numerical Model : Fullynon-linear hydrodynamic - No PTO - Including friction. . . . . . . . . . . . . . . 77

100 Wave State WS60 zoom on result between 60s and 90s. Numerical Model : Fullynon-linear hydrodynamic - No PTO - Including friction. . . . . . . . . . . . . . . 77

101 Wave State WS10 full calculation run for 150 seconds. Numerical Model : Linearhydrostatic - No PTO - No friction. . . . . . . . . . . . . . . . . . . . . . . . . . 77

102 Wave State WS10 zoom on result between 60s and 90s. Numerical Model : Linearhydrostatic - No PTO - No friction. . . . . . . . . . . . . . . . . . . . . . . . . . 77

103 Wave State WS10 full calculation run for 150 seconds. Numerical Model : Fullynon-linear hydrodynamic - No PTO - Including friction. . . . . . . . . . . . . . . 78

104 Wave State WS10 zoom on result between 60s and 90s. Numerical Model : Fullynon-linear hydrodynamic - No PTO - Including friction. . . . . . . . . . . . . . . 78

105 Wave State WS60 full calculation run for 150 seconds. Numerical Model : Linearhydrostatic - With 6N pre-loaded PTO - No friction. . . . . . . . . . . . . . . . . 78

106 Wave State WS60 zoom on result between 60s and 90s. Numerical Model : Linearhydrostatic - With 6N pre-loaded PTO - No friction. . . . . . . . . . . . . . . . . 78

107 Wave State WS60 full calculation run for 150 seconds. Numerical Model : Fullynon-linear hydrodynamic - With 6N pre-loaded PTO - Including friction. . . . . 79

108 Wave State WS60 zoom on result between 60s and 90s. Numerical Model : Fullynon-linear hydrodynamic - With 6N pre-loaded PTO - Including friction. . . . . 79

109 Wave State WS10 full calculation run for 150 seconds. Numerical Model : Linearhydrostatic - With 6N pre-loaded PTO - No friction. . . . . . . . . . . . . . . . . 79

110 Wave State WS10 zoom on result between 60s and 90s. Numerical Model : Linearhydrostatic - With 6N pre-loaded PTO - No friction. . . . . . . . . . . . . . . . . 79

111 Wave State WS10 full calculation run for 150 seconds. Numerical Model : Fullynon-linear hydrodynamic - With 6N pre-loaded PTO - Including friction. . . . . 79

112 Wave State WS10 zoom on result between 60s and 90s. Numerical Model : Fullynon-linear hydrodynamic - With 6N pre-loaded PTO - Including friction. . . . . 79


List of Tables

1 Summuary of the different PTO loading tested . . . . . . . . . . . . . . . . . . . 24

2 Center of gravity position evaluated through experiment . . . . . . . . . . . . . . 28

3 Rotor and setup mass details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Center of gravity position evaluated through experiment . . . . . . . . . . . . . . 29

5 Sum up of the different calculation run for the inertia . . . . . . . . . . . . . . . 31

6 Summary of the mass, center of gravity and inertia considered for the rotor. . . 33

7 Summary of the irregular wave state used . . . . . . . . . . . . . . . . . . . . . . 34

8 Summary of the delays, for each wave periods, between the wave gauges 1 and 2and between the wave gauges 2 and 3 . . . . . . . . . . . . . . . . . . . . . . . . 35

9 List of the WaveLab time series analysis on the irregular wave elevation measure-ments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

10 Comparison of the meshes in matter of hydrostatic stiffness and infinite added mass 48

11 Wave period used for the comparison between the linear numerical model and theexperimental one. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

12 Pitch amplitude experimentally obtained for the different regular wave states.Results given in degree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

13 Pitch amplitude experimentally obtained for the different regular wave statesnormalized by the wave amplitude (RAO). Results given in degree/cm. . . . . . . 50

14 Pitch amplitude numerical obtained for the different regular wave states. Resultsgiven in degree, for the linear numerical model. . . . . . . . . . . . . . . . . . . . 50

15 Relative pitch amplitude difference between the numerical model and the exper-imental measurement. Results given in terms of relative errors, for the linearnumerical model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

16 Wave frequency, length and number for all the wave periods run during the ex-periment. All the calculation have been done for a water depth h = 0.81 m . . . 53

17 Sum up of the linearity coefficient H/L for all the regular wave tests. . . . . . . . 54

18 Pitch amplitude numerical obtained for the different regular wave states. Resultsgiven in degree, for the linear numerical model considering the structural friction. 58

19 Relative pitch amplitude difference between the numerical model and the ex-perimental measurement.Results given in terms of relative errors, for the linearnumerical model considering the structural friction. . . . . . . . . . . . . . . . . . 58

20 Non linear restoring moment evaluated for 14 angles and 5 water depths aroundthe rest position (angle : 50.5◦ and depth : 107mm above axle). . . . . . . . . . . 59

21 Pitch amplitude numerical obtained for the different regular wave states. Resultsgiven in degree, for the linear numerical model implemented with a non-linearrestoring moment and considering the structural friction. . . . . . . . . . . . . . . 60

22 Relative pitch amplitude difference between the numerical model and the exper-imental measurement. Results given in terms of relative errors, for the linearnumerical model implemented with a non-linear restoring moment and consider-ing the structural friction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

23 Pitch amplitude numerical obtained for the different regular wave states. Re-sults given in degree, for the a full non-linear numerical model, considering thestructural friction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67


24 Relative pitch amplitude difference between the numerical model and the exper-imental measurement. Results given in terms of relative errors, for the a fullnon-linear numerical model, considering the structural friction. . . . . . . . . . . 67

25 Pitch amplitude numerical obtained for the different regular wave states. Resultsgiven in degree, for the linear numerical model considering a 6N pre-loaded PTO. 72

26 Relative pitch amplitude difference between the numerical model and the exper-imental measurement. Results given in terms of relative errors, for the linearnumerical model considering a 6N pre-loaded PTO. . . . . . . . . . . . . . . . . . 72

27 Pitch amplitude numerical obtained for the different regular wave states. Resultsgiven in degree, for the fully non-linear numerical model considering a 6N pre-loaded PTO and the structural friction. . . . . . . . . . . . . . . . . . . . . . . . 73

28 Relative pitch amplitude difference between the numerical model and the ex-perimental measurement. Results given in terms of relative errors, for the fullynon-linear numerical model considering a 6N pre-loaded PTO and the structuralfriction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

29 Pitch amplitude numerical obtained for the different regular wave states. Resultsgiven in degree, for the linear numerical model considering a 15N pre-loaded PTO. 74

30 Relative pitch amplitude difference between the numerical model and the exper-imental measurement. Results given in terms of relative errors, for the linearnumerical model considering a 15N pre-loaded PTO. . . . . . . . . . . . . . . . . 74

31 Pitch amplitude numerical obtained for the different regular wave states. Resultsgiven in degree, for the fully non-linear numerical model considering a 15N pre-loaded PTO and the structural friction. . . . . . . . . . . . . . . . . . . . . . . . 75

32 Relative pitch amplitude difference between the numerical model and the ex-perimental measurement. Results given in terms of relative errors, for the fullynon-linear numerical model considering a 15N pre-loaded PTO and the structuralfriction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75


1 Introduction

This report has for main purpose to compare the results of a numerical model against exper-imental measurements. This numerical model is based on the linear theory (i.e. linear waves,small wave amplitude compare to the rotor and small motions). Due to the specificities of therotor this model has been improved by introducing non-linear hydrostatic and hydrodynamicmoments. Some extra structural frictions (due to some friction inside the liaison between theaxle and the structure) and a Power Take Off (PTO) moment have also been implemented intothe numerical model. Indeed, one should keep in mind that considering a wave energy devicethe goal is to generate electricity by damping the system, thus this damping must be includedinto the calculation.

At first, it has been tried out to fit a numerical model with the measurement made by ArthurPecher in 2010 (see Report [1]). However, after a couple of months of calculations and im-provements of the numerical model it appeared that the wave used during these experimentswere too high to be either small compare to the rotor dimension or linear. Therefore, it hasbeen decided to run new experiments in the Aalborg University’s flume using more or less thesame setup than the one used in 2010. This time mainly small waves have been considered andthe experiment has been focused on regular waves. Moreover, the rotor has been fixed to theaxle. This fixation does not allow to have the same PTO system as the one installed on the fullWEPTOS prototype, where the rotor is damped only when it goes up. However, fixing the rotorto its axle allows to measure with precision the pitch motion of the rotor when it goes up andwhen it goes down. A reliable pitch motion measurement is very important to ensure a goodpossible comparison between experimental measurements and a numerical model.

Since it has been necessary to go to the laboratory, tests to experimentally determine themass, center of gravity and inertia have been run as well.

1.1 Generalities

In this report the gravity has been taken constant, g = 9.806 m · s−2, as well as the waterdensity, ρ = 1035kg ·m−3. All the data given in this report have been given in the InternationalUnits System. Among others, the angles have been given in degrees (◦) and the moment innewton meter (Nm).

1.2 Coordinate system definitions

Due to the use of different softwares and due to different needs 3 coordinate systems havebeen used:

• Body-fixed coordinate system is centered in O and defined through (x, y, z). This coordi-nate system has been used has reference system, as it is centered on the axle of rotationof the rotor with the z-axis vertical and x-axis directing face to the waves.

• Global coordinate system is centered in O′ and defined through (x, y, z). This is the globalWamit coordinate system. Under WAMIT, in this coordinate system, the water line isat z=0. Therefore the body-fixed coordinate system need to be adjust by the use of theparameters XBODY (1 : 4) (i.e. XBODY (3) = −107, and XBODY (1) = XBODY (2) =XBODY (4) = 0).

• SolidWork system: (x0, y0, z0) this is the coordinate system that has been used to numer-ically evaluate center of mass and the inertia. However, this coordinate system has neverbeen directly used (neither in the report nor the calculation). However, if one would liketo check the coordinates/values taken from the SolidWork ”.pdf” file, he should be awarethat these values are given in this latter coordinate system.


Figure 3: Scheme representing the different coordinate systems used during the calculations

Note : All the pitch angles have been given in the Body-fixed coordinate system, with 0◦ whenthe rotor is aligned with the x-axis.


2 Experimental study of the Weptos Rotor

Experiments have been run on the single WEPTOS floater. These experiments have beenrun with the same setup that the ones run by A. Pecher, J. P. Kofoed and T. Marchalot (seereport [1]). This time only the configuration “W5” has been considered in this report. Thesetup (i.e. rotor, structure, PTO system) has been conserved. The main idea behind this newexperiments is to generate wave conditions that would be closer to the linear model hypothesis.This has been done through the use of smaller wave amplitude, and also by focusing on the testson regular waves that are easier to compare with a numerical calculation. Furthermore, fromthese experiments it is possible to see under which conditions the linear numerical model fits theexperimental measurements and to determine the limits of the model. Finally the mass, inertiaand hydrostatic restoring moments have also been re-evaluated experimentally.

2.1 Experimental Issue

As for most experiments, before being able to run wave sequences in good conditions differentchallenges had to be faced. Among them the wave maker had to be fixed and the beach volumehad to be increased. More details about these 2 issues are given in this section.

2.1.1 Wave maker issue

When the flume became available for these experiments, the main wave maker panel wasbroken. It appeared that it had been broken few weeks before. As one can see from Figure 4the last 25cm of the bottom of the wave maker panel was missing.

Figure 4: Broken wave maker panel Figure 5: Fixed wave maker panel

At first, it was decided, after discussion with Jens Peter Kofoed, to try to generate waves withthe panel in this state and see if it was possible to manage something. However, after a couple ofdays of runnings and adjustments it appeared that the influence of the “missing” part was quiteimportant. For example to get regular waves of 4cm height with 1s period it was necessary toask to generate a waves of 4.79cm height. This need for 20% extra wave height could have beenokay to deal with if it was constant for all the wave heights or at least all the wave periods butit was not the case. Indeed, to obtain a wave of 6cm height with 1.8s period it was necessary togenerate a wave of 5.2cm height (i.e. a wave 13% less high than what is expected). Such largedeviance made very difficult the generation and the analysis of regular waves and impossible thegeneration of accurate irregular waves.

However, thanks to the efficiency and the availability of AAU technicians it was possible to


fix the wave maker (i.e. replacement of the wave maker panels) during the week 24 and the testshave been run in good condition during the week 25.

2.1.2 Beach issue

Due to the setup of the system (i.e. rotor fixed to a 2 piles structure) it has hardly beenpossible to play with the level of the still water level. Indeed, to get the right still water levelcompared to the rotor axle (i.e. 107mm above it), the total water depth needed to be h=0.82mat the rotor position. Considering the flume such water level is quite large and thus it has beennecessary to increase the beach size/height. About 1 cubic meter of rocks has been added tothe existing beach. One could see the beach with and without the water from Figures 6 and 7.The emerged part of the beach is not very large (about 15cm), however, since the wave heightsused during this experiment were fairly small, the emerged part of the beach was sufficient toavoid too much reflection.

Figure 6: Rock beach without water Figure 7: Rock beach with water

2.2 Experimental setup

The following section has for purpose to describe the different elements of the experimentalsetup. This goes from the methodology followed to run the tests, to the wave maker, flume androtor specification. Finally, some details on the measurement setup used during the tests arealso given.

2.2.1 Overview

The system setup is very similar to the one used during the first experiment session madeby Arthur Pecher during the summer 2011 (see Report [2]). However, as one might notice,this time it has been possible to fix the rotor (orange part on the pictures) to the axle whilebefore the rotor was only driving the axle to one direction. This “fixed setup” allows the PTOto work on both sides but more importantly it allows to have a very accurate measurement ofthe pitch motion in both directions and not only when the pitch is going up, as it was duringthe experiments made during the summer 2011.


2.2.2 Methodology

In this idea of comparing a numerical model to the experiment results, an extensive work hasbeen done on the measurement accuracy. Furthermore, in order to optimize the results accuracy,the following methodology has been followed :

1. A setup of 3 wave gauges has been setup and placed at the exact position of the rotor (therotor was therefore not in position yet). The wave sequences are launched and monitoredby these 3 wave gauges. Moreover, a webcam/ruler setup has been placed on the side ofthese wave gauges. This system helped to validate the calibration of the wave gauges.

2. The wave gauges setup was then taken out and replaced by the rotor with its PTO andpitch measurement. Then the same wave sequences were re-launched again.

This methodology allows to be sure to exactly measure the wave hitting the rotor at the rotorposition and not 1m or so before. One could hope that using such a methodology would givemore accurate results.

Note :

• More details on the calibration of the wave gauges, pitch and PTO measurement toolsmay be found in section 2.2.6.

• The repeatability of the wave sequences has been validated for different sea states. Andthus it has been proven that the wave maker is able to reproduce the exact same wavecondition twice.

• It is assumed that the reflection of the rotor does not affect the wave hitting the rotor.This can be insured in terms of reflection on the side walls next to the rotor considering thesize, the shape and the freedom of the rotor. However, it appeared during the experiencethat during some tests involving large waves, some significant reflection phenomena mighthave affected the accuracy of the results. (see Section 2.5.2)

2.2.3 Wave Maker specification

The wave maker is controlled by the software AwaSys V.6 made by AAU (see [7]).

AwaSys allows to generate, among others, regular wave of different wave heights and periods,as well as irregular waves using a JONSWAP Spectrum parametrized by a peak period, asignificant wave height and an enhancement peak. Moreover, through AwaSys it is possibleto generate a “replay file”. This file records the exact motion of the wave maker during a testand it is then possible to reproduce the exact same wave state. This file has been used in thecase of irregular waves.

2.2.4 Flume specification

As one could see from the figure 8 the flume is about 18m long, 1.2m wide and 1.15m deep.However, the bottom of the flume is not flat, the slope is about 6% over 5.5m (i.e. the waterdepth difference is about 0.34m). A “simple” numerical calculation has been run to evaluate theinfluence of this slope on the wave height. This numerical calculation is based on energy fluxconservation. This model showed that the slope should not generate more than 2% of realitvedifference between the asked wave height and the actual wave height. Thus any slope effect canlargely be neglected.


Figure 8: Geomerty flume details

2.2.5 Rotor setup Specification

2.2.5.1 Rotor ConfigurationAs said in the introduction the rotor used during the experiment is configured in “W5”. Thismeans that no mass has been added in the central hole and that 10 “small” and the 5 “large”weights have been placed in the 2 dedicated cylindrical holes in the rotor. In order to be asaccurate as possible all the elements composing the rotor have been weighted and the globalcenter of gravity and inertia have been experimentally evaluated. The details on the evaluationof the center of gravity and inertia of the rotor are given in the section 2.3

2.2.5.2 Still water level and Rotor rest angleThe still water level has been place 0.107m above the axle of the rotor. This water depthhas been determined from the previous marks (tape placed on the setup structure). One couldnotice that this 0.107mm are about the same distance as the distance between the axle and therear side of the rotor (i.e. the radius of the cylindrical rear part of the rotor is about 10cm).For such water depth the rotor rest angle as been evaluated around 50.6◦ (with +/- 0.5◦

of accuracy). To measure the angle with a good accuracy the rotor has been fixed with thePTO at a position close from the stable one. Then the angle has been evaluated by measuringthe distance between the tip of the rotor and above structure (since the rotor is fixed by thePTO it has been possible to measure this distance with a good accuracy). Through a simpletrigonometry function the angle of the rotor at this position has been determined. Finally,through the potentiometer record and by releasing the PTO and launching some small wave toget rid of any issue due to the friction in the axle it has been possible to determine with a fairlygood accuracy the rotor angle.

Details on the exact calculation :

• Distance from axle to top structure : 0.45m. Distance from axle to tip : 0.251m. Distancefrom tip to top structure (with fixed PTO): 0.286m.

• Angle of the rotor (with fixed PTO) : 54.16◦

• Measured difference angle when rotor freed (through potentiometer) : -3.56◦

• Rest angle (for SWL of 0.107m above axle) : 50.60◦

Note : During the week of experiment it has been noticed that the SWL was slightly decreasingover time. This might be due to some pipe leaks or just to the water evaporation. Anyhow,the SWL was getting down for about 5 to 10 cm every 24 hours. Thus, during one batch testsrunning the regular waves for example (about 2h) the water level was getting down for about0.5-1cm. This difference of SWL may have some effect on the hydrodynamic response of therotor.


2.2.5.3 Power Take Off (PTO) specificationThe PTO used during these tests is the same one as the one used during the 2010 experiment(i.e. the PTO is simulated by the friction of a pre-tensioned cable around a metallic wheel, see“2” on Figure 13 and report [1]). From the Coulomb friction definition (see website [3]), whentwo surfaces are sliding on each other, Ff ≤ νFn. Where, Ff is the friction force exerted by eachsurface on the other; ν is the coefficient of friction and Fn is the normal force exerted from eachsurface on the other, directed perpendicularly (normal) to the surface. This inequality becomesan equation when the two surface are sliding one on the other. The case of the used PTO isslightly more complicated. The pretension in the cable will lead to some friction around thebottom half of the wheel (see Figure 9). When the rotor is moving, the metallic wheel generatessome friction due to the contact with the cable. This will generate a tangent force inducing adelta of force in the load cells. From this difference of loading one can determine the tangentforce applied on the wheel and then the moment. The scheme Figure 9 and the equation Eq.1explain exactly how the PTO moment has been determined from the forces measured by theload cells.

Note :

• The sign of the moment is then chosen in function of the angular velocity

• The force→

Ffric from Figure 9 is given as the one from the wheel acting on the cable (i.e.

the cable generates a force −→

Ffric on the wheel).

Figure 9: Simple Scheme explaining the PTO principle.

MPTO = RWheel ·Ffriction= RWheel ·

((T0 +

Ffric2

)− (T0 −Ffric2

)

)= RWheel · (FPTO,1 − FPTO,2)

(1)

The metallic wheel, which is fixed to the axle has a radius of 0.050m, a mass of 0.7kg and aninertia of 8.84e-04 kg ·m2.

It, however, appears that the PTO was not exactly constant for all the wave states. Indeed,Table 1 gives the mean, maximal and minimum PTO value averaged over one period for different


loadings (i.e. the maximal value corresponds to the maximal averaged PTO of one wave state).Therefore, to stay as consistent as possible, it has been chosen to use the PTO value found foreach wave state, these values can be seen in Annex.

Case PTO Pretension Delta Force Induced PTO Moment Comment[N ] [N ] [Nm]

"Down" "Up" "Down" "Up"1 0 - 0.00 0.00 0.00 0.00 Free motion

2 6

Mean 6.03 3.82 0.30 0.19Max 7.92 5.26 0.40 0.26Min 4.55 2.20 0.23 0.11

3 15

Mean 8.39 6.83 0.42 0.34Max 11.15 9.02 0.56 0.45Min 4.68 3.88 0.23 0.19

4 25 - - - - - No motion

Table 1: Summuary of the different PTO loading tested

0 2 4 6 8 10−15

−10

−5

0

5

10

15

Pitc

h [d

eg] /

For

ce [N

]

Time [s]

Pitch motionLoad PTO 1Load PTO 2

Figure 10: Experimental measurement : Pitch, PTO 1 and PT0 2 representing the two load sensor(on each side of the metallic wheel). This is done for the case of H=4cm and T=1.20s andPTO pre-tension of 6 N .

Note :

• One should not confuse the PTO pre-tension, which is the force applied by the cable onthe metallic wheel, with the PTO moment, which is the actual PTO induced moment seenby the rotor.

• The given values in terms of force and moment should be considered as reliable with more


or less 5%-10% error. This fairly large error comes from some drift issue that appearedduring the test. Bascially the measured loads were sightly lower at the end of the teststhan before. This might be due to a “simple” electrical drift or because the cable getsslightly less tighten after some wave tests.

• During the experiment a third PTO loading has been used (10N pre-loading). However,after analysis of the results it appears that the loading varied quite a lot during the batchtests. It has, therefore, been choose to not consider this loading in this report.

• The strongest PTO used during the test, case “4” with 25N pre-loading, has been usedto experimentally evaluate the exciting moment obtain on the rotor under regular waves.However, it appears that this methodology was not precise enough to get reliable results.One may see in annex some details about these calculations.

2.2.6 Measurement setup specification

2.2.6.1 Wave gauges setup

Three wave gauges have been used. They have been aligned on a common structure. The firstand the second wave gauges are separated by 15cm while the second and the third wave gaugesare separated by 25cm (see Figure 11).

At first the wave have been run without the presence of the rotor and with the wave gaugesplaced at the exact rotor position. While this has been done and in order to validate themeasurement made by the wave gauges a ruler/webcam system has been setup next to the wavegauges (see Figure 12). This system allows, by watching the recorded video image by imageto estimate the wave height with less than 1-2mm errors. This “verification system” helped tosee that a “simple” 2 points calibration (using the 10cm pistons setup) was not precise enoughand leads to heterogeneous measurements. The 3 wave gauges were giving wrong and differentresults, the measurements were varying from 10% to 20% (i.e. more than 5mm) compare towhat the webcam/ruler system was recording. Therefore a 3 points calibration has been used.This calibration has been done with good accuracy thanks to the metallic bar fixed to all thewave gauges (see Figure 12). Indeed, this bar, fixed to the wave gauges structure, is drilled every5 cm. Therefore by placing a small nail in each 2mm holes it has been possible to move the 3wave gauges within + 5 cm and - 5 cm compared to the main position. This 3 points calibrationappeared to be more precise.

Then, when the rotor was in position, the wave gauges were moved 1.18m in front of therotor position. Their presence does not affect at all the water elevation, and these measurementhelped to evaluate the reflexion of the rotor.

2.2.6.2 Potentiometer and Load cells measurement setup

On the Figure 13, one may see the setup used to measure both the pitch motions (poten-tiometer “3” on the picture) and the forces acting on the PTO (the two load cells “1”).

1. Potentiometer : Pitch motion measurement

The amplification factor of the potentiometer for the rotor has been calibrate through 2different methods :

• measure of the voltage evolution of the potentiometer when the rotor turns +360◦

and then -360◦.

• measure of the voltage limits of the potentiometer by making 360◦ with it. Then bycomparing the axle radius and the potentiometer wheel radius it is possible to make


Figure 11: Simple scheme of the positioning ofthe wave gauges

Figure 12: Picture of the full setup used tomeasure the wave sequences

a correlation between the voltage and the actual rotor angle.

This two methods led to a fairly similar amplification factor of 117 deg/V. To determinethe offset, the rotor has been placed in still water and its angle has been measured asexplained in section 2.2.5.2. Moreover, before and during each batch test the rest positionof the rotor has been measured (by making a break during the different sea states). Suchfile allowed to compensate the difference of still water level compare to the 50.5◦ measuredfor 107 mm above the axle.

2. Load Cells : PTO applied forces

To evaluate the forces applied on the PTO wheel, two load cells have been placed, see “1”on Figure 13. Each of these load cells is connected to a single amplifier box (see “3” inFigure 14). The load cells have been calibrated through the use of weights of 0.5kg, 1kgand 2kg. Due to some drifting the load cells have been calibrated before each batch tests.

Note : In order to connect the different weights to the load cells, a metallic “S” of 40g hasbeen used. If someone check the calibration files he should be careful about it.

2.2.6.3 Acquisition and amplification boxes

The following paragraph will briefly describe the amplifier boxes used to acquire all the mea-surements made during the experiments. The following points refer to the Figure 14 :


Figure 13: Rotor setup use during all the wave states.

1. Wave Recorder (Wave gauges amplifier) : Amplify all the measurement made from the 3wave gauges. (Input : raw signal from Wave gauges ; Output : USB data acquisition box“4”)

2. Channel transducer modifier (Potentiometer amplifier) : Amplify the results from thepotentiometer. The port “F” has been use since a simple amplification was necessary here.The port A, B, C... were offering the opportunity to get as output the sum, the difference,the product or the division of two input.(Input : raw signal from the potentiometer ;Output : USB data acquisition box “4”)

3. Load cells single amplifier : Amplify the measurement made from the load cells. Atfirst, the issue was to properly configure the filter/amplifier integrated into each amplifier.Indeed, inside of each boxes it is possible to configure it through the use of 0/1 button.The idea was to configure them to be able to properly amplify the load cells signal. (Input: raw signal from the load cells ; Output : USB data acquisition box “4”)

4. USB Data Acquisition DT9804 : Transform all the digital amplified signals into a numer-ical signal transfered through a USB cable to a computer and analyzed by the softwareWaveLab [8]. (Input : all the amplified signals ; Output : numerical signal to Computer)

Note :

• All the boxes have been grounded to avoid any extra drifting.

• The software used to acquire all the signal was WaveLab 6 ([8]).

• To avoid too much noise and to be closer to the rotor, all the boxes and the computerhave been placed on top of the flume, i.e. about 2-3m from the rotor.


Figure 14: Acquisition and amplification boxes used to acquire the measurement from the wavegauges, potentiometer and PTO load cells.

2.3 Mass and inertia measurement

2.3.1 Theorical values

Theoretical values of the mass, center of gravity and inertia of the rotor have been given byThommy Larsen. The Table 2 sum up all these theoretical values.

Theoretical results

Mass m 4.421 [kg]

Center of Gravity xg 50.75 [mm]zg -27.94 [mm]

Inertia I22 4.980E-02 [kg.m2]

Table 2: Center of gravity position evaluated through experiment

of the rotor have been evaluated experimentally to validate the theoretical values given byThommy Marchalot and used in the report [1].

2.3.2 Mass of the Rotor

The rotor has been dismantled and weighed piece by piece. The Table 3 details the weightof each piece. One could note that in the original report the configuration W5 was evaluatedat 4.4kg. This value is not found back here, the W5 rotor configuration is evaluated at 4,29kg.This non negligible difference confirms that the theoretical values given in terms of the centerof gravity and inertia may not be accurate enough.

2.3.3 Center of Gravity

To find the "exact" position of the center of gravity once the rotor is in position into thewater, the flume has been slowly filled in and/or emptied. While the water level was slowlychanging the rotor angle and the water depth were recorded. From these measurements andRhin3D ([6]), it has been possible to evaluate the moment of buoyancy at different times (i.e.for different water depths and rotor angles). Since, for each of these positions, the rotor is inan equilibrium position the moment of buoyancy should be balanced by the moment of gravity.Knowing the mass it is then possible to evaluate the exact position of the center of gravity. Thistest was run twice and for each of these tests 5 different time/position have been considered.The detailed results from these tests might be seen from tables in placed in annex.


Experimental results

Object Details mass [kg]

Large hole weights 5 weights 1.09Small hole weights 10 weights 0.54

Rotor (W0) No weight 2.66Rotor W5 With all weights 4.29

Axle (without PTO Wheel) 1.38PTO Wheel 0.70

Extra Tightening rings... 0.35

Total Weight 6.72

Table 3: Rotor and setup mass details

Moreover, this calculation has been validated with the “in-air” rest angle of the rotor. Indeed,it appears that out of the water the rotor with the “W5” configuration finds a rest angle of-58.5◦. One could already note that if the CoG given is used then then in-air rest angle shouldbe -61◦. The Table 4 gives the coordinates of the experimentally evaluated Center of Gravity.


Angle 0.00 −58.5 degreexg 48.66 0.00 mmzg −29.82 −57.07 mm

Table 4: Center of gravity position evaluated through experiment

Note :

• These Center of Gravity has been evaluated for a mass of 4.29kg. One could wonder whythe mass of the axle has not been considered. This can be explained by the fact that sincethe center of gravity of the axle is (0,0,0) it is pointless to consider it while evaluating theCoG of the rotor. If one wants to consider the full system (i.e. with a mass of 6.72kg, thenthe CoG should be considered at the position (31.06,0,-19.04)mm.

• It has not been easy to run these tests since it appears that the smallest mistake in termsof angle or depth may lead to non negligible errors. And due to some “friction” betweenthe axle and the two arms of the structures, the angle of the rotor might be about 1◦

off. For example the in-air rotor is stable for angles between -59.5◦ and -57.5◦ (for suchangle the moment generated by the gravity is not enough to move the rotor to its real restposition). This is why it has been chosen to base the calculation on a in-air rest angle of-58.5◦.

2.3.4 Inertia Measurement

To experimentally evaluate the rotor inertia, it has been chosen to place the rotor in the airand to drop it from a certain angle. Then by considering its natural period of motion it wasexpected to be able to determine the inertia of the rotor through the use of a numerical modelsolving equation 2.

I22 · α =Mg(α) = −m · g ·xg(α) (2)

Where :


• I22 is the inertia of the rotor.

• α is the rotor angle and α its acceleration.

• Mg the gravity moment acting on the rotor.

• m the rotor mass.

Moreover, since the rotor can easily be setup in W5 and W0 configuration, both configurationshave been tried out. As a reminder, the only difference between these 2 configurations is thepresence or not of cylindrical weights placed into 2 cylindrical holes in the rotor side. It istherefore easy to evaluate the difference of inertia with and without the presence of these extraweights.

However, as one may see from the Figure 15 and 16 the rotor motion is quite largely damped.Considering that the friction of the air on the structure should be negligible. This friction mustcome from the structural friction between the axle and the structure.

0 1 2 3 4 5 6 7 8 9 10−200

−150

−100

−50

0

50

100

Time [s]

Pitc

h m

otio

n [d

eg]

Figure 15: Inertia test with configuration W0.Initial angle 65◦.

0 1 2 3 4 5 6 7 8 9 10−200

−150

−100

−50

0

50

100

Time [s]

Pitc

h m

otio

n [d

eg]

Figure 16: Inertia test with configuration W5.Initial angle 90◦

It is therefore necessary to add a friction moment into the equation 2,see equation 3. Noticethat the addition of this friction moment has an effect on both the amplitude and the period ofthe oscillations. It is therefore necessary to find a “coupled” inertia/friction that fits these twocurves.

I22 · α =Mg(α) +MFriction(α) (3)

This friction is assumed to follow the Coulomb friction definition, see Section 2.2.5.3. Inthe case of the WEPTOS rotor, the normal force can be approximate by two different forces.The first one could be the “pre-tension” due to the pressure of the structure around the axle.This force would therefore generate a constant friction moment. Another force that might beconsidered is the one coming from the gravity and the movement of the rotor, the force Faxledefined by the Equation 4 express this second non-fixed force.

~Faxle = ~Fgravity + ~Fcentrifugal

| ~Faxle| =√

(Fcentrifugal cos(α))2 + (Fcentrifugal sin(α) + Fgravity)

2

(4)

Note : the expression of the Faxle stays true only when the rotor is in the air. When the rotoris in the water and under wave forces then this formulas must be re-evaluate.

Finally three different kind of friction moments have been tried out to confirm the theory andcheck the importance of the each of this different friction moments. At first the model has been


run with a constant friction only Mfric = ν1, then only a moment using the Faxle has been triedout Mfric = ν2 ∗ Faxle and finally the two different friction moments have been cumulated (seeEq.5).

Mfric = Mfric1,Constant +Mfric1,V ariable

= ν1 + ν2Faxle

(5)

The methodology followed to run this numerical model has been to run in the time domainthe model for 10 friction coefficients and 10 inertias. Run after run the range of the inertia andfriction coefficient array have been reduced to finally get the coefficients allowing to get the bestfitting for both case “W0” and “W5”. The following table 5 sums up all these results and theiraccuracy.

Best Fit for I22 [kg.m2] ν1[Nm] ν2[−] Error [%]

Test with 1 Fixed Friction

Common 4.800E-02 1.371E-01 43.89W0 4.900E-02 1.014E-01 7.93W5 4.870E-02 1.514E-01 9.22

Test with 1 Non fixed Friction

Common 4.940E-02 2.000E-03 17.03W0 4.900E-02 2.000E-03 7.86W5 4.990E-02 2.000E-03 6.83

Test with 2 Non fixed Friction

Common 4.960E-02 1.800E-03 0.01 8.4417W0 4.950E-02 1.800E-03 0.01 8.4754W5 4.980E-02 1.900E-03 0.01 6.8579

Table 5: Sum up of the different calculation run for the inertia

From the table 5 one could see that the use of a fixed plus a non fixed PTO leads to betterresult. Thus the bold values are the ones that will be kept for all the rest of this report. Thepitch evolution with these coefficients for both “W0” and “W5” cases is displayed in Figures 17and 18.

0 1 2 3 4 5 6 7 8 9 10−200

−150

−100

−50

0

50

100

Time [s]

Pitc

h m

otio

n [d

eg]

Experimental MeasurementNumerical model fit W0Numerical model fit W5Numerical model fit Common

Figure 17: Numerical model results and exper-imental measurements of the inertia testfor the configuration configuration W0.The numerical model has been run for the3 couples of coefficients given in the Table5

0 1 2 3 4 5 6 7 8 9 10−200

−150

−100

−50

0

50

100

Time [s]

Pitc

h m

otio

n [d

eg]

Experimental MeasurementNumerical model fit W0Numerical model fit W5Numerical model fit Common

Figure 18: Numerical model results and exper-imental measurements of the inertia testfor the configuration configuration W5.The numerical model has been run for the3 couples of coefficients given in the Table5

To give an idea of the precision of the results, the scatter diagrams of the latest tests run for


a constant plus a non constant friction moment are given in Figures 19, 20 and 21. As one maysee from these scatter diagrams the accuracy of the results should be below 1% error.

1.71.75

1.81.85

1.91.95

2

x 10−3

5.5

6

6.5

7

7.5

x 10−3

0.0495

0.0496

0.0497

0.0498

MB551

MB552

I22

10

12

14

16

18

20

22

Figure 19: Scatter diagram of the error, for theconfiguration W0, of 1000 numerical mod-els run for 10x10 friction coefficients andfor 10 inertias.

1.71.75

1.81.85

1.91.95

2

x 10−3

5.5

6

6.5

7

7.5

x 10−3

0.0495

0.0496

0.0497

0.0498

MB551

MB552

I22

8

10

12

14

16

18

20

22

24

26

Figure 20: Scatter diagram of the error, for theconfiguration W5, of 1000 numerical mod-els run for 10x10 friction coefficients andfor 10 inertias.

1.71.75

1.81.85

1.91.95

2

x 10−3

5.5

6

6.5

7

7.5

x 10−3

0.0495

0.0496

0.0497

0.0498

MB551

MB552

I22

20

25

30

35

40

45

Figure 21: Scatter diagram of the mean error of both configurations run of 1000 numerical modelsrun for 10x10 friction coefficients and for 10 inertias.

Finally to show the influence of the “non-fixed” part of the friction moment the figures 22 and23 show the time evolution of the friction moment for both cases W0 and W5 with the coefficientgiven by the Table 5. This last figure confirms the importance of both friction moment.


0 2 4 6 8 100.095

0.1

0.105

0.11

Time [s]

|Mfr

ic| [

Nm

]

Figure 22: Time evolution of the friction mo-ment for the case W0, with I22 =4.960E-02 andMfric = 1.800E−03∗Faxle+0.01.

0 2 4 6 8 100.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

Time [s]

|Mfr

ic| [

Nm

]

Figure 23: Time evolution of the friction mo-ment for the case W5, with I22 =4.960E-02 andMfric = 1.800E−03∗Faxle+0.01.

Note: More details about the two other solutions are available in appendix.

2.3.5 Summary of the final values

Finally the Table 6 summarized all the definitive values that are used in all the other sectionsof this report.


Mass m 4.29 [kg]

Center of Gravity xg 48.66 [mm]zg -29.82 [mm]

Inertia I22 4.960E-02 [kg.m2]

Table 6: Summary of the mass, center of gravity and inertia considered for the rotor.

Note : the mass is only considering the Rotor, its weight and all the components integratedinto it (i.e. everything excepted the axle, the PTO wheel and the structure, for a total mass of4.29 kg). Moreover, the position of the center gravity is given considering the rotor with a 0◦

angle. On the other hand in terms of inertia all the moving part have been considered and aregiven at the center of the body fixed coordinate system.

This makes sense since the inertia needed to be including all the moving parts (i.e. for atotal weight of 6.72 kg), while the mass/center of gravity just needed to be in accordance. Ifone wants to consider the axle, and the PTO into the mass and the center of gravity a simplecalculation may allow to do it. But considering the given problem it does not seem relevant.

2.4 Wave tests

2.4.1 List of the tests run

In terms of regular waves the following wave heights have been tested : H=0.02 ; 0.03 ; 0.04; 0.05 ; 0.06 ; 0.08 m. All these wave heights have been run for the following periods : T=0.75 ;0.90 ; 1.00 ; 1.05 ; 1.10 ; 1.20 ; 1.40 ; 1.80 s. In terms of irregular wave a JONSWAP spectrum hasbeen used to generate the wave sequences. The sea states that have been run are part of the onesthat have been used during the experiments described by the report [1] (i.e. only the smallestwave states have been considered in this report). A proper JONSWAP peak enhancement has


been evaluated through the formula given by the DNV Standard [10], page 30. Finally, theseare wave states described in Table 7 have been considered.

Wave State Hs [m] Tp [s] Gamma Scale

WS05 0.040 0.900 1.777 8.33WS10 0.060 1.040 2.381 8.33WS15 0.090 1.210 3.039 8.33WS20 0.120 1.390 3.113 8.33WS25 0.150 1.560 3.058 8.33WS60 0.043 1.158 1.000 23.4WS70 0.085 1.447 1.043 23.4WS80 0.128 1.736 1.185 23.4

Table 7: Summary of the irregular wave state used

One could notice from the “scale” factor that all the waves do not come from the same “site”.The wave states WS05 to WS25 have been evaluated for Anholt P2 scaled at a ratio of 1 : 8.33,while the wave states WS60 to WS80 represent the Danish North Sea scaled at a ratio of 1 : 23.4.

All the regular wave states have been run in a batch mode : 2 minutes run for 1 minutebreak. Only 1 minute of each test has been recorded (from 45s to 1m45s), this had for purposeto obtain a fairly clean measurement data. All the irregular wave state have been run for 25min.This allowed to get about 1000 waves per test, which is the minimum value necessary to have areliable representation of a sea state.

2.4.2 Evaluation of the wave measurement

2.4.2.1 Regular waveFor each wave state the 3 aligned wave gauges were measuring the wave elevation. Each of themeasurements shows a slight time delay (due to the distance between each, see Figure 26). Thisdelay can be evaluated and solved (in order to get a mean of the three measurement) throughtwo different methods.

2.4.2.1.1 First crest MethodA first fairly “crude” solution would consist to consider one crest and align all the signal throughthis crest. The Figures 24 and 25 are showing, in the case of H=0.04m and T=1.00 the resultsof this calculation.

2.4.2.1.2 Linear Theory MethodAnother solution, would consists of using the linear theory : tdelay = distance/celerity. Thewave celerity, c = L/T (with L the wave length and T the wave period), can be obtain throughthe wave period and the water depth by solving the linear dispersion relation (see Eq. 6).

ω2 = gk · tanh(kh)

With ω =2π

T

k =2π

L

(6)


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

Time [s]

Wav

e E

leva

tion

WG1 Non AlignedWG1 Non AlignedWG1 Non Aligned

Figure 24: Non aligned wave elevation mea-surements for H=0.04m and T=1.00s.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

Time [s]

Wav

e E

leva

tion

WG1 AlignedWG1 AlignedWG1 Aligned

Figure 25: Aligned wave elevation measure-ments, through the 1st crest after 10s, forH=0.04m and T=1.00s.

This calculation leads to the time delays showed in Table 8.

T Delay WG1 -> WG2 Delay WG2 -> WG3

[s] distance =0.15m distance =0.25m0.75 0.128 0.2130.90 0.107 0.1781.00 0.096 0.1611.05 0.092 0.1531.10 0.088 0.1471.20 0.082 0.1361.40 0.073 0.1211.80 0.064 0.106

Table 8: Summary of the delays, for each wave periods, between the wave gauges 1 and 2 andbetween the wave gauges 2 and 3

This, elegant, solution seems to work fairly fine for the case H=0.04m and T=1.00s, as onecould see from Figure 27).


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

Time [s]

Wav

e E

leva

tion

WG1 Non AlignedWG1 Non AlignedWG1 Non Aligned

Figure 26: Non aligned wave elevation mea-surements for H=0.04m and T=1.00s.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

Time [s]

Wav

e E

leva

tion

WG1 AlignedWG1 AlignedWG1 Aligned

Figure 27: Aligned wave elevation measure-ments, through the linear wave theory, forH=0.04m and T=1.00s.

Finally, considering that both methods lead to fairly equivalent results the second solutionusing the linear theory is used. Once all the signals have been aligned (from a time point of view),it is possible to get an average signal from them. This final signal will be the one considered toget the actual wave height obtained at the rotor level.

To obtain an average wave height, the height of 18 wave periods is evaluated. Then theaverage value of these 18 wave heights is calculated.

In annex it is possible to see a detailed evaluation of the wave accuracy for each wave states.It appears that excepted maybe for T=1.80s all the measured wave height are fitting the expec-tations.

2.4.2.2 Irregular wave

2.4.2.2.1 First crest MethodIn terms of irregular waves the 3 measurements made by the different wave gauges have beensynchronized through the first top crests (see Figure 28 and Figure 29. Then the 3 synchronizedsignals have been averaged to get the average wave elevation of each sea state, see Figure 30.

83 84 85 86 87 88 89

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Time[s]

Wat

er e

leva

tion

[m]

Wg 1Wg 2Wg 3

Figure 28: Non aligned wave elevation mea-surements for WS10.

82 83 84 85 86 87 88 89−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Time[s]

Wat

er e

leva

tion

[m]

Wg 1Wg 2Wg 3

Figure 29: Manual aligned wave elevationmeasurements for WS10.


82 83 84 85 86 87 88 89−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Time[s]

Wat

er e

leva

tion

[m]

Figure 30: Manual averaged wave elevationmeasurements for WS10.

82 83 84 85 86 87 88 89−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Time[s]

Wat

er e

leva

tion

[m]

Figure 31: AwaSys averaged wave elevationmeasurements for WS10.

2.4.2.2.2 AwaSys MethodHowever, this a simple averaging might seem not accurate enough. Indeed, considering thatirregular waves are the sum of regular waves with different wave periods and wave heights, itseems “too simple” to just synchronize them all as such. Moreover, the software AwaSys offersthe possibility to do a “reflection” analysis. This analysis allows (among other possibilities) toevaluate the mean total signal. From this analysis, an extract from the wave elevation might beseen in Figure 31. One could notice from the figures 30 and 31 even if the two figures are not toodifferent the signal is not exactly similar. Therefore, considering that the AwaSys calculationmethod is more advance the output signal from AwaSys is the one that will be used during theexperiments.

Furthermore, considering the flume slop and the different disturbance phenomena it seemsinteresting to evaluate the significant wave height and the peak period for each wave state andalso for each channels. For the Table 9 one can see that for most of the wave state the 3 channelshave recorded equivalent signal. Moreover, the measured see state characteristics appear to beclose to the asked theoretical values.

2.4.3 Repeatability of the tests

2.4.3.1 Regular waveTo validate the repeatability of the wave maker, all the experimental tests (without the rotorand only the wave gauges) have been run twice, with at least one day difference. Then it hasbeen possible to compare the wave height obtained for each test to the one that was asked for.The results of this tests can been seen in Appendix. One could see that the repeatability of thewave maker seems fair enough for most of the cases. Moreover, this test also gives an idea ofthe measurement accuracy and their calibration.

Note : From the tables in annex one should note the limitation of the wave maker for waveperiod of 1.8 s.

2.4.3.2 Irregular waveTo make sure that the wave elevation generated through a JONSWAP spectrum was always thesame, the “replay” files made through the software AwaSys [7] have been used. Such files allowto save the exact motion of the panel made during a given state. Then by running directly thisfile, through AwaSys, it is possible to generate the exact same wave elevation sequence.


File Ch. No. waves Hs Tp Hs (Average) Tp (Average) Hs (theoric) Tp (theoric)

[m] [s] [m] [s] [m] [s]WS05 1 1805 0.036 0.898

0.036 0.893 0.040 0.900WS05 2 1817 0.036 0.914WS05 3 1805 0.036 0.868

WS10 1 1561 0.057 1.0670.057 1.067 0.060 1.040WS10 2 1561 0.057 1.067

WS10 3 1559 0.057 1.067

WS15 1 1301 0.089 1.2490.089 1.249 0.090 1.210WS15 2 1301 0.089 1.249

WS15 3 1295 0.089 1.249

WS20 1 1152 0.118 1.3840.118 1.410 0.120 1.390WS20 2 1154 0.118 1.384

WS20 3 1140 0.120 1.463

WS25 1 1015 0.155 1.5520.156 1.585 0.150 1.560WS25 2 1008 0.156 1.552

WS25 3 1004 0.156 1.652

WS60 1 1461 0.041 1.3470.041 1.347 0.043 1.158WS60 2 1462 0.041 1.347

WS60 3 1462 0.042 1.347

WS70 1 1166 0.086 1.5060.087 1.477 0.085 1.447WS70 2 1166 0.087 1.463

WS70 3 1169 0.088 1.463

WS80 1 972 0.135 1.7070.136 1.689 0.128 1.736WS80 2 970 0.137 1.707

WS80 3 976 0.137 1.652

Table 9: List of the WaveLab time series analysis on the irregular wave elevation measurements.

2.5 Recorded pitch motion

2.5.1 Wave maker delay

As said, all the tests for regular waves have been run for 2 minutes. The motions and loadhave then been recorded for the last 60 s of each of these tests. All this tests have been runin batch mode (for both the WaveLab ([8]), recording software) and AwaSys ([7], generatingsoftware). However, it appeared that even by calculating the same break time for both batchmode the one from AwaSys was slightly delayed after each wave sequence. This delay is due tothe wave maker warm up time before each wave sequence. It has been evaluated that betweeneach wave height sequence (8 tests) the delay is about 8 s. Therefore, for example, at the end ofthe 48 wave tests the recorded file was not recording from 45 s after the launching of the wavesequence but slightly before its launching - then few seconds are needed for the waves to reachthe rotor. Such delay might be seen from the Figures 32 and 33.

Furthermore, the reader should keep in mind another phenomenon of delay. Considering twowave periods, the time necessary for both waves to reach the rotor will not be the same. Indeed,for a given depth the group celerity is directly linked to the wave period. Therefore, consideringa wave hight of 6 cm, for example, one could note from the figures 34 and 35 that the regularwave with a wave period of 1.80 s took more time to reach the rotor than the one with a wave


Figure 32: Full pitch motion recorded file from WaveLab. For H = 02 cm and T = 0.75 s.


period of 0.75 s.

Note : One should keep in mind while analyzing the figures 34 and 35, that considering thewave maker delay the record made for the wave sequence with a period of 1.80s starts about6s later (relatively to the wave maker starting point) than the one made for the wave sequencewith a period of 0.75s.



All these delay issues might generate difficulties to compare the numerical model with theexperiments. Thus, since the startup of the rotor is not interesting here, these delay issues havebeen avoid by considering only the last 20 s of the recorded signal.

2.5.2 Reflexion issue

For the case T = 1.10 s and H = 5 cm the pitch is in a stable oscillating motion for morethe first 10 s and then without obvious reason its amplitude start to grow to reach about 35%oscillating amplitude more than previously (see Figure 37). This phenomeon seems to be dueto reflexion of the waves on the beach. The same phenomeon appears, quite less strongtly, for


T = 1.10 s and H = 4 cm, see Figure 36, about 16% reflexion.



At the light of this phenomenon, a brief study of the necessary time for the wave to reachthe beach and come back, has been made, see Eq.7. Where dreflected represents the traveleddistance by the wave before coming back (i.e. in this case two times the distance between the

rotor and the beach) and cg represent the group velocity with :cgc

=1

2

(1 +

2kh

sinh(2kh)

).

treflexion =dreflected

cg(7)

In the case of a wave period of 1.00 s (k = 3.36 m−1) the time needed for the wave group(after having reach the rotor) to reach the beach and come back is evaluated to 22.62 withdreflected = 20.2 m and cg = 0.89 m/s. Moreover, considering the wave maker delays and fromFigure 38 it is assumed that the waves have reach the rotor about 5 s before record starts. Thereflection should then happen about 17 s after the beginning of the recorded file and it is indeedwhat it can be seen from figures 37. While for H=0.04 m the same reflexion phenomenon appearsabout 8 s, which fits the theory considering the wave maker delays. However, even with all thesesigns which seem to confirm some reflexion issue for T = 1.10 s and H = 0.05m it has to benoted that while the wave tests were run without the rotor (i.e. only wave measurement at therotor position) no reflexion phenomenon appeared. Moreover, considering that the rotor shouldabsorb most of the wave elevation is seems surprising that a so large reflexion phenomenonappears on the beach.


From the Figures 38 and 39 one could notice that no reflexion issue seems to be there. Fromsmaller wave considering the recording times (i.e. more than 35 s after the launching of the



wave states) it is not possible to evaluate if there was any reflexion issue. Finally from Figure40 and 41 one can see that the reflexion phenomenon only appears for T=1.10.




3 Linear study of the rotor

3.1 Theory

In this section only a “pure” linear system solution is studied. To be able to compare thenumerical model with the experimental measurements it is necessary to run the numerical cal-culations in the time domain. Indeed, only in the time domain it is possible to had the PTOmoment has describe in section 5.1 and to run irregular waves. However, to find the final linearequation used in the model (Equation Eq.15), it is necessary to switch from the frequency do-main to the time domain. This section has for main purpose to describe the method used to doso.

3.1.1 Second Newton’s law

The main theory used in this report is based on the “simple” second law of Newton. Moreover,considering that the WEPTOS rotor has only one degree of freedom, only pitch motions andmoments will be studied, see Eq.8. In this equation I55 represents the inertia of the rotor andz5 its pitch angle. One might also recognize Mhy as the sum of hydrodynamic moments : Mex

the excitation moment, Mrad the radiation moment and Mhs the hydrostatic moment. Finallyother extra moment such as the one generated by the PTO, or some extra friction might haveto be considered as well.

I22 z5(t) = Mhy(t) +MPTO(t) +Mothers(t)

I22 z5(t) = (Mex(t) +Mrad(t) +Mhs(t)) +MPTO(t) +Mothers(t)(8)

3.1.2 Frequency domain

In the frequency domain the linear equation driving pitch motion, for only 1 degree of freedomand under regular waves can be written as shown in Eq.9.

[−ω2(I22 +A55(ω)) + iωB55(ω) + C55

]Z5 = |X5(ω)| (9)

Where :

• A55 represents the added mass (dependent of the wave frequency)

• B55 represents the damping coefficient (dependent of wave frequency)

• C55 = ρg(∫waterplane x

2dS + V ∗ zb)−mgzg represents Hydrostatic restoring coefficient

• z5 represents the Pitch motion

• X5 represents the exciting forces (dependent of the wave frequency)

All these coefficient have been evaluated through the use of WAMIT ([9]) and by consideringonly the wetted surface of the rotor at its rest position. In the configuration W5 and at 107mmof depth (above the axle) the rest position of the WEPTOS rotor is 50.5◦. However, to obtainthese coefficient with a good accuracy all the following steps have been studied :

• Geometric analysis of the system

• Meshing of the rotor

• Hydrostatic calculation

• Hydrodynamic calculation through WAMIT


3.1.3 Time domain

From this linear equation in the frequency domain it is now necessary to go back to the timedomain using these coefficients. The fundamental relations between the time- and frequency-domain are the relations between the impulse-response functions K5,rad(t) and K5,diff (t) andthe “frequency domain coefficients” A55(ω), B55(ω) and X5(ω), see Eq.11 and Eq.12.

L55(t) =2

π

∞∫0

[A55(ω)−A55(∞)]cos(ωt)dω

=2

π

∞∫0

B55(ω)

ωsin(ωt)dω

(10)

K55,rad(t) =∂

∂tL55(t)

=2

π

∞∫0

B55(ω)cos(ωt)dω

(11)

K5,diff (t) =1

2π

∞∫−∞

X5(ω)eiωtdω (12)

Where :

• L55 represents the radiation impulse-response function due to an impulsive motion of thebody.

• K55,rad represents the radiation impulse-response function due to an impulsive velocity ofthe body.

• K5,diff represents the diffraction impulse-response function (Fourier transform of the ex-citing force X5(ω)).

From there, the excitation and radiation moments, in the time domain, can be evaluated asshown in Eq.14 and Eq.13.

Mex(t) =∞∫−∞

K5,diff (t− τ)η(τ)dτ

=∞∫−∞

K5,diff (τ)η(t− τ)dτ(13)

Mrad(t) = A∞z5 (t) +t∫0

K55,rad(t− τ) z5(τ)dτ (14)

Finally by replacing the hydrodynamic moments (Eq.14 and Eq.13) into the 2nd Newton law(Eq.8), the “final” linear equation (Eq.15) is found.

M55z5 (t) +Mrad(t) + C55 z5(t) = Mex(t)

(M55 +A∞) z5 (t) +t∫0

K55,rad(t− τ) z5(τ)dτ + C55 z5(t) =∞∫−∞

K5,diff (τ)η(t− τ)dτ(15)

Note :

• All the equations and the definitions that have been given in this section have been deter-mined through both manuals of WAMIT (Chap 13.6, [11]) and TiMIT (Chap 7, [4]).


• More details on how the different Fourier transforms have been implemented into Matlab([5]) throught the use of the ifft function, may be found in Annexe.

3.2 WAMIT Calculation

3.2.1 Geometric analysis of the system

3.2.1.1 Generalities on the geometric filesFor the mesh and the WAMIT calculations all the units have been kept as given by ThommyLarsen and used in the report Experimental study on a rotor for WEPTOS Wave Energy Con-verter [2]:

• Distance in millimeters

• Mass in grams

• Time in seconds

• Angle in degree

Moreover, keeping the distances in millimeters allow to have a detailed mesh definition. In-deed, WAMIT approximate the mesh values after the 6th decimals. Therefore, the followingparameters have been set up as:

• Gravity : g = 9806.65mm/s2

• Water density ρ = 0.001035g/mm3

Then, under Matlab, the results of WAMIT have been changed into the international unitsystem. In this report, all the results are given in the international unit system.

3.2.1.2 Analysis of the main holeThe meshes have been made under Rhino3D and then exported to WAMIT format. The originalshape comes from a .step file given by Thommy Larsen. From there the rotor has been movedand trimmed to fit WAMIT calculation. Indeed, WAMIT requires that the body-fixed coordinatesystem is placed at the center of the axle of rotation. (this will then allow the pitch rotation tobe around the right axis). Moreover, only the wetted surface needs to be meshed for WAMIT.Thus the rotor has been trimmed at 107mm above the plane z=0. The Figures 42 to 44 give anidea of shape of the rotor wetted surface at its rest position.

One could note from these figures that the hole of the rotor is completely under water. Con-sidering this, and the fact that it is a cylindrical hole centered around the axis of rotationof the rotor it is pointless to consider it in WAMIT calculations (no influence on neither theadded mass, damping, exciting force nor the hydrostatic moment). The mesh used for WAMITcalculation has therefore been made without considering this hole, see Figure 44.

Note : One could wonder if this hole has any effect on the hydrostatic stiffness C55 evaluatedby WAMIT. However, from the definition of WAMIT C55 = ρg(

∫waterplane x

2dS + V zb)−mgzgthe determination of this coefficient is independent of the presence of the hole. This makes sense,since considering the shape of this hole, the moment of buoyancy ρgV zb is clearly independentof the presence of the hole.


Figure 42: Shape of the rotor without the holeat 50.64 degrees, trimmed

Figure 43: Shape of the rotor with the mainhole at 50.64 degrees, trimmed

Figure 44: Meshing of the rotor without the hole at 50.64 degrees, trimmed

3.2.2 Main WAMIT Characteristics

WAMIT calculations are mainly parameterized using 3 different files :Geometric, Poten, Forceand Config file. A more detailed explanation of the function of each file can be found in theWamit manual [9].

Note : All the option managed by the config file have been let null.

3.2.2.1 Poten File.potThis first file has for purpose to define:


• the position of the floating body into the global coordinate system. For the case of theWEPTOS rotor it has been placed at (0, 0,−107 mm)

• which degree of freedom will be study in terms of both radiation and diffraction. In thiscase only the pitch motion has been studied for both radiation and diffraction.

• the number of waves that are going to be studied, and their periods. It has been chosento evaluate 256 waves for wave periods going from 31.4s to 0.21s. This array has beendetermine from the wave frequency. Indeed, the idea was to have a constant wave frequencystep with a maximal wave frequency large enough to be able to see the coefficients goingto their asymptotes. It was therefore needed that ω goes from 0.2 rad/s to 29.90rad/s bystep of 0.12 rad/s.

• the wave heading orientation. In this case, only frontal waves have been considered(Beta=180◦).

3.2.2.2 Force File.frcThe Force file has for purpose to define:

• which hydrodynamic parameter is going to be studied. In this case only the option 1 to 4have been setup to 1. (i.e. only the added mass, damping coefficients, exciting forces, andbody motion have been studied)

• the water density. In this case, ρ has been set up to 0.001035 g.mm−3

• the position of the center of gravity. From the experimental tests the center of massposition is (53.91 mm, 0.0 , 18.71 mm) in terms of the body coordinate system for therotor at 50.64◦ (see 2.3.3).

• the mass and the inertia. In this case only the inertia around the axle of rotation isinteresting. Therefore only this one has been input.

Mass : 4290.69 g

Inertia, I55 : 4.94E07 g.mm2

Note: The Force can be made with 2 different form, the calculation made for this report havebeen made with the second form force file.

3.2.2.3 Geometric file File.gdfThe geometric file defines the shape of the rotor. While meshing under Rhino3D, different pointhave been workout carefully :

• the panel orientation

• a coherent repartition of the panel around the shape

• a reasonable maximal aspect ratio

• no mesh point can be above the still water surface

Moreover, it is possible with WAMIT to define a plane of symmetry with respect to the planex = 0 and/or y = 0. In the case of the WEPTOS rotor a plane symmetry with respect of y=0has been used. (i.e. ISX = 0 and ISY = 1).

Finally, the geometric file also define the scale length and the gravity :

• ULEN = 1 mm

• g = 9806.65 mm/s2


3.3 WAMIT hydrodynamic coefficient

All the WAMIT hydrodynamic coefficients have been evaluated with three different meshesof 800, 1200 and 1600 panels for 256 wave frequencies between 0.2 and 30 rad/s. The rotor tiphas been placed in “positive” x and the Betah wave angle has been set at 180◦. This allows tohave waves coming from the positive x, directly hitting the tip of the rotor. As one may seefrom Figure 46 the damping coefficient has been extended to 1024 wave frequencies (i.e. to 120rad/s) through the use of a 2nd order polynomial function. The main of this extension is toextend the damping coefficient to large enough frequencies that it is almost zeros. Otherwisethe inverse Fourier transform (Eq.10) would not be accurate enough.

0 5 10 15 20 25 300

0.005

0.01

0.015

0.02

0.025

0.03

ω [rad/s]

A55

[kg.

m2 ]

A

55 for 800p

A55

for 1200p

A55

for 1600p

A55

for ω−>∞, 800p

A55

for ω−>∞, 1200p

A55

for ω−>∞, 1600p

Figure 45: WAMIT added mass in function ofthe wave frequency and the number ofpanels. With the rotor at the rest anglefor 107mm of water depth.

0 20 40 60 80 1000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

ω [rad/s]

B55

[kg.

m2 .s

−1]

B

55 for 800p

B55

for 1200p

B55

for 1600p

B55

Extended, 800p

B55

Extended 1200p

B55

Extended 1600p

Figure 46: WAMIT and extended damping co-efficient in function of the wave frequencyand the number of panels. With the ro-tor at the rest angle for 107mm of waterdepth.

0 5 10 15 20 25 300

10

20

30

40

ω [rad/s]]

|X5| [

N]

0 5 10 15 20 25 30−40

−20

0

20

40

ω [rad/s]]

Re[

X5] [

N]

0 5 10 15 20 25 30−40

−20

0

20

ω [rad/s]]

Im[X

5] [N

]

Figure 47: WAMIT exciting force (absolute,real and imaginary part) in function of thewave frequency and the number of pan-els. With the rotor at the rest angle for107mm of water depth.

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

20

25

30

35

Period [s]

Pitc

h [d

eg/c

m]

RAO for 800pRAO for 1200pRAO for 1600p

Figure 48: RAO calculated by WAMIT infunction of the wave frequency and thenumber of panels. With the rotor atthe rest angle for 107mm of water depth.At this angle the resonance frequency isabout 0.87rad/s.

From these graphs one could already see that the different meshes seem to lead to a very goodconvergence. Anyhow, the Table 10 gives the relative errors between the different mesh in termsof infinite added mass and hydrostatic stiffness.


Number of Panels A55 for ω →∞ C55

Value Relative Error Value Relative Error

800 1.07e− 02 0.54% 3.867 0.01%1200 1.07e− 02 0.01% 3.866 0.00%1600 1.07e− 02 − 3.866 −

Table 10: Comparison of the meshes in matter of hydrostatic stiffness and infinite added mass

3.4 Impulse response function

The two time dependent functions (see Eq.13 and Eq.14) have been plotted in Figure 49 and50. One could notice that the radiation impulse response function has only been plotted forpositive time values since it is an even function. About the diffraction impulse response it is notan even function thus it has been plot for negative time values as well. From these figures onecould note that the different meshes seems to “perfectly” converge.

−4 −3 −2 −1 0 1 2 3 4−150

−100

−50

0

50

100

time [s]

IRF

Exc

itatio

n

Excitation IRF for 800pExcitation IRF for 1200pExcitation IRF for 1600p

Figure 49: Impulse response of the excitingforce K55, as a function of the number ofPanels

0 0.5 1 1.5 2 2.5 3 3.5 4−1

−0.5

0

0.5

1

1.5

time [s]

IRF

Rad

iatio

n

Radiation IRF for 800pRadiation IRF for 1200pRadiation IRF for 1600p

Figure 50: Comparison of the retardationfunctionK55,rad, as a function of the num-ber of Panels

Finally, as one may see, both of the impulse response function tend to 0 for large |ω|. Thusit has been chosen, from these figures to limit the convolutions done for excitation moment(see Eq.13) and for the radiation moment (see Eq.14), respectively, to τ ∈ [−2.5 ; 2.5]s andτ ∈ [0 ; 2.5]s.

3.5 Comparison with the experimental data

3.5.1 Experimental and numerical Results

To compare the numerical results with the experimental ones, 8 different wave periods (seeTable 11) and 6 different wave height have been considered : 0.02 cm, 0.03 cm, 0.04 cm, 0.05 cm,0.06 cm, 0.08 cm. The results of the linear numerical model might be seen from the Figure 51.As one would have expected, the different results obtained by the linear model in the timedomain are all identical (main principle of the linear model). Moreover, the RAOs numericallyevaluated in the time domain seem to fairly well fit the one given by WAMIT in the frequencydomain. Only at the resonance one could notice that the pitch amplitude evaluated in the timedomain are lower. This seems to be the results of some lack of discretization in the IRF. It hasbeen tried to improve it and to run the matlab “ode45” function for smaller time steps but itdid not improve much the result quality.


Experiment Numerical modelWave Period Wave frequency Wave Period Wave frequency

[s] [rad/s] [s] [rad/s]

0.75 8.38 0.75 8.380.09 6.98 0.90 6.981.00 6.28 1.00 6.281.05 5.98 1.06 5.931.10 5.71 1.10 5.691.20 5.24 1.20 5.231.40 4.49 1.39 4.521.80 3.49 1.81 3.47

Table 11: Wave period used for the comparison between the linear numerical model and the exper-imental one.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

30

35

Wave Period [s]

RA

O [d

eg/c

m] (

Pitc

h A

mpl

itude

[deg

] / (

Wav

e A

mpl

itude

[m] *

100)

Wamit RAONum with H=0.02Num with H=0.03Num with H=0.04Num with H=0.05Num with H=0.06Num with H=0.08Exp with H=0.02Exp with H=0.03Exp with H=0.04Exp with H=0.05Exp with H=0.06Exp with H=0.08

Figure 51: Comparison of the experimental and numerical (in frequency and time domain) RAO.The results are given in terms of pitch amplitude degree for 1 cm wave amplitude.

Thus it is now interesting to compare the pitch motion obtained in the time domain withthe experimental calculation. The Figure 51 shows the 6 experimentally evaluated ResponseAmplitude Operator (RAO : pitch amplitude, in degree, divided by the wave amplitude, incentimeter.). Each of these RAOs correspond to one the considered wave height.

Note : The wave height used to normalize the experimental value of the RAO are the onegiven in annex from the experiment of the 14/06.

From this figure different conclusion maybe deduced:

• The resonance period around 0.90s evaluated by WAMIT is fairly different from the oneexperimentally obtained (around 1-1.05 s).

• The different experimental RAOs are largely varying in function of the wave height. Thismeans that the rotor does not react linearly in function of the wave height.

• The RAO obtained for H=0.02m and H=0.03m, this might indicate that for these smallwave height the rotor seems to react linearly.

• All the RAOs (excepted maybe the on for H=0.08m) are equivalent for the wave periodof 1.20s, 1.40s.

• For T=0.75s and T=1.80s, the 5 RAOs obtained from the 5 smallest wave height are alsoalmost equivalent, even if it is less clear than for the 2 previously called wave periods.


The two tables 12 and 13 detail the values behind the Figure 51. From the second table it iseven more clear that for T=1.20, T=1.40 and also T=0.75 the RAO obtained for the differentwave height are very close.

T 0.75 0.90 1.00 1.05 1.10 1.20 1.40 1.80H Units [s] [s] [s] [s] [s] [s] [s] [s]

0.02 [m] 8.97 20.68 30.12 16.69 13.68 7.43 5.75 4.550.03 [m] 14.37 29.34 47.40 31.10 22.33 11.50 8.55 7.840.04 [m] 18.36 35.07 54.01 57.12 34.56 15.64 11.83 11.310.05 [m] 24.05 39.95 57.87 60.88 43.55 19.56 16.17 16.200.06 [m] 26.41 46.12 60.29 63.24 58.16 26.40 19.86 22.740.08 [m] 29.82 52.00 69.48 71.76 70.20 37.42 27.34 33.61

Table 12: Pitch amplitude experimentally obtained for the different regular wave states. Resultsgiven in degree.


0.02 [m] 8.97 20.68 30.12 16.69 13.68 7.43 5.75 4.550.03 [m] 9.58 19.56 31.60 20.73 14.89 7.66 5.70 5.230.04 [m] 9.18 17.53 27.00 28.56 17.28 7.82 5.91 5.660.05 [m] 9.62 15.98 23.15 24.35 17.42 7.82 6.47 6.480.06 [m] 8.80 15.37 20.10 21.08 19.39 8.80 6.62 7.580.08 [m] 7.45 13.00 17.37 17.94 17.55 9.35 6.84 8.40

Table 13: Pitch amplitude experimentally obtained for the different regular wave states normalizedby the wave amplitude (RAO). Results given in degree/cm.

Finally the pitch amplitude of the numerical calculation are given in the Table 14 and therelative difference between these results and the experimental ones are given by the Table 15.


0.02 [m] 12.01 29.17 18.24 13.18 10.85 7.83 5.79 4.370.03 [m] 17.95 43.75 27.34 19.75 16.28 11.75 8.69 6.560.04 [m] 24.01 58.33 36.48 26.35 21.73 15.68 11.59 8.740.05 [m] 29.98 72.92 45.53 32.93 27.13 19.57 14.49 10.930.06 [m] 35.91 87.50 54.66 39.49 32.56 23.50 17.38 13.110.08 [m] 47.96 116.66 72.96 52.72 43.41 31.35 23.18 17.48

Table 14: Pitch amplitude numerical obtained for the different regular wave states. Results givenin degree, for the linear numerical model.

Note: The Table 15 gives the relative error between the pitch amplitude measured duringthe experiments and the one obtained through the linear numerical model (i.e. Error =Apitch,num −Apitch,exp

Apitch,exp∗ 100. Thus considering this relative error one should understand that

the 7.75% error from the case H=0.02m and T=1.20s actually represents a pitch amplitudedifference of 0.5◦.

From these tables one may conclude :



0.02 [m] 34.68% 22.03% -39.07% -17.85% -18.37% 5.78% -1.12% -4.62%0.03 [m] 26.35% 34.27% -43.10% -34.39% -26.71% 1.81% -1.37% -13.69%0.04 [m] 28.26% 56.27% -34.13% -52.86% -37.37% -0.68% -4.19% -20.53%0.05 [m] 23.63% 80.71% -27.40% -43.93% -37.19% -3.15% -11.94% -28.61%0.06 [m] 35.09% 84.77% -11.00% -37.52% -44.03% -11.50% -12.11% -37.56%0.08 [m] 54.02% 116.54% 4.62% -26.40% -37.42% -14.61% -13.27% -42.34%

Table 15: Relative pitch amplitude difference between the numerical model and the experimentalmeasurement. Results given in terms of relative errors, for the linear numerical model.

• As expected the numerical model fits particularly well the experimental measurementobtained for T=1.20s and T=1.40s. It seems, indeed, that for these wave frequencies therotor fits the linear theory.

• The main errors between the numerical model and the experimental measurements areobtained near the resonance period. This might be explained by the fact that for thisperiod the pitch amplitude becomes very large and thus the linear theory reaches itslimits.

• Moreover, one should be careful to consider all the tables in their ensemble and not value byvalue. Indeed, just by considering the Table 15 for T=1s and H=0.08m, one might concludethat for this wave period and height the numerical model fits very well the experimentalmeasurement, which is true. However, by looking at the Figure 51 one should understandthat it is more by “luck” than anything else that the numerical model fits the measurementfor this wave amplitude and period. Furthermore, this issue point out the main interestof having being able to run new experiment with more regular wave states tested.

3.5.2 Analysis of the results in the time domain

To give an idea of the result solution and to show some surprising effect, four figures repre-senting the results in the time domain have been plotted here. One may find all the time domainresults in Annex.

0 2 4 6 8 10 12 14 16 18 20−8

−6

−4

−2

0

2

4

6

8

Time [s]

Pitc

h [d

eg]

ExperimentalNumerical

Figure 52: Comparison between the numeri-cal model and the experimental measure-ment. For H=0.02m and T=1.20s, con-sidering a linear numerical model.

0 2 4 6 8 10 12 14 16 18 20−30

−20

−10

0

10

20

30

Time [s]

Pitc

h [d

eg]



From the figures Figure 52 and Figure 53 one could confirm that for T=1.20s the numericalmodel seems to perfectly fit the experimental pitch measurement for all the wave height fromH = 0.02m to H = 0.06m. While from the figures Figure 54 and Figure 55 one could see how forthe same wave heights the numerical largely diverge from the experimental model. Furthermore,from the latest figure it might be seen the surprising pitch motion of the rotor (visually confirmedduring the experiment). This non sinusoidal signal is probably due to a large and quick restoring


0 2 4 6 8 10 12 14 16 18 20−100

−50

0

50

100

Time [s]

Pitc

h [d

eg]



0 2 4 6 8 10 12 14 16 18 20−30

−20

−10

0

10

20

30

Time [s]

Pitc

h [d

eg]



0 2 4 6 8 10 12 14 16 18 20−40

−30

−20

−10

0

10

20

30

40

Time [s]

Pitc

h [d

eg]



0 2 4 6 8 10 12 14 16 18 20−60

−40

−20

0

20

40

60

80

Time [s]P

itch

[deg

]



moment of the rotor compare to the wave period. This phenomenon can not be simulate by asimple linear numerical model, it is therefore necessary to improve it and to add some non-lineareffects into the calculation.

Finally from Figures 56 and 57 and considering the reflexion phenomenon foreseen in section2.5.2 one could see that the numerical underestimate the pitch motion even without consideringthe reflexion. Indeed, from Figures 36 and 37 one would notice that before any phenomenonof reflexion the pitch motion were respectively contains between [-25◦ ; +29◦] and [-34◦ ; +39◦]which is still larger than the numerically evaluated pitch motion. For so large wave amplitude it isnot very surprising that the linear numerical model does not fit the experimental measurements.

3.6 Limitation of the numerical model

As it might have been seen so far, the numerical model is far from perfectly fitting theexperimentally measured pitch motion. This large difference might be explained in certain casesby the following limitations of the linear model.

3.6.1 Wave Amplitude and Non linearity

The numerical calculations run by WAMIT have for hypothesis to consider small wave am-plitude compare to the considered floating object. In this case the rotor is about 326 mm long,214 mm wide, 240 mm deep, thus a wave height of 80 mm is large compare to the rotor di-mensions. While wave height of 20 mm can be considered as reasonably small compare to therotor dimension. Furthermore, from the snapshot display in Figure 58 one could see that indeedthe 8 cm height incoming waves seems very large compare to rotor. This wave amplitude willnecessarily generates strong non-linear effects not considered in the linear theory assumption.

Furthermore, while comparing the numerical simulation with the experimental measurements,the non linearities of the laboratory waves might also have an influence.


Figure 58: Snapshot from video during the experiments at AAU. The condition are for regular wavewith a wave height of 8 cm

To evaluate how linear/non linear is a given regular wave state (defined by its wave height,wave period and depth) two linearity factors might be found in the literature.

• A “simple one” is HL with H the wave height and L the wave length. This coefficient should

be smaller than 2-3% in order to “guaranty” linear waves.

• A more detailed one, which defines regular waves as linear if kAtanh(kh) << 1 (with k = 2π

L ,L the wave length, A the wave amplitude and h the water depth).

One could already note that for large water depth (i.e. tanh(kh)→∞) this second coefficientis equivalent to πH

L . There is in this case only a factor π between the two coefficients.

During the experiment the water depth was about 0.81 m and the wave amplitude is from0.02 m to 0.04 m. The Table 17 summarize the linearity coefficients obtained for different waveperiods. The second linearity coefficient lead to the same kind of results.

Test with h = 0.81 m 1 2 3 4 5 6 7 8

Wave period T [s] 0.75 0.90 1.00 1.05 1.10 1.20 1.40 1.80Wave frequency ω [rad/s] 8.38 6.98 6.28 5.98 5.71 5.24 4.49 3.49Wave length L [m] 0.88 1.26 1.56 1.71 1.87 2.20 2.89 4.22Wave number k [m−1] 7.15 4.97 4.04 3.67 3.35 2.85 2.18 1.49

Table 16: Wave frequency, length and number for all the wave periods run during the experiment.All the calculation have been done for a water depth h = 0.81 m


H [m] / T[s] 0.75 0.90 1.00 1.05 1.10 1.20 1.40 1.80

0.02 2.28% 1.81% 1.16% 0.68% 0.36% 0.16% 0.06% 0.01%0.03 3.42% 2.71% 1.74% 1.02% 0.54% 0.25% 0.09% 0.02%0.04 4.55% 3.61% 2.32% 1.36% 0.72% 0.33% 0.11% 0.03%0.05 5.69% 4.52% 2.90% 1.69% 0.90% 0.41% 0.14% 0.03%0.06 6.83% 5.42% 3.48% 2.03% 1.09% 0.49% 0.17% 0.04%0.08 9.11% 7.23% 4.64% 2.71% 1.45% 0.66% 0.23% 0.05%

Table 17: Sum up of the linearity coefficient H/L for all the regular wave tests.

3.6.2 Large motion angle amplitude

One of the first phenomenon that appears has obviously non-linear, when looking at the rotorin action, is its very large pitch motion amplitude. As it might have been notice from thefigure section 3.5 even for “small” wave amplitude the rotor pitch amplitude are very large. Oneshould keep in mind that the WAMIT calculation have been run for the wetted surface at therest position only.

Figure 59: Snapshot of the extreme down rotorposition for H=0.05m and T=1.00 s

Figure 60: Snapshot of the extreme up rotorposition for H=0.05m and T=1.00 s

To get a better understanding on the influence of this large pitch amplitude the Figures 59and 60 show the rotor close to its extreme angle for H = 0.05 m and T = 1.00 s. From thesetwo graphs it is easy to understand why for large wave amplitude and/or large pitch amplitudethe simple linear model can’t converge to reliable results.

Moreover, the Figure 61 shows the evolution of the infinite added mass in function of thepitch angle(i.e. through WAMIT the added mass for an infinite frequency has been evaluatedfor different rotor angles). One could notice from this figure that :

• The infinite added mass largely varies with the rotor angle. There is a factor of almost5 between the infinite added mass at the rest position and the maximal extreme oneevaluated for pitch angle = -120◦.

• The minimum infinite added mass is evaluated for an pitch angle of 40◦ (where the rotoris vertical). Moreover, considering the shape of the rotor it is not surprising to find asymmetric evolution of the added mass compare to this angle.

• The value used by the numerical model (i.e. the one evaluated for Pitch=0◦) is fairly smallcompare to the infinite added mass evaluated for most of the other angle.


−150 −100 −50 0 50 100 1500

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Angles [deg]

A55

,∞ [k

g.m

2 ]

Figure 61: Evolution of the added mass at infinite frequency in function of the rotor pitch angle.Pitch equal zero degree is equivalent to the rest position.

3.6.3 Slamming phenomenon

Another phenomenon that appears while watching the video of the WEPTOS rotor, is thatfor some wave states the oscillations are such that the rotor starts slamming while falling downback into the water. Indeed, it seems that at least the amplitude taken by the pitch are soimportant that they lead to slamming phenomenon (see snapshot displayed in Figure 62). Thisphenomenon is difficult to integrate into the numerical model but one should keep it in mindwhile analyzing time domain results (espescially for large wave height and at the resonanceperiod).

Figure 62: Snapshot of the rotor slamming. Wave height 5cm, wave period 1s, regular waves

3.6.4 Experimental wave reflection

The experimental tests have been run in a wave flume detailed in section 2.2.4 which has beenvery much filled in for this tests. Therefore, even if the beach has been improved, some reflectionmight appeared and deteriorated the reliability of the tests. Indeed, from the Section 2.5.2 onewould notice that some reflexion issue have been notice for some wave states. It seems that this


issue mainly appears for wave period around 1.10 s.Therefore, the reader might keep in mindthat this reflexion might have affected the quality of the measurement.


4 Solutions to improve and/or extend the numerical model

As it has been seen in the previous section, the linear numerical model shows some divergencefrom the measurements made during the free motion under regular waves experiments. However,the model used so far is fairly simple and some improvement can be made. For this master thesisthree main improvements have been added to the linear model :

• Addition of a friction moment

• Addition of non-linear hydrostatic restoring moment (function of both the pitch angle andthe water elevation)

• Addition of non-linear hydrodynamic moments (function of the pitch angle only)

4.1 Addition of a friction moment

4.1.1 Details on the friction moment

As it as been seen through the inertia experimental study, there is quite some friction inthe liaison axle/structure. This friction played an important role while the rotor was in air.Therefore one might wonder if it is negligible or not while the rotor moves under regular waves.

To run the calculation with a friction moment, a momentMfric (see Eq.5), has been evaluatedusing (ν1, ν2) = (1.8e-03,1.00e-02) and the force Faxle used is described by the Eq.16.

~Faxle = ~Fgravity + ~Fcentrifugal + ~Fbuoyancy (16)

With :

• Fgravity and Fcentrifugal defined as detail in Eq.4.

• Fbuoyancy = ρg ·V olumesubmerged(z5, h).The submerged volume has been evaluated for 14angles and 5 water depth (see Tables 20 en annex NL).

Note : Some extra hydrodynamic forces in surge and heave have been considered as negligible.

4.1.2 Analysis of the RAO

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

30

35

Wave Period [s]

RA

O [d

eg/c

m] (

Pitc

h A

mpl

itude

[deg

] / (

Wav

e A

mpl

itude

[m] *

100)


Figure 63: Comparison of the experimental and numerical (in frequency and time domain) RAO.The results are given in term of pitch amplitude [deg] for 1 cm wave height.

The pitch amplitude of the numerical calculations are given in the Table 18 and the relative


difference between these results and the experimental one are given by the Table 19.


0.02 [m] 10.30 25.51 17.06 12.65 10.49 7.69 5.72 4.280.03 [m] 16.39 39.57 26.46 19.23 16.16 11.65 8.78 6.580.04 [m] 22.38 54.68 35.47 26.06 21.41 15.63 11.58 8.650.05 [m] 28.33 69.09 45.17 32.66 27.04 19.68 14.52 10.940.06 [m] 34.27 84.55 53.59 39.24 32.57 23.66 17.55 13.090.08 [m] 46.11 111.06 71.93 52.80 43.36 31.52 23.33 17.68

Table 18: Pitch amplitude numerical obtained for the different regular wave states. Results givenin degree, for the linear numerical model considering the structural friction.


0.02 [m] 15.44% 6.75% -43.02% -21.12% -21.12% 3.80% -2.31% -6.59%0.03 [m] 15.34% 21.44% -44.94% -36.09% -27.25% 0.92% -0.43% -13.29%0.04 [m] 19.54% 46.49% -35.95% -53.37% -38.28% -0.99% -4.23% -20.87%0.05 [m] 16.81% 71.22% -27.98% -44.40% -37.41% -2.62% -11.74% -28.88%0.06 [m] 28.90% 78.54% -12.74% -37.91% -44.02% -10.89% -11.29% -38.41%0.08 [m] 48.07% 106.14% 3.13% -26.29% -37.51% -14.15% -12.74% -41.68%

Table 19: Relative pitch amplitude difference between the numerical model and the experimentalmeasurement.Results given in terms of relative errors, for the linear numerical model consideringthe structural friction.

Different conclusions might be deduced from the figure 63 and the Tables 18 and 19 :

• As one would have notice from the RAO (see Figure 65) the numerical results are varyingin function of the wave height. This is due to the non linearity of the structural frictionwhich is dependent of the pitch velocity.

• Even if this friction moment is not very important compare to the different hydrodynamicmoment, it seems that it has some effect on the pitch motions. These damping effects areespecially visible for small wave period (i.e. large pitch velocity) and for these periods thenumerical pitch motions were larger than the experimentally measured ones. Therefore, itseems that indeed this friction helps the numerical model to globally fit the actual rotormotions.

This structural friction moment is now always considered in the calculations presented in thefollowing sections of this report. All the following calculation have also been run without it,however it always appears that this friction helps to get more accurate results.

4.2 Addition of a non linear hydrostatic restoring moment

A first solution to improve the solution could be to evaluate the hydrostatic restoring momentof the rotor for different pitch angles instead of using a purely linear restoring moment −C55z5(t)(see Eq.15). Indeed, one of the main limitations of our model is that the rotor describes quitelarge pitch amplitude. Therefore the assumption of a linear hydrostatic moment might not bevery accurate.


4.2.1 Details of the solutions used

The main idea here is to exchange the linear hydrostatic moment “−C55z5(t)” with a nonlinear version of the hydrostatic moment. The hydrostatic restoring moment (in the case ofpitch motion) is the sum of the moment generated by the gravity and the moment generatedby the buoyancy. These two moments have been evaluated for 14 different angles and 5 waterdepths. The gravity moment has simply been evaluated through trigonometric function, whilefor the buoyancy one, it has been necessary to use Rhino3D to evaluate the submerged volumeand the center of the position of buoyancy for each angle. The results for these 14 angles canbe seen in Table 20.

h / deg -120 -100 -80 -60 -40 -20 0

-04 2.69 3.79 4.38 4.22 2.83 0.34 -0.96-02 2.69 3.79 4.43 4.47 3.53 1.13 -0.5000 2.69 3.79 4.44 4.55 4.01 2.00 0.0202 2.69 3.79 4.44 4.55 4.11 2.85 0.5704 2.69 3.79 4.44 4.55 4.11 3.18 1.15

linear 8.10 6.75 5.40 4.05 2.70 1.35 0.00

h / deg 20 40 60 80 100 120 140

-04 -1.35 -1.24 -0.98 -0.95 -1.62 -3.31 -3.79-02 -1.14 -1.24 -1.20 -1.42 -2.42 -3.99 -4.0400 -0.90 -1.24 -1.44 -1.95 -3.30 -4.46 -4.1102 -0.64 -1.25 -1.71 -2.51 -4.14 -4.55 -4.1104 -0.37 -1.25 -1.98 -3.09 -4.44 -4.55 -4.11

linear -1.35 -2.70 -4.05 -5.40 -6.75 -8.10 -9.45

Table 20: Non linear restoring moment evaluated for 14 angles and 5 water depths around the restposition (angle : 50.5◦ and depth : 107mm above axle).

Note : The equivalent linear hydrostatic moment (used so far) has also been placed intothe table to compare its values with the non-linear ones. The linear hydrostatic moment is :−C55 ·Z5. With C55 the hydrostatic coefficient evaluated in WAMIT for the rotor at 55◦ andZ5 the pitch angle in radians.

The Figure 64 show the linear and the non linear hydrostatic moment evolution in functionof the pitch angles. As one might see from this figure the non-linear restoring moment issubstantially smaller than the linear one for most of the pitch angles and depths. Thereforeone could imagine that using a non linear hydrostatic restoring moment will engender largeramplitudes than the linear ones.


Moreover the pitch amplitude of the numerical calculations are given in the Table 21 and therelative difference between these results and the experimental one are given by the Table 22.

Different conclusions might be deduced from the figure 65 and the Tables 21 and 22:

• The first interesting thing to point out is that the peak frequency has moved from about0.90 s to 1.05 s. This is a first indication that to numerically analyze the rotor pitchmotion it is necessary to use a non-linear model.

• As said and seen from the Figure 65 the non-linear hydrostatic restoring moment is sub-stantially smaller than the linear one for “large amplitude”. This weak restoring moment


−150 −100 −50 0 50 100 150−10

−8

−6

−4

−2

0

2

4

6

8

10

Angles [deg]

Hyd

rost

atic

res

torin

g m

omen

t [N

m]

Linearh=−0.04mh=−0.02mh=0.00mh=0.02mh=0.04mSign limit

Figure 64: Linear and non-linear hydrostatic restoring moment in function of the rotor pitch angle

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

10

20

30

40

50

60

Wave Period [s]

RA

O [d

eg/c

m] (

Pitc

h A

mpl

itude

[deg

] / (

Wav

e A

mpl

itude

[m] *

100)


Figure 65: Comparison of the experimental and numerical (in frequency and time domain) RAO.The results are given in term of pitch amplitude [deg] for 1 cm wave height. The time domaincalculation have been run with the linear model including a non-linear hydrostatic restoringmoment and structural friction moment


0.02 [m] 12.00 24.37 21.33 12.00 9.50 6.55 4.00 1.990.03 [m] 17.06 30.99 39.64 44.30 15.52 10.15 5.94 3.090.04 [m] 21.28 36.09 44.85 107.03 99.26 14.06 8.05 4.230.05 [m] 24.88 39.55 51.62 118.80 127.01 18.30 10.24 5.930.06 [m] 29.33 44.06 95.95 127.90 144.22 23.34 12.47 6.990.08 [m] 49.73 52.59 108.71 135.26 186.97 231.81 17.22 10.23

Table 21: Pitch amplitude numerical obtained for the different regular wave states. Results givenin degree, for the linear numerical model implemented with a non-linear restoring moment andconsidering the structural friction.

explains the very large pitch motion obtained for wave heights larger than H = 0.04 mnear the peak period.



0.02 [m] 34.56% 1.96% -28.77% -25.21% -28.58% -11.00% -31.80% -56.23%0.03 [m] 20.09% -4.88% -17.47% 46.95% -30.14% -12.06% -32.57% -59.31%0.04 [m] 13.64% -3.31% -19.17% 91.37% 186.13% -10.81% -33.44% -61.54%0.05 [m] 2.60% -1.98% -17.41% 102.27% 193.99% -8.63% -37.78% -61.31%0.06 [m] 10.35% -6.96% 56.25% 102.27% 145.78% -11.11% -36.94% -66.73%0.08 [m] 59.72% -2.43% 55.90% 88.94% 169.48% 531.38% -35.72% -66.28%

Table 22: Relative pitch amplitude difference between the numerical model and the experimentalmeasurement. Results given in terms of relative errors, for the linear numerical model imple-mented with a non-linear restoring moment and considering the structural friction.

• Finally, for H = 0.02 m H = 0.03 m and H = 0.04 m, considering wave periods differentof 1.05 s and 1.10 s (i.e. “far” from the resonance) the numerical model compute pitchmotions fairly close to the experimental ones.


To give an idea of the time domain solutions and to show some surprising effect, eight figuresrepresenting the pitch motions in the time domain have been plotted here. One may find all thetime domain results in Annex.

0 2 4 6 8 10 12 14 16 18 20−40

−20

0

20

40

60

Time [s]

Pitc

h [d

eg]


Figure 66: Comparison between the numeri-cal model and the experimental measure-ment. For H=0.03m and T=1.00s, consid-ering a linear numerical model includingnon-linear hydrostatic restoring momentand the structural friction.

0 2 4 6 8 10 12 14 16 18 20−60

−40

−20

0

20

40

60

80

Time [s]

Pitc

h [d

eg]



0 2 4 6 8 10 12 14 16 18 20−30

−20

−10

0

10

20

30

Time [s]

Pitc

h [d

eg]



0 2 4 6 8 10 12 14 16 18 20−100

−50

0

50

100

150

Time [s]

Pitc

h [d

eg]




0 2 4 6 8 10 12 14 16 18 20−15

−10

−5

0

5

10

15

Time [s]

Pitc

h [d

eg]



0 2 4 6 8 10 12 14 16 18 20−20

−15

−10

−5

0

5

10

15

20

Time [s]

Pitc

h [d

eg]



0 2 4 6 8 10 12 14 16 18 20−10

−5

0

5

10

Time [s]

Pitc

h [d

eg]



0 2 4 6 8 10 12 14 16 18 20−20

−15

−10

−5

0

5

10

15

20

Time [s]

Pitc

h [d

eg]



Different point maybe raised from the time domain Figures 66 to 73 :

• From the six first figures one could confirm that for smaller wave height than H=0.04mthe numerical calculations lead to equivalent results than the ones measured during theexperiment, for wave periods different from the resonance frequency.

• For T=1.10s the numerical results are almost twice larger than the experimental measure-ments. This can be explained by the large pitch amplitude of the rotor at this frequency.This remark tend to confirm the need of considering a full non-linear hydrodynamic com-putation model.

• From the two lasts figures one could notice that even with addition of a non-linear hydro-static restoring moment the numerical model does still not fit the very non-linear shape ofpitch motions obtain for T=1.80s.


4.3 Fully non linear numerical model

4.3.1 Theory

Finally, considering that the model is still not able to properly fit the experimental mea-surements, probably due to the too large change of hydrodynamic response of the actual rotorin function of its pitch angle, it has been chosen to run a numerical model using non-linearhydrodynamic coefficient.

The idea is to use WAMIT to evaluate the hydrodynamic coefficients for different angle (i.e.every 20◦ from -120◦ to 120◦ compare to the rest position). And to evaluate the infinite addedmass as well as the both excitation and radiation impulse response function for each angle.

Finally, it would be possible to evaluate the two hydrodynamic moments for any angle througha linear extrapolation of the impulse response, see Equation 19.

Mex(t, z5(t)) =∞∫−∞

K5,diff (τ, z5(t))η(t− τ)dτ (17)

Mrad(t, z5(t)) = A∞(z5 (t)) · z5 (t) +t∫0

K55,rad(t− τ, z5(t)) z5(τ)dτ (18)

One could notice from Eq.17 and Eq.18 that the impulse response are extrapolated to beevaluated for a given angle z5(t) and then the convolution is made as previously. One wouldconsider than an improvement of this calculation could be to use varying excitation and radiationimpulse response into the convolution. For example for the radiation instead of using :∞∫−∞

K5,diff (τ, z5(t))η(t− τ)dτ ,

this expression could have been used :∞∫−∞

K5,diff (τ, z5(τ))η(t− τ)dτ .

However, the latest expression is much more complicated to implement into the numericalmodel and its mathematical reliability is not clear. It has therefore been chosen to focus on thefirst one.

z5(t) =Mex(t, z5(t))−Mrad(t, z5(t))−Mhs(z5t))

M55

z5(t) =

∞∫−∞

K5,diff (τ, z5(t)) η(t− τ)dτ −t∫0

K55,rad(t− τ, z5(t)) z5(τ)dτ −Mhs(z5(t))

M55 +A∞(z5(t))(19)

Note : Obviously in this calculation the restoring moment is kept non-linear as seen Section4.2

4.3.2 WAMIT hydrodynamic coefficients

It appears to be very difficult to make WAMIT able to compute certain rotor shape. Indeed,for pitch angles of -60◦, -40◦ and 120◦ it appears that WAMIT had great difficulties to obtain aconverging results for both the radiation and the excitation. As one would notice from Figures75, 74 and 76 the mesh have been very finely discretized (about 8000 panels). However, evenwith this discretization WAMIT could not give a converging results for wave periods smaller


than T = 0.40 s (while for the others angles WAMIT has been run until T = 0.20 s). Differentoptions (such as ILOG, IDIAG...) and meshes have been tried. It appears that the best resultis obtained with an fine and homogeneous meshing all around the rotor and fairly large on theside (in order to limit the number of panels used by the meshes). One conclusion from this lackof accuracy could come from the fact that the smaller the wave the less accurate/converging arethe calculation. And for these angles, the smallest error in the calculation of the hydrodynamicpressure on the top or the bottom side of the rotor tip would lead to a large difference of momentat the center of rotation.

Figure 74: Meshing of the rotor at -10◦ (-60◦compare to the rest position), trimmed

Figure 75: Meshing of the rotor at 10◦(-40◦compare to the rest position), trimmed

Figure 76: Meshing of the rotor at 160◦ (+120◦ compare to the rest position), trimmed

Note : The rotor at -60◦ see Figure 74 is actually fully underwater. A finer discretization ofthe top of the rotor has been tried out without much better results of convergence.

It appears that for T = 0.40 s (ω = 15.15rad/s) both the exciting moment and the dampingare close from 0, see Figures 77, 78, 79 and 79. It has therefore been chosen to consider themhas null for wave frequencies above ω = 15.15rad/s.

Finally the impulse response function are given in function of the time and the pitch anglesin Figures 81,82, 83 and 84.


0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

ω [rad/s]

B55

[kg.

m2 .s

−1 ]

Angle : 0°Angle : −20°Angle : −40°Angle : −60°Angle : −80°Angle : −100°Angle : −120°

Figure 77: Damping B55 in function of thewave frequency for the rotor angles from-120◦ to 0◦ compare to the rest position.

0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

ω [rad/s]

B55

[kg.

m2 .s

−1 ]

Angle : 0°Angle : 20°Angle : 40°Angle : 60°Angle : 80°Angle : 100°Angle : 120°

Figure 78: Damping B55 in function of thewave frequency for the rotor angles from0◦ to +120◦ compare to the rest position.

0 5 10 15 20 250

50

100

150

ω [rad/s]]

|X5| [

Nm

.m−

1 ]

0 5 10 15 20 25−50

0

50

100

150

ω [rad/s]]

Re[

X5] [

Nm

.m−

1 ]

0 5 10 15 20 25−100

−50

0

50

100

ω [rad/s]]

Im[X

5] [N

m.m

−1 ]

Figure 79: Exciting force X5 in function of thewave frequency for the rotor angles from-120◦ to 0◦ compare to the rest position.Note: the colors are identical to the onegiven Figure 78

0 5 10 15 20 250

20

40

60

80

ω [rad/s]]

|X5| [

Nm

.m−

1 ]

0 5 10 15 20 25−50

0

50

100

ω [rad/s]]

Re[

X5] [

Nm

.m−

1 ]

0 5 10 15 20 25−50

0

50

100

ω [rad/s]]

Im[X

5] [N

m.m

−1 ]

Figure 80: Exciting force X5 in function of thewave frequency for the rotor angles from0◦ to +120◦ compare to the rest position.Note: the colors are identical to the onegiven Figure 78

0 2 4 6 8 10 12−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time [s]

Kra

d [Nm

]


Figure 81: K55,rad the radiation impulse-response function due to an impulsive ve-locity of the body, given for the rotor an-gles from -120◦ to 0◦ compare to the restposition.

0 2 4 6 8 10 12−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time [s]

Kra

d [Nm

]


Figure 82: K55,rad the radiation impulse-response function due to an impulsive ve-locity of the body, given for the rotor an-gles from 0◦ to +120◦ compare to the restposition.


−4 −2 0 2 4 6 8 10 12−200

−150

−100

−50

0

50

100

Time [s]

KD

iff [N

m.m

−1 .s

−1 ]


Figure 83: K5,diff the diffraction impulse-response function, given for the rotor an-gles from -120◦ to 0◦ compare to the restposition.

−4 −2 0 2 4 6 8 10 12−150

−100

−50

0

50

100

Time [s]

KD

iff [N

m.m

−1 .s

−1 ]


Figure 84: K5,diff the diffraction impulse-response function, given for the rotor an-gles from 0◦ to +120◦ compare to the restposition.

Note :

• From the Figures 81 and 84 one could see that the IRF obtained for the rotor at -60◦, -40◦

and 120◦ are larger than the other ones. This can be explained by two reasons :

– The lack of evaluated frequencies lead to unreliable results and thus these IRF arenot correct.

– For such angle the IRF are actually very large. This may make sense considering theunderwater shape of the rotor for these angles.

• One would notice that for the non-linear IRF the convolutions will respectively be donefor τ ∈ [0.0, 12.0]s and τ ∈ [−12.0, 12.0]s.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

30

35

Wave Period [s]

RA

O [d

eg/c

m] (

Pitc

h A

mpl

itude

[deg

] / (

Wav

e A

mpl

itude

[m] *

100)


Figure 85: Comparison of the experimental and numerical (in frequency and time domain) RAO.The results are given in term of pitch amplitude [deg] for 1 cm wave height. The time domaincalculation have been run with full non-linear hydrodynamic coefficients.

The pitch amplitude of the numerical calculations are given in the Table 21 and the relativedifference between these results and the experimental one are given by the Table 22.

Different conclusions might be deduced from the Figure 85 and the Tables 23 and 24:

• For small and large wave periods the results are fairly equivalent to the one previouslyobtained. For H = 0.02 m it also seems that adding the full non-linear hydrodynamic



0.02 [m] 11.56 26.35 17.99 12.11 9.62 6.63 3.98 2.030.03 [m] 16.19 34.27 41.27 22.06 16.14 10.74 6.12 2.920.04 [m] 19.85 39.46 52.99 48.41 25.01 15.27 8.35 3.890.05 [m] 22.77 42.89 61.96 58.43 51.89 20.06 10.76 5.100.06 [m] 27.22 46.10 61.65 68.31 53.30 86.92 13.49 6.610.08 [m] 49.14 53.47 108.24 102.40 62.06 92.79 19.25 10.34

Table 23: Pitch amplitude numerical obtained for the different regular wave states. Results givenin degree, for the a full non-linear numerical model, considering the structural friction.


0.02 [m] 29.60% 10.26% -39.94% -24.53% -27.68% -9.86% -32.07% -55.24%0.03 [m] 13.97% 5.17% -14.08% -26.70% -27.35% -6.96% -30.60% -61.61%0.04 [m] 6.05% 5.71% -4.30% -13.34% -27.90% -3.17% -30.94% -64.63%0.05 [m] -6.09% 6.30% -0.86% -0.52% 20.11% 0.17% -34.58% -66.68%0.06 [m] 2.40% -2.64% 0.39% 8.02% -9.17% 227.31% -31.80% -68.54%0.08 [m] 57.61% -1.78% 55.23% 42.96% -10.55% 152.74% -28.01% -65.88%

Table 24: Relative pitch amplitude difference between the numerical model and the experimentalmeasurement. Results given in terms of relative errors, for the a full non-linear numerical model,considering the structural friction.

coefficient does not change much the accuracy of the numerical model results. For suchsmall waves the main errors may come from the experimental accuracy.

• From wave height of 0.03 m to 0.05 m the accuracy of the results seems constant andfairly good. The remaining errors are probably due to the fact that the hydrodynamicnon-linear coefficient should have also been evaluated for different water level (i.e. as ithas been done for the restoring hydrostatic moment), see Section 4.4.


0 2 4 6 8 10 12 14 16 18 20−40

−20

0

20

40

60

Time [s]

Pitc

h [d

eg]


Figure 86: Comparison between the numeri-cal model and the experimental measure-ment. For H=0.03m and T=1.00s, consid-ering a fully non-linear numerical modelincluding the structural friction.

0 2 4 6 8 10 12 14 16 18 20−60

−40

−20

0

20

40

60

80

Time [s]

Pitc

h [d

eg]



Different point maybe raised from the time domain Figures 86 to 91 :

• From the Figures 88 and 89 one could notice that the large motion amplitude noted with thelinear model including non-linear hydrostatic have disappeared. Therefore, a substantial


0 2 4 6 8 10 12 14 16 18 20−30

−20

−10

0

10

20

30

Time [s]

Pitc

h [d

eg]



0 2 4 6 8 10 12 14 16 18 20−40

−30

−20

−10

0

10

20

30

40

Time [s]

Pitc

h [d

eg]



0 2 4 6 8 10 12 14 16 18 20−60

−40

−20

0

20

40

60

80

Time [s]

Pitc

h [d

eg]



0 2 4 6 8 10 12 14 16 18 20−60

−40

−20

0

20

40

60

80

100

Time [s]

Pitc

h [d

eg]



improvement of the model has been made there.

• For the two first figures where T = 1.00 s the numerical results appears to be fairlyirregular. As one can see from Figure 91 equivalent irregular phenomenon appears fordifferent wave state. It seems that if the pitch motions are too large and vary too quicklythe lack of angle discretization for the non-linear hydrodynamics coefficient starts to beimportant.

• Finally, by looking at the RAO one could consider that for H=0.05m and T=1.10s thenumerical model perfectly its the experimental pitch motions. However, one should re-member that for this exact wave state it has been proven that there are most probablysome issue with the reflexion and that for this wave state the pitch motion of the WEPTOSrotor would have been about 30% lower than these ones.

4.4 Remaining limitation of the models

4.4.1 Discretization of the non-linear hydrodynamic coefficients

In the previous section the hydrodynamic coefficients have been setup as a function of thepitch angle. Two critics can be made conserning this calculation :

• as it has already been said, for certain pitch angles (i.e. -60◦, -40◦ and 120◦) it has beendifficult to evaluate the hydrodynamic coefficients. For these angles, the exciting momentand damping has been crudely estimate for large wave frequencies. This approximationmay have lead to unreliable results.

• in the same idea it has not been possible to do a check of the convergence of the resultsfor different meshes, as it has been done for the rest position in section 3.3.

• this calculation might have necessitated a more detailed angle discretization (evaluation


of the hydrodynamic coefficients every 10◦ for example). However, considering the giventime for this master thesis and the lack of convergence of WAMIT calculations for certainrotor shape it has not been possible to go into a more detailed discretization.

• last but not least, to reach better results it would have been necessary to evaluate thesehydrodynamic coefficients for all these angles and for different water levels. Indeed, ideallythe same kind of linearization than the one made for the restoring hydrostatic momentwould have been necessary. As one may see from the Figure 92 the added mass for infinitefrequency has been evaluated for 4 water level around the still water level. From this figureone may appreciate the water level dependence of the added mass. This large variationmay be explained by the fact that adding one cm of water on top of still water level willgenerate more force far from the axle. These forces will then lead to a very large additionof moment. However, it has not been possible to go to such detailed discretization fordifferent reasons :

– mathematical reliability : it is clear that in terms of added mass and damping this“depth discretization” should not have deteriorate the reliability of the results. How-ever, in terms of the exciting force it not obvious that this discretization in terms ofwater depth is actually reliable.

– lack of convergence : the difficulties that appears to compute some of the rotor shapes,with the tip of the rotor under water, close from the surface, would have necessarilybeen more numerous.

– lack of time : this double discretization of the hydrodynamic coefficients would haverequired about 5 times more meshes and WAMIT calculations.

−60 −40 −20 0 20 40 600.005

0.01

0.015

0.02

Water level [mm]

A55

,∞[k

g.m

2 ]

Figure 92: Evolution of the added mass at infinite frequency in function of the water level for therotor at its rest position.

4.4.2 Retardation function convolution

In terms of the retardation function it has been considered that taking the radiation impulseresponse at the instant “t” and using it for the convolution was enough. However, one mightwonder if it was not necessary - or at least more accurate - to consider an history of the impulseresponse and to do the convolution with :∞∫−∞

K5,diff (τ, z5(τ))η(t− τ)dτ .)

As said in the section 4.3.1, this technique appears as not necessarily reliable and definitely


not easy to add into the numerical model.

4.4.3 Other phenomenon

One should remember, that considering the previous limitation, even with all the improvementmade in this section the numerical model still does not take into consideration :

• The slamming effect : As seen in the section 3.6 for certain wave states with wave heightslarger than H=0.04m some important slamming appears. Considering the amplitude ofthe oscillations and the shape of the rotor for these wave height, an apparition of slammingis not very surprising.

• The viscous effect : For the same reason than the slamming one could expected to havesome viscous resistance at the tip of the rotor, when this one goes under water. Even ifthe viscous forces applying on the rotor would probably not be very large considering thatthey take place at the tip of this one, i.e. very far from the axle, the generated momentfrom this phenomenon might be important.


5 Extension of the numerical model

5.1 Addition of a Power Take Off moment

In the case of the wave converter WEPTOS, the electricity is generated from a Power TakeOff generator which will damp the system while its generating power. This extra damping hasbeen considered for some of the experiment by using constant PTO moment system.

Details on the PTO solution used for these test might be found in section 2.2.5.3. The twoPTO load (6N and 10N, pre-loading) have been implemented into the purely linear model andfor the fully non-linear model including the friction moment. The PTO moment have beenadded in the time domain calculation as shown in Eq.20.

I22 z5(t) = Mex(t) +Mrad(t) +Mhs(t) +MPTO(t) (20)

Note : During the experiments a third PTO with 10N pre-loading has been tried. However,after analyzing the results it appears that the PTO loading varied quite a lot during the batchtest. It has, therefore, been choose to not consider this loading in this report.

5.1.1 Results for 6N pre-loading

5.1.1.1 Linear modelAt first, the PTO has been implemented into the “simple” linear model. This PTO has beenevaluated from the experiment for each sea states. One could note that the Tables 25 and 26do not contain results for H = 0.02m. This is due to the fact that for so small waves the rotordid not move with the PTO. Therefore no comparison was relevant.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

Wave Period [s]

RA

O [d

eg/c

m] (

Pitc

h A

mpl

itude

[deg

] / (

Wav

e A

mpl

itude

[m] *

100)

Num with H=0.03Num with H=0.04Num with H=0.05Num with H=0.06Num with H=0.08Exp with H=0.03Exp with H=0.04Exp with H=0.05Exp with H=0.06Exp with H=0.08

Figure 93: Comparison of the experimental and numerical (in frequency and time domain) RAO.The results are given in term of pitch amplitude [deg] for 1 cm wave height. For the linearnumerical model considering a 15N pre-loaded PTO.



0.03 [m] 9.43 20.08 17.78 14.24 12.00 10.06 7.90 5.940.04 [m] 15.24 33.62 25.92 20.90 17.50 13.74 10.95 7.990.05 [m] 20.91 47.31 34.25 27.64 23.31 17.77 13.83 9.950.06 [m] 26.78 62.55 44.48 33.70 28.25 21.96 16.69 12.020.08 [m] 37.58 90.41 62.76 47.07 39.29 29.61 22.28 16.27

Table 25: Pitch amplitude numerical obtained for the different regular wave states. Results givenin degree, for the linear numerical model considering a 6N pre-loaded PTO.


0.03 [m] 58.59% 198.27% 95.86% 124.51% 56.62% 39.91% 30.74% 20.69%0.04 [m] 21.32% 58.61% 16.79% 33.58% 19.85% 21.15% 13.47% 4.70%0.05 [m] 18.74% 42.39% 3.43% 18.45% 9.41% 16.83% 0.35% 1.29%0.06 [m] 11.53% 55.84% 6.17% -2.35% 6.30% 15.65% -1.86% -7.54%0.08 [m] 34.12% 93.84% 14.15% -14.13% -8.95% 10.62% -0.27% -21.35%

Table 26: Relative pitch amplitude difference between the numerical model and the experimentalmeasurement. Results given in terms of relative errors, for the linear numerical model consid-ering a 6N pre-loaded PTO.

5.1.1.2 Non Linear modelThen the same calculations have been run with the “most advance” model including the structuralfriction and fully non-linear hydrodynamic coefficients.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

Wave Period [s]

RA

O [d

eg/c

m] (

Pitc

h A

mpl

itude

[deg

] / (

Wav

e A

mpl

itude

[m] *

100)


Figure 94: Comparison of the experimental and numerical (in frequency and time domain) RAO.The results are given in term of pitch amplitude [deg] for 1 cm wave height. For the fullynon-linear numerical model considering a 6N pre-loaded PTO and the structural friction.



0.03 [m] 10.27 18.69 17.06 12.97 10.51 7.48 3.44 1.370.04 [m] 15.19 26.95 29.00 20.73 16.45 12.32 6.56 0.230.05 [m] 18.99 32.71 37.09 32.53 22.61 16.63 9.28 1.560.06 [m] 22.18 38.02 41.79 44.47 28.01 20.60 11.77 3.540.08 [m] 26.24 45.53 46.20 48.50 51.02 27.47 17.51 6.70

Table 27: Pitch amplitude numerical obtained for the different regular wave states. Results givenin degree, for the fully non-linear numerical model considering a 6N pre-loaded PTO and thestructural friction.


0.03 [m] 72.70% 177.63% 88.04% 104.49% 37.25% 5.23% -43.08% -72.49%0.04 [m] 20.96% 27.12% 30.66% 32.45% 12.69% 8.53% -31.96% -96.93%0.05 [m] 7.84% -1.56% 11.99% 39.40% 6.14% 9.32% -32.68% -84.08%0.06 [m] -7.63% -5.27% -0.30% 28.85% 5.40% 8.49% -30.79% -72.74%0.08 [m] -6.36% -2.38% -15.97% -10.07% 18.24% 2.65% -21.61% -67.60%

Table 28: Relative pitch amplitude difference between the numerical model and the experimentalmeasurement. Results given in terms of relative errors, for the fully non-linear numerical modelconsidering a 6N pre-loaded PTO and the structural friction.

5.1.1.3 Analysis of the ResultsAs one may see both model give better results than when the rotor was free. This can beexplained by the fact that adding a PTO extremely reduces the pitch motions. Thus all thenon-linearities linked to large pitch motions (including slamming, viscous effect...) are largelyreduced. Moreover, excepted for T=1.80s the results appears to be fairly good. The exceptionat T=1.80s can be once more explained by the difficulties of the wave maker to produce accuratewave at this frequency (see wave accuracy table in annex).

5.1.2 Results for 15N pre-loading

5.1.2.1 Linear modelFor the 15N pre-loading PTO, only the wave heights larger than 0.04 m have been evaluated.As previously said, this is due to the fact that for smaller waves (i.e. H=0.02 m, H=0.03m andeven H=0.04) the rotor was not able to move.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

Wave Period [s]

RA

O [d

eg/c

m] (

Pitc

h A

mpl

itude

[deg

] / (

Wav

e A

mpl

itude

[m] *

100)


Figure 95: Comparison of the experimental and numerical (in frequency and time domain) RAO.The results are given in term of pitch amplitude [deg] for 1 cm wave height. For the linearnumerical model considering a 15N pre-loaded PTO.


0.05 [m] 21.04 47.31 34.25 27.64 23.31 17.77 13.83 9.950.06 [m] 26.56 62.33 44.48 33.70 28.25 21.96 16.69 12.020.08 [m] 37.52 90.16 62.59 46.88 39.49 29.74 22.29 16.30

Table 29: Pitch amplitude numerical obtained for the different regular wave states. Results givenin degree, for the linear numerical model considering a 15N pre-loaded PTO.


0.05 [m] 258.60% 981.10% 294.95% 249.66% 114.41% 107.11% 42.70% 123.79%0.06 [m] 151.84% 370.62% 112.40% 62.22% 59.08% 46.14% 18.01% 24.92%0.08 [m] 75.17% 171.75% 53.44% 26.93% 22.35% 24.67% 7.47% 0.36%

Table 30: Relative pitch amplitude difference between the numerical model and the experimentalmeasurement. Results given in terms of relative errors, for the linear numerical model consid-ering a 15N pre-loaded PTO.

5.1.2.2 Non Linear modelThen the same calculations have been run with the “most advance” model including the structuralfriction and fully non-linear hydrodynamic coefficients.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

Wave Period [s]

RA

O [d

eg/c

m] (

Pitc

h A

mpl

itude

[deg

] / (

Wav

e A

mpl

itude

[m] *

100)

Num with H=0.05Num with H=0.06Num with H=0.08Exp with H=0.05Exp with H=0.06Exp with H=0.08

Figure 96: Comparison of the experimental and numerical (in frequency and time domain) RAO.The results are given in term of pitch amplitude [deg] for 1 cm wave height. For the fullynon-linear numerical model considering a 15N pre-loaded PTO and the structural friction.


0.05 [m] 18.99 32.71 37.09 32.53 22.61 16.63 9.28 1.560.06 [m] 22.18 38.02 41.79 44.47 28.01 20.60 11.77 3.540.08 [m] 26.24 45.53 46.20 48.50 51.02 27.47 17.51 6.70

Table 31: Pitch amplitude numerical obtained for the different regular wave states. Results givenin degree, for the fully non-linear numerical model considering a 15N pre-loaded PTO and thestructural friction.


0.05 [m] 223.55% 654.97% 327.66% 311.53% 108.00% 93.80% -4.23% -65.62%0.06 [m] 110.30% 187.08% 99.88% 114.06% 57.73% 37.10% -16.79% -63.18%0.08 [m] 22.50% 37.23% 13.89% 31.33% 58.06% 14.56% -15.57% -58.75%

Table 32: Relative pitch amplitude difference between the numerical model and the experimentalmeasurement. Results given in terms of relative errors, for the fully non-linear numerical modelconsidering a 15N pre-loaded PTO and the structural friction.

5.1.2.3 Analysis of the Results

As one may have notice, this PTO of 15N pre-loading appears to be very strong, and to blockmost of the motions. The fairly large errors shown by the previous tables for both models maycome from two main reasons :

• Considering that in this calculation the PTO moment is fairly strong compare to the restof the hydrodynamic moments, the smallest error (let’s says 5% error) in the evaluationof the actual PTO forces would necessarly lead to important differences in the resultingpitch motions.

• The non consideration of the influence of the water level on the hydrodynamic coefficientsmay also lead to large errors in this configuration. Indeed, in this case, the pitch motionare quite small (less than 15 degrees of amplitude) therefore one can imagine that there


are almost no slamming nor other non-linear effects due to these motions. Thus the useof hydrodynamic coefficient varying in function of the pitch motion might not be veryuseful. However, the water level is still importantly varying in comparison of the bodyfixed coordinate system (especially for the considered wave height). Therefore, one couldimage that the remaining errors between the numerical calculation and the experimentalmeasurements come from the non-consideration water level influence on the hydrodynamiccoefficients.

5.2 Results for irregular waves

During the experiment 8 irregular wave states have been run. These wave states have beenrun for about 30 minutes in order to have reliable statistic on it. However, the numerical modelshave only been run over the first 150 s. This appears to be enough to allow a good comparisonof the numerical model and the experiment.

Note : One should remember that the wave states WS05 to SW25 have been determined con-sidering one site and one scaling coefficient while the tests WS60 to SW70 have been determinedfrom another site and other scaling coefficient. Therefore, the two “smallest” wave states actuallyare WS05 and WS60.

5.2.1 Without PTO

0 50 100 150−0.05

0

0.05

Time [s]

Wav

e el

evat

ion

[m]

0 50 100 150−60

−40

−20

0

20

40

60

Time [s]

Pitc

h [d

eg]


Figure 97: Wave State WS60 full calculationrun for 150 seconds. Numerical Model :Linear hydrostatic - No PTO - No friction.

60 65 70 75 80 85 90−0.04

−0.02

0

0.02

0.04

Time [s]

Wav

e el

evat

ion

[m]

60 65 70 75 80 85 90−40

−30

−20

−10

0

10

20

30

40

50

Time [s]

Pitc

h [d

eg]


Figure 98: Wave State WS60 zoom on resultbetween 60s and 90s. Numerical Model :Linear hydrostatic - No PTO - No friction.


0 50 100 150−0.05

0

0.05

Time [s]

Wav

e el

evat

ion

[m]

0 50 100 150−50

0

50

100

Time [s]

Pitc

h [d

eg]


Figure 99: Wave State WS60 full calculationrun for 150 seconds. Numerical Model :Fully non-linear hydrodynamic - No PTO- Including friction.

60 65 70 75 80 85 90−0.04

−0.02

0

0.02

0.04

Time [s]

Wav

e el

evat

ion

[m]

60 65 70 75 80 85 90−40

−30

−20

−10

0

10

20

30

40

50

Time [s]

Pitc

h [d

eg]


Figure 100: Wave State WS60 zoom on resultbetween 60s and 90s. Numerical Model: Fully non-linear hydrodynamic - NoPTO - Including friction.

0 50 100 150−0.05

0

0.05

0.1

0.15

Time [s]

Wav

e el

evat

ion

[m]

0 50 100 150−80

−60

−40

−20

0

20

40

60

80

100

Time [s]

Pitc

h [d

eg]


Figure 101: Wave State WS10 full calculationrun for 150 seconds. Numerical Model :Linear hydrostatic - No PTO - No fric-tion.

60 65 70 75 80 85 90−0.05

0

0.05

Time [s]

Wav

e el

evat

ion

[m]

60 65 70 75 80 85 90−60

−40

−20

0

20

40

60

Time [s]

Pitc

h [d

eg]


Figure 102: Wave State WS10 zoom on resultbetween 60s and 90s. Numerical Model: Linear hydrostatic - No PTO - No fric-tion.


0 50 100 150−0.05

0

0.05

0.1

0.15

Time [s]

Wav

e el

evat

ion

[m]

0 50 100 150−80

−60

−40

−20

0

20

40

60

80

100

120

Time [s]

Pitc

h [d

eg]


Figure 103: Wave State WS10 full calculationrun for 150 seconds. Numerical Model: Fully non-linear hydrodynamic - NoPTO - Including friction.

60 65 70 75 80 85 90−0.05

0

0.05

Time [s]

Wav

e el

evat

ion

[m]

60 65 70 75 80 85 90−60

−40

−20

0

20

40

60

80

100

120

Time [s]

Pitc

h [d

eg]


Figure 104: Wave State WS10 zoom on resultbetween 60s and 90s. Numerical Model: Fully non-linear hydrodynamic - NoPTO - Including friction.

5.2.2 With 6N pre-loaded PTO

0 50 100 150−0.05

0

0.05

Time [s]

Wav

e el

evat

ion

[m]

0 50 100 150−30

−20

−10

0

10

20

30

Time [s]

Pitc

h [d

eg]


Figure 105: Wave State WS60 full calculationrun for 150 seconds. Numerical Model :Linear hydrostatic - With 6N pre-loadedPTO - No friction.

60 65 70 75 80 85 90−0.04

−0.02

0

0.02

0.04

Time [s]

Wav

e el

evat

ion

[m]

60 65 70 75 80 85 90−25

−20

−15

−10

−5

0

5

10

15

20

25

Time [s]

Pitc

h [d

eg]


Figure 106: Wave State WS60 zoom on resultbetween 60s and 90s. Numerical Model :Linear hydrostatic - With 6N pre-loadedPTO - No friction.


0 50 100 150−0.05

0

0.05

Time [s]

Wav

e el

evat

ion

[m]

0 50 100 150−30

−20

−10

0

10

20

30

40

Time [s]

Pitc

h [d

eg]


Figure 107: Wave State WS60 full calculationrun for 150 seconds. Numerical Model: Fully non-linear hydrodynamic - With6N pre-loaded PTO - Including friction.

60 65 70 75 80 85 90−0.04

−0.02

0

0.02

0.04

Time [s]

Wav

e el

evat

ion

[m]

60 65 70 75 80 85 90−25

−20

−15

−10

−5

0

5

10

15

20

25

Time [s]

Pitc

h [d

eg]


Figure 108: Wave State WS60 zoom on resultbetween 60s and 90s. Numerical Model: Fully non-linear hydrodynamic - With6N pre-loaded PTO - Including friction.

0 50 100 150−0.05

0

0.05

0.1

0.15

Time [s]

Wav

e el

evat

ion

[m]

0 50 100 150−60

−40

−20

0

20

40

60

Time [s]

Pitc

h [d

eg]


Figure 109: Wave State WS10 full calculationrun for 150 seconds. Numerical Model :Linear hydrostatic - With 6N pre-loadedPTO - No friction.

60 65 70 75 80 85 90−0.05

0

0.05

Time [s]

Wav

e el

evat

ion

[m]

60 65 70 75 80 85 90−50

−40

−30

−20

−10

0

10

20

30

40

Time [s]

Pitc

h [d

eg]


Figure 110: Wave State WS10 zoom on resultbetween 60s and 90s. Numerical Model :Linear hydrostatic - With 6N pre-loadedPTO - No friction.

0 50 100 150−0.05

0

0.05

0.1

0.15

Time [s]

Wav

e el

evat

ion

[m]

0 50 100 150−40

−20

0

20

40

60

80

Time [s]

Pitc

h [d

eg]


Figure 111: Wave State WS10 full calculationrun for 150 seconds. Numerical Model: Fully non-linear hydrodynamic - With6N pre-loaded PTO - Including friction.

60 65 70 75 80 85 90−0.05

0

0.05

Time [s]

Wav

e el

evat

ion

[m]

60 65 70 75 80 85 90−40

−20

0

20

40

60

80

Time [s]

Pitc

h [d

eg]


Figure 112: Wave State WS10 zoom on resultbetween 60s and 90s. Numerical Model: Fully non-linear hydrodynamic - With6N pre-loaded PTO - Including friction.


5.2.3 Analysis of the results

From the figures from Figure 97 to Figure 112, different conclusions can be made :

• First of all it appears to be very difficult to make the numerical model fit the experimentalmotions for irregular wave states. This might be explained by different reasons :

– Large wave amplitude : even when considering the smallest irregular wave states thewave amplitude easily reach 0.04m and more. And it has already been proven thatfor so large wave amplitude the model does not work properly. This tend to provethat there are still some strong model limitations in tems of wave amplitude.

– Large peak : on top of the fact that the wave amplitude are fairly large for thesewave states, the fact that they come by “group” in an irregular way (normal patentfor irregular waves) may explain why the numerical model shows some difficulties tofit the rotor motions. Indeed, it seems fair to think that such model would moreeasily fit a regular patent than an irregular one.

– Reflexion : a reflexion analysis has been run for these irregular wave states thoughtWaveLab. It appears that in average the reflexion is about 20%. However, this is thereflexion obtained for the wave states run without the presence of the rotor. And itis difficult to evaluate the influence of the rotor on these results.

• All these said, from Figure 98 and Figure 100 one could consider that even if both modelsseem to be close from the actual peach motion, for WS60, the fully non-linear one seemsto give closer results.

• From Figure 103 one can appreciate the fact that having wave height reach 0.05m leadsto very large errors while considering a fully non-linear model.

• Finally, using a PTO does not not seem to be very helpful. This might be due to thenecessity, for irregular waves to consider the PTO mean values (see Table 1) instead ofmore accurate values as it has been done for regular waves. Indeed, these mean valuesmight not be very accurate or at least not accurate enough to obtain a good convergence ofthe numerical calculations with the experimental measurements. Moreover, while lookingat the time evolution of the PTO forces in the case of irregular wave states, one maysee that, for a lot of “small” and/or “short” waves, the PTO does not reach it maximalconstant value (used by the numerical model). And the fact that the PTO is almost neverconstant may largely explain the errors seen in the previous figures. Therefore, for futureexperiments one may consider a more classical PTO system linearly linked to the pitchvelocity. This PTO damping solution may show better results.


6 Conclusion

6.1 Results

During this master thesis different models have been run to simulate the motion of the WEP-TOS rotor. It appears that only fairly advance model considering a non-linear hydrostaticrestoring moment (function of the water depth and the pitch angle) as well as non-linear hydro-dynamic coefficients (function of the pitch angle) may lead to reasonable results for small regularwaves. However, it appears that even considering this model the wave height acceptability wasstill too low to ensure a good convergence of the results for irregular wave states.

It may be noted that software as Wamit and the linear theory have been mainly made forvessels (and large offshore platform). And even if vessels may show some pitch motions (amongother) these motions are very small compare to the ones undertake by the rotor. Moreover inthe case of a vessel there is almost no variation of the water level compare to the body-fixedreference. While for the Rotor, considering that it is fixed around the axle, the water levellargely vary). This double point may largely explains the difficulties that have been faced toobtain to obtain a fairly acceptable numerical model for the WEPTOS rotor.

Finally, it is still very interesting to see that the latest model obtains the same resonanceperiod as the one found during the experiment. This result is important since it has beenexplained in the report [1] that one of the main issue with the WEPTOS project would be to“move” the resonance period to larger wave period (for a given scale). As one may know, it isindeed very important for wave energy converter to have a resonance period (which is most thetime quite narrow) adapted to the most powerful wave period.

6.2 Future possible study

From the previous sections of this report, it seems that to make this model to have a betterconvergence with the experimental measurement, one should consider to evaluate the hydrody-namic coefficients with an even more detailed angle discretization and also in function of thewater level. In the same idea the restoring hydrostatic moment could also be evaluated bytaking into consideration an water level angle. Indeed, the length of the rotor is not that smallcompare to some of the wave length. Therefore the water level is not exactly horizontal fromthe tip to the rear of the rotor. Another option to improve the numerical model could be tocompute a retardation function taking into consideration the evolution of the radiation impulseresponse inside the convolution. After this done, if someone is interested with really precisenumerical model and to be able to consider “large” waves, one should also include slamming andviscous effects. Finally, it might be interesting to see how would fit this numerical model whileconsidering other PTO solution. Thus one might be interested in running new experiment witha linearly damped PTO solution and then compare the results with a numerical model.

On top of this study, on could also be interested in implementing a numerical model that wouldconsider the rotor for different wave angles. Indeed, while considering the full WEPTOS device,the rotors are never facing the wave (at least the wave heading angle is 30◦). Moreover, theinteraction between the different rotor and the WEPTOS structure may also be very interestingto study in order to get a good approximation of the total power capacity of the device.

Finally, it has been shown in this report that a simple linear model was not able to determinethe resonance period of the rotor and that it was necessary to use a non-linear model to get itwith an acceptable accuracy. Therefore, through an iterative use of the non-linear model, onecould determine how to improve this resonance period, with relatively minor changes on themass, center of mass, inertia, or even the shape of the rotor in order to obtained an resonanceperiod adapted to the selected site and scale of the device.


References

[1] T. Marchalot A. Pecher, J. P. Kofoed. Experimental study of WEPTOS Wave EnergyConverter. Departement of Civil Engineering of Aalborg University, Wave Energy ResearchGroup, 2011.

[2] T. Marchalot A. Pecher, J. P. Kofoed. Experimental study on a rotor for WEPTOS WaveEnergy Converter. Departement of Civil Engineering of Aalborg University, Wave EnergyResearch Group, 2011.

[3] Raúl Baragiola. Friction. University of Virginia, 2002. http://www.virginia.edu/ep/SurfaceScience/friction.html.

[4] J. N. Newman F. T. Korsmeyer, H. B. Bingham. TiMIT User Manual V4.0. MassachusettsInstitute of Technology, 1999.

[5] Mathworks. Matlab. http://www.mathworks.fr/products/matlab/.

[6] McNeel. Rhinoceros 3d. http://www.rhino3d.com/.

[7] University of Aalborg. AwaSys 6. http://www.hydrosoft.civil.aau.dk/AwaSys/.

[8] University of Aalborg. WaveLab 3. http://www.hydrosoft.civil.aau.dk/wavelab.

[9] Massachusetts Institute of Technology. WAMIT v5.4. http://www.wamit.com/.

[10] Det Norske Veritas. Design of Offshore Wind Turbine Structures. Offshore Standard DNV-OS-J101, October 2010. http://www.dnv.com/industry/energy/segments/wind_wave_tidal/standards_guidelines/.

[11] Inc. WAMIT. WaMIT User Manual V7.0. Massachusetts Institute of Technology, 2011.

http://www.virginia.edu/ep/SurfaceScience/friction.html

http://www.virginia.edu/ep/SurfaceScience/friction.html

http://www.mathworks.fr/products/matlab/

http://www.rhino3d.com/

http://www.hydrosoft.civil.aau.dk/AwaSys/

http://www.hydrosoft.civil.aau.dk/wavelab

http://www.wamit.com/

http://www.dnv.com/industry/energy/segments/wind_wave_tidal/standards_guidelines/

http://www.dnv.com/industry/energy/segments/wind_wave_tidal/standards_guidelines/

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