UNIVERSITATISACTA UPSALIENSIS
UPPSALA
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1130
Buoy Geometry, Size and
Hydrodynamics for Power Take Off Device for Point Absorber Linear Wave Energy Converter
HALVAR GRAVRÅKMO
ISSN 1651-6214 ISBN 978-91-554-8902-1
Dissertation presented at Uppsala University to be publicly examined in Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Monday, 28 April 2014 at 13:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Dr. Luke Myers (University of Southampton).
Abstract
Gravråkmo, H. 2014. Buoy Geometry, Size and Hydrodynamics for Power Take Off Device for Point Absorber Linear Wave Energy Converter. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1130. 71 pp. Uppsala:
Acta Universitatis Upsaliensis. ISBN 978-91-554-8902-1.
Wave energy converters of point absorber type have been developed and constructed. Full scale experiments have been carried out at sea and electricity has been successfully delivered. Linear permanent magnet generators together with a subsea substation and buoys of various geometric shapes have been investigated theoretically and experimentally. The design has in large extent an electronic approach, keeping the mechanical part of it as simple as possible, due to the long life span and reliability of electric components.
Because of the nature of a linear generator, the internal translator with permanent magnets has a limited stroke length which will be reached when the buoy is exposed to large wave heights.
Internal springs at the top and bottom of the generator prevent the translator from hitting the generator hull. Inertial forces due to the mass and velocity of the translator and the buoy and its heave added mass compresses the spring. The added mass is a rather large part of the total moving mass. Simulations of a converter with a vertical cylindrical buoy and with a toroidal buoy were conducted, as well as real sea experiments with converters with cylindrical buoys of two different sizes and a toroidal buoy. The overloads are likely to affect the design and service life of the generator, the buoy and the wire which interconnects them.
Buoy shapes with as much excitation force as possible and as little heave added mass as possible were sought. A toroidal buoy caused less overloads on the generator at sea states with short wave periods and relatively large wave height, but for sea states with very long wave periods or extremely high waves, the magnitude of the overloads was mainly determined by the maximum displacement of the buoy.
Snap loads on the interconnecting wire, as the slack wire tensed up after a very deep wave trough, were found to be greater but of the same order of magnitude as forces during the rest of the wave cycle.
During a 4 day period at various wave conditions, two converters with cylindrical buoys proved efficiency between 11.1 % and 24.4 %. The larger buoy had 78 % larger water plane area than the other buoy which resulted in 11 % more power production. Short wave period was beneficial for the power production.
Infinite frequency heave added mass was measured for a cylindrical buoy at real sea and found to be greater than the linearly calculated theoretical added mass.
Keywords: point absorber, wave, energy, converter, ocean, wec, toroidal, buoy, torus, cylindrical, cylinder, experiment, full scale, trial, sea, energy, renewable
Halvar Gravråkmo, Department of Engineering Sciences, Electricity, Box 534, Uppsala University, SE-75121 Uppsala, Sweden.
© Halvar Gravråkmo 2014 ISSN 1651-6214
ISBN 978-91-554-8902-1
urn:nbn:se:uu:diva-220344 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-220344)
to Gabriel Olav Gravråkmo-Moreno
List of Papers
This thesis is based on the following papers, which are referred to in the text by their Roman numerals.
I S. Tyrberg, M. Stålberg, C. Boström, R. Waters, O. Svensson, E.
Strömstedt, A. Savin, J. Engström, O. Langhamer, H. Gravråkmo, K. Haikonen, J. Tedelid, J. Sundberg, M. Leijon, ”The Lysekil Wave Power Project: Status Update”, WREC X Glasgow, july 21-25, 2008 II S. Tyrberg, H. Gravråkmo and M. Leijon, ”Tracking a Wave Power
Buoy Using a Network Camera: System Analysis and First Results”, Proceedings of the 28th International Conference on Ocean, Offshore and Arctic Engineering, Honolulu, Hawaii, May 31-June 5, 2009.
III C. Boström, O. Svensson, M. Rahm, E. Lejerskog, A. Savin, E. Strömst- edt, J. Engström, H. Gravråkmo, K. Haikonen, R. Waters, D. Björklöf, T. Johansson, J. Sundberg and M. Leijon, ”Design proposal of electrical system for linear generator wave power plants”, Proc. of IECON 2009, 35th annual conference of the IEEE Industrial Electronics and Society, Porto, Portugal, no. PD-027448, pp. 4429-4434, Nov 3-5, 2009.
IV H. Gravråkmo, M. Leijon, E. Strömstedt, J. Engström, S. Tyrberg, A.
Savin, O. Svensson, R. Waters, ”Description of a Torus Shaped Buoy for Wave Energy Point Absorber”, Proceedings of Renewable Energy, Pacifico Yokohama, Yokohama, Japan, June 27-July 2, 2010.
V M. Leijon, R. Waters, M. Rahm; O. Svensson, C. Boström, E. Ström- stedt, J. Engström, S. Tyrberg, A. Savin, H. Gravråkmo, H. Bern- hoff, J. Sundberg, J. Isberg, O. Ågren, O. Danielsson, M. Eriksson, E.
Lejerskog, B. Bolund, S. Gustafsson, K. Thorburn, ”Catch the wave to electricity”, IEEE Power and Energy Magazine, Volume 7, Issue 1, Page(s):50 - 54, Jan-Feb, 2009.
VI E. Lejerskog, H. Gravråkmo, A. Savin, E. Strömstedt, S. Tyrberg, K.
Haikonen, R. Krishna, C. Boström, M. Rahm, R. Ekström, O. Svens- son, J. Engström, B. Ekergård, A. Baudoin, V. Kurupath, L. Hai, W. Li, J. Sundberg, R. Waters, M. Leijon, ”Lysekil Research Site, Sweden: A Status Update”, Proceedings of the 9th European Wave and Tidal En- ergy Conference, Southampton, Sept 5-9, 2011.
VII B. Ekergård, H. Gravråkmo, M. Leijon, ”Theory and Simulations of an End Stop Solution in a Linear Wave Power Generator”, Manuscript, Jan, 2014.
VIII H. Gravråkmo, E. Strömstedt, J. Engström, O. Svensson, M. Leijon,
”Toroidal Buoy for Point Absorver WEC with Focus on Added Mass and Overloads”, Submitted to Journal of Ocean Engineering and Marine Energy, Mar, 2014.
IX H. Gravråkmo, E. Lejerskog, E. Strömstedt, A. Savin, M. Leijon,
”Power Production by Linear Wave Energy Converters of Point Absorber Type with Vertical Cylindrical Buoys during Various Sea States”, Submitted to Journal of Marine Science and Engineering, Mar, 2014.
X H. Gravråkmo, E. Strömstedt, A. Savin, O. Svensson, M. Leijon,
”Measurements of Extreme Forces on a Wave Energy Converter of Point Absorber Type and Estimation of Added Mass of Cylindrical Buoy”, Accepted by the 33rd International Conference on Ocean, Offshore and Arctic Engineering, Jun 8-13, 2014.
Reprints were made with permission from the publishers.
Contents
1 Introduction . . . 11
2 Theory and Background . . . 13
2.1 The Wave Energy Converter . . . 13
2.1.1 Overview of The System . . . 13
2.1.2 Reliability . . . 13
2.1.3 Keep it Simple . . . 13
2.1.4 Mechanical Reliability . . . 14
2.1.5 Electrical Reliability . . . 14
2.1.6 Generator Dynamics . . . 16
2.2 Oscillating System - Equation of Motion . . . 16
3 Wave Theory . . . 19
3.1 Linear Wave Theory . . . 19
3.1.1 The Energy within Ocean Gravity Waves . . . 19
3.1.2 Ocean Waves . . . 19
3.1.3 Group Velocity - Phase Velocity . . . 20
3.2 Wave Climate . . . 21
3.2.1 Wave Power Spectral Density . . . 21
3.2.2 Wave Climate Influence on Converter Economy . . . 23
3.3 Wave-Body Interaction . . . 27
3.4 The Excitation Force . . . 30
3.4.1 Interpretation of Added Mass . . . 31
3.4.2 Added Mass Calculation . . . 31
3.4.3 Hydrodynamic Damping . . . 32
3.5 Frequency Dependency of Excitation Force and Added Mass . . 33
3.5.1 Size Dependency of Excitation Force and Added Mass . . . 35
3.5.2 Geometry Dependency of Excitation Force and Added Mass 35 3.5.3 Note on the History of Torus Added Mass Estimation . . . . 35
4 Simulations and Experiments . . . 39
4.1 Testing of Converters . . . 39
4.1.1 Buoys . . . 39
4.1.2 The Research Site . . . 40
4.1.3 Experiments and Simulations . . . 44
5 Summary of Results . . . 47
5.1 End Stop Forces on Toroidal and Cylindrical Buoy . . . 47
5.2 Buoy Size . . . 49
5.3 Power Production and Wave Period, Height and Direction and
Tide Oscillations . . . 50
5.4 Measurement of Added Mass on Cylindrical Buoy . . . 50
5.5 Elongated Toroidal Buoy . . . 52
6 Conclusions and Further Work . . . 55
6.1 More Further Work . . . 56
7 Acknowledgment . . . 59
8 Summary of Papers . . . 61
9 Svensk Sammanfattning . . . 67
Bibliography . . . 69
Nomenclature & Abbreviations
¨z Acceleration of the buoy-translator body [m/s2]
˙z Velocity of the buoy-translator body [m/s]
η Surface elevation [m]
µ Viscosity [Pa s]
∇ Displacement [kg]
ω Angular frequency of wave [rad/s]
φ Velocity potential [m2/s]
ρ Density of sea water 1024 [kg/m3] ζA Amplitude of wave [m]
a Radius of torus element or cylinder [m]
b Radius of torus; center to element center [m]
Bb Hydrodynamic damping [N s/m]
Bt Electromagnetic damping [N s/m]
Cb Hydrostatic stiffness [N/m]
Ct Translator spring stiffness, LG2 and LG3 [N/m]
d Characteristic size, e.g. buoy diameter. Unless stated otherwise [m]
E Energy density [J/m2] Ek Kinetic energy density [J/m2] Ep Potential energy density [J/m2]
f Array of all wave frequencies of the FFT [Hz]
fh Hydrostatic approximation heave excitation force [N]
fi Inertial force [N]
fx Heave excitation force [N]
fxFK Froude-Krylov heave force [N]
g Gravitational acceleration 9.81 [m/s2] h Water depth [m]
Hm0 Significant wave height [m]
k Angular repetency of wave (wave number) [m-1] mb Mass of buoy [kg]
mn n-th moment of S mt Mass of translator [kg]
maz Heave added mass of buoy [kg]
ns Normal unit vector of the wet surface P Pressure [Pa]
P Wave power density [W/m]
Pd Dynamic pressure [Pa]
Ps Static pressure [Pa]
PFK Froude-Krylov pressure [Pa]
S The FFT power spectral density of η [m2/Hz]
s The wet surface [m2] t Time [s]
Te Wave energy period [s]
Tmean Mean wave period [s]
Tzero Zero crossing wave period [s]
x Horizontal distance from reference point, in direction of wave propa- gation [m]
z Buoy and translator upwards distance from equilibrium [m]
BEM Boundary element method FEM Finite element method WEC Wave energy converter
1. Introduction
Today electricity is taken for granted and it’s used in industry and households all over the world. Society today would simply not be the same without elec- tricity. Electricity is relatively easy to transport and is the power source of a wide range of equipment. Easy to forget is that the history of electricity as a high capacity power source is rather short. Hungarian Anyos Jedlik finished the construction of the worlds first electric motor about 1853, the same year as the Dutch painter Vincent van Gogh was born, and soon formulated the con- cept of a dynamo. Now, electricity for enormous amounts of money is traded every day. Electricity is a central part of the world as we know it today. The search for sources which economically can be converted to electricity seems to be never ending. Commercial large scale hydroelectric, nuclear, coal, wind and fossil fuel power plants have existed for some time. Still the hunt goes on, driven by the highly valued electric energy.
Wave energy converters (WECs) also described as wave power plants have been developed for a very long time. An early patent on a WEC was presented by P. Wright during 1898, utilizing a buoy moving mainly in the up and down direction as a wave approaches [1, p. 3]. During the oil crisis in the early 70’s the interest in, and in some countries the governmental investments in, wave power research increased and later after the oil crisis decreased. Salter’s Duck, officially named the Edinburgh Duck by Stephen Salter is an example of a wave concept of the 70s. The concept never reached the commercial market, the reasons for this can be discussed but this device was highly advanced.
Nevertheless, a growing demand exists for electricity produced without the bi products carbon dioxide or radioactive waste.
Our society wants electricity, but people don’t want visual or audible dis- turbances. Planning of windmills have in some cases shown to be difficult due to resistance among local residents. Wave power plants have the advantage of not interfering with residential or recreational areas if located at some distance from shore. Also windmills and ocean current power plants might be located offshore. However, such devices, including wave power plants might interfere with territories used for fishing or military operation purpose.
The price of electricity produced is of course of the utmost importance. In the end, for wave power to be useful, the cost of energy must be comparable with the cost of energy produced by other sources. Alternatively, clean energy can also be given benefits by political means, if let’s say, a political decision is made to prefer clean energy to non-clean energy. In capitalistic nations this
will usually lead to the implementation of economic benefits to clean energy production or economic restrictions to non-clean energy or a combination of both.
This far, no company has proven large scale electricity production to be fully economically sustainable without subventions, and huge effort is being focused on development of wave power technology and investors believe in wave energy. The great interest in wave energy is mainly caused by the high energy flux density of waves. Energy radiated from the sun reaches the earth surface producing heat. The heat causes, through heat convection, winds and the winds generate ocean waves. In each step energy is converted to a dif- ferent form of energy which has some power density through some given 2 or 3 dimensional space like the air or the ocean surface. In each step the to- tal energy might decrease, but also the energy flux density might increase. If typical power density in various media are to be compared, say the media is sunlight, wind and waves, the power density of ocean waves just beneath the ocean surface is typically five times larger than the power density of wind 20 m above the ocean surface. Compared to sunlight, the power density of waves just beneath the surface is 20-30 times greater [2, p. 1].
As shown, there exists a good potential for electricity produced by ocean waves. Prototypes of wave energy converters have several times and in vari- ous shapes been built, still electricity production by WECs have not yet fully reached the commercial market. Reasons for this might be the high cost of off- shore technology and offshore operations. One major challenge of wave power development is the great time variation of energy flux density of waves. The wave power plants must be dimensioned for the most harsh wave power, but the year average electricity production depends mainly on the most common wave power of the site. It’s important to notice that the cost of and the pro- duction of a wave power plant depends on two different variables. This will be further discussed in this thesis. Simply speaking, one might say the task of engineers and scientists is to develop WECs in such a way that they become economically competitive with existing electric power sources. There is a limit how much money the electricity from wave power can cost, for anyone to buy it, which economically restricts commissioning, operation, maintenance and demolition of such power plants.
It’s reasonable to assume that the cost of produced electricity from new technology such as wave power plants will decrease as the technology devel- ops, as was the case with wind power.
The main aim of this study was to do investigate the influence from the choice of buoy for the wave energy converter of point absorber type of the Lysekil project, with regard to overloads and power production. Experiments and simulations enlighten
2. Theory and Background
2.1 The Wave Energy Converter
2.1.1 Overview of The System
As mentioned in the introduction, the economical cost of the electricity pro- duced is of major importance. A wave energy power plant must be economi- cally sustainable to make success on the market, even if clean energy sources to some extent might be politically supported by instances promoting clean energy sources, like a government. A huge number of wave energy projects exist today [3].
There are several ways to lower the cost. Maintenance of an offshore plant involves offshore operations which are significantly more expensive than on- shore operations, traditionally. The price of offshore operations might be low- ered, but for any electric power plant, onshore or offshore, lowering the num- ber of hours needed for maintenance might lead to reduced cost of electricity delivered.
2.1.2 Reliability
Any component, in the case of a WEC, mechanical or electrical component, has some reliability. That is, each component has some stochastic probability of failing. A WEC is dependent on a number of components to operate, the probability of the WEC to fail to operate during a specific period of time is the product of failure probability of each component, when not taking redun- dancy in consideration. According to this, it’s desirable to incorporate as low number of parts as possible in a WEC or use components with high reliability or practice a combination of these two suggestions.
2.1.3 Keep it Simple
Uppsala University started a wave energy project in 2002. Paper I describes the history of this project. The generator used in this wave energy project is a linear generator connected, through a synthetic or steel wire, to a floating buoy which makes it a point absorber WEC as seen on fig. 2.1c. Figure 2.1a and 2.1b show examples of buoys used. The linearity of this generator refers to a linear movement of the translator of the generator. This is not a rotating machine, the internal movement of the machine is vertical, up and down. The
generator is described in Paper V. Taking a step backward, looking at the big picture, wave particles move in circular or almost circular patterns. Thus the dynamic wave pressure causing the Froude-Krylov force is also a periodic function. One might say that wave particles move about an imaginary hub.
Not many WECs convert this circularly moving force directly to a rotating electrical generator, the reasons for this is an interesting topic to discuss but will not be carried through in this thesis.
2.1.4 Mechanical Reliability
The generator of a point absorber is driven by the two vertical component of the Froude-Krylov force, permitting the buoy to move freely in the two hori- zontal modes of freedom; sway and surge. Rotational movement in any of the 3 modes of such a buoy will not be considered in this thesis. The yaw rota- tion is unlikely to have any significant influence on the system. Roll and pitch of the buoy will not be discussed in this thesis. A buoy axisymmetric about a vertical line is usually chosen for a point absorber WEC. Shortly summa- rized one component of a rotating force drives the linear movement of a linear generator.
The reason of the choice of such a linear generator is the simplicity of it.
There will mainly be a linear force acting on the generator and no great mo- ment, as huge moments on a machine might introduce significant engineering challenges. In order to keep the reliability of the system high, the number of parts, especially moving parts, is kept down to a minimum.
2.1.5 Electrical Reliability
The reliability of each component of the system will not be described, how- ever, generalizations of the reliability of mechanical and electrical compo- nents might be done. In general, mechanical moving parts have a tendency to wear out, while electrical components don’t within reasonable operating lifetime of the device in which it’s embedded [4]. There are some exceptions to this, like some semiconductors and capacitors which might handle a lim- ited number of charge-discharge cycles, and electrical parameters of devices might drift over time. However, when not exposed to excessive heat which might significantly shorten the life span of electrical components or to tran- sient electrical stresses, the life span of electrical components might be very long. Figure 2.2 shows a typical curve of failure rate density, so called bath- tub curve, of a general component. The curve shows that in early life there is an infant mortality and after running in, components tend to have about con- stant reliability throughout its life and will sooner or later wear out. The sec- tion burn in is similar for electrical and mechanical components, but electrical components tend to have a longer period of useful life than moving mechani- cal components, and many electrical components don’t wear out or wear out
(a) A cylindrical buoy under operation connected to a generator on the seabed, near Lysekil.
(b)The toroidal buoy R2 used in the Ly- sekil project.
(c) Representation of the wave energy converters.
(d) Picture of the linear generator LG3, before deployment.
(e)The force transducer at the bottom of the cylindrical buoy. The signal ca- ble was protected by several layers of tube and curled to increase flexibility.
(f) Steel chains connected to force transducers on the hexagonal approx- imated toroidal buoy R2. Signal ca- bles are protected inside plastic tubes.
Figure 2.1:Some of the buoys, generators and measurement equipment used in the study.
Figure 2.2: Reliability of an arbitrary component. The curve shows failure rate through the components life cycle. Courtesy of Michael Carchia/ Carnegie Mellon University
much more slowly than moving mechanical components [5, 6]. Thus, elec- trical components are preferred to mechanical components. The wave energy is converted to electrical energy in an early stage, utilizing as few mechanical parts as possible in the power train between the waves and the stator. This way the power flow from the waves is converted from a mechanical media (buoy, wire and translator) to electrical media (cable) and can be dealt with by the use of electrical components in the major part of the system, this way avoiding excessive use of mechanical components.
2.1.6 Generator Dynamics
Fig. 2.3a shows the principle of the generator dynamics. Generators LG2 and LG3 have translator springs which pull the translator downwards. A scetch of LG2 and LG3 is shown on fig. 2.3c. They are built with a frame, while LG9 and LG12 have no frame. LG9 and LG12 also does not have translator springs, like shown on fig. 2.3b, thus the translators are solely pulled down by gravity.
All generators have an upper end stop spring which prevents the translator from hitting the generator hull when the translator reaches high velocity or when the wave excitation force is very large. This spring is an essential part of this study. All generators also have a lower end stop spring which prevents the translator from hitting the bottom of the generator hull. Fig. 2.1d shows a picture of generator LG3.
2.2 Oscillating System - Equation of Motion
The buoy of the point absorber converter is connected to the linear genera- tor with a steel or synthetic wire. For simplicity the only movement in heave mode is considered, vertical movement only. Further, the wire is assumed to be always straight, the weight of the translator pulls the wire straight due to
(a)Representation of lin- ear generator LG2 and LG3.
(b)Representation of lin- ear generator LG9 and LG12.
(c) Scetch of LG2 and LG3. LG9 and LG12 have no frame and are supported by the hull.
Figure 2.3:Illustrations of the dynamics of the generators and the mechanical struc- ture of LG2 and LG3.
gravity or the translator is forced downwards by springs connected between the translator and the generator frame. This may in real life not always be true since the damping of the generator is significant so the possibility of a situa- tions where the buoy moves downwards more quickly than the translator can’t be excluded. The wire is assumed to be infinitely stiff, that is it will not stretch, which might be a reasonable assumption since the wire is thick thus very stiff, also the synthetic wires used are braided and the steel wires used have strands twisted clockwise and anticlockwise, this restricts rotational movement and coupling between different modes of motion. Linear wave theory is used and the buoy is assumed to be small compared to the wave length. The diameter of the buoys used is about 3 to 6 meters.
Also the influence of air is neglected since the air density is much lower than the density of water. Non-linear effects of the generator are not consid- ered. Ideal fluid is assumed, which excludes friction between the buoy and the water. This is reasonable when the Reynolds number is high, which means the inertial forces are large compared to the viscous forces (drag forces). The magnitude of inertial force is
O fi= (ρ ˙z2d2) (2.1)
where ρ is the water density, ˙η is the velocity of the body in question and d is the physical size of the body, in this case the diameter of the buoy. The magnitude of viscous force is
O fv= (µ ˙zd) (2.2)
where µ is the viscous coefficient, in this case of sea water. Thus the ra- tio of inertial forces to viscous forces, which is the Reynolds number,Re , is dimensionless
Re=ρ ˙zd
µ (2.3)
Large structures like ships, rigs and large buoys have a high reynolds num- ber and their movements are dominated by inertial forces, while small bodies like bacterias have a small reynolds number and their movement is much more dominated by viscous forces than inertial forces. A buoy of the size used in this experiment is a small ocean structure where viscous forces might have some influence, but these effects are not accounted for in this thesis [7]. Large bodies have high reynolds number and their movements are dominated by in- ertial forces, while small bodies like bacterias have a small reynolds number and their movement is much more dominated by viscous forces than inertial forces. A buoy of the size used in this experiment is a small ocean structure where viscous forces might have some influence, but these effects are not ac- counted for in this thesis. [7].
Because of the relatively stiff connection between the translator and the buoy and the exclusively vertical movement, it’s considered one moving body with one common damping, stiffness and mass, which will all be explained.
This equation governs the movement of the buoy.
(Cb−Ct)z + (Bb+ Bt)˙z + (mb+ maz+ mt)¨z = fx(t) (2.4) This is the governing equation of this oscillating system in the time domain.
The calculation of this is divided into two subproblems [8].
The right hand side, the excitation force is found by calculating the force on a static body, the buoy is fixed as the waves act on it.
The left hand side of the equation is solved by forcing the buoy to oscillate in still water, with no incoming waves, due to the force previously calculated, thus radiating waves.
3. Wave Theory
3.1 Linear Wave Theory
Waves which are long compared to the amplitude are well described by the lin- ear wave theory or in other words linearized equations of motion. Tall waves are less well described by this theory and breaking waves can not at all be described by linear theory. Another example of non-linear phenomena is am- plitude dispersion which involves that wave velocity depends on wave height, not only wave frequency. Linear waves can be described by potential flow.
Non-linear effects are barely considered in this study.
Water waves are dispersion, meaning that the velocity of the wave depends on its frequency, as can be seen in table 3.6. The table also shows other proper- ties of harmonic, or regular, linear waves. Harmonic waves are characterized by a single frequency. Diffraction is the phenomena of objects in the water changing the wave, e.g. an island will change the direction of a wave. Also the topography of the seabed will change influence the wave and its direction.
Also a wave power converter will disturb the incoming wave. Objects which are small compared to the wavelength will disturb the wave in a smaller extent than objects that are large compared to the wavelength.
3.1.1 The Energy within Ocean Gravity Waves
Considering single wave particles in a wave, figure 3.1 shows how the particles move in circles. Notice that as the wave moves on the surface of deep water, the wave particles move approximately in circles with a period equal to one wave period and diameter equal to the wave height. In shallow water the wave particles move in ellipses. On the bottom of deep water the water particles don’t move at all, and on the bottom of shallow water, the wave particles move purely horizontal. Table 3.6 shows how deep water and shallow water can be defined. Deep water is in general depths larger than half a wavelength.
From the particle movement we might conclude that waves don’t transport much water over longer distances, the water is transported locally in circles or ellipses.
3.1.2 Ocean Waves
This means the water mass transported, or momentum, by the waves is in av- erage close to zero. The reason why the momentum is not precisely zero, is
Figure 3.1:The path wave particles move in, approximately, in deep water (left) and shallow water (right). Note that the particles travel in circles, but they do not rotate.
that the loops in which water particles travel, for non-linear reasons, are not completely closed, during one wave period, the particle ends up almost where it started, but not precisely at the same spot. The wave particle has drifted slightly due to Stokes drift [9, p. 79]. This gives rise to some net momen- tum causing, among other contributions such as the Coriolis effect, tempera- ture and salinity variations and tides, not described here, surface water in the oceans to circulate.
This mass transport is small compared to the local rotational mass transport of waves, so the time average net mass transport is neglected in this context.
Given the water particles have some velocity, it’s clear that the water parti- cles also have some kinetic energy. As the water particles in addition to move in the horizontal plane, also move in the vertical plane, it’s given that water particles have some potential energy caused by gravity. The magnitude of po- tential power of a wave is equal to the magnitude of kinetic energy [10]. For those familiar to electromagnetics, this can be thought of as analogous to the oscillations of magnetic and electrical energy in an electromagnetic wave.
Ocean waves can be sinusoidal or be superpositioned by a set of waves, each of them having its own height and frequency. Waves are created as wind stretches the water surface, producing very irregular waves. With time, as these waves travel, they develop into more regular waves [9, p. 83].
3.1.3 Group Velocity - Phase Velocity
As explained, ocean gravity waves do not transport much mass, however, they do transport a significant amount of energy. Also, waves often travel in groups of a number of a few waves, so called wave trains. Looking at the ocean, on relatively deep water or preferably not on a beach, spotting such a wave train, say after a boat or ship recently passed by, the last and the first waves of the wave train are small. Looking carefully, the waves in the back are growing higher as the waves in the front of the wave train decrease in height. Con-
cluding from this the velocity of waves and wave trains are not equal. Not being aware of this, and not paying attention to the evanescent nor the grow- ing waves one will get fooled; the waves will approach land much later than what one might intuitively estimate on basis of the wave’s velocity. More pre- cisely, in deep water the velocity of one wave is twice that of the wave train as shown in table 3.6. The total energy E stored in a horizontal square meter of wave is [9, p. 77]
E= Ek+ Ep= 2Ek= 2Ep=ρ g
2 |ζA|2 (3.1)
depending on in order of appearance kinetic and potential energy, water density, gravity and the wave amplitude squared. This energy of the waves is transported with the velocity of the wave train also called the group velocity.
As seen, waves transport some energy, an energy flux exists. J denotes the en- ergy E [kW] transported per time [s] and meter of wave crest, in other words, kilowatts per meter of wave crest, or energy flux density. Fig. 3.4 shows the yearly average energy flux of ocean waves in the world.
3.2 Wave Climate
3.2.1 Wave Power Spectral Density
As described in chapter 3, real ocean waves are irregular and might contain more than one prevalent frequency at once. Often, a broad spectrum of fre- quencies is found during one period of time. To characterize the ocean waves at a location, the surface elevation is recorded for a short time for example for about 20-30 minutes, as can be seen on fig. 3.2. If this interval is shortened, it might poorly represent the present wave condition, as wave activity might fluctuate greatly from minute to minute, and waves might travel in groups. A longer elevation series can be recorded but will in less extent catch the wave activity at the moment. Several techniques are available to record surface el- evation, from satellites, buoys, subsea equipment or overhead equipment, uti- lizing laser, ultrasound or accelerometer to record the surface elevation as it changes with time.
The recorded surface elevation might be transformed by Fast Fourier Trans- form to produce a wave power spectral density S as can be seen on fig. 3.3 which illustrates how the wave power is spread among the various frequen- cies which were detected during the transform.
The frequency or frequencies characterizing the present sea state can be seen as peaks of the power spectral density chart. The spectrum is charac- terized by the wave height Hm0, eqn. 3.2, also referred to as characteristic or significant wave height, which is calculated from the zeroth moment. The n-th
Figure 3.2:30 minutes of surface elevations measured by Waverider buoy at the Ly- sekil research site. The period of individual waves are not clearly visible, but the figure shows the amplitudes of the larger waves and the minute to minute fluctuations in wave amplitudes.
Figure 3.3:A wave power density spectrum produced by Fast Fourier Transform of 30 min. surface elevations recorded at the Lysekil research site.
moment is calculated by eqn 3.6. The zeroth moment is also the variance of the spectrum. The mean wave period Tmean, of the spectrum is 3.3. The zero crossing period Tzerois shown in eqn. 3.4 and the energy period Teis described by eqn. 3.5 [11].
Hm0= 4√
m0 (3.2)
Tmean=m0
m1 (3.3)
Tzero=r m0
m2 (3.4)
Te= m−1
m0 (3.5)
mn= Z ∞
0
fnS( f ) d f (3.6)
The total energy flux density P of the present sea state is described in eqn.
3.7 [9].
P= ρ g2
64πHm02 Te (3.7)
3.2.2 Wave Climate Influence on Converter Economy
By calculating the wave power repeatedly, the average power for an extended period of time can be obtained, say for one year. This yearly average greatly depends on the most common wave climate of the region in question. The presence of occasional high power sea states or a 10 or 100 year wave, that is the highest wave likely to appear statistically during 10 and 100 years, does not in great extent contribute to the yearly average wave power. As mentioned in the introduction, a wave power plant must be constructed to withstand se- vere sea, say a 10 or 100 year wave. Extreme waves will on the other hand not contribute much to the year average electricity production of a converter, which might even shut down during harsh sea, like windmills do during ex- treme winds.
Because it’s assumed that the cost of a wave power plant depends on the extreme conditions at the site, and the income based on electricity production depends to some extent on the wave resources occurring most frequent at the site, some sites are well suited for electricity production from ocean waves.
Such sites are those with minor variations of wave power. The perfect site would be a site with constant wave energy flux, but such sites may not exist.
Looking at map 3.5 [12] this ratio varies greatly from one region of the world to another. Regions close to equator, say between 5 degrees north and south of equator the wave power is stable. Notice also that far from the equator some regions have a more stable wave climate than other regions.
This makes it clear, the available wave resources at a site is alone not so interesting, to most wave energy devices. Stability of the wave climate is a significant attribute in the design of a wave power plant.
Figure 3.4:Offshore energy transported by ocean waves, kW per meter of wave crest.
The data originate from the ECMWF (European Center for Medium-Range Weather Forecasts) WAM model archive and are calibrated and corrected (by OCEANOR) against a global buoy and Topex satellite altimeter database.
Figure 3.5:Ratio of the local 100 year wave to the local mean wave height. The data originate from the ECMWF WAM model archive and are calibrated and corrected by Fugro OCEANOR against a global buoy and Topex satellite altimeter database
3.3 Wave-Body Interaction
The generator operates by the wave excitation force pushing the buoy up- wards. The gravity and the translator springs 2.3a in the generator pull the buoy and translator downwards. As the buoy oscillates the water surround- ing the buoy accelerates, causing an added mass, or imaginary mass which influences the dynamics of the system, such as the natural heave period of the buoy-generator body and especially studied, the inertial force from the translator as it hits the upper end stop. As the buoy oscillates, it produces waves, causing a damping force, or wave radiation force on the buoy. Vis- cous forces are neglected, due to large bodies have high Reynolds numbers, and only heave movement is considered as the horizontal movements of the buoy are small compared to the water depth at the site. Added mass, damping force and the excitation force are the important hydrodynamic properties of the system. These properties are described here and further in textbooks like Newman [13].
Diffraction is the influence on a wave as the wave reaches any kind of obsta- cle. The presence of a body in water influences the wave pressure, the dynamic pressure, usually increasing the pressure. For large bodies, this must be con- sidered and can be done by potential theory on a computer. A common method is numerical boundary element software in the frequency domain which de- termines the potential flow around a body at a given wave frequency and unit wave height, height of 1, say meter. Added mass, damping and excitation force can be found. The force is calculated by integrating the Froude-Krylov and diffraction pressure over the body wet surface of the body, giving the force on the body at the given frequency and unit wave. This force scales directly with wave height [14]. The calculated coefficients depends on the frequency of the incoming wave since the potential flow depends on the shape of the wave.
Wave length and wave frequency is connected through the dispersion relation described earlier.
In the boundary element software, the fluid motion is described by Laplace’s equation under several conditions. The dynamic boundary condition states that a fluid particle on the surface stays on the surface, as the surface can not hold a pressure, also named as free surface condition. Further, the fluid can not penetrate the hull of submerged body. Greens theorem is used to convert a volume integral to surface integral, calculating the fluid motions on all surfaces, the hull, the seabed, the surface and imaginary boundary surfaces which enclose the given fluid volume. The kinematic condition ensures that fluid particles at the boundary stay at the boundary at all times.
The boundary element code used in this project was commercial. Alter- native potential flow code is also described by Hulme [15], Fernandes [16]
and Mavrakos [17] on axisymmetric bodies. Also to mention, is the alterna- tive method of calculating excitation force on a body, the Haskind relations,
/LQHDU7KHRU\RI5HJXODU:DYHV5HYLHZ Wave propertySHALLOW WATER (d/ < 1/20)λINTERMEDIATE WATER (1/20 < d/ < 1/2)λDEEP WATER (d/ > 1/2)λ Velocity potential (X=∇φ)
φ ωω=agk(z+d) kd(t-kx)cosh coshcosφ ωω=agk(z+d) kd(t-kx)cosh coshcosφω ω=−ag etkxkz cos() Dispersion relation JNGω2 = gk kdωtanh JNω Wave length - wave period relationλ = Tgdλ π
π λ = g 2T2d 2tanhλ π = g 2T ( 1.56 T)22≈ Wave profileηω = a(t-kx)sinηω = a(t-kx)sinηω = a(t-kx)sin Dynamic pressuredp =ga(t-kx)ρωsindp=gak(z+d) kd(t-kx)ρωcosh coshsinpgaetkxdkz =−ρωsin() Horizontal particle velocityu =a kd (t-kx)ω ωsinu =ak(z+d) kd(t-kx)ωωcosh sinhsinuaetkxkz =−ωωsin() Vertical particle velocityw =a z+d d(t-kx)ωωcosw =ak(z+d) kd(t-kx)ωωsinh sinhcoswaetkxkz =−ωωcos() Horizontal particle acceleration&cosu =a kd(t-kx)
2 ω ω&cosh sinhcosu =ak(z+d) kd(t-kx)2 ωω&cos()uaetkxkz =−ωω2 Vertical &sinw = -az+d (t-kx)2 ωω&sinh sinw=-ak(z+d) (t-kx)2 ωω&sin()waetkxkz =−−ωω2
Figure 3.6:Basic equations of linear wave theory. Part of booklet Linear Wave Theory.
Note that symbols might differ from the rest of this text. See nomenclature on next page. Property of NTNU and H. E. Krogstad and Ø. A. Arntsen [10]
/LQHDU7KHRU\RI5HJXODU:DYHV5HYLHZ Wave propertySHALLOW WATER (d/ < 1/20)λINTERMEDIATE WATER (1/20 < d/ < 1/2)λDEEP WATER (d/ > 1/2)λ Velocity potential (X=∇φ)φ ωω=agk(z+d) kd(t-kx)cosh coshcosφ ωω=agk(z+d) kd(t-kx)cosh coshcosφω ω=−ag etkxkz cos() Dispersion relation JNGω2 = gk kdωtanh JNω Wave length - wave period relationλ = Tgdλ π
π λ = g 2T2d 2 tanhλ π = g 2T ( 1.56 T)22 ≈ Wave profileηω = a(t-kx)sinηω = a(t-kx)sinηω = a(t-kx)sin Dynamic pressuredp =ga(t-kx)ρωsindp=gak(z+d) kd(t-kx)ρωcosh coshsinpgaetkxdkz =−ρωsin() Horizontal particle velocityu =a kd (t-kx)ω ωsinu =ak(z+d) kd(t-kx)ωωcosh sinhsin uaetkxkz =−ωωsin() Vertical particle velocityw =a z+d d(t-kx)ωωcosw =ak(z+d) kd(t-kx)ωωsinh sinhcoswaetkxkz =−ωωcos() Horizontal particle acceleration&cosu =a kd(t-kx)
2 ω ω&cosh sinhcosu =ak(z+d) kd(t-kx)2 ωω&cos()uaetkxkz =−ωω2 Vertical particle acceleration&sinw = -az+d d(t-kx)2 ωω&sinh sinhsinw=-ak(z+d) kd(t-kx)2 ωω&sin()waetkxkz =−−ωω2- 27 -
Dynamic pressuredp =ga(t-kx)ρωsindp=ga kd(t-kx)ρω coshsinpgaetkxd=−ρωsin() Horizontal particle velocityu =a kd (t-kx)ω ωsinu =ak(z+d) kd(t-kx)ωωcosh sinhsinuaetkxkz =−ωωsin() Vertical particle velocityw =a z+d d(t-kx)ωωcosw =ak(z+d) kd(t-kx)ωωsinh sinhcoswaetkxkz =−ωωcos() Horizontal particle acceleration&cosu =a kd(t-kx)
2 ω ω&cosh sinhcosu =ak(z+d) kd(t-kx)2 ωω&cos()uaetkxkz =−ωω2 Vertical particle acceleration&sinw = -az+d d(t-kx)2 ωω&sinh sinhsinw=-ak(z+d) kd(t-kx)2 ωω&sin()waetkxkz =−−ωω2 Group velocityccg=FFNG NGJ=+1 212 2( sinh)ccg=1 2 ωππλ=7N T = wave period λ = wave length a = wave amplitude g = acceleration of gravity c = λ/T = phase speed t = time x = direction of propagation z = vertical co-ordinate positive upward, origin at still water level d = water depth pd = dynamic pressure GSJ] RSρ= total pressure in the water ( -ρgz = hydrostatic pressure, op = atmospheric pressure). E =ga
1 2
2 ρ = wave energy (per unit surface area) PEcg= = wave energy flux (per unit width along the wave crest)
which in the most common form describes the relation between the excitation force and the damping coefficient [18].
3.4 The Excitation Force
Excitation force is divided up in Froude-Krylov force which is the force from the undisturbed wave on the body, such as a buoy. However, the presence of the buoy affects the wave, causing it to change, or diffract. This might cause an additional force. The sum of these forces, the excitation force can be calculated by the BEM method, or other numerical methods. For small objects compared to the wave length, the excitation force can be approximated as being equal to solely the Froude-Krylov force.
For a sinusoidal wave eqn. 3.8 is a wave potential, which is the spatially integrated velocity field, when not considering diffraction.
φ =gζA
ω ekzsin(kx − ωt) (3.8)
The dynamic term of Bernoulli’s equation gives the dynamic pressure Pd= −ρ∂ φ
∂ t (3.9)
Combining eqn 3.8 and 3.9 results in the dynamic pressure Pd= −ρgζAcosh k(h + z)
cosh(kh) cos(kx − ωt) (3.10) In addition, the water pressure increases with distance from the mean water surface, might also be referred to as static pressure, since for some floating structures the draft might be about constant.
Ps= −ρgz (3.11)
Which leads to the following equation of the Froude-Krylov pressure
PFK= Pd+ Ps (3.12)
In the case of oscillating structures in heave mode, like in this study, the dynamic pressure PBd makes up a small part of the total pressure and may be neglected. Integrating the pressure and the normal vector of the wet surface, over the entire surface gives the Froude-Krylov force.
fxFK= Z
PFKnsds (3.13)
The vertical component of this force is the heave excitation force. With no dynamic pressure taken into consideration, the heave excitation force of the body only depends on the magnitude of water volume it displaces. This is the hydrostatic approximation which is a good approximation for the wave energy converter buoys, resulting in the force
fh= Z
ρ g(η − z)nsds (3.14)
More accurate excitation force calculations can be done by taking diffrac- tion force in consideration, using numerical code.
3.4.1 Interpretation of Added Mass
The added mass is not a mass, but behaves like a mass in an oscillating system, it may be treated as a mass in the governing equation. The cause of added mass is water particles accelerating as they pass an obstacle. Say, an infinitely thin plate with its length axis parallel to its own direction of oscillation will have no added mass since the water particle’s path will not be disturbed by the presence of the plate. Any other body, with some thickness perpendicular to the direction of oscillation will have added mass since the water particles are forced to accelerate around the body.
3.4.2 Added Mass Calculation
Added mass can be calculated numerically, for example by the BEM method or for canonical geometries analytically, like a fully submerged infinite long cylinder or a fully submerged sphere [13]. In both cases potential flow theory might be used. Added mass can also be experimentally measured and scaled.
Traditional added mass calculations on slender structures, such as ships, as- sume no flow in the length direction of the ship as it oscillates in heave, thus for each section of the ship the 2-dimensional added mass is found numeri- cally, or empirically as for most ship designs there exist an older hull design similar to the new one. For more or less simple axisymmetric bodies, such as a buoy, scientists have calculated added mass numerically. A convenient ta- ble of added mass values is available in the public document [19]. Empirical equations predicting added mass exist as well.
For any calculation of added mass, proximity of a wall, bottom or free sur- face must be considered. Think of a cube oscillating up and down in infinite water, the water particles will have to accelerate when near the accelerating body. Then think of a cube oscillating up and down very close to the ocean
floor. In the latter case the water particles will have to accelerate with a higher magnitude, since the floor blocks the movement of water, forcing the water to move through a narrow slot. The water must accelerate more if there is less of it, if it’s ought to fill the same volume displaced by the cube during the period of time equal to the first case. Through conservation of energy the added mass must be higher in the latter case since high accelerations lead to high velocities and kinetic energy is proportional to the square of velocity [7, p. 141].
Close to a free water surface, the added mass is very frequency dependent, and also complicated to calculate [19, p. 19]. Some bodies, like slender hor- izontal cylinders oscillating vertically are frequency independent just a short distance beneath the water surface [8, p. 3.27]. The heave added mass of a fully submerged body close to the free surface is influenced by the proximity of the surface, and the heave added mass of a floating body is affected by the draft. For some geometries, the added mass of a floating body is half of the value as for the same body submerged in infinite water [19, p. 40] [20, p. 22].
In the Lysekil project both vertical cylindrical buoys and toroidal buoys have been used. The added mass of them are fundamentally different. The added mass of a cylinder depends on the frequency, but the added mass of a toroidal buoy do so much more, with extreme positive and negative peaks [17, p. 393] [7]; a toroidal floating buoy might have negative added mass for some frequencies, but for long waves compared to the buoy length, which are the waves of interest in this project, the added mass stays positive at all times.
The toroidal buoys used in the Lysekil experiment are approximated with 6 cylindrical steel sections welded and bolted together. One of those used are made of 6 cylindrical steel sections of 2705 mm centerline length and 1016 mm diameter. It’s comparable with a toroidal buoy with a length from center to the center of the hull elements 5 times the radius of the elements. Mavrakos has described the hydrodynamic characteristics of such a buoy [17].
3.4.3 Hydrodynamic Damping
Damping of an oscillating buoy in ideal fluid is caused by radiation of waves.
Any floating object oscillating in heave mode in water radiates waves, thus dissipates energy. As an analogy this can be thought of as electric resistance.
Hydrodynamic damping is also referred to as radiation resistance. This topic will not be emphasized since the electric damping of the translator in the gen- erator might dominate the damping of the system. Damping coefficients may be numerically calculated by software, calculating the radiation of the floating body. This might be the same computer code, radiation-diffraction problem software, used to determine the excitation forces or the response amplitude operator (ROA) which models the relation between water elevation and the elevation of the body.
