energy from w aves and tidal currents towards 20yy · energy from w aves and tidal currents towards...
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ENERGY FROM WAVES AND TIDAL CURRENTS Towards 20yy ? Emile Baddour Institute for Ocean Technology National Research Council August 2004 Acknowledgements: The author wishes to thank IC for their support of the project, Geoff Lewis of IC for numerous discussions through phone, emails and meetings, CISTI for their collection and their help in gathering related information and NRC for allowing the time spent on the project.
TABLE OF CONTENTS Summary
1. Introduction 2. Ocean waves and marine currents
2.1 Origins, generation and propagation of wind waves 2.2 Description and modification of waves 2.3 Real sea characteristics 2.4 Wave energy resource distribution 2.5 Data about waves 2.6 Marine currents origins 2.7 Marine currents resource 2.8 Marine currents characteristics and resource distribution
3. Ocean energy harnessing systems 3.1 Wave energy harnessing technology 3.2 Wave energy developments and activities 3.3 Ocean currents energy technology 3.4 Ocean currents developments and activities
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4. Transfer of power systems 4.1 Interfaces 4.2 Storage of energy 4.3 Electric power generation and conversion 4.4 Power transmission to the grid
5. Ocean power economics 5.1 Factors affecting the evaluation of costs 5.2 Capital and Operating costs of Ocean Energy systems 5.3 Generation costs of wave and current energy 5.4 Comparison with costs of other renewables and electric
power prices 6. R&D programmes
6.1 In Canada 7. Timeline proposal
7.1 Ocean energy influential events 7.2 Timeline 7.3 Scientific, technical, societal and economic challenges 7.4 Constraints and opportunities
8. References 9. Recent scientific and technical research bibliography
Appendices: Appendix 1. Formulation and computation of wave power in long-crested regular and irregular seas Appendix 2: Useful wave & tidal power/energy links Appendix 3: Recent publications from IOT
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Summary: Renewable energy sources have been known for centuries. It is only recently in the past 30 years that modern technology and investment got together to produce viable alternatives to fossil fuel that can produce energy on a useful, economical scale. The stimulus to develop renewables is being driven by several factors. It is found that fossil fuels are causing global warming and there is pressure to reduce the amount of carbon dioxide produced in energy production. There is prospect of rising pricing of oil and gas as well as depletion of the reserves and relying on importing fuel. Now formally on accepting the Kyoto protocol Canada has committed towards reducing gas emitions that cause house effects. In its Millennium Statement, "Energy for Tomorrow’s World – Acting Now!", the report of the World Energy Council, presents three principles for energy development: Accessibility is the provision of reliable, affordable modern energy services. Availability addresses the quality and reliability of the service. Acceptability addresses environmental goals and public attitudes, specifically local pollution and global climate change. These principles are underpinned by ten policy actions which include: keeping all energy options open and ensuring adequate and appropriate research funding. (World Energy Council, 2000) Keeping with the recommendation of keeping all energy options open, the present document reports on Energy from the Ocean. To put this energy source in perspective we have to note that presently this new renewable is part of the Other sources making the 0.5% of the World Total Primary Energy Supply. See Figure 1.1 below. This report presents an overview of the main developments in ocean energy from waves and currents and could be considered as an introduction to the topic of Ocean Energy Systems, specifically from ocean waves and currents. Limited time and not being in the scope of this review precluded verification of some of the claims herein, using different or independent sources from the ones reported. However, it can indicate the trend. Further development and assessment of the resource are needed. The report concludes with a list of tasks classified under 6 main possible threads for an action plan for research and development in ocean energy systems and a proposal to identify the Institute of Ocean Technology of the National Research Council, http://iot-ito.nrc-cnrc.gc.ca/about.html as the Center for Testing, Evaluation, and Research in Ocean Energy Systems. The center is proposed as a node within an Ocean Energy Technology Network that would allow Industry, Academia and Government a much-needed collaboration within their identified respective roles.
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1. Introduction The consumption in energy around the world is estimated to increase over the next decades. The traditional methods of energy production are contributing to serious environmental effects that are still unknown. The energy sector is looking into a potential opportunity in the renewable energy. In the dynamic evolution of the renewable energy industry the harnessing of the energy from the oceans in its waves and currents forms is emerging. Relatively the technology of energy from the sea is new. Presently it is not competitively economic in comparison to wind energy for example, however the interest is there and is steadily increasing, as we shall show in this report. It is imperative to sensitize at the grass root level and increase the interest of government, industry and academia from the present level. An important feature of sea waves is their high energy density and it is recognized by some as being the highest among renewable energy sources. The idea of converting the energy whether of ocean waves or ocean currents into useful energy forms is not new. Leishman and Scobie (1976) have documented the development of wave power devices. According to them the first British patent dates back to 1855, while Girard and Son in France patented their ideas back as early as 1799. They have also counted about 340 patents in the period from 1855 up to 1973. It would be interesting to survey the patents of the last 30 years alone. Several configurations and set ups have been actually designed and tested at model scale. As it will be shown in this report some have been even deployed and operated in the sea. Recent impetus to modern research and development of ocean energy conversion happened after the sudden increase in oil prices in 1973. Earlier the harnessing of the energy of the tides and its energy have been considered and followed the principle of accumulating the water movements in creating a small head behind a dam. The La Rance power generation was built around 1969 followed by the Annapolis valley plant in 1984. Both are still running today and using the motion of the waters activated like clockwork by the tides. Several research programs started then and are still running on an on and off basis since that date in Europe depending on the support available from governments, national research centers, universities and private sources. The recent activities in the field come from the UK, Portugal, Norway, Denmark, Ireland and Sweden. These programs aim at developing wave power conversion that could be industrially exploitable in the medium and long term. Not only specific European governments public sector funding have been involved but also an increasing interest from the European Commission has been observed. Research Programs sponsored by the commission started around 1994 significantly contributing to stimulating and coordinating the activities carried out by universities, research centers and the industry. It is difficult right now to mention a precise dollar figure on the funds involved in ocean systems developments, however estimates could be found in the literature and will be reported. As mentioned above in the past thirty years energy from the ocean has gone through cycles of enthusiasm in concentrated effort of development, disappointment and back to the drawing board reconsideration stages. However, a vast amount of experience has been gained and added to improved designs and performance of ocean power techniques. This effort in Research and Development is bringing wave/current energy
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ever closer to commercial utilization. A number of commercial plants are being built in Europe, Australia and elsewhere. A number of devices have proven their applicability on a large scale in harsh operational environments and other are in different stages of their Research and Development cycle with different levels of their implementation. These will be reported herein. From studying the experts reports it is concluded that extensive R&D work is needed at both fundamental and application levels with the objective of improving the cost estimation, performance, feasibility of ocean energy systems to establish their position in the renewable energy market. The societal, political, industry and academic sectors are converging towards a need and demand for more "green", renewable energy. This trend is clearly seen in the Kyoto agreement and expressed in action plans in Europe, Canada, USA and other parts of the world. The following excerpt from of the Speech from the Throne in BC speaks for itself: Speech from the Throne, The Honourable Iona Campagnolo, Lieutenant Governor at the Opening of the Fifth Session, Thirty-Seventh Parliament of the Province of British Columbia, February 10, 2004
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…. Renewable energy has lately been receiving a lot of attention all over the world, in particular, from the media, government policy makers, energy industry, environmental and other interest groups. However, despite expectations and many efforts by governments to promote and subsidise the use of renewable energy resources, renewables still face high entry barriers in energy markets. There are many reasons for this, but the most important barrier in many cases appears to be the perceived poor economics of renewables compared to fossil fuels. This is often due to the traditional pricing structures, which do not internalise social and environmental costs and other externalities of energy provision and use. There also are financial and institutional barriers related to the typically small size of renewable installations, etc. The recent International Energy Agency Fact Sheet, "Renewables in Global Energy Supply" published in November 2002, shows the share of renewables of the World Total Primary Energy Supply (TPES) as 13.8%. This includes both commercial and
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non-commercial energy and covers all major renewable energy resources. Combustible renewables and waste account for nearly 80% of the renewables share, hydro for 16.5% and "new" renewables: geothermal, solar, tidal, wave, wind and other, together account for 0.5%. See Figure 1.1 below.
Figure 1.1 Fuels Shares of World Total Primary Energy Supply (source: IEA, 2002)
The domination of the energy supply by fossil fuels is set to continue for a foreseeable future, since their resource base remains adequate and the adverse environmental impacts are attracting a considerable effort to identify and deploy cleaner fossil fuel technologies, an area in which World Energy Council (WEC) has been active for several years. In its Millennium Statement, "Energy for Tomorrow’s World – Acting Now!", the report of the World Energy Council, presents three principles for energy development: Accessibility is the provision of reliable, affordable modern energy services for which payment is made under policy specifically targeted on meeting the needs of the poor. Availability addresses the quality and reliability of the service Acceptability addresses environmental goals and public attitudes, specifically local pollution and global climate change. These principles are underpinned by ten policy actions which include keeping all energy options open and ensuring adequate and appropriate research funding. (World Energy Council, 2000). See also International Energy Agency, Ocean Energy Systems Annual Report (2003). Keeping with the recommendation of keeping all energy options open, the present document reports on Energy from the Ocean. To put this energy source in perspective
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we note that this new renewable is part of the Other sources making the 0.5% of all the renewables, See Figure 1.1.
2 Ocean waves and marine currents 2.1 Origins and generation of wind waves 2.2 Wave energy resource distribution 2.3 Description and modification of waves 2.4 Real sea characteristics 2.5 Data about waves 2.6 Marine currents origins 2.7 Marine currents resource 2.8 Marine currents characteristics and resource distribution 2.1 Origins and generation of wind waves There are many kinds of waves in the ocean. They differ in form, velocity and origin. Some waves are too long and low to see, also there exist waves that travel on density interfaces below the sea surface. Waves may be generated by ships or landslides or the passage of the moon or by earthquakes or changes in the atmospheric pressure. The waves which are of interest in this report are those mainly raised by the winds. The energy in the waves comes from the sun through the winds as they blow over the oceans due to the differential heating of the earth. The winds transfer their energy to the surface of the sea creating waves. See Kinsman (1984). Wave energy is hence considered as a concentrated form of solar energy. The mechanism of the transfer of energy from the atmosphere to the surface of the sea is complex. The main phenomena can be reduced to: Air flowing over the free surface of the water activates a tangential stress on the water surface and result in the formation and growth of the waves. Variable shear stresses and pressure fluctuations are created by turbulent air flowing on the sea surface. Further wave increase and development happens when these oscillations and fluctuations are in phase with the waves. Some kind of resonance effect. Wave reaching a certain height will be directly affected by the wind forcing on the upwind face of the wave causing further growth. At each of the above steps energy in transferred to the water. The amount of energy transferred and hence the size of the resulting waves is a function of the wind speed, the length of time it blows and the distance over which it blows called the fetch. As the wave continues to grow the surface facing the wind becomes higher and steeper and the process of wave building becomes more efficient. However up to a point because there is a limit on how steep a wave can be. Steepness is the ratio of the height of the wave (distance between a crest and the following trough) to its length
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(distance between a crest and the following one) is approximately 1:7 in deep water. In the generating area, often a storm, wind waves form what is called a "sea". At the upwind end of the fetch the waves are small but with distance they develop i.e. their period and height increase and eventually they reach maximum dimensions possible for the wind that is raising them. The sea is then said to be fully developed. The waves have absorbed then as much energy as they can from wind of that velocity. An extension of the fetch or a lengthening of the time would not produce larger waves. (Kinsman 1987, Bascom 1976) See also Sverdrup and Munk: Wind, Sea and Swell. See Figure 2.1 for a schematic of waves development and propagation stages.
Figure 2.1. Concept of wave generation and propagation. The fetch is within the dashed line. Source: R. Silvester.
2.2 The wave energy resource distribution At each of the above steps energy in transferred to the water. The amount of energy transferred and hence the size of the resulting waves is a function of the wind speed, the length of time it blows and the distance over which it blows called the fetch. It is found that at each step power is concentrated and that solar power levels of about 100W/m2
can eventually be transformed into waves with power levels of over 1000 kW per meter of wave crest . The distribution of the wave energy resource over the globe and its daily and seasonal variability during the year are dependent on the major wind distributions and systems that are the main cause for the generation of ocean waves. The main wind systems are due to extra-tropical storms and trade winds. Ocean currents like the Gulf Stream in the North Atlantic and the Kuroshio current in the pacific feed energy into extra-tropical cyclones creating low pressure systems with wind speed that can reach up to 25 m/s and blow over a thousand kilometer fetch for two to four consecutive days before subsiding by hitting the coast. The storms are most frequent during the winter. Extra-tropical cyclones follow a north-easterly track and continually build the waves in
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the southern sector of the storm. These waves will travel in the same direction as the storm that generates them. However, waves generated in the northern part of a northern-hemisphere cyclone travel opposite to the direction of the storm and get less exposure to the storm’s wind energy. The result is that one gets comparatively less wave energy in the western sides of an ocean basin than in the eastern side. This feature is found in the northern hemisphere sectors of the Atlantic as well as the Pacific Ocean. The annual average of the wave-power levels is estimated along the edge of North America from 10 to 20 kW/m reaching a level of 50 kW/m along Newfoundland cost while along the edge of the eastern continental shelf of the north Atlantic is estimated to vary from 40kW/m off Portugal up to 75kW/m off the Irish and Scottish coast dropping to about 30kW/m off the Northern part of the Norwegian coast. Figure 2.2 shows the geographical distribution of coastal wave power levels. Along the coast of California the level is estimated as 30 kW/m and increasing to 60 kW/m along the coast of Northern British Colombia. Local variation in the estimate of wave power availability is certain and more accurate estimates are needed. Accurate knowledge of the directionality of the wave regimes is also needed. Similar analysis has been done for the southern hemisphere and is not presented here. (See for example Wavegen 2000) As for the trade winds systems off tropical coasts they carry an annual offshore wave power levels of the order of 10 to 20 kW/m. In comparison to the extra-tropical storm winds the trade-winds are more persistent and the variations in these winds intensity between the seasons are smaller. Some of these tropical areas might also be affected by swells generated in storm regions further north or south. Therefore some believe that this sustained source of wave energy around the equator is qualitatively a potential area for wave energy systems deployment since these areas avoid the extreme conditions found in higher density locations. (See for example Wavegen 2000)
Figure 2.2 Approximate global distribution of time-average deep water wave power. Wavepower is given in kW/m of wave front. Source: Thorpe (1998) and World Energy Council
Organisation (2004).
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2.3 Description and modification of waves Waves in the ocean are generally irregular. They are short-crested as opposed to long-crested, directional and more or less random in nature. No two waves have exactly the same height and they travel across the surface at different speeds and in different directions. Techniques for the coping with the chaotic nature of these waves on the real sea surface are discussed in the next section. It is first necessary to give an overview of the characteristics of ideal regular waves. Such waves rarely occur in the real ocean environment although they can be produce in laboratory wave tanks. They are important also because of the fact that the theory of irregular waves is based on the assumption that they can be represented by superposing or adding together a suitable number of regular waves. In order to discuss waves we use a standard set of definitions and terms for parts of the wave. The principal ones are defined as follows (Figure 2.3): Crest : The high point of a wave Trough: The low point of a wave. Wave height: Vertical distance from trough to crest. Wave length: Horizontal distance between adjacent crests. Wave period: The time in seconds for a wave crest to travel a distance equal to
one wave length. Wave frequency: The inverse of the wave period. Wave celerity: The ratio of wave length and wave period. Note that there is a direct relationship between wave period and wavelength but wave height is independent of either.
Figure 2.3 The important features of water surface waves.
After © Craton 1992, Brock Univ. In the offshore deep water regions waves vary slowly over space. However, as they approach towards the coast interaction with the seabed and currents could lead to significant changes in the characteristics of these waves. There also exist wave-coast line interactions resulting in what is called focusing, defocusing and sheltering of the waves. These changes in the wave climate will affect the energy densities and characteristic. In general near shore wave systems carry less energy than their offshore counterparts.
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Shallow water phenomena are generally classified according to their features in maintaining energy or not. Hence we classify wave changes phenomena as conservative or non-dissipative and dissipative. The main conservative processes include:
i- shoaling and refraction ii- diffraction and reflection
Shoaling is defined as the variation of wave height due to changes in water depth. As the water depth decreases the wave height first decreases then would increase very rapidly. The sudden increase in wave height causes the wave to break. Refraction like shoaling is caused by a varying depth seabed resulting in the focusing and scattering of waves and the turning of the wave crests becoming parallel to the bottom contours lines. Refraction can also be caused by the interaction of the waves with currents. Refraction in the form of focusing of the waves can be a positive factor in wave energy extraction. Diffraction is a negative factor. In wave energy extraction applications diffraction affect the smoothing of the distribution of wave energy in space. The main dissipative processes of interest include a reduction of the total amount of wave energy by converting it into current, water turbulence, sediment transport or heat. Also included in such dissipative phenomena are wave breaking, bottom friction, wave reflection from sloping or rough surfaced structures or beds and percolation. Waves loose energy by friction at the sea bottom. This is more important once the waves travel in shallower waters. The losses increase with travel distance and bottom roughness. 2.4 Real sea characteristics Waves range in size from the ripples in a pond to the great storm waves of the ocean and the tides whose wavelength is half the distance around the earth. Waves are classified according to their period (or frequency) that ranges from less than one second to more than hundred thousand seconds (tides). The energy spectrum diagram of Prof. Munk shows that the energy in the ocean is distributed among several major groups of waves each with a characteristic range of periods. At the lower end of the spectrum with the very short period waves we have: Ripples: of periods of fractional seconds Wind chop: of periods between 1 to 4 seconds Fully developed seas: of periods ranging between 5 to 12 seconds Swells: of periods ranging between 6 to 22 seconds Surf beats: of periods ranging between 1 to 3 minutes Tsunamis: of periods ranging between 10 to 20 minutes Tides: with periods of 12 or 24 hours.
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Each are generated and developed in a special way. See section 2.1 for the mechanisms generating wind waves. The above waves are also called gravity waves, since once they are created gravity is the force that drives them by attempting to restore the original flat surface. Also the small tiny ripples of one or two millimeters in height, generated by a small breeze are sometimes also called capillary waves because they are controlled by surface tension. Tsunamis and tides are generated by different mechanisms than the wind. See Figure 2.4.
Figure 2.4 Types of waves that occur on the oceans classified by their wave period. The red bars indicate the mean wave period for each type and the yellow bars show the range of wave periods. After © Craton 1992, Brock University.
Figure 2.5 after W. Munk shows the relative amount of energy in each of the wave systems described above. See also Bascom (1976).
Figure 2.5 The Ocean Wave Spectrum (Source Munk. See Bascom 1976)
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Real seas include waves that are random in height, period and direction. It is usually assumed that within a short length of time the characteristics of real seas remain the same hence defining what is called a sea state. Statistical parameters derived from the wave spectrum are used to describe such sea states and characteristics relevant to their energy content. See Appendix 1. The following wave height and period parameters are most often used: The significant wave height Hs : is the average height of the highest one-third waves. The energy period Te : is the mean wave period with respect to the spectral distribution of transport of energy. Tp the peak period is defined as the period corresponding to the peak in the variance density spectrum of sea surface elevation. It is the harmonic frequency component having the greatest amount of energy at a place passed by a random wave system. Some ocean energy systems can be tuned to this frequency. It resembles the tuning of a radio circuit to an electromagnetic field. (Thorpe, 1998). In deep water the power in each sea state P is given by as:
P = 0.5 Hs2Te kW/m
Where Hs is expressed in meters and Te in seconds and density of water taken as 1000 kg/m3. See Thorpe, 1998. In Thorpe, the annual variation in sea states is represented in a scatter diagram indicating how often a sea state with a combination of characteristic parameters occur annually. The annual average wave power level can be determined from the scatter diagram in the form:
Pave=∑ Pi Wi /∑ Wi
Where sea states of power level Pi occur Wi times per year. Several models for the formulation of the wave energy density distribution in terms of the frequency have been proposed. They are based on the velocity of the wind that generated the waves. The most notable are the Pierson-Moskowitz, the JONSWAP spectral models (Joint North Sea Wave Project) and others. Reference is here made to, for example, Kinsman, 1984 for further details. Some of the properties of wind waves are shown in Figure 2.6. In the figure the wave period is plotted against the amount of energy contained for three wind velocities. Each curve (spectrum) represents the distribution of energy between various periods in a fully developed sea. The area under each curve gives an estimation of the total energy. As an example we can consider a 20 knot wind (10 m/s). A knot is about 0.5 m/s. This relatively modest wind raises waves whose average is 5 feet or 1.52m and whose energy is spread over a range of periods ranging between 7 to 10 seconds (or frequencies between 0.15 Hz to 0.1 Hz) . If the wind increases to 30 knots (15 m/s) the waves increases substantially and the period gets longer. There is more energy available and these longer waves store it better, (Kinsman 1984). The average height of the waves are now 13.6 feet or 4.14 m and the maximum energy is centered around a period of 12 seconds or frequency of 0.08 Hz. See Figure 2.6. For a 40 knots ( or 20 m/s) wind the spectrum shows a sharp peak at 16.2 seconds ( or 0.062 Hz) and the average height of the waves are now in this case 28 feet (or 8.53 m).
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Figure 2.6 Wave spectra for fully developed seas for winds of 20, 30 and 40 knots.
After Pierson, Neumann and James see Bascom 1984. The following Table 2.1 shows the important characteristics of seas that are fully developed for winds of various velocities. An important point to note here is that a particular wind at a certain speed must blow for at least some time (shown in the table) along a minimum fetch length to raise fully the waves it is capable to generate. From the table for a 50-knot or 25 m/s wind blowing for 3 days over a 1500 miles fetch the highest tenth of the waves would average about 100 feet (about 30 m) high. Storms rarely reach such dimensions or durations.
Table 2.1 Conditions in fully developed seas After Munk , Kinsman 1984.
Wind Distance Time Waves
Velocity (knots)
Length of fetch in
nautical miles (km)
(hours)
Average
height in ft (m)
Hs Significant height (ft)
H10 Average of the highest
10% (ft)
Period where most of
energy is concentrated
(sec) 10 10 (16.7km) 2.4 0.9 (0.27m) 1.4 (0.43m) 1.8 (0.55m) 4 15 34 (57.8km) 6 2.5 (0.76m) 3.5 (1.07m) 5 (1.53m) 6 20 75 (127.5km) 10 5 (1.52m) 8 (2.44m) 10 (3.05m) 8 25 160 (272.1km) 16 9 (2.74m) 14 (4.27m) 18 (5.50m) 10 30 280 (476.2km) 23 14 (4.26m) 22 (6.71m) 28 (8.54m) 12 40 710 (1207.5km) 42 28 (8.52m) 44 (13.41m) 57 (17.4m) 16 50 1420 (2415.1km) 69 48 (14.63m) 78 (23.77m) 99 (30.17m) 20
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2.5 Data about ocean waves For the evaluation and estimation of long-term series of wave data two methodologies have been suggested. The first is based on measurement and observations and the second on building time series with numerical wind-wave models. A wide variety of on site and remote sensing measuring methods are available that produce accurate wave data. Visual observations made from sea-going ships are the earliest type of wave data for the oceans. Wind-wave models are mathematical algorithms encapsulated in computer programs that numerically generate and propagate wave energy based on input wind data or other relevant data. The accuracy is good for open ocean resource assessment in large basins such as the North-Atlantic and Pacific Oceans. These models are implemented at most meteorology centers. The available wave data is not an easy task to collect since the data are archived at several institutions that have different procedures to access it. 1) Measurements and Observations of ocean (wave/current) climate: Measurements A wide choice of measuring systems exists. The choice of a system depends on a number of parameters, namely: on depth, access and wave conditions of the measurement site and of the required details, for example directionality. Figure 2.7 shows a diagram of several wave measurement systems. Figure 2.8 shows a sketch of an in situ measuring system based on a buoy with local data storage as well as transmission to an on-shore processing data station.
Figure 2.7: Diagrammatic Sketch of several measurement systems
Source: Pontes T. Department of Renewable Energies, Lisbon, Portugal. Original in Earle and Bishop (1984). See also Brooke (2003).
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Figure 2.8: Diagrammatic sketch of an in situ wave measuring system.
Source: Pontes T. Department of Renewable Energies, Lisbon, Portugal and Laboratorio nacional de Engenharia e Tecnologia, Portugal . See also Brooke (2003).
Some devices can provide wave directional information by their own or when coupled with other devices. In situ devices can store information or transmit by cable or telemetry to an on-shore data processing stations. Remote sensing could use laser or radar devices mounted on a satellite, aircraft, ship, or could be based on land. Studies comparing device accuracy have been performed and could be found in the literature. Generally, the types of measurements are classified as:
i) In situ measurements ii) Remote sensing measurements
Wave-recording buoys are used extensively in the open-sea. Their data are generally in the form of time series of sea surface elevation from which wave height, period and direction parameters could be calculated. Spectral and direct analyses of the time series are used for this purpose. Beside buoys other devise are used. For example in coastal areas submerged or suspended pressure and acoustic probes, wave staffs, current meters could be used to obtain non-directional information about wave activities characteristic. When such probes are used in arrays directional information could be calculated. Energy devices utilizing Oscillating Water Column principle would generally include some measuring system of the water surface inside the chamber. Difficulties could arise in the presence of water spray. For resource assessment long period measurements are needed. However these are not easy to find. Short-term wave data acquisition is usually performed for coastal engineering projects as well as for offshore oil platforms. Remote sensing systems are used to provide spatial information about the sea surface in contrast to the information gathered at one point (in situ) as described above. The simplest remote-system measuring devices rely on aerial photography. These could be
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used for studies of wave refraction and diffraction near and along coastlines. They would be important for optimizing near-shore and shoreline plants sites. Satellites have also been advanced as a means to remote sensing wave on the surface of the ocean. The main limitation with this technology is that the data would only be intermittent. This strongly limits the data available in a specific zone. The positive side of it is that the method is statistically unbiased since the satellite sampling method is not under the effects of the sea conditions they try to measure. Altimeter data values have been used. The cost of performing and collecting wave data is not discussed in this report. This is by no means an exhaustive review and evaluation of the methods used for wave data collection. Observations Earliest types of wave data resulted from visual observations made from traveling ships. These have been archived from 1850s onward. Presently the visual observations are performed using well-defined procedures and techniques. Reference is here made to the world Meteorological Organization publications and standards. Several authors studied the accuracy of visual data. The report on these studies is outside the scope of this report. However, a number of reports conclude that in general, visually wave directions are most reliable, wave heights are considered satisfactory but wave periods are much less accurate. Visual data are considered to supplement the data obtained from measurements. Several global and regional atlases of visual wave climates could be found in the literature. They would also be relied upon wherever measurements are not available. 2) Theoretical and/or Computational ocean wave models Theoretical and computational models have been used to assess shoreline, near-shore or offshore wave power resource. Deepwater models are available to simulate and compute the propagation of deepwater waves and their energy transport. Calculations of the transformation of deepwater wave systems when they approach shallower depth are also required. The size of the coastal area in these studies can vary depending on the seabed topography and the length the coastal line. Scaled physical models are supplemented with computational computations that are based on mathematical models of wave propagation and transformation. This is an ongoing area of research in offshore and coastal engineering and the results could well suit the application discussed in this report. Such models except perhaps wave breaking and some other complex interactions can satisfactorily describe a wide range of wave propagation phenomena. Simplified representations are hence used in these cases. The purpose being to evaluate and determine the amount of ocean (wave/current) energy transported and dissipated. Without being exhaustive Table 2.2 and Table 2.3 present an example of an attempt to classify computational shallow water wave models that are found to be appropriate for the assessment of ocean energy resource. The information in the tables follow the criteria presented by Southgate (1987, 1993). These tables help to illustrate only the vast amount of studies on waves and currents, their generation, propagation, forecasting and modeling. A separate study of the resource evaluation methodology is needed to assess the progress in this field of research and development.
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Table 2.2 Wave processes incorporated in shallow water computational models (source after Southgate 1987, 1993)
Wave processes modeled
Computational model
Internal diffraction
External diffraction
Reflections Bottom friction
Wave breaking
Forward tracking
Not modeled, but numerical smoothing by ray averaging
Only for special
situations
Yes
Yes (1)
Yes (1,2)
Ray models
Backtracking
Same as above
No
Yes
No
No, apart from check at inshore points
Refraction Same as above
No No (3) Yes Yes (2)
Hyperbolic refraction-diffraction
Yes
Yes
No
Yes
Yes (2)
Parabolic refraction-diffraction
Yes
Yes, but difficult in
general
No (3)
Yes
Yes (2)
Elliptic refraction-diffraction
Yes
Yes
Yes
Yes
Yes
Grid models
Nonlinear Yes Yes Yes Yes Yes
Note: All models require reasonable gentle depth variations
(1) Except for intersecting wave trains (2) Approximate energy loss in shallow water only (3) Backscattered waves cannot be modeled
Table 2.3 Shallow water computational models: model suitability parameters
(source after Southgate 1987) Computational
model Bathymetry
(special requirements)
Extent of sea area
Type of coastline
Number of inshore points
Offshore wave
conditions Forward tracking
Regular depth variation;
poor for shoal systems
Unlimited
Any
Covering entire modeled area
Regular wave (1)
Ray models
Backtracking
No
Unlimited
Any, except where depth variation is important
One
Full
spectrum
Refraction
Regular depth variation
Unlimited
Reasonably straight coast
facing open sea
Covering entire modeled area
Regular wave (1)
Hyperbolic refraction-diffraction
Same as above
Unlimited
Same as above
Same as above
Regular wave (1)
Parabolic refraction-diffraction
No
Limited to a few kilometers
at most
Same as above Same as above Regular wave (1)
Elliptic refraction-diffraction
No
Same as above
Same as above Same as above Regular wave (1)
Grid models
Nonlinear No Same as above Same as above Same as above Surface elevation
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Note: All models require reasonable gentle depth variations (1) Full spectrum S(fθ ) may be covered by multiple runs. Wave-current interaction models have also been studied. The review of this topic is beyond the scope of this report. However, the work of Baddour et al (1990, 1991) on this particular subject of wave-current interaction and more recently in 2003- 2004 on nonlinear wave generation and propagation is worth noting, see Appendix 3. These models also allow the development of a computational tool for energy flux estimations in deep and shallow water for specified sea states. Other CFD tools and computer programs could theoretically be used for such purposes. An evaluation campaign of these tools are perhaps in order. 2.6 Marine currents origins Currents within the oceans are determined, for the most part, by the large scale structure of atmospheric circulation; currents at the surface of the oceans are dictated by the prevailing winds that blow over the water surface. Surface waters move due to friction between the moving air and the water surface. Figure 2.9 shows the major surface currents of the world’s oceans. This displacement of water at the surface, in turn, contributes to the generation of currents that extend to great depth within the oceans. See: Craton 1993, Brock university.
We are here interested in locally generated currents due to tides that are simply the rhythmic rising and falling of the surface of the ocean over the course of a day. Tides are not currents, themselves, but as the water surface rises and falls along a coast the water must flow to accommodate the geometry and topography of the coastline. The rising and falling of the ocean surface is due to the rotation of the approximately spherical Earth that is covered by a slightly elliptical ocean. The form of the surface of the oceans is dictated by gravitational interaction between the Earth, the Sun and the Moon.
The centrifugal force about the earth (due to its rotation) acts outward from the centre of the Earth and is equal in all directions. The presence of the Moon, in a circular orbit around the Earth, results in a gravitational attraction between the two bodies. Gravitational attraction is strongest at the closest point between the two bodies and diminishes away from that point. Thus, the moon’s gravitational attraction is strongest on the side of the Earth that faces the moon and weakest on the opposite side of the Earth. The average gravitational force exerted on the Earth by the moon is balanced exactly by the centrifugal force of the spinning Earth. On the side of the Earth facing the moon the gravitational force is strongest and exceeds the centrifugal force so that the ocean surface is pulled slightly towards the moon (creating a bulge on the ocean surface). On the opposite side of the Earth the centrifugal force exceeds the gravitational force, pushing the ocean surface slightly away from the Earth (creating a second bulge on the ocean surface). The result is an ocean surface with two bulges at 180 degrees to each other (and shallower water at 90 degrees to each of the bulges).
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The position of these two bulges are fixed with respect to the position of the Moon while the Earth rotates about its polar axes and the ocean’s ellipsoidal axis is the line that joins the moon and the Earth. The tides are produced as the Earth rotates through these two bulges on the ocean surface. See Craton 1992. Every 24 hours a point on the ocean will rotate through the bulge twice, creating two high tides, and through the relatively shallow areas at 90 degrees to the axis of the bulge, creating two low tides. The water level at any point varies in a constant manner from high to low tide and back again twice over every day.
Fig 2.9: Major surface currents (Source © Craton 1992, Brock University)
2.7 Marine current resource The global marine current energy resource that we are interested in, is mostly driven by the tides. Other technologies would be interested in currents driven by thermal and density effects and are not presented here. As discussed above the tides cause water to flow inwards twice each day (flood tide) and seawards twice each day (ebb tide) with a period of approximately 12 hours and 24 minutes (a semi-diurnal tide), or once both inwards and seawards in approximately 24 hours and 48 minutes (a diurnal tide). In most locations the tides are a combination of the semi-diurnal and diurnal effects, with the tide being named after the most dominant type. The strength of the currents varies, depending on the proximity of the moon and sun relative to Earth. The magnitude of the tide-generating force is about 68% moon and 32% sun due to their respective masses and distance from Earth (Open University, 1989). Where the semi-diurnal tide is dominant, the largest marine currents occur at new moon and full moon (spring tides) and the lowest at the first and third quarters of the moon (neap tides). With diurnal tides, the current strength varies with the declination of the moon (position of the moon relative to the equator). The largest currents occur at the extreme declination of the moon and lowest currents at zero declination. Further differences occur due to changes between the distances of the
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moon and sun from Earth, their relative positions with reference to Earth and varying angles of declination. These occur with a periodicity of two weeks, one month, one year or longer, and are entirely predictable (Bernstein et al, 1997, and World Energy Council 2000). 2.8 Marine currents characteristics and resource distribution Generally the marine current resource follows a sinusoidal curve with the largest currents generated during the mid-tide. The ebb tide often has slightly larger currents than the flood tide. At the turn of the tide (slack tide), the marine currents stop and change direction by approximately 1800. The strength of the marine currents generated by the tide varies, depending on the position of a site on the earth, the shape of the coastline and the bathymetry (shape of the sea bed). Along straight coastlines and in the middle of deep oceans, the tidal range and marine currents are typically low. Generally, but not always, the strength of the currents is directly related to the tidal height of the location. However, in land-locked seas such as the Mediterranean, where the tidal range is small, some sizeable marine currents exist. There are some locations where the water flows continuously in one direction only, and the strength is largely independent of the moon’s phase. These currents are dependent on large thermal movements and run generally from the equator to cooler areas. The most obvious example is the Gulf Stream, which moves approximately 80 million cubic metres of water per second (Gorlov, 1997). Another example is the Strait of Gibraltar where in the upper layer a constant flow of water passes into the Mediterranean basin from the Atlantic (and a constant outflow in the lower layer). Areas that typically experience high marine current flows are in narrow straits, between islands and around headlands. Entrances to lochs, bays and large harbours often also have high marine current flows (EECA, 1996). Generally the resource is largest where the water depth is relatively shallow and a good tidal range exists, Bay of Fundy is an excellent example. In particular, large marine current flows exist where there is a significant phase difference between the tides that flow on either side of large islands. There are many sites world-wide with velocities of 5 knots (2.5 m/s) and greater. Countries with an exceptionally high resource include the UK (E&PDC, 1993), Ireland, Italy, the Philippines, Japan and parts of the United States. Few studies have been carried out to determine the total global marine current resource, although it is estimated to exceed 450 GW (Blue Energy, 2000). See World Energy Council reports. A recommendation is here hence made to initiate a study to estimate the energy in ocean currents in Canada. We all know of the currents available in the Bay of Fundy for example. We need an accurate estimate of the energy due to currents and its distribution along the coast. Similar to a wave energy-prospecting project we need a marine current energy detailed survey. See Figure 2.10.
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Figure 2.10 Variation in Tidal range along the coast of North America. Tidal heights are
proportional to the currents induced. Ranges given are the differences in water levels from low tide to high tide, in meters. The bare map is courtesy of Theodora.com.
3 Ocean energy harnessing systems 3.1 Wave energy harnessing technology 3.2 Wave energy developments and activities 3.3 Ocean currents energy technology 3.4 Ocean currents developments and activities 3.1 Wave energy harnessing technology Wave power developments face a number of difficulties. Basically, they include the following: Large loading in extreme, harsh weather conditions, corrosive environment, randomness in power input or low transmission frequencies. The design of a wave energy harnessing system to be efficient and competitive has to deal with these difficulties in an efficient way. This means that the system must be beneficial and economically reasonable. Starting with a conceptual idea a wave energy device goes through a long evolution: usually starting with theoretical analyses and design the project goes through extensive
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experimental R&D work in the wave tanks at small and intermediate scales. This R&D work is required before the first prototype can be deployed in the sea. Freak loads in the sea may exceed the estimated values and are difficult to predict. High degree of knowledge and sophistication are needed so that the design of a wave energy system may operate safely in extreme conditions and be economically viable. This knowledge and sophistication could be found nowadays in the offshore engineering industry. In ocean energy resource utilization, in contrast to other renewables, there are a large number of ideas and concepts for wave energy harnessing or conversion. In Japan, North America and Europe there are over a 1000 wave energy harnessing techniques patents. Harnessing wave energy could involve three levels: The first level called primary conversion of wave energy is gained by an oscillating system. These systems include for example: a floating body, an oscillating solid element or oscillating water within a structure. The system will then be able to store some kinetic and/or potential energy extracted from the wave. A second level called secondary conversion may be required to convert the stored energy into some useful form. In this level devices for control and power take off involve controllable valves, hydraulic rams and various hydraulic and pneumatic components as well as electronic hardware and software. This secondary conversion is usually obtained by means of a turbine through rotation of a shaft. Tertiary conversion could be needed if electric generators are used for the conversion of the harnessed power into electricity. Brooke 2003. Classification: In the literature a number of ways are found to classify wave energy converters (WEC). According to their horizontal size and orientation: If the size of the system is small compared to the typical wavelength then the WEC is called a point absorber. See Budal and Falnes (1975). On the other hand if the extension is large and comparable to the typical wavelength then the WEC is called a line absorber. Terminator and attenuator have also been used to denote these WECs. A WEC is a terminator if it is aligned along the prevailing direction of wave crests and is an attenuator if aligned normal to the prevailing wave crests. According to their different location with respect to the coastline: WECs may be located onshore, nearshore or offshore. Onshore WEC’s are on the coast line, however nearshore is the designation given to WEC’s if located in shallow waters and within 10-15 km distance from the coastline, while offshore systems are the ones which would be developed beyond that. According to their locations with respect to the mean water level: WECs are found partly above and partly below the mean water level. They may be completely submerged and placed on the seabed below the mean water level. Devices may be moored in a floating on the free surface or partly submerged either nearshore or offshore. Some systems could be called hybrid in the sense that nearshore units could be pumping fluid in a closed loop to an elevated reservoir on the shore from which energy would be extracted.
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Figure 3.1 from Hagerman (1995) identifies twelve distinct process variations. The main features that distinguish one concept from another are the mode of oscillation for energy absorption, type of absorber, and type of reaction point. Hence energy can be absorbed from heave motion, surge, pitch and yaw or combinations of these as shown in Figure 3.1. Table 3.1 is a modification of Hagerman (1995) classification and presented in Brooke (2003). The wave energy conversion is described as follows. See Brooke (2003). The wave force acts on a movable absorbing member which reacts against a fixed point on land or sea-bed based structure, or against another movable, but force–resisting structure. Heave forces may be reacted against a submerged horizontal plate. Wave forces may also be reacted against a long spine. The wave force results in oscillatory motion of the absorbing member. The product of wave force and corresponding motion represents absorbed wave energy.
Figure 3.1 Classification of Wave Energy Converters Systems. Hagerman(1995)
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Table 3.1 Classification of wave energy devices processes
Source Brooke (2003) with reference to Hagerman (1995) of Fig. 3.1
Wave Energy Conversion Process Primary Location of System
# Description
Reference to Hagerman Fig 3.1
Onshore 1.1 Fixed oscillating water column 2 1.2 Reservoir filled by wave surge 1 1.3 Pivoting flaps 4 Nearshore 2.1 Freely floating oscillating water column 3 2.2 Moored floating oscillating water column 3 2.3 Bottom mounted oscillating water column 2 2.4 Reservoir filled by direct wave action 1 2.5 Flexible pressure device 11 2.6 Submerged buoyant absorber with sea floor reaction 12 2.7 Heaving float in bottom mounted or moored floating caisson 5 2.8 Floating articulated cylinder with mutual force reaction Offshore 3.1 Freely heaving float with sea floor reaction point 6 3.2 Freely heaving float with mutual force reaction 7 3.3 Contouring float with mutual force reaction 8 3.4 Contouring float with sea floor reaction point 9 3.5 Pitching float with mutual force reaction 10 3.6 Flexible bag with spine reaction point 11 3.7 Submerged pulsating-volume body with sea floor reaction point 3.8 Reservoir filled by direct wave action 1 The main generic types of wave energy harnessing schemes could hence be listed as: See Brooke (2003) and Hagerman (1995).
1. The Oscillating Water Column systems: which in general include a partially submerged, hollow structure open to the sea below the water line.
2. Overtopping systems: that collect the water of incident waves to create a head to drive one or more low head turbines.
3. Point absorbers systems: these are either floating or mounted on the sea bed and provide a heaving motion that is transformed by mechanical and/or hydraulic subsystems into linear or rotational motion to drive electric generators.
4. Surging devices : these devices use the particle velocity in a wave to drive a deflector or to generate pumping effect of a flexible bag facing the wave. 5. Other devices:
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that do not fall under the above 1-4 classes. These include important developments of which we find: the Salter duck, the Cockerell raft and the McCabe Wave Pump. 3.2 Wave energy developments and activities Recently there has been a renewed interest in ocean wave and tidal current energy. Recent WATTS conferences attest to that. See WATTS 2004 conference proceedings for more details and status of recent developments. New ocean energy companies have been involved in the development of new technologies. Examples of these are the Pelamis, the Archimedes Wave Swing and the Limpet. The plan is to increase the worldwide ocean energy capacity to 6 MW in the near future. See also Thorpe (2000). As of 2002 the installed capacity (around the world) was about 1 MW, mainly from demonstration projects. Source Thorpe (2000) and Brooke (2003). See the following tables from the same sources. Shoreline wave energy systems: Energy Conversion Process
Country and location Site Status
Fixed OWC Australia, Port Kembla Breakwater Adv. Stage Develop. " China, Dawanshan 1 Shoreline Operational " China, Shanwei Adv. Stage Develop. " India, Vizhinjam, Harbour Operational " Japan, Sanze Shoreline gully Operational " Japan, Sakata port Breakwater Operational " Japan, Kujukuri-Cho Breakwater Operational " Japan, Haramachi Operational " Mexico Operational, sea water pump " Norway, Toftoy Cliff wall Operational " Portugal, Pico, Azores Rocky gully Operational " UK, Isle of Islay Shoreline gully Operational " UK, Isle of Islay Cliff face Operational
Reservoir filled by wave surge
Norway, Toftoy Gully and interior bay
Operational
Pivoting flaps Japan, Muroran Port Seawall Operational Pivoting flaps Japan, Wakasa Bay Seawall Operational Nearshore wave energy systems
Energy Conversion Process
Device name Country and location Status
Freely floating OWC - China, various Op., navigation buoy " - Japan, various Operational " Kaimei floating Japan, Yura Operational
Fixed floating OWC Mighty Whale Japan, Gokasho Bay Opearational " Sperbuoy UK, Plymouth Adv. Stage Dev.
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" Shim wind-wave syst. South Korea Adv. Stage Dev. Bottom mounted OWC Osprey UK, Thurso - Reservoir filled by wave Floating wave power Sweden Adv Stage Dev. Flexible pres. device SEA Clam UK Adv Stage Dev. Submerged buoyant absorber sea-floor RP
- - -
Heaving float in bottom-mounted or moored caisson
ConWEC Norway Adv. Stage Dev.
Floating articulated cylinder with inertial RP
Pelamis UK, Shetland/Isle of Islay Adv. Stage Dev.
Offshore wave energy systems Energy Conversion Process
Device name Country and location Status
Freely heaving float with sea floor RP
OPT Wave Power System
USA, Australia, Portalnd, Australia
Adv. Stage Dev.
" Danish Heaving buoy Denmark, Hanstholm Adv. Stage Dev. " Phase-controlled
Power Buoy Norway, Trondheim Fjord Adv. Stage Dev.
" DELBUOY US Operational Freely heaving float with inertial RP
Hosepump Sweden Adv. Stage Dev.
" IPS Buoy Sweden Adv. Stage Dev. Contouring float with inertial RP
McCabe Wave Pump Ireland, Shannon River Estuary
Adv. Stage Dev.
" Wave Energy Module US Adv. Stage Dev. Contouring float with sea-floor RP
Kaiyo Jack-up Rig Japan, Iriomote Island Okinawa
Adv. Stage Dev.
" Contouring Raft UK Adv. Stage dev. " Contouring Raft US Adv. Stagre Dev.
Pitching float with inertial reaction point
- - -
Flexible bag with spine reaction point
- - -
Submerged pulsating-volume body with sea-floor RP
Archimedes Wave Swing
Netherlands, Viano do Castello, Portugal
Adv. Stage Dev.
3.3 Ocean currents energy technology There are basically two ways to convert tidal currents energy into useful power. They rely on the types of turbines to be used.
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1) Turbines harnessing the potential energy in a low hydrostatic head The first type needs a hydrostatic head for it to transformation the stored potential energy into power. The technology is very similar to the technology used in traditional hydroelectric power plants. For this purpose arises the requirement of a dam or barrage across a tidal bay or estuary. Example of this technology is found in the Annapolis Royal Tidal Power plant and the one at La Rance in France.
Fig. 3.2 The phases of operation at La Rance, France. The darkened areas in the graph indicate the amount of head or quantity of energy available. The upper diagram shows the cycle without
pumping. The lower one shows more energy available in sipite the expenditure on pumping. © After Clancy (1968) The tides : Pulse of the Earth
Building dams is an expensive process and its footprint and environmental impact could be large. In any case, at certain points along the dam gates and turbines are installed. When there is an adequate difference in the elevation of the water on the different sides of the barrage the gates are opened. The hydrostatic head causes the water to flow through the turbines to produce electricity. Technically, power can be generated by water flowing both into and out of the bay. Basically the barrage holds the water in the estuary as the tide falls. Then, gates are opened and the water rushes
29
seaward through the turbine(s). Later, the rising tide will be held back by the barrage then released to flow through another turbine (or the same if it is designed to do so) into the river estuary. See Figure 3.2. The major factors in determining the cost effectiveness of a tidal power site are the size (length, height) of the required dam, and the difference in height between high and low tides. These factors are expressed in what is called a site Gibrat ratio. The Gibrat ratio is the ration of the length of the dam in meters to the annual energy production in Kilowatt hours. (1 Kilowatt hour = 1 KWH = 1000 watts used for 1 hour). The smaller the Gibrat site ratio the more desirable is the site. Very little is understood about how altering the tides can affect complex aquatic and shoreline ecosystems. Unfortunately one of the only methods of increasing our knowledge about how tidal barrages affect the environment may be the study of the effects before and after such plants have been built. This strategy perhaps might then be too late!! 2) Turbines harnessing the kinetic energy of the flow Useful energy can be generated from marine currents using completely submerged turbines comprising of rotor blades and a generator. They are sometimes called water turbines. Water turbines work on the same principle as wind turbines by using the kinetic energy of moving fluid and transferring it into useful rotational and electrical energy. The velocities of the currents are lower than those of the wind, however owing to the higher density of water (835 times that of air) water turbines are smaller than their wind counterparts for the same installed capacity. The power that can be extracted from the currents is dependent on the velocity of the water flow and the area and efficiency of the water turbine, and can be calculated as follows:
Where ρ is the density of sea water 1025 kg/m3 A is the area of the rotor blades in m2 v is the marine current velocity in m/s and Cp is the power coefficient, a measure of the efficiency of the turbine. Two types of turbines have been proposed. i) Horizontal axis turbines (axial flow turbine). ii) Vertical axis turbines (cross flow turbine). See Figure 3.3 for the conceptual designs and possible configurations.
P3 C 21 vAP ρ=
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Figure 3.3 Conceptual diagrams of types of current turbines and their potential configurations. Source © World Energy Council In order for marine current energy to be utilised, a number of potential problems will need to be addressed, including: Avoidance of cavitation by reducing tip speeds to approximately 8 m/s. This suggests a turbine with a higher solidity than a wind turbine; Prevention of marine growth building up on the blades or ingress of debris; Proven reliability, as operation and maintenance costs are potentially high; Corrosion resistance, bearing systems and sealing; Turbines may be suspended from a floating structure or fixed to the seabed. In large areas with high currents, it will be possible to install water turbines in groups or clusters to make up a marine current farm, with a predicted density of up to 37 turbines per square km. This is to avoid wake-interaction effects between the turbines and to allow for access by maintenance vessels (DTI, 1999).
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As there are currently no commercial turbines in operation, it is difficult to assess the cost of energy and competitiveness with other energy sources. Initial studies suggest that for economic exploitation, velocities of at least 2 m/s (4 knots) will be required, although it is possible to generate energy from velocities as low as 1 m/s. As the technology matures and with economies of scale, it is likely that the costs will reduce substantially, Rudkin, 2001. Future of Marine Current Energy Compared with other renewable technologies, there has been little research into utilising marine current energy for power generation. However, in principle marine current energy is technically straightforward and may be exploited using systems based on proven engineering components (FMP, 1999). In particular, knowledge gained from the oil and gas industry, the existing hydro industry and the emerging wind energy industry can be used to overcome many of the hurdles facing marine current energy. The global marine current energy resource is very large, and it has a number of advantages over other renewables. The table below shows a comparison of the ocean waves and marine current energy resource with other renewables and conventional energy sources. It is clear that there are many benefits to utilising current energy, including : (Vortec Energy)
The resource has four times the energy density of a good wind site, so the diameter of water turbines can be less than half that of a wind turbine for the same energy output.
The water velocities and therefore power outputs are completely predictable, once accurate site measurements have been taken.
Water turbines will not need to be designed for extreme atmospheric fluctuations as required with wind turbines, meaning that the design can be better cost-optimised.
With increased conflicts over land use, water turbines offer a solution that will not occupy land and has minimal or zero visual impact.
The greatest resource is in close proximity to coastlines and many areas with high population densities.
The technology is potentially modular and avoids the need for large civil engineering works.
The environmental impact resulting from marine current energy use is likely to be minimal. Project planning will need to be cognisant of species protection including fish and marine mammals, although since the blade velocities and pressure gradients are low this is unlikely to cause any serious problems (Fraenkel, 1999). In the process of locating turbines, consideration of shipping routes and present recreational uses such as fishing and diving will be required. Fishery exclusion zones might be necessary to be established.
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Table 3.2: Comparison of wave and marine current energy with other energy resources Source: Emily Rudkin, Vortec Energy (2001).
The table (Rudkin et al, 2001) shows that marine current energy is one of the most promising new renewable energy sources, and is deserving of further investment. Furthermore, the know-how is now available to combine existing technologies to utilise marine current energy for power generation. It is likely that water turbines will initially be deployed in island or coastal communities with strong marine currents and which are isolated from national grid systems, where they are most likely to offer a cost-effective alternative. However, marine currents have the potential to supply significant quantities of energy into the grid systems of many countries. As interest grows, marine current energy is likely to play an increasing role in complementing other energy technologies and contributing to the future global energy supply mix. See World Energy Council report. 3.4 Ocean currents developments and activities The applications and activities in this sector are also divided according to the technology. We have two types: the applications that use the tidal energy in generating a hydrostatic head hence needing a dam and the applications that use water turbines to harness the marine current kinetic energy.
1) Tidal energy plants i) La Rance in Bretagne, France: This is a major tidal generating station in operation. It is a 240 MW (1megawatt= 1MW=1 million watts) at the mouth of La Rance River estuary on the northern coast of France. Note that a large coal or nuclear power plant generates about 1000 MW of electricity. It has been in operation since 1966 and has been a very reliable source of electricity. The energy resource is exactly predictable relying of course on the tides. This plant was supposed to be one of many tidal power plants in France, until the nuclear program in that country was greatly expanded in the late 1960’s. ii) Annapolis Royal Tidal Power plant, on the Annapolis river, Bay of Fundy, NS. This was designed as an experimental facility and is of the order of 20 MW. It is in
33
operation since 1984. It would have been interesting to know the economics of the plant if it had more than one generating unit. iii) Murmansk tidal power plant: of the order of 0.4 MW. Not much is known about this one. iv) Studies have been undertaken to examine the potential of several other tidal power sites worldwide. It has been estimated that a barrage across the Severn River in the west of England could supply as much as 10% of that country’s electricity needs. (12 GW). The proposed facility on the Severn would have a construction cost of $15 Billion . Similarly, several sites in Cook Inlet in Alaska and the White Sea in Russia have been found to have potential to generate large amounts of electricity.
2) Water turbines plants Marine current energy is at an early stage of development, with only a small number of prototypes and demonstration units having been tested to date. There are no commercial grid-connected turbines currently operating. A number of configurations have been tested on a small scale that are essentially wind turbines adapted for the marine environment. Generally speaking, turbines are either of horizontal axis or vertical axis type. Variants of these two types have been investigated, including turbines using concentrators or shrouds, and tidal fences.© World Energy Council. i) Horizontal axis turbines (axial flow turbine). This is similar in concept to the widespread horizontal axis wind turbine. Prototype turbines of up to 10 kW have been built and tested using this concept. There are currently plans to install a demonstration machine of 300 kW off the south coast of the United Kingdom (MCT, 2000). Concentrators (or shrouds) may be used around the blades to increase the flow and power output from the turbine. This concept has been tested on a small scale in a number of countries, including New Zealand (Rudkin, 2001). More information on horizontal axis turbines and their use could be found in: http://www.europeanenergyfair.com/download/marine_current_turbines.pdf See also Marine Current Turbines Ltd. information and their web site at: http://www.marineturbines.com/ . The company has been formed to develop technology for exploiting flowing water in general and tidal streams in particular. The goal is both to arrive at cost-effective and reliable power systems and to develop these commercially on a large scale. This is being achieved through what they call a phased R&D programme and in partnership with an industrial consortium together with various strategic partners who are shareholders in this company. According to their plan Marine Current Turbines Ltd has no intention of manufacturing the technology; this function is achieved by its partners and "third parties", but the company "does research the resource with a view to developing future projects". The company will develop, own and deliver the technology and the resource to go with it. The company's shareholders consist of the management, private investors and several corporate investors - namely Seacore Ltd., London Power Company plc, Carrs Milling plc and IT Power Ltd. Further details could be found from the company information site. From early December 2003, a tidal turbine or underwater windmill started running a power-generating propeller mounted on the seabed of the Kvalsund channel. The
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turbine is equipped with blades of 15-16m and has been connected to the nearby town of Hammerfest’s power grid in Northern Norway via a shore connecting cable. See Figure 3.4 and http://www.e-tidevannsenergi.com/index.htm. It will be imperative to follow up on the study of the economics of the project.
Fig. 3.4 Diagrammatic sketch of the underwater tidal current mill operating in Kvalsund channel (~70 0 N) in Northern Norway since the fall of 2003.
© World Energy Council ii) Vertical axis turbines (cross flow turbine). Both drag and lift turbines have been investigated, although the lift devices offer more potential. The best-known example is the Darrieus turbine with three or four thin blades of aerofoil cross-section. Some stand-alone prototypes have been tested, including a 5 kW Darrieus turbine in the Kurushima Straits, Japan. The concept of installing a number of vertical axis turbines in a tidal fence is being pursued in Canada, with plans to install a 30 MW demonstration system in the Philippines (Blue Energy, 2000). For more information see: www.bluenergy.com
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4. Transfer of power systems 4.1 Interfaces 4.2 Storage of energy 4.3 Electric power generation and conversion 4.4 Power transmission to the grid Waves or currents action has to be converted from one form to another form that is amenable for transmission and/or storage. Usually the harnessed energy is converted into electricity that is then either fed into a grid system or transformed into some other form for storage purposes. Hence to achieve the first transformation, power transfer systems have to turn the slowly varying oscillating forces of incoming waves into the fast, unidirectional forces required to drive generators that will produce electricity. With very few exceptions most of the systems consists of a two-stage operation: a mechanical rotary device coupled to an electrical generator. A wide range of options is available to convert and transfer energy. After the initial energy-harnessing device most systems will consist of a mechanical interface an electrical generator and a way for transmission of power and deliver it to the grid. 4.1 Interfaces The following interfaces have been proposed, adapted or used. Direct mechanical interface Only few designs for a pure mechanical power transfer have been proposed. They have not been tested in practice. Haggerman (1995) and the European Commission Review (1993) do not report these as options. Devising mechanical components to convert the oscillating variable forces into RPM unidirectional output is difficult. Thorpe (1992) indicates that to deal with such large force will require large size components making the option uneconomic. Recently the following has been reported: The Wave Rotor: see Retzler (1996). This concept uses the Magnus effect on two contra-rotating cylinders to absorb wave power. It has been tested on a small-scale model. The extrapolation to a full scale capability is unknown. The OLASS system: see Rebello et al (1995). This is an Oscillating Water Column device in which the energy is extracted through a float system on the oscillating water column. This is then mechanically coupled to a generator. This is done by means of what is called a mechanical rectifier and speed multiplier. A small prototype developed about 15 kW. Few details are available. Air turbines interface A simple way of transforming the low velocities and high forces of air compressed by sea waves into high speeds and low forces required by conventional generators is
36
provided by air turbines. They seem to fulfil the requirements of a power transfer system. The combination of air column and turbine seem to provide a cost efficient approach to gearing. The most popular air turbine is found to be the Wells turbine. To have the property to rotate in the same direction even when the direction of the air flow changes the turbine uses symmetrical air foils with their chords being in the plane of rotation. No pitch angles are proposed. Blades are hence cheap to manufacture in comparison and the losses are small during idling. Some of the configurations that are found in the literature include the following: Monoplane turbines: In steady air flow these turbines were found to have a maximum efficiency of about 60% in small scale testing in steady air flow. Monoplane turbines with guide vanes: their efficiency can reach 70% in small scale testing and steady airflows. They have poorer behaviour against stall. Contra-rotating turbines: this configuration is made up of two Wells turbines placed close together with their blades rotating in opposite directions. The efficiency about 70% was found for small scale testing in steady airflows. A wider operating range is found in this case in comparison to the ones with vanes. See Gato et al (1996, 1997). Efficiency measurements on Wells turbines (and any other turbines in that matter see below) under realistic oscillating flows is a priority area for further research work. Other air turbines configurations that have been considered: Variable pitch turbines: See European Commission pilot OWC in the Azores. Russel et al (1996). Impulse turbines: with adjustable guide vanes Use of valves: to allow some of the airflow to bypass the turbines hence avoiding stall. Valves can be used to delay the movement of the water column hence allowing its velocity to come into phase with the wave force. Better suited for hydraulic conversion. See Korde (2002). Water Turbines: In this sector of the industry these are considered to be a well-developed technology. The surrounding fluid provides a good supply of working fluid. The various designs proposed claim to offer control over the volume of water flow allowing the devise to manage variations in wave power levels enabling relatively conventional electrical generators to be used. According to the European Commission report (1994) several types of turbine could be found specifically designed for different working heads or pressures. A summary of some of these water turbines is as follows: Pelton wheels: This configuration is suitable for high pressure operations. These include for example: The original Bristol Cylinder with high pressure Pelton wheel
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The Hose-pump Swedish design with a high pressure Pelton wheel. Francis turbines: This configuration is suitable for medium pressure operations Kaplan turbines: This configuration is suitable for low pressure operations. These include for example: The Danish water pump with low pressure propeller, The Tapchan with low pressure Kaplan turbine. Two of these schemes The Tapchan and the Hose-pump have been successfully demonstrated in pilot plants. In a sense showing that water turbines can be used in wave power devices with little development providing well known problems in the use of these turbines are taken care of. Cavitation is the first that would come to the front. Hydraulic systems For several wave energy devices high pressure oil systems have been proposed and used. As part of power take-off technology these systems have the following advantages: They are capable of handling high power levels in a small volume. They can be made to adapt to various types of input motions (circular or linear) and so could be utilised on a range of types of devices. They can be computer controlled hence allowing optimization of the device on a wave by wave basis. They can accommodate a wide range of input power levels. The reliability of hydraulic systems has been proven in their wide use in many applications. In wave energy harnessing systems they will be subject to much different environments, namely: Hydraulic systems will be in contact with seawater. The design will have to prevent and make sure that seawater does not come in contact with the oil in the system. Corrosion problems and ways to mitigate with underwater systems specific problems in general have been very well researched and studied within the Offshore Engineering community. Use of new materials like ceramics and ceramic coatings have proven successful in such applications. Hydraulic rams might be affected with high loads due to large displacements in extreme conditions. High dynamic loadings in general will have to be taken into account. Some wave energy applications are predicting their operations under high torque and variable displacement. In general, current rotary hydraulic machinery applications operate under relatively low torque and fixed displacement. Nebel (1992) describes the problems with control systems to optimize the capture and conversion efficiencies of wave energy schemes. Current methods of controlling hydraulic mechanisms could involve much power losses. The European Energy Programme (Russel and Diamantaras 1996, Eshan et al 1996) predicted the need to design tailor-made hydraulic systems and is an intrinsic part of the programme. However current hydraulic systems can be used in wave energy devices. Research in their efficiency and reliability will be needed.
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4.2 Storage of energy Power is proportional to velocity squared and the surface velocity of waves varies with wave height and period of waves. Wave power subsequently is not steady and constant but varies significantly. This variation has an impact on the capital cost of a wave power installation and its power losses. In general it is found that to facilitate the integration of the system to the grid short-term energy storage is needed. For ocean waves systems the three main short-term energy storage are: the flywheel, pressure accumulators and water reservoirs. Flywheels provide energy storage in the form of rotational kinetic energy. They incur some losses. Found to be more efficient in large systems. Gas accumulators have been suggested and would use an inert gas contained in steel accumulators to store energy in oil hydraulic systems. Gas is stored at high pressure and low volume and the energy is released as the gas expands to a greater volume and lower pressure. Water reservoirs as in particular the Norwegian Tapchan systems could be used as a short –term energy storage. Further details could be found in Edinburgh-Scopa-Laing 1979, Hotta 1996, Brook 2003. There may also be the need to longer-term energy storage or energy transformation for future use. Use of batteries and Hydrogen technology might then be included in the equation. This topic is beyond the scope of our discussion. 4.3 Electric power generation and conversion Power generation Electrical generators are used to convert mechanical power at the shaft to electrical power. For high conversion efficiency from wave energy to electrical energy without storage large rotation differences are used in the turbine-generator system. Special generators can be found to satisfy these requirements. Examples of these include multi-pole generators, linear generators and generators with reluctance changes. Linear generators are suitable for direct connection to the mechanical reciprocating motion of wave energy systems. Research and Development in the field of linear generators and suitable electrical generation systems for the feasible applications here discussed are needed. Power conversion To get grid-quality power from ocean energy systems it might be required to convert from alternating current to direct current and vice-versa. A number of electric power conversion methodology and designs have been put forward in projects, proposals and pilot schemes and cannot be extensively reviewed in the present report. The techniques are based on previous developments in the electrical and electronics fields. This is an issue that must be addressed and reviewed by any future proposal and scheme. Agreements and discussions with hydro-power authorities and grid operators are also a must during the feasibility and development stages.
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See for example SEASUN (1988) and Beattie et al (2000) for detailed outlines and discussions of alternatives. Briefly we can classify the types of power conversion as follows: See Brooke (2003).
i) Constant or nearly constant RPM power conversion and they are of two types: Synchronous generators and Asynchronous generators. See Brooke (2003).
ii) Electronic power conversion High power electronics has been used to convert electrical power between AC and DC. AC systems of different frequencies can in general be connected. Configurations could include a direct converter (AC/AC) or pass via a DC interconnection (AC/DC/AC).
iii) Variable or free generator rpm power conversion Large variations in RPM are possible through power electronics and using a DC interconnection between the generator and the power grid. For low speeds or RPM a generator is heavy and consequently expensive. The generator produces voltage that will be frequency and amplitude modulated by the random wave field and cannot be directly connected to the grid. The voltage variations are reduced in the DC interconnection ig the DC voltage from several generators running at random RPM are added in series. 4.4 Power transmission to the grid Through power collecting systems the converters are connected to the electrical grid on land. Transmissions losses will have to be accounted for and are proportional to the distance between the converters and the grid. The transmission to the grid is through sea-floor cables. Hydraulic transfer may be possible depending on the distance not being large. Electrical transmission Electrical transmission to the grid is a large topic and has been the object of study in electrical engineering. Reviews could also be found for the application here discussed in SEASUN (1988). See also Brooke (2003) for more details. Also the reviews in Scott (2001) and Thorpe (1992) of the grid connection of large-scale ocean power projects and the transmission systems pertaining to UK specific projects are worth noting. Extensive experience exists in electrical cables laying in the sea using specially designed vessels for this purpose. Power can be transmitted as AC or DC. The losses in the cable transmission must be estimated and accounted for. Maintenance of the cables are also issues to be studied. The need of electrical transformers to transfer power between grids of different voltage levels might also arise. The design of the electrical transmission system is a techno-economical problem of optimization, Brooke (2003). Ocean power generation and transmission would gain from the experience of wind power systems. In this case major short cuts in research and development could be saved and gained from cross-fertilization between the two applications. Worth checking for example the international company ABB and others for their experience and development of systems of power transmission for wind farms. See ABB (2002).
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Hydraulic transmission In some ocean energy schemes the secondary energy transformation is realized by either water or hydraulic oil, Brooke (2003), CONA (2003). Hydraulic oil is normally used internally in a wave power device while water can be pumped long distances in open or closed systems. The Swedish hose-pump system is an example using sea water in an open system The losses are proportional to velocity squared. Hence preference to low flow velocities combined with high pressure. The point of the discussion is that those problems and issues of generation and conversion of the electrical power have to be discussed in the design and development stages of any project.
5. Ocean (wave/current) Power Economics 5.1 Factors affecting the evaluation of costs 5.2 Capital and Operating costs of Ocean Energy systems 5.3 Generation costs of wave and current energy 5.4 Comparison with costs of other renewables and electric
power prices The commercial deployment of systems for harnessing wave energy is only just recently tentatively beginning to happen. This is in spite of extensive work in research and development since the late seventies. In general it is considered that experience to enable the accurate assessment of the costing of such systems is lacking, Thorpe (1992), Brooke (2003). Hence an actual assessment of the degree of their competitiveness is difficult. No large-scale offshore ocean energy systems have been deployed. Only prototypes have been installed giving also a much needed experience in making the economic evaluation. This chapter reviews some of the factors recognized by industry to evaluate costs. 5.1 Factors affecting the evaluation of costs of ocean energy systems. Usually developers of ocean energy systems define the cost of energy in cents per kilowatt-hour. The method for calculating this cost is, most of the time, not identified. Important financial assumptions are not reported. For example: rates of return, debt to equity ratio, discount rates if any, are not mentioned. Potential venture capital investors find it difficult to evaluate a project and viability in particular if no evaluation of competing technologies or other appropriate benchmarks are available, Brooke (2003). A detailed analysis is beyond the scope of the present review. However the following points are provided. The US Electric Power Research Institute in its Technical assessment Guide (EPRI TAG), see Electric Power Institute 1987, 1993 and Hagerman 1995, provides what is believed, an accurate methodology for evaluating the costs of energy. They introduce what is called a levelized cost of energy index that will allow also the comparison between alternative designs or technologies.
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5.2 Capital and Operating costs of Ocean Energy systems Capital costs usually include costs of construction, assembly and installation of the plant. Unfortunately no large-scale devices have been built. Currently available are prototypes and include all the additional costs involved in such a stage. Sources of cost data are hence difficult to find. Other similar or close to areas of activities in offshore engineering could provide data that would help to calculate the capital costs of ocean energy systems. Three approaches are found in that respect and are summarized below, see for example Brooke (2003) and Thorpe (1992). Costing by analogy: approximate costs estimates could be made from similar projects. Adjustments must obviously be made to take care of relative sizes or other characteristics. Finding remotely related systems is more likely because of the scarcity of ocean energy systems. Conventional costing estimate: this is sometimes defined as a bottom-up costing method. In this case detailed information about the project are needed, from drawings and construction plans. Hence a complete work breakdown structure could be developed with units rates for every component. In general some aspects of the system using standard current technology in civil and offshore engineering would be amenable for such costing. Compatibility between costs for different systems and by different evaluators would be difficult. Parametric costing estimate: this method is intermediate between the two described above. It relies on that some functional relationship between characteristics of an item of the system and its cost. This direct dependency is generally derived from past experience and/or engineering practice. These relationships are similar to those used also in methodology ii) above. This means that it requires also outline drawings and specifications together with rates for materials, labor and transport. Information is available in general except perhaps for some aspects of the wave energy device components. Thorpe 1992 suggests that this is the method suitable for and employed to make estimates of costing of ocean energy related projects.
Table 5.1 Cost and Performance characteristics of generic wave energy systems (Source: Thorpe 2000)
Cost and Performance Shoreline Near-shore Offshore Unit Costs ($/kW) 1800-2100 1500-1800 1500-3000 O&M & Insurance Costs ($/kW/year) 30-45 42-48 30-90 Availability 94-96 93-96 90-95 Annual Output (kWh/kW) 2000-2500 2200-2500 3000-4000
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5.3 Generation costs of wave energy For any application the determining factors are the cost per kW of the delivered system or the cost per kW-hour of the delivered energy whether in the form of electrical energy production or in kind, McCormick and Kraemer 2002. Thorpe (1998, 2000) has compared the cost efficiencies of various wave energy conversion technologies. As mentioned above he has divided the systems into shoreline, nearshore and offshore systems. He found that the more recent designs both shoreline and offshore devices can produce electricity at less than 0.1 US$ per kW. Figure 5.1 shows an unraveled design spiral as suggested by Thorpe. The figure demonstrates the cost cycling of the design process in time.
1 Initial concept 2 Technical and Economic Feasibility Study 3 Preliminary Design and Cost Analysis 4 Design Alterations 5 Cost Analysis 6 Design Adoptions
Figure 5.1 Unraveled Design-Cost Spiral for an Ocean Energy System Source: Thorpe 2000 Also it shows the improvement in the cost estimates as time and more analysis is performed. Table 5.1 shows the predicted costs and performance for generic types of wave energy devices, Thorpe 2000. Thorpe 1998, also shows independently predicted electricity generating costs of nearshore (including onshore) and offshore systems against the year in which the device was designed. These costs are shown to be site specific and show reduction to approximately 7 cents/kWh in 2000 for both categories at 8% discount rate over the lifetime of the scheme.
1
2
3
4
5
6 1
2
3
4
5
6
2
3
time
4
5
6
Cost estimates
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Figure 5.2 Examples of Evolution of electricity cost for onshore and nearshore systems. Source Thorpe 2000.
Figure 5.3 Examples of Evolution of electricity cost for offshore systems.
Source Thorpe 2000. Apart from the schemes analyzed in Figure 5.2 and 5.3 there are also other systems that claim to be able to produce electricity at similar costs levels. This indicates that with a suitable climate generating costs of 3.5-8 c/kWh should be achievable. These
Evolution of electricity cost for onshore and nearshore systems. Source: Thorpe (2000)
0
5
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20
1980 1985 1990 1995 2000 2005
Design year
Co
st o
f el
ectr
icit
y (p
/kW
h)
NEL OWC Wavegen’s OSPREY Wavegen’s LIMPET
Evolution of electricity costs for offshore systems. Source: Thorpe (2000)
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1982
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ost
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Edinburgh Duck
Bristol Cylinder
SEA Clam
PS Frog
Mcabe Wave Pump
Sloped IPS Buoy
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systems have not been evaluated independently or else they are still in the early stages of R&D. Therefore their costs and performance are subject to considerable uncertainty. Brooke (2003), Sjostrom et al 1996, Thorpe (1997). In Figure 5.4 below we present a plot of the resulting costs against the year in which the design of a device was completed. The graph is provided by the European Thematic Network on Wave Energy. See their report Wave Energy Utilization (2002). It complements the information in Figures 5.2 and 5.3 above in that it shows for comparison the average electricity price of wind generated electricity in the EU. At best the improvement in wave energy cost is similar to improvement of generating costs of wind turbines.
Figure 5.4 Predicted Electricity cost for Wave Energy Technologies (Source: European Thematic
Network on Wave Energy and Wave Energy Utilization in Europe report 2002) In general, the economics of wave energy have shown a gradual improvement with time. See Thorpe (1998), Brooke (2003) and the figures above. Developing more understanding due to continual R&D results in reduced capital costs. This trend is also found in other emerging technologies. Falnes (1996) shows that a funding for R&D is required to bring wave energy converters to a commercial level. When this level has been achieved then the potential for selling ocean energy systems in a huge market largely increases. Recently there are systems that propose the utilization of wave energy in conjunction with wind energy. These systems will then be capable of harnessing two sources of energy giving more flexibility in the power production. Multi device approach to ocean energy exploitation might be the way of the future. It must be noted that these are still very much site dependent. Generating costs estimates will have in this case to be higher than individual devices. Figure 5.5 from Hagerman (1995), shows a sketch of a capital cost learning curve and how R&D and experience gained in successive developments affect in time capital costs reducing it to a mature plant cost level.
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Figure 5.5 Hagerman (1995) sketch of a capital cost learning curve.
In general the capital costs of the first individual scheme will be much higher because: See Hagerman (1995), Brooke (2003).
• of technical innovation and immaturity. However, following a learning curve and gaining the experience the benefits follow.
• of the perception of initial risk. Again this perception will initially inflate the costs.
• of the large mobilization and demobilization costs. These are accounted for on the costs of a single device. These costs would be defrayed over a number of following up schemes.
• initially the economies of scale are lacking. Hagerman (1995) The generating costs could, for the above reasons, be as much as twice or more than the costs mentioned in Figures 5.2, 5.3 and 5.4. 5.4 Comparison with costs of other renewables and electricity prices The costs of electricity from renewable resources are function of many factors including:
• The type of energy source and its availability. • The type of device that harnesses the energy • The efficiency of the plant • The site location.
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Unless a particular plant or project and device are specified it is very difficult to have definitive costs for the electricity generation. Table 5.2 below presented at the 1993 World Energy Congress (Brooke, 2003) is a representing of some estimates of costs per kWh for a number of alternative renewable energy resources. The installations that combine both wind and ocean energy whether from waves or currents should be economically competitive with systems harnessing only onshore wind.
Table 5.2 Typical costs of electricity from some renewable sources Source: World Energy Congress 1993, Thorpe 1995, Brooke 2003.
System Location Date Cost (c/kWh)
Solar thermal;parabolic trough New Mexico, USA 2020 7.5-11 Solar Thermal; parabolic dish New Mexico, USA 2020 6.0-10 Solar thermal; Central receiver New Mexico, USA 2020 5.0-9.0 Photovoltaic New Mexico, USA 2020 5.0-14 Photovoltaic, thin film New Mexico, USA 2020 6.0-10 Photovoltaic, multiple thin film New Mexico, USA 2020 4.0-7.0 Wind turbine (6-9 m/s wind spd.) - 1995 3.6-6.5 Wind turbine (6-9 m/s wind spd.) - 2000 3.0-5.5 Wind turbine (6-9 m/s wind spd.) - 2010-2020 2.0-4.5 Combined wind-wave system South Korea 1995 11.0-18.0 Combined wind-wave system UK 1995-1999 6.0-9.0 Comparison with electricity prices Another way of evaluating the economics and competitiveness of ocean energy systems is to compare the costs of electrical generation from ocean systems against the prices that customers pay for electricity. It might become increasingly relevant especially that local communities might be interested to invest in small local systems. The location of such communities and the lack there off of grid hook up while proximity of ocean energy resource might be a decisive factor. The ocean energy could then compete with diesel generation or might in the presence of the latter complement it. Table 5.3 shows some examples of electricity prices in the USA, Japan and the UK. They are representative of prices in the industrialized countries. Note the prices of electricity to households. These prices would be higher in remote communities. Energy from the ocean might be competitive with electricity purchased by households.
Table 5.3 Electricity prices in some industrialized countries. (Source: Brooke 2003)
Country & Date Source reference Categories Cost (cent/kWh) USA – 1994 IEA 1994 Price to industry 4.7 USA 1994 IEA 1994 Price to households 8.4 Japan 1993 IEA 1994 Price to households 14-24 UK 1995 DTI 1996 Price to industry 6.4 UK 1995 DTI 1996 Price to households 8-13.4
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6. R&D programmes R&D in ocean energy harnessing is ongoing in several countries around the world. See for example the International Energy Agency Annual Report (2003) on Implementing an Agreement on Ocean Energy Systems for further details. An overview is here presented of some Canadian activities that came to the attention of the present reviewer. This chapter should be upgraded for up to date information and the inclusion what other countries are specifically doing. Section 3.2 above lists related activities found around the world. 6.1 Canada Ocean Wave Energy As reported by the International Energy Agency (2003) Canada’s coastlines have favourable wave energy resources. In 1995 Powertech Labs Inc. carried out a wave energy resource assessment for the coast of BC based on the wave records obtained by Marine Environment data Services. See Bhuyan et al 1995. The result of the assessment recognizes that the most promising resources would be found in the Queen Charlotte Sound and on the West Coast of Moresby and Graham Island. The "Green Electricity Resource Map" recently issued by BC Hydro shows an average wave power level of 33 kW/m along the west coast of Vancouver Island. (International Energy Agency 2003) The total incident wave power for the west coast of Vancouver Island is estimated to be 8.25 GW. A wave energy resource map for BC can be viewed at the Canadian Cartographics Ltd. web site: http://www.canmap.com/index.htm. BC Hydro has initiated a pre-feasibility assessment of the potential for the development of wave energy resources in 2000. This has been done with the encouragement obtained by the rising electricity demand and inline with a voluntary commitment to meeting 10% of increased demand through a variety of new green energy sources. Two specific sites in Ucluelet and Winter Harbour have been identified each with over 200MW of potential wave power capacity. In 2001BC Hydro selected Ucluelet site as the initial as the initial site for the wave demonstration projects in Vancouver Island. The International Energy Agency reports that presently wave parameters near the Ucluelet site are being monitored using a moored tri-axis buoy. Also in the report is that BC Hydro has signed memorandum of understanding with Energetech of Australia and Ocean Power Delivery (UK) to build two demonstration plants in 2004, both through joint ventures. Each of the plants will have installed capacity of 2MW. It is intended that both of these demonstration plants be connected to the existing grid of BC Hydro. The report goes on to say that on the east coast in the Maritimes, NS, the Wavemill Energy Corp is marketing a wave energy converter called "wavemill" with a patented suction chamber capable of being factory produced as an off the shelf unit. See http://www.wavemill.com/products.htm . A series of performance trials of a wave mill device were conducted in 1998 at the Hydraulic Laboratory of NRC.
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Presently at the NRC Institute for Ocean Technology computational tools are being developed for modelling the generation and propagation of non-linear, regular and irregular steep waves in infinite and finite depths. The computations also allow the accurate calculation of wave energy flux in various sea states and the modeling of the interactions between the component waves. See Appendix 3 for examples of recent publications on that subject. Tidal current energy As part of BC Hydro’s initiatives on green energy technologies, the feasibility of exploiting tidal current as one of the energy resources has been examined by Triton Consultants in 2002. See Triton Consultants 2002. The report is available on BC hydro web site. See http://www.bchydro.com/. A tidal resource map again could be found at Canadian Cartograhics Ltd. The resource assessment identified 55 sites with current speeds over 2m/s, which would yield a gross annual energy potential in the order of 20,000 GWh. Twelve specific sites identified considering the deployment feasibility with a total energy production of 2,700 GWh per year. Blue Energy Canada a BC company is looking for financing for a tidal current demonstration project using heir technology particularly for a tidal fence concept. The International Energy Agency report that they have through their previous association with another company undertaken some laboratory trials on Darrieus-type underwater vertical–axis turbines, often called Davis turbines. http://www.bluenergy.com/oceanenergy.html Another BC company called Clean Current has undertaken numerical and experimental hydrodynamic design on the ducted horizontal axis turbines and has developed two innovative ideas related to electricity generation and turbine design. A model of the turbine has been tested. NRC's Institute for Ocean Technology (IOT) plans to develop a capability for testing, performance evaluation and for R&D work on Ocean Energy Systems. The Design and Fabrication, testing facilities and numerical modeling capabilities makes it a one-stop shop for testing concepts and designs of such systems. http://www.nrc-cnrc.gc.ca/
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7. Timeline Proposal 7.1 Ocean energy influential events 7.2 A Timeline 7.3 Scientific, technical, societal and economic challenges 7.4 Constraints and opportunities 7.1 Ocean energy influential events: • Kyoto 1997: to target to reduce GHG emissions. Agreement that set out those targets and the options available to countries to achieve them is known as the Kyoto Protocol: -6 percent goal below 1990 levels by the period between 2008 and 2012. • Investment : Canada is investing $500M in Action Plan 2000. Plus $625M over five years announced in Budget 2000 results a total commitment of $1.1Billion to address climate change over the next five years and builds on the $850M that Canada spent during the previous five years. Focus on climate change issues increases • Price of oil: Large increase in the price of oil and 9/11 syndrome • Environmental concerns: Societal changes and environmental concerns for the future are increasing. Awareness of related issues is increasing.
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7.2 A Timeline
A T imeline
S horeline model
S horeline prototypedemons trator
2004
2005
2006
National Coordinator,E valuator & T es t Centre (NR C IOT )
T echnology R es earch Programme& T echnology Development
Nears hore model
Nears hore prototypedemons trator
Phase I
2006
2009
HydrodynamicsHydrographic map: waves ,currents and iceHydrogen technologyDeployment, recoveryP ower technology: take off
convers ioncontroltrans mis s iontrans formation
E conomicsF eas ibility and market analys isR eliability, Maintenance techniquesNavigation
Phase II
A T imeline
Nearshore and Offshore s ites prospectingWeather PredictionPower take off prediction Power storageMarket analysisSurvivabilityMooringCivil engineering, FoundationsSynergy with offshore industry…
Offshore model
Offshore prototypedemonstrator
2010
2008
20yy
Phase III
T echnology Development
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Working Group on environmental issues
Ocean Energy T echnologyNetwork
Working Group on Policy S tatement on Ocean Energy Generation
Refined Strategy on Impact Assessment
Public Education and Acceptability
Market Expansion
NRC IOTIndustry, Government ,Academia
Significant contribution to x% of T otal Energy Generation
Resource assessment Prediction of wind, wave and current resource; presence of iceL onger term weather forecast? Ocean energy prospecting.
T echnologies neededData Acquisition T echniquesCable Network, Grid connectionSurface treatments, coatings, CorrosionNon-electrical outputs: hydrogenDesalination, others?
Environmental Impact Research
2004
20yy
output
2009
Working Group on R&D & T echnology development
National Coordinator, Evaluator & T est Centre (NRC IOT )NL WAVE POWER?
The above Timeline is a proposal for the development of an Ocean Energy Technology Network that would include Industry, Academia and Government departments in Canada. Of the latter it is proposed that the National Research Council and Industry Canada play a significant role. The Institute for Ocean Technology of the National Research Council http://iot-ito.nrc-cnrc.gc.ca/about.html is well positioned to be considered as the National Coordinator for the Evaluation and Testing of Ocean Systems. Working groups should be developed to focus on the following tasks. The list is not intended to be exhaustive in any way and the list is not suggesting any priority. In fact some of the tasks will have to be worked upon in parallel. The following is adapted in part from the Wave Energy Utilization report (2002) and the European Wave Energy Network on Wave Energy. See International Energy Agency Annual Report (2003). The purpose of the present exercise is to initiate discussions among the proponents of such developments in Canada including industry, academia and government and regulatory departments. 7.3 Scientific, Technical, Societal and Economic challenges of an Ocean Energy Industry: The following list of issues to be addressed are necessary for the development of Ocean Energy harnessing activities. The following is adapted from the European
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Thematic Network on Wave Energy report: Wave Energy Utilization in Europe: Current Status and Perspective (2002). Included also is an expanded list of R&D issues and further comments on computational tools to complement any experimental physical modelling activities. 1) R&D in Ocean Energy devices:
- Waves and currents energy systems and devices status and R&D requirements. - Tidal energy systems status and devices and R&D requirements.
- Strategy and action plan for above. - Lessons learned from Offshore Industry (technology transfer) would
benefit ocean energy technology and its applications. Status and requirements. See also 2) below.
- Scaled model and/or demonstration project developments requirements.
- Development of performance evaluation procedures. - Implement performance evaluation on specific concepts.
2) Generic Technologies development:
- Plant control and power output prediction - Plant monitoring and performance assessment - Loads and survivability studies - Maintenance and reliability studies - Modelling and standardised design methods and calculations - Experimental and mathematical modelling of systems and resource
estimation - Hydrographic analysis and mapping - Harsh environment in the ocean studies: waves, currents, tides and
ice - Data base development and power prospecting - Experimental and/or numerical work of testing concepts modeling of
arrays of multiple wave and other energy devices, wave and energy generation and propagation simulation.
– Wave focusing and rogue wave simulation. – Real-time wave behavior forecasting – Mooring and long term fatigue of lines and connections – Cable construction, production, testing and laying offshore – Couplings for quick release and reattachment of mooring and cables – Flexible electrical connectors – Environmentally acceptable fluid for hydraulic systems – Power smoothing systems - Follow the advancement in hydrogen technology - International collaboration: attract prototype models to our shores, responding to Canadian-specific environment (ice). - Electrical power storage technology.
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- Control and transmission systems - Flexible electrical connectors - Environmentally acceptable fluid for hydraulic systems - Development of more efficient power generation units - Power conversion - Offshore control systems - Maintenance-free systems. Is it attainable?
3) Cooperation with the Power Generation and Distribution Industry:
- Development of safety standards. - Assessment of procedures, costs and facilities for power generation
and transmission. - Development of power quality standards. - New regulatory and energy transfer regimes.
4) Financing and Economic Issues: - Market status of ocean energy: feasibility and market analysis. - Economics of Ocean Energy whether from waves, wind or currents. - Financing of ocean energy projects issues. - Economic impact on environment and local communities.
5) Social, institutional and environmental impact: - Planning considerations. - Environmental impact studies. - Institutional barriers studies and regulatory regimes. - Industrial benefit and job creation studies.
6) Promotion of Ocean Energy: - Support for Ocean Energy events and meetings. - Publications in International Journals. - Dissemination of information and relevant material - Development of relevant web site. - Involvement with education curricula in schools, promotion of
renewables in general and Ocean Energy in particular. Below are some comments on the need of accurate and robust computational models for the optimization of the structural configurations and designs of Ocean energy Systems. The need for resource estimation techniques was discussed in Sections 2 and 3 of this report. The optimal structural configuration of an ocean energy system could be studied and is achievable by taking into consideration the complex interaction of the structure with the environmental effects. These should include the effects in extreme and harsh conditions of waves, currents, winds and ice (if present) as well as the seabed. This objective can be achieved with good information on those environmental effects and robust analytical/computational techniques. Synergy with offshore engineering and the vast experience gained by this industry are critical. Fatigue of structural components and of the foundations/anchoring as well as the ultimate capacities are some of the issues. The dynamic excitation forces will have to consider the environmental loading effects due the wind, waves, currents, and possibly ice on the structure, the interaction
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between the devices, their dynamic loading and the structure supporting it and the soil or seabed. Computational techniques of offshore or onshore structures in general require the modeling of the structural responses, including dynamics of the structure, the evaluation and modeling of the environment and the accurate modeling of the interaction of the structure with the soil through the foundation, whether it is gravity based or flexibly moored. These are presently issues in offshore engineering R&D. 7.4 Constraints and opportunities: Technological:
In general the technology for wave energy and stream systems has yet to be developed and proven at full size, particularly for the offshore wave systems and the variable-pitch vertical axis turbine tidal stream systems. There are no insurmountable technological barriers to the deployment of such systems. Opportunities abound for innovation, R&D and synergy with the offshore engineering industry.
Financial, economical: There is a need for proven economics and resource assessment methodologies. Opportunities for investors are infinite and the market for renewables in general and the ocean energy resource in particular, will keep increasing.
Institutional:
A number of statutory bodies are involved in our coastline and surrounding waters. The development of an Ocean Energy Industry will require an extensive consultation process.
55
8. References [1] ABB (2002) http://www.abb.se (in Swedish)
[2] Baddour, R.E. and Song, S.W. (1998). The rotational flow of finite amplitude periodic water waves on shear currents. Applied Ocean Research, 20, 163-171.
[3] Baddour, R. E. and Song, S. (1990). On the interaction between waves and currents. Ocean Engineering, 17, 1-21.
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[8] Blue Energy Canada Inc. (2000); http://www.bluenergy.com ; Canada.
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[11] Budal, K and Falnes, J. (1975). A resonant point absorber of ocean wave power. Nature, 256, 478-479.
[12] Bhuyan et al (1994). Powertech Inc. Assessment of Wave Energy Resources for the West Coast of Canada, June 1994. Also http://www.powertech.bc.ca
[13] Clancy, E.P. (1968) The tides: Pulse of the Earth, Doubleday and Co. New York.
[14] Clement, A.H. et al (2002) Wave Energy in Europe- Current status and perspectives, Renewable and Sustainable Energy reviews, 6, 405-431.
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[22] FMP (Foresight Marine Panel) (1999); Energies from the Sea – Towards 2020; A Marine Foresight Report, United Kingdom.
[23] Falnes. J. (1996). A crude estimate of benefit-cost ratio for utilization of the wave enrgy of the world’s oceans. Proc. Sec. European Wave Power Confernece, Lisbon, Portugal, 102-104.
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[28] Gato, L.M.C. and Curren, R. (1997). The energy conservation performance of several types of Wells turbine designs. Proc. I. Mech. E. Part A, J. Power Engineering, 211, 133-145.
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[30] Hagerman, G. (1992) Wave resource assessment review and evaluation. In comprehensive review and analysis of Hawaii’s renewable Energy assessments. State of Hawaii Department of Business, Economic Development and Tourism, Honolulu, Hawaii.
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[42] Nebel, P.(1992) Maximizing the efficiency of wave energy plants using complex-conjugate control. Proc. I. Mech E. Journal of Systems and Control, 206, 225-236.
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[45] Rebello, L. et al (1996). 1996 Project OLASS 1000-experiences, modelling and results. Proc. Second European Wave power Conference, Lisbon, Portugal, 312-319.
[46] Retzler, C.H. (1996). The wave rotor.
58
Proc. Second European Wave power Conference, Lisbon, Portugal, 312-319. [47] Rudkin, E.J. and Loughman, G.L. (2001). Vortec – the marine enrgy solution. Marine Renewable Energy Conference 2001; Newcastle, United Kingdom. [48] Russel, A. and Diamantaras, K. (1996) The European Commission Wave Energy
R&D Programme. Proc. Second European Wave Power Conference, Lisbon, Portugal, 8-11. G. Elliot and Diamantaras, K. editors.
[49] SEASUN Power Systems (1988). Power conditioning and transmission, wave
energy resource and technology assessment for coastal North Carolina. Final Report. North Carolina Alternative Energy Corporation & Virginia Power.
[50] Scott, N.C. (2001) Grid connection of large-scale wave energy projects. Proc.
Fourth European Wave Power Conf., Aalborg, Danemark. 88-92. [51] Southgate, H.N. (1987) Wave prediction in deepwater and at the coastline, HR
report #SR114. [52] Southgate, H.N. (1993) The use of wave transformation models to evaluate
inshore wave energy resource. Proc. First European Symposium on Wave Energy, Edinburgh, UK.
[53] Thorpe, T.W. (1999) An overview of wave enrgy technologies: status,
performance and costs. Proceedings, International One day Seminar, Institution of Mechanical Engineers, London UK.
[54] Thorpe, T.W. (1999) A brief overview of wave energy. ETSU report R-120.
Department of Trade and Industry, UK. [55] Thorpe, T.W. (1992) A review of wave energy. ETSU Report R-72, 1, Department
of Trade and Industry, UK. [56] Thorpe, T.W. (1999) A brief overview of wave energy. ETSU Report R-120.
Department of Trade and Industry, UK. [57] Thorpe, T.W. (1999) An evaluation of wave energy. ETSU Report AEAT/
R0400. Dpartment of Trade and Industry, UK [58] University of Edinburgh, Wave Power Group. 2000.
www.mech.ed.ac.uk/research/wavepower/ [59] Various authors. 1993 Preliminary design and model test of a wave power
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59
[60] WATTS (2003, 2004). Wave and Tidal Technology Symposium Proceedings. [61] Wavegen. (2000) <http// www.wavegen.co.uk> [62] Wave Mill Energy Corp (2004) http://www.wavemill.com/products.htm [63] World Energy Council (2004) http://www.worldenergy.org/wec-geis [64] World Energy Council (2000) http://www.worldenergy.org/wec-
geis/publications/reports/etwan/execsummary/exec-summary.asp
9. Recent scientific and technical research bibliography Theme: Ocean Energy related topics Ocean Engineering - From 1995-2004 D. C. Hong, S. Y. Hong and S. W. Hong, Numerical study of the motions and drift force of a floating OWC device, Ocean Engineering, Volume 31, Issue 2, February 2004, Pages 139-164. E. Vijayakrishna Rapaka, R. Natarajan and S. Neelamani, Experimental investigation on the dynamic response of a moored wave energy device under regular sea waves, Ocean Engineering, In Press, Corrected Proof, Available online 2 December 2003. Y. M. C. Delauré and A. Lewis, 3D hydrodynamic modelling of fixed oscillating water column wave power plant by a boundary element methods, Ocean Engineering, Volume 30, Issue 3, February 2003, Pages 309-330. Paolo Boccotti, On a new wave energy absorber, Ocean Engineering, Volume 30, Issue 9, June 2003, Pages 1191-1200. Umesh A. Korde, Latching control of deep water wave energy devices using an active reference, Ocean Engineering, Volume 29, Issue 11, September 2002, Pages 1343-1355. W. E. Rogers, J. M. Kaihatu, H. A. H. Petit, N. Booij and L. H. Holthuijsen, Diffusion reduction in an arbitrary scale third generation wind wave model, Ocean Engineering, Volume 29, Issue 11, September 2002, Pages 1357-1390. A. Brito-Melo, L. M. C. Gato and A. J. N. A. Sarmento, Analysis of Wells turbine design parameters by numerical simulation of the OWC performance, Ocean Engineering, Volume 29, Issue 12, September 2002, Pages 1463-1477. A. F. de O. Falcão, Wave-power absorption by a periodic linear array of oscillating water columns, Ocean Engineering, Volume 29, Issue 10, August 2002, Pages 1163-1186. D. J. Wang, M. Katory and Y. S. Li, Analytical and experimental investigation on the hydrodynamic performance of onshore wave-power devices, Ocean Engineering, Volume 29, Issue 8, July 2002, Pages 871-885.
60
S. Neelamani and M. Vedagiri, Wave interaction with partially immersed twin vertical barriers, Ocean Engineering, Volume 29, Issue 2, February 2002, Pages 215-238. Ruo-Shan Tseng, Rui-Hsiang Wu and Chai-Cheng Huang, Model study of a shoreline wave-power system, Ocean Engineering, Volume 27, Issue 8, August 2000, Pages 801-821. A. F. de O. Falcão and P. A. P. Justino, OWC wave energy devices with air flow control, Ocean Engineering, Volume 26, Issue 12, December 1999, Pages 1275-1295. U. A. Korde, Efficient primary energy conversion in irregular waves, Ocean Engineering, Volume 26, Issue 7, July 1999, Pages 625-651. Applied Ocean Research - 1995-2004 José Perdigão and António Sarmento, Overall-efficiency optimisation in OWC devices, Applied Ocean Research, Volume 25, Issue 3, June 2003, Pages 157-166. Umesh A. Korde, Systems of reactively loaded coupled oscillating bodies in wave energy conversion, Applied Ocean Research, Volume 25, Issue 2, April 2003, Pages 79-91. A. F. de O. Falcão and R. J. A. Rodrigues, Stochastic modelling of OWC wave power plant performance, Applied Ocean Research, Volume 24, Issue 2, April 2002, Pages 59-71. A. F. de O. Falcão, Control of an oscillating-water-column wave power plant for maximum energy production, Applied Ocean Research, Volume 24, Issue 2, April 2002, Pages 73-82. U. A. Korde, On providing a reaction for efficient wave energy absorption by floating devices, Applied Ocean Research, Volume 21, Issue 5, October 1999, Pages 235-248. H. Eidsmoen, Tight-moored amplitude-limited heaving-buoy wave-energy converter with phase control, Applied Ocean Research, Volume 20, Issue 3, June 1998, Pages 157-161. M. Greenhow and S. P. White, Optimal heave motion of some axisymmetric wave energy devices in sinusoidal waves, Applied Ocean Research, Volume 19, Issues 3-4, June-August 1997, Pages 141-159. Umesh A. Korde, Performance of a wave energy device in shallow-water nonlinear waves: part I, Applied Ocean Research, Volume 19, Issue 1, February 1997, Pages 1-11. Umesh A. Korde, Performance of a wave energy device in shallow-water nonlinear waves: part II, Applied Ocean Research, Volume 19, Issue 1, February 1997, Pages 13-20. K. Thiruvenkatasamy and S. Neelamani, On the efficiency of wave energy caissons in array, Applied Ocean Research, Volume 19, Issue 1, February 1997, Pages 61-72. S. A. Mavrakos and P. McIver, Comparison of methods for computing hydrodynamic characteristics of arrays of wave power devices, Applied Ocean Research, Volume 19, Issues 5-6, October-December 1997, Pages 283-291. J. R. Chaplin and C. H. Retzler, Predictions of the hydrodynamic performance of the wave rotor wave energy device, Applied Ocean Research, Volume 17, Issue 6, December 1995, Pages 343-347. P. McIver and M. McIver, Wave-power absorption by a line of submerged horizontal cylinders, Applied Ocean Research, Volume 17, Issue 2, 1995, Pages 117-126.
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Coastal Engineering - 1995-2004 Jaak Monbaliu, Roberto Padilla-Hernández,Julia C. Hargreaves, Juan Carlos Carretero Albiach, Weimin Luo, Mauro Sclavo and Heinz Günther, The spectral wave model, WAM, adapted for applications with high spatial resolution, Coastal Engineering, Volume 41, Issues 1-3, September 2000, Pages 41-62. T. C. Lippmann, A. H. Brookins and E. B. Thornton, Wave energy transformation on natural profiles, Coastal Engineering, Volume 27, Issues 1-2, May 1996, Pages 1-20. Renewable Energy - 1995-2004 A. Thakker and T. S. Dhanasekaran, Computed effects of tip clearance on performance of impulse turbine for wave energy conversion, Renewable Energy, Volume 29, Issue 4, April 2004, Pages 529-547. Ajit Thakker and Fergal Hourigan, Modeling and scaling of the impulse turbine for wave power applications, Renewable Energy, Volume 29, Issue 3, March 2004, Pages 305-317. T. Setoguchi, Y. Kinoue, T. H. Kim, K. Kaneko and M. Inoue, Hysteretic characteristics of Wells turbine for wave power conversion, Renewable Energy, Volume 28, Issue 13, October 2003, Pages 2113-2127. A. S. Bahaj and L. E. Myers, Fundamentals applicable to the utilisation of marine current turbines for energy production, Renewable Energy, Volume 28, Issue 14, November 2003, Pages 2205-2211. Mats Leijon, Hans Bernhoff, Marcus Berg and Olov Ågren, Economical considerations of renewable electric energy production--especially development of wave energy, Renewable Energy, Volume 28, Issue 8, July 2003, Pages 1201-1209. T. Setoguchi, S. Santhakumar, M. Takao, T. H. Kim and K. Kaneko, A modified Wells turbine for wave energy conversion, Renewable Energy, Volume 28, Issue 1, January 2003, Pages 79-91. Wibisono Hartono, A floating tied platform for generating energy from ocean current, Renewable Energy, Volume 25, Issue 1, January 2002, Pages 15-20. T. Setoguchi, S. Santhakumar, H. Maeda, M. Takao and K. Kaneko, A review of impulse turbines for wave energy conversion, Renewable Energy, Volume 23, Issue 2, June 2001, Pages 261-292. Fernando Ponta and Gautam Shankar Dutt, An improved vertical-axis water-current turbine incorporating a channelling device, Renewable Energy, Volume 20, Issue 2, June 2000, Pages 223-241 Apparatus for dissipating wave energy : Dorrell Donald E Rarotonga 01, Cook Island, Renewable Energy, Volume 11, Issue 2, June 1997, Page 273. V. S. Raju and M. Ravindran, Wave energy : potential and programme in India, Renewable Energy, Volume 10, Issues 2-3, 3 February 1997, Pages 339-345. P. R. S. White, The European programme to develop the Wells air turbine for applications in wave energy, Renewable Energy, Volume 9, Issues 1-4, September-December 1996, Pages 1207-1212. F. Peter Lockett, Mathematical modelling of wave energy systems, Renewable Energy, Volume 9, Issues 1-4, September-December 1996, Pages 1213-1217. Kunal Ghosh, Cascade wind turbines for the oscillating water column wave energy device: Part 1, Renewable Energy, Volume 9, Issues 1-4, September-December 1996, Pages 1219-1222.
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Hitoshi Hotta, Yukihisa Washio, Hitoshi Yokozawa and Takeaki Miyazaki, R&D on wave power device "Mighty Whale", Renewable Energy, Volume 9, Issues 1-4, September-December 1996, Pages 1223-1226. Wibisono Hartono, A floating tied platform for generating energy from ocean current, Renewable Energy, Volume 25, Issue 1, January 2002, Pages 15-20. R. C. McGregor and W. E. R. Desouza, On the analysis of tidal energy schemes with large diurnal variations with application to Singapore, Renewable Energy, Volume 10, Issues 2-3, 3 February 1997, Pages 331-334. H. Maeda, S. Santhakumar, T. Setoguchi, M. Takao, Y. Kinoue and K. Kaneko, Performance of an impulse turbine with fixed guide vanesfn2 for wave power conversion, Renewable Energy, Volume 17, Issue 4, 1 August 1999, Pages 533-547. V. N. M. R. Lakkoju, Combined power generation with wind and ocean waves, Renewable Energy, Volume 9, Issues 1-4, September-December 1996, Pages 870-874. Renewable and Sustainable Energy Reviews - 1997-2004 Alain Clément, Pat McCullen, António Falcão, Antonio Fiorentino, Fred Gardner, Karin Hammarlund, George Lemonis, Tony Lewis, Kim Nielsen, Simona Petroncini et al., Wave energy in Europe: current status and perspectives, Renewable and Sustainable Energy Reviews, Volume 6, Issue 5, October 2002, Pages 405-431. Roger H. Charlier, A "sleeper" awakes: tidal current power, Renewable and Sustainable Energy Reviews, Volume 7, Issue 6, December 2003, Pages 515-529. Roger H. Charlier, Re-invention or aggorniamento? Tidal power at 30 years, Renewable and Sustainable Energy Reviews, Volume 1, Issue 4, December 1997, Pages 271-289. Journal of Energy Engineering - 1995-2004 Performance Prediction of Contrarotating Wells Turbines for Wave Energy Converter Design. R. Curran, T. J. T. Whittaker, S. Raghunathan, and W. C. Beattie, J. Energy Engrg. 124, 35 (1998) Productivity of Ocean-Wave-Energy Converters: Turbine Design R. Curran, J. Energy Engrg. 128, 13 (2002)
63
Appendices Appendix 1: Mathematical formulation and computation of wave power/energy in irregular seas The local behaviour of waves is determined by what is defined as the spectrum of the VHDVWDWH7KLVLVJLYHQPDWKHPDWLFDOO\E\WKHIXQFWLRQ6I WKDWVSHFLILHVKRZWKH
ZDYHHQHUJ\LVGLVWULEXWHGLQWHUPVRIIUHTXHQF\IDQGGLUHFWLRQ 6HHIRr example Kinsman (1984) for further details on this topic. The spectrum is usually summarized by a small number of wave parameters. They are WKHZDYHKHLJKW+PWKHSHULRG7VHFWKHIUHTXHQF\I 7+]DQGGLUHFWLRQ
(rad). For the wave height the most widely used parameter is the significant wave height which can be computed from the wave directional spectrum as follows:
2/104mH s =
where 0m is the zero-th spectral moment, the n-th moment being defined as
∫ ∫∞
=π
θθ2
0 00 ),( ddffSfm n
The convention of considering S as the distribution of the variance of the sea surface elevation has been used. Thus the total variance is 0m (m2 ) and the actual energy per
unit area is 0gmρ . Also the significant wave height is defined (followed convention) as 2/1
04mH s = . (See Mollison 1986 or other references)
The energy mean period is defined as
0
1
m
mTe
−=
and the peak period pT is the inverse of the peak frequency pf which corresponds to
the highest spectral density. For sea states with only one wave system pT is useful in
providing a measure of the period of the waves with the highest energy density. The significant wave period sT which is the average period of the highest one third of the
waves is generally less used. The wave power level or flux of energy per unit wave front can be computed from
θθρπ
ddffShfcgP g ),( ),( 2
0 0∫ ∫∞
=
where ρ is the water density. In deep water the group velocity which is the velocity at which the energy propagates is given by
f
gcg π4
=
64
Thus the wave power is given by
1
212
0 0
2
4 ),(
4 −−∞
== ∫ ∫ mg
ddffSfg
Pπ
ρθθπ
ρ π
which can be expressed in terms of sH and eT as
es THP 5.0 2≅
When sH is expressed in meters and eT in seconds the power level is given in kW/m. Appendix 2: Useful wave & tidal power links
Universities, National and Government sites Australian Renewables including Wave Energy : http://www.greenhouse.gov.au/renewable/index.html Caddet renewable energy website : http://www.caddet.org/index.php Danish Wave Energy : http://www.waveenergy.dk/ DTI Renewables (UK Government) : http://www.dti.gov.uk/energy/renewables/ European Commission ’Thermie’ wave energy site : http://europa.eu.int/comm/energy_transport/atlas/htmlu/wave.html European Marine Energy Centre, Orkney (test centre for marine energy) : http://www.emec.org.uk/ European Wave Energy Research Network (EWERN) : http://www.ucc.ie/ucc/research/hmrc/ewern.htm European Wave Energy Thematic Network : http://www.wave-energy.net/ Japan Marine Science & Technology Center, JAMSTEC : http://www.jamstec.go.jp/jamstec/MTD/Whale/
Marine Institute, Cork Ireland: http://www.marine.ie/rnd+projects/index.htm
Norwegian Wave Energy Site : http://www.phys.ntnu.no/instdef/prosjekter/bolgeenergi/index-e.html Open University, UK : http://www.openuniversity.edu/ Scottish Executive Energy website: http://www.scotland.gov.uk/about/ELLD/EN-CS/00017058/energyhome.aspx World Wave Atlas : http://seawatch.mg.uoa.gr/
ZZZPHFKHGDFXNUHVHDUFKZDYHSRZHU
65
Device Developers and Data Providers AquaEnergy Group Ltd (USA) : http://www.aquaenergygroup.com/home.htm Daedalus Informatics Greece: Hybrid wave and wind system : http://www.daedalus.gr/DAEI/PRODUCTS/RET/General/What%20is%20Wave%20Power EMU Consult, Denmark (Wave Dragon) : http://www.spok.dk/consult/waves.shtml Energetech Australia Pty Ltd (includes Denniss-Auld Turbine) : http://www.energetech.com.au/ The Engineering Business Ltd (Stingray tidal stream device) : http://www.engb.com/ Interproject Service AB (IPS OWEC Buoy): http://www.ips-ab.com/ Marine Current Turbines Ltd : http://www.marineturbines.com/home.htm Oceanor, Norway (Wave data) : http://www.oceanor.no/ Ocean Power Delivery Ltd., Scotland (Pelamis) : http://www.oceanpd.com/ Ocean Power Technologies, USA : http://www.oceanpowertechnologies.com/ Sea Power International AB, Sweden : http://www.seapower.se/indexeng.html SeaVolt Technologies, USA (Wave Rider) under construction Strom AS (Tidal Stream generator at Hammerfest in Norway) : http://www.tidevannsenergi.com/ Teamwork Technology Bv (Archemides Wave Swing): http://www.waveswing.com/ Verdant Power (tidal current demonstration in NYC): http://www.verdantpower.com/Initiatives/eastriver.shtml Wave Dragon ApS (Danish) : http://www.wavedragon.net/ Wavegen, Scotland (Limpet) : http://www.wavegen.co.uk/ Wavemill Energy Corporation (Cape Breton, Nova Scotia) : http://www.wavemill.com/
Lobbying, Promotional & Trade Organisations Practical Ocean Energy Management Systems (US) : http://www.poemsinc.org/ Renewable Power Association (UK) : http://www.r-p-a.org.uk/home.fcm Seapower - Marine Renewable Energy Association (UK): http://www.bwea.com/marine/resource.html Scottish Coastal Forum : http://www.scotland.gov.uk/environment/coastalforum/ Scottish Energy Environment Foundation : http://www.mecheng.strath.ac.uk/feature-seef.htm Scottish Renewables Forum: http://www.ipa-scotland.org.uk/
66
National Review documents A brief review of Wave Energy : http://www.research-innovation.ed.ac.uk/expertise/physical-sciences/energy.pdf
Also see http://www.research-innovation.ed.ac.uk/flashindex.html
Options for the development of wave energy in Ireland : http://www.irish-energy.ie/uploads/documents/upload/publications/wave.pdf
Other Links Edinburgh Designs Ltd (Test tanks and absorbing wave maker systems): http://www.edesign.co.uk/
http://www.edesign.co.uk/
McGraw-Hill Higher Education virtual wave tank : http://www.mhhe.com/physsci/physical/giambattista/wave_tank/wave_tank.html
Appendix 3: Recent publications from IOT
1 Copyright © 2004 by ASME
Proceedings of OMAE ’ 04 23rd International Conference on Offshore Mechanics
and Arctic Engineering June 20-25, 2004, Vancouver, Canada
OMAE2004-51044
THE NONLINEAR INTERACTION AND RESONANCE OF STEEP LONG-CRESTED BICHROMATIC SURFACE WAVES IN A NUMERICAL WAVE TANK
R. E. Baddour National Research Council -Canada
Institute for Ocean Technology P.O.Box 12093, A1B 3T5 St John’s, NL, CANADA
W. Parsons College of the North Atlantic
Ridge road, Box 1150, A1C 6L8
St John’s, NL, CANADA
ABSTRACT
We are studying numerically the problem of generation and propagation of long-crested gravity waves in a tank containing an incompressible inviscid homogeneous fluid initially at rest with a horizontal free surface of finite extent and of infinite depth. A non-orthogonal curvilinear coordinate system, which follows the free surface is constructed which gives a realistic ’’continuity condition’’, since it tracks the entire fluid domain at all times. A depth profile is assumed and employed to perform a waveform relaxation algorithm to decouple the discrete Laplacian along dimensional lines, thereby reducing its computation over this total fluid domain. In addition, the full nonlinear kinematic and dynamic free surface conditions are utilized in the algorithm. A bichromatic deterministic wave maker using a Dirichlet type boundary condition and a suitably tuned numerical beach is utilized. This paper pays special attention to satisfying the full nonlinear free surface conditions and presents the nonlinear interaction of the higher order components, especially near resonance.
INTRODUCTION
A two-dimensional rectangular basin containing an incompressible inviscid homogeneous fluid initially at rest with a horizontal free surface of finite extent is considered to generate and propagate long-crested waves. On the left vertical boundary a wavemaker is positioned while at the right hand side is the radiation boundary. By this we mean, the right hand side boundary is designed to avoid reflections, which complicates the flow field. This can be accomplished by implementing a radiation condition, hence the name, which allows the waves to pass through the boundary, or by placing a numerical beach near this boundary. We choose the latter. As in Parsons and Baddour (2002), a depth profile for the potential is assumed, giving us a
waveform relaxation method, and thereby drastically reducing the computational cost of solving Laplace’s equation. A numerical beach is also used to absorb the wave energy at the radiation boundary. A bichromatic wavemaker is employed using a Dirichlet type boundary condition, a non-orthogonal curvilinear coordinate system, which follows the free surface, and the full nonlinear kinematic and dynamic free surface boundary conditions are employed; see Parsons and Baddour (2003). Although, these ideas can be extended to finite depth tanks, we presently restrict our attention to the infinite depth case. This method is a volume-discretization method, which uses waveform relaxation to reduce the “computational dimension” of the problem. As such, it compares favorably with the boundary-discretization method that reduces the dimension of the problem by solving it only on the boundary. See the review by Tsai and Yue (1996). Concerning the phenomenon of resonance in surface gravity waves, see Debnath (1994) for some of the classical results and the references there in. Note that these methods usually involve series methods where a number of high order terms are included in the expansion of the equations. We point out that our model utilizes the full nonlinear model, and therefore should contain these results, as special cases. In fact it should be remembered that resonance is a purely nonlinear effect, and therefore requires a nonlinear model to be seen. Also, the results can be quite dramatic, so it is important to be aware of them when solving ocean engineering problems.
PROBLEM FORMULATION Consider a two-dimensional rectangular basin containing an incompressible inviscid homogeneous fluid of constant density ρ , in a Cartesian coordinate system ),( zx , with the
2 Copyright © 2004 by ASME
origin at the still water level, the positive z-axis pointing upwards. The horizontal extent of the basin is Lx ≤≤0 , so
0>L is the length of the tank. The depth of the basin is infinite. See Figure 1 for the coordinate system configuration. This is an initial-value problem, since the surface is initially at rest for time, 0<t and is disturbed at 0≥t giving rise to surface waves. See Debnath (1994). If the ),,( tzxΦ is the
velocity potential and ),( txη is the free surface elevation, the
problem is defined by the following equations, where g is the
acceleration due to gravity. The conservation of mass equation is Laplace's equation:
0 ,0 ),,( 02
2
2
2
≥≤≤≤≤∞−=∂
Φ∂+∂
Φ∂tLxtxz
zxη (1)
For ,0≥t the full nonlinear kinematic and dynamic free surface boundary conditions are given, respectively, by:
0 =∂Φ∂−
∂Φ∂
∂∂+
∂∂
zxxt
ηη (2)
and 0])()[(2
1 22 =+∂Φ∂+
∂Φ∂+
∂Φ∂ ηg
zxt (3)
on ),( txz η= , and
z as 0 −∞→→∂Φ∂z
(4)
Introducing the following function: )),,(,(),( ttxxtx ηχ Φ= (5)
which represents the potential on the free surface, we have:
tztt ∂
∂∂Φ∂+
∂Φ∂=
∂∂ ηχ
(6)
implying:
tztt ∂
∂∂Φ∂−
∂∂=
∂Φ∂ ηχ
(7)
Also, we have that
. xzxx ∂
∂∂Φ∂+
∂Φ∂=
∂∂ ηχ
(8)
See Wehausen and Laitone (1960). The initial conditions are given by:
)()0,( (x);)/1()0,0,( );()/1()0,0,( 0 00 xxxxx ηηρρ =Φ=ΦΦ=Φ••
where )()/1( 0 xΦρ and )()/1( 0 x•Φρ represent the given free
surface impulse and )(0 xη the initial displacement, where the
“dot” represents a time derivative. Since we are considering a fluid in which the initial velocities are zero and the initial free surface is at rest at 0=z , we take
Lx0for 0)( ;0)( ;0)( 000 ≤≤==Φ=Φ•
xxx η (9)
Therefore we are left with two lateral boundary conditions for 0≥t at 0=x and Lx = , which we call the LHS lateral
boundary condition and the RHS lateral boundary condition, respectively. The LHS boundary condition involves a wavemaker, which we assume is the sum of two sinusoid waves, each of the general form
∞<<∞≤<∞−= t- 0,x- ),cos(),( txkAtx iiii ωη (10)
where, 2/ii HA = is the amplitude of each wave, ik is the
wavenumber and iω is the angular frequency of each wave;
2,1=i . The steepness of the wave is π/iii AkS = . Using
standard water wave mechanics, the associated linear velocity potential gives the LHS boundary condition, and is the sum of two terms of the general form
),sin(),,( txkegA
tzx iizk
i
ii
i ωω
−=Φ (11)
for ,0- 0, t0,x ≤<∞≥= z
where the linear dispersion relation is
gkii =ω , 2,1=i . (12)
Note that in the implementation of the wavemaker algorithm we assume the validity of (11) over the range ),0( tz η≤<−∞ i.e. up to the free surface. It is prudent to point that it is not necessary, but only a convenience and for completeness to make the assumption of the existence of the sinusoids in Equation (10), since we are only inputting the potential given by Equation (11). Therefore conditions outside the interval [ ]L,0 have no effect on the solution. This progressive wave at 0=x and 0≥t is the sole source of the disturbance that gives rise to water waves in the initially calm basin. If we assume a wall at Lx = so the RHS lateral boundary condition becomes
0, t,- ,0),,( ≥≤<∞=
∂Φ∂ ηz
x
tzL (13)
we will get reflections from waves going from the wavemaker at 0=x toward the RHS wall. To absorb these waves we place a “numerical beach” on the RHS. This is accomplished by including a damping term µ in the dynamic free surface
boundary condition equation applied at the damping zone. This is accomplished by adding a damping term µχ to the right
hand side of the dynamic free surface boundary condition; see Equation (33). However, an abrupt introduction of this damping term causes some reflection at the boundary between the damping zone and the non-damping zone. The following third-order polynomial distribution of the damping coefficient ensures a smooth transition between these two zones:
,ˆ2ˆ3
)ˆ(3
1
3
21
2
kgL
x
L
xx
−=νµ where 1L is the length of the
damping zone, k is the representative wave number, which will be discussed later, and ν is the normalized damping coefficient.
Note that x is a local variable where, ,0ˆ =x when 1LLx −= ,
and ,ˆ 1Lx = when Lx = .
This numerical beach is found to be most effective when, 0.18.0 ≤≤ν .
Clearly, setting 0=ν is equivalent to removing the beach.
Also, 1L is usually taken to be about two representative
wavelengths, λ21 ≈L , where λ is the wavelength associated
3 Copyright © 2004 by ASME
with the representative wavenumber, and is given by k/2πλ = . See Parsons and Baddour (2002).
THE LAPLACIAN ON THE FREE SURFACE COORDINATE SYSTEM We define the Free Surface Coordinate System (FSCS), ),( ws ,
where 0≥s is the arclength along the free surface ),( txz η= and 0≤w is the vertical depth to any point in the
fluid from the free surface. This is a non-orthogonal moving coordinate system that follows the free surface. Clearly, the coordinates satisfy the following equations (over the complete fluid domain),
Lx0 , ),(
10
2≤≤
∂∂+= ∫ τ
ττη
dt
sx
, (14)
0. wL,x0 ),,( ≤≤≤−= txzw η (15)
If we define the unit vector s along the arclength, and therefore tangent to the free surface ),( txz η= , then
k sin i cosˆ θθ +=s (16)
k tan- s secˆ θθ=i (17) where
.tan 1
∂∂
= −
x
ηθ (18)
The arclength along the coordinate curve s is given by
dx 12
∂∂+=
xds
η (19)
or dx secθ=ds . (20)
Clearly we have the following differential operator formulas that allow us to transform easily between rectangular coordinates ),( zx and free surface coordinate system ),( ws :
w tan-
s sec
∂∂
∂∂=
∂∂ θθx
(21)
z sin cos
∂∂+
∂∂=
∂∂ θθ
xs (22)
wz ∂∂=
∂∂
. (23)
The Laplacian for this non-orthogonal coordinate system can be constructed from the second-order covariant fundamental metric tensor
=
1 sin
sin 1
θθ J (24)
where its associated second-order contravariant tensor is given by
sec sectan
sectan sec 2
2
−
−=
θθθ
θθθ J (25)
where clearly θ2cos) == GHWJJ .
The Laplacian is given by the following tensor equation
2,1, ,1 =
∂Φ∂
∂∂=∆Φ λαα
αλλ xx
JJJ
(26)
where . x, 21 wsx == See Kaplan (1993). We get
]sectan
tansec[ sec
∂Φ∂+
∂Φ∂−
∂∂+
+
∂Φ∂−
∂Φ∂
∂∂=∆Φ
wsw
wss
θθ
θθθ (27)
THE DYNAMIC AND KINEMATIC BOUNDARY CONDITIONS ON THE FREE SURFACE COORDINATE SYSTEM Using equation (8) and equation (18), the kinematic free surface boundary condition (KFSBC) given by equation (2), becomes
0sectan 2 =∂Φ∂−
∂∂+
∂∂
zxtθχθη
. (28)
The dynamic free surface boundary condition (DFSBC), given by equation (3), can be rewritten using equations (7) and equation (8). After some algebra and using equation (18) and the KFSCB, equation (28), we get the following final form:
.0sec2
12
22
=+
∂Φ∂−
∂∂
+∂∂ ηθχχ
gzxt
(29)
where all values in (28) and (29) are evaluated on the free surface. Since the free surface coordinate system tracks the free surface, these values are readily obtained and allow for the integration in time t .
SPATIAL APPROXIMATION AND WAVEFORM RELAXATION In this paper, we employ a curvilinear grid that follows the free surface. In light of equation (18), this grid is taken to be ( ) ,,,0,1j ;,,1,0 ,, Lji NMiws == (30)
where xss iii ∆+= −− 11 secθ , and
ixx
i x=
∂∂
+=2
1secηθ ,
where we take ,00 =s and wjw j ∆−= , where ,/ MLx =∆ so
,xixi ∆= and LNhw /=∆ , where h is the depth to which we
solve Laplace’s equation, taken to be greater than or equal to one wavelength of the wavemaker. We allow the possibility that
;xw ∆≠∆ see equation (23). We must solve Laplace’s equation, (27) over this grid. The semi-discretized approximation of this potential
),,( tws jiΦ is written as )(, tjiΦ ;,,1,0 Mi =
;,,1,0 LNj = .0≥t
To implement the wavemaker, we assume the sum of the two sinusoids in (10), for 0≤x , ∞<<−∞ t , and their associated (linear) velocity potentials, (11) and dispersion relation (12). The LHS boundary condition is implemented as a Dirichlet boundary condition.
4 Copyright © 2004 by ASME
The Dirichlet wavemaker:
[ ])sin(][
)sin()(
222
2
111
1,0
2
1
tckek
gA
tckek
gAt
j
j
wk
wkj
−
−=Φ
(31)
where, i
i k
gc = is the velocity of each wave, 2,1=i ., and
;,,1,0 LNj = 0≥t . The horizontal bottom boundary
condition, given by (4), will be automatically satisfied by the relaxation method, as we shall soon see. The boundary condition at the free surface, ),,( txz η=
0 ;1,,2,1 : )(0, ≥−=Φ tMiti (32)
is given by the kinematic and dynamic boundary conditions, (28) and (29), respectively, and involve integration over time. This will be discussed in the next section. The RHS boundary condition for the numerical beach is incorporated in the calculation of the free surface potential over the damping zone, and involves modification of equation (29) to,
.0)(sec)(2
1 222 =++
∂Φ∂−
∂∂
+∂∂ ηµχθχχ
gzxt
(33)
Therefore, the potential should vanish everywhere at the wall, and as a convenience, we may employ equation (13) to get the RHS boundary condition; that is,
0. t;,,1,0 ),()( ),1(, ≥=Φ=Φ − LjMjM Njtt (34)
Even with a uniform x∆ , the curvilinear grid will have non-constant .s∆ Therefore, to apply finite difference formulas to discretize Laplace's equation (27), we will need appropriate finite difference operators. Clearly, applying these discrete operators to Laplace's equation (27), is best done using computer algebra, and the authors utilized Maple. We refer to the resulting equation, as the "semi-discretized Laplace equation". The details are to be given elsewhere. Consider its application over any s -coordinate curve for fixed w in the interior of the domain. That is, we evaluate the semi-discretized Laplacian at ),( ji ws , to calculate
;1,,2,1 )(, −=Φ Mitji 1,,2,1 −= LNj (fixed); .0≥t This
will involve nine potentials, namely:
1−js -coordinate curve: )(1,1 tji −−Φ )(1, tji −Φ )(1,1 tji −+Φ
js -coordinate curve: )(,1 tji−Φ )(, tjiΦ )(,1 tji+Φ
1+js -coordinate curve: )(1,1 tji +−Φ )(1, tji +Φ )(1,1 tji ++Φ
See Figure 1. The idea behind waveform relaxation is to follow the lead of separation of variables methods and assume that the potential can be written as,
jiji tt φχ ~)()(, =Φ ,,,1,0 Mi = 0 ,,1,0 ; ≥= tNj L (35)
where )(tiχ is the potential at ),( txz η= (i.e. the free surface
potential) and therefore satisfies Equations (28) and (33), and since we are considering the infinite depth basin,
Lkw
j Nje j ,,2,1,0 ,~
==φ (36)
where k is a representative wavenumber, which we take “based” of the wavenumbers of the wavemaker given in Equation (11). For example k can be taken to be the average of
the two wavenumbers, 21 , kk . Therefore the nine potentials that
occur in the discretized Laplace equation become,
1−js -coord.curve: 11~
)( −− ji t φχ 1~
)( −ji t φχ 11
~)( −+ ji t φχ
js -coord.curve: )(,1 tji−Φ )(, tjiΦ )(,1 tji +Φ
1+js -coord.curve: 11~
)( +− ji t φχ 1~
)( +ji t φχ 11~
)( ++ ji t φχ
So clearly the only "unknowns" are the three potentials along the js -coordinate curve, and we have successfully decoupled
the system along dimensional lines, by "relaxing" the w dependence. Note that, in light of equation (36), the bottom boundary condition (4) is automatically satisfied. Since the
terms ji t φχ ~ ),( , ;,,2,1 ;,,1,0 LNjMi == 0≥t are
known and become incorporated into the vector fsb&
, this gives
rise to the iterated matrix equation
( ) ( ) )()()(1
tbtbtA wm
q
fsq
j
&&&
+=Φ+
, (37)
for Maxqq ,...,1,0= where ( )0fsb&
is obtained from equation (36)
and all subsequent ( )qfsb&
, ,...,1,0=q are obtained from
( )qjΦ&
. We refer to this equation as the semi-discretized
Laplace equation, since )(, tjiΦ is discrete over space but
continuous over time. In general, two types of iteration schemes are possible, Gauss-Jacobi in which "new values" are used only after a complete iteration has been completed and Gauss-Seidel in which "new values" are used as soon as they are available. Clearly, the Gauss-Jacobi method is fully parallel, but presently we employ the Gauss-Seidel method, since it generally converges faster. The matrix A is a square sparse tridiagonal matrix of dimension )1()1( −×− MM and
[ ] ,)(,),(),()( ,1,2,1T
jMjjj tttt −ΦΦΦ=Φ
&
where 0 t;,,2,1 ≥= LNj .
Note that the matrix A does not depend on the iteration parameter q , and therefore must be inverted only once per time step. The terms ,)(,0 tjΦ where 0 ;,,2,1,0 ≥= tNj L are known
since they are given by the wavemaker condition (31)
incorporated into the vector ).(tbwm
&
The inversion of the complete ×−− )1)(1( LNM )1)(1( −− LNM system reduces to )1( −LN
inversions of the )1()1( −×− MM linear system given by (37).
Furthermore, if w∆ is a constant, then the matrix A is identical for any s -coordinate curve, and the Gaussian elimination
5 Copyright © 2004 by ASME
operations necessary to invert each system, using LU-decomposition, need be performed only once. This effectively reduces the dimension of the problem by one. Such reduction in computation is essential, since the matrix A changes at each time-step, and therefore the system must be inverted at each time-step.
TIME INTEGRATION Equation (37) must be fully discretized to complete the numerical model. For T a positive finite real number, we solve our initial-value problem over the time interval [ ]T,0 . For a
natural number TN , we let TNTt /=∆ , which gives the
sequence of time steps , nt where ,tntn ∆= TNn ,,1,0 = .
The approximation of )( ),( i, nnji tt χΦ and the free surface
elevation ),( ni txη is denoted by ,,ni
nji χΦ and n
iη ,
respectively, for ,,1,0 Mi = , LNj ,,1,0 = , TNn ,,1,0 = .
To solve this linear system we require the vector )( nfs tb&
which
contains the terms niχ , ,1,,2,1 −= Mi ,,,2,1 TNn =
which involve integration over time. The differential Equations
(28) and (33) are used to generate the approximations niη and
niχ , respectively for ,1,,2,1 −= Mi TNn ,,2,1 = . In
both cases, we use a four-point one-sided finite difference to
approximate z∂Φ∂
. To calculate nix
)(∂∂η
and nix
)(∂∂χ
, we spline
niη and n
iχ , respectively, with a piecewise polynomial of
degree three, and then calculate it's derivative in closed form.
This also allows us to evaluate )(tan 1 ni
ni x∂
∂= − ηθ .
Since the fluid in the basin is initially at rest, we have the initial
conditions, 00 =iη , ,00 =iχ and ,00, =Φ ji where
,1,,2,1 −= Mi .1,,2,1 −= LNj A fourth-order (explicit)
Runge-Kutta method gets the additional starting values niη ,
niχ , where 3,2,1=n and .1,,2,1 −= Mi At each of these
time-steps, the current value for niχ and the fully discretized
version of equation (37) is used to get ,n
jiΦ ,3,2,1=n
1,,2,1 −= Mi and .1,,2,1 −= LNj
After this initial phase, for TNn ,,6,5,4 = , ,1,,2,1 −= Mi
,1,,2,1 −= LNj an Adams fourth-order predictor-corrector
method is used. Using the starting values from the Runge-Kutta method, a fourth-order Adams-Bashford method is used as the
predictor to advance niχ . Then, using this current value and
the fully discretized version of equation (37), we advance
,n
jiΦ . Next a fourth-order Adams-Moulton corrector method
is used, with these values for niχ and ,
njiΦ , to advance
niη . This is followed by another application of the Adams-
Moulton corrector method to advance niχ using this
“corrected” value of niη . See Baddour and Parsons (2003).
TEST PROBLEMS AND CONCLUSIONS Monochromatic Wavemaker To test and illustrate the “high order” of nonlinearity of the model in the monochromatic case, we consider the following test problem: Starting from “calm water”, we input a monochromatic sinusoid at 0=x , with moderately high steepness, 06.0=S . The horizontal extent of the tank is )(0.1 mL = , and we input a total
of 4 waves including a damping zone of 2 wavelengths. This sets the wavelength of the wavemaker )(25.0 m=λ , with
corresponding wavenumber )/1(1.25 mk = , and by using the
linear dispersion relation, Equation (12), the angular frequency )/1(7.15 s=ω . The discretization parameters are
/15z ,25/ λλ =∆=∆x , with 10, == Maxqh λ , and
40/Tt =∆ . In Figure (2), we plot the free surface elevation at a fixed location in the tank, namely )1.0( Lη after 38 periods, after
which the model became unstable. Note that the instability is not visible in the “generation” over time at a fixed location in space. It is the “propagation” over space that is subject to this instability. See Baddour and Parsons (2003). Figure (3) gives the spectrum of this time series using the Matlab “fft”, and employing a Gaussian window to minimize leakage. In fact all the spectrum plots we present in this paper utilizes this windowing technique. See Briggs and Henson (1995). Note that the model is generating (at least) the first five orders in the classical Stokes expansion. Bichromatic Wavemaker In this test, we employ a bichromatic wavemaker, again, starting from “calm water”, with steepness, 03.0,015.0 21 == SS . The
horizontal extent of the tank is )(0.1 mL = , and the wavelengths
of the wavemaker are )(25.01 m=λ and )(125.02 m=λ , where
the damping zone is 22λ . This sets the wavenumbers of the
wavemaker, to be )/1(1.251 mk = , and )/1(3.502 mk = ,
respectively, and using the linear dispersion relation Equation (12), the corresponding angular frequencies are )/1(7.151 s=ω ,
and )/1(3.222 s=ω , respectively. The discretization parameters
are /20z ,40/ λλ =∆=∆x , with 10, == Maxqh λ , and
50/Tt =∆ . Figure (4) shows the spectrum after 100 periods of simulation, relative to the second frequency, 2ω . There are “at
least 25 interactions” visible in the plot. This is further evidence of the nonlinear interactions generated by the model. Since the wavemakers are simple linear sinusoids, all these nonlinear interactions are the result of nonlinearities in the problem; recall that we are solving the full nonlinear problem. Clearly, it is these interactions that can give rise to resonance.
6 Copyright © 2004 by ASME
Resonance in the Bichromatic Wavemaker Case We are interested in showing resonance in the “generation over time” at a fixed location in the tank, namely at Lx 1.0= .
Therefore, for these tests, we define 2
1
ωω
=r , where 1ω and 2ω
are the angular frequencies of the bichromatic wavemaker employed, and the corresponding wavenumbers can be calculated using the linear dispersion relation, given by Equation (12). The resonant interactions occur when 2≈r , since the interactions given by )1(221 +=+ rωωω and
)12(2 221 −=− rωωω , coincide.
For the first test we take 5.1=r , and show that the solution does not show resonance. Let )(0.1 mL = , and
/10z ,20/ λλ =∆=∆x , with 10, == Maxqh λ , and 30/Tt =∆ .
We choose 7.151 =ω 5.102 =ω , and we set the steepness
03.01 =S and 009.02 =S , respectively. Figure (5) shows the
spectrum after 50 periods of simulation, relative to the first frequency, 1ω . Note that the components given by
2.2621 =+ ωω , and 9.202 21 =− ωω are generated, but of
course, do not coincide, and therefore both are significantly smaller than 1ω and 2ω . This represents a non-resonant
solution. In the second test, we set 0.2=r , and show that we get resonance. In particular, the setup is identical to the first test, above, except 7.151 =ω 9.72 =ω , and we set the steepness
03.01 =S and 015.02 =S , respectively. To add emphasis to the
resonance phenomenon, we give the spectrum of the free surface elevation of the wavemaker (at 0=x ) in Figure (6). The spectrum of the free surface elevation generated by the model at (at Lx 1.0= ), is shown in Figure (7). Note that we get a very significant resonant effect at
2121 26.23 ωωωωω −==+=r . In fact the amplitude of this
resonant wave is slightly greater than the amplitude generated for 1ω , and slightly less than the amplitude generated for 2ω .
Comparing these amplitudes with the corresponding amplitudes for the wavemaker, given in Figure (6), it is clear that most of the “energy” that goes into creating this resonant component comes from the energy associated with 2ω , since the amplitude
generated by the model for 2ω is significantly less than the
amplitude corresponding to 2ω in the wavemaker. The
corresponding comparison for 1ω shows that the resonant effect
on this component is much smaller, but nonzero, nonetheless. We can calculate an “effective steepness” for this resonant wave, using the linear dispersion relation, given by Equation (12). We calculate the wavenumber of this resonant wave to be
)/1(8.56 mk r = , and measure its amplitude to be
)(00368.0 mAr = , giving a steepness of 067.0=rS , which is
more than twice the maximum steepness of the component waves in the wavemaker, 03.01 =S . This warns us of an
instability, which we call “resonance induced instability” caused by the increase of steepness of the resonant wave, relative to the steepness of the waves in the wavemaker. In fact, in the next
test, it was necessary to choose the steepness of one of the waves in the wavemaker, very small. An initial attempt at this run with a larger steepness wavemaker resulted in almost immediate overflow. The final test shows that resonance decreases, but is not lost, if we increase the value of r to 0.3=r . As before, the setup is identical to the first test, above, except 7.151 =ω 24.52 =ω ,
and we set the steepnesses 03.01 =S and 003.02 =S ,
respectively. See Figure (8) for the spectrum of the free surface elevation (at Lx 1.0= ), after 50 periods. Note that we get a strong interaction at 05.262 21 =−ωω , which does not coincide
with a weaker interaction at 94.2021 =+ωω .
We conclude with a general observation of the value of using the full nonlinear model over the simpler linear model. Although, the higher order components are smaller than the first order linear component, for each wave generated by the model, for example, see Figure (4), the nonlinear interaction of these components can be quite significant. In this paper, we have considered cases in which the discrepancies between the linear and nonlinear models were extreme. Clearly, in any of these runs, the difference between a linear and our nonlinear model is quite significant. In fact it should be remembered that this resonance is a purely nonlinear effect, and does not occur in a purely linear solution. Since these resonance effects can be quite dramatic, it is important to be aware of them when solving ocean engineering problems. They can be a nuisance in some cases, and a hazard in other cases, but they should never be blindly ignored. This requires a fully nonlinear approach to the modeling.
REFERENCES Baddour, RE and Parsons, W (2003) A comparison of Dirichlet and Neumann wavemakers for the numerical generation and propagation of transient long-crested surface waves, Proc 22nd Int. Conf. on Offshore Mechanics and Arctic Engineering, OMAE, Cancun, Mexico, paper 37281. See also Jour. OMAE, accepted for publication. Briggs, WL and Henson, VE (1995). The DFT, An owner’s manual for the discrete Fourier transform, SIAM, Philadelphia. Debnath, L. (1994) Nonlinear water waves, Academic Press, New York. Kaplan, W (1993).Advanced calculus, Fourth Edition, Addison-Wesley, Reading, Massachusetts. Parsons, W. and Baddour, R. E. (2002), The generation and propagation of transient long-crested surface waves using a waveform relaxation method, Proceedings Advances in Fluid Mechanics 2002, Wessex Institute of Technology Press. Parsons, W. and Baddour, R. E. (2003), A numerical wave tank for the generation and propagation of bi-chromatic nonlinear long-crested surface waves, Proceedings Fluid Structure Interaction 2003, Wessex Institute of Technology Press.
7 Copyright © 2004 by ASME
Tsai, W. and Yue, D. K. (1996), Computation of nonlinear free-surface flows, Annual Review of Fluid Mechanics, Vol. 28, Annual Reviews Inc., Ca. The MathWorks Inc. (1996), MATLAB 5.2 user’s guide, Natick, Mass. Wehausen, J.V. and Laitone, E.V. (1960) Surface Waves. In Handbuch der Physik Ed. S. Flugge, Vol. IX, Fluid Dynamics III, Springer-Verlag, Berlin.
ACKNOWLEDGMENTS To College of the North Atlantic, Newfoundland, Canada, for giving Wade Parsons a one year sabbatical. To Natural Sciences and Engineering Research Council of Canada and the National Research Council’s Institute for Marine Dynamics, Newfoundland, Canada, for awarding Wade Parsons a Visiting Fellowship.
jw
1+jw
1−jw
0=i i 1+i1−i
LN1−ix
ix 1+ix
zN
0=j
j
1−j
1+j
FS
ix∆ 1+∆ ix
jz∆
1+∆ jz
),( txz η=
x
z
0s
js
Figure 1. Free surface coordinate system
Figure 2. Free surface elevation at x=0.1L as a function of time in (s), for a monochromatic wavemaker at x=0.0.
Figure 3. Spectrum of the time series of the surface elevation given in Figure 2 at x=0.1L
8 Copyright © 2004 by ASME
Figure 4. Spectrum of free surface elevation at x=0.1L after 100 periods of simulation relative to 2ω , with a bichromatic
wavemaker: )/1(7.151 s=ω and )/1(3.222 s=ω
Figure 5. Spectrum of the free surface elevation at x=0.1L after 50 periods of simulation relative to 1ω , for a
bichromatic wavemaker: )/1(7.151 s=ω and
)/1(5.102 s=ω , 5.1=r
Figure 6. Spectrum of the free surface elevation of the wavemaker at x=0 : )/1(7.151 s=ω and )/1(9.72 s=ω ,
0.2=r
Figure 7. Spectrum of the free surface elevation at x=0.1L generated by the model : )/1(7.151 s=ω and )/1(9.72 s=ω ,
0.2=r
9 Copyright © 2004 by ASME
Figure 8. Spectrum of the free surface elevation at x=0.1L after 50 periods of simulation : )/1(7.151 s=ω and
)/1(24.52 s=ω ; 0.3=r
The Generation and Propagation of Deep Water Multichromatic Nonlinear Long-crested Surface Waves
W . Parsons 1 and R. E. Baddour 2
1 College of the North Atlantic, Box 1150, Ridge Road, St John’s, NF, A1C 6L8, Canada.
2 National research Council - Canada
Institute for Ocean technology St. John’s, NL, Canada
ABSTRACT. A two dimensional rectangular basin containing an incompressible inviscid homogeneous fluid, initially at rest with a horizontal free surface of finite extent is considered to generate and propagate nonlinear, long-crested waves. A depth profile for the potential is assumed, giving us a waveform relaxation method, thereby drastically reducing the computational cost of solving Laplace’s equation. A multichromatic stochastic wavemaker employing a Dirichlet type boundary condition is applied, with the latter following a standard wave energy spectrum. Laplace's equation is solved using a non-orthogonal boundary fitted curvilinear coordinate system, which follows the free surface, and the full nonlinear kinematic and dynamic free surface boundary conditions are employed. The behavior of this model is studied using standard signal processing tools and a discussion of the results is given. In addition, statistical properties of the output of the model are related to the corresponding statistical properties of the input. KEY WORDS: Nonlinear-waves; regular; irregular waves; nonlinear free surface conditions; multichromatic; non-orthogonal coordinates; waveform relaxation. INTRODUCTION A two-dimensional rectangular basin containing an incompressible inviscid homogeneous fluid initially at rest with a horizontal free surface of finite extent is considered to generate and propagate long-crested waves. On the left vertical boundary a wavemaker is positioned while at the right hand side is the radiation boundary. The same numerical beach is used as in Parsons and Baddour (2002), and a depth profile for the potential is assumed, giving us a waveform relaxation method, and thereby drastically reducing the computational cost of solving Laplace's equation. A multichromatic deterministic and stochastic wavemaker employing a Dirichlet type boundary condition is applied, see Baddour and Parsons (2003) for a comparison of Dirichlet and Neumann monochromatic wavemakers and Parsons and Baddour (2003) for the bichromatic Dirichlet wavemaker model. This method can easily be extended to the case of a wavemaker utilizing a Neumann type boundary condition. Laplace's equation is solved using a non-orthogonal boundary fitted curvilinear coordinate system, which
follows the free surface. This gives a more realistic "continuity condition", since it involves the entire fluid domain. Also, the full nonlinear kinematic and dynamic free surface boundary conditions are employed. Although, these ideas can be extended to finite depth tanks, this requires a rewriting of the coordinate system, so we restrict our attention to the infinite depth case. The extension of these ideas to the finite depth case will be presented elsewhere. To place our work in perspective, see the review by Tsai and Yue (1996) "of the recent advances in computations of incompressible flows involving a fully nonlinear free surface". The present work advances the field of volume-discretization methods, using finite differences, by applying the relatively new method of waveform relaxation to reduce the "computational dimension" of the problem. This puts volume-discretization methods on a similar footing with boundary-discretization methods, which reduce the dimension of the problem by solving it on the boundary. The purpose of the present paper is to show that the method presented could be applied to the multichromatic case of excitation with both deterministic and statistical wavemakers. A complete review of the literature concerning sea-states reproduction is outside the scope of the present objective. PROBLEM FORMULATION Consider a two-dimensional rectangular basin containing an incompressible inviscid homogenHRXVIOXLGRIFRQVWDQWGHQVLW\ LQD
Cartesian coordinate system (x,z), with the origin at the still water level, the positive z-axis pointing upwards. The horizontal extent of the basin is Lx ≤≤0 , so 0>L is the length of the tank. The depth of the basin is infinite. See Figure 1 for a diagram of the numerical wave tank and the rectangular coordinate system configuration. This is an initial-value problem, since the surface is initially at rest for time, 0<t and is disturbed at 0=t giving rise to surface waves. See Debnath (1994).
If ),,( tzxΦ is the velocity potential and ),( txη is the free surface
elevation, the problem is defined by the following equations, where g is the acceleration due to gravity. The conservation of mass equation is Laplace's equation:
0 ,0 ),,( 02
2
2
2
≥≤≤≤≤∞−=∂
Φ∂+∂
Φ∂tLxtxz
zxη (1)
For 0≥t , the full nonlinear kinematic and dynamic free surface
boundary conditions are given, respectively, by:
0 =∂Φ∂−
∂Φ∂
∂∂+
∂∂
zxxt
ηη (2)
and 0])()[(2
1 22 =+∂Φ∂+
∂Φ∂+
∂Φ∂ ηg
zxt (3)
on ),( txz η= , and
z as 0 −∞→→∂Φ∂z
(4)
Introducing the following function: )),,(,(),( ttxxtx ηχ Φ= (5)
which represents the potential on the free surface, we have:
tztt ∂
∂∂Φ∂+
∂Φ∂=
∂∂ ηχ
(6)
implying:
tztt ∂
∂∂Φ∂−
∂∂=
∂Φ∂ ηχ
(7)
Also, we have that
. xzxx ∂
∂∂Φ∂+
∂Φ∂=
∂∂ ηχ
(8)
See Wehausen and Laitone (1960). The initial conditions are given by:
)()0,( (x);)/1()0,0,( );()/1()0,0,( 0 00 xxxxx ηηρρ =Φ=ΦΦ=Φ••
where )()/1( 0 xΦρ and )()/1( 0 x•Φρ represent the given free
surface impulse and )(0 xη the initial displacement, where the “dot”
represents a time derivative. Since we are considering a fluid in which the initial velocities are zero and the initial free surface is at rest at
0=z , we take
Lx0for 0)( ;0)( ;0)( 000 ≤≤==Φ=Φ•
xxx η (9)
Therefore we are left with two lateral boundary conditions for 0≥t at 0=x and Lx = , which we call the LHS lateral boundary condition
and the RHS lateral boundary condition, respectively. The LHS boundary condition involves a wavemaker, which we assume is the superposition of a finite number of sinusoid waves as follows:
,)sin()cos(),(1
∑Ω
=
−−−=n
nnnnnn txkBtxkAtx ωωη (10)
∞<<∞≤<∞ t- 0,x- ; where nA and nB is the amplitude, n
k the
wave number and nω the angukar frequency of each wave, for
.,...,2,1 Ω=n . Using standard water wave mechanics, the associated velocity linear potential is given by
∑Ω
=
−+
−=Φ
1 )cos(
)sin(
),,(n
nnzk
n
n
nnzk
n
n
txkegB
txkegA
tzxn
n
ωω
ωω
(11)
for 0 t0,x- ≥≤<∞ and ),0( tz η≤<−∞ , where the dispersion relation is
gknn =ω (12)
The free surface elevation is at the wavemaker is then given by
∑Ω
=
+=1
)sin()cos(),0(n
nnnn tBtAt ωωη (13)
and its associated potential is
∑Ω
=
+−=Φ1
)cos()sin(),,0(n
nzk
n
nn
zk
n
n tegB
tegA
tz nn ωω
ωω
(14)
It is prudent to point that it is not necessary, but only a convenience and for completeness to make the assumption of the existence of the sinusoids in Equation (10), since we are only inputting the potential given by Equation (14). Therefore conditions outside the interval [ ]L,0
have no effect on the solution. This progressive wave at 0=x and 0≥t is the sole source of the disturbance that gives rise to water
waves in the initially calm basin. If we assume a wall at x=L so the RHS lateral boundary condition becomes
0),,( =
∂Φ∂
x
tzL for η≤<−∞ z , 0≥t (15)
we will get reflections from waves going from the wavemaker at 0=x toward the wall. This distracts from the objective of the present model and is to be avoided. To absorb these waves we place a "numerical beach" on the RHS. This is accomplished by including a damping term
LQWKHG\QDPLFIUHHVXUIDFHERXQGDU\FRQGLWLRQHTXDWion. See Parsons and Baddour (2002) for details. THE LAPLACIAN ON THE FREE SURFACE COORDINATE SYSTEM We define the free surface coordinate system (FSCS), ),( ws , where
0≥s is the arclength along the free surface ),( txz η= and 0≤w is the vertical depth to any point in the fluid from the free surface. This is a non-orthogonal moving coordinate system that follows the free surface. Clearly, the coordinates satisfy the following equations (over the complete fluid domain),
Lx0 , ),(
10
2≤≤
∂∂+= ∫ τ
ττη
dt
sx
, (16)
0. wL,x0 ),,( ≤≤≤−= txzw η (17)
If we define the unit vector s along the arclength, and therefore tangent
to the free surface ),( txz η= , then
k sin i cosˆ θθ +=s (18)
k tan- s secˆ θθ=i (19) where
.tan 1
∂∂
= −
x
ηθ (20)
The arclength along the coordinate curve s is given by
dx 12
∂∂+=
xds
η (21)
or dx secθ=ds . (22)
Clearly we have the following differential operator formulas that allow us to transform easily between rectangular coordinates ),( zx and free
surface coordinate system ),( ws :
w tan-
s sec
∂∂
∂∂=
∂∂ θθx
(23)
z sin cos
∂∂+
∂∂=
∂∂ θθ
xs (24)
wz ∂∂=
∂∂
. (25)
The Laplacian for this non-orthogonal coordinate system can be constructed from the second-order covariant fundamental metric tensor
=
1 sin
sin 1
θθ J (26)
where its associated second-order contravariant tensor is given by
sec sectan
sectan sec 2
2
−
−=
θθθ
θθθ J (27)
where clearly θ2cos) == GHWJJ .
The Laplacian is given by the following tensor equation
2,1, ,1 =
∂Φ∂
∂∂=∆Φ λαα
αλλ xx
JJJ
(28)
where . x, 21 wsx == See Kaplan (1993). We get
]sectan
tansec[ sec
∂Φ∂+
∂Φ∂−
∂∂+
+
∂Φ∂−
∂Φ∂
∂∂=∆Φ
wsw
wss
θθ
θθθ (29)
We note that equation (29) can be derived directly, by carrying out the differentiations in (1), using the differential operators (23) and (24) and recalling that the unit vector (19) is non-constant and must be differentiated. THE DYNAMIC AND KINEMATIC BOUNDARY CONDITIONS ON THE FREE SURFACE COORDINATE SYSTEM Using equation (8) and equation (20), the kinematic free surface boundary condition (KFSBC) given by equation (2), becomes
0sectan 2 =∂Φ∂−
∂∂+
∂∂
zxtθχθη
. (30)
The dynamic free surface boundary condition (DFSBC), given by equation (3), can be rewritten using equations (7) and equation (8). After some algebra and using equation (20) and the KFSCB, equation (30), we get the following final form:
.0sec2
12
22
=+
∂Φ∂−
∂∂
+∂∂ ηθχχ
gzxt
(31)
where all values in (30) and (31) are evaluated on the free surface. Since the free surface coordinate system tracks the free surface, these values are readily obtained. The above derivatives allow for the integration in time t . SPATIAL APPROXIMATION AND WAVEFORM RELAXATION In this paper, we employ a curvilinear grid that follows the free surface. In light of equation (23), this grid is taken to be ( ) ,,,0,1j ;,,1,0 ,, Lji NMiws == (32)
where xss iii ∆+= −− 11 secθ , and
ixx
i x=
∂∂
+=2
1secηθ ,
where we take ,00 =s and wjw j ∆−= , where ,/ MLx =∆
so ,xixi ∆= and LNhw /=∆ , where h is the depth to which we
solve Laplace’s equation, taken to be greater than or equal to one wavelength of the wavemaker. We allow the possibility that ;xw ∆≠∆ see equation (25). We must solve Laplace's equation, (29) over this grid. The semi-discretized approximation of this potential ),,( tws jiΦ is
written as )(, tjiΦ ;,,1,0 Mi = ;,,1,0 LNj = .0≥t
To implement the wavemaker, we assume the sinusoid (10), for 0≤x , ∞<<−∞ t , and associated (linear) velocity potential, (11) and
dispersion relation (12). The LHS boundary condition can be implemented as a Dirichlet or Neumann condition. We refer to these as the Dirichlet wavemaker and Neumann wavemaker, respectively; see Baddour and Parsons (2003). In this paper we restrict our attention to the Dirichlet condition, and assure the reader that the extension to the Neumann wavemaker is stright forward. In light of Equation (14) the Dirichlet wavemaker is then given by: The Dirichlet wavemaker:
∑Ω
=+
+
+
−
=Φ1
)),0((
)),0((
,0
)cos(
)sin(
)(n n
wtk
n
n
nwtk
n
n
j
tegB
tegA
tjn
jn
ωω
ωω
η
η
(33)
for ;,,1,0 LNj = 0≥t and where nA and nB is the amplitude, nk
the wave number and nω the angular frequency of each component
wave, Ω= ,...,2,1n , and we recall the linear dispersion relation (12). There is no limitation in the present model that precludes modeling a physical wavemaker. The present choice given in (33) is a convenient way that would allow us to compare with the standard linear theory and to the Stokes expansion expression for a nonlinear free surface. See Baddour and Parsons (2003) and Parsons and Baddour (2003). The construction of a coordinate system with a moving left boundary is in progress. Together with (33) a ramping function over two mean periods is utilized to satisfy the initial conditions of the problem as well as to minimize any initial transient impulses in the free surface elevation and velocities. The horizontal bottom boundary condition, given by (4), will be automatically satisfied by the relaxation method, as we shall soon see. The boundary condition at the free surface, ),,( txz η=
0 ;1,,2,1 : )(0, ≥−=Φ tMiti (34)
is given by the kinematic and dynamic boundary conditions, (30) and (31), respectively, and involve integration over time. This will be discussed in the next section. The RHS boundary condition for the numerical beach is incorporated in the calculation of the free surface potential over the damping zone, and involves modification of equation (31) to,
.0)(sec)(2
1 222 =++
∂Φ∂−
∂∂
+∂∂ ηµχθχχ
gzxt
(35)
see Parsons and Baddour (2002). Therefore, the potential should vanish everywhere at the wall, and as a convenience, we may employ equation (15) to get the RHS boundary condition; that is,
0. t;,,1,0 ),()( ),1(, ≥=Φ=Φ − LjMjM Njtt (36)
(YHQZLWKDXQLIRUP [WKHFXUYLOLQHar grid will have non-FRQVWDQW V
Therefore, to apply finite difference formulas to discretize Laplace’s equation (29), we will need appropriate finite difference operators. Clearly, applying these discrete operators to Laplace’s equation (29), is best done using computer algebra, and the authors utilized Maple. We refer to the resulting equation, as the "semi-discretized Laplace equation". The details are to be given elsewhere. Consider it’s application over any s -coordinate curve for fixed w in the interior of the domain. That is, we evaluate the semi-discretized Laplacian at ),( ji ws , to calculate ;1,,2,1 )(, −=Φ Mitji
1,,2,1 −= LNj (fixed); .0≥t This will involve nine potentials,
namely:
1−js -coordinate curve: )(1,1 tji −−Φ )(1, tji −Φ )(1,1 tji −+Φ
js -coordinate curve: )(,1 tji−Φ )(, tjiΦ )(,1 tji+Φ
1+js -coordinate curve: )(1,1 tji +−Φ )(1, tji +Φ )(1,1 tji ++Φ
See Baddour and Parsons (2003). The idea behind waveform relaxation (WR) is to follow the lead of separation of variables methods and assume that the potential can be written as,
)(~
)()( ,, ttt jiiji φχ=Φ ,,,2,1 Mi = 0 ,,1,0 ; ≥= tNj L (37)
where )(tiχ is the potential at ),( txz η= (i.e. the free surface
potential) and therefore satisfies equations (30) and (35), and since we are considering the infinite depth basin,
,~ *
,jwk
ji e=φ ,,,2,1 Mi = 0 ,,1,0 ; ≥= tNj L (38)
where *k is the representative wavenumber, which for the monochromatic case we take as the wavenumber of the wavemaker; see equation (10) and Baddour and Parsons (2003). In the multichromatic case, however, there are many wavenumbers to choose from. In this
case we use an iterative approach and choose some ’’reasonable’’ *k ,
where *k [ ]ΩΩ∈ kkkkkk ,...,,max,,...,,min 2121 and use equation (38)
as a ’’first guess’’ only. We then iterate to converge on the depth profile
),,()(~
, txt ijji γφ = ,,...,2,1 Mi = 0 ,,1,0 ; ≥= tNj L (39)
where .0,,...,2,1,1),(0, ≥== tMitxiiγ Therefore the nine potentials
that occur in the discretized Laplace equation become,
1−js -coord.curve: 1,11~
)( −−− jii t φχ 1,~
)( −jii t φχ 1,11~
)( −++ jii t φχ
js -coord.curve: )(,1 tji−Φ )(, tjiΦ )(,1 tji+Φ
1+js -coord.curve: 1,11~
)( +−− jii t φχ 1~
)( +ji t φχ 1,11~
)( +++ jii t φχ
Clearly the ’’unknowns’’ are the three potentials along the js -coordinate
curve and we have successfully decoupled the system along dimensional lines, by ’’relaxing’’ the w -dependence. Note that, in light of equations (38) and (39), the bottom boundary condition (4) is
automatically satisfied. Since the terms ,)(~
),( , tt jii φχ ,,...,2,1 Mi =
0 ,,1,0 ; ≥= tNj L are known and become incorporated into the
vector fsb&
this semi-discretized Laplace equation gives rise to the
iterated matrix equation
( ) ( ) )()()(1
tbtbtA wmq
fsq
j
&&&
+=Φ+
, (40)
for ,...,1,0=q where ( )0fsb&
is obtained from equation (38) and all
subsequent ( )qfsb&
, ,...,1,0=q are obtained ( )qjΦ&
. IN general, two
types of iteration schemes are possible, Gauss-Jacobi in which "new values" are used only after a complete iteration has been completed and Gauss-Seidel in which "new values" are used as soon as they are available. Clearly, the Gauss-Jacobi method is fully parallel, but presently we employ the Gauss-Seidel method, since it generally converges faster. The matrix A is a square sparse tri-diagonal matrix of dimension )1()1( −×− MM , and
[ ]TjMjjj tttt )(),...,(),()( ,1,2,1 −ΦΦΦ=Φ&
, where 1,,2,1 −= LNj ;
.0≥t Note that the matrix A does not depend on the iteration parameter q , and therefore must be inverted only once per time step.
The terms )(,0 tjΦ where 1,,2,1 −= LNj , .0≥t are known
since they are given by the wavemaker condition (33) and are
incorporated into the vector )(tbwm
&
.
The inversion of the complete )1)(1()1)(1( −−×−− LL NMNM system
reduces to )1( −LN inversions of the )1()1( −×− MM linear system
given by (40). Furthermore, if w∆ is a constant, then the matrix A is identical for any js -coordinate curve, and the Gaussian elimination
operations necessary to invert each system, using LU-decomposition need be performed only once. This effectively reduces the dimension of the problem by one. Concerning speed our method belongs to the class of very efficient iterative methods for solving large systems. This becomes very significant when we move to 3-D. There is a vast literature on techniques for accelerating these schemes in the area of iterative matrix methods and are not discussed here. TIME INTEGRATION Equation (40) must be fully discretized to complete the numerical model. For T a positive finite real number, we solve our initial-value problem over the time interval [ ]T,0 . For a natural number TN , we let
TNTt =∆ , which gives the sequence of time steps nt , where
tntn ∆= , TNn ,...,1,0= . The approximation of )(, nji tΦ , )( ni tχ and
the frees surface ),( ni txη is denoted by jin
,Φ , )( ni tχ and niη ,
respectively, for ,,1,0 Mi = , LNj ,,1,0 = , TNn ,,1,0 = .
To solve this linear system we require the vector )( nfs tb&
which
contains the terms niχ , ,1,,2,1 −= Mi ,,,2,1 TNn = which
involve integration over time. The differential equations (30) and (35)
are used to generate the approximations niη and n
iχ , respectively
for ,1,,2,1 −= Mi TNn ,,2,1 = . In both cases, we use a four-
point one-sided finite difference to approximate z∂Φ∂
. To calculate
nix
)(∂∂η
and nix
)(∂∂χ
, we spline niη and n
iχ , respectively, with a
piecewise polynomial of degree three, and then calculate it’s derivative
in closed form. This also allows us to evaluate )(tan 1 ni
ni x∂
∂= − ηθ .
Since the fluid in the basin is initially at rest, we have the initial
conditions, 00 =iη , ,00 =iχ and ,00, =Φ ji where
,1,,2,1 −= Mi .1,,2,1 −= LNj A fourth-order (explicit)
Runge-Kutta method gets the additional starting values niη , n
iχ ,
where 3,2,1=n and .1,,2,1 −= Mi At each of these time-steps,
the current value for niχ and the fully discretized version of
equation (41) is used to get ,n
jiΦ ,3,2,1=n 1,,2,1 −= Mi and
.1,,2,1 −= LNj
After this initial phase, for TNn ,,6,5,4 = , ,1,,2,1 −= Mi
,1,,2,1 −= LNj an Adams fourth-order predictor-corrector
method is used. Using the starting values from the Runge-Kutta method, a fourth-order Adams-Bashford method is used as the
predictor to advance niχ . Then, using this current value and the
fully discretized version of equation (40), we advance ,n
jiΦ . Next a
fourth-order Adams-Moulton corrector method is used, with these
values for niχ and ,
njiΦ , to advance n
iη . This is followed by
another application of the Adams-Moulton corrector method to advance
niχ using this “corrected” value of n
iη . See Baddour and Parsons
(2003) for a flow chart. IRREGULAR WAVES AND OCEAN SPECTRA The wavemaker (33), can be rewritten in terms of a sum of "phase shifted cosine functions", as follows:
∑Ω
=
+=Φ1
0,0 )cos()(n
nnnj tt εωφ (41)
where the coefficients are
220 nnn BA +=φ and the phase angles are
= −
n
nn B
A1tanε , (42)
.,...,2,1 Ω=n .
To analyze the irregular wave generated when 2≥n in (41), we recall the following results; see Goda (2000). An energy spectrum
)( nS ωη corresponding to any irregular time history can be defined,
WKH DUHD RI HDFK UHFWDQJOH RI ZLGWK WKH IUHTXHQF\ LQWHUYDO LV
proportional to the energy attributed to that frequency band and represented by the corresponding single sinusoid wave component. The amplitude of each component sine wave in (41) is therefore given by
δωωφ η )(20 nn S= (m) (43)
It is also necessary to specify the phase angles nε , and we will choose
them randomly. This will will give different time histories for different (otherwise identical) runs, but where the energy spectrum will be the same. Note that if the spectrum is defined at equally spaced frequencies it will be necessary to interpolate the spectral ordinates at the randomly spaced frequencies. It was found that the randomly chosen frequencies case does not reproduce the spectrum very well. We choose to use the uniform frequencies discretization of the spectrum with random phases
nε in equation (41). Note that the period of the simulation 1T before
repetition can be increased by choosing larger number of frequencies, i.e. larger Ω . The advantage of this is that the spectrum is well preserved as discussed below. In the test problem to follow, we consider the multichromatic, uniform frequencies, random phases case and document how the model performs.
Open Ocean Conditions: ITTC has adopted the Bretschneider spectrum as the standard wave energy spectrum to represent the conditions which occur in the open ocean. It is called the ITTC two-parameter spectrum, and is defined by
−=
45exp)(
ωωωη
BASB m²/(rad/s) (44)
where 4
23175.172
T
HA = 42 / sm and
4
691
TB = 4−s
Clearly, the two parameters that are input to the wavemaker, (41), are
the significant wave height 31H and the average period T , and we
define T
πω 2= , (45)
and using the linear dispersion relation (12),
g
k2ω= , (46)
and so k
πλ 2= . (47)
In analogy to regular waves, we can define the significant steepness:
λ31
31H
S = . (48)
In all cases, we choose a real number 1>WMα , and set
ωαω *WM=Ω (49)
For the uniform frequencies case mentioned above, we choose
1
*
11
1 +Ω=
+Ω=
Ω−== ΩΩ ωαωωωδωω WM . (50)
We then define the period
TTWM
+Ω==αω
π 12
11 , (51)
and note that the resulting irregular wave will repeat with this period, since this is the lowest frequency in the spectrum and all the frequencies are harmonically related. If we also choose the simulation time
1*TNTS = , (51)
for some natural number N , then all frequencies will be harmonically related to the "window" [ ]ST,0 , used to evaluate the discrete Fourier
transform. One big advantage of this setup is that the problem of "leakage" in the discrete Fourier transform can be eliminated for the wavemaker. Clearly, this can never be strictly attained for the output from the model. However, even this inevitable leakage can be reduced by using special types of windows. This will be discussed in the next section concerning the tests. Also, see Briggs and Henson (1995). TEST PROBLEM AND CONCLUSIONS We present the following tests designed to show that our model can generate and propagate multichromatic waves with reasonable accuracy. This will be judged by comparing the mean period and significant waveheight, and the spectrum of the free surface elevation at a fixed point inside the tank, with that for the wavemaker. In our first test, we take 29=Ω uniformly distributed frequencies,
with 3=WMα , and the parameters, sT 0.4= and mH 25.031 = in
the spectrum (44). Therefore, m0.25=λ , and we set mL 0.250= , with "10 waves" in the tank (not counting the beach), at steady-state, which gives the required mean wavelength, and significant steepness,
01.031 =S . The discretization parameters are 40λ=∆x ,
20λ=∆z , and 200Tt =∆ , and we plot the numerical
approximation of the free surface elevation LxTx S ≤≤0:),(η after
10.10100 TTTS == . See Figure (2). In Figure (2) the above
parameters are shown in the figure for convenience. In Figures (3) and (6) we plot the free surface elevation of the wavemaker for this full simulation and it’s spectrum, respectively. Note that the wavemaker is periodic with exactly 10 periods during the simulation and the spectrum given by (47) is well reproduced, see Figure (6). Recall the discussion above on leakage of the Fourier transform. From Figure (3) the measured mean period is 3.98 s with significant wave height 0.244 m. Again, we choose to generate periodic irregular waves to show that the model was behaving as expected. To generate longer irregular nonperiodic waves we need to take more frequencies. Similarly, we plot the free surface elevation Tn NntL ,...,2,1:),*1.0( =η generated by the model for this full
simulation and it’s spectrum, respectively in Figures (4) and (7). The measured mean period is found to be 3.26 s with significant wave height 0.246 m. Unlike the wavemaker which is exactly periodic, the model is "almost periodic" with 10 "periods", as expected. More interestingly, the spectrum given by (47) is reasonably well reproduced over most of the frequency range, with a couple of exceptions. We get a little "overshoot" around the peak frequency, and the lower frequency modes are lacking. This latter deficiency can be alleviated by taking a longer tank and a longer simulation time. The most important difference, however is that we get a bimodal spectrum, with the appearance of a second peak in the spectrum. We believe this is due to the nonlinear generation of higher order modes and their interactions. See Parsons and Baddour (2003), where this is reported for the case of a monochromatic and bichromatic wavemaker. Again, recalling the discussion above on leakage, we point out that a "Gaussian" window was employed to reduce leakage. See Figure (5). In the second test, we take 299=Ω uniformly distributed frequencies chosen for the above spectrum. Conditions are set such that the simulation will not be periodic for 400 seconds. If a longer nonperiodic simulation is required more frequencies may be chosen. The free surface elevation at the wavemaker is shown in Figure (8) with the resulting free surface elevation generated by the model at the fixed location in the tank at Lx 1.0= given in Figure (9). The spectrum of the wave maker is shown in Figure (10) along with the Bretschneider spectrum. However, we purposely ran the model for more than an integral number of periods to show the effect of leakage. The spectrum of the free surface elevation generated by the model again at
Lx 1.0= is shown after 740 sec in Figure (11). Note that we still get the "bimodal" spectrum as in the 29 frequencies case shown in Figure (7). Fig (12) shows the free surface elevation elevation after 1.85 1T .
FUTURE WORK We showed in this paper that the waveform relaxation method that we successfully applied to the generation and propagation of monochromatic transient long-crested nonlinear surface gravity waves on a two-dimensional rectangular basin of finite extent and infinite depth containing an incompressible inviscid homogeneous fluid initially at rest with a horizontal free surface can be extended to the multichromatic case. This involved the use of the full nonlinear dynamic and kinematic boundary conditions, and the solution of Laplace’s equation on a non-orthogonal curvilinear coordinate system that follows the free surface. Despite the fact that this required the Laplace matrix to be inverted at each time-step, we get an efficient accurate method. Clearly, this efficiency becomes more significant for larger problems. In particular for short-crested surface waves, our problem becomes fully three dimensional and such efficiency becomes
essential. In addition, we can employ the dimensional decoupling of waveform relaxation methods to reduce the three dimensional problem to two iterated two dimensional problems, Parsons (1999). This dynamic Laplace solver permits us to generate and propagate transient multichromatic waves, which are then completely absorbed by the same numerical beach given in Parsons and Baddour (2002). Finally, we stated in the introduction of this paper, that these ideas can be extended to the finite depth case as well as to physical wavemakers at the LHS boundary. In Parsons and Baddour (2002) we used the appropriate "hyperbolic profile" to facilitate the extension on a Cartesian coordinate system for the linear proble. Using methods from Riemannian geometry, as extension of the free surface coordinate system used in this paper is being developed to construct a curvilinear coordinate system for a domain with finite depth and irregular shaped dynamic bottom and side boundaries. ACKNOWLEDGEMENTS To College of the North Atlantic, Newfoundland, Canada, for giving Wade Parsons a one year sabbatical. To Natural Sciences and Engineering Research Council of Canada and the National Research Council’s Institute for Ocean Technology, Newfoundland, Canada, for awarding Wade Parsons a Visiting Fellowship. REFERENCES Baddour, RE and Parsons, W (2003). ’’A comparison of Dirichlet and
Neumann wavemakers for the numerical generation and propagation of transient long-crested surface waves,’’ Proc 22nd Int Conf on Offshore Mechanics and Arctic Engineering, OMAE, Cancun, Mexico, paper 37281. Also in Jour. of OMAE, accepted for publication.
Briggs, WL and Henson, VE (1995). The DFT, An owner’s manual for the discrete Fourier transform, SIAM, Philadelphia. Debnath, L (1994). Nonlinear water waves, Academic Press, New York Goda, Y (2000). Random seas and design of marine structures, World Scientific, Singapore. Kaplan, W (1993). Advanced calculus, Fourth Edition, Addison-
Wesley, Reading, Massachusetts. Parsons, W (1999). “Waveform relaxation methods for Volterra
integro-differential equations,” Ph.D. Thesis, MUN, St. John's, NL. Parsons, W and Baddour, RE (2003). “A numerical wavetank for the
generation and propagation of bichromatic nonlinear long-crested surface waves”, Fluid Structure Interaction, Proc of Second Int Conf, Wessex Institute of Technology Press, pp 407-420 .
Parsons, W and Baddour, RE (2002). “The generation and propagation
of transient long-crested surface waves using a waveform relaxation method, Advances in Fluid Mechanics, Proc Int Conf, Wessex Institute of Technology Press, pp 683-693.
Tsai, W and Yue, DK (1996). Computation of free-surface flows,
Annual Review of Fluid Mechanics, Vol. 28, Annual Reviews Inc., Ca.
Wehausen, JV and Laitone, EV (1960). Encyclopedia of Physics, Vol.
IX, Fluid Dynamics III, Springer-Verlag, Berlin.
Figure 1 System configuration and coordinate system
Figure 2 Free surface elevation after 100 periods
Figure 3 Free surface elevation in (m) of wavemaker at x=0
Figure 4 Free surface elevation in (m) at x=0.1L
Figure5 Gaussian window of free surface elevation at x=0.1 L
Figure 6 FFT (eta at wavemaker x=0) with Guassian window
)),,(,(),( ttxxtx ηχ Φ= xzt ∂∂−
∂Φ∂=
∂∂ χθθη
tansec 2
∂Φ∂−
∂∂−−=
∂∂ 2
22
sec2
1
zxg
tθχηχ
02
2
2
2
=∂
Φ∂+∂
Φ∂zx
Figure 7 FFT (eta at x=0.1L) with Guassian window
Figure 8 Free surface elevation at x=0
Figure 9 Free surface elevation at x=0.1L
Figure 10 Spectrum of surface elevation at x=0
Figure 11 Spectrum of Surface elevation at x=0.1L
Figure 12 Free Surface Elevation after 1.85 T1