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
1
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 longcrested regular and irregular seas
Appendix 2: Useful wave & tidal power/energy links
Appendix 3: Recent publications from IOT
2
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.nrccnrc.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.
3
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
4
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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VXVWDLQDEOHVRXUFHVRISRZHU
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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
5
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
6
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
2.2
2.3
2.4
2.5
2.6
2.7
2.8
Origins and generation of wind waves
Wave energy resource distribution
Description and modification of waves
Real sea characteristics
Data about waves
Marine currents origins
Marine currents resource
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
7
(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
8
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).
9
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.
10
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:
ishoaling and refraction
iidiffraction 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.
11
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)
12
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).
13
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)
10
15
20
25
30
40
50
10 (16.7km)
34 (57.8km)
75 (127.5km)
160 (272.1km)
280 (476.2km)
710 (1207.5km)
1420 (2415.1km)
2.4
6
10
16
23
42
69
0.9 (0.27m)
2.5 (0.76m)
5 (1.52m)
9 (2.74m)
14 (4.26m)
28 (8.52m)
48 (14.63m)
Hs
Significant
height (ft)
H10
Average of
the highest
10% (ft)
Period where
most of
energy is
concentrated
(sec)
1.4 (0.43m)
3.5 (1.07m)
8 (2.44m)
14 (4.27m)
22 (6.71m)
44 (13.41m)
78 (23.77m)
1.8 (0.55m)
5 (1.53m)
10 (3.05m)
18 (5.50m)
28 (8.54m)
57 (17.4m)
99 (30.17m)
4
6
8
10
12
16
20
14
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).
15
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
16
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.
17
Table 2.2 Wave processes incorporated in shallow water computational models
(source after Southgate 1987, 1993)
Computational model
Forward
tracking
Ray models
Backtracking
Refraction
Grid models
Hyperbolic
refractiondiffraction
Parabolic
refractiondiffraction
Elliptic
refractiondiffraction
Nonlinear
Internal
diffraction
Not modeled,
but numerical
smoothing by
ray averaging
Wave processes modeled
External
Reflections
Bottom
diffraction
friction
Only for
special
Yes
Yes (1)
situations
Wave
breaking
Yes (1,2)
Same as
above
Same as
above
No
Yes
No
No
No (3)
Yes
No, apart
from check at
inshore points
Yes (2)
Yes
Yes
No
Yes
Yes (2)
Yes
Yes, but
difficult in
general
No (3)
Yes
Yes (2)
Yes
Yes
Yes
Yes
Yes
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)
Extent of sea
Bathymetry
Type of
Number of
Offshore
Computational
(special
area
coastline
inshore points
wave
model
requirements)
conditions
Ray models
Forward
tracking
Regular depth
variation;
poor for shoal
systems
Backtracking
No
Unlimited
Refraction
Regular depth
variation
Unlimited
Grid models
Hyperbolic
refractiondiffraction
Parabolic
refractiondiffraction
Elliptic
refractiondiffraction
Nonlinear
Same as
above
Unlimited
Any
Any, except
where depth
variation is
important
Reasonably
straight coast
facing open sea
Covering entire
modeled area
Regular
wave (1)
One
Full
spectrum
Covering entire
modeled area
Regular
wave (1)
Regular
wave (1)
Unlimited
Same as above
Same as above
Limited to a
few kilometers
at most
Same as above
Same as above
No
Regular
wave (1)
Same as above
Same as above
No
Same as above
Regular
wave (1)
No
Same as above
Same as above
Same as above
Surface
elevation
18
Note: All models require reasonable gentle depth variations
(1) Full spectrum S(f θ ) 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).
19
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
20
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.
21
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
3.1
3.2
3.3
3.4
Ocean energy harnessing systems
Wave energy harnessing technology
Wave energy developments and activities
Ocean currents energy technology
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
22
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.
23
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)
24
Table 3.1 Classification of wave energy devices processes
Source Brooke (2003) with reference to Hagerman (1995) of Fig. 3.1
Primary
Location
System
Onshore
Nearshore
Offshore
Wave Energy Conversion Process
of
#
1.1
1.2
1.3
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
Reference
to
Hagerman
Fig 3.1
Description
Fixed oscillating water column
Reservoir filled by wave surge
Pivoting flaps
Freely floating oscillating water column
Moored floating oscillating water column
Bottom mounted oscillating water column
Reservoir filled by direct wave action
Flexible pressure device
Submerged buoyant absorber with sea floor reaction
Heaving float in bottom mounted or moored floating caisson
Floating articulated cylinder with mutual force reaction
Freely heaving float with sea floor reaction point
Freely heaving float with mutual force reaction
Contouring float with mutual force reaction
Contouring float with sea floor reaction point
Pitching float with mutual force reaction
Flexible bag with spine reaction point
Submerged pulsating-volume body with sea floor reaction point
Reservoir filled by direct wave action
2
1
4
3
3
2
1
11
12
5
6
7
8
9
10
11
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:
25
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
"
"
"
"
"
"
"
"
"
"
"
"
Reservoir filled
by wave surge
Pivoting flaps
Pivoting flaps
Australia, Port Kembla
China, Dawanshan 1
China, Shanwei
India, Vizhinjam,
Japan, Sanze
Japan, Sakata port
Japan, Kujukuri-Cho
Japan, Haramachi
Mexico
Norway, Toftoy
Portugal, Pico, Azores
UK, Isle of Islay
UK, Isle of Islay
Norway, Toftoy
Breakwater
Shoreline
Adv. Stage Develop.
Operational
Adv. Stage Develop.
Operational
Operational
Operational
Operational
Operational
Operational, sea water pump
Operational
Operational
Operational
Operational
Operational
Japan, Muroran Port
Japan, Wakasa Bay
Harbour
Shoreline gully
Breakwater
Breakwater
Cliff wall
Rocky gully
Shoreline gully
Cliff face
Gully
and
interior bay
Seawall
Seawall
Operational
Operational
Nearshore wave energy systems
Energy
Process
Conversion
Freely floating OWC
"
"
Fixed floating OWC
"
Device name
Country and location
Status
Kaimei floating
Mighty Whale
Sperbuoy
China, various
Japan, various
Japan, Yura
Japan, Gokasho Bay
UK, Plymouth
Op., navigation buoy
Operational
Operational
Opearational
Adv. Stage Dev.
26
"
Bottom mounted OWC
Reservoir filled by wave
Flexible pres. device
Submerged
buoyant
absorber sea-floor RP
Heaving float in bottommounted
or
moored
caisson
Floating
articulated
cylinder with inertial RP
Shim wind-wave syst.
Osprey
Floating wave power
SEA Clam
-
South Korea
UK, Thurso
Sweden
UK
Adv. Stage Dev.
Adv Stage Dev.
Adv Stage Dev.
-
ConWEC
Norway
Adv. Stage Dev.
Pelamis
UK, Shetland/Isle of Islay
Adv. Stage Dev.
Device name
Country and location
Status
OPT Wave Power
System
Danish Heaving buoy
Phase-controlled
Power Buoy
DELBUOY
Hosepump
USA, Australia, Portalnd,
Australia
Denmark, Hanstholm
Norway, Trondheim Fjord
Adv. Stage Dev.
Adv. Stage Dev.
Adv. Stage Dev.
US
Sweden
Operational
Adv. Stage Dev.
IPS Buoy
McCabe Wave Pump
Adv. Stage Dev.
Adv. Stage Dev.
Contouring Raft
Contouring Raft
-
Sweden
Ireland, Shannon River
Estuary
US
Japan, Iriomote Island
Okinawa
UK
US
-
-
-
-
-
Offshore wave energy systems
Energy
Conversion
Process
Freely heaving float with
sea floor RP
"
"
"
Freely heaving float with
inertial RP
"
Contouring float with
inertial RP
"
Contouring float with seafloor RP
"
"
Pitching float with inertial
reaction point
Flexible bag with spine
reaction point
Submerged
pulsatingvolume body with seafloor RP
Wave Energy Module
Kaiyo Jack-up Rig
Archimedes
Swing
Wave
Netherlands, Viano
Castello, Portugal
do
Adv. Stage Dev.
Adv. Stage Dev.
Adv. Stage dev.
Adv. Stagre Dev.
-
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.
27
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
28
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:
P = 1 2 ρ A v3 CP
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.
29
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).
30
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 costoptimised.
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.
31
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
32
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
33
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
34
4. Transfer of power systems
4.1
4.2
4.3
4.4
Interfaces
Storage of energy
Electric power generation and conversion
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
35
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
36
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.
37
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 multipole 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.
38
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).
39
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
5.2
5.3
5.4
Factors affecting the evaluation of costs
Capital and Operating costs of Ocean Energy systems
Generation costs of wave and current energy
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.
40
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
Unit Costs ($/kW)
O&M & Insurance Costs ($/kW/year)
Availability
Annual Output (kWh/kW)
Shoreline
1800-2100
30-45
94-96
2000-2500
Near-shore
1500-1800
42-48
93-96
2200-2500
Offshore
1500-3000
30-90
90-95
3000-4000
41
5.3 Generation costs of wave energy
For any application the determining factors are the cost per kW of the delivered
system or the cost per kW-hour of the delivered energy whether in the form of electrical
energy production or in kind, McCormick and Kraemer 2002. Thorpe (1998, 2000) has
compared the cost efficiencies of various wave energy conversion technologies. As
mentioned above he has divided the systems into shoreline, nearshore and offshore
systems. He found that the more recent designs both shoreline and offshore devices can
produce electricity at less than 0.1 US$ 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.
3
3
4
2
4
2
3
5
Cost
estimates
1
1
2
5
4
6
5
6
6
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.
42
Evolution of electricity cost for onshore and nearshore
systems. Source: Thorpe (2000)
Cost of electricity (p/kWh)
NEL OWC
Wavegen’s OSPREY
Wavegen’s LIMPET
20
15
10
5
0
1980
1985
1990
1995
2000
2005
Design year
Figure 5.2 Examples of Evolution of electricity cost for onshore and nearshore systems. Source
Thorpe 2000.
100
Edinburgh Duck
Bristol Cylinder
10
SEA Clam
PS Frog
Mcabe Wave Pump
1
05
98
00
20
20
19
95
90
92
19
19
86
19
19
82
85
19
19
80
Sloped IPS Buoy
19
Cost of electricity (p/kWh)
Evolution of electricity costs for offshore 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
43
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.
44
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.
45
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
Solar thermal;parabolic trough
Solar Thermal; parabolic dish
Solar thermal; Central receiver
Photovoltaic
Photovoltaic, thin film
Photovoltaic, multiple thin film
Wind turbine (6-9 m/s wind spd.)
Wind turbine (6-9 m/s wind spd.)
Wind turbine (6-9 m/s wind spd.)
Combined wind-wave system
Combined wind-wave system
Location
New Mexico, USA
New Mexico, USA
New Mexico, USA
New Mexico, USA
New Mexico, USA
New Mexico, USA
South Korea
UK
Date
2020
2020
2020
2020
2020
2020
1995
2000
2010-2020
1995
1995-1999
Cost (c/kWh)
7.5-11
6.0-10
5.0-9.0
5.0-14
6.0-10
4.0-7.0
3.6-6.5
3.0-5.5
2.0-4.5
11.0-18.0
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
USA 1994
Japan 1993
UK
1995
UK
1995
IEA 1994
IEA 1994
IEA 1994
DTI 1996
DTI 1996
Price to industry
Price to households
Price to households
Price to industry
Price to households
4.7
8.4
14-24
6.4
8-13.4
46
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.
47
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.nrccnrc.gc.ca/
48
7.
7.1
7.2
7.3
7.4
Timeline Proposal
Ocean energy influential events
A Timeline
Scientific, technical, societal and economic challenges
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.
49
7.2
A Timeline
A T imeline
2004
National Coordinator,
E valuator & T es t Centre
(NR C IOT )
S horeline model
S horeline prototype
demons trator
Nears hore model
Nears hore prototype
demons trator
2005
T echnology R es earch P rogramme
& T echnology Development
P has e I
2006
2006
2009
P has e II
Hydrodynamics
Hydrographic map: waves ,
curr ents and ice
Hydrogen technology
Deployment, recover y
P ower technology: take off
convers ion
control
trans mis s ion
trans formation
E conomics
F eas ibility and market analys is
R eliability, Maintenance
techniques
Navigation
A T imeline
Offshore model
Offshore prototype
demonstrator
2008
2010
20yy
Phase III
Nears hore and Offshore s ites
pros pecting
Weather Prediction
Power take off prediction
Power s torage
Market analys is
S urvivability
Mooring
Civil engineering, F oundations
S ynergy with offs hore
indus try
…
T echnology Development
50
Ocean E nergy T echnology
Network
NR C IOT
Indus try, Government ,
Academia
2004
National Coordinator, E valuator & T es t
Centre (NRC IOT )
NL WAVE POWE R?
Working Group on
Policy S tatement on
Ocean E nergy Generation
R esource assessment
Prediction of wind, wave and current
resource; pres ence of ice
L onger term weather forecast?
Ocean energy prospecting.
Working Group on
environmental issues
output
R efined S trategy on
Impact Ass essment
Public E ducation and
Acceptability
Market E xpansion
S ignificant contribution to x% of
T otal E nergy Generation
Working Group on R &D & T echnology
development
2009
T echnologies needed
Data Acquisition T echniques
Cable Network, Grid connection
S urface treatments, coatings, Corrosion
Non-electrical outputs: hydrogen
Desalination, others?
E nvironmental Impact R esearch
20yy
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
51
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.
52
- 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
53
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.
54
8.
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geis/publications/reports/etwan/execsummary/exec-summary.asp
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.
59
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.
60
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.
61
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 12231226.
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)
62
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 6HHIR
r 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:
H s = 4m10 / 2
where m0 is the zero-th spectral moment, the n-th moment being defined as
m0 = ∫
2π
∫
0
∞
f n S ( f ,θ ) df dθ
0
The convention of considering S as the distribution of the variance of the sea surface
elevation has been used. Thus the total variance is m0 (m2 ) and the actual energy per
unit area is ρgm0 . Also the significant wave height is defined (followed convention) as
H s = 4m0 . (See Mollison 1986 or other references)
The energy mean period is defined as
m
Te = −1
m0
and the peak period T p is the inverse of the peak frequency f p which corresponds to
1/ 2
the highest spectral density. For sea states with only one wave system T p is useful in
providing a measure of the period of the waves with the highest energy density. The
significant wave period Ts 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
P = ρg ∫
2π
0
∫
∞
0
c g ( f , h) S ( f ,θ ) df dθ
where ρ is the water density. In deep water the group velocity which is the velocity at
which the energy propagates is given by
cg =
g
4πf
63
Thus the wave power is given by
ρg 2
P=
4π
2π
∞
0
0
∫ ∫
f
−1
ρg 2
S ( f ,θ ) df dθ =
m−1
4π
which can be expressed in terms of H s and Te as
P ≅ 0.5 H s Te
2
When H s is expressed in meters and Te 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
64
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%20
Wave%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/
65
National Review documents
A brief review of Wave Energy : http://www.researchinnovation.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.irishenergy.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_ta
nk.html
Appendix 3: Recent publications from IOT
66
Proceedings of OMAE ’ 04
rd
23 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
W. Parsons
College of the North Atlantic
Ridge road,
Box 1150, A1C 6L8
St John’s, NL, CANADA
R. E. Baddour
National Research Council -Canada
Institute for Ocean Technology
P.O.Box 12093, A1B 3T5
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 ( x, z ) , with the
1
Copyright © 2004 by ASME
origin at the still water level, the positive z-axis pointing
upwards. The horizontal extent of the basin is 0 ≤ x ≤ L , so
L > 0 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, t < 0 and is disturbed at t ≥ 0 giving rise to
surface waves. See Debnath (1994). If the Φ( x, z , t ) is the
velocity potential and η ( x, t ) 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:
∂2Φ ∂2Φ
(1)
+
= 0 − ∞ ≤ z ≤ η( x, t), 0 ≤ x ≤ L, t ≥ 0
∂x 2 ∂z 2
For t ≥ 0, the full nonlinear kinematic and dynamic free surface
boundary conditions are given, respectively, by:
∂η ∂η ∂Φ ∂Φ
+
−
=0
(2)
∂t ∂x ∂x ∂z
and
∂Φ 1 ∂Φ 2 ∂Φ 2
+ [(
) +(
) ] + gη = 0
∂t 2 ∂x
∂z
(3)
on z = η ( x, t ) , and
∂Φ
→ 0 as z → −∞
(4)
∂z
Introducing the following function:
χ ( x, t ) = Φ( x, η ( x, t ), t )
(5)
which represents the potential on the free surface, we have:
∂χ ∂Φ ∂Φ ∂η
=
+
(6)
∂t
∂t
∂z ∂t
implying:
∂Φ ∂χ ∂Φ ∂η
=
−
(7)
∂t
∂t ∂z ∂t
Also, we have that
∂χ ∂Φ ∂Φ ∂η
=
+
.
(8)
∂x
∂x
∂z ∂x
See Wehausen and Laitone (1960).
The initial conditions are given by:
•
•
Φ(x,0,0) = (1/ ρ)Φ0(x);Φ(x,0,0) = (1/ ρ)Φ0 (x); η(x,0) = η0(x)
•
where (1 / ρ )Φ 0 (x) and (1 / ρ ) Φ 0 ( x) represent the given free
surface impulse and η 0 ( x) 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 z = 0 , we take
•
Φ 0 ( x) = 0; Φ 0 ( x) = 0; η 0 ( x) = 0 for 0 ≤ x ≤ L
(9)
Therefore we are left with two lateral boundary conditions for
t ≥ 0 at x = 0 and x = L , 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
η i ( x, t ) = Ai cos(k i x − ω i t ), - ∞ < x ≤ 0, - ∞ < t < ∞
(10)
where, Ai = H i / 2 is the amplitude of each wave,
k i is the
wavenumber and ω i is the angular frequency of each wave;
i = 1,2 . The steepness of the wave is S i = k i Ai / π . 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
A g
(11)
Φ i ( x , z , t ) = i e k i z sin( k i x − ω i t ),
ωi
for
x = 0, t ≥ 0, - ∞ < z ≤ 0,
where the linear dispersion relation is
ω i = k i g , i = 1,2 .
(12)
Note that in the implementation of the wavemaker algorithm
we assume the validity of (11) over the range −∞ < z ≤ η (0, t ) 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 [0, L] have no effect on the
solution. This progressive wave at x = 0 and t ≥ 0 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
∂Φ ( L, z , t )
= 0, - ∞ < z ≤ η , t ≥ 0,
(13)
∂x
we will get reflections from waves going from the wavemaker
at x = 0 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:
 3xˆ 2 2 xˆ 3 
µ ( xˆ ) = ν  2 − 3  kg , where L1 is the length of the
L
L1 
 1
damping zone, k is the representative wave number, which will
be discussed later, and ν is the normalized damping coefficient.
Note that x̂ is a local variable where, xˆ = 0, when x = L − L1 ,
and xˆ = L1 , when x = L .
This numerical beach is found to be most effective when,
0.8 ≤ ν ≤ 1.0 .
Clearly, setting ν = 0 is equivalent to removing the beach.
Also, L1 is usually taken to be about two representative
wavelengths, L1 ≈ 2λ , where λ is the wavelength associated
2
Copyright © 2004 by ASME
with the representative wavenumber, and is given by
λ = 2π / k . See Parsons and Baddour (2002).
THE LAPLACIAN ON
COORDINATE SYSTEM
THE
FREE
SURFACE
We define the Free Surface Coordinate System (FSCS), ( s, w) ,
where s ≥ 0 is the arclength along the free surface
z = η ( x, t ) and w ≤ 0 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),
s=
 ∂η (τ , t ) 
1+ 

 ∂τ 
0
∫
x
w = z − η ( x, t ),
2
dτ , 0 ≤ x ≤ L ,
0 ≤ x ≤ L,
w ≤ 0.
(14)
 ∂η 
ds = 1 + 

 ∂x 
or
ds = sec θ dx .
where clearly
J
= GHWJ ) = cos 2 θ .
∂ 
∂Φ
∂Φ 
− tan θ
 sec θ
+
∂s 
∂s
∂w 
∂ 
∂Φ
∂Φ 
+
+ sec θ
 − tan θ
]
∂w 
∂s
∂w 
(27)
THE DYNAMIC AND KINEMATIC BOUNDARY
CONDITIONS ON THE FREE SURFACE COORDINATE
SYSTEM
SPATIAL
APPROXIMATION
RELAXATION
(18)
∂
∂
∂
= sec θ
- tanθ
(21)
∂x
∂s
∂w
∂
∂
∂
= cosθ
+ sinθ
(22)
∂s
∂x
∂z
∂
∂
=
.
(23)
∂z ∂w
The Laplacian for this non-orthogonal coordinate system can be
constructed from the second-order covariant fundamental metric
tensor
sin θ 
1

(24)
J
= 
1
 sin θ

where its associated second-order contravariant tensor is given
by

sec 2 θ
− tan θ sec θ 
J
=
(25)
 − tan θ sec θ
sec 2 θ 

[
(20)
(17)
Clearly we have the following differential operator formulas
that allow us to transform easily between rectangular
coordinates ( x, z ) and free surface coordinate system ( s, w) :
∆Φ = sec θ
x 2 = w. See Kaplan (1993). We get
(19)
(16)
2
dx
where x 1 = s,
(26)
Using equation (8) and equation (18), the kinematic free surface
boundary condition (KFSBC) given by equation (2), becomes
∂η
∂χ
∂Φ
+ tan θ
− sec 2 θ
=0.
(28)
∂t
∂x
∂z
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:
2
2
∂χ 1  ∂χ 
 ∂Φ  
+ 
(29)
 − sec 2 θ 
  + gη = 0.
∂t 2  ∂x 
 ∂z  

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 .
(15)
If we define the unit vector ŝ along the arclength, and therefore
tangent to the free surface z = η ( x, t ) , then
sˆ = cos θ î + sinθ k̂
iˆ = sec θ ŝ - tanθ k̂
where
 ∂η 
θ = tan −1  .
 ∂x 
The arclength along the coordinate curve s is given by
The Laplacian is given by the following tensor equation
1 ∂ 
∂Φ 
∆Φ =
 JJαλ α , α , λ = 1,2
λ
∂x 
J ∂x

AND
WAVEFORM
In this paper, we employ a curvilinear grid that follows the free
surface. In light of equation (18), this grid is taken to be
s i , w j , i = 0,1, , M ; j = 0,1, , N L ,
(30)
{(
)
where s i = s i −1 + sec θ i −1 ∆x , and
}
2 

 ∂η  
sec θ i =  1 + 
,


 ∂x  

 x = xi
where we take s 0 = 0, and w j = − j∆w , where ∆x = L / M , so
x i = i∆x, and ∆w = h / N L , 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
∆w ≠ ∆x; see equation (23). We must solve Laplace’s equation,
(27) over this grid.
The semi-discretized approximation of this potential
Φ( s i , w j , t )
is written as Φ i , j (t )
i = 0,1, , M ;
j = 0,1, , N L ; t ≥ 0.
To implement the wavemaker, we assume the sum of the two
sinusoids in (10), for x ≤ 0 , −∞ < t < ∞ , and their associated
(linear) velocity potentials, (11) and dispersion relation (12).
The LHS boundary condition is implemented as a Dirichlet
boundary condition.
3
Copyright © 2004 by ASME
~
kw
φj =e j,
The Dirichlet wavemaker:
Φ 0, j (t ) = − A1
− A2
[ ]
g k1w j
e
sin(k1 c1t )
k1
(31)
g k2 w j
[e
] sin( k 2 c 2 t )
k2
where, ci =
g
ki
is the velocity of each wave, i = 1,2 ., and
t ≥ 0}
(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,
∂χ 1  ∂χ 2
∂Φ 
+ ( ) − sec 2 θ ( ) 2  + µχ + gη = 0.
(33)
∂t 2  ∂x
∂z 
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,
Φ M , j (t ) = Φ ( M −1), j (t ), j = 0,1, , N L ; t ≥ 0.
(34)
Even with a uniform ∆x , the curvilinear grid will have nonconstant ∆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 semidiscretized
Laplacian
at
(s i , w j ) ,
to
calculate
Φ i , j (t ) i = 1,2, , M − 1; j = 1,2, , N L − 1 (fixed); t ≥ 0. This
will involve nine potentials, namely:
s j −1 -coordinate curve: Φ i −1, j −1 (t )
Φ i , j −1 (t )
Φi +1, j −1 (t )
s j -coordinate curve: Φ i−1, j (t )
Φ i , j (t )
Φ i +1, j (t )
s j +1 -coordinate curve: Φ i −1, j +1 (t )
Φ i , j +1 (t )
Φi +1, j +1(t )
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,
~
Φ i , j (t ) = χ i (t )φ j i = 0,1, , M , j = 0,1, , N L; t ≥ 0 (35)
where χ i (t ) is the potential at z = η ( x, t ) (i.e. the free surface
potential) and therefore satisfies Equations (28) and (33), and
since we are considering the infinite depth basin,
(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, k1 , k 2 . Therefore the nine potentials that
occur in the discretized Laplace equation become,
j = 0,1, , N L ; t ≥ 0 . 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, z = η ( x, t ),
{Φ i,0 (t ) : i = 1,2, , M − 1;
j = 0,1,2, , N L
~
s j −1 -coord.curve: χ i −1 (t )φ j −1
~
χ i (t )φ j −1
~
χ i +1 (t )φ j −1
s j -coord.curve: Φ i−1, j (t )
Φ i , j (t )
Φ i + 1, j ( t )
~
χ i +1 ( t )φ j +1
~
~
χ i (t )φ j +1
s j +1 -coord.curve: χ i −1 (t )φ j +1
So clearly the only "unknowns" are the three potentials along
the s j -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
χ i (t ), φ j , i = 0,1, , M ; j = 1,2, , N L ; t ≥ 0 are
&
known and become incorporated into the vector b fs , this gives
{
}
rise to the iterated matrix equation
( )
( )
( )
q +1
&
&
q
( t ) = b fs
( t ) + b wm ( t ) ,
(37)
& 0
for q = 0,1,..., qMax where b fs is obtained from equation (36)
&
A Φ
j
(b& ) ,
q
and all subsequent
(Φ& j )q .
fs
q = 0,1,..., are obtained from
We refer to this equation as the semi-discretized
Laplace equation, since Φ i , j (t )
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 ( M − 1) × ( M − 1) and
&
Φ j (t ) = Φ1, j (t ), Φ 2, j (t ), , Φ M −1, j (t ) T ,
[
]
where j = 1,2, , N L ; t ≥ 0 .
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 Φ 0, j (t ) , where j = 0,1,2, , N L ; t ≥ 0 are known
{
}
since they are given by the wavemaker condition (31)
&
incorporated into the vector bwm (t ).
The
inversion
of
the
complete
( M − 1)( N L − 1) × ( M − 1)( N L − 1) system reduces to ( N L − 1)
inversions of the ( M − 1) × ( M − 1) 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
4
Copyright © 2004 by ASME
operations necessary to invert each system, using LUdecomposition, 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.
Moulton corrector method to advance {χ in } using this
“corrected” value of {η in } . See Baddour and Parsons (2003).
TEST PROBLEMS AND CONCLUSIONS
Monochromatic Wavemaker
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 [0, T ] . For a
natural number N T , we let ∆t = T / N T , which gives the
sequence of time steps {t n }, where t n = n∆t , n = 0,1, , N T .
The approximation of Φ i , j (t n ), χ i (t n ) and the free surface
elevation
η ( xi , t n )
is
denoted
by
Φ in, j , χ in and
η in ,
respectively, for i = 0,1, , M , j = 0,1, , N L , n = 0,1, , N T .
&
To solve this linear system we require the vector b fs (t n ) which
contains the terms {χ in } , i = 1,2, , M − 1, n = 1,2, , N T ,
which involve integration over time. The differential Equations
(28) and (33) are used to generate the approximations {η in } and
{χ in } , respectively for i = 1,2, , M − 1, n = 1,2, , N T . In
both cases, we use a four-point one-sided finite difference to
∂η
∂χ
∂Φ
approximate
. To calculate ( ) in and ( ) in , we spline
∂z
∂x
∂x
{η in } and {χ in } , respectively, with a piecewise polynomial of
degree three, and then calculate it's derivative in closed form.
∂η
This also allows us to evaluate {θ in = tan −1 ( ) in } .
∂x
Since the fluid in the basin is initially at rest, we have the initial
conditions,
η i0 = 0 ,
χ i0 = 0,
and
Φ i0, j = 0, where
i = 1,2, , M − 1, j = 1,2, , N L − 1. A fourth-order (explicit)
Runge-Kutta method gets the additional starting values {η in } ,
{χ in } ,
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 x = 0 , with moderately high steepness, S = 0.06 . The
horizontal extent of the tank is L = 1.0(m) , and we input a total
of 4 waves including a damping zone of 2 wavelengths. This
sets the wavelength of the wavemaker λ = 0.25(m) , with
corresponding wavenumber k = 25.1(1 / m) , and by using the
linear dispersion relation, Equation (12), the angular frequency
ω = 15.7(1 / s) .
The
discretization
parameters
are
∆x = λ / 25, ∆z = λ/15 ,
h = λ , q Max = 10 ,
with
and
∆t = T / 40 .
In Figure (2), we plot the free surface elevation at a fixed
location in the tank, namely η (0.1L) 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, S1 = 0.015, S 2 = 0.03 . The
horizontal extent of the tank is L = 1.0(m) , and the wavelengths
where n = 1,2,3 and i = 1,2, , M − 1. At each of these
of the wavemaker are λ1 = 0.25(m) and λ 2 = 0.125(m) , where
{χ in }
the damping zone is 2λ 2 . This sets the wavenumbers of the
time-steps, the current value for
and the fully discretized
version of equation (37) is used to get {Φ in, j } n = 1,2,3,
i = 1,2, , M − 1 and j = 1,2, , N L − 1.
After this initial phase, for n = 4,5,6, , N T , i = 1,2, , M − 1,
j = 1,2, , N L − 1, 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 {χ in } . Then, using this current value and
the fully discretized version of equation (37), we advance
{Φ in, j } . Next a fourth-order Adams-Moulton corrector method
is used, with these values for {χ in } and {Φ in, j } , to advance
{η in } . This is followed by another application of the Adams-
wavemaker, to be k1 = 25.1(1 / m) , and k 2 = 50.3(1 / m) ,
respectively, and using the linear dispersion relation Equation
(12), the corresponding angular frequencies are ω1 = 15.7(1 / s ) ,
and ω 2 = 22.3(1 / s ) , respectively. The discretization parameters
are
∆x = λ / 40, ∆z = λ/20 ,
with
h = λ , q Max = 10 ,
and
∆t = T / 50 . Figure (4) shows the spectrum after 100 periods of
simulation, relative to the second frequency, ω 2 . 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.
5
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 x = 0.1L .
ω1
, where ω1 and ω 2
ω2
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 r ≈ 2 ,
since the interactions given by ω1 + ω 2 = ω 2 (r + 1) and
2ω1 − ω 2 = ω 2 (2r − 1) , coincide.
For the first test we take r = 1.5 , and show that the solution
does not show resonance. Let
L = 1.0(m) , and
∆x = λ / 20, ∆z = λ/10 , with h = λ , q Max = 10 , and ∆t = T / 30 .
We choose ω1 = 15.7 ω 2 = 10.5 , and we set the steepness
S1 = 0.03 and S 2 = 0.009 , respectively. Figure (5) shows the
spectrum after 50 periods of simulation, relative to the first
frequency, ω1 . Note that the components given by
ω1 + ω 2 = 26.2 , and 2ω1 − ω 2 = 20.9 are generated, but of
course, do not coincide, and therefore both are significantly
smaller than ω1 and ω 2 . This represents a non-resonant
solution.
In the second test, we set r = 2.0 , and show that we get
resonance. In particular, the setup is identical to the first test,
above, except ω1 = 15.7 ω 2 = 7.9 , and we set the steepness
S1 = 0.03 and S 2 = 0.015 , respectively. To add emphasis to the
resonance phenomenon, we give the spectrum of the free
surface elevation of the wavemaker (at x = 0 ) in Figure (6).
The spectrum of the free surface elevation generated by the
model at (at x = 0.1L ), is shown in Figure (7). Note that we get
a
very
significant
resonant
effect
at
ω r = ω1 + ω 2 = 23.6 = 2ω1 − ω 2 . In fact the amplitude of this
resonant wave is slightly greater than the amplitude generated
for ω1 , and slightly less than the amplitude generated for ω 2 .
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 ω 2 , since the amplitude
generated by the model for ω 2 is significantly less than the
amplitude corresponding to ω 2 in the wavemaker. The
corresponding comparison for ω1 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
k r = 56.8(1 / m) , and measure its amplitude to be
Ar = 0.00368(m) , giving a steepness of S r = 0.067 , which is
more than twice the maximum steepness of the component
waves in the wavemaker, S1 = 0.03 . 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
Therefore, for these tests, we define r =
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 r = 3.0 . As before, the setup is
identical to the first test, above, except ω1 = 15.7 ω 2 = 5.24 ,
and we set the steepnesses S1 = 0.03 and S 2 = 0.003 ,
respectively. See Figure (8) for the spectrum of the free surface
elevation (at x = 0.1L ), after 50 periods. Note that we get a
strong interaction at 2ω 1 − ω 2 = 26.05 , which does not coincide
with a weaker interaction at ω 1 + ω 2 = 20.94 .
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, AddisonWesley, 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.
6
Copyright © 2004 by ASME
Tsai, W. and Yue, D. K. (1996), Computation of nonlinear freesurface flows, Annual Review of Fluid Mechanics, Vol. 28,
Annual Reviews Inc., Ca.
z
z = η ( x, t)
j =0
s0
FS
x
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.
j−1
∆z
w j −1
wj
i=0
sj
j
∆ z j +1 i − 1
i
j
j+1
i+1
w j +1
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.
Figure 2. Free surface elevation at x=0.1L as a function of
time in (s), for a monochromatic wavemaker at x=0.0.
N
∆ xi
z
x i −1
∆ x i +1
xi
x i +1
N
L
Figure 1. Free surface coordinate system
Figure 3. Spectrum of the time series of the surface
elevation given in Figure 2 at x=0.1L
7
Copyright © 2004 by ASME
Figure 4. Spectrum of free surface elevation at x=0.1L after
100 periods of simulation relative to ω 2 , with a bichromatic
wavemaker: ω 1 = 15.7(1 / s ) and ω 2 = 22.3(1 / s )
Figure 5. Spectrum of the free surface elevation at x=0.1L
after 50 periods of simulation relative to ω 1 , for a
bichromatic wavemaker: ω 1 = 15.7(1 / s ) and
Figure 6. Spectrum of the free surface elevation of the
wavemaker at x=0 : ω 1 = 15.7(1 / s ) and ω 2 = 7.9(1 / s ) ,
r = 2 .0
Figure 7. Spectrum of the free surface elevation at x=0.1L
generated by the model : ω 1 = 15.7(1 / s ) and ω 2 = 7.9(1 / s ) ,
r = 2 .0
ω 2 = 10.5(1 / s) , r = 1.5
8
Copyright © 2004 by ASME
Figure 8. Spectrum of the free surface elevation at x=0.1L
after 50 periods of simulation : ω 1 = 15.7(1 / s ) and
ω 2 = 5.24(1 / s) ; r = 3.0
9
Copyright © 2004 by ASME
Proceedings of The Fourteenth (2004) International Offshore and Polar Engineering Conference
Toulon, France, May 23−28, 2004
Copyright © 2004 by The International Society of Offshore and Polar Engineers
ISBN 1-880653-62-1 (Set); ISSN 1098-6189 (Set)
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
KEY WORDS: Nonlinear-waves; regular; irregular waves; nonlinear
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 volumediscretization methods on a similar footing with boundarydiscretization 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.
free surface conditions; multichromatic; non-orthogonal coordinates;
waveform relaxation.
PROBLEM FORMULATION
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.
INTRODUCTION
Consider a two-dimensional rectangular basin containing an
incompressible inviscid homogenHRXV IOXLG RI FRQVWDQW GHQVLW\ LQ D
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 0 ≤ x ≤ L , so L > 0 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 initialvalue problem, since the surface is initially at rest for time, t < 0 and
is disturbed at t = 0 giving rise to surface waves. See Debnath (1994).
If Φ ( x, z , t ) is the velocity potential and η ( x, t ) 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:
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 longcrested 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 nonorthogonal boundary fitted curvilinear coordinate system, which
∂ 2Φ
∂ 2Φ
(1)
= 0 − ∞ ≤ z ≤ η ( x, t ), 0 ≤ x ≤ L, t ≥ 0
∂x2 ∂z 2
For t ≥ 0 , the full nonlinear kinematic and dynamic free surface
249
+
boundary conditions are given, respectively, by:
∂η ∂η ∂Φ ∂Φ
+
−
=0
∂t ∂x ∂x ∂z
(2)
∂Φ 1 ∂Φ 2 ∂Φ 2
+ [(
) +(
) ] + gη = 0
∂t 2 ∂x
∂z
on z = η ( x, t ) , and
and
∂Φ
→0
∂z
z → −∞
as
η (0, t ) =
Ω
(3)
Φ (0, z , t ) =
Also, we have that
(8)
See Wehausen and Laitone (1960).
The initial conditions are given by:
•
Φ(x,0,0) = (1/ ρ)Φ0(x);Φ(x,0,0) = (1/ ρ) Φ0 (x); η(x,0) = η0(x)
(1 / ρ ) Φ 0 ( x) represent the given free
(13)
(9)
Therefore we are left with two lateral boundary conditions for t ≥ 0 at
x = 0 and x = L , 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:
s=
n =1
(14)
dτ , 0 ≤ x ≤ L ,
0 ≤ x ≤ L,
w ≤ 0.
(16)
(17)
(18)
(19)
where
n = 1,2,..., Ω. . Using standard water wave mechanics, the associated
velocity
linear
potential
is
given
by
 ∂η 
θ = tan −1  .
 ∂x 
(20)
The arclength along the coordinate curve s is given by
(11)
 ∂η 
ds = 1 + 

 ∂x 
2
dx
or
and −∞ < z ≤ η (0, t ) , where the dispersion
ω n = kn g
∫
2
sˆ = cos θ î + sinθ k̂
iˆ = sec θ ŝ - tanθ k̂
wave number and ω n the angukar frequency of each wave, for
∑
n

Bn g k n z
e cos(ω nt )
ωn

If we define the unit vector ŝ along the arclength, and therefore tangent
to the free surface z = η ( x, t ) , then
(10)
- ∞ < x ≤ 0, - ∞ < t < ∞ ; where An and Bn is the amplitude, k n the
 An g k n z

e sin( k n x − ω n t ) 
Ω 
 ωn

Φ ( x, z , t ) =


B
g
n =1 + n e k n z cos( k x − ω t ) 
n
n
 ω n

 ∂η (τ , t ) 
1+ 

 ∂τ 
0
x
w = z − η ( x, t ),
Ω
∑{An cos(kn x − ωnt) − Bn sin(kn x − ωnt )},
e k n z sin(ω nt ) +
We define the free surface coordinate system (FSCS), ( s, w) , where
s ≥ 0 is the arclength along the free surface z = η ( x, t ) and w ≤ 0 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),
•
Φ 0 ( x) = 0; Φ 0 ( x) = 0; η 0 ( x) = 0 for 0 ≤ x ≤ L
for - ∞ < x ≤ 0, t ≥ 0
relation is
n
THE LAPLACIAN ON THE FREE SURFACE COORDINATE
SYSTEM
•
surface impulse and η 0 ( x) 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
z = 0 , we take
η ( x, t ) =
n
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 [0, L ]
have no effect on the solution. This progressive wave at x = 0 and
t ≥ 0 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
∂Φ ( L, z , t )
=0
for −∞ < z ≤ η , t ≥ 0
(15)
∂x
we will get reflections from waves going from the wavemaker at x = 0
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.
(7)
where (1 / ρ )Φ 0 ( x) and
 An g
∑ − ω
n =1
χ ( x, t ) = Φ( x, η ( x, t ), t )
(5)
which represents the potential on the free surface, we have:
∂χ ∂Φ ∂Φ ∂η
=
+
(6)
∂t
∂t
∂z ∂t
implying:
•
n
n =1
Introducing the following function:
∂χ ∂Φ ∂Φ ∂η
=
+
.
∂x
∂x
∂z ∂x
n
and its associated potential is
(4)
∂Φ ∂χ ∂Φ ∂η
=
−
∂t
∂t ∂z ∂t
Ω
∑{A cos(ω t ) + B sin(ω t )}
(21)
ds = sec θ dx .
(22)
Clearly we have the following differential operator formulas that allow
us to transform easily between rectangular coordinates ( x, z ) and free
(12)
The free surface elevation is at the wavemaker is then given by
surface coordinate system ( s, w) :
250
∂
∂
∂
= sec θ
- tanθ
∂x
∂s
∂w
∂
∂
∂
= cosθ
+ sinθ
∂s
∂x
∂z
∂
∂
=
.
∂z ∂w
(23)
where s i = s i −1 + sec θ i −1 ∆x , and
(24)
where we take s 0 = 0, and
sin θ 

1

1
= 
 sin θ
(26)
where its associated second-order contravariant tensor is given by
J

sec 2 θ
=
 − tan θ sec θ

where clearly
J
− tan θ sec θ 
sec 2 θ 
written as Φ i , j (t ) i = 0,1, , M ; j = 0,1, , N L ;
The Laplacian is given by the following tensor equation
∆Φ =
1
J
∂ 

∂x λ 
where x = s,
αλ
∂Φ 
, α , λ = 1,2
∂x α 
(28)
x 2 = w. See Kaplan (1993). We get
1
∆Φ = sec θ
JJ
[
∂ 
∂Φ
∂Φ 
− tan θ
 sec θ
+
∂s 
∂s
∂w 
∂ 
∂Φ
∂Φ 
+
+ sec θ
 − tan θ
]
∂w 
∂s
∂w 
∑
(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.
for j = 0,1, , N L ; t ≥ 0 and where An and Bn is the amplitude, kn
the wave number and ω n the angular frequency of each component
wave, n = 1,2,..., Ω , 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, z = η ( x, t ),
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
∂η
∂χ
∂Φ
+ tan θ
− sec 2 θ
=0.
∂t
∂x
∂z
(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:
∂χ 1  ∂χ 
 ∂Φ 
+ 
 − sec 2 θ 

∂t 2  ∂x 
 ∂z 

2
2
 + gη = 0.

{Φ i,0 (t ) : i = 1,2, , M − 1;
(31)
SPATIAL APPROXIMATION AND WAVEFORM RELAXATION
∂χ 1  ∂χ 2
∂Φ 
+ ( ) − sec 2 θ ( ) 2  + µχ + gη = 0.
∂t 2  ∂x
∂z 
In this paper, we employ a curvilinear grid that follows the free surface.
In light of equation (23), this grid is taken to be
s i , w j , i = 0,1, , M ; j = 0,1, , N L ,
(32)
)
t ≥ 0}
(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,
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 .
{(
t ≥ 0.
To implement the wavemaker, we assume the sinusoid (10), for x ≤ 0 ,
−∞ < 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:
 An g  k n (η ( 0,t ) + w j ) 

e
sin(ω nt ) 
Ω −




 ωn

Φ 0 , j (t ) =
(33)


B
g
η
k
(
(
0
,
t
)
+
w
)
j 
n =1 + n e n
cos(ω nt )

 ω n 

(27)
= GHWJ ) = cos 2 θ .
w j = − j∆w , where ∆x = L / M ,
so x i = i∆x, and ∆w = h / N L , 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 ∆w ≠ ∆x;
see equation (25). We must solve Laplace'
s equation, (29) over this
grid.
The semi-discretized approximation of this potential Φ ( s i , w j , t ) is
(25)
The Laplacian for this non-orthogonal coordinate system can be
constructed from the second-order covariant fundamental metric tensor
J
2 

 ∂η  
sec θ i =  1 + 
,


 ∂x  

 x = xi
}
(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,
Φ M , j (t ) = Φ ( M −1), j (t ),
251
j = 0,1, , N L ; t ≥ 0.
(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 ( s i , w j ) , to calculate Φ i , j (t ) i = 1,2, , M − 1;
( )
& 0
for q = 0,1,..., where b fs
is obtained from equation (38) and all
& q
& q
subsequent b fs , q = 0,1,..., are obtained Φ j . 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
and
of
dimension
( M − 1) × ( M − 1) ,
&
T
Φ j (t ) = Φ1, j (t ), Φ 2, j (t ),..., Φ M −1, j (t ) , where j = 1,2, , N L − 1 ;
j = 1,2, , N L − 1 (fixed); t ≥ 0. This will involve nine potentials,
[
namely:
]
t ≥ 0. Note that the matrix A does not depend on the iteration
s j −1 -coordinate curve: Φ i −1, j −1 (t )
Φ i , j −1 (t )
Φ i +1, j −1 (t )
s j -coordinate curve: Φ i−1, j (t )
Φ i , j (t )
Φ i+1, j (t )
s j +1 -coordinate curve: Φ i −1, j +1 (t )
Φ i , j +1 (t )
Φ i +1, j +1 (t )
parameter q , and therefore must be inverted only once per time step.
{
} where
The terms Φ 0, j (t )
j = 1,2, , N L − 1 , t ≥ 0. are known
since they are given by the wavemaker condition (33) and are
&
incorporated into the vector bwm (t ) .
The inversion of the complete ( M − 1)( N L − 1) × ( M − 1)( N L − 1) system
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,
~
Φ i , j (t ) = χ i (t )φi , j (t )
i = 1,2, , M , j = 0,1, , N L; t ≥ 0 (37)
reduces to ( N L − 1) inversions of the ( M − 1) × ( M − 1) linear system
given by (40). Furthermore, if ∆w is a constant, then the matrix A is
identical for any s j -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.
where χ i (t ) is the potential at z = η ( x, t ) (i.e. the free surface
potential) and therefore satisfies equations (30) and (35), and since we
are considering the infinite depth basin,
~
k *w
φi , j = e j ,
i = 1,2, , M ,
j = 0,1, , N L; t ≥ 0
(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
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 [0, T ]. For a natural number NT , we let
case we use an iterative approach and choose some ’’reasonable’’ k * ,
where k * ∈ [min{k1, k2 ,..., kΩ }, max{k1, k2 ,..., kΩ }] and use equation (38)
as a ’’first guess’’ only. We then iterate to converge on the depth profile
~
φ i , j (t ) = γ j ( xi , t ), i = 1,2,..., M ,
j = 0,1, , N L; t ≥ 0
∆t = T NT , which gives the sequence of time steps
{tn },
where
t n = n∆t , n = 0,1,..., NT . The approximation of Φ i , j (tn ) , χ i (tn ) and
(39)
the frees surface η ( xi , t n ) is denoted by Φ ni , j , χ i (tn ) and η in ,
where γ i ,0 ( xi , t ) = 1, i = 1,2,..., M , t ≥ 0. Therefore the nine potentials
respectively, for i = 0,1, , M , j = 0,1, , N L , n = 0,1, , N T .
that occur in the discretized Laplace equation become,
&
To solve this linear system we require the vector b fs (t n ) which
s j −1 -coord.curve: χ i −1(t )φi −1, j −1
~
χi (t)φi, j −1
~
χ i +1 (t )φi +1, j −1
s j -coord.curve: Φ i−1, j (t )
Φ i , j (t )
Φ i+1, j (t )
contains the terms {χ in } , i = 1,2, , M − 1, n = 1,2, , N T , which
involve integration over time. The differential equations (30) and (35)
~
χ i (t )φ j +1
~
χ i +1 (t )φi +1, j +1
are used to generate the approximations {η in } and {χ in } , respectively
~
~
s j +1 -coord.curve: χ i −1(t )φi −1, j +1
for i = 1,2, , M − 1, n = 1,2, , N T . In both cases, we use a four-
Clearly the ’’unknowns’’ are the three potentials along the s j -coordinate
point one-sided finite difference to approximate
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 χ i (t ),φi , j (t ) , i = 1,2,..., M ,
{
(
}
( )
( )
∂χ
∂η n
) i and ( ) in , we spline {η in } and {χ in } , respectively, with a
∂x
∂x
piecewise polynomial of degree three, and then calculate it’s derivative
j = 0,1, , N L; t ≥ 0 are known and become incorporated into the
&
vector b fs this semi-discretized Laplace equation gives rise to the
iterated matrix equation
& q
&
& q +1
AΦj
(t ) = b fs (t ) + bwm (t ) ,
∂Φ
. To calculate
∂z
in closed form. This also allows us to evaluate {θ in = tan −1 (
∂η n
)i } .
∂x
Since the fluid in the basin is initially at rest, we have the initial
conditions,
(40)
252
η i0 = 0 ,
χ i0 = 0,
and
Φ i0, j = 0, where
i = 1,2, , M − 1, j = 1,2, , N L − 1.
A
fourth-order
Runge-Kutta method gets the additional starting values
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
A
 −B 
S Bη (ω ) = 5 exp 4 
m²/(rad/s)
(44)
ω
ω 
(explicit)
{η in } ,
{χ in } ,
where n = 1,2,3 and i = 1,2, , M − 1. At each of these time-steps,
the current value for {χ in } and the fully discretized version of
equation (41) is used to get {Φ i , j } n = 1,2,3, i = 1,2, , M − 1 and
n
After this initial phase, for n = 4,5,6, , N T , i = 1,2, , M − 1,
j = 1,2, , N L − 1, 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
{χ in } .
Then, using this current value and the
k =
fourth-order Adams-Moulton corrector method is used, with these
values for {χ in } and {Φ i , j } , to advance {η in } . This is followed by
n
and so
another application of the Adams-Moulton corrector method to advance
(2003) for a flow chart.
IRREGULAR WAVES AND OCEAN SPECTRA
+ εn )
(41)
n =1
where the coefficients are
A 
(42)
φn0 = An 2 + Bn 2 and the phase angles are ε n = tan −1 n  ,
 Bn 
n = 1,2,..., Ω. .
To analyze the irregular wave generated when n ≥ 2 in (41), we recall
the following results; see Goda (2000). An energy spectrum
Sη (ω n ) corresponding to any irregular time history can be defined,
WKH DUHD RI HDFK UHFWDQJOH RI ZLGWK
(m)
691
2π
k
.
(45)
(46)
(47)
for some natural number N , then all frequencies will be harmonically
related to the "window" [0, TS ] , 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).
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
φ n 0 = 2Sη (ω n )δω
λ =
ω2
,
g
B=
(49)
ωΩ = αWM * ω
For the uniform frequencies case mentioned above, we choose
*ω
ω − ω1
ω
α
= Ω = WM
.
(50)
ω1 = δω = Ω
Ω
Ω +1
Ω +1
We then define the period
2π  Ω + 1 
T ,
T1 =
=
(51)
ω1  αWM 
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
(51)
TS = N *T1 ,
The wavemaker (33), can be rewritten in terms of a sum of "phase
shifted cosine functions", as follows:
n 0 cos(ω nt
and
In analogy to regular waves, we can define the significant steepness:
H1 3
S1 3 =
.
(48)
λ
In all cases, we choose a real number αWM > 1 , and set
{χ in } using this “corrected” value of {η in } . See Baddour and Parsons
∑φ
m2 / s 4
2π
,
T
and using the linear dispersion relation (12),
n
Ω
2
3
4
define ω =
fully discretized version of equation (40), we advance {Φ i , j } . Next a
Φ 0 , j (t ) =
H1
s −4
T4
T
Clearly, the two parameters that are input to the wavemaker, (41), are
the significant wave height H1 3 and the average period T , and we
where A = 172.75
j = 1,2, , N L − 1.
(43)
It is also necessary to specify the phase angles ε n , 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
ε n in equation (41). Note that the period of the simulation T1 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.
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 Ω = 29 uniformly distributed frequencies,
with αWM = 3 , and the parameters, T = 4.0 s and H1 3 = 0.25m in
the spectrum (44). Therefore, λ = 25.0m , and we set L = 250.0m ,
with "10 waves" in the tank (not counting the beach), at steady-state,
which gives the required mean wavelength, and significant steepness,
S1 3 = 0.01 . The discretization parameters are ∆x = λ 40 ,
253
∆z = λ 20 ,
and
∆t = T 200 ,
and
we
plot
the
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.
numerical
approximation of the free surface elevation {η ( x, TS ) : 0 ≤ x ≤ L} after
TS = 100T = 10.0T1 .
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
{η (0.1 * L, tn ) : n = 1,2,..., NT } 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 Ω = 299 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 x = 0.1L 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
x = 0.1L 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 T1 .
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, AddisonWesley, 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 .
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
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.
254
Figure 1 System configuration and coordinate system
χ ( x, t ) = Φ( x,η ( x, t ), t )
Figure 4 Free surface elevation in (m) at x=0.1L
∂η
∂Φ
∂χ
= sec 2 θ
− tan θ
∂t
∂z
∂x
2
2
∂χ
1  ∂χ 
2  ∂Φ 
= − g η − 
 − sec θ 
 
∂t
2  ∂ x 
 ∂z  

∂ 2Φ ∂ 2Φ
=0
+
∂x 2
∂z 2
Figure5 Gaussian window of free surface elevation at x=0.1 L
Figure 2 Free surface elevation after 100 periods
Figure 6 FFT (eta at wavemaker x=0) with Guassian window
Figure 3 Free surface elevation in (m) of wavemaker at x=0
255
Figure 7 FFT (eta at x=0.1L) with Guassian window
Figure 10 Spectrum of surface elevation at x=0
Figure 11 Spectrum of Surface elevation at x=0.1L
Figure 8 Free surface elevation at x=0
Figure 12 Free Surface Elevation after 1.85 T1
Figure 9 Free surface elevation at x=0.1L
256