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CFD Optimisation of an Oscillating Water
Column Wave Energy Converter
A Thesis submitted in Fulfilment of the Requirements for the
Degree of Master of Engineering Science
By
Michael Horko
School of Mechanical Engineering
Faculty of Engineering, Computing and Mathematics
2007
ii
ABSTRACT
Although oscillating water column type wave energy devices are nearing the stage of
commercial exploitation, there is still much to be learnt about many facets of their
hydrodynamic performance. This research uses the commercially available FLUENT
computational fluid dynamics flow solver to model a complete OWC system in a two
dimensional numerical wave tank. A key feature of the numerical modelling is the focus
on the influence of the front wall geometry and in particular the effect of the front wall
aperture shape on the hydrodynamic conversion efficiency. In order to validate the
numerical modelling, a 1:12.5 scale experimental model has been tested in a wave tank
under regular wave conditions. The effects of the front lip shape on the hydrodynamic
efficiency are investigated both numerically and experimentally and the results
compared. The results obtained show that with careful consideration of key modelling
parameters as well as ensuring sufficient data resolution, there is good agreement
between the two methods. The results of the testing have also illustrated that simple
changes to the front wall aperture shape can provide marked improvements in the
efficiency of energy capture for OWC type devices.
iii
ACKNOWLEDGMENTS
First and foremost I would like to thank my beautiful wife, Linda for her immeasurable
patience and encouragement whilst I ‘played around’ preparing this thesis over what
I’m sure must have seemed to be a lifetime. To my sons, Halden and Sebastian whose
naïve yet profound banter brought me back down to earth on many occasions.
I am indebted to my supervisor, Krish Thiagarajan, whose guidance and assistance to
satisfy my ambition to understand wave energy will always be appreciated. This work
would also not have been possible without the support of the Australian Research
Council, Small Grants Program and the support of the Western Australian, Alternative
Energy Development Board.
Finally, I sincerely thank all those who provided me friendship and support throughout
the preparation of this research, in particular the staff and students from the Faculty of
Engineering, particularly Michael Morris-Thomas and Koroush Abdolmaleki.
iv
v
CONTENTS Abstract ........................................................................................................................ ii
Contents ........................................................................................................................v
List of Figures ............................................................................................................viii
Greek Symbols ..........................................................................................................xiii
Acronyms .................................................................................................................. xiv
1 Renewable Energy from Waves............................................................................1
1.1 General ..............................................................................................................1
1.2 OWC Type Wave Devices ................................................................................3
1.3 Previous Research .............................................................................................4
1.4 CFD Modelling .................................................................................................5
1.5 Thesis Statement and Outline............................................................................6
2 Theory ...................................................................................................................8
2.1 Wave Theory.....................................................................................................8
2.1.2 Small Amplitude Theory.................................................................................. 9
2.1.3 Wave Velocity and Wave Classification........................................................ 10
2.2 Higher Order Theories ....................................................................................13
2.2.1 General ........................................................................................................13
2.2.2 Extension to Stokes 2nd Order Waves .........................................................14
2.2.3 Wave Kinematics above Mean Water Level...............................................15
2.3 OWC Efficiency..............................................................................................15
2.3.1 Incident Wave Power ..................................................................................16
2.3.2 Hydrodynamic Power .................................................................................16
2.3.3 Pneumatic Power.........................................................................................18
2.3.4 Efficiency Calculation.................................................................................18
2.4 Resonance .......................................................................................................19
2.5 Theoretical Efficiency.....................................................................................23
3 CFD Analysis .....................................................................................................26
3.1 Introduction.....................................................................................................26
3.2 Numerical Wave Tank ....................................................................................27
3.2.1 Domain Setup..............................................................................................27
vi
3.2.2 Wave Generation......................................................................................... 31
3.2.3 Numerical Model Set Up ............................................................................ 33
3.3 Validation of Wave Propagation..................................................................... 38
3.3.1 Wave Profile Study ..................................................................................... 38
3.3.2 Velocity Profile ........................................................................................... 42
3.4 OWC Numerical Analysis .............................................................................. 45
3.5 Front Lip Analysis .......................................................................................... 58
3.5.1 Introduction................................................................................................. 58
3.5.2 Geometry..................................................................................................... 58
4 Experimental Program ........................................................................................ 68
4.1 Introduction..................................................................................................... 68
4.2 Experimental Testing ...................................................................................... 68
4.2.1 Experimental Overview .............................................................................. 68
4.2.2 Model Geometry ......................................................................................... 69
4.2.3 Model Scaling and Test Regime ................................................................. 71
4.2.4 Instrumentation and Measurement.............................................................. 72
4.2.5 Wave Height Measurement......................................................................... 73
4.2.6 Pressure Measurement ................................................................................ 73
4.2.7 Data Analysis .............................................................................................. 74
4.2.8 Front Lip Testing ........................................................................................ 76
4.2.9 Results and Discussion ............................................................................... 79
4.3 Detailed Frequency Experiment...................................................................... 87
4.3.1 Introduction................................................................................................. 87
4.3.2 Experimental Setup..................................................................................... 87
4.3.3 Experimental Data and Analysis................................................................. 88
4.3.4 Discussion of Detailed Frequency Experiment........................................... 91
5 Comparative Evaluation...................................................................................... 93
5.1 Introduction..................................................................................................... 93
5.2 OWC Front Lip Analysis ................................................................................ 93
5.3 Detailed Frequency Comparison..................................................................... 97
5.4 Chapter Summary ......................................................................................... 101
6 Site Modelling................................................................................................... 102
vii
6.1 Introduction...................................................................................................102
6.2 Site Characteristics........................................................................................102
6.3 Wave Resource..............................................................................................103
6.4 Gross Power Capture ....................................................................................106
6.4.1 Annual Energy Available..........................................................................106
6.4.2 OWC Performance ....................................................................................110
6.4.3 OWC Annual Output ................................................................................112
6.5 Discussion .....................................................................................................114
7 Conclusion & Recommendations......................................................................116
7.1 Numerical Wave Tank ..................................................................................116
7.2 OWC Parametric Simulations .......................................................................117
7.3 OWC Efficiency Modelling ..........................................................................118
7.4 Site Modelling...............................................................................................118
7.5 Recommendations.........................................................................................119
References .....................................................................................................................120
Appendices....................................................................................................................128
Appendix A: Additional Experimental Results ....................................................... 129
Appendix B: Experimental Photographs.................................................................. 133
Appendix C: Additional CFD Plots ......................................................................... 136
Appendix D: Typical User Defined Function for Numerical Wave-Maker ............ 138
Appendix E: Typical Fluent Input Data ................................................................... 140
Appendix F: University of Western Australia Wave Tank Details.......................... 145
viii
LIST OF FIGURES
Figure 1.1 OWC Power Curve (The Carbon Trust, 2006)............................................... 2
Figure 1.2 Principals of an OWC type wave energy device ............................................. 3
Figure 2.1 Basic Parameters of a Sinusoidal Wave ......................................................... 9
Figure 2.2 Wave Theory Application ............................................................................ 13
Figure 2.3 Nomenclature for Ma (1995)........................................................................ 21
Figure 2.4 Hydrodynamic efficiency versus Kh for b/h=1 and ..................................... 24
Figure 2.5 Hydrodynamic efficiency versus Kh for a/h=0.5 and b/h=0.125 ( ——),
b/h=0.25 (– – –), b/h=0.5 (– – – –), b/h=1.0 (············) ................................. 24
Figure 3.1 NWT Schematic ........................................................................................... 29
Figure 3.2 NWT grid in the region of the wave making boundary................................ 30
Figure 3.3 Numerical wave-maker set-up....................................................................... 33
Figure 3.4 Geo-Reconstruction and Donor-Acceptor Scheme Approximations ........... 34
Figure 3.5 Non-Iterative Time Advancement flow chart (Fluent User Guide, 2005) ...36
Figure 3.6 Wave Elevation and Difference in Wave Elevation between....................... 40
Figure 3.7 Difference in Wave Elevation between Theory and Simulation .................. 41
Figure 3.8 Velocity Plots for x=299 to x=363................................................................ 43
Figure 3.9 Comparison of Simulated and Stokes 2nd Order x & y velocity profiles
for a wave located at x=300m from wave maker for positions (i) 0 L (ii)
¼ L (iii) ½ L (iv) ¾ L. ................................................................................... 44
Figure 3.10 Numerical Domain for Replication Study .................................................. 46
Figure 3.11 OWC Dimensions....................................................................................... 46
Figure 3.12 OWC Grid Details ...................................................................................... 49
Figure 3.13 Wave Theory Applicability ........................................................................ 52
Figure 3.14 Typical Numerical Wave Probe Output ..................................................... 53
Figure 3.15 Input Wave Height versus Calculated CFD Wave Height ......................... 53
Figure 3.16 Input Wavelength versus Calculated CFD Wavelength ............................. 55
Figure 3.17 Efficiency Variation with Vent loss coefficient ......................................... 56
Figure 3.18 CFD Efficiency Analytical Sequence......................................................... 57
Figure 3.19 OWC Efficiency as a function of kh........................................................... 58
ix
Figure 3.20 OWC Front Lip Numerical Domain ...........................................................59
Figure 3.21 Efficiency of Various Lip Shapes...............................................................60
Figure 3.22 Relative Efficiency of Various Lip Shapes ................................................61
Figure 3.23 Velocity Vectors at time step t=40seconds for T=9secs, H=2m. ...............63
Figure 3.24 OWC Efficiency: Basecase Lip Vs Rounded Lip........................................64
Figure 3.25 Relative Efficiency increase between the Rounded and Basecase Lip........65
Figure 3.26 Results of Basecase Lip Vs Rounded Lip 2 - Key OWC Performance
Monitors for Case 9, H=1.5m, T=6s. ............................................................67
Figure 4.1 Model Schematic ..........................................................................................70
Figure 4.2 Experimental Front Lip Variations...............................................................71
Figure 4.3 Experimental Wave Tank Set-up..................................................................72
Figure 4.4 Experimental Flow Chart...............................................................................75
Figure 4.5 Typical Experimental Results. (Basecase lip H=2m, T=6 seconds).............79
Figure 4.6 Experimental Vs Theoretical wave profiles for H=2m, T=6s case................81
Figure 4.7 Basecase Efficiency Vs Vent Opening (cm) .................................................82
Figure 4.8 Wave Height Vs Efficiency Results ..............................................................82
Figure 4.9 Efficiency as a function of lip submergence..................................................84
Figure 4.10 Lip Efficiency Vs Kh Results ......................................................................85
Figure 4.11 Efficiency (Ratio to Basecase) Vs Kh Results.............................................86
Figure 4.12 OWC Experimental Setup ..........................................................................88
Figure 4.13 Frequency Analysis Flowchart ...................................................................90
Figure 4.14 Detailed Analysis: Frequency Domain Vs Time Domain Test Results .....91
Figure 5.1 Basecase Lip Efficiencies: Experimental Vs CFD .......................................94
Figure 5.2 Basecase Lip Experimental Vs CFD (detailed CFD) ...................................95
Figure 5.3 Average Efficiency: CFD and Model Test ...................................................96
Figure 5.4 Average Efficiency Relative to Basecase: CFD and Model Test .................96
Figure 5.5 CFD Vs Experimental efficiency results ......................................................98
Figure 5.6 Evans and Porter Theory for b/h=0.7 and varying a/h .................................98
Figure 5.7 Comparison of CFD, Experimental and Theoretical Efficiencies for the
Basecase configuration. ................................................................................99
Figure 5.8 Comparison of CFD lip variations with Evans and Porter Theory.............101
Figure 6.1 Port Kembla Wave Data .............................................................................105
x
Figure 6.2 Rottnest Wave Data – Annual Energy Scatter Diagram............................. 108
Figure 6.3 Port Kembla Wave Data – Annual Energy Scatter Diagram...................... 108
Figure 6.4 Annual wave energy as a function of Tp..................................................... 109
Figure 6.5 Energy Distribution for a Pierson Moskowitz spectrum (Hs=2m, Tp=9s) .110
Figure 6.6 Basecase OWC Efficiency Curves for Regular Waves (Tw) and Irregular
Waves (Tp) .................................................................................................. 111
Figure 6.7 OWC Efficiency Curves: Basecase and Rounded Lip ............................... 112
Figure 6.8 Basecase Lip: Port Kembla CFD Conversion Summary............................ 113
Figure 6.9 Basecase Lip: Rottnest CFD Conversion Summary.................................... 114
xi
LIST OF TABLES
Table 2.1 Wave Classification ........................................................................................11
Table 2.2 Wave Energy Propagation ..............................................................................12
Table 2.3 Natural Periods using McCormick (1981) .....................................................21
Table 2.4 Natural Periods using Ma (1995) for h=11.5m .............................................22
Table 2.5 Natural Periods using Ma (1995) for h=6m ..................................................22
Table 3.1 Material Properties..........................................................................................31
Table 3.2 Convergence Criteria ......................................................................................36
Table 3.3 NWT Analysis Cases ......................................................................................39
Table 3.4 Wave Height Results.......................................................................................41
Table 3.5 Analysis Duration Results...............................................................................42
Table 3.6 CFD Monitors .................................................................................................50
Table 3.7 Wave Conditions used for CFD Analysis .......................................................51
Table 3.8 Numerical to input wave height results comparison.......................................54
Table 3.9 Front Lip Variations Analysed.......................................................................59
Table 3.10 CFD Lip Test Regime..................................................................................60
Table 3.11 CFD Lip Test Efficiencies ...........................................................................61
Table 4.1 Experimental Parameters ...............................................................................71
Table 4.2 Preliminary Testing Wave Properties ............................................................76
Table 4.3 Preliminary Testing Regime ..........................................................................77
Table 4.4 Error Budget Estimate....................................................................................80
Table 4.5 CFD Lip Test Efficiencies .............................................................................86
Table 4.6 Detailed Frequency Test Program .................................................................89
Table 6.1 Rottnest Wave Data – Hs, Tp Occurrence Percentages ...............................104
Table 6.2 Port Kembla Wave Data – Hs, Tp Occurrence Percentages ........................105
Table 6.3 Rottnest Wave Data – Annual Energy Scatter Diagram..............................107
Table 6.4 Port Kembla Wave Data – Annual Energy Scatter Diagram.......................107
Table 6.5 Annual Power Summary as a function of location and OWC lip ................114
xii
NOMENCLATURE
Latin Symbols a depth of submergence of OWC front lip m
A wave amplitude m
ATw amplitude of spectral component wave m
Awp water plane area m2
b width of the OWC chamber inline with the incident wave direction m
c wave celerity m.s-1
cg group velocity of a wave m.s-1
d depth of OWC lip submergence m
Ei incident average wave energy per unit width J.m-1
f wave frequency Hz
g acceleration due to gravity 9.81 m.s-1
h water depth m
H wave height m
Hs significant wave height m
k wave number m-1
kA wave steepness parameter
kh water depth parameter
Kh infinite water depth parameter
L wavelength m
LR length scale ratio
MWL mean water level
p(t) pressure inside the chamber with time Pa
pd dynamic Bernoulli pressure Pa
Pi incident wave power per unit width Wm-1
Po converted power per unit width Wm-1
Re Reynolds number
S surface area of an arbitrary plane m2
S(ω) spectral wave energy m2s
xiii
T wave period s
Tp peak wave period s
tR time scale ratio
Tw period for spectral component regular wave s
Tz average zero up-crossing wave period s
u(x,z,t) water particle velocity in the x direction m.s-1
vR velocity scale ratio
V(t) vertical velocity of the internal free surface in time m.s-1
w(x,z,t) water particle velocity in the z direction m.s-1
w width of the model m
x coordinate system
y coordinate system
z coordinate system
Z wave surface elevation at wave-maker m
Greek Symbols
π constant pi
� wave frequency rad.s-1
� velocity potential
∇ first partial derivative
υ kinematic viscosity m2s-1
� phase angle rad
η free surface elevation m
ρ density kg.m-3
εhyd hydrodynamic efficiency
xiv
Acronyms
2D Two dimensional
3D Three dimensional
CAD Computer Aided Drafting
CFD Computational Fluid Dynamics
FFT Fast Fourier Transform
GR Geometric Reconstruction
HRIC High Resolution Interface Capturing
JONSWAP Joint North Sea Wave Atmosphere Program
NITA Non-Iterative Time Advancement
NWT Numerical Wave Tank
OWC Oscillating Water Column
O&M Operation and Maintenance
PV Present Value
RAM Random Access Memory
SWL Still Water Level
TWH Terra Watt Hours
UDF User Defined Function
VOF Volume of fluid
1
1 RENEWABLE ENERGY FROM WAVES
1.1 General The oceans contain enormous amounts of energy that is dissipated along the world’s
coastlines. It has been estimated that the practical world wave energy resource is
somewhere between 2000 TWH and 4000 TWH annually (The Carbon Trust, 2006). To
put this into perspective, this equates to a value of approximately 20% of the world’s
electricity production in the year 2003 (International Energy Agency, 2005).
Consequently, energy from waves can be considered on the world stage as a power
producing means. Perhaps even more importantly, given recent scientific understanding
of the effects and drivers behind the greenhouse effect, wave power has the potential to
play an important role as a carbon free energy resource.
In general, the research and development for power utility scale wave energy devices is
still in the elementary stages such that the net cost of energy is still somewhat higher
than conventional or other forms of renewable power generation. Given this cost
penalty there has been only limited commercial exploitation of this significant wave
energy resource (Pelamis, Ocean Power Delivery, 2007). The net cost of energy can be
principally summarised as sum of capital costs, operating and maintenance costs
divided by the amount of energy produced reduced to a common cost base by using
present value techniques (equation 1).
Capital Cost + PV (O&M Costs) [2.1]Cost of Energy =
PV (Energy Production)
Although all three components to the equation are important drivers, studies such as
those by The Carbon Trust (2006) have indicated that significant gains can be made in
the short to medium term in the area of energy production, whereas capital and O&M
costs improvements are more likely to occur in the medium term. Thus, during this
infancy stage in wave energy, it is suggested that design improvements to increase
energy production may be an enabling mechanism to raise wave energy from a research
area to a mainstream electricity generation discipline.
2
Energy Production from wave energy devices is a function of a number of areas
including:
• The extent to which an energy capture device is matched to the wave resource.
• The efficiency of the systems of energy conversion.
• The proportion of the time the device is able to generate.
These areas are illustrated in Figure 1.1 which illustrates a typical power conversion
multi-dimensional graph that plots power output against axes of wave period and wave
height. The feature of prime importance in energy production is the ‘Power Captured by
Device’ which is simply the efficiency with which a device can capture the incoming
wave energy. This figure also identifies key operational aspects such as minimum cut-in
conditions whereby no power is generated and power shedding conditions whereby the
device either moves into a survival mode in large waves or the power absorbed exceeds
the capacity of the generating equipment. It is however the accurate modelling and
optimisation of this power capture efficiency that is the prime motivation for this
present investigation.
Figure 1.1 OWC Power Curve (The Carbon Trust, 2006)
3
1.2 OWC Type Wave Devices Vantorre et al (2004) categorise wave energy devices into two main groups: “Active
devices where the interface element responds to the wave action and produces
mechanical work, and Passive devices where the device remains stationary and the
water movement relative to the structure is made to work”. The Oscillating Water
Column (OWC) device can be considered the closest to maturity of the latter group.
This type of device consists of a land-backed chamber in which the front wall has an
opening to let waves pass into the device whilst the rear wall extends down to the
seabed. The wave action makes the water level in the chamber oscillate causing the air
in the chamber to flow in and out through a turbine to generate electrical energy (Figure
1.2).
Figure 1.2 Principals of an OWC type wave energy device
These types of device are the most common type of wave energy device currently in
operation with at several prototype plants currently operating worldwide (for example,
in Scotland, Portugal, Sweden, Australia and India).
The optimum design of an OWC is based upon the idea of inducing resonant motion of
the water chamber oscillations by tuning the device parameters to the ambient waves.
This is a complex phenomenon and involves the energy transfer between the incoming
4
wave and the hydrodynamic, pneumatic, aerodynamic and electrical power take-off
attributes of the device.
1.3 Previous Research The research and development work on OWC type wave energy plants has been
primarily based on the simplified analytical models. In this area, there have been several
studies, Evans (1982), Evans and Porter (1995) and Ma (1995), which characteristically
assume a simplified ‘rigid piston’ approach to the modelling of the free surface inside
the OWC device and / or the interface below the OWC lip. Sarmento and Falcao (1985)
extended these to develop models that allowed for spatial variation of the free surface
shape within the OWC chamber. Overall though, the numerical modelling has been
performed in 2-Dimensions and using first order assumptions combined with simplified
geometries.
Prior to the full-scale fabrication of existing prototype plants, mathematical modelling
has typically been used to specify initial OWC design parameters (i.e. key geometric
dimensions, turbine parameters) which have then been experimentally tested, verified
and optimised prior to implementation in the prototypes (Ravindran and Swaminanthan,
1989 and Joyce et al, 1993).
Purely numerical studies have been performed using a numerical wave tank (Clément,
1996) to determine the influence of geometric parameters on the non-linear radiation
response of an OWC plant. Evans and Porter (1995) demonstrated an approach for
optimising a two-dimensional device with respect to OWC front lip submergence while
Tindall and Xu (1996) have performed an optimisation study modelling the effects of
the power take-off turbine.
3-Dimensional numerical studies (Lee et al, 1996 and Brito-Melo et al, 1999) have been
performed using commercially available hydrodynamic radiation-diffraction codes
developed for the analysis of floating bodies. The codes accurately predicted the
hydrodynamic behaviour of the OWC; however the implementation of the OWC
5
analysis required considerable modification to the software requiring the researchers to
have an extensive knowledge of computer programming in addition to access to the
source code.
1.4 CFD Modelling As discussed, several numerical methods have been used to compute two-dimensional
and unsteady surface flows for wave energy devices. The majority of these use first
order simplifications and although this benefits the numerical effort it can also limit the
validity of the results. More recently codes have been developed to cater for the
simulation of fully non-linear attributes in flow (eg CANAL code, Clément 2004). The
ability of these research derived numerical codes to be utilised in real-world situations is
somewhat limited. This is generally because, rather than be of a general nature, they are
often task specific codes and thus not readily adapted to variances in geometry,
environmental conditions or other variations such as power take-off influences.
Over the last 10 years or so, a number of general computational fluid dynamic (CFD)
codes have been commercialised- eg FLUENT®, Star-CD®, CFX. In many high
technology industries such as Formula 1 motor-sport or Americas Cup yachting, these
programs these have become an accepted part of life. In parallel with the software
development, computing power continues to increase according to Moore’s Law
(Wikipedia, 2007) and become cheaper. These cumulative effects have enabled what
was once only available to high end research to become a somewhat mainstream
engineering tool.
Application to ocean engineering problems has been somewhat limited and typically
results have been used in a qualitative manner rather than quantitatively. The use of
commercial CFD has been used in relation to OWC devices, but only in relation to the
airflow through the upper structure, White et al (1997).
The numerical code FLUENT was chosen for this research because of its general
modelling capabilities which allows for any user defined geometry, the ability to model
6
properties such as viscosity and compressibility as well as the ability to utilise the
geometric reconstruction (GR) technique with volume of fluid (VOF) method for
tracking the air-water interface which has been shown to be superior to the donor-
acceptor (DA) or high resolution interface capturing (HRIC) schemes (Rhee et al,
2005). Other CFD software such as CFX or COSMOS are also potentially suitable,
however the inability to access these codes using the university software precluded their
use.
1.5 Thesis Statement and Outline Whilst physical experimentation can provide accurate, realistic results of hydrodynamic
flow, it is often time consuming, expensive and laborious. Numerical techniques do
exist to model OWC efficiency however many of these require simplifications that may
result in modelling errors or the inability to model certain geometric effects. With recent
advances in computing power, using complex CFD solvers for the numerical simulation
of the OWC efficiency has become a viable alternative that may not only provide
accurate solutions potentially quicker and cheaper than model testing, but may also
allow the numerical modelling of complex geometries not previously possible.
The motivation for this work was to assess the use of CFD for this class of problem
whilst gaining an increased understanding as to features that may affect OWC
efficiency.
The primary stages of the research effort can be described as follows;
• Firstly, development of a 2D numerical wave tank using CFD that can represent
the physical model to an appropriate order of accuracy whilst maintaining
realistic computational effort.
• Secondly, extend the numerical wave tank to include a detailed OWC model to
determine energy capture efficiencies
• Thirdly, determine the effect of various OWC geometric parameters on
efficiency
7
• Fourthly, develop matching experimental results to allow comparison with and
validation against the numerical work
• Finally, perform a real world comparison of the annual power prediction using
the numerical results obtained to demonstrate the effects of site characteristics
and the benefits optimisation may have on device design.
8
2 THEORY In this chapter some fundamental concepts and definitions are presented. First, the
mathematical equations relating to description of linear wave theory are outlined and
then these are extended to form Stokes 2nd Order theory. Secondly the formulae used to
define and calculate OWC performance and efficiency are illustrated. Finally, a brief
description of resonance is explained with respect to OWC type wave energy
converters.
2.1 Wave Theory This section describes the equations that are required to define the wave conditions.
Firstly, simple linear theory is explained and is then extended to describe higher order
theories.
2.1.1 Theoretical Considerations
Real waves propagate in a viscous fluid over an irregular seabed of varying
permeability. Viscous effects are usually concentrated in a thin “boundary” layer near
the surface and the seabed and the main body of fluid motion is nearly irrotational.
Since water can also be considered to be effectively incompressible, a velocity potential
and a stream function should exist for waves.
In order to illustrate some of the theoretical considerations it is convenient to initially
discuss a simplified small-amplitude water wave problem with the following
assumptions:
• Fluid is incompressible and inviscid (i.e. ρ = constant)
• Flow is irrotational (i.e. lacks viscosity)
• Uniform density
• Waves are planar (ie 2-Dimensional)
• Monochromatic waves
9
2.1.2 Small Amplitude Theory
The small amplitude wave theory can be developed by the introduction of a velocity
potential, ( )),, tzxφ . Horizontal and vertical particle velocities are defined in the fluid as
dxdφ
=u and dzdw φ
= . By combining the velocity potential, Laplace’s equation,
Bernoulli’s equation and appropriate boundary conditions (eg dzdφ =0 at the seabed) the
small amplitude formulas may be developed as per Dean and Dalrymple, (1966).
Wave Profile:
The elevation of the free surface is given by:
)22cos(2
),(T
tL
xHtx ππη −= [2.1]
Figure 2.1 Basic Parameters of a Sinusoidal Wave
L
10
This may be rewritten as:
)cos(),( tkxAtx ωη −= [2.2]
Where:
k = wave number
ω= angular frequency (s-1)
t = time (s)
and wave number is given by:
Lk π2= [2.3]
The dispersion relationship which relates wavelength to frequency can be given by
khgk tanh2 =ω [2.4]
Velocities
Using the known form of the free surface and velocity potential, expressions for
horizontal, u(x,z,t) and vertical, w(x,z,t) velocities can be given by the follow equations
)cos(cosh
)(cosh),,( tkxkh
hzkgAkx
tzxu ωω
φ−
+=
∂∂
= [2.5a]
)sin(cosh
)(sinh),,( tkxkh
hzkgAkz
tzxw ωω
φ−
+=
∂∂
= [2.5b]
2.1.3 Wave Velocity and Wave Classification
The speed at which a waveform propagates is termed the phase velocity or wave
celerity, c. Since the distance travelled by a wave in one wave period is one wavelength,
the wave celerity can be written as:
11
TLc = [2.6]
Or alternatively re-arranged and written as:
kc ω= [2.7]
( )khgTc tanh2π
= [2.8]
The speed at which a wave train travels is generally not identical to the speed at which
individual waves travel. This speed is defined as the group velocity, cg, and is typically
less than the celerity, c, in deep or transitional water depths. Given these differences in
wave properties it is also useful to classify waves according to the water depth in which
they travel. Standard classifications by the US Army Corps, Shore Protection Manual
(1984) have been made according to the magnitude of the ratio h/L and the non-
dimensional wave number Kh, are shown in Table 2.1.
Classification Deep Transitional Shallow
h/L >1/2 1/25 < h/L < 1/2 < 1/25
kh > π 1/4< kh < π < 1/4
Tanh kh ≈ 1 Tanh kh ≈ kh
c π2
gT ( )khgTc tanh2π
= gh
cg 22c
TLcg ==
+=
)2sinh(21
2 khkh
TLcg
ccg =
Table 2.1 Wave Classification
Group velocity, cg, is important as it is this velocity at which energy is propagated. This
can be illustrated by describing how a wave propagates in calm water which is
analogous to wave generation within a model test tank. This has been described by
Ippen (1966) and is relevant to the understanding of kinetic and potential energy and
wave propagation within a NWT.
12
From Dean and Dalrymple (1966), the total wave energy, E, is equal to the sum of the
potential energy and the kinematic energy which can be shown to be of equal magnitude
for small amplitude wave theory. If a wave generator starts generating waves, it can
only impart kinetic energy into the domain. As such it imparts energy equal to E/2 to the
water during the first stroke. That is, after one first stroke, a wave will be present with a
total energy equal to E/2. One period later, this wave has advanced one wavelength but
it has left half of its energy, E/4 behind in the form of potential energy. The kinetic
energy (E/4) now occupies a previously undisturbed area immediately ahead of the
previous wave. In the meantime, a second wave was generated, occupying the position
of the original wave. This wave now has an energy of E/2 + E/4 = 3E/4. Repeating this
simple analogy gives the distribution of wave heights as a train progresses as shown in
Table 2.2.
Series Wave number, m Total
1 2 3 4 5 Energy
1 1/2 E - - - - 1/2 E
2 3/4 1/4 E - - 2/2 E
3 7/8 4/8 1/8 E - - 3/2 E
4 15/16 11/16 5/16 1/16 E - 4/2 E
5 31/32 26/32 16/32 6/32 1/32 E 5/2 E
Table 2.2 Wave Energy Propagation
For a large number of waves, the leading wave effectively becomes insignificant and the
energy increases very close to the centre of the wave group. For deep water the energy
front is located at the centre of the wave group. For shallow water, this front would have
been at the front of the group. For any depth, the ratio of group velocity to celerity
13
defines this energy front and hence confirms the previous statement that wave energy is
transported with the group velocity rather than celerity.
2.2 Higher Order Theories
2.2.1 General The solution of the hydrodynamic equations for gravity wave phenomena can be
improved by extending the theories and hence increasing the agreement between the
theoretical and the observed behaviour. Improving the accuracy of theories invariably
comes at computational expense and hence engineering judgement is used to define the
limiting conditions for various wave theories. Figure 2.2 illustrates the approximate
limits of validity for several wave theories and has become recognised by its use in
several texts (US Army Corps, 1984).
Figure 2.2 Wave Theory Application
(Trend line for tests considered in this work shown ——— )
14
A trend line that approximates the tests considered in this work is plotted onto the
graph. It can be seen that the applicability of linear theory is somewhat restricted.
Stokes 2nd Order expansion provides almost full coverage of the chosen conditions and
has been chosen given the relative simplicity in application to the problem at hand. A
derivation of the 2nd Order Stokes correction is given in the following sections.
2.2.2 Extension to Stokes 2nd Order Waves During derivation of linear wave theory, small quantities such as the higher order
expansion values within the Taylor expansion of the free surface elevation were
neglected in order to simplify the calculations. Stokes 2nd Order Theory is a well
known variation to Linear Theory and includes an additional ‘higher order’ component
in the formulation. The details can be described as per Dean and Dalrymple, (1980):
Surface Profile:
( ) ( )( ) )(2cos)2cosh2(
sinhcosh
16cos
2 3
2
tkxkhkhkhkHtkxH ωωη −++−=
[2.9]
x-velocity:
( )( ) )(2cos
sinh2cosh
163)cos(
cosh)(cosh
2u 4
2
tkxkh
zhkkHtkxkh
zhkgkH ωωωω
−+
+−+
= [2.10]
z-velocity:
[2.11]
Where:
Wave period = T
Frequency = ω = 2 π/T
Wave length = L
Wave number: = k = 2 π /L
( )( ) )(2sin
sinh2sinh
163) sin(
cosh ) ( sinh
2 w 4
2
tkxkh
zhkkHt kx kh
z h k gk H ωωω ω
−+
+ − + =
15
Celerity: = c = L/T = ω/k
Wave Height = H
Water Depth = h
Time = t
2.2.3 Wave Kinematics above Mean Water Level Linear wave theory in principle applies to very small waves so it does not predict
kinematics for points above the MWL as they are not in the ideal ‘fluid’. A common
practice consists of using linear wave theory in conjunction with empirical wave
stretching techniques to provide a more realistic representation of near-surface water
kinematics. The empirical wave stretching techniques popular in the offshore industry
include vertical stretching, linear extrapolation, Wheeler stretching and delta stretching
(Couch and Conte, 1997). Although not necessarily the most accurate technique,
vertical stretching (Marshall and Inglis, 1986) is computationally more efficient and
hence was adopted for this study. Vertical Stretching has the effect of setting the
particle velocities above MWL equal to those calculated for the MWL. Although in
general, stretching techniques are applied only to linear waves, in this study it will be
applied to Stokes 2nd Order waves.
2.3 OWC Efficiency The primary purpose of the research is to investigate the efficiency with which an OWC
can generate power when subject to the influence of gravity waves. Efficiency can be
determined by comparing the output power against the theoretical input wave power for
a particular condition. Thus to determine efficiency we need to develop expressions for
the theoretical power of the incident wave train as well as the power absorbed by the
OWC.
16
2.3.1 Incident Wave Power It can be shown (McCormick, 1981) that the expressions for the total wave energy (Ei)
and the average incident wave power (Pi) over a wave cycle using Stokes 2nd Order
Theory are, respectively,
(J) [2.12]
(W) [2.13]
Where;
Wave period = T
Frequency = ω = 2 π/T
Wave length = L
Device width: = b
Celerity: =cg
Wave Height = H
Water Depth = h
Time = t
2.3.2 Hydrodynamic Power As with most wave energy applications, the horizontal dimension of the interior
chamber is assumed to be small compared to the prevailing wavelength thus the internal
water surface may be assumed to be sufficiently plane in long waves to be considered as
a body in heave. In the instance of waves with shorter wavelengths, the internal surface
may be non-planar, the water column may experience both pitching and heaving
motion. Although this may have consequences for the system’s natural period(s), the
mean power absorbed by the OWC device is primarily dependent only on the heave
motion of the water column and the dynamic air pressures inside the device. It has been
shown by Brendmo et al, (1996) that although an approximation, it will provide good
+ = 6 4
2 2
64 9 1
8 . Pi
h k H b c H g
g ρ
+ = 6 4
2 2
64 91
8 . Ei
h k H H g ρ
17
agreement when the wavelengths under consideration are long when compared to the
characteristic horizontal dimension of the inner OWC surface.
Hence, assuming that the inner surface of the OWC behaves as a piston the
hydrodynamic power, Phyd, absorbed by the OWC can be computed from the simple
formulation for power
TimentDisplacemeForce.Power = Watts [2.14]
Re-arranging this formulation and substituting Force = Pressure x Area and assuming
that the hydrodynamic power is transferred to the air column, we can write this as:
(W)
[2.15]
That is, for the OWC wave energy converter:
Pressure = the pressure inside the chamber (Pa)
Area = surface area of the water surface inside the chamber (m2/m width)
dTdS = differentiated position of the water surface inside the chamber
= velocity of the OWC “piston” (m/s)
Thereby, given knowledge of the pressure developed inside the chamber and the surface
oscillations inside the chamber we can determine the hydrodynamic power transmitted
to the OWC.
Since this is a forced vibration problem, the oscillations of the OWC will have the same
frequency as that of the input wave. The pressure developed inside the device will have
the same frequency too albeit there may be a phase difference between the pressure and
flow rate. This multiplication of pressure and flow rate (2.15) will give the absorbed
power with respect to time. The average of this absorbed power over an integer multiple
dTdS
Area .. Pressure Phyd =
18
of wave periods provides the average Phyd of the OWC for the particular conditions
considered.
2.3.3 Pneumatic Power In practical terms, the OWC efficiency following the next stage of conversion in an
OWC Power Plant can be determined by investigating the energy flux of the air flow
across the turbine. For the case of this present study, due to scalar difficulties in
modelling a turbine, a simple vent has been utilised to provide representative similarity
to the pressure drop associated with the air flow passing through a turbine. This is
considered adequate given this studies focus on hydrodynamic rather than pneumatic
conversion efficiencies.
The Power at the vent, Pvent , can be determined from the following:
[2.16]
Pressure = the pressure inside the chamber
dAvelocity.∫ = integral of the air velocity profile across the vent
This determination is somewhat academic for experimental work as the ability to
accurately measure the air velocity profile across the vent is practically impossible. It is
still however of interest when performing numerical analysis as it is possible to
determine all necessary variables and a comparison with the hydrodynamic power may
give an indication as to the pneumatic losses in the system.
2.3.4 Efficiency Calculation The ratio of the power absorbed by the device to the incident power provides a measure
of the efficiency of the device. This is defined as ε and can be simply calculated as
follows:
dA velocity Pressure . . Power ∫ =
19
in
out
PP
Efficiency = [2.17]
wave
hydhyd P
P=ε [2.18]
2.4 Resonance As briefly described in the introduction, Section 1.2, optimum OWC designs are based
upon the idea of obtaining resonant motion of the water column within the device. Such
motion implies that this will provide peak wave energy capture efficiencies. Designers
of OWC converters will thus endeavour to optimize the design of the OWC system to
create systems that resonate in the ambient conditions to achieve highest possible
energy production.
As resonance can be primarily related to device geometry (McCormick 1981) a designer
will aim tune the various parameters of the device to not only excite resonance from a
particular set of ambient wave conditions, but to maximise energy production by
providing a broad efficiency peak such that significant energy capture also occurs even
during ‘off-peak’ conditions.
To accurately determine the resonant peaks analytically is very difficult due to the
number of parameters that affect its value. Historically these peaks have been
determined from model testing or estimated using simplified formulations. Two such
simplified formulations shall be discussed below:
(A) Natural Period using Cavity Resonance Dynamics
As discussed by McCormick (1981), the natural heave period of an undamped floating
body may be described as:
[2.19]
w
wp
mmgA+
=ρ
π21fz
20
Where:
fz = natural frequency
Awp = area of water plane
m = mass of the heaving system
mw = added mass of water excited by heaving motion
Using a ‘piston’ analogy between the buoy and the water column of the device the
natural frequency of an OWC chamber with solid back wall may be estimated. To do
this, the following formulations will be adopted
Awp = Breadth x Width = B x W
m = mass of water column
= Breadth x Width x Submergence x density = (B x W x d1ip)ρ
mw = added mass of an equivalent rigid body equivalent to the water column
= 0.25πρ(W)(2B)2/2
Notes to added mass:
1. Added mass formulation is taken for a prism from table 3.2 of McCormick
(1981)
2. 2B has been used for added mass effective width as OWC chamber has a back
wall and thus 2B is the appropriate width. The total added mass has been divided
by 2 to reflect we are considering half the 2B width only.
That is, the formulae can thus be re-written:
[2.20]
As will be discussed in later chapters, for the ‘prototype’ OWC under consideration in
this study which has a width, B=8m, the natural frequency and period may be estimated
as a function of lip submergence as per Table 2.3:
2 2 1 f
z B dlip
g ππ
+
=
21
Case Lip Submergence
Natural Heave Period
Natural Heave Frequency
1 0m 7.11 0.141
2 1m 7.39 0.135
3 2m 7.66 0.131
4 3m 7.91 0.126
Table 2.3 Natural Periods using McCormick (1981)
(B) Natural Period using an approximate non-linear model
Ma (1995) has developed an approximate non-linear model for an OWC with a side
opening that was shown to provide reasonable agreement with experimental validation
and will be briefly described herein.
The formulation for the resonant frequency of the device was proposed as follows:
1
2
SHg
TT +=ω [2.21]
Where:
ωT = resonant frequency
g = acceleration due to gravity
HT = depth of submergence of front lip
S1 = length of the stream line that runs between the internal OWC water surface and a
point between the OWC lip and the seabed. The streamline is a quarter of an ellipse.
Figure 2.3 Nomenclature for Ma (1995)
HT
S1
OWC
22
Using the formulation, the resonant periods and frequencies are presented for the
nominal prototype OWC under consideration in this study with a width, B=8m and
water depth, h=6m and h=11.5m:
Case Lip Submergence
Natural Heave Period
Natural Heave Frequency
1 0m 4.11 0.24
2 1m 4.79 0.21
3 2m 5.44 0.18
4 3m 6.07 0.16
Table 2.4 Natural Periods using Ma (1995) for h=11.5m
Case Lip Submergence
Natural Heave Period
Natural Heave Frequency
1 0m 5.7 0.175
2 1m 6.35 0.157
3 2m 7.1 0.141
4 3m 8.05 0.124
Table 2.5 Natural Periods using Ma (1995) for h=6m
Both these simplified analyses take no account of the OWC damping, lip shape etc in
the formulations and as such may be considered ‘rough’ guidance only. The results of
the formulation in Table 2.3 can be considered an upper bound to the natural period.
The results do demonstrate an increase in natural period with lip submergence due to the
increase in the oscillating mass. Also, the formulation by Ma (1995) also suggests that
the natural period may increase when an OWC is placed in shallower conditions
although the exact reasoning behind this is not fully understood.
23
2.5 Theoretical Efficiency The theory developed by Evans and Porter (1995) considers the efficiency of an OWC
and its relationship to front wall submergence. The theory assumes an OWC device
immersed in an ideal fluid in the presence of linear, progressive waves. The key
parameters of interest are the front wall submergence depth (a) and the chamber length
(b) normalised with respect to water depth (h). This theory also considers that the
turbine is modelled as a linear pressure drop with damping set to optimal conditions and
takes no account of the front lip shape or thickness.
The theory has been previously presented by Morris-Thomas et al (2005) for analogous
situation to the cases considered in this work. A plot of efficiency versus Kh is
presented in Figure 2.4 for a general case where b/h=1 and for a variety of front wall
submergence values (a/h). Kh represents the infinite water depth parameter, and can be
determined as follows:
Kh = kh tanh(kh) [2.21]
As a full description of the development of this theory can be found in Evans and Porter
(1995) it will not be elaborated further here. Kh will be utilised in lieu of kh to provide
comparative analysis with these results.
The efficiency curves using the theory exhibit the following characteristics of
significance to this study:
• Of interest in wave energy extraction is when the fluid between the back wall and the
front walls is excited into a resonant, piston-like motion. As described in Evans and
Porter (1995), if one assumes b/a is small, the fluid in between the walls may be
considered to act as a solid body. Simple modelling of this situation gives rise to the
condition that Ka≈1 for resonance which equates to Kh=2 for the case a/h=0.5 as
shown in Figure 2.5. The curves do illustrate trends towards this behaviour,
particularly for smaller values of b/h.
24
Figure 2.4 Hydrodynamic efficiency versus Kh for b/h=1 and
a/h=0.125 ( ——), a/h=0.25 (– – –), a/h=0.5 (– – – –), a/h=0.75 (··········)
(Evans and Porter, 1995)
Figure 2.5 Hydrodynamic efficiency versus Kh for a/h=0.5 and
b/h=0.125 ( ——), b/h=0.25 (– – –), b/h=0.5 (– – – –), b/h=1.0 (············)
(Evans and Porter, 1995)
25
• Larger values of a/h cause the resonance frequency to decrease. This can be
explained physically as a result of the additional distance a fluid particle must travel
during the period of motion due to the increased front lip submergence. This directly
causes a decrease in the value of Kh at which resonance occurs. A similar effect can
also be had by increasing the value of a/h as shown in Figure 2.4.
• Large motions can also occur within the OWC when the fluid in the OWC is excited
into an asymmetric sloshing mode as though it were a closed tank. In a closed tank it
can be shown that this mode occurs for values of kb = nπ. For the case shown with
b/h = 1, this occurs at a value of Kh ≈ π and is clearly demonstrated by the spikey
behaviour near this value of Kh. For shallow lip submergence, the motion acts less
like a closed tank and this resonant mode moves away from Kh ≈ π. Increasing
submergence, moves the resonance closer to this mode and the peak becomes more
spiked.
26
3 CFD ANALYSIS
3.1 Introduction CFD is a computer-based mathematical modelling tool that incorporates the solution of
the fundamental equations of fluid flow, the Navier-Stokes equations, in combination
with other allied equations. The Navier-Stokes equations represent the laws of
conservation of mass, momentum and energy in differential form. These partial
differential equations in integral form are then approximated as finite-volume
expressions and reformed into algebraic equations to allow for numerical computation
within a specified domain. The FLUENT software used for this study uses the finite
volume method to solve the Navier-Stokes equations and has several features for multi-
phase flows applicable to the problem at hand. Among these features is the ability to
implement the VOF method to track the air-water interface within the domain. This is
not only important as a means to delineate the interface but is also critical for the correct
modelling of the hydro-pneumatic interaction within the OWC chamber
As with any numerical modelling, simplifications and approximations need to be made
to allow finite analytical durations or to explain phenomena not yet fully understood (eg
turbulence). It is therefore prudent to perform systematic validation of any numerical
work against either known theoretical or experimental solutions prior to acceptance as a
valid method. Experimental validation is of particular importance as it may reveal real-
world conditions that were not envisaged during the numerical development that may
require incorporation into the chosen modelling tool.
This chapter is concerned with the CFD modelling of OWC type wave energy devices
with particular focus on the energy absorption ability of a device. The development of a
CFD model involves the creation of a domain, generation of waves and the
hydrodynamic and pneumatic modelling of the interaction of these waves with the
OWC. The work in this chapter firstly details the development of a Numerical Wave
Tank (NWT) to validate the wave generation. Secondly, an OWC is modelled within the
NWT and the device efficiency is investigated. Thirdly, it has also been known that
27
parameters such as front lip shape and submergence depth affect efficiency (Sarmento,
1995; Thiruvenkatasamy, Neelami and Sato, 1998). Studies are performed to evaluate
these effects for certain geometric configurations. These analyses are described in this
chapter.
3.2 Numerical Wave Tank The NWT is the basic building block to which various features that warrant
consideration (eg an OWC) may be added. It is thus of fundamental importance that the
NWT provide results with an appropriate degree of accuracy to ensure that results from
subsequent modelling are not distorted or diminished.
The analysis of any fluid flow using CFD is an iterative process consisting of three
basic steps:
1. Numerical Domain Setup
2. Modelling and Computation
3. Evaluation of the Results
These steps applied to the development of the NWT are described in the following
sections.
3.2.1 Domain Setup As part of the pre-processing, one must define a geometry to which the CFD will be
applied. The geometry chosen needs to take into account the size of the device and the
surrounding volume that needs to be modelled in order to create a realistic response
without significant ‘boundary effects’ (eg reflection). This model setup also includes the
generation of the mesh to define the individual volumes that make up the computational
domain. In addition to the creation of the mesh, boundary conditions such as a wave
generator need to be carefully considered in order to accurately reproduce real world
situations.
28
3.2.1.1 Geometry
A schematic of the NWT proposed is presented in figure 3.1. In this model, the tank size
is L=960m and H=20m. At the left hand side of the tank a wave generation boundary is
created whilst the bottom and right hand side of the tank are represented by walls.
As with experimental testing, techniques to allow a sufficient number of waves to be
analysed prior to potential contamination from reflected waves is required. Numerical
techniques such as numerical damping (Koo et al, 2006) or active wave absorption
paddles (Sarmento and Brito Melo, 1995) may be applied to minimise the domain size
but both require considerable effort to calibrate and ensure satisfactory application. For
this study an approach was utilised whereby the length of the tank was based upon a
length equal to two times the experimental wave tank dimensions used in later
experimental validation. This length allowed for several of waves to progress past the
point of interest prior to any reflections contaminating with the results. The simplicity
of the proposed technique was deemed sufficient for the purposes of this study
3.2.1.2 Numerical Results
The calculation of wave profiles and OWC device efficiency require a number of
parameters to be monitored, however these are only some of the many variables, data
sets and graphical representations can be extracted both during analysis and post
processing. For a full description of the available information refer to the FLUENT
User Manual (2005).
Free surface elevations are determined at particular instant using inbuilt functions
within the software that plots a contour for a particular quantity. To obtain the free
surface plots, the user requests that data be extracted for the VOF fraction=0.5 which
defines the interface between the air phase (VOF=1) and the water phase (VOF=0) at
each air-water interface cell.
Velocity measurements in the domain can be extracted by the definition of a “line” such
that any properties of the flow along the line may be extracted for a particular time.
29
Both the water surface and the velocity profiles will be used during the validation
exercise.
Figure 3.1 NWT Schematic
3.2.1.3 Mesh Generation
To discretise the Navier Stokes equations, the domain must be covered by a
computational mesh. The numerical domain and mesh were initially created using
GAMBIT geometry and mesh generation software that is a companion programme to
the FLUENT CFD software. GAMBIT can be used to create the domain using
modelling-based geometry tools or to import geometry created in standard CAD
programs. Following creation of the geometry, the model is then meshed using a variety
of different tools depending upon the problem at hand. Following creation of the
geometry and mesh, the boundaries are defined and the model can then exported to a
*.msh file for later import directly into FLUENT. For further detail on GAMBIT refer
to FLUENT Users Guide, 2005.
The relatively simple geometry of the NWT allows for efficient modelling of the
domain using quadrilateral cells. A base grid dimension of 1m x 1m was chosen which
provides in the order of 50 cells per wavelength for typical test conditions.
20m
Stokes 2nd order wave maker
11.5mWater Phase
Air Phase
Walls
Pressure Inlet
960m
30
Preliminary analytical runs did identify that the resolution of the mesh at the air-water
interface region was insufficient to satisfactorily model the wave shape. The model
mesh was adapted in these locations by halving the cell dimensions. This made the
wave height to cell dimension ratio to be in the order of four (4). To assist with the
definition of the flow velocities in the region of the wave maker (x = 0 to 5m) and the
seabed interface (z = 0 to 1m) were adapted whereby existing density is increased by
creating new nodes between existing cell faces. This ‘grid adaptation’ effectively halves
the nominal cell dimensions. Figure 3.3 shows the initial 30m of the NWT domain
illustrating these refinements to the mesh.
Figure 3.2 NWT grid in the region of the wave making boundary
3.2.1.4 Boundary conditions
To define a problem that results in a unique solution it is necessary to specify the
information on the flow variables at the domain boundaries. It is important to define
these correctly as they can have a significant impact on the numerical solution.
The base of the tank and right hand wall are set as wall boundaries in order to bound the
domain. Tangential and normal fluid velocities are set to zero for the cells adjacent to
the wall boundaries.
31
The NWT top is set as a pressure inlet in order to mimic a “free” boundary such that air
flows can occur, if required, either into or out of the domain.
The interior of the domain is set to a “Fluid Zone” for which all active equations will be
solved. Fluid material input is required and the following has been used:
Material Phase Type Density
(kg/m3)
Dynamic Viscosity
(kg m-1 s-1)
Temperature (ºC)
Air Primary 1.225 1.7894 x 10-5 20
Water (fresh) Secondary 998.2 0.001003 20
Table 3.1 Material Properties
A ‘velocity inlet’ has been used to describe the wave maker. The inputs required are the
magnitude for each of the velocity components. Although the ‘velocity inlet’ is ideally
intended for incompressible flows it is used here to input the flow of water only and
does not include air within the prescribed flow conditions. The formulation for the
velocity components is described in the following sections.
3.2.2 Wave Generation To provide accurate NWT simulations of OWCs, the generation of realistic waves is
crucial. A number of techniques are available to generate waves in FLUENT. Firstly,
waves can be generated by a moving flap that can move horizontally mimicking the
wave generation techniques commonly used in experimental wave tanks. In general, this
option is not very convenient in a numerical wave tank as the computational effort
required for mesh regeneration at each time-step can be significant. The second option,
prescribing the flow conditions and wave height is a much more convenient technique
and simply requires the velocity and wave height to be imposed as determined by the
appropriate theory.
32
In this study a wave is generated by dynamically linking a User Defined Function
(UDF) to the inlet velocity boundary within the FLUENT analysis module. The inlet
velocity boundary condition allows the user to define the u and w velocity components
as well as the phase type for each cell at the boundary. The UDF is a separate function
written in C programming language that determines the Stokes 2nd Order wave velocity
components for each cell on the boundary at each timestep during the simulation. The
process can be described as per the following steps and as illustrated in Figure 3.3:
I. The UDF function accepts the timestep and the cell coordinate information from
FLUENT.
II. The UDF calculates the wave surface elevation according to theory (equation 2.9)
and defines this as Z.
III. For fluid cells with: cell mean z coordinate ≤ water depth; the u and w velocity
coordinates are calculated according to Stokes 2nd Order wave theory (equations
[2.10] and [2.11]).
IV. For fluid cells with: Z ≥ cell mean z coordinate ≥ h; the u and w velocities are set
to those for a value of z=h (i.e. linear stretching).
V. For fluid cells with mean z coordinate > Z, the u and w velocities are set to zero.
For each water depth, wave height, wave period and wavelength a unique UDF is
created by modifying the problem parameters. A typical UDF is provided for
information in Appendix D.
Note that at the inflow boundary, positive and negative velocities can occur so fluid can
flow in and out. As the Stokes 2nd Order velocity integrated over a wave period will be a
non-zero number, there is a net flow into the domain. Given the duration of the
simulations and the size of the numerical domain this has been evaluated to be
acceptable.
33
Figure 3.3 Numerical wave-maker set-up
3.2.3 Numerical Model Set Up
3.2.3.1 Multiphase
The numerical wave tank problem involves multiple phases – that is, air and water. The
definition and monitoring of this interface is of primary importance to the analysis of
OWCs as air/water interface within the OWC creates the ‘piston’ that compresses the
air that drives the turbine to ultimately convert the wave energy into electricity.
FLUENT has a number of techniques to cater for multi-phase flows. The Volume of
Fluid (VOF) method chosen for this study has been shown to be the most applicable and
sufficiently accurate to capture the essential flow features around free surface wave
flows (Rhee et al, 2005).
The VOF method places the free surface in cells that are partially filled with water and a
volume fraction is calculated which represents the portion of the cell that is filled with a
Velocity=Velocity(h) Wave Height
Velocity=0
Velocity=Stokes 2nd Order
Velocity Inlet Boundary
Water Depth
Pressure Inlet Boundary
34
pre-determined fluid type. Once the volume fraction is known, the actual phase interface
can be resolved.
The modelling of the phase interface using VOF can be modelled in FLUENT using
methods such as the geo-reconstruction or the donor-acceptor techniques. The
geometric reconstruction technique is used for the analysis as according to the FLUENT
Use Manual (2005), it models the interface more accurately than the other methods. The
geometric reconstruction scheme assumes that the interface between two fluids has a
linear slope within each cell and uses the linear shape to calculate the advection of fluid
through the cell faces. An example of how the air-water interface is approximated can
be seen in Figure 3.4.
Figure 3.4 Geo-Reconstruction and Donor-Acceptor Scheme Approximations
3.2.3.2 Initial Solution
Prior to performing analysis an initial solution is applied to the domain. Air was defined
as the primary phase (Table 3.1). FLUENT automatically assumes the primary phase
species is present in every cell unless otherwise defined. To create the 2nd phase (water)
within the domain, it needs to be ‘patched’ over the lower portion to provide constant
fill level equal to the chosen water depth. As verification studies against experimental
modelling are performed later with an experimental tank that is initially still, velocities
for all cells within the domain were set to zero as an initial condition to provide like-for-
like similarity.
35
3.2.3.3 Solver Controls
FLUENT offers a wide variety of solvers, discretisation schemes and variables which
can affect the solution quality and convergence. A summary of the key solution
parameters chosen are discussed below, however much more detail may be found in the
Fluent User Guide (2005).
The segregated solver is the solution algorithm used by FLUENT for this class of
problem. Using this approach, the governing equations are solved sequentially (i.e.
segregated from one another). Because the governing equations are non-linear (and
coupled), several iterations of the solution loop must be performed before a converged
solution is obtained. Non-Iterative Time Advancement (NITA) technique is also
available and has been shown to speed up the iteration process significantly (FLUENT
Users Guide, 2005).
The idea underlying the NITA scheme is that, in order to preserve overall accuracy, it is
not necessary to reduce the error from each sequential solution step to zero, but only
have to make it the same order as the time discretisation error. The computational flow
of the NITA scheme, as seen in Figure 3.5 illustrates that only a single global iteration
per time-step is performed. Sub-iterations are performed within each time-step, but the
outer, velocity-pressure iteration is performed just once (hence the term "non-iterative")
within a given time-step which significantly speeds up transient simulations. This
approach effectively drives the error in each sub-iteration to the time discretisation
error, not zero and has the net effect of allowing a computation in the order of 3 to 4
times faster than standard iterative techniques. It should be noted that certain analytical
runs did not converge satisfactorily using the NITA technique. Rather than reduce
timestep or increase mesh resolution, the analysis was performed using the default
iterative method albeit with significantly more computational effort.
Convergence criterion have been set using the default factors which will, in general,
provide 2nd order accuracy (FLUENT User Guide, 2005). These parameters are shown
in Table 3.2.
36
Figure 3.5 Non-Iterative Time Advancement flow chart (Fluent User Guide, 2005)
Maximum Number of Initial Iterations
Correction Tolerance
Residual Tolerance
Relaxation Factor
Pressure 10 0.25 0.0001 1
Momentum 5 0.05 0.0001 1
Energy 5 0.05 0.0001 1
Table 3.2 Convergence Criteria
Because of the nonlinearity of the equation set being solved it is necessary to control the
change of variables. This is achieved by a process called under-relaxation. This process
37
effectively reduces the change of each variable, φ, during each iteration. In a simple
form, the new value of the variable within a cell depends upon the old value φ old plus
the computed change in the variable multiplied by under-relaxation factor, α, as
follows:
φαφφ ∆+= old [3.1]
Typically the default under-relaxation values were used during all analyses. Certain
wave height – wave period combinations did involve numerical instabilities particularly
during analysis involving the OWC chamber which required the relaxation factor to be
lowered to provide convergence. These runs also typically required the number of
intermediate corrections to be increased to allow convergence (up to 2 times)
3.2.4 Analysis
3.2.4.1 Time Step and Convergence
In CFD analysis the numerical model is not only discretised in space, but also in time.
In time we always have to make an approximation to the ideal solution because it is not
possible to use infinitely small timesteps. That is, we try to use the largest timestep
possible in order to reduce simulation time commensurate with achieving an acceptable
level of accuracy.
For transient solutions within FLUENT the solver iterates to convergence at each time
step such that the residuals defined by the user are achieved, then advances
automatically. The time step ∆t must be small enough to resolve time dependent
features and to ensure convergence within the maximum number of iterations set by the
user.
For nearly all the analysis performed, the default convergence criterion in FLUENT was
sufficient. This criterion requires that the scaled residuals decrease to 10-4 for all
equations except the energy equations, for which the criterion is 10-6. Certain cases did
38
not converge and hence trial and error alterations were made to parameters such as the
convergence criteria, under-relaxation or the number of sub-iterations per timestep in
order to complete computations.
3.3 Validation of Wave Propagation For the validation of wave propagation, several tests have been performed using a
personal computer with a 2.2 GHz processor and 1 GB or RAM running FLUENT 6.2
software. Firstly, a series of runs was performed to evaluate the wave elevation has been
investigated. Special attention was paid to the size of the grid and the timestep
necessary for an accurate simulation of the waves. Secondly, given the importance of
the flow velocities into and out of the OWC, the accuracy of the simulated velocity
profiles has also been examined.
3.3.1 Wave Profile Study In this section, waves have been simulated in a NWT and the surface profiles compared
to wave theory. Attention was paid to the number of cells used and the time steps
required in order to achieve reliable results. As previously discussed, simulations have
been performed using Stokes 2nd Order wave theory as the input wave profile.
The wave simulations used a wave typical of what may have a high occurrence of being
experienced by OWC wave converters. The wave chosen for this study has a period of
7.1 seconds in a water depth of 11.5m resulting in a wavelength of 63.4m. The wave
height was set at 1.5m.
In the simulations performed, the timestep and the mesh size have been varied. These
are presented as a ratio of the wave period and the wavelength in Table 3.3. The
Basecase timestep utilised is 0.01 seconds which equates to T/710 and the Basecase
mesh dimension was 1m which equates to L/64.
.
39
Case Timestep (s)
Timestep Ratio Typical Mesh Dimension (m)
Ratio to L
Basecase (a) 0.01 T/710 1 L/64 b 0.01 T/710 0.5 L/128 c 0.01 T/710 0.25 L/264 d 0.005 T/1420 1 L/64 e 0.05 T/142 1 L/64 f 0.002 T/3550 1 L/64 g
(model scale) 0.01 T/710 1 L/64
Table 3.3 NWT Analysis Cases
The results of the simulation are shown in Figures 3.6 and 3.7 as well as Tables 3.4 and
3.5.
In Figure 3.6 the Basecase (a) wave profile at t=150 seconds is plotted against the
Linear and Stokes 2nd Order wave theories. Also shown using the right hand axis is the
error between the generated wave profile and the input Stokes 2nd order profile as a
percentage of wave height. Figure 3.7 shows the error between Stokes 2nd Order
Theory and a number of the variations simulated including the magnitude of the 2nd
Order Correction. It is clear that Case (f) using the 0.02 second timestep has
significantly more error in the generated waves compared to the other cases. The other
cases typically show good agreement with theory with the order of magnitude of the
error being less than the correction applied to linear theory to obtain a 2nd order wave
profile. This is what one expects – the wave shape should eventually evolve from the
generated wave profile into a steady state ‘natural wave profile’ and that this profile will
have a profile somewhere between 1st Order (i.e. Linear) theory and 2nd order theory.
(Note that as this is a ‘snapshot’ of the wave profiles, waves generated later in the
simulation are closer to the x-axis origin – i.e. it can be seen that wave height error
decreases as simulation time increases)
Table 3.4 summarises the wave elevation data for a set of 9 wave profiles starting from
the wavemaker. The results illustrate that there is some minor benefit from either
decreasing the timestep or increasing the mesh density however this offers only of a
40
minor benefit. Table 3.5 demonstrates the analytical duration for each of the analysis
runs for a 200 second simulation which matches later experimental study durations.
Practical analytical durations would be expected to be limited to less than 24 hrs
duration with an ideal duration that would allow a number of runs to be performed in a
working day. Using this criterion one may expect that Cases a, d and e are practically
acceptable. However given the wave profile error one would expect that a practical
timestep / grid combination is case (d) whereby the grid to wavelength ratio is 1/64 and
1/1400 for the ratio of timestep to wave period. This result for the wavelength ratio is
similar to that given by Kleefsman (2005).
Wave Profile at t=150 seconds
10
10.5
11
11.5
12
12.5
13
0 63.4 126.8 190.2 253.6 317 380.4 443.8
Distance from Wavemaker (m)
Wav
e H
eigh
t Ele
vatio
n (m
)
-50%
-40%
-30%
-20%
-10%
0%
10%
20%
30%
40%
50%
% e
rror
1st order2nd orderBasecase t=0.01error basecase
Figure 3.6 Wave Elevation and Difference in Wave Elevation between
Theory and Simulation for case (a)
41
Wave Theory Vs CFD Wave Height Difference
-80%
-60%
-40%
-20%
0%
20%
40%
60%
80%
0 63.4 126.8 190.2 253.6 317 380.4 443.8
Distance from Wavemaker (m)
Diff
eren
ce a
s %
Wav
ehei
ght
1st Vs 2nd OrderBasecasedef
Figure 3.7 Difference in Wave Elevation between Theory and Simulation
9 Period Averages Case
Ratio to Theory
Maxima Error (Ave.)
Minima Error (Ave)
CPU Duration for 200s
Simulation (hrs)
Analysis Duration Ratio
to Basecase
a 102.7% 2.2% 3.4% 10.9 100%
b 103.8% 2.1% 3.9% 27.8 256%
c 103.2% 2.0% 3.3% 123.5 1136%
d 103.2% 2.1% 3.5% 15.7 145%
e 117.4% 12.6% 4.9% 2.4 22%
f 102.8% 2.3% 3.5% 38.7 356%
g 102.6% 2.2% 3.5% 10.9 100%
Table 3.4 Wave Height Results
42
Case Timestep Typical Mesh Dim.
(m)
CPU Duration for 200s Simulation (hrs)
Analysis Duration Ratio
to Basecase
a 0.01 1 10.9 100%
b 0.01 0.5 27.8 256%
c 0.01 0.25 123.5 1136%
d 0.005 1 15.7 145%
e 0.05 1 2.4 22%
f 0.002 1 38.7 356%
g 0.01 1 10.9 100%
Table 3.5 Analysis Duration Results
3.3.2 Velocity Profile The total energy in a wave consists of two kinds: the potential energy resulting from the
displacement of the free surface and the kinetic energy, due to the water particle
movement within the fluid. The previous section dealt with the investigation into the
modelling of the potential energy (i.e. free surface elevation), whilst this section will
consider the accuracy in the modelling of internal kinematics.
A snapshot of the particle velocities at ¼ points along a single wave at time t=100
seconds for case (d) is shown at the approximate location of the proposed OWC in
figure 3.8. The location of this wave which has a wavelength of 63.4m, starts at x=299m
and extends to x=363.4m. Figure 3.9 also illustrates the simulated velocity profiles, but
these are now compared to the nominal Stokes 2nd order velocity profiles at ¼ points
along the wave under consideration. Note that the numerical analysis also shows the
velocity components for the air particles and as such the profile extends to the limit of
the domain whereas the theoretical profile only extends to the water surface.
The velocities show good agreement with theory. It was thus concluded that the
numerical analysis which, under these conditions, does produce representative wave
43
profiles of sufficient accuracy to enable further correlation work and model testing to
proceed.
(a) Velocity Contour
(b) X-Velocity profiles for a Wave (i) 0 L (ii) ¼ L (iii) ½ L (iv) ¾ L (v) L
(c) Y-Velocity profiles for a Wave (i) 0 L (ii) ¼ L (iii) ½ L (iv) ¾ L (v) L
Figure 3.8 Velocity Plots for x=299 to x=363
x=299 x=363.4
Top of Meshed domain
Seabed
Air Velocity Profile
Water Velocity Profile
Water Surface
One Wavelength
x=299 x=363.4
Top of Meshed domain
Seabed
Air Velocity Profile
Water Velocity Profile
Water Surface
One Wavelength
44
Vx(1)
0
2
4
6
8
10
12
14
16
18
20
-1.5 -1 -0.5 0 0.5 1 1.5Velocity (m/s) (i) 0 L
Vy(1)
0
2
4
6
8
10
12
14
16
18
20
-1.5 -1 -0.5 0 0.5 1 1.5Velocity (m/s)
Vx(2)
0
2
4
6
8
10
12
14
16
18
20
-1.5 -1 -0.5 0 0.5 1 1.5Velocity (m/s) (ii) ¼ L
Vy(2)
0
2
4
6
8
10
12
14
16
18
20
-1.5 -1 -0.5 0 0.5 1 1.5Velocity (m/s)
Vx(3)
0
2
4
6
8
10
12
14
16
18
20
-1.5 -1 -0.5 0 0.5 1 1.5Velocity (m/s) (iii) ½ L
Vy(3)
0
2
4
6
8
10
12
14
16
18
20
-1.5 -1 -0.5 0 0.5 1 1.5Velocity (m/s)
Vx(4)
0
2
4
6
8
10
12
14
16
18
20
-1.5 -1 -0.5 0 0.5 1 1.5Velocity (m/s) (iv) ¾ L
Vy(4)
0
2
4
6
8
10
12
14
16
18
20
-1.5 -1 -0.5 0 0.5 1 1.5Velocity (m/s)
CFD (1) 2nd Order Theory
Figure 3.9 Comparison of Simulated and Stokes 2nd Order x & y velocity profiles for a wave
located at x=300m from wave maker for positions (i) 0 L (ii) ¼ L (iii) ½ L (iv) ¾ L.
45
3.4 OWC Numerical Analysis
3.4.1 Introduction
Subsequent to the NWT analysis, the numerical work was extended to include the
hydrodynamic analysis of an OWC type wave converter under a variety of wave
conditions. The aim of the work was to determine the efficiency of energy capture and
to map the efficiency of the OWC as a function of non-dimensional wave number, kh.
The efficiency profile thus determined allows researchers to determine the annualised
power output of an OWC and is thus of upmost importance in the effective design of a
wave energy plant.
3.4.2 Methodology
The numerical modelling of the OWC system was achieved using the same fundamental
setup as described for the NWT in the previous section including using the optimised
mesh and timestep refinements. In addition to the basic NWT setup, a number of
additional parameters are required for the modelling of the OWC system, the main
features of which are described in the following sections.
3.4.3 Geometry
A new geometry was created and meshed using GAMBIT. A schematic of the
numerical model of the OWC – NWT system is shown in Figure 3.10. In this model the
tank size is 470m between the wave maker and the front of the OWC. The tank size is
based upon the experimental wave tank described in later chapters used for the
experimental work in this thesis.
The modelling utilised a 2-dimensional OWC that was representative of a device chosen
to suit typical Australian wave conditions. The full scale prototype is to have a main
OWC chamber 10m wide by 8m long and was initially proposed for a water depth of
6m but later changed to 11.5 due to various siting considerations of various possible
locations. The change in water depth is reflected in the analysed water depths as well as
46
variations in the experimental front lip submergence. Other simplifications to the OWC
geometry included simply squaring off the OWC chamber at the top and utilising a
simple rectangular front lip.
Figure 3.10 Numerical Domain for Replication Study
Figure 3.11 OWC Dimensions
3.4.4 Boundary Conditions
OWC devices utilise a turbine to extract the energy from the internal air column flow
that is caused by waves acting on the chamber. A number of turbine types have
typically been utilised for the power take-off such as the Wells turbine (Curran,
Lip thickness =0.5m
vent
Unit width
Lip submergence =1.75m
h=11.5m
8m
8m 470m
Wave Probe
Pressure Probe
OWC Model
Stokes 2nd order wave maker
11.5m
Vent
Water Phase
Air Phase
20m
47
Raghunathan and Whittaker, 1997) or the Denniss-Auld turbine (Curran and Gato,
1997). As the intention of the study is to perform experimental validation a method of
utilising a squared off vent was used for this study.
The idea behind using a vent is twofold; Firstly, it is relatively simple to replicate in an
experimental campaign whereas a real-life turbine such as the typically used Wells
turbine, requires a complicated damping scenario to be matched to the performance
characteristics of the device. Secondly, modelling an OWC using a vent to provide the
load has been successfully used previously (Thiruvenkatasamy, Neelami and Sato,
1998). As this project focuses on the hydrodynamic efficiency, avoiding the turbine in
the study was considered a well-suited compromise.
In addition to the vent modelling, the OWC numerical model will utilise the same
pressure inlet and velocity inlet boundary conditions as used in the NWT analysis to
model the upper domain interface and the wave-making boundary.
3.4.5 Numerical Mesh
Typically the same hexahedral mesh as used in the NWT study was created. This was
based on a standard 1m x 1m grid with adaption to halve the mesh size about the mean
water surface and the first 5m extending from the wave maker apart from mesh (refer to
Figure 3.12).
The presence of the OWC, and in particular the OWC vent, requires special
consideration to the mesh creation. Mesh sizing requires satisfactory definition with
respect to the dimensions of key geometric parameters such at the front lip and the
OWC vent. To allow computationally smooth transitions between cells of differing
sizes a tetrahedral mesh has been used for the OWC region and for a section extending
10m towards the wave maker. An interface was created at this location such that a
hexahedral mesh could be used for the remainder of the domain as a tetrahedral mesh
would have significantly increased the number of cells with a proportional increase in
computational effort. The tetrahedral mesh does however allow for good transition
48
where there are significant changes in geometry and mesh size as is the case with an
OWC.
The flow under the OWC front lip is a critical aspect of OWC design given it is the
“entry” point for the energy into the OWC. Previous studies have also identified this as
an area where turbulence may be generated (Curran, 1992). As the typical mesh size
was 1m whilst the lip dimension was only 0.5m, refinement to adequately capture the
flow was performed such that a number of cells were created across the bottom of the
lip (note: qualitative analysis of the flow patterns showed that at least five divisions
provided satisfactory flow transitions). This resulted in a grid dimension of
approximately 0.1m as can be shown in Figure 3.12 (b).
The OWC vent presented similar issues to provide suitable flow definition. As the vent
dimension is only 62.5mm in the full size prototype, this is very small compared to the
typical mesh dimension of 1m. As per the front lip, to provide a good transition between
the mesh outside the chamber and the flow in the vent region a transitional mesh was
created. A fine mesh could have been used throughout the region, but this would have
increased analytical duration excessively.
The following steps were performed:
1. Prior to meshing the OWC in two dimensions, the grid requirement across the
vent was qualitatively assessed to provide a satisfactory flow transition. At
minimum of four cells were used across the opening prior to later refinement
that doubled the number of cells (i.e. initial cell dimension was ~15mm)
2. The OWC wall grid adjacent to the vent (i.e. +/- 2m either side) was defined
such that the grid dimension was no greater than 75mm.
3. A two dimensional tetrahedral grid was generated in the OWC.
4. Following initial grid generation the upper 1m of the OWC chamber was
adapted to effectively halve the initial grid dimensions in this zone.
The final mesh produced can be seen in figures 3.12 (c) and (d).
49
(a) OWC Mesh (b) Detail of mesh under OWC lip
(c)Mesh detail near OWC Vent (d) Mesh detail at OWC Vent
Figure 3.12 OWC Grid Details
3.4.6 Monitors
To provide suitable data for post processing a number of flow parameters are monitored
during the analysis and recorded to file. This is easily achieved within FLUENT by
using ‘surface monitors’ which are points, lines or areas defined within the domain from
which the user can extract and save information during each time step. As per equations
(2.14) and (2.15) we require the motion of the OWC water surface and the associated
pressure within the chamber to determine the efficiency of the OWC. The internal air
pressure can be determined by a simple point monitor that records the pressure. The
OWC chamber motion cannot be determined directly but can be deduced by placing a
pressure probe a suitable distance below the water surface and measuring the static head
50
of water with a pressure monitor (note: the monitor must always be submerged). This
pressure recording can than be converted during post processing to a head of water after
subtracting the internal air pressure. The velocity calculation can then performed using
Euler differentiation average over between three to five time steps in order to smooth
the calculated response.
In addition to the monitors of the OWC piston motion and pressure, the pneumatic
energy passing the vent can be determined directly by creating a monitor that
automatically integrates the pressure multiplied by the velocity profile over the vent.
This can be created using ‘user defined monitors’ (FLUENT User Guide, 2005).
A summary of the monitors used is given in Table 3.6.
Item Description Monitor Location
1. Pressure inside Chamber Pressure monitor x=470, y=17
2. Water Level in Chamber Pressure Monitor x=470, y=10
3. Vent
Monitor of the Integral of
pressure x velocity across the
vent
OWC vent
Table 3.6 CFD Monitors
3.4.7 Wave Generation
A testing regime was performed that aimed to cover the expected range of conditions
that may be encountered, but especially to provide detail at the peak efficiency.
Previous work (Irvin, 2004) has shown that OWC efficiency is sensitive to frequency
steps and as such suitable detail is required in order to adequately describe trends and
maxima.
To maintain the wave conditions such that it by and large suited Stokes 2nd Order
theory validity (refer to Figure 3.13), a near constant wave height was chosen as
appropriate for this study. Experimental equipment limitations meant that in reality, this
51
was unable to be achieved; hence the theoretical waves were altered to match the
experimental program as given in Table 3.7 below:
Case T (s)
Frequency (Hz)
ω (rad/s)
H (m) kh Kh
1 23.19 0.04 0.27 0.77 0.30 0.09
2 16.05 0.06 0.39 0.93 0.44 0.18
3 12.89 0.08 0.49 1.02 0.55 0.28
4 9.47 0.11 0.66 1.18 0.79 0.52
5 8.59 0.12 0.73 1.22 0.88 0.63
6 8.26 0.12 0.76 1.36 0.93 0.68
7 7.07 0.14 0.89 1.49 1.14 0.93
8 6.52 0.15 0.96 1.54 1.27 1.09
9 6.04 0.17 1.04 1.51 1.42 1.27
10 5.54 0.18 1.13 1.58 1.63 1.51
11 5.26 0.19 1.20 1.64 1.78 1.68
12 4.92 0.20 1.28 1.62 1.99 1.91
13 4.74 0.21 1.33 1.60 2.12 2.06
14 4.55 0.22 1.38 1.57 2.29 2.24
15 4.31 0.23 1.46 1.57 2.52 2.49
16 4.17 0.24 1.51 1.54 2.69 2.66
17 3.98 0.25 1.58 1.49 2.94 2.92
18 3.82 0.26 1.65 1.44 3.19 3.18
19 3.66 0.27 1.72 1.39 3.47 3.46
20 3.53 0.28 1.78 1.37 3.72 3.72
Table 3.7 Wave Conditions used for CFD Analysis
52
Figure 3.13 Wave Theory Applicability
3.4.8 Wave Height Calibration
The numerically generated wave heights were calculated by converting the pressure
probe readings to a static head using the same technique described previously for the
OWC internal motion. The nominal input wave height was then extracted from this data
using two methods. Firstly a simple ‘maxima minus minima’ technique was used
whereby the data was examined and the maximum pressure head for each was
subtracted from the subsequent minimum pressure head to produce a series of wave
heights for the time period of interest. The data sets were averaged over several waves.
Secondly, an FFT formulation was applied to the wave probe time trace to determine
the first order amplitude of the wave trace. Unfortunately due to data size limitations
only part of the same trace could be analysed using the FFT technique which limited the
accuracy of the calculation, particularly the calculation of the wave period. Even given
the limitations of the FFT code used, the match-up for wave heights was typically
H/gT2
(x10-2)
h/gT2 (x10-2)
Region of Testing
53
within 3% between the two methods. A typical time trace of the wave probe is shown
in Figure 3.14 whilst Figure 3.15 illustrates the ratio of the calculated wave height
compared to the theoretical wave heights as a function of Kh.
-1.00
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
104.6 110.14 115.68 121.22 126.76 132.3 137.84 143.38 148.92
Figure 3.14 Typical Numerical Wave Probe Output
0.00
0.50
1.00
1.50
2.00
2.50
3.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00
Kh
Wav
e H
eigh
t (m
)
Actual Wave Height at Wave Probe
Theoretical Input Wave Height
Figure 3.15 Input Wave Height versus Calculated CFD Wave Height
Initial results highlighted that errors in wave height were significantly larger than the
minor variation expected from the earlier NWT study. As wave power is a function of
the wave height squared, errors in wave height can have a significant effect on the wave
energy conversion efficiency calculation. Upon reflection, it was identified that the
standard grid and timestep size, did in fact vary sufficiently from the earlier NWT
54
validation case to cause issue. Previous work (Kleefsman, 2005) has identified that the
numerical wave generation, particularly with respect to wave heights, will vary with
both the wavelength to grid dimension ratio and the wave period to timestep ratio, with
higher ratios producing better results. Table 3.8 illustrates the ratio of measured wave
height (at OWC) versus theoretical wave height applied at the wave generation
boundary as well as the ratios of wavelength and period previously mentioned. The
results demonstrate comparable results to Kleefsman (2005).
Rather than re-mesh the domain and vary the numerical time step for each case in order
to generate ‘correct’ height waves, which would have been a laborious task, it was
decided to calibrate the waves to correct the wave height variation.
Case Ratio of measured to input wave height
Wavelength/ Grid dimension
Period/Timestep
1 1.01 165 3210
2 0.97 131 2578
3 0.90 92 1894
4 0.88 82 1718
5 0.86 78 1652
6 0.85 63 1413
7 0.82 57 1303
8 0.79 51 1208
9 0.74 44 1108
10 0.69 41 1051
11 0.64 36 984
12 0.61 34 948
13 0.58 32 909
14 0.54 29 863
15 0.50 27 834
16 0.46 25 796
17 0.41 23 763
18 0.38 21 732
Table 3.8 Numerical to input wave height results comparison
55
This re-calibration used the NWT described earlier, but this time the full range of the
conditions Table 3.7 were simulated and wave heights were measured by a numerical
wave probe. A wave probe was placed at the location of the measurement wave probe as
used in the experimental work described in later chapters. The reasoning for this was
that the numerical and experimental simulations should be matched to provide the same
wave heights as close as practically possible encountering the OWC to provide accuracy
in efficiency calculations. Additionally, given this wave probe was very close to the
OWC minimal wave height decay was expected between this reading and the OWC,
even though there may have been reduction from the wave height initially generated.
To assist confirmation that the actual waves generated were satisfactory,
notwithstanding they were smaller than the wave height generated at the boundary, the
wavelengths were compared to theory. Results of this exercise using the same wave
probe data as the wave height analysis are provided in Figure 3.16. The results show a
good correlation between the input wavelength and the calculated wavelengths with
only small errors (i.e. <5%) even for large values of Kh. This provided confidence that
the method utilised would provide acceptable results.
Wave Length Variance
70%
80%
90%
100%
110%
120%
130%
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00
kh
Act
ual W
avel
engt
h: T
heor
etic
al R
atio
Figure 3.16 Input Wavelength versus Calculated CFD Wavelength
56
3.4.9 Vent Calibration
As described earlier, the vent modelled in FLUENT is a simple vent with a pressure
drop across the vent proportional to the square of the velocity. Due to the shape of the
vent to be used later in the experimental campaign would not be tested to determine the
vent loss coefficient, a test as to the sensitivity of the numerical analysis to the loss
coefficient was performed. In this analysis, Case 10 (ref Table 3.7) was tested using a
number of loss coefficients, with unity being the considered the base. The result of the
effect on efficiency is shown in Figure 3.17. As little difference is demonstrated the
coefficient of 1.0 was maintained in all analyses.
Vent Calibration
50%
60%
70%
80%
90%
100%
110%
0 1 2 3 4 5
Vent Loss Coefficient
Rel
ativ
e Ef
ficie
ncy
Figure 3.17 Efficiency Variation with Vent loss coefficient
3.4.10 Analysis
Following each of the numerical runs the data was processed to determine the efficiency
of the OWC under the particular wave conditions. The numerical analyses were carried
out as per the following flow chart in Figure 3.18 using Excel spreadsheets.
Figure 3.19 summarised the results of the numerical testing by plotting the efficiency
against Kh. To aid in the visualisation, a least squares 3rd order polynomial has been
fitted to the data. For a full description and analysis of the results refer to discussions
later in Chapter 5.
57
Figure 3.18 CFD Efficiency Analytical Sequence
OWC Model Setup - Import mesh - Compile Wave UDF - Set solver parameters - Set results output
Fluent Analysis
Pressure Probe Data
Differentiate Wave Probe = Water Surface Velocity
OWC Wave Probe
OWC Absorbed Power =Velocity x Pressure
Efficiency of OWC =
werIncidentPowerAbsorbedPo
NWT Wave Probe
Calibrated Incident Wave Height
Incident Wave Power
58
Basecase OWC Efficiency (CFD)
R2 = 0.95650%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
KH
Effic
ienc
y
Figure 3.19 OWC Efficiency as a function of kh
(Note: 5% error bars included for information)
3.5 Front Lip Analysis
3.5.1 Introduction
Previous work (Clemente 1996) has identified that front lip geometry may have a
significant influence on the efficiency of an OWC. This work however was limited to a
unique geometry. As an extension of the analytical work variations in geometry of the
front lips and the corresponding effects on device efficiency were examined. The work
was performed in two phases; the first was a screening study using a simplified domain
to facilitate the number of analytical runs and secondly a detailed analysis was
undertaken on one of the more promising configurations.
3.5.2 Geometry
A series of CFD simulations was performed to investigate the effect of the front lip
shape on the efficiency of wave power absorption. An initial screening study was
59
performed with a reduced domain size to reduce the computational effort required. An
illustration of the screening configuration is shown in Figure 3.20.
Figure 3.20 OWC Front Lip Numerical Domain
The simulations have been performed with a variety of different front lip sizes and
shapes are described in Table 3.9.
Basecase Lip
Square Lip#2
Rounded Lip #1
Rounded Lip #2
Rounded Lip #3
0.5m thick 1.0m thick 0.5m Thick + rounding
0.5m thick + 0.5m dia semi-
circles
0.5m thick + 1.125m dia semi-
circles
Table 3.9 Front Lip Variations Analysed
The wave environment was restricted to Stokes 2nd order, monochromatic, progressive
waves with frequencies and wave heights chosen to provide a parametric variation
Wave Probe
Pressure Probe
OWC Model
Stokes 2nd order wave maker
150m
6m
Vent
Water Phase
Air Phase
60
similar to the likely operating conditions of a prototype wave energy device with a peak
efficiency at approximately T=8seconds. As discussed earlier, at the time of the analysis
the representative modelling of this study was reflecting a proposed prototype – as such
there was some variation in the water depth compared to the previous OWC analysis.
These screening simulations were thus performed for a water depth of h=6m, a lip
submergence of d=2m and waves with wave amplitude H=2m for 5 wave periods as
given in Table 3.10.
T Frequency ω H Case (s) (Hz) (rad/s) (m)
kh Kh
1 12.0 0.08 0.73 2.0 0.42 0.17
2 9.0 0.11 0.66 2.0 0.57 0.29
3 7.5 0.13 0.49 2.0 0.71 0.43
4 6.5 0.15 0.39 2.0 0.84 0.58
5 4.0 0.25 1.57 2.0 1.63 1.44
Table 3.10 CFD Lip Test Regime
Results of the CFD testing are shown below in Figure 3.21 and the average of the test
efficiencies summarised in Table 3.11.
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
- 0.4 0.8 1.2 1.6
Kh
Effic
ienc
y
CFD: Basecase Square Lip CFD: Square Lip #2
CFD: Round lip #1 CFD: Round lip #2
CFD: Round lip #3
Figure 3.21 Efficiency of Various Lip Shapes
61
0%
20%
40%
60%
80%
100%
120%
140%
160%
- 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Kh
Out
put R
elat
ive
to B
asec
ase
Model Test: Square Lip #2 Model Test: Round lip #1
Model Test: Round lip #2 Model Test: Round lip #3
Figure 3.22 Relative Efficiency of Various Lip Shapes
Case Average Efficiency % Increase to Basecase
Basecase Square Lip#1 24.0% 0 %
Square Lip #2 24.4% 2%
Round lip #1 26.1% 9%
Round lip #2 29.5% 23%
Round lip #3 30.1% 25%
Table 3.11 CFD Lip Test Efficiencies
3.5.3 Results and Discussion
Figure 3.22 shows that variation in efficiency for each of the different lip shapes. Initial
examination shows that the efficiencies, overall for the 6m water depth are significantly
less than the 11.5m water depth. This may be partly attributed to the depth of lip
62
submergence as a function of water depth which has been shown (Evans and Porter,
1995) to be a significant influence on efficiency. For example, in detailed analysis
performed in Section 3.4 this lip submergence to water depth ratio was 0.15 and the
peak efficiencies were in the order of 70% whereas here it is 0.33 and the efficiencies
are less than 40%.
In addition to the numerical post-processing, flow visualisation plots can also be used to
demonstrate differences in the cases analysed. Figures 3.23 (a) to (e) show the flow
pattern mid-way through an in-stroke for the case of T=9seconds, H=2m for each the lip
configurations. The flow demonstrates the development of an area of recirculation just
behind the front lip as the fluid flows into the OWC chamber. The formation of these
flows is a good indicator of energy loss, particularly when the counter flow extends into
the domain away from a wall.
The flow patterns for the smoother lips (Round Lip#2 and Round Lip#3) illustrate well
formed flow patterns with smaller area of opposing flows and where there are flows
they are inclined to be a good deal closer to the lip walls. This marries with the
analytical results which indicate a higher efficiency of conversion of the incoming wave
energy.
This may be understood by effect of the greater viscous losses as particle velocities
increase as well as the quadratic loss coefficient presented by the OWC vent as wave
height increases.
Also illustrated clearly in the flow visualisations are the air flow circulation patterns
within the OWC. The circulation pattern appears to be reflected in the vent outflow
which is shown to be directional even though the original intention was to have the flow
simply orthogonal to the OWC water column. This was perhaps due to the offset vent
location and this too may be an area of significant energy loss and may be of interest for
further research.
63
(a) Basecase lip
(c) Round Lip#1
(b) Square Lip #1
(d) Round Lip #2
(e) Round Lip #3
Figure 3.23 Velocity Vectors at time step t=40seconds for T=9secs, H=2m. Note: The free surface is also shown for information
64
3.5.4 Supplementary Lip Detailed Analysis
The preliminary front lip study indicated the efficiency varies considerably depending
upon the type of front lip chosen. Following this work a variation to the detailed OWC
analysis, again for a water depth of h=11.5 was performed. For this work, instead of the
simple 0.5m wide squared off front lip, Round Lip 2 was utilised for the geometry of
the OWC. Round Lip 2 was chosen as it had provided a more general increase in
efficiencies over the entire range of kh values tested in the screening study compared to
other variants.
The analysis was performed using the ‘full domain’ for a water depth of 11.5m and the
same input wave conditions as per table 3.8. The actual incoming wave heights were
again re-calibrated as per previous work and the results of the analysed efficiency are
shown in figure 3.24 where the results have been plotted against the Basecase lip shape.
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
KH
Effic
ienc
y
Basecase Roundlip
Basecase Trendline Roundlip Trendline
Figure 3.24 OWC Efficiency: Basecase Lip Vs Rounded Lip
65
0%
100%
200%
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00KH
Effic
ienc
y R
atio
Rounded Lip Efficiency Ratio to Basecase Linear Trendline
Figure 3.25 Relative Efficiency increase between the Rounded and Basecase Lip
As per the screening study, there is a marked increase in the converted efficiency of the
incoming wave – on average the increase is in the order of 18%. Figure 3.25 illustrates
the increase in efficiency compared to the Basecase as a function of Kh. This shows that
as Kh increases, so too does the effect of the rounding on the lip. This may to be
attributed to the increase in both the flow velocities for waves of increasing Kh which
would intuitively suggest more likelihood of turbulence.
Figures 3.26 (a) to (c) illustrate the comparison between the Basecase and the Rounded
lip monitor outputs. These indicate that there is an increase in both the pressure (b) and
water column (c) results, thus providing in an increase in the energy transferred through
the OWC vent (c). These results also show that the power absorbed on the up-stroke of
the OWC water column is significantly higher than on the downstroke of the water
column. This result has been highlighted previously by Raju, Jayakurna and Neelami
(1992).
66
0
5000
10000
15000
20000
25000
30000
35000
40000
100 110 120 130 140 150time (s)
Pow
er (W
)
Basecase
Round Lip 2
(a) Vent Energy Reading (x-axis=time, y axis=Watts)
-8000
-6000
-4000
-2000
0
2000
4000
6000
8000
10000
100 110 120 130 140 150
time (s)
Pre
ssur
e (P
a)
BasecaseRound Lip 2
(b) OWC Pressure Reading
67
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
100 110 120 130 140 150
time (s)
OW
C W
ater
Lev
el (m
)
Basecase Lip
Round Lip 2
(c) Wave Probe Reading
Figure 3.26 Results of Basecase Lip Vs Rounded Lip 2 - Key OWC
Performance Monitors for Case 9, H=1.5m, T=6s.
68
4 EXPERIMENTAL PROGRAM
4.1 Introduction Explicit analytical solutions for the hydrodynamic – pneumatic interactions within an
OWC are non-existent. In order to test the validity of wave energy system designs or
concepts, designers must typically perform dedicated experimental campaigns.
Experimental programs are a costly exercise and may require several iterations prior to
completion of the final design, thus incurring large costs. Advances in computer
hardware combined with the accessibility of commercially available CFD software
offers the possibility to create complete numerical models with relative ease. These
models however still require validation over a systematic range of conditions to
ascertain and demonstrate their accuracy.
Due to the infancy of the wave energy industry and the large variety of competing
technologies, no comprehensive experimental data sets is as yet available for the
modelling of generic OWC type devices.
This chapter describes the experimental work aimed at determining the efficiency of an
OWC device under a variety of conditions to provide data for comparison with the
numerical analysis previously described. Firstly, an overview of the common
experimental procedure is described including an outline of the wave environment and
theoretical considerations. Secondly, the key experimental results obtained are
discussed.
4.2 Experimental Testing
4.2.1 Experimental Overview Two sets of experiments will be discussed in this study: Firstly, an investigation of
OWC efficiency for a variety of different wave and model configurations. Secondly, an
69
investigation of a single OWC geometry subjected to a very detailed range of wave
conditions.
The series of experiments were performed in the University of Western Australia’s,
Shenton Park facility. The wave tank measures 49m long x 1.5m wide x 1.5m deep
wave tank and is equipped with a piston-type wave-maker capable of generating two-
dimensional regular and irregular waves. At the downstream end of the tank, a
corrugated beach with baffles is installed for damping purposes. Details of the wave
tank are provided in Appendix F.
A false bottom remained in-place in the wave tank during the first set of OWC
experiments due to a concurrent experimental programme. The false bottom was
removed prior to the detailed OWC experimental work.
4.2.2 Model Geometry Typically, model dimensions are governed by the width and depth of the experimental
wave tank facility. These dimensions must be large enough to allow clear understanding
of the wave-structure interaction yet small enough to enable representative waves to be
generated by the wave making facility. The scale factor decided upon for these
experiments was 1:12.5 this giving a model length of 640mm to represent the 8m length
at full scale. Figure 4.1 shows the principal model dimensions.
As previously described in the numerical modelling discussion this study is based upon
and OWC prototype under investigation for use in Australian waters. As this research
occurred whilst the Australian prototype effort was still in the development phase, the
full water depths varied between 6m and 11.5 due to various final siting changes. This
variation is reflected in the experimental work whereby a water depth of either 480mm
or 920mm is used.
70
Figure 4.1 Model Schematic
The model was constructed from timber framing with Perspex panels to allow viewing
of internal water movement during test. The joints between the Perspex and the timber
framing were filled with silicone sealant to prevent air leakage when the chamber is
pressurised under wave action. The chamber spanned between the sides of the wave
tank to effectively prevent wave transfer beyond the device. The base case model
utilised a 0.04m thick vertical face (prototype scale=0.5m). Removable front walls to
the lower part of the OWC were fabricated to allow for easy changing of key parameters
such as lip shape and submergence depth during the tests. The lips included rubber seals
to prevent air leakage. The front lip alternatives tested are provided in Figure 4.2.
The vent which represented the turbine load was created by fabricating the model with a
30mm slot in the roof with an adjustable width cover of Perspex that allowed
adjustment of the effective slot width from fully closed up to fully open.
Lip thickness
vent
1.37mLip submergence
0.48-0.92m
0.64m
71
Basecase Lip
Square Lip#2
Rounded Lip #1
Rounded Lip #2
Rounded Lip #3
40mm thick (0.5m)
80mm thick (1.0m)
40mm thick + rounding
40mm thick 40mm dia semi-
circles
40mm thick 90mm dia semi-
circles
Figure 4.2 Experimental Front Lip Variations
4.2.3 Model Scaling and Test Regime With the length scale factor set at 1:12.5, the hydraulic aspects of the model were
governed by Froudian similitude, which provides for the following ratios:
Length LR = 12.5
Time tR = LR0.5 ≈ 3.536
Velocity vR = LR/tR ≈ 3.536
A summary of the key experimental parameters is given in Table 4.1.
Environmental / Physical Variable Prototype Experimental Model
Wave height: (H) 2 m 160 mm
Wave period: (T) 8 s 2.83 s
Water depth: (h) 6 m, 11.5m 480 mm, 920mm
Front Wall distance below sea level: (d) 2 m 160 mm
Vent Opening Turbine 5mm slot (0.8% of roof area)
Table 4.1 Experimental Parameters
72
4.2.4 Instrumentation and Measurement The model was located approximately 30m from the piston type wave maker. Wave
probes were positioned along the centreline of the tank and also within the OWC model
to measure incident wave heights and OWC internal water levels. Two pressure
transducers were fitted to the inside back wall of the OWC between the still water level
and the OWC roof. The signal readings for both wave probes and pressure transducers
were amplified and then read into a personal computer using Labview data acquisition
software.
A schematic of the model is shown below in figure 4.2. Photographs of the setup are
provided in Appendix B.
Figure 4.3 Experimental Wave Tank Set-up
External
Wave
Probes
Wave
Probes
OWC
Vent Pressure
Transducers
False Bottom
Piston
Wave
maker
Piston wave
maker
Beach Wave probes
(WP1 & WP2)
30m
73
4.2.5 Wave Height Measurement Wave heights in the model were measured with wave probes that were logged via a
signal amplifier, through an analogue to digital signal converter to a data acquisition
computer. The probes measured instantaneous water depth at each point with a
sampling rate of 20 Hz. The wave probes were of the resistance type utilising two wires
approximately 10 mm apart.
Two wave probes were placed at distances of 3m and 9.5m from the face of the device
to measure the incoming wave heights and three wave probes were placed inside the
chamber at quarter points to measure the movements of the water column inside the
OWC chamber.
The wave probes required manual calibration to account for changes in resistance due to
ambient water temperature variations. The calibration of the wave probes was checked
twice daily during testing, such that the accuracy of crest to trough distances (wave
heights) presented is expected to be within ± 2 to 4 mm at model scale (i.e. ± 25 to
50 mm at full scale ≈ 1-2%).
4.2.6 Pressure Measurement Pressure inside the OWC was measured by pressure transducers mounted on the back
wall of the chamber. The pressure transducers were positioned between the still water
level and the top of the OWC chamber to minimise the effect of either the water surface
or the air flow in and out of the vent. The pressure transducers were also fed to the data
acquisition computer using a signal amplifier and analogue to digital converter. The
pressure transducers did not require calibration as they were provided complete with
certified calibration coefficients valid to an accuracy of ±1 to 2% within the pressure
ranges tested.
74
4.2.7 Data Analysis
4.2.7.1 Experimental Data
The raw data from each of the measurement devices (wave probes & pressure
transducers) during each individual run were captured by a data acquisition computer.
The computer utilised Labview software and combined with the calibration coefficients,
converted the data directly into actual pressures and water surface positions. The results
were saved as a series of data files.
In addition to the data sets from testing, digital photographs and analogue video for
most of the model tests has also been recorded (Refer to photographs in Appendix B).
4.2.7.2 Analysis
The efficiency of an OWC can be defined as the average energy imparted into the
internal OWC water column divided by the average power of the input wave.
The input wave power was determined from theory (formulae 2.13) using wave heights
and wave periods extracted from the experimental results. The wave probe farthest from
the OWC was used to determine the wave heights as statistically few wave height
recordings were available from the nearer wave probe prior to also capturing waves
reflected from the OWC. The wave heights were determined by averaging the maxima
minus minima recordings over several waves.
The OWC energy absorption was determined by the integral over time of the internal
OWC water column velocity multiplied by the internal pressure. The water column
velocity was resolved by differentiating the OWC internal wave probe recordings with
respect to time. The velocity calculation was performed using Euler differentiation
average over 3 time steps in order to smooth the calculated response. Time domain
integration of the experimental data was then performed to determine the power
absorbed by the OWC internal water column. A flow diagram of the analytical steps is
provided in Figure 4.2.
75
Figure 4.4 Experimental Flow Chart
Experimental Model Setup
- Calibrate Wave Probes - Calibrate Wavemaker - Calibrate Pressure Probes - Set up Data Acquisition
Computer
Experimental Runs- Data Acquisition - Convert raw data
to Real data
Pressure Probe Data output
Differentiate Wave Probe = Water Surface Velocity
OWC Internal Wave Probe
output
OWC Absorbed Power =Velocity x Pressure
Efficiency of OWC =
werIncidentPowerAbsorbedPo
External Wave Probe output
Determine Incident Wave
Height
Incident Wave Power
76
4.2.8 Front Lip Testing The wave environment was restricted to monochromatic progressive waves with
frequencies and wave heights chosen to provide a parametric variation similar to the
likely operating conditions of a real wave energy device for water depth of 6m which
was to have a nominal efficiency peak in the region of T = 7 to 8 seconds. The wave
periods were targeted to correlate with the CFD analysis data set given in Table 3.9.
Due to sensitivities with the wave generation equipment, the resulting test conditions
are marginally different as given in Table 4.2.
T Frequency ω H Wave Condition (s) (Hz) (rad/s) (m)
kh
1 12.1 0.08 0.73 2.0 0.42
2 9.1 0.11 0.66 2.0 0.57
3 7.5 0.13 0.49 2.0 0.70
4 6.5 0.15 0.39 2.0 0.84
5 4.1 0.24 1.53 2.0 1.57
Table 4.2 Preliminary Testing Wave Properties
In total, four different series of tests were carried out in this phase of the experimental
work. These are summarised as follows:
(1) Vent Calibration: Using the basecase model, tests were carried out for a range of
vent positions in order to ascertain highest efficiency setting. This ‘optimised’ vent
setting was then used for subsequent tests.
(2) Basecase Wave Height Testing: Basecase model tested for 3 wave heights. Vent set
at maximum efficiency from Stage 1.
(3) Lip Submergence: Basecase model tested with alternate 1m lip and 3m lip
submergence.
77
(4) Alternate Lip Configurations: 4 x alternate front lip shapes tested for 3 x wave
heights.
This testing regime is summarised in Table 4.3.
Prototype Model Test Description Lip Type, Depth
# Tests
T (s) H (m) T (s) H (m)
1 Vent Calibration
Base, 2m 20 4.1-12.06 2 1.13-3.39 0.16
2 Base Case Base, 2m 15 4.1-12.06 1,2,3 1.13-3.39 0.08-0.24
3 Submergence Base, 1-3m 30 4.1-12.06 1,2,3 1.13-3.39 0.08-0.24
4 Alternative Lip Types
4 variations, 1-3m
60 4.1-12.06 1,2,3 1.13-3.39 0.08-0.24
Table 4.3 Preliminary Testing Regime
Acceptable wave height repeatability was shown throughout the testing program. This
was demonstrated by measuring the wave probes with and without the OWC in the tank.
A variation in wave height of less than 5% was noted.
For each set of conditions tested, the water surface was still prior to initiating the wave
maker. Data acquisition commenced manually just prior to the first waves approaching
the wave probes OWC and continued for approximately 10-20 wave periods.
Measurements were completed prior to the waves reflected from the OWC model, being
reflected back from the wave maker.
A typical set of experimental data plotted against measured time step (∆t=0.05 secs) is
given in Figure 4.5.
78
OWC Chamber Pressure
-400
-300
-200
-100
0
100
200
300
400
0 250 500 750 1000 1250
Timestep (0.05 secs each)
Pres
sure
(Pa)
Far Pressure Probe(Pa)
Near Pressure Probe(Pa)
(a) Pressure Recording (Pa)
OWC Internal Wave Probes
-6
-4
-2
0
2
4
6
0 600
Timestep (0.05 secs each)
Wav
e H
eigh
t (cm
)
Near Internal
Mid Internal
Far Internal
(b) Incoming Wave Height (cm)
Far Wave Wave Probe
-10
-5
0
5
10
15
20
0 200 400 600 800 1000 1200 1400 1600
Timestep (0.05 secs each)
Wav
e H
eigh
t (cm
)
c) OWC Chamber Wave (cm)
79
OWC Chamber Inner Surface Velocity
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
1 46 91 136 181 226 271 316 361 406 451 496 541 586 631 676 721 766 811 856 901 946 991 1036 1081 1126 1171
Time Step (.05s)
Velo
city
m
/s
Velocity (average depth) m/s
(d) Calculated OWC Chamber Surface Velocity (cm/s)
Absorbed Power per period(=pi x vi x Area) delta T
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
1 51 101 151 201 251 301 351 401 451 501 551 601 651 701 751 801 851 901 951 1001 1051 1101 1151
Time Step (e) Absorbed Power (W)
Figure 4.5 Typical Experimental Results. (Basecase lip H=2m, T=6 seconds)
4.2.9 Results and Discussion
4.2.9.1 Preliminary Review
The initial results from the experimental work show the maximum efficiencies obtained
using the Basecase lip were in the region of 23% compared to the 30% obtained using
the numerical analysis. Although it is not entirely unexpected for there to be
discrepancy between testing techniques this is a significant difference. It is believed that
some effect from the false bottom in the experimental tank may have had some affect on
the experimental wave quality. To evaluate this, the wave probe used to determine input
wave energy is compared with the corresponding 2nd order theoretical wave for the case
80
of H=2, T=6s in Figure 4.6. Both wave plots are shown at full scale. The curves show
good agreement for the wave peaks, however there is a significant secondary
component visible in the wave troughs which may explain some of the differences
between the experiment and the simulation and is likely to have been influenced by the
false bottom in the tank.
To further assist in the determination of experimental uncertainty the following estimate
of error budget is presented:
Item Error Estimated Min Estimated Max
Incoming Wave Power
a) Incident Wave Probe Error 1% 2%
b) False Bottom Effect on Incident Wave Probe 2.5% 10%
c) Incident Wave Power Calculation Error ( Note: Power is proportional to H2xP)
4% 26%
Incoming Wave Power
d) Pressure Transducer Error 1% 2.5%
e) OWC Wave Probe Error 1% 2%
f) Absorbed Wave Power Calculation Error ( Note: Power is proportional to H2xP)
3% 7%
Efficiency Calculation
g) Efficiency Calculation Error 7% 34%
Table 4.4 Error Budget Estimate
It should be noted that this additional hump in the wave troughs varied in prominence
for different wave conditions. For the case illustrated in Figure 4.6, the error in wave
height is estimated at approximately 10% which is used as the maximum error effect for
the false bottom in Table 4.4. Using these values, the error on efficiency has been
determined and has an estimated upper bound error of +/- 34%. A lower bound estimate
is in the order of 7%. This would imply significant experimental variations may occur
81
principally due to the effect of the false bottom and as such the results should be
evaluated with this in mind.
-1
-0.5
0
0.5
1
1.5
2
109.5 117.4 125.3 133.2
Time (secs)
Am
plitu
de (m
)Exp H2 T6
Stokes 2nd Order Theory
Figure 4.6 Experimental Vs Theoretical wave profiles for H=2m, T=6s case
4.2.9.2 Stage 1: Vent Testing
The purpose of this testing was to determine the vent setting on the experimental OWC
that provided the greatest average efficiency over the range of test cases for the basecase
lip configuration. The vent (or turbine in a prototype OWC) provides both load and
damping to the water column motion and thus has a significant effect on efficiency and
is an important design consideration during development of a prototype OWC. That is,
OWC designers would aim to match the turbine characteristics to the hydrodynamic
conversion properties in an effort to maximise energy extracted (Curran, Stewart and
Whittaker, 1998).
The experimental results are shown in Figure 4.7 whereby the hydrodynamic efficiency
hydroη is plotted versus the infinite water depth parameter KH where Kh=kh tanh kh.
This clearly illustrates that the efficiency is greatest for a vent opening of 0.5 cm. This
has been the value used for all subsequent experiments. This gap of 0.5cm is effectively
82
0.8% of the OWC plan area and is a similar ratio as obtained to testing by Koola (1990)
whereby a nominal opening ratio of 1% was identified as the most efficient.
Phase 1 : Efficiencies as a function of Vent Gap (cm)
0%
5%
10%
15%
20%
25%
0.0 0.5 1.0 1.5 2.0Kh
Effic
ienc
y
0.0
0.5
1.0
2.0
Figure 4.7 Basecase Efficiency Vs Vent Opening (cm)
4.2.9.3 Stage (2) Wave Height Testing
The second stage tested the Basecase model for a range of wave heights and wave
periods. The results of the testing are shown in Figure 4.8 with the efficiency plotted
against the infinite water depth parameter Kh.
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
0 0.4 0.8 1.2 1.6 2
Kh
Effi
cien
cy H=1mH=2mH=3m
Figure 4.8 Wave Height Vs Efficiency Results
83
Figure 4.8 shows that wave height does influence the efficiency of the OWC device
with a general trend of efficiencies decreasing with increasing wave height. This may be
a reflection of the greater viscous losses expected as particle velocities increase as well
as the increase in presented by the OWC vent as wave height increases given that losses
will be proportional to velocity squared.
4.2.9.4 Stage (3) Lip Submergence Testing
The third stage tested the Basecase model for a range of front lip submergence {d/h =
0.17 (d=1m), d/h = 0.33 (d=2m) and d/h=0.5 (d=3m)}. The results of the testing are
shown in Figure 4.9 with the efficiency plotted against the infinite water depth
parameter Kh.
Phase 3 Testing ComparisonH=2m
0%5%
10%15%20%25%30%35%40%
0 0.5 1 1.5 2Kh
Effic
ienc
y
Lip Submergence = 1mLip Submergence = 2mLip Submergence = 3m
(a) Wave height =2m
84
Phase 3 Testing ComparisonH=3m
0%5%
10%15%20%25%30%35%40%
0 0.5 1 1.5 2Kh
Effic
ienc
y
Lip Submergence = 1mLip Submergence = 2mLip Submergence = 3m
(b) Wave height =3m
Figure 4.9 Efficiency as a function of lip submergence
These results indicate the important influence of lip submergence on the efficiency of
the OWC. The current trend indicates efficiencies increasing with decreasing
submergence. Obviously there must be a limit to the increase as lip submergence
reduces to zero whereby it is expected that we would encounter internal OWC air
escaping under the front lip as the seal. This may have been expected for the h=1m lip
submergence and H=3m incident wave height but observations during the testing
showed that the impingement of the wave trough on the front lip was not in tune with
the internal water column motion which lagged behind the outside water movement.
Though there was potential for the water seal between the inside and outside of the
OWC to be broken, as the OWC water surface was higher this was prevented from
occurring by the flow of water from inside to outside of the OWC which effectively
“filled the gap”.
The results for the higher wave heights indicate a possible increase in efficiency with
wave height that may have been a function of the effect whereby the presence of the
wave trough outside of the OWC in fact aided the downstroke of the OWC water
column thus increasing energy extraction in this part of the wave cycle.
85
4.2.9.5 Stage (4) Alternative Lip Testing
The fourth stage tested alternative lip shapes as presented in Figure 4.2. The results of
the testing for H=2m are shown in Figure 4.10 and include error bars with the maximum
error of 34% for the Basecase condition for information.
Phase (4) Testing H=2m
0%
5%
10%
15%
20%
25%
30%
0.10 0.50 0.90 1.30 1.70Kh
Effic
ienc
y
Model Test: Basecase Square Lip Model Test: Square Lip #2
Model Test: Round lip #1 Model Test: Round lip #2
Model Test: Round lip #3
Figure 4.10 Lip Efficiency Vs Kh Results
(Note: Error Bars included for Basecase at +/- 34%)
Although it is clear to see that the alternate lips have higher efficiencies, to assist with
the clarity of discussion, the results in Figure 4.10 have also been re-plotted as a ratio of
the efficiency of the Alternate lip efficiency divided by the Basecase lip efficiency.
These results are plotted in Figure 4.11.
Figure 4.11 clearly shows all alternatives proposed show increases of 20% to 30% for
Kh < 0.5. For increasing Kh (and hence decreasing wave period) the effects are reduced
for the wider lip and the largest of the rounded lips suggesting that they behave in much
the same manner. The two smaller rounded lips maintained the efficiency gains
consistently across the Kh values tested. It should be noted that although the potential
error within the results due to experimental inaccuracies may be up to 34% as stated,
86
given the general consistency within the results one could assume that the increases in
efficiency for the lip variants are genuine effects.
Phase (4) Testing H=2m
90%
100%
110%
120%
130%
140%
150%
0.10 0.50 0.90 1.30 1.70Kh
Effic
ienc
y
Model Test: Square Lip #2 Model Test: Round lip #1
Model Test: Round lip #2 Model Test: Round lip #3
Figure 4.11 Efficiency (Ratio to Basecase) Vs Kh Results
To further illustrate the testing, an average efficiency for the tested Kh values has been
summarised in Table 4.5. This illustrates that the effects of the best alternative lip is
similar in magnitude to the benefits obtained from reducing the front lip submergence.
Case Average Efficiency % Increase to Basecase
Basecase Square Lip 15.1% -
Basecase Lip (1m submergence) 19.3% 28%
Basecase Lip (3m submergence) 13.0% -14%
Square Lip #2 17.6% 16.9%
Round lip #1 18.6% 22.8%
Round lip #2 18.4% 22.2%
Round lip #3 18.8% 24.6%
Table 4.5 CFD Lip Test Efficiencies
87
Further discussion and comparison will be made between the experimental and
numerical analysis in the following chapter.
4.3 Detailed Frequency Experiment
4.3.1 Introduction The front lip experimental work highlighted several areas interest to OWC efficiency
such as the importance of load (i.e. vent) matching, wave height, front lip submergence
and front lip shape effects. Similarly to the numerical studies, these results
demonstrated the efficiency effects are frequency dependant although the exact nature
of this is perhaps unclear due to the limited number of frequency steps used in the
testing. This was the motivation for this second set of testing – that is, to develop a data
set that included much finer resolution of the frequency dependencies.
4.3.2 Experimental Setup A series of tests was performed using much the same experimental methodology and
equipment including the same OWC model using a water depth of 11.5m to match the
detailed CFD work. Other minor changes to the setup were that the Basecase front lip
submergence was set to 1.75m rather than 2m, the vent was set at 5mm. In addition, the
false tank bottom present in the previous testing was removed. This series of tests was
carried out by Irvin, 2004. A schematic of the experimental setup used is shown in
figure 4.12.
88
Figure 4.12 OWC Experimental Setup
4.3.3 Experimental Data and Analysis
4.3.3.1 Testing Regime
The raw experimental data was made available from the Irvin (2004) experimental
campaign which represents the Basecase square lip configuration. The tests performed
as a function of frequency and wave height are set out in Table 4.6.
4.3.3.2 Analysis
The previous experimental analysis for the OWC lip study was performed using a time
domain combination of the OWC velocity and pressure which was averaged and
compared to the average input wave energy. This analytical method was repeated for
this study, but additionally, a frequency domain analysis was also developed and
applied. The aim of this frequency domain analysis was to determine whether the OWC
outputs had any combinations from higher order frequencies other than the wave
frequency.
Wave Probe 1 Wave Probes 3
& 42 Pressure Transducers
Beach
Model
Piston wave
maker
18.10m 15.01m 4.50m
Wave Probe 2
89
Prototype Experimental Data Test
f(Hz) H (m) f(Hz) H (m) kh
1. 0.04 0.77 0.15 0.062 0.2977
2. 0.06 0.93 0.22 0.074 0.437
3. 0.08 1.02 0.27 0.081 0.5536
4. 0.11 1.18 0.37 0.094 0.7865
5. 0.12 1.22 0.41 0.098 0.8845
6. 0.12 1.36 0.43 0.108 0.9287
7. 0.14 1.49 0.50 0.119 1.1388
8. 0.15 1.54 0.54 0.123 1.2749
9. 0.17 1.51 0.59 0.121 1.4241
10. 0.18 1.58 0.64 0.126 1.6292
11. 0.19 1.64 0.67 0.131 1.7744
12. 0.20 1.62 0.72 0.129 1.9855
13. 0.21 1.60 0.75 0.128 2.1188
14. 0.22 1.57 0.78 0.125 2.2855
15. 0.23 1.57 0.82 0.125 2.5193
16. 0.24 1.54 0.85 0.123 2.6851
17. 0.25 1.49 0.89 0.119 2.9403
18. 0.26 1.44 0.93 0.115 3.1883
19. 0.27 1.39 0.97 0.111 3.4648
20. 0.28 1.37 1.00 0.109 3.7249
Table 4.6 Detailed Frequency Test Program
A Matlab routine was developed to perform the frequency domain analysis. A flowchart
of the analysis steps is provided in Figure 4.13 and can be summarised as follows:
Output Power
i. The internal OWC Wave Probe readings were converted to velocity using simple
Euler differentiation. This was averaged over a 5 time steps to reduce the numerical
noise.
ii. A data set of the converted power in the time domain was obtained by multiplying
the OWC water column velocity by the Internal OWC pressure
90
iii. A FFT analysis of this data set was taken to reveal the magnitude of the converted
power
Input Wave Power
i. The Wave Probe furthest from the OWC was passed through a FFT algorithm to
reveal the input wave height and frequency
ii. The Input wave power was then calculated using formula 2.13
Figure 4.13 Frequency Analysis Flowchart
Experimental Data
Pressure Probe Output
Differentiate Wave Probe = Water Surface Velocity
OWC Internal Wave Probe Output
OWC Absorbed Power =Velocity x Pressure
Efficiency of OWC =
werIncidentPowerAbsorbedPo
External Wave Probe Output
Perform FFT on wave probe data to
determine Hi, fi
Incident Wave Power
Pin
Perform FFT on absorbed Power to
determine Poutput
91
4.3.4 Discussion of Detailed Frequency Experiment Two different analysis techniques were used to determine the output power from the
OWC. Figure 4.14 shows the efficiency plotted as a function of non-dimensional wave
number Kh. To assist with the interpretation of the results, 3rd order polynomial least
squares regression curves have been added to each data set.
Firstly, it can be observed that the curves show an almost identical regression curve
with peak efficiency being of approximately 70%. The locations of peaks are almost
identical with a value of Kh~1.4. The closeness of match between the two curves
indicates that the calculation of efficiency is validly modelled by either methodology.
Accordingly it can be said that the displacement of the OWC water column is a linear
response.
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Kh
Effic
ienc
y
Frequency Domain Analysis
Time Domain Analysis
Poly. (Frequency Domain Analysis)
Poly. (Time Domain Analysis)
Figure 4.14 Detailed Analysis: Frequency Domain Vs Time Domain Test Results
Notes
1. Both trendlines are 3rd order polynomials and have R2 value >0.95.
2. Error Bars equal to 5% shown for information.
92
Also of interest is that if one assumes that the regression curves model reflect the ideal
situation, it can be seen that individual data points for even adjacent test conditions can
vary within +/-10%. This infers that actual individual wave efficiencies may be
sensitive to small changes in Kh and that one should aim to test sufficient cases in order
to define the curve sufficiently such that it may on average represent the real life
situation. This is of particular importance near the peak efficiency which will typically
be a major point of interest to researchers.
As with the initial experiments, further discussion and comparison will be made
including comparison between the experimental and numerical results in the following
chapter.
93
5 COMPARATIVE EVALUATION
5.1 Introduction The previous chapters outlined the numerical simulations and the experimental
programme conducted on a generic OWC type device under a variety of comparative
wave conditions. In this chapter, a combined evaluation of the numerical and
experimental results will be performed. Firstly, the preliminary studies which focused
on efficiency effects of the front lip will be examined and secondly the results of the
detailed OWC frequency analysis will be compared.
5.2 OWC Front Lip Analysis The OWC was initially analysed numerically using a simplified computational domain
to determine the efficiency of the OWC with a variety of front lip shapes. It was also
subsequently experimentally tested under similar conditions.
The efficiencies of the experimental and numerical simulations for the Basecase square
front lip are shown in Figures 5.1 and 5.2. In Figure 5.1(a) the efficiency predicted by
CFD is compared to the efficiency predicted by the experimental campaign for a
nominal wave height of 2m. The results demonstrate a rather poor correlation between
the two methods. In an effort to explain this mismatch, further CFD analysis was
performed to provide a number of intermediate data points between the previous results
of Figure 5.1 (b). These results, shown in Figure 5.2, highlight the significant efficiency
variations that may occur with only minor changes in the value of Kh. This emphasizes
the sensitivity of this type of analysis and hence one must carefully consider the step
size between adjacent data points in order to sufficiently capture the correct results.
Furthermore the previous detailed numerical analysis for the deeper water depth
(Section 3.4) identified that it is evident that the efficiency, in general, follows a curve
even though there may be significant error ‘noise’ in individual data point’s results.
Given the detailed results have efficiency offsets from the fitted curve of up to 10%
efficiency value, the efficiencies for the shallow water OWC is typically in the region of
25%, and the limited number of data points it is of no surprise there is some difficulty in
94
comparing the two methods even without considering the errors as described in Table
4.4.
The overall trends do show some similarity to the trends described by the Evans and
Porter (1995) theory previously described in Figure 2.4 and Figure 2.5. Particularly for
the cases of a/h=0.5 the efficiency curve is asymmetric with a peak at smaller values of
Kh, tapering off as Kh increases. As the value of b/h is 1.33 one may expect that the
resonance will decrease even further and as such the peak at Kh≈0.3 is not inconsistent
with the theory albeit at significantly lower peak efficiency value. This same effect of
reduced efficiency for shallower water depths is also confirmed by studies by Wang,
Katory and Li (2002).
The magnitude of the peak efficiencies for CFD analysis approached 50% whereas the
experimental efficiencies averaged approximately 20%. These are significantly lower
than the ideal 100% efficiency theoretically possible. Although the lack of experimental
data points may have accounted for ‘missing the peak’, this is only expected to account
for some of the reduction. As shown in the lip submergence tests, decreasing
submergence from a basecase value of a/h=0.33 to 0.17 provides a significant (i.e. 50%)
increase in the magnitude of efficiency and such may account for a major portion of the
reduced efficiency. In addition, it is expected that losses through the air vent and the
coarse optimisation of the vent will also have significant impacts.
0%
10%
20%
30%
40%
50%
60%
- 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Kh
Effic
ienc
y
CFD:BasecaseSquare Lip
Experimental
(a) CFD & Experiment for similar Kh
0%
10%
20%
30%
40%
50%
60%
- 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Kh
Effic
ienc
y
CFD:BasecaseSquare LipDetailedBasecaseAnalysisExperimental
(b) CFD with detailed Kh results
Figure 5.1 Basecase Lip Efficiencies: Experimental Vs CFD
95
Another area of potential difference between the numerical work and the experimental
work may be attributable to scale effects. Initially, this was thought to be from the
compressibility of the air. To determine if this has a significant effect, a matching series
of numerical simulations was performed, but this time the numerical domain
dimensions were scaled down to model scale. The efficiencies of the refined numerical
analysis for both prototype and experimental scale are shown in figure 5.2. The curves
do show very similar results and as such it is suggested that the effect of scaling
contributes only relatively minor variations in efficiency. That is, this also suggests that
OWC modelling at experimental scale will provide valid efficiency results that will
have validity at full scale.
A number of other effects also may have combined to limit the effectiveness of a direct
comparison from this aspect of the study. For example, the false bottom present in the
experimental tank is likely to have had a measurable effect on efficiency. A study by
Wang, Katory and Lee (2002) investigated OWC performance with a similar situation
of varying step size as waves approached an OWC. This study identified significant
effects due to the step with decreasing efficiency as water depth decreased. On top of
these effects, the numerical analysis domain was of a reduced length and it is possible
that this may have increased discrepancy between the experimental and numerical
results.
0%
10%
20%
30%
40%
50%
60%
- 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Kh
Effi
cien
cy
Detailed BasecaseAnalysis
Experimental
CFD: Basecase @Experimental Scale
Figure 5.2 Basecase Lip Experimental Vs CFD (detailed CFD)
96
Although there may be difficulty in providing a direct comparison between the CFD and
the experimental lip analysis results, a simple average efficiency for each of the OWC
front lip variations has been plotted in Figure 5.3. Also plotted in Figure 5.4 is the
relative efficiency when compared to the Basecase lip. These results reveal similar
trends of increasing efficiency relative to the Basecase. For example, the Basecase lip
exhibited the lowest efficiency, whereas the rounder lips as one may expect showed a
considerable efficiency increase in the order of 30%.
23.9%
24.4%
26.0%
28.8%
30.1%
15.1%
17.6%
18.6%
18.4%
18.8%
0% 5% 10% 15% 20% 25% 30% 35%
Basecase SquareLip#1
Square Lip #2
Round lip #1
Round lip #2
Round lip #3
CFD
ModelTest
Figure 5.3 Average Efficiency: CFD and Model Test
100%
111%
115%
132%
141%
100%
120%
127%
126%
132%
0% 25% 50% 75% 100% 125% 150% 175%
Basecase SquareLip#1
Square Lip #2
Round lip #1
Round lip #2
Round lip #3
CFD
ModelTest
Figure 5.4 Average Efficiency Relative to Basecase: CFD and Model Test
97
Summarising, it can be said that even with the limited experimental efficiency
resolution with respect to Kh; the uncertainties as a result of the effect of the false tank
bottom and the limited domain length used in the numerical analysis, the results are still
useful, when used for comparative purposes. The results will be particularly helpful in
initial screening studies where several alternatives may need to be considered within a
limited time frame and the aim is to identify which is superior rather than for
determination of absolute values. In addition, the CFD visualisations can highlight flow
patterns that may warrant further investigation and elaboration in later, more detailed
studies.
5.3 Detailed Frequency Comparison
In the previous section it was identified that OWC efficiency is particularly sensitive to
data point spacing and that small differences in say Kh can change individual efficiency
values by the order of ±10% when compared to the trend curves. To minimise this
aspect on the results, further analyses was performed with significantly increased
number of data points for both the numerical and experimental techniques for an OWC
device at the deeper water depth (i.e. h=11.5m).
Figure 5.5 plots the results of the CFD analysis with the results from the experimental
analysis. The hydrodynamic efficiency, ηhyd is plotted versus the infinite water depth
parameter, Kh. To assist interpretation of the results, 3rd order polynomial regression
curves have been fitted to each data set.
Firstly, the results demonstrate good agreement between location of the peak efficiency
with both results showing maximum efficiencies at a value of Kh ≈ 1.4 (i.e. kh ≈ 1.58
and T ≈ 5.7 secs). The numerical analysis had a peak efficiency ≈ 79% whereas the
experimental peak is ≈ 69%. Although there is some difference in the magnitude of the
peak efficiencies, the regression curves are well matched in shape and location of the
peaks.
98
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Kh
Effic
ienc
y
Experimental Results
CFDResults
Figure 5.5 CFD Vs Experimental efficiency results
To further validate the results, analytical theory developed by Evans and Porter (1995)
and previously presented by Morris-Thomas et al (2005) for this same case is presented.
Figure 5.6 shows the Evans and Porter (1995) theory applied to the case at hand (a/h =
0.163) and additionally two other cases for increasing submergence. Figure 5.7 shows
the case at hand compared to the polynomial regression curves for both the
experimental and numerical studies.
0%
20%
40%
60%
80%
100%
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0Kh
Effic
ienc
y
a/h=0.163
a/h=0.185
a/h=0.25
Figure 5.6 Evans and Porter Theory for b/h=0.7 and varying a/h
99
The agreement in terms of the general form between all the curves in Figure 5.7 is
generally very good. All results show a broad banded efficiency arc centred about a
resonant peak. The efficiency for the Evans and Porter (1995) theory for the
corresponding value of a/h exhibits a peak at approximately Kh≈1.7 (i.e. corresponding
to T ≈ 5.3 secs) whilst both the CFD and the experimental work exhibit peaks at Kh≈1.4
(i.e. corresponding to T ≈ 5.7 secs). This may be at least partly explained by the Evans
and Porter (2005) assumption of an infinitely thin front wall, which of course, differs
from the comparative studies. As with increasing the front wall depth, increasing wall
thickness will increase the distance a typical fluid particle will travel during a period of
motion within the OWC and thus tend to lower the value of Kh at which efficiency
peaks for the CFD and experimental studies.
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
Kh
Effic
ienc
y
Evans and Porter
Experimental Results
CFDResults
Figure 5.7 Comparison of CFD, Experimental and Theoretical
Efficiencies for the Basecase configuration.
The magnitude of the CFD and experimental efficiencies are also somewhat lower than
the theory. Realistically, one could never achieve the theoretical value of ηhyd = 100% as
determined by the Evans and Porter (1995) theory. This variation will be at least in part
100
explained by the ideal fluid assumptions of the theory as these would not account for
any viscous losses previously identified in the region of the front lip as fluid enters or
leaves the OWC.
The theoretical formulation also assumes optimised damping with a linear loss turbine
whereas the model used for the experimental and CFD work corresponded to a
quadratic loss vent. Moreover, this exploration of efficiency made some attempt to
optimise the damping efficiency of the vent; however this was far from a rigorous
examination. In addition, flow visualisation of the air velocities within the chamber
showed significant swirl patterns suggesting energy losses within the chamber itself.
The experimental testing of wave height versus efficiency (eg Figure 4.7) also identified
that efficiency decreased with increasing wave height. Small wave linear assumptions
would not account for this effect. Further analysis would be required to quantify these
suggestions.
The revised ‘rounded’ front lip efficiencies may also been compared to the theoretical
formulation and these are presented in Figure 5.8. Again there is broad agreement with
the theory, this time however the resonant peak is closer to theory at Kh ≈1.5 and the
peak efficiency too is significantly higher at approximately 90%. The greater similarity
to theory compared to the Basecase front lip may be explained by the reduced viscous
losses in the region of the front lip.
101
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
KH
Effic
ienc
yEvans & Count
Basecase
Roundlip
Figure 5.8 Comparison of CFD lip variations with Evans and Porter Theory
5.4 Chapter Summary With careful consideration of the model set-up and testing regime, it can be seen that the
efficiency curves for an OWC device can be developed by either numerical or
experimental work. The efficiencies demonstrate good agreement to the ideal theory
from Evans and Porter and any variations can in general, be explained by the
simplifying assumptions used in the theoretical derivation.
In addition, it can be seen that even with a basic analysis with considerable
uncertainties, the flows region under the front lip of the OWC can be identified as
having a significant effect on the power generating efficiency. Both Front lip shape and
submergence impact significantly on the efficiency and alternative configurations using
experimental or numerical techniques can be ranked by even relatively basic
approaches.
102
6 SITE MODELLING
6.1 Introduction The case for taking any wave energy project from the research phase to commercial
reality can only be made if the basis for the projected cost of energy generated and
supplied to the electrical distribution network is fully understood. Of equal importance
with the capital and O&M costs to the projected cost of energy is the annualised power
production for a particular project, as previously described in equation [2.1].
Understanding the energy output is important not only for the economic evaluation of a
particular project, but also allows power utilities to compare the performance of
competing technologies. Similarly, wave energy device designers need to be mindful of
all these factors when deciding on key geometric parameters or assessing the benefit of
any potential design enhancements.
In this chapter the background to the evaluation for the annualised energy output of a
wave energy converter will be presented. Following on from the general formulation,
the energy production using the actual wave climates for two potential sites will be
established using the previous numerical analysis results for the prototype OWC device.
A comparison between the output power curves will be discussed to demonstrate the
application of the numerical work to OWC design including the implications of the
front lip studies.
6.2 Site Characteristics In practice, the near-shore wave energy resource, which fixed OWC type collectors are
designed to tap into, is usually less than the deep, offshore resources. Many factors
contribute to this directly and are characteristically related to the specifics of the chosen
site.
103
These factors can include:
• Local Seabed topography
• Tidal range
• Currents
• Directionality of the incoming waves
• Refraction effects
6.3 Wave Resource In order to analyse wave power production from any device an average of the long-term
wave conditions is required. Wave data records for particular sites will be typically
determined using Waverider buoys. The Waverider buoys are generally moored at the
point of interest and both calculate and log the encountered seastates using in-built
specialist electronics. The seastates encountered by Waverider buoys are averaged over
typically a 30 minute period and the average represented by an energy spectrum
described by the significant wave height, Hs, and the peak energy period, Tp which are
used to define commonly used spectral representation of seastates such as the
JONSWAP spectrum. Each of these individual seastate spectra over the measuring
period will be sorted into bins according to the value of Hs and Tp to create a scatter
diagram wherein each bin is given a probability of occurrence.
Information that may affect OWC performance, such as directionality or tidal effects,
can also be included within the scatter diagram. The scatter diagrams may be broken
down into time frames such as monthly statistics in order to evaluate seasonal output,
however annual statistic are sufficient given the nature of this investigation.
Wave energy researchers must also consider that Waverider buoy data is not always
available for a proposed site. Data can however be obtained by transforming the wave
resource data from a nearby measurement location to the device location by using
standard 2-Dimensional wave models (eg Mike21). In performing this transformation
due consideration is usually given to issues such as; changes in water depth, bathymetry
induced refraction and tidal induced refraction.
104
For this study, measured wave data for the following sites was considered:
a) Port Kembla (New South Wales): the Waverider buoy is located approximately
100m offshore the harbour groyne (Lawson and Treloar, 2002)
b) Rottnest Island (Western Australia): the Waverider buoy is located approximately
1000m south-west of the island (Department of Planning and Infrastructure, 2002).
The data available for Port Kembla was in the form of hourly Hs and Tp records
averaged over 20 minute periods for an 8 year period whereby the Rottnest Data records
were similarly formatted but for a period of 4 years. The data for both sites was then
reduced to a common annual occurrence format using a system of bins described earlier.
The peak period bins chosen were 1 second wide and the wave bins define wave heights
in 0.5m increments. The data was then divided by the total number of records to
produce Hs, Tp percentage occurrence matrices as shown in Tables 6.1 and 6.2.
Hs versus Tp Occurrence Matrix
Hs Tp 5.50 6.50 7.50 8.50 9.50 10.50 11.50 12.50 13.50 14.50 15.50 16.50 17.50 18.50 19.50 20.50 Total0.75 0.1% 0.0% 0.0% 0.1% 0.1% 0.5% 1.0% 0.9% 0.9% 0.3% 0.4% 0.1% 0.1% 0.1% 0.0% 0.0% 4.8%1.25 0.6% 0.2% 0.1% 0.5% 0.9% 2.4% 3.7% 4.3% 4.1% 2.3% 2.3% 0.4% 0.5% 0.2% 0.0% 0.0% 22.6%1.75 0.8% 1.2% 0.7% 0.8% 0.9% 2.1% 3.3% 4.5% 6.2% 2.4% 3.0% 0.5% 0.8% 0.3% 0.0% 0.1% 27.5%2.25 0.5% 1.4% 0.9% 0.5% 0.5% 1.1% 1.4% 2.7% 4.8% 3.2% 2.8% 0.5% 0.7% 0.3% 0.0% 0.0% 21.1%2.75 0.0% 0.4% 0.6% 0.3% 0.3% 0.4% 0.6% 0.8% 2.0% 1.7% 2.4% 0.6% 0.5% 0.2% 0.1% 10.9%3.25 0.1% 0.3% 0.4% 0.3% 0.3% 0.4% 0.3% 0.8% 0.8% 1.4% 0.4% 0.3% 0.1% 0.0% 5.9%3.75 0.0% 0.1% 0.2% 0.2% 0.3% 0.3% 0.1% 0.4% 0.3% 0.8% 0.2% 0.3% 0.1% 0.0% 3.1%4.25 0.0% 0.1% 0.1% 0.1% 0.1% 0.1% 0.2% 0.1% 0.4% 0.1% 0.3% 0.1% 1.7%4.75 0.0% 0.0% 0.0% 0.0% 0.1% 0.1% 0.1% 0.1% 0.2% 0.1% 0.2% 0.0% 0.9%5.25 0.0% 0.0% 0.1% 0.0% 0.1% 0.0% 0.2% 0.1% 0.1% 0.7%5.75 0.0% 0.0% 0.0% 0.0% 0.0% 0.1% 0.1% 0.1% 0.0% 0.0% 0.0% 0.4%6.25 0.0% 0.0% 0.0% 0.0% 0.0% 0.1% 0.0% 0.1% 0.0% 0.3%6.75 0.0% 0.0% 0.0% 0.0% 0.0% 0.1%7.25 0.0% 0.0%7.75 0.0% 0.0%
2.0% 3.3% 2.7% 3.0% 3.3% 7.3% 10.9% 14.0% 19.7% 11.3% 14.2% 2.8% 4.0% 1.3% 0.3% 0.2% 100% Table 6.1 Rottnest Wave Data – Hs, Tp Occurrence Percentages
105
Hs versus Tp Occurrence Matrix Hs Tp 4.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00 12.00 13.00 14.00 15.00 Total0.25 0.1% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.2%0.50 0.1% 0.0% 0.1% 0.2% 0.2% 0.4% 0.8% 1.1% 0.6% 0.8% 0.4% 0.1% 0.0% 0.0% 0.0% 4.7%0.75 0.3% 0.3% 0.9% 1.4% 1.8% 2.6% 3.0% 3.3% 1.8% 2.0% 0.9% 0.3% 0.2% 0.1% 0.0% 19.0%1.00 0.1% 0.6% 1.4% 2.5% 3.4% 4.9% 4.9% 4.8% 1.9% 1.8% 0.7% 0.3% 0.2% 0.0% 0.0% 27.5%1.25 0.0% 0.3% 0.9% 1.9% 2.7% 3.9% 4.1% 4.3% 1.6% 1.5% 0.5% 0.2% 0.1% 0.0% 0.0% 22.1%1.50 0.0% 0.3% 0.9% 1.5% 2.2% 2.5% 2.7% 1.0% 0.9% 0.4% 0.1% 0.1% 0.0% 0.0% 12.6%1.75 0.0% 0.3% 0.7% 1.0% 1.2% 1.5% 0.6% 0.5% 0.2% 0.1% 0.1% 0.0% 0.0% 6.3%2.00 0.0% 0.1% 0.3% 0.5% 0.6% 0.9% 0.4% 0.3% 0.2% 0.1% 0.0% 0.0% 0.0% 3.2%2.25 0.0% 0.1% 0.2% 0.4% 0.6% 0.2% 0.2% 0.1% 0.0% 0.0% 0.0% 0.0% 1.7%2.50 0.0% 0.0% 0.1% 0.2% 0.3% 0.2% 0.1% 0.0% 0.0% 0.0% 0.0% 1.1%2.75 0.0% 0.1% 0.1% 0.1% 0.1% 0.1% 0.0% 0.0% 0.0% 0.0% 0.5%3.00 0.0% 0.0% 0.0% 0.1% 0.1% 0.1% 0.0% 0.0% 0.0% 0.0% 0.3%3.25 0.0% 0.0% 0.0% 0.1% 0.1% 0.1% 0.0% 0.0% 0.0% 0.0% 0.2%3.50 0.0% 0.0% 0.0% 0.0% 0.1% 0.0% 0.0% 0.0% 0.0% 0.2%3.75 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.1%4.00 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.1%4.25 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%4.50 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%4.75 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%5.00 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%5.25 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%5.50 0.0% 0.0% 0.0% 0.0% 0.0%5.75 0.0% 0.0% 0.0%
0.4% 1.3% 3.5% 7.2% 10.6% 15.9% 18.0% 19.9% 8.7% 8.6% 3.4% 1.1% 0.8% 0.3% 0.0% 100.0% Table 6.2 Port Kembla Wave Data – Hs, Tp Occurrence Percentages
The wave data may also be presented in various ways. Figure 6.1 for example,
illustrates the same data as Table 6.1, however in graphical form. Using this form,
designers may often find it easier to visualise the wave characteristics that should be
targeted in order to optimise energy production.
0.5
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
0.0 2.0 4.0 6.0 8.0
10.0 12.0 14.0 16.0 18.0 20.0
0.0%
1.0%
2.0%
3.0%
4.0%
5.0%
6.0%
7.0%
8.0%
9.0%
10.0%
11.0%
% O
ccur
ance
Wave Height
Wave Period
Rottnest Wave Height Occurrence Table
Figure 6.1 Port Kembla Wave Data
106
6.4 Gross Power Capture The power capture of the OWC for a particular seastate is determined by multiplying
the efficiency of the wave energy device with the energy available. The annualised
energy for a particular location can then be determined by combining the occurrence
matrix by the wave height and period power capture matrix and summing the individual
bin values.
6.4.1 Annual Energy Available Each of the Hs, Tp wave bins in a scatter diagram define a particular wave energy. The
mean incident power, per metre of wave crest, for irregular seas can be determined as
follows (Josset and Clement, 2006):
πρ
64
22esTHgPm = [6.1]
Where;
Pm = wave power (W/m)
Tp = peak spectral wave period (s)
Tz = mean zero up-crossing wave period (Tz ≈ 1.41 Tp)
Te = peak energy wave period (Te ≈ 0.828 Tz)
Using this formulation the temporal sea-state distribution of energy for all combinations
of Hs, Tp can be determined by multiplying the percentage occurrence for a particular
bin with the energy of the seastate for that particular bin.
Tables 6.3 and 6.4 describe the annual energy scatter matrices for Rottnest and Port
Kembla whilst Figures 6.2 and 6.3 displays the data graphically. The annual average
power, per metre of wavefront, can be determined by the sum of all the components. For
Port Kembla, the average power is 4.99 kW/m and likewise for Rottnest the average
power is calculated as 19.98 kW/m. This examination shows that the wave energy
available for the Rottnest location is far greater than the Port Kembla location.
107
Displaying this energy information graphically is particularly useful for designers as it
enables the easy understanding of where the available energy for a particular location
resides and thus is useful for guiding the designer to understanding where particular
device efficiency characteristics should be tuned towards.
Annual Energy as function Tp, Hs (kW/m)
Hs Tp 6.50 7.50 8.50 9.50 9.50 10.50 11.50 12.50 13.50 14.50 15.50 16.50 17.50 18.50 19.50 20.50 Total (kW/m)0.75 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.10 1.25 0.0 0.0 0.0 0.0 0.0 0.1 0.2 0.2 0.2 0.1 0.2 0.0 0.0 0.0 0.0 0.0 1.28 1.75 0.0 0.1 0.0 0.1 0.1 0.2 0.3 0.5 0.7 0.3 0.4 0.1 0.1 0.0 0.0 0.0 3.07 2.25 0.0 0.1 0.1 0.1 0.1 0.2 0.2 0.5 0.9 0.7 0.6 0.1 0.2 0.1 0.0 0.0 3.96 2.75 0.0 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.6 0.5 0.8 0.2 0.2 0.1 0.0 - 3.21 3.25 - 0.0 0.1 0.1 0.1 0.1 0.1 0.1 0.3 0.3 0.7 0.2 0.2 0.0 0.0 - 2.40 3.75 - 0.0 0.0 0.1 0.1 0.1 0.1 0.1 0.2 0.1 0.5 0.1 0.2 0.1 0.0 - 1.74 4.25 - - 0.0 0.0 0.0 0.1 0.0 0.1 0.2 0.1 0.3 0.1 0.2 0.1 - - 1.24 4.75 - - 0.0 0.0 0.0 0.0 0.1 0.1 0.1 0.1 0.2 0.1 0.2 0.0 - - 0.88 5.25 - - - - 0.0 0.0 0.1 0.0 0.1 0.0 0.2 0.1 0.2 - - - 0.80 5.75 - - - 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.1 0.1 0.1 0.0 - - 0.59 6.25 - - - - - 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.2 0.0 - - 0.52 6.75 - - - - - - 0.0 0.0 0.0 0.0 0.0 - - - - - 0.14 7.25 - - - - - - - - - - - - 0.0 - - - 0.03 7.75 - - - - - - - - - - 0.0 - - - - - 0.03 8.25 - - - - - - - - - - - - - - - - - SUM 0.11 0.32 0.39 0.50 0.48 0.91 1.37 1.89 3.56 2.57 4.31 1.11 1.80 0.51 0.10 0.03 19.98
Table 6.3 Rottnest Wave Data – Annual Energy Scatter Diagram
Annual Energy as function Tp, Hs (kW/m)
Period Tp (secs) 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 Total (kW/m)0.50 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.04 0.75 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.31 1.00 0.0 0.0 0.0 0.1 0.1 0.1 0.1 0.2 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.77 1.25 0.0 0.0 0.0 0.1 0.1 0.2 0.2 0.2 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.98 1.50 - 0.0 0.0 0.0 0.1 0.1 0.2 0.2 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.83 1.75 - - 0.0 0.0 0.0 0.1 0.1 0.1 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.58 2.00 - - 0.0 0.0 0.0 0.0 0.1 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.40 2.25 - - - 0.0 0.0 0.0 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.28 2.50 - - - 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 - 0.22 2.75 - - - - 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 - 0.14 3.00 - - - - 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 - 0.09 3.25 - - - - 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 - 0.08 3.50 - - - - - 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 - 0.07 3.75 - - - - - 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 - 0.05 4.00 - - - - - 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 - 0.04 4.25 - - - - - - 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 - 0.03 4.50 - - - - - - 0.0 0.0 0.0 0.0 0.0 - - 0.0 - 0.03 4.75 - - - - - - 0.0 0.0 0.0 0.0 0.0 0.0 - - - 0.03 5.00 - - - - - - 0.0 0.0 0.0 0.0 0.0 0.0 - - - 0.02 5.25 - - - - - - - 0.0 0.0 0.0 0.0 0.0 - - - 0.01 5.50 - - - - - - - - 0.0 0.0 0.0 0.0 - - - 0.01 5.75 - - - - - - - - - 0.0 - 0.0 - - - 0.01
0.00 0.02 0.07 0.19 0.37 0.65 0.85 1.14 0.60 0.66 0.25 0.09 0.07 0.04 0.00 4.99
Table 6.4 Port Kembla Wave Data – Annual Energy Scatter Diagram
108
0.751.75
2.753.75
4.755.75
6.757.75
8.50
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 210
0.2
0.4
0.6
0.8
1
Average Annual Energy Available
(kW/m)
Wave Height, Hs (m)
Wave Period Tp (s)
Figure 6.2 Rottnest Wave Data – Annual Energy Scatter Diagram
0.501.00
1.502.00
2.503.00
3.504.00
4 5 6 7 8 9 10 11 12 13 14 15 16 17 180
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Average Annual Energy Available (kW)
Wave Height Hs (m)
Wave Period Tp (s)
Tp
Figure 6.3 Port Kembla Wave Data – Annual Energy Scatter Diagram
109
The current study has only touched on OWC efficiency variations as a function of wave
height. Although for full scale devices this is of interest, for the purpose of this analysis
the simplification will be made that the power generation efficiency is constant for all
wave heights and is as determined by the numerical analysis described in earlier
chapters. Given this, we can simplify these power charts from 3-dimensional to a simple
2-dimensional graph by summing the available energy for each wave height for each
wave period. These are graphically represented in Figure 6.4.
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
5 10 15 20
Peak Wave Period (secs)
Annu
alis
ed A
vera
ge E
nerg
y (k
W)
Rottnest Power Distribution
Port Kembla Power Distribution
Figure 6.4 Annual wave energy as a function of Tp
The graph of available energy as a function of wave period for both Rottnest and Port
Kembla clearly demonstrates that Rottnest is a more energetic site. The graph also
illustrates the different site characteristics such as periods of peak energy. For example,
the peak energy available at the Rottnest site has a Tp ≈ 15s whilst Port Kembla has a
Tp ≈ 11s.
110
6.4.2 OWC Performance Whilst the energy curves described in the previous section are a function of Hs and Tp,
the efficiency determined in the numerical analysis of the OWC is a function of regular
wave height and period. For purposes of this study we can relate the spectral wave
conditions to the regular wave results by decomposing the spectral seastate into a set of
monochromatic waves (Faltinsen, 1990). A Pierson-Moskowitz spectrum [6.2] has been
chosen as this provides the amplitude of the incident wave field for each wave
component knowing two basic parameters (Hs and Tz):
Irregular Seastate:
Defined by Hs, Tz
( )54)2(152 241)(
−−−
= ωππωπ
πω zT
z
eT
HsS
[6.2]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.0 0.5 1.0 1.5 2.0
ω (s-1)
S(ω
) (m
2 s)
Figure 6.5 Energy Distribution for a Pierson Moskowitz spectrum (Hs=2m, Tp=9s)
Equivalent Monochromatic Seastates
To recover the monochromatic incident wave set, one can use formulae [6.3] below to
determine the amplitude of each of the component wave trains.
( ) ( )ωω dSATw 2= [6.3]
Where d(ω) is the width of the discretised frequency step.
Note: For this study, each seastate was discretised into 50 equal frequency steps.
111
Thus knowing the amplitude of each monochromatic wave (ATw) and its period (Tw),
one can determine the component incident wave power using [2.13]. The incident wave
power is then multiplied by the efficiency transfer function for each component and
total energy converted for the seastate is the sum of the individual ATw,Tw efficiencies.
The device efficiency for a particular Hs & Tp is determined as per equation [2.17].
Figure 6.7 graphs the individual regular wave efficiency as a function of Tw, is
compared the spectral efficiency as a function of Tp in. This graph illustrates how that
even though the efficiency curves for a spectral seastate may have a lower peak
efficiency, the combination of the individual wave components tends to broaden the
peak over a large range of wave periods.
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Spectral Period (T p) orRegular Wave Period (T w) (s)
Effic
ienc
y
Spectral EfficiencyRegular Wave Efficiency
Figure 6.6 Basecase OWC Efficiency Curves for Regular Waves (Tw) and Irregular Waves (Tp)
The efficiency regression curves previously determined for the Basecase and Rounded
Front Lips using the CFD analysis for monochromatic waves have been re-calculated
using the methods described above to be a function of a spectral seastate as a function of
Hs and Tp and are presented in Figure 6.6.
112
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Period T p (s)
Effic
ienc
y (%
)
Basecase Efficiency (%) Rounded Lip Efficiency
Figure 6.7 OWC Efficiency Curves: Basecase and Rounded Lip
6.4.3 OWC Annual Output The output of the OWC can be simply determined by multiplying the efficiency for
particular Hs, Tp as calculated in the Section 6.4.2 by the annual energy occurrence.
Figures 6.6 and 6.7 illustrate a summary of this calculation for the Basecase lip for Port
Kembla and Rottnest, showing the device efficiency, the available energy and the
annualised energy converted as a function of wave period.
The annual output of the devices for all locations and front lip types is simply calculated
as the sum of all the individual annual wave energy occurrence bins and is provided in
Table 6.5. Also included is a column, denoted ‘Power Factor’ that represents the annual
amount of energy converted as a function of the annual power available. This term is
commonly used for renewable wind power where the factor is typically in the region of
40% for the latest generation devices. The power factor is shown to vary between 50%
and 60% which is high compared to expected norms, however this may be attributable
to a number of factors. Firstly, only the hydrodynamic efficiency is considered and the
113
efficiencies from the pneumatic, the mechanical and the electrical / electronic
conversions have been assumed to be 100%. In reality, these are likely to combine to a
figure of approximately 80% for optimised OWC wave devices (Department of Trade
and Industry, 2002) bringing this factor down to between 40% and 50% which is still
above that for comparable wind turbines. Secondly, it is known that wave conversion is
affected by wave height, directionality and tidal effects (eg tide affects lip
submergence). These factors will also reduce the final output efficiencies.
-
0.2
0.4
0.6
0.8
1.0
1.2
1.4
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Period Tp
Pow
er (k
W)
0%
10%
20%
30%
40%
50%
60%
70%
Annual Power Distribution (kW) Output Spectrum (kW)
Wave Device Efficiency (%)
Effic
ienc
y
Figure 6.8 Basecase Lip: Port Kembla CFD Conversion Summary
114
-
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5 7.5 9.5 11.5 13.5 15.5 17.5 19.5Period T p (s)
Pow
er (k
W)
0%
10%
20%
30%
40%
50%
60%
70%
Annual Power Distribution (kW) Output Spectrum (kW)
Wave Device Efficiency (%)
Effic
ienc
y
Figure 6.9 Basecase Lip: Rottnest CFD Conversion Summary
Case Front Lip Configuration
Location Annual Average Energy Available
Annual Average Energy Converted
Power Factor
1 Basecase Lip Port Kembla 4.99 kW/m 2.54 kW/m 50.8%
2 Rounded Lip Port Kembla 4.99 kW/m 2.73 kW/m 54.6%
3 Basecase Lip Rottnest 19.98 kW/m 7.98 kW/m 40.3%
4 Rounded Lip Rottnest 19.98 kW/m 7.94 kW/m 40.1%
Table 6.5 Annual Power Summary as a function of location and OWC lip
6.5 Discussion Predictions for the wave energy produced on an annualised basis demonstrate that
although more power is produced at the Rottnest Island site, the OWC system in
question has been better tuned to the Port Kembla site given the higher Power Factor.
That is, it can be seen that although neither site ideally matches the annualised power
available as a function of period to the peak of the OWC efficiency curve, there is a
significant mismatch in the Rottnest peak energy to the OWC peak. Using results of this
115
form a designer would aim to tune the geometric design of the OWC to closer match the
available energy – this may be achieved by a number of ways such as changing the
OWC chamber dimensions, altering lip geometry and depth or turbine matching
techniques. For example, one would expect that a similar device designed specifically
for the Rottnest climate may expect to produce in the order of 10% to 15% more energy
than is currently showing all other parameters remaining equal.
Also of interest is the comparison in output power for the Basecase compared to the
Rounded Lip. The Port Kembla analysis illustrates that a significant increase (≈ 10%)
can be achieved with this minor structural modification. The Rottnest analysis shows
negligible effect due to the fact that there at higher wave periods, away from the peak
efficiencies, there was little difference between the two lip types.
Discussion on areas such as analytical efficiency as a function of wave height, seastate
directionality, cut-in and shut-down conditions and tidal variations are all potentially
significant effects that would require attention for a more detailed power capture
evaluation. Validation of the assumption is using two-dimensional CFD analysis
compared to full three dimensional numerical modelling and the effect of irregular
waves on the OWC efficiency are also worthy of further consideration. In addition,
turbine machinery and electrical losses are also of importance and would require
inclusion in any detailed analysis.
116
7 CONCLUSION & RECOMMENDATIONS
In this research, a commercially available computational fluid dynamics code has been
used to perform a fluid-structure analysis of an oscillating water column type wave
energy converter to determine the efficiency of energy absorption. The wave energy
device efficiency has been compared to the results of an experimental programme and
in certain cases, theoretical approximations. The focus of this work has been on the
simulation of the interaction of the incident wave on the OWC and in particular the
effects from varying the parameters associated with the front lip of the device such as
aperture shape and depth.
Rather than using bespoke modelling tools developed in a research environment, the
numerical simulations have been performed using FLUENT which is a commercially
available, general CFD code. Prior to numerical modelling of the entire OWC system,
wave generation within a numerical wave tank has been examined with particular
attention paid to the free surface modelling and internal wave kinematics. Following
this study, a systematic numerical investigation was then carried out on an OWC system
to model the interaction between the incoming waves and the complex geometries
affecting fluid entry into the OWC chamber including the interaction with the
pneumatic power take-off device.
7.1 Numerical Wave Tank With continual advances in computing power, the simulation of fluid dynamics using
the numerical methods that iteratively solve the Navier Stokes equations and using
Volume-of-Fluid techniques the free surface is now seen as a practical alternative to
model testing. Wave propagation through the numerical wave tank under consideration
has been validated for the simulation of Stokes 2nd Order waves. A user-defined-
function defining the inlet velocity components to simulate a wave-maker provided a
satisfactory method for simulating a wave-maker was successfully developed.
117
The wave surface profile was evaluated referencing a number of key parameters
including discretisation of the time steps with respect to wave period and the number of
cells per wave length / wave height. These ratios are shown to have a significant
influence on the quality of the output and the duration of the analysis. A suitable
combination of cell size and time step increment that provided both sensible analytical
durations combined with an accurate representation of the free surface elevation was
determined. Even though the generated wave heights required some recalibration the
free surface error was shown to be smaller than the magnitude of the 2nd order
correction to 1st order wave theory.
Attention has also been paid to the internal wave kinematics and this was evaluated
several wave periods downstream of the wave maker. The u and w fluid velocity
profiles along a wavelength were evaluated and showed very good correlation with
wave theory.
7.2 OWC Parametric Simulations An investigation into the efficiency implication for OWC design parameters of front lip
shape, thickness and submergence depth has been investigated both numerically and
experimentally. The studies show that the efficiency is indeed significantly affected by
these parameters.
The lip shapes tested illustrated that the flow under the front lip as the fluid enters and
exits the OWC chamber is particularly sensitive. By altering the lip shape either by
increasing the thickness or providing rounding the efficiency could be increased to
provide greater than 20% more power capture compared to the Basecase lip. The CFD
flow visualisation provided confirmation that the variations tested allowed for smoother
flows by reducing the abrupt change in flow direction between the external and internal
chamber fluid reducing turbulent back-flow.
The initial numerical and experimental modelling was performed utilising only a limited
number of wave periods. This was followed with a detailed CFD analysis that identified
118
that insufficient resolution of the wave conditions can lead to an inadequate
understanding of OWC hydrodynamics.
In addition, front lip submergence tests during the experimental studies identified the
decrease in overall efficiency as lip submergence increased and CFD modelling at both
model and full-scale identified little variation in calculated efficiencies justifying the
suitability of scale model testing.
7.3 OWC Efficiency Modelling Subsequent to the recognition that OWC modelling requires significant resolution of
wave periods, numerical testing of different OWC configurations was carried out.
Whereas the earlier experimental testing employed only 5 periods, the subsequent
testing utilised 20 separate periods. The comparison with equivalent experimental
results showed good agreement albeit that the numerical analysis showed a small
differential increase in peak efficiency of some 10%. The numerical and experimental
work was also compared to a simplified 2-dimensional theory that assumed an ideal
fluid, linear power take-off and 1st order progressive waves. The general trends in the
theoretical results are captured by the numerical and experimental work however the
theory overestimated the magnitude of energy capture. The theoretical period of
maximum efficiency was captured satisfactorily by the experimental and numerical
work. Differences with theory are assumed to be caused by the linear assumptions
including the disregard for front lip shape and the vent modelling technique.
7.4 Site Modelling Using the results of the numerical modelling, an estimate of the average annualised
power that could be developed at two sites was investigated. The power output
modelling highlighted the effects of needing to match the OWC efficiency
characteristics with the wave power characteristics for a particular site to enable the
most efficient generation of wave power.
119
7.5 Recommendations The research has revealed that it is possible to take an off-the-shelf numerical CFD and
apply it to the complex problem of oscillating water column efficiencies with great
effect. Given the global interest in marine renewables the optimal design of OWCs is a
research area that requires significant attention and thus the ability to utilise commercial
CFD codes to further this work should benefit both the theoretical analyst and the wave
energy developer alike.
This work has covered a large range of topics and several areas of significance to OWC
design have only just been touched upon. Further extensions to this work could be to:
• Extended CFD modelling to a full 3-dimensional domain
• Perform the numerical modelling under irregular wave conditions
• Introduce real turbine characteristics to allow complete wave to wire
efficiency modelling
• Investigate whether commercial CFD can be used investigate other key areas
of interest to OWC designers such as extreme wave loading.
• Investigate the pneumatic modelling of the chamber shape, in particular, the
effect of swirl as demonstrated in the CFD studies.
• Perform further experimental and numerical modelling to investigate venting
under the front lip as function of wave height / lip submergence and the phase
lag between the incident wave and OWC motion.
• Extend the analysis to investigate design parameters such as maximum and
minimum chamber air pressures for various OWC configurations and
seastates.
These additional investigations may assist to the continued development of OWC
systems such that one day they may at least partly contribute to providing power for the
global energy demand.
120
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APPENDICES
129
Appendix A: Additional Experimental Results
Experimental Efficiency Results
0
10
20
30
40 11.5
22.5
33.5
10
20
30
40
50
60
Period (s)
Base Case - Efficiency
Wave Height (cm)
Effi
cien
cy (%
)
5 10 15 20 25 30 35
10
15
20
25
30
35
40
45
50
55
60
Wave Height (cm)
Effi
cien
cy (%
)
Base Case - Efficiency
T=1.16sT=1.83sT=2.13sT=2.56sT=3.41s
(a) Basecase (Lip submergence 2m)
0
10
20
30
40 11.5
22.5
33.5
10
20
30
40
50
60
70
Period (s)
1m Lip - Efficiency
Wave Height (cm)
Effi
cien
cy (%
)
5 10 15 20 25 30 35
10
20
30
40
50
60
70
Wave Height (cm)
Effi
cien
cy (%
)
1m Lip - Efficiency
T=1.16sT=1.83sT=2.13sT=2.56sT=3.41s
(b) Lip Submergence 1m
0
10
20
30
40 11.5
22.5
33.5
0
20
40
60
80
Period (s)
3m Lip - Efficiency
Wave Height (cm)
Effi
cien
cy (%
)
5 10 15 20 25 30 35
0
10
20
30
40
50
60
70
Wave Height (cm)
Effi
cien
cy (%
)
3m Lip - Efficiency
T=1.16sT=1.83sT=2.13sT=2.56sT=3.41s
(b) Lip Submergence 2m
Figure A1 : Efficiency Plots as a function of Wave Period and Wave Height
130
0
10
20
30
40 11.5
22.5
33.5
0
20
40
60
80
Period (s)
Square Lip #2 - Efficiency
Wave Height (cm)
Effi
cien
cy (%
)
0
10
20
30
40 11.5
22.5
33.5
10
20
30
40
50
60
70
Period (s)
Rounded Lip #1 - Efficiency
Wave Height (cm)
Effi
cien
cy (%
)
(a) Square Lip#2 (a) Round Lip#1
0
10
20
30
40 11.5
22.5
33.5
0
20
40
60
80
Period (s)
Rounded Lip #2 - Efficiency
Wave Height (cm)
Effi
cien
cy (%
)
0
10
20
30
40 11.5
22.5
33.5
0
20
40
60
80
Period (s)
Rounded Lip #3 - Efficiency
Wave Height (cm)
Effi
cien
cy (%
)
(a) Round Lip#2 (a) Round Lip#3
Figure A2: Efficiency Plots as a function of Wave Period and Wave Height
131
OWC Phase Lag
The phase lag between the wave immediately outside the OWC and inside the OWC has
been calculated. By inspection, there is negligible phase lag between the internal probes.
It is likely then that no lateral standing waves were produced.
The phase lag was calculated using the following procedure:
1. The near wave probe signal is offset to yield the wave profile at the OWC front face
(which was 3.275m further downstream of the near wave probe). The magnitude of
the time shift was calculated using the wave speed (calculated from the wave length
and period as determined by the wave calibration) and the distance from the probe to
the OWC front face (3.275m).
2. One wave period is extracted from the signal and the peaks (maxima) for the near
wave and near internal signals are found.
3. The corresponding times these peaks occur are determined and the phase lag
calculated by dividing this time period by the wave period and multiplying by 2π.
4. This procedure is repeated and an average determined over several periods.
(Note: The near internal signal is assumed to always lag the near wave signal – if the
time period between peaks indicates a lead rather than a lag, the actual time period
between peaks is then the wave period added to the negative time period).
132
Phase Lag Plots
5 10 15 20 25 30 350
50
100
150
200
250
300
Wave Height (cm)
Pha
se L
ag (d
egre
es)
Base Case - OWC Phase Lag
T=1.16sT=1.83sT=2.13sT=2.56sT=3.41s
5 10 15 20 25 30 350
50
100
150
200
250
300
Wave Height (cm)
Pha
se L
ag (d
egre
es)
1m Lip - OWC Phase Lag
T=1.16sT=1.83sT=2.13sT=2.56sT=3.41s
5 10 15 20 25 30 350
50
100
150
200
250
300
W ave Height (cm)
Pha
se L
ag (d
egre
es)
3m Lip - OW C Phase Lag
T=1.16sT=1.83sT=2.13sT=2.56sT=3.41s
5 10 15 20 25 30 350
50
100
150
200
250
300
W ave Height (cm)
Pha
se L
ag (d
egre
es)
Square Lip #2 - OW C Phase Lag
T=1.16sT=1.83sT=2.13sT=2.56sT=3.41s
5 10 15 20 25 30 350
50
100
150
200
250
300
Wave Height (cm)
Pha
se L
ag (d
egre
es)
Rounded Base Lip #1 - OWC Phase Lag
T=1.16sT=1.83sT=2.13sT=2.56sT=3.41s
5 10 15 20 25 30 350
50
100
150
200
250
300
W ave Height (cm)
Pha
se L
ag (d
egre
es)
Rounded Base Lip #2 - OW C Phase Lag
T=1.16sT=1.83sT=2.13sT=2.56sT=3.41s
5 10 15 20 25 30 350
50
100
150
200
250
300
Wave Height (cm)
Pha
se L
ag (d
egre
es)
Rounded Base Lip #3 - OWC Phase Lag
T=1.16sT=1.83sT=2.13sT=2.56sT=3.41s
Figure A3: Phase Lag Plots as a function of Wave Height and Wave Period
133
Appendix B: Experimental Photographs
Photo 1: Front view of OWC model in wave tank
Photo 2: Close-up of OWC model in tank showing lip submergence
134
Photo 3: Waves moving towards camera past wave height probes
Photo 4: Data acquisition computer during experimental run
135
Photo 5: Alternative OWC model front lips
136
Appendix C: Additional CFD Plots
(a) OWC Grid &
Air-water interface
shown red (i.e.
VOF=0.5)
(b) OWC Grid &
Air shown red, water
shown yellow.
(c) Phase contours
at wave maker
Figure C1: Example Plots for case with H=1.54m, T=6.52s at t=100s
137
(a) Velocity vectors under OWC lip
(a) Velocity vectors in OWC region
(a) Velocity vectors at wave maker
(a) Velocity contours at wave maker
(a) Velocity contours in Domain
Figure C2: Example Plots for case with H=1.54m, T=6.52s at t=100s
138
Appendix D: Typical User Defined Function for Numerical Wave-
Maker
UDF Example (Written in C-Language)
/***********************************************************************/ /* WAVE PROFILE */ /* UDF for specifying a transient velocity profile boundary condition */ /* for a Stokes 2nd Order wave theory (origin is at seabed level) */ /***********************************************************************/ #include "udf.h" /* Define Constants, Wave Properties & Domain Variables */ #define PI 3.14159265 /* constant */ #define GRAV 9.81 /* acc'n due to gravity */ #define H 1.355 /* waveheight */ #define D 11.5 /* water depth */ #define T 8.26184 /* wave period */ #define L 77.80158 /* wavelengh */ DEFINE_PROFILE(x_velocity, thread, position) { real x[ND_ND]; /* this will hold the position vector */ real y; real AA,BB,CC,DD,EE,FF,GG,LL,ZZ,K; /* y=position vector, AA,BB,etc are temporary stores */ face_t f; begin_f_loop(f, thread) { real t = RP_Get_Real("flow-time"); F_CENTROID(x,f,thread); y = x[1]; ZZ= y - D; K = 2*PI/L; AA= cosh (K*(ZZ + D)); BB= cosh (K* D); CC = sinh (K*D); DD= cos (PI/2.0 -2.0 *PI*(t/T)); EE= cos (2.0*(PI/2.0 -2.0 *PI*(t/T))); FF= cosh (2.0*K*D); GG= cosh (2.0*(K*(ZZ + D))); /* this defines the wave height (LL) at time=t */ LL= D+(H*DD/2.0)+(PI*H*H/(8.0*L))*(BB/pow (CC,3.0))*(2.0+FF)*EE; if (y <= D) /* this defines the profile below water level= D */ F_PROFILE(f,thread,position)=H/2.0*(GRAV*T/L)*AA*DD/BB+0.75*PI*PI*H*H/(L*T)*GG*EE/pow(CC,4.0);
139
else if (y <= LL) /* this stretches the velocity above mean water level*/ F_PROFILE(f,thread,position)=H/2.0*(GRAV*T/L)*BB*DD/BB+0.75*PI*PI*H*H/(L*T)*FF*EE/pow(CC,4.0); else /* This sets the velocity above the wave height to zero */ F_PROFILE(f, thread, position) = 0; } end_f_loop(f, thread) } DEFINE_PROFILE(y_velocity, thread, position) { real x[ND_ND]; /* this will hold the position vector */ real y; real K,MM,NN,OO,PP,QQ,RR,SS,TT,UU,VV,WW,ZZZ; /* y=position vector, AA,BB,etc are temporary stores */ face_t f; begin_f_loop(f, thread) { real t = RP_Get_Real("flow-time"); F_CENTROID(x,f,thread); y = x[1]; ZZZ= y - D; K = 2*PI/L; MM= sinh (K*(ZZZ + D)); NN= cosh (K* D); OO= cos (PI/2.0 - (2.0 *PI* t/T)); WW= cos (2.0*(PI/2.0 -2.0 *PI*(t/T))); VV= sin (PI/2.0 - (2.0 *PI* t/T)); QQ= sin (2.0*(PI/2.0 -2.0 *PI*(t/T))); RR= sinh (K*D); SS= cosh (2.0*K*D); TT= sinh (2.0*(K*(ZZZ + D))); UU= sinh (2.0*K*D); PP= D+(H*OO/2.0)+(PI*H*H/(8.0*L))*(NN/pow (RR,3.0))*(2.0+SS)*WW; if (y <= D) /* this defines the profile below water level */ F_PROFILE(f,thread,position)=H/2.0*(GRAV*T/L)*MM*VV/NN+0.75*PI*PI*H*H/(L*T)*TT*QQ/pow(RR,4.0); else if (y <= PP) /* This stretches the velocity above mean water level */ F_PROFILE(f,thread,position)=H/2.0*(GRAV*T/L)*RR*VV/NN+0.75*PI*PI*H*H/(L*T)*UU*QQ/pow(RR,4.0); else /* This sets the velocity above the wave height to zero */ F_PROFILE(f, thread, position) = 0; } end_f_loop(f, thread) }
140
Appendix E: Typical Fluent Input Data
FLUENT Version: 2d, segregated, vof, lam, unsteady (2d, segregated, VOF, laminar, unsteady) Release: 6.2.16 Title: Models ------ Model Settings --------------------------------------------------------- Space 2D Time Unsteady, 1st-Order Implicit Viscous Laminar Heat Transfer Enabled Solidification and Melting Disabled Radiation None Species Transport Disabled Coupled Dispersed Phase Disabled Pollutants Disabled Soot Disabled Boundary Conditions ------------------- Zones name id type -------------------------------------- fluid 2 fluid wall 3 wall pressure_inlet 4 pressure-inlet velocity_inlet 5 velocity-inlet default-interior 7 interior Boundary Conditions fluid Condition Value ---------------------------------------- Material Name water-liquid Specify source terms? no Source Terms () Specify fixed values? no Fixed Values () Motion Type 0 X-Velocity Of Zone 0 Y-Velocity Of Zone 0 Rotation speed 0 X-Origin of Rotation-Axis 0 Y-Origin of Rotation-Axis 0 Deactivated Thread no Porous zone? no
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Porosity 1 Solid Material Name aluminum wall Condition Value ------------------------------------------------------------- Wall Thickness 0 Heat Generation Rate 0 Material Name aluminum Thermal BC Type 1 Temperature 300 Heat Flux 0 Convective Heat Transfer Coefficient 0 Free Stream Temperature 300 Wall Motion 0 Shear Boundary Condition 0 Define wall motion relative to adjacent cell zone? yes Apply a rotational velocity to this wall? no Velocity Magnitude 0 X-Component of Wall Translation 1 Y-Component of Wall Translation 0 Define wall velocity components? no X-Component of Wall Translation 0 Y-Component of Wall Translation 0 External Emissivity 1 External Radiation Temperature 300 Rotation Speed 0 X-Position of Rotation-Axis Origin 0 Y-Position of Rotation-Axis Origin 0 X-component of shear stress 0 Y-component of shear stress 0 Surface tension gradient 0 Specularity Coefficient 0 pressure_inlet Condition Value ------------------------------------------- Gauge Total Pressure 0 Supersonic/Initial Gauge Pressure 0 Total Temperature 300 Direction Specification Method 1 X-Component of Flow Direction 1 Y-Component of Flow Direction 0 X-Component of Axis Direction 1 Y-Component of Axis Direction 0 Z-Component of Axis Direction 0 X-Coordinate of Axis Origin 0 Y-Coordinate of Axis Origin 0 Z-Coordinate of Axis Origin 0 is zone used in mixing-plane model? no velocity_inlet Condition Value -------------------------------------------------------------- Velocity Specification Method 1
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Reference Frame 0 Velocity Magnitude 0 X-Velocity (profile udf x_velocity) Y-Velocity (profile udf y_velocity) X-Component of Flow Direction 1 Y-Component of Flow Direction 0 X-Component of Axis Direction 1 Y-Component of Axis Direction 0 Z-Component of Axis Direction 0 X-Coordinate of Axis Origin 0 Y-Coordinate of Axis Origin 0 Z-Coordinate of Axis Origin 0 Angular velocity 0 Temperature 300 is zone used in mixing-plane model? no default-interior Condition Value ----------------- Solver Controls --------------- Equations Equation Solved ------------------------ Flow yes Volume Fraction yes Energy yes Numerics Numeric Enabled --------------------------------------- Absolute Velocity Formulation yes Unsteady Calculation Parameters -------------------------------------------- Time Step (s) 0.0049999999 Max. Iterations Per Time Step 20 Relaxation Variable Relaxation Factor ------------------------------- Pressure 0.30000001 Density 1 Body Forces 1 Momentum 0.69999999 Energy 1 Linear Solver Solver Termination Residual Reduction
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Variable Type Criterion Tolerance -------------------------------------------------------- Pressure F-Cycle 0.1 X-Momentum Flexible 0.1 0.7 Y-Momentum Flexible 0.1 0.7 Energy Flexible 0.1 0.7 Discretization Scheme Variable Scheme ------------------------------ Pressure Body Force Weighted Momentum Second Order Upwind Energy Second Order Upwind Solution Limits Quantity Limit --------------------------------- Minimum Absolute Pressure 1 Maximum Absolute Pressure 5e+10 Minimum Temperature 1 Maximum Temperature 5000 Material Properties ------------------- Material: water-liquid (fluid) Property Units Method Value(s) -------------------------------------------------------------- Density kg/m3 constant 998.2 Cp (Specific Heat) j/kg-k constant 4182 Thermal Conductivity w/m-k constant 0.6 Viscosity kg/m-s constant 0.001003 Molecular Weight kg/kgmol constant 18.0152 Standard State Enthalpy j/kgmol constant 0 Reference Temperature k constant 298.15 L-J Characteristic Length angstrom constant 0 L-J Energy Parameter k constant 0 Thermal Expansion Coefficient 1/k constant 0 Degrees of Freedom constant 0 Speed of Sound m/s none #f Material: air (fluid) Property Units Method Value(s) ------------------------------------------------------------------- Density kg/m3 constant 1.225 Cp (Specific Heat) j/kg-k constant 1006.43 Thermal Conductivity w/m-k constant 0.0242 Viscosity kg/m-s constant 1.7894001e-05 Molecular Weight kg/kgmol constant 28.966 Standard State Enthalpy j/kgmol constant 0 Reference Temperature k constant 298.15 L-J Characteristic Length angstrom constant 3.711 L-J Energy Parameter k constant 78.6 Thermal Expansion Coefficient 1/k constant 0
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Degrees of Freedom constant 0 Speed of Sound m/s none #f Material: aluminum (solid) Property Units Method Value(s) --------------------------------------------------- Density kg/m3 constant 2719 Cp (Specific Heat) j/kg-k constant 871 Thermal Conductivity w/m-k constant 202.4
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Appendix F: University of Western Australia Wave Tank Details
UWA Wave Tank Schematic
Wave-current flume • 49 m x 1.5 m with a maximum water depth of 1.2 m. • Piston wave maker:
o Regular or spectral waves o Height range 0.2 – 0.7 m, o Period range 1 – 5 sec.
• Recirculating current: Via two tandem pumps. • Viewing windows available.
Equipment: • Wave and pressure probes. • Load cells and current profiler. • PC-based data acquisition. • Time series and Fourier analysis
1.5
33 m
14.6
Wav
e flu
me
Shallow
Water
Test
Cottesloe Beach model
Office space
Wav
e
Con
trol r
oom
49m
Shed areas for
storage