Analysis of Near Field in Front of a
Piston Wavemaker and MIKE 21
BW’s Handling of Surfzone.
MEK, DTU and DHI
by Anders Wedel Nielsen
Hans Jacob Simonsen
June, 2002
Analysis of Near Field in Front of a Piston Wavemaker
and MIKE 21 BW’s Handling of Surfzone.
by
Anders Wedel Nielsen, c973610
and
Hans Jacob Simonsen, c973782
Polytechnical Midway Project, MEK, DTU, Lyngby, spring of 2002,
in cooperation with DHI Water and Environment.
June, 2002
Abstract
English
The purpose of this project is to investigate the wave field near the wavemaker and the influence
of the evanescent modes. A linear description of the evanescent modes is investigated. MIKE 21
Boussinesq Wave Module (M21BW) is tested with focus on the handling of breaking waves and
moving shoreline and compared to laboratory experiments.
A wide spectrum of waves was tested in the wave flume during the project. Different values of the
wave height and period was tested as and different types of wavemaker signals: sinusoidal, irregular,
cnoidal and solitary.
The phenomenons of the waves moving towards shallower waters is looked into in connection with
the experimental results. The theories behind M21BW and the linear theory describing evanescent
modes are investigated.
By comparing laboratory measurements performed in the near field close to a piston wavemaker, the
validity of the linear description of evanescent modes is tested. Amplitudes of the evanescent modes
are described sufficiently by use of linear theory, while the phases of the evanescent modes are not
well described using the same theory.
Laboratory experiments of breaking monochromatic waves and breaking of irregular waves are compared to the M21BW modelling of the laboratory experiments. Breaking is modelled well using
M21BW. Moving shoreline is handled sufficiently in cases of irregular waves, but in cases of monochromatic waves M21BW does not produce an acceptable result.
Danish
Formålet med projektet er at undersøge bølgefeltet nær en stempelbølgegenerator og undersøge i
hvilket område en lineær beskrivelse er anvendelig. MIKE 21 Boussinesq Waves (M21BW) håndtering
af brydende bølger er testet og sammenlignet med laboratorieforsøg.
Mange forskellige bølger er blevet undersøgt eksperimentielt i en bølgerende. Forsøg med varierende
bølgehøjder og -perioder samt forskellige typer bølger: sinusformede, uregelmæssige, cnoidale og
solitære er blev udført.
Ligeledes indeholder rapporten en teoretisk gennemgang af bølgernes udvikling når de bevæges mod
lavere vande.
Resultater fra laboratorieeksperimenter sammenlignes med den lineær beskrivelse af evanescent modes,
hvor det findes, at fasen ikke er beskrevet godt, men amplituden af evanescent modes er godt beskrevet
ved brug af lineær teori.
Laboratorie resultater med fokus på brydningszonen, området umiddelbart før samt runup på en
hældende bund er sammenlignet med M21BW model resultater af samme forsøgsopstilling. Brydende
bølger bliver godt beskrevet med M21BWs computer model. Runup er kun godt beskrevet ved
uregelmæssige bølger mens runup af regelmæssige monokromatiske bølger ikke er godt beskrevet af
M21BW.
3
Preface
Analysis of Near Field in Front of a Piston Wavemaker and MIKE 21 BW’s Handling of Surfzone.
is written as Polytechnical Midway Project on MEK, DTU, Lyngby. The work load of this report is
nominally set to 27 hours per week for 13 weeks.
The large project has been welcomed as a great opportunity to try our technical competence. The
great extend of the project has made it very attractive as this gave us an opportunity to absorb in
the subjects.
Three tutors was connected to the project, Bjarne Büchmann from MEK, DTU, Kgs. Lyngby and
Hemming Schäffer, DHI, Hørsholm and Henrik Kofoed-Hansen, DHI, Hørsholm. The two latter
functioning as external tutors.
Acknowledgements
First of all we would like to give our thanks to DHI, Water and Environment for kindly letting us
use the experimental facilities and rebuilding the wave flume.
A special thanks to our tutors for the time spent explaining theoretical concepts, lending of software,
programming code and finding relevant articles.
A special thanks to all the friendly people at DHI who were always willing to help us with all sorts
of problems.
Guide for reading the report
The model is divided into three main parts. The first part contains the experimental work, the
second part focusing on the near field close to the wavemaker and the third part concentrating on
the surf zone.
In the report we may designate a differentiation by the notation xy which is equivalent to ∂x
∂y .
Due to the text editing software the figures in the report are placed rather randomly, and we apologise
for the inconvenience this may cause. In part III we have tried to make the curves on the figures as
useful as possible in a non-color report situation. This is done to make the report ”copy-friendly”.
The laboratory measurements are represented in the figures by black solid lines and the MIKE 21
BW computer modelled results by blue dashed lines.
4
Contents
1 Introduction
8
1.1 Problem specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
I
Laboratory Experiments
11
2 Physical setup of experiments
2.1 The wave flume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The wave gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Placing of the wave gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Setup of wave gauges focusing on the local field in front of the wavemaker .
2.3.2 Setup of wave gauges focusing on the surf zone . . . . . . . . . . . . . . . .
2.4 Presentation of the analysed waves . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Waves analysed in investigation of the near field in front of the wavemaker
2.4.2 Waves analysed in investigation of the surf zone . . . . . . . . . . . . . . . .
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II Wavemaking.
Analysis of Near Field in Front of a Piston Wavemaker
3 Theory concerning evanescent modes
3.1 First order wavemaking . . . . . . . . . . .
3.2 Analysing the results . . . . . . . . . . . . .
3.2.1 Theoretical analysis . . . . . . . . .
3.2.2 Practical application of the analysis
4 Program to analysis of evanescent modes
4.1 Determination of analysed time interval .
4.2 The complex wave number . . . . . . . . .
4.3 Principles in the Fast Fourier Transform .
4.4 Determining the phase . . . . . . . . . . .
4.4.1 The constant phase method . . . .
4.4.2 The variable phase method . . . .
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5 Finding useful time window
27
5.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.3 Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5
6
CONTENTS
6 The result of the evanescent modes analysis
33
6.1 Result of analysis with constant phase method . . . . . . . . . . . . . . . . . . . . . . 33
6.2 Result of analysis with changing phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7 Statistical results
40
7.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7.3 Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
III Surfzone.
Analysis of MIKE 21’s Boussinesq Wave Module Handling of Breaking Waves
and Runup.
45
8 Analysis of laboratory experiments in the surfzone
8.1 Time series analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Analysis of time series from runup gauge . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Analysis of FFT analysis in different distances from the wavemaker . . . . . . . . .
8.4 Analysis of time series measured on the runup gauge in the cases of irregular waves
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9 Introduction to MIKE 21 Boussinesq wave module
62
9.1 Handling breaking using surface roller concept . . . . . . . . . . . . . . . . . . . . . . 62
9.2 Handling moving shoreline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
10 Setup of MIKE 21 Boussinesq wave module
64
10.1 Physical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
10.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
11 Calibrating Breaker and Moving Shoreline Parameters
67
11.1 Calibrating parameters used in wave breaking . . . . . . . . . . . . . . . . . . . . . . . 67
11.2 Calibrating parameters used in moving shoreline . . . . . . . . . . . . . . . . . . . . . 70
12 Analysis of MIKE 21 BW results
12.1 Analysis of time series in different distances from the wavemaker, sinusoidal wavemaker
signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.1 Case 1, T = 1.0 s, H = 0.1 m . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.2 Case 2, T = 1.5 s, H = 0.08 m . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.3 Case 3, T = 2.0 s, H = 0.13 m . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 Analysis of time series measured on the runup gauge, sinusoidal wavemaker signal . .
12.3 FFT analysis of time series in different distances from the wavemaker, irregular waves
12.4 Analysis of time series measured on the runup gauge, irregular waves . . . . . . . . . .
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13 Conclusions
93
13.1 Conclusion of the investigation of the near field in front of a wavemaker . . . . . . . . 93
13.2 Conclusion of the investigation of MIKE 21 BW handling of the surf zone . . . . . . . 93
A Preliminary calculations
95
A.1 Assuming no shoaling in the near field near the wavemaker . . . . . . . . . . . . . . . 95
A.2 Changing of wave length from wavemaker to wave gauge 10 . . . . . . . . . . . . . . . 95
B FFT results
96
C Program testing
98
7
CONTENTS
D The
D.1
D.2
D.3
D.4
D.5
D.6
code for the evanescent analysis program
Program to analysis evanescent modes (constant phase method) . . . . . . . . . . . .
Program to analysis evanescent modes (changing phase method) . . . . . . . . . . .
readfile.m: open the file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
bolgelaengde.m: calculating the wave length . . . . . . . . . . . . . . . . . . . . . . .
timeWindow.m: calculating the time window . . . . . . . . . . . . . . . . . . . . . .
bolgehojdeEvan.m: calculating the evanescent modes amplitude and the progressive
modes amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D.7 ksiteration.m: calculating the wave numbers . . . . . . . . . . . . . . . . . . . . . . .
D.8 runfft.m: doing the FFT analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D.9 Hevan2.m: calculating the evanescent mode amplitudes and the progressive modes
amplitude from the complex result from FFT analysis . . . . . . . . . . . . . . . . .
D.10 stat.m: calculating the statistic results . . . . . . . . . . . . . . . . . . . . . . . . . .
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E Suggestions to improvements of MIKEZero
118
E.1 Fejl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
E.2 Forslag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
F Experiments performed during the project
124
F.1 All experiments performed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
F.2 CD-ROM containing all the timeseries . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Chapter 1
Introduction
Making a construction in water is a costly affair. A lot of money can be saved by avoiding over
dimensioning of the construction. We need knowledge about how a given situation will affect a
construction in order to make the optimal dimensions of the structure, e.g. making the structures
strong enough to withstand storms, tides amongst others, but on the same time decrease the expenses.
An estimate of the forces can be given either by laboratory modelling of the situation or by making
a computer model of the scenario.
If the scenario is artificially recreated in the laboratory we wish to make the modelled situation as
close to the situation in nature/real size as possible. This is difficult in a laboratory though, because,
among others, the boundaries can not be seen as infinite as in nature and therefore reflections will
occur in laboratory.
To avoid re-reflections, i.e. simulate the nature in the best possible way, active absorbtion has
been developed with the purpose to absorb the energy reflected back to the wavemaker. This is done
by measuring the wave height on the wave gauge and comparing with the expected.
Because we have a mismatch between the profile of the wavemaker and the profile of the natural
particle velocity in a wave evanescent modes will be created. Evanescent modes are standing waves
decreasing exponentially with the distance to the wavemaker and therefore it will usually have no
influence on the water level near the model structure. When measuring the water level on the
wavemaker to use in the active absorbtion, the additional wave will create unwanted disturbances.
This is especially pronounced during the generation of irregular waves where the high frequency
waves have relative large evanescent modes.
Another way of testing a structure is by making a computer model of the scenario. Computer
models are under constant development where new features and capability ranges are introduced.
The computer models keep achieving better and better results and thus takes over much of the
assignments previously carried in laboratory experiments.
But computer models can not be applied without comparing the output from the computer model
with the results from a ”real” problem. E.g. comparison to measurements in a wave flume.
1.1
Problem specification
In this report we will investigate both the evanescent modes in the near field in front of a piston
wavemaker. We will also investigate MIKE 21 Boussinesq Waves module in the breaking zone and
just before the point of breaking through comparison of the computer simulations and experiments
carried out in the laboratory.
Evanescent modes will be investigated focussing on the both the phases of the evanescent modes
and the amplitudes of the progressive wave and the evanescent modes, but the same results will
also be investigated not focusing on the phases but concentrating on the size of the wave amplitude.
Measurements in the near field close to a piston wavemaker will be investigated with focus on the
8
CHAPTER 1. INTRODUCTION
9
evanescent modes. The theoretical tool is linear theory.
MIKE 21 Boussinesq Waves module is in the current version able to handle moving shoreline.
The model is also able to handle values of kh up to 3.1; kh expressing how deep the water is compared
to the wavelength. MIKE 21 Boussinesq Waves module is tested and compared to the measurements
we perform near the breaking zone. Especially the modelling of the moving shoreline is tested. In
addition we will see how M21BW describes the situation, which has been setup in the laboratory,
before the point of breaking, i.e. a general testing of the MIKE 21 software.
CHAPTER 1. INTRODUCTION
1.2
H
T
L
h
ω
f
k
c
cg
x
z
a
α
t
φ
X0
S
g
γ
R
R2%
ζ
W
∇
List of symbols
wave height, m
wave period, s
wave length, m
water depth
angular wave frequency (= 2π
t ), Hz
frequency (= 1/T ), Hz
−1
wave number(= 2π
t ), m
wave celerity, m/s
wave group celerity, m/s
cartesian x coordinate, direction of the wave propagation, m
vertical cartesian coordinate, positive upwards, zero in mean water level, m
bottom slope
bottom slope, ◦
time, s
velocity potential, m2 /2
X on deep water (e.g. L0 deep water wave length)
wavemaker stroke, m
gravitational acceleration, m/s2
peak enhancement factor
vertical wave runup, m
2% wave runup exceedence value, m
surf similarity parameter
Statistical weighting parameter
∂
∂
nabla operator = ( ∂x
, ∂z
)
10
Part I
Laboratory Experiments
11
Chapter 2
Physical setup of experiments
In the following the experiments carried out in DHI laboratory in the spring of 2002 are described.
For a complete list of experiments done in the period please refer to appendix F. As previously
mentioned we investigate two different areas but we use the same wave flume and equipment in both
parts. Thus if nothing else is mentioned the description is valid for both the investigation of the near
field in front of the wavemaker and the investigation of the surf zone.
2.1
The wave flume
The experiments are setup in a 2D wave flume with an approximate length of 22 m. The bottom of
the flume is horizontal the first 1.3 m from the wavemaker. From 1.3 m to around 8 m it is has a
slope of 0.032, changing to 0.010 and from 15 m to 22 m the slope is 0.063, see figure 2.1.
It is assumed that the slope of the bed is so small, that in the near field close to the wavemaker we
can neglect the changes in wave heights and wave lengths due to changes in water depth (shoaling),
see appendix A.1. The specified depth varies from experiments. A water depth of 0.55 m at the
wavemaker, lead to a distance of 19.2 m from the coastline to the mean position of the piston
wavemaker.
0.1
0
−0.1
Depth, m
a = 0.063
−0.2
−0.3
a = 0.010
−0.4
a = 0.032
−0.5
−0.6
0
2
4
6
8
10
12
Distance from the wavemaker, m
14
16
18
Figure 2.1: Bathymetry (bottom profile). The depths are at a water depth of 0.55 m at the wavemaker. ’a’ indicates the approximate bottom slope in each region.
12
CHAPTER 2. PHYSICAL SETUP OF EXPERIMENTS
13
The wavemaker is placed in the deepest end of the wave flume, on figure 2.1 is this x = 0. The
wavemaker is of a piston type. On the wavemaker is mounted two wave gauges used for active
absorbtion. The active absorbtion is in principle the comparison of the expected and the measured
water level on the wavemaker. The system used for active absorbtion is AWACS. AWACS and other
active absorbtion systems are applied with the purpose of avoiding re-reflection of wave. Active
absorbtion is applied for the experiments in the surf zone but not in the experiments investigating
the near field in front of the wavemaker.
By visually inspection of our time series it is our experience that the wave height specified as
input to the wavemaker is not always the same as the wave height measured in a wave gauge a certain
distance from the wavemaker.
2.2
The wave gauges
The surface elevation was measured by wave gauges (two parallel conductors) which measure the
change in resistance by sending electricity through. The type of wave gauges used did not compensate
for changes in temperature. Before we started a series of experiments we calibrated the wave gauges
because of changes in temperature (among others). The accuracy is 1 cm.
Measuring the runup was done using the same principle as the other (normal) wave gauges;
sending electricity through two parallel conductors. Two parallel conductors were glued to a plastic
board. This plastic board will in the following be referred to as the runup gauge. The runup gauge
was put into the concrete bottom in a way that the top side of the runup gauge intersects with the
top side of the concrete bottom, see figure 2.2. In this way the runup gauge will disturb the water
flow in the least possible way. lowest part of the runup gauges was placed approximately 18.4 m from
the wavemaker, i.e. where the bottom slope is a = 0.063, α = 3.6◦ .
Figure 2.2: Sketch of the runup gauge. a) shows a section around the coastline. The small rise is
the runup gauge b) shows the section of the gauge. The two small rises to the left and to the right
of the round object are the parallel conductors. The round object mounted in the middle between
the to conductors is mounted with tape. c) show the runup gauge from above. The black solid lines
are the parallel conductors and the dotted line the round object mounted in the middle. The square
underneath the metal conductors and the round object is the plastic bar on which the conductors
are glued on to and the round object is taped to.
The runup gauge measures a spatially much wider range because it was mounted on a sloping
bottom: a vertical runup of 1 cm corresponds to a total runup, measured on the runup gauge, of
1cm/ sin 3.6◦ = 15 cm. Because of the wider spatial range we had to send a higher voltage through the
conductors in order to cover all of the range of the runup gauge. Because the accuracy is measured on
the voltage and we send twice the voltage through the accuracy is twice the accuracy of the normal
CHAPTER 2. PHYSICAL SETUP OF EXPERIMENTS
14
wave gauges, i.e. 2 cm. The vertical accuracy is also the total accuracy projected on the vertical.
The vertical accuracy of the runup gauge is therefore 0.13 cm.
A problem we encountered was a very thin layer of water sticking to the runup gauge during rush
down. The problem was reduced by mounting a round object between the two metal plates with
non-conductive tape, as shown on figure 2.2. In this way the runup gauge was only conductive when
the water level was above the round object. The height of the round object was approximately 2 mm.
2.3
Placing of the wave gauges
We divided the laboratory experiments into two parts. On measuring in the near field in front of the
wavemaker and the other focusing on the surf zone. With the limitation of wave gauges we had to
split the experiments up in two parts in order to get as many wave gauges as possible in the area we
wished to investigate.
In both the part concerning the local field near the wavemaker and the part focusing on the surf
zone we did two different set of experiments using different setup of the wave gauges. In the following
we concentrate on only one of the setups in each of the parts.
On some of the graphs we referred to each wave gauge as a channel. Therefore channel 1 is the
signal coming from wave gauge number 1.
During the experiments near the wavemaker the water depth was set to 0.52 m, whereas the
water depth was set to 0.55 m during the investigation of the surf zone.
2.3.1
Setup of wave gauges focusing on the local field in front of the wavemaker
Because of the fact that the evanescent modes decrease exponential with the distance from the
wavemaker, the wave gauges is placed so it is also decreasing from the wavemaker, but not exactly
exponential. The position of the wave gauges in from the middle position of the piston is seen in
figure 2.1.
Table 2.1: The position of the wave gauges and the depths at the wave gauges used for investigation
of the near field in front of a piston wavemaker. The position is the distance to the mean position of
the wavemaker.
Gauge no. position [m] depth [m]
0
0.550
1
0.14
0.550
2
0.22
0.550
3
0.32
0.550
4
0.44
0.550
5
0.59
0.550
6
0.73
0.549
7
0.92
0.549
8
1.20
0.549
9
1.37
0.546
10
1.56
0.539
11
1.73
0.532
12
2.56
0.501
All the wave gauges is fixed in the distance given by table 2.1 expect the gauge number which is
the wave gauges mounted on the wavemaker. The long distance between first wave gauge and the
wavemaker is caused by the stroke of the piston up to approximately 0.1 m.
15
CHAPTER 2. PHYSICAL SETUP OF EXPERIMENTS
2.3.2
Setup of wave gauges focusing on the surf zone
In order to focus on the breaking zone in the best possible way we chose to place as many wave
gauges near the point of breaking as possible but on the same time covering both the runup and all
of the breaking zone.
Table 2.2: The position of the wave gauges in distance from the wavemaker during the investigation
of the surf zone.
Channel position, [m] depth, [m]
1
02.30
0.509
2
09.68
0.324
3
14.99
0.258
4
15.40
0.234
5
15.80
0.211
6
16.20
0.187
7
16.60
0.163
8
17.02
0.136
9
17.40
0.112
10
17.83
0.084
11
18.21
0.059
12
18.60
0.034
13
19.05
0.006
14
runup
-
2.4
Presentation of the analysed waves
As the purpose of the two parts are very different the waves analysed are also different. We have
done a lot of different experiments with a lot of different wave characteristics. We have chosen to
focus on a few cases in each part. For an overview of all the waves tested please refer to appendix F.
2.4.1
Waves analysed in investigation of the near field in front of the wavemaker
We have chosen 3 cases of the evanescent modes, listed in table 2.3.
Case no.
1
2
3
Table 2.3: The 3 cases for evanescent modes
Wave height, H [m] Period, T , [s] Wave length, L [m]
0.011
0.5
0.39
0.035
0.7
0.76
0.035
1.5
2.86
k [m−1 ]
16.11
8.27
2.20
ω [Hz]
12.57
8.98
4.19
Case 1 is a small short wave with steepness H
L = 0.030, the relative low steepness give a more
constant form of the wave with the distances to the wavemaker, see chapter 5. The short period give
a pronounced evanescent mode.
Case 2 is longer and steeper than the first, the steepness being 0.049. The larger wave amplitude
gives a relative smaller uncertainty caused by the accuracy of the wave gauges.
Case 3 is a small amplitude, long wave with steepness 0.011, the low steepness give a wave of
more constant form. The long period give a small evanescent mode.
16
CHAPTER 2. PHYSICAL SETUP OF EXPERIMENTS
2.4.2
Waves analysed in investigation of the surf zone
We have analysed three different regular monochromatic waves using a sinusoidal wavemaker signal
and three irregular waves. The regular waves are in the following denoted case 1-3 and the irregular
case IR 1-3.
Presentation of regular waves, cases 1-3
The cases analysed are shown in table 2.4:
Table 2.4: Analysed regular waves
T [s]
Case 1 1.0
Case 2 1.5
Case 3 2.0
in the investigation of
H [m] kh
H/L
0.10
2.3 0.065
0.08
1.2 0.027
0.13
0.82 0.031
the surf zone, cases 1-3.
ζ
0.25
0.40
0.43
L being the wave length at the wavemaker (h = 0.55 m) calculated by linear theory. H is the wave
height used for input to the wavemaker signal (later referred to as H specified on the wavemaker).
Due to technical problems the specified wave height is not always the same height as measured in
the flume. kh and the wave steepness, H/L calculated at the wavemaker are also listed.
p
The calculated surf similarity parameters, ζ, are also listed. ζ is defined by a/ H0 /L0 , where
a is the slope in the breaking zone, L0 = gT 2 /(2π) is the deep water wave length, H0 is the deep
water wave height, g is the gravitational acceleration and T the wave period. ζ is used as an estimate
of the breaking type; spilling breakers occur for ζ < 0.5 and plunging breaker for 0.5 < ζ < 3.3
[Madsen, P. A. et al., (1997)]).
Case 1 is probably the most difficult case to analyse. Because of the large value of kh we can expect
that MIKE 21 Boussinesq Wave Module (M21BW) will have difficulties in modelling the higher harmonics. The maximum kh value M21BW can model is 3.1 [MIKE 21 BW, Scientific Documentation].
All of the higher harmonics of case 1 will for sure exceed that limit. On the other hand the surf
similarity parameter, ζ indicates that this case is a spilling breaker. The theory behind M21BW
handling of breaking waves is founded on spilling breakers and M21BW should therefore model the
breaking of this case well.
Case 2. With kh of only 1.2 the bound second harmonic (khbound = 2kh) is also in the range
of MIKE 21’s validity. An approximate wave number of the free second harmonics (the existence
of these will be commented in section 8.1) is found by solving the linear dispersion relation with
ω = 2ωfirst harmonic . The kh value of these free second harmonics is 3.9 and is therefore out of the
range.
The surf similarity parameter is a little bit high which correspond to a breaking type closer to
a spilling type. The limit of the spilling breaker is, as previously mentioned ζ = 0.5. By visually inspection of the wave we found that the breaker type was very close to a plunging breaker.
But experience show that plunging breakers also are fairly successful modelled by use of M21BW
[Madsen, P. A. et al., (1997)].
Case 3 has an even smaller value of kh. It is expected that this case can be simulated fairly
accurate using MIKE 21. Even the kh of the free second harmonics are in the range of M21BW’s
capabilities (khfree second ≈ 2.3. The breaker type is, as in case 2, theoretically a spilling breaker but
according to visual inspection close to a plunging breaker.
Presentation of Irregular Waves, Cases IR 1-3
The irregular wave was made from a JONSWAP spectrum [WS User Manual]. We tested different
values of the peak enhancement factor, γ. γ specified the width of the energy spectrum; low values
17
CHAPTER 2. PHYSICAL SETUP OF EXPERIMENTS
corresponds to a wide spectrum i.e. the energy is more spread to a wider spectrum of frequencies.
Opposite is high values of γ which correspond to the energy being concentrated near the peak
frequency. The significant wave height, Hs is defined as the mean value of the highest one third of
all the waves in the wave train. Tp is defined as the period of the wave component with the most
energy.
By testing different values of γ we will test M21BWs capabilities in the kh range. The cases
analysed are shown in table 2.5. Notice that case IR 3 has a significant wave height of 0.08 m as
oppose to case IR 1 and IR 2 with a significant wave height of 0.03 m.
Table 2.5: Irregular waves, cases IR 1-3.
Tp [s] Hs [m] γ
kh [ ] H/L [ ]
Case IR 1
1.5
0.03
1
1.2
0.01
Case IR 2
1.5
0.03
3.3
1.2
0.01
Case IR 3
1.5
0.08
10
1.2
0.027
ζ
0.65
0.65
0.40
The values of kh are fairly small, but as γ becomes small the spectrum expands to include higher
kh values.
Part II
Wavemaking.
Analysis of Near Field in Front of a
Piston Wavemaker
18
Chapter 3
Theory concerning evanescent modes
The evanescent modes are standing waves caused by a mismatch between the particle velocity profile
of a specified wave and the shape of the wavemaker. The evanescent modes decrease exponentially
with the distance from the wavemaker and have the same frequency as the specified wave.
When making a construction in a model scale and testing it in a wave flume the evanescent modes
will not have a direct influence on the experiments as long as the model is placed sufficiently far from
the wavemaker. But to avoid re-reflection, i.e. the wave are reflected from the construction back
to the wavemaker and back to the construction again, an active absorber is designed. The active
absorber works in principle by comparing the actual water level and the water level specified water
level on the wavemaker. When the wavemaker detect a difference between the specified water level on
the wavemaker and the actual measured water level on the wavemaker the position of the wavemaker
is adjusted.
The evanescent modes, which are of course largest at the wavemaker, will have an influence on the
water level at the wavemaker. This change in water level will affect the active absorbtion. Especially
in generation of and active absorbtion of irregular wave trains where the high frequency components
makes relatively larger evanescent modes.
3.1
First order wavemaking
We setup a two dimensional system as seen in figure 3.1 and assume that all motion is in these two
dimensions. In the near field in front of the wavemaker horizontal bottom is assumed. The fluid is
assumed incompressible and the effects of viscosity turbulence and surface tension are neglected.
We define the velocity potential φ as
µ
¶
∂φ ∂φ
(u, w) =
(3.1)
,
∂x ∂y
By assuming that the wave height, H, is much smaller than the wave length, L we can setup the
following linearized boundary value problem:
∇2 φ = 0
φz = 0
1
φt − η = 0
g
φtt + gφz = 0
φx − Xt (t) = 0
everywhere
(3.2)
z = −h
(3.3)
z=0
(3.4)
z=0
(3.5)
x=0
(3.6)
∂
∂
∇ is the nabla operator = ( ∂x
, ∂z
), g is the gravitational acceleration and X(t) is the piston
wavemaker position to time t. x and z are he coordinates as defined on figure 3.1
19
20
CHAPTER 3. THEORY CONCERNING EVANESCENT MODES
Figure 3.1: Sketch of the near field in front of a piston wavemaker. The solid vertical line is the
piston wavemaker and the dashed line is the wavemaker (piston) to time t.
The first equation, the Laplace equation (3.2), expressing conservation of mass and because
we assume incompressible fluid, also conservation of volume. Equation 3.3 the bottom boundary
condition expresses how nothing flow through the bottom which corresponds to no vertical velocity
at the bottom. Equation 3.4 is the dynamic free surface condition expressing how the pressure in
the surface is equal to the atmospheric pressure. Equation 3.5 states that a particle in the surface
stays in the surface. The last boundary condition is what separates wavemaking from waves in open
water. This equation (3.6) expresses that the particle velocity at the wavemaker equals the velocity
of the wavemaker.
Solving the equations 3.2 to 3.6 we obtain the solution, [Schäffer, H. A. (1994)]:
φ(x, z, t) = Re
And the dispersion relation:
Ã
∞
cosh(kn (h + z)) i(ωt−kn x)
igS X
Bn
e
ω
cosh(kn h)
n=0
ω 2 = gkn tanh(kn h)
!
(3.7)
(3.8)
Where n ∈ Z
Given ω and h equation 3.8 has one real solution which we designate k0 . k0 corresponds to the
wave number of propagating wave. An infinite number purely imaginary solutions (k n , n 6= 0) are
also solutions to equation 3.8, these corresponds to the wave number of the evanescent modes.
The surface elevation is according to the free dynamic surface elevation:
η(x, t) = −
1 ∂φ ¯¯
g ∂t z=0
Ã
η(x, t) = Re SB0 e
(3.9)
i(ωt−k0 x)
+S
∞
X
n=1
Bn e
i(ωt−kn x)
!
(3.10)
Bn , n ∈ Z are found by satisfying the boundary condition at the wavemaker, equation 3.6. For a
piston flap the coefficient reads [Schäffer, H. A. (1994)]:
Bn =
1
ω2 2
g kn 1 + G n
(3.11)
21
CHAPTER 3. THEORY CONCERNING EVANESCENT MODES
where
Gn =
2kn h
sinh(2kn h)
n≥0
(3.12)
kn is found from equation 3.8. Notice that Bn is purely imaginary for n 6= 0 and real for
n = 0. This means that the complex amplitude of the evanescent modes are π2 out of phase with the
complex amplitude of the progressive wave on the wavemaker, x = 0. By substituting k n , n 6= 0 into
equation 3.10 we can see that, as kn is purely imaginary, the evanescent modes will not fluctuate
with the distance from the wavemaker, x. The x-dependence is only a factor in the amplitude of the
evanescent modes. Therefore the evanescent modes are fluctuating with time but not with distance
to the wavemaker, x. The amplitude of the evanescent modes varies exponentially with the distance
to the wavemaker. To satisfy the physical conditions we exclude all solutions to equation 3.8 where
Im (kn ) > 0 ensuring that the evanescent modes will decrease exponentially with x. In the following
kn , n > 0 will denote the kn where Im(kn ) < 0 (not excluded).
Inserting 3.11 into 3.10 we have:
ω2 S
η(x, t) = 2
Re
g
Ã
∞
X 1
1
1
1
ei(ωt−k0 x) +
ei(ωt−kn x)
k0 1 + G 0
kn 1 + G n
n=1
!
(3.13)
For a piston wavemaker, as in our case S(z) = S. The stroke, S is calculated from the condition
that we define H as the wave height where the evanescent modes are zero.
3.13 the first
¡ In equation
¢
term must the equal the surface elevation described as: ηfar field = H2 Re eωt−k0 x
S=
3.2
3.2.1
H(sinh(2k0 h) + k0 h)
2 sinh2 (k0 h)
(3.14)
Analysing the results
Theoretical analysis
The complex vectors representing the waves are sketched in the complex plane on figure 3.2. The
coordinate system and the resulting vector rotates with the frequency of the waves, ωt. At a certain
x the situation on the figure is therefore stationary.
From equation 3.13 it is also seen, that the phase shifting is ϕ = k0 x, where ϕ refers to figure 3.2.
We wish to solve the system sketched in figure 3.2. From FFT analysis of the laboratory experiments we determine the resulting vector in the complex plane. If linear theory is assumed valid and
we on the same time assume flat bottom (see appendix A) we can calculate ϕ at any distance from
the wavemaker(see figure 3.2). Refereing to figure 3.2 and equation 3.13 we know that θ = π2 . The
direction of the evanescent modes and the direction of the progressive part are by the assumption
of linear theory both known. The size and direction of the resulting vector is know from FFT analyzes of the measurements. The only unknown in the system are the sizes of the amplitudes for
the progressive complex vector and the amplitude of the complex representation of the evanescent
modes.
During this study we intend to find the sizes of the amplitudes of the progressive waves and the
evanescent modes and compare these to theoretical values.
Writing the system sketched in figure 3.2 in matrix form:
CHAPTER 3. THEORY CONCERNING EVANESCENT MODES
22
Figure 3.2: Sketch of the complex representation of the evanescent modes and the progressive wave.
The system rotates with ωt. Because of the rotation the system is stationary with time.
~res = V
~evan + V
~prog
V
= Aevan (x)~vevan + Aprog ~vprog
¸·
¸
·
Re(~vevan ) Re(~vprog ) Aevan (x)
=
Aprog
Im(~vevan ) Im(~vprog )
·
¸·
¸
cos θ cos ϕ Aevan (x)
=
sin θ sin ϕ
Aprog
¸
·
¸·
0 cos k0 x Aevan (x)
=
1 sin k0 x
Aprog
(3.15)
~res is measured from FFT analysis of the time series in the different
It is assumed that the vector V
distances from the wavemaker. The matrix containing the information of the vectors direction is
assumed known from theory. If the determinant of this is zero there is an infinite number of solutions
to the system. In practice the solution becomes very inaccurate when the determinant is close to
zero. Determinant equals or close to zero corresponds to the progressive wave and the evanescent
modes being in phase. On 3.2 this means that ϕ − θ = pπ, p ∈ N and it is therefore impossible to
determine any of the vectors as the two vectors are linear dependent.
The solved system looks like this:
¸ ·
·
¸−1
0 cos k0 x
Aevan (x)
~res
=
V
(3.16)
Aprog
1 sin k0 x
3.2.2
Practical application of the analysis
Above is describes how the theory is applied ”in theory”. In practice a great deal of factors makes
influence on the complexity of the system and the solving of the system.
The wavemaker uses a warm up period of 5 seconds. When this warm up period is not a multiple of
the wave period then the wave will have an initial phase shifting. We also discovered a synchronising
CHAPTER 3. THEORY CONCERNING EVANESCENT MODES
23
error between the logging of the signal from the wavemaker and the wave gauges. This will also
give an unknown initial phase shifting. It is also mentioned that the wave gauges is placed with an
accuracy of approximately 2 cm in the horizontal distance from the wavemaker.
All above mentioned has an influence on the initial phase of the resulting wave (or the direction
of the complex resulting vector). This again leads to that in practice our phases of the evanescent
modes, θ and and the phase of the progressive wave ϕ are influenced by all above mentioned factors.
Therefore the two phases has an unknown phase shifting.
A way to come around these problems in an elegant and easy to program way is: Making a
time series from theory in section 3.1 a theoretical phase in a given point is calculated. The phase
difference between the measured time series and the artificially constructed time series are added to
the measured time series. In this way we ”force” the phase to be the theoretical phase as deduced
in previous section 3.2.1. The focus of the analysis is in this way led to the sizes of the amplitudes
of the progressive waves and the evanescent modes, as we previously have stated was the purpose of
this study.
Chapter 4
Program to analysis of evanescent
modes
The code for the program can be seen in appendix D.
4.1
Determination of analysed time interval
It is necessary to find the time interval where none of the wave gauges are influenced by waves
generated under the gain up of the flap and reflected waves.
The flume is divided in small parts ∆x(=0.1m), the depth is calculated for each x and the wave
length L is found using the linear dispersion relation.
It is now assumed that the period T for the wave is equal to the specified value, the measuments
show that this is a very good assumption, see appendix B. The wave velocity is now calculated:
c=
L
T
(4.1)
and the time it take the wave to move the distance x =
τ=
n
end
X
∆tn =
P
∆x is:
n=n
end
X
nstart
nstart
∆x
cn
(4.2)
The start time for the time window is the time it takes for the fully gained wave to reach the
wave gauge furthest from the wavemaker. The length of the time interval is the time it takes for the
wave to move from the last wave gauge to the end of the flume and back again.
It is assumed that the long waves generated during the warm up period is small, as seen in chapter
5 this is not right. It is also assumed that the reflection only will take place in the coastline.
The method above is not a sufficiently accurate method. A more correct method would be to
use the group velocity cg as the travelling speed of the waves. We know from theory that the waves
travel with the speed c but the wave heights travel with the group velocity. Using the group velocity
will give the result:
τ=
n=n
end
X
∆tn =
n=n
end
X
nstart
nstart
∆x
cg
(4.3)
The group velocity is slower than the phase velocity, so this can explain why the time window
start and end too early, see chapter 5.
24
CHAPTER 4. PROGRAM TO ANALYSIS OF EVANESCENT MODES
4.2
25
The complex wave number
The wave numbers are calculated from equation 3.8. The solution to the equation can be split up in
two parts, a real part, where the wave number is found by solving:
ω 2 = gk tanh(kh)
,
k∈R
(4.4)
This has one solution which corresponds to the propagating wave. The purely imaginary solutions
are calculated from the equation:
ω 2 = −g k˜n tan k˜n h ,
kn = −ik˜n
k˜n ∈ R+
(4.5)
√
where i is the imaginary unit (i = −1). Equation 4.5 is solved for n = {1, 2, . . . , 50}, where
n = 1 is the minimum absolute value of kn and n = 2 is the second lowest absolute value of kn and
so on. For practical reasons the equation is solved for kn h and not for kn .
4.3
Principles in the Fast Fourier Transform
The FFT is used for finding the phase and the size of the resulting vector representing the sum of
the progressive wave and the evanescent modes, for details see page 19.
In order to being able to find the phase of the time series we did not apply zero padding or
windowing. The consequence on applying zero padding and windowing is that the phase information
”in the ends” of the time series are eliminated. A rule of thumb is that the treated (e.g. by
zero padding and windowing) time series should be continues and differentiable if the time series is
extended by the time series itself in the time direction if the FFT should give reasonable results.
Because we do not apply windowing and/or zero padding and the previous stated condition must be
fulfilled we need to use an equal number of waves as input signal to the FFT analysis.
The FFT will output a number of complex numbers, each corresponding to a frequency. From
each complex number two pieces of information are given: the absolute value of the complex number
corresponds to the wave height and the direction of the complex number corresponds to the phase
in t = 0 of the input time series.
4.4
Determining the phase
All our theoretical derivations are based on the assumption, that in x = 0 the phase of the evanescent
mode is π/2 compared to the progressive wave. Because we want to calculate the phase from FFT
analysis of the time series we also need to assume that the time series in x = 0 starts in t = 0.
Because we do not know the time when the wavemaker starts gaining up we have one more
unknown: the initial phase. As mentioned above it is very important that the time series in x = 0
starts at t = 0 and therefore the initial phase must be found.
We have chosen two different methods to the determination of the initial phase:
• Using the position signal from the wavemaker to determination of the initial phase, constant
phase method.
• Assuming the phases calculated from linear theory are correct. From the theoretical phases
calculate the initial phase, variable phase method.
CHAPTER 4. PROGRAM TO ANALYSIS OF EVANESCENT MODES
4.4.1
26
The constant phase method
From the time window, previously discussed, we find the start time to which the waves are undisturbed.
The position of the wavemaker is sampled and we know from the horizontal boundary condition,
equation 3.6, that the wavemaker position in time and the surface elevation of the progressive wave
are π/2 out of phase. By making a zero crossing analysis of the wavemaker position signal we can
determine the initial phase. The same initial phase is used in all the time series from the different
wave gauges.
Using this method we have made the assumption, that the position of the piston and the surface
elevation of the progressive wave are π/2 out of phase. The terms we have neglected are of second
order and higher.
4.4.2
The variable phase method
In principle the phases at each wave gauge is assumed accurate calculated from linear theory. From
this information we can create an artificial time series (with evanescent modes) and compare the
phase of the measured time series to the artificial time series using e.g. zero crossing analysis.
By this method we can concentrate the analysis to the determination and comparison of the
amplitudes of the evanescent modes and the progressive wave. Opposite the previous method where
both the phase and the amplitude contribute to uncertainties.
Chapter 5
Finding the useful interval of the
surface elevation time series
The aim for this chapter is to find the useful interval of the surface elevation time series. The useful
time interval is the interval where none of the wave gauges are disturbed by the waves caused by
warm up or the reflected waves. For details about the calculations, see section 4.1. For practical
reasons the time windows are the same length for all the wave gauges, i.e. we would sample the same
number of waves. In realty could the first wave gauge have a longer time window than the second
and so on. The measured wave heights in this chapter is found using zero crossing. The wave height
is defined as the mean of the differes between the maximum and the minimum value for each wave
inside the time window. By using zero crossing we are able to split the time series into single waves
and then make an average wave height.
5.1
Case 1
0.015
Wave elevation
Measured mean wave heigth
Specified wave heigth
Start and end time
0.01
Elevation [m]
0.005
0
−0.005
−0.01
0
10
20
30
40
50
60
70
Time [s]
Figure 5.1: The measured time serie for case 1 measured in a distance of 0.14 m from the wavemaker.
Figures 5.1 and 5.2 show the measured time series for the first and the last wave gauge for case 1.
27
28
CHAPTER 5. FINDING USEFUL TIME WINDOW
−3
8
x 10
Wave elevation
Measured mean wave heigth
Specified wave heigth
Start and end time
6
4
Elevation [m]
2
0
−2
−4
−6
−8
0
10
20
30
40
50
60
70
Time [s]
Figure 5.2: The measured surface elevation time serie for case 1 in a distance of 1.73 m from the
wavemaker.
The measured wave height is larger than the specified near the wavemaker (figure 5.1), the specified
wave height was Hspec = 0.011 m and the measured was Hmes = 0.014 m. This must be because of
the evanescent modes. If we look at the wave height far away from the wavemaker is the measured
wave height smaller (H = 0.009 m) than the specified (figure 5.2), this is characteristic for all the
waves that the measured progressive waves are smaller than the specified. The crest amplitude is
larger than the trough amplitude, which indicate nonlinearities. The measured steepness of the case
1 wave is close to the wavemaker Hmes
= 0.036 (distance to the wavemaker, x = 0.14 m ) and only
L
0.023 in x = 1.73 m. The linear theory can normally be used until H
L = 0.05, and as seen at figure
5.3 the nonlinearities are fairly small, nearly invisible.
A closer look at figure 5.1 show that the first approximately 5 s are disturbed which cause a little
larger wave (approx. 0.5 mm) and the last approximately 10 s are disturbed which cause a little
smaller wave (approx. 0.4 mm) this is smaller than the expected uncertainty of the wave gauges, but
it look very consistent and is probably caused by the error in the calculation of the time window,
where we use c instead of cg see 4.1. The time window contain 70 waves, this should be enough to
make the FFT analysis.
5.2
Case 2
The figures 5.4 and 5.5 show the measured time series for the first and the last wave gauge for case
2. The specified wave height was Hspec = 0.036 m, the measured wave height at first wave gauge
(0.14 m) is Hmes = 0.041 m and at the last wave gauge the wave height is Hmes = 0.033 m and like
case 1 this decreasing wave height due to evanescent modes.
There seems to be some nonlinearities already at the first wave gauge, see 5.4
. The steepness of the wave at the first wave gauge is Hmes
= 0.053 and at the last wave gauge
L
Hmes
= 0.043, in other words it is not a very good assumption to use linear theory in this case. At
L
figure 5.6 is a short part of the surface elevation time serie showed. It is possible to see a small
29
CHAPTER 5. FINDING USEFUL TIME WINDOW
−3
x 10
4
Wave elevation
Measured mean wave heigth
Specified wave heigth
Start and end time
Elevation [m]
2
0
−2
−4
−6
17.6
17.7
17.8
17.9
18
18.1
Time [s]
18.2
18.3
18.4
18.5
Figure 5.3: Close look at the measured surface elevation time series for case 1 in a distance of 1.73 m
from the wavemaker.
nonlinaerities, but it is still not heavy nonlinear.
Due to the same problem as case 1 the time window is starting and ending around 5 seconds too
early at wave gauge 1 (x = 0.14 m), but it is hard see the same at wave gauge 10 (x = 1.73 m)
because it seem to be disturbed by a long wave in all of the time interval from t = 0 s to t = 60 s.
The time window contain 33 waves, this should be enough waves to make a good FFT analysis.
5.3
Case 3
The figures 5.7 and 5.8 show the the measured time series for the first and the last wave gauge for
case 3. The specified wave height was Hspec = 0.035 m. The measured at wave gauge 1 (x = 0.14 m)
is Hmes = 0.035 m and at wave gauge 10 (x = 1.73 m) it is Hmes = 0.032 m This wave has nearly
no differences between the amplitudes of the crests and the toughs. The steepness of the wave is at
Hmes
wave gauge 1 Hmes
L = 0.012 and at wave gauge 10
L = 0.011. This indicate that linear theory will
be a very good approximation.
The wave height seem to be nearly constant inside the time window, this indicate that the intervals
are not influenced by the warm up period or the reflections from the end of the flume.
The time window contain 11 waves it is a small number of waves for FFT analysis.
30
CHAPTER 5. FINDING USEFUL TIME WINDOW
0.025
0.02
Wave elevation
Measured mean wave heigth
Specified wave heigth
Start and end time
0.015
0.01
Elevation [m]
0.005
0
−0.005
−0.01
−0.015
−0.02
−0.025
0
10
20
30
40
50
60
70
Time [s]
Figure 5.4: The measured surface elevation time series for case 2 in a distance of 0.14 m from the
wavemaker.
0.025
0.02
0.015
Wave elevation
Measured mean wave heigth
Specified wave heigth
Start and end time
Elevation [m]
0.01
0.005
0
−0.005
−0.01
−0.015
−0.02
0
10
20
30
40
50
60
70
Time [s]
Figure 5.5: The measured surface elevation time series for case 2 in a distance of 1.73 m from the
wavemaker.
31
CHAPTER 5. FINDING USEFUL TIME WINDOW
0.015
Wave elevation
Measured mean wave heigth
Specified wave heigth
Start and end time
0.01
Elevation [m]
0.005
0
−0.005
−0.01
−0.015
14
14.2
14.4
14.6
Time [s]
14.8
15
15.2
Figure 5.6: Close look at the measured surface elevation time series for case 2 in a distance of 1.73 m
from the wavemaker.
0.025
Wave elevation
Measured mean wave heigth
Specified wave heigth
Start and end time
0.02
0.015
0.01
Elevation [m]
0.005
0
−0.005
−0.01
−0.015
−0.02
−0.025
0
10
20
30
40
50
60
70
Time [s]
Figure 5.7: The measured surface elevation time series for case 3 in a distance of 0.14 m from the
wavemaker.
32
CHAPTER 5. FINDING USEFUL TIME WINDOW
0.02
Wave elevation
Measured mean wave heigth
Specified wave heigth
Start and end time
0.015
0.01
Elevation [m]
0.005
0
−0.005
−0.01
−0.015
−0.02
0
10
20
30
40
50
60
70
Time [s]
Figure 5.8: The measured surface elevation time series for case 3 in a distance of 1.73 m from the
wavemaker.
Chapter 6
The result of the evanescent modes
analysis
6.1
Result of analysis with constant phase method
The method to analyse the results is the constant phase method. The Principle in the method is
that the phase at all the wave gauges are related to the position of the piston, for more details see
section 4.4.1. The results of the constant phase method are plotted on figures 6.1 to 6.3. The figures
also contain the theoretical results (linear theory) of the specified wave height. As seen in chapter 5
the measured waves will probably be smaller than the specified, so the amplitude of the theoretical
wave will be too large compared to what is expected.
0.01
Evanescent mode (linear theory)
Progressive mode (linear theory)
Measured evanescent mode
Measured progressive mode
Amplitude [m]
0.005
0
−0.005
−0.01
−0.015
0
0.2
0.4
0.6
0.8
1
1.2
Distance from wavemaker [m]
1.4
1.6
1.8
Figure 6.1: The result of the constant phase method, case 1.
The figures 6.1 to 6.3 show clearly that the results are not good enough. The results are absolutely
unphysical with a heavy fluctuating amplitude. Only far away from the wavemaker the result are
starting to look better, with nearly constant amplitude of the progressive wave and the evanescent
modes (last point at figure 6.2 may be disturbed by a low determinant, see table 6.1).
The table 6.1 show the determinats for the 3 cases in the different measure points. A small
33
34
CHAPTER 6. THE RESULT OF THE EVANESCENT MODES ANALYSIS
0.2
Evanescent mode (linear theory)
Progressive mode (linear theory)
Measured evanescent mode
Measured progressive mode
0.15
0.1
Amplitude [m]
0.05
0
−0.05
−0.1
−0.15
−0.2
−0.25
0
0.2
0.4
0.6
0.8
1
1.2
Distance from wavemaker [m]
1.4
1.6
1.8
Figure 6.2: The result of the constant phase method, case 2.
0.6
Evanescent mode (linear theory)
Progressive mode (linear theory)
Measured evanescent mode
Measured progressive mode
0.4
Amplitude [m]
0.2
0
−0.2
−0.4
−0.6
−0.8
0
0.2
0.4
0.6
0.8
1
1.2
Distance from wavemaker [m]
1.4
1.6
Figure 6.3: The result of the constant phase method, case 3.
1.8
CHAPTER 6. THE RESULT OF THE EVANESCENT MODES ANALYSIS
35
Table 6.1: The determinats for the 3 cases.
Distance from wavemaker [m] Case 1 Case 2 Case 3
0.14
0.63i
-0.41i
-0.95i
0.22
0.92i
0.23i
-0.89i
0.32
-0.42i
0.87i
-0.76i
0.44
-0.70i
0.89i
-0.57i
0.59
1.00i
-0.13i
-0.27i
0.73
-0.69i
-0.96i
0.03i
0.92
0.62i
-0.29i
0.43i
1.20
-0.89i
0.91i
0.87i
1.56
-1.00i
-0.97i
0.96i
1.73
0.91i
0.08i
0.79i
determinat can explain some of the large deviations, but specially case 1 have some very large
deviations caused by other factors, probably inaccurately determined phases.
Table 6.2: The differences between the measured and the linear theory phases.
Distance from wavemaker [m] Case 1 Case 2 Case 3
0.14
0.96
1.50
0.66
0.22
0.94
1.53
0.68
0.32
0.64
1.43
0.73
0.44
0.82
1.49
0.72
0.59
1.31
1.32
0.65
0.73
1.29
1.30
0.65
0.92
1.34
1.28
0.64
1.20
0.17
0.51
0.42
1.56
0.25
0.46
0.43
1.73
0.36
0.47
0.45
Table 6.2 show the differences between the measured and the linear theory phases. Generally the
differences are smaller far away from the wavemaker, this is in agreement with the result plotted at
figure 6.1 to 6.3, where the 3 last point is nearly as expected, see table 6.2. The conclusion is that
the phases are wrong and a phase error has a high influence on the amplitude.
There is several things there can cause errors on the phases, the most common is:
• A synchronisation error, for example a synchronisation error between the sampling of the
wavemaker position and the sampling of the wave gauges.
• An error of the position of the wave gauges.
• An error of the wave number k.
A synchronisation error will be constant in time, most likely also in space, because it is expected
that the sampling of the wave gauges is well synchronised, because it is logged by the same eqiupment
to the same file, so the synchronisation error will only take place between the sampling of the
wavemaker position and the sampling of the elevation. The size of this error of the synchronisation
is unknown, but probably smaller than 0.1 s.
An error of the position of the wave gauges will be random for each wave gauge. The uncertainty
of the position of the wave gauges is around ±2 cm of course in the horizontal direction.
CHAPTER 6. THE RESULT OF THE EVANESCENT MODES ANALYSIS
36
A constant error of the wave number will result in a increasing phase error, because the wave
number will be multiplied with the increasing distance to the wavemaker. The wave number error
can also be changing, for example because of the gently slopping bed, but the difference of depth is
only 1.8 cm and will have no influence of the linear wave number in case 1 and 2 (the difference in
the wave number, k is lower than 0.001 m−1 in case 1 and 0.001 m−1 in case 2). Case 3 has a little
bigger change of the wave number at k̃ = 0.03 m−1 . In a distance to the wavemaker at x =1.73 m
it will cause a phase error at ϕ̃ = k̃x ≈ 0.06. Therefore we can not explain the errors in phases by
inaccurately determined wave numbers.
Table 6.3: The relation between phase error and time or position error.
Case no. Time error t̃ [s/rad] Position error x̃ [m/rad]
1
0.08
0.06
2
0.11
0.12
3
0.24
0.46
To give an idea about the size of the error, table 6.3 shows how large an error in time or position
should be for make a specific phase error. The error of case 1 can be explain as errors in the phase
caused by synchronisation and position errors, by comparing table 6.2 and 6.3. The errors of case 2
and case 3 can not only be caused of the synchronisation and position error, but higher order effects
can also be a part of the explanation. In table 6.2 are the differences between the phase error close
to the wavemaker and the phase error at wave gauge 10, smallest for case 3 (the most linear wave)
∆ϕ = 0.21. The difference in case 1 is ∆ϕ = 0.60 and for the most nonlinear, case 2, is the difference
∆ϕ = 1.21. This indicate that some of the errors are caused by higher order effects.
6.2
Result of analysis with changing phase
To get a more useful result the method of changing phase will be used. The principle of the changing
phase method is to find the phase using theoretical (in this case linear) theory and adjust the measured
wave to match the theoretical phase, see section 4.4.2. The results for changing phase method are
plotted on figure 6.4 to 6.6. The determinats are the same as for the constant phase method, see
table 6.1.
The results using changing phase method are much better than the results using constant phase
method. Table 6.4 show the results for case 1.
Table 6.4: Results using chaning phase method, case 1. ∆aprog = aprog,mes − aprog,theo and ∆aevan =
aevan,mes − aevan,theo .
Ch. no. aprog,mes [m] aevan.mes ∆aprog [m] ∆aevan [m]
1
0.0095
0.0039
0.0040
0.0016
2
0.0039
0.0011
-0.0016
-0.0006
3
0.0032
0.0009
-0.0023
-0.0003
4
0.0059
0.0009
0.0004
0.0002
5
0.0050
0.0004
-0.0005
-0.0001
6
0.0043
0.0004
-0.0012
0.0001
7
0.0049
0.0000
-0.0006
-0.0001
8
0.0046
0.0001
-0.0009
0.0000
9
0.0045
0.0000
-0.0010
0.0000
10
0.0045
0.0000
-0.0010
0.0000
37
CHAPTER 6. THE RESULT OF THE EVANESCENT MODES ANALYSIS
0.01
Evanescent mode (linear theory)
Progressive mode (linear theory)
measured evanescent mode
Measured progressive mode
0.009
0.008
Amplitude [m]
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0
0
0.2
0.4
0.6
0.8
1
1.2
Distance to wavermaker [m]
1.4
1.6
1.8
Figure 6.4: The result with a changing phase, case 1.
0.03
Evanescent mode (linear theory)
Progressive mode (linear theory)
measured evanescent mode
Measured progressive mode
0.025
Amplitude [m]
0.02
0.015
0.01
0.005
0
0
0.2
0.4
0.6
0.8
1
1.2
Distance to wavermaker [m]
1.4
1.6
Figure 6.5: The result with a changing phase, case 2.
1.8
38
CHAPTER 6. THE RESULT OF THE EVANESCENT MODES ANALYSIS
0.02
Evanescent mode (linear theory)
Progressive mode (linear theory)
measured evanescent mode
Measured progressive mode
0.018
0.016
Amplitude [m]
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
0
0.2
0.4
0.6
0.8
1
1.2
Distance to wavermaker [m]
1.4
1.6
1.8
Figure 6.6: The result with a changing phase, case 3.
Table 6.4 show results as expected. The progressive wave height at wave gauge 10 is equal to the
wave height found from the surface elevation time serie. The total wave height is 1 mm lower than
the specified as seen on the surface elevation time serie section 5.1.
The evanescent modes goes to zero as expected, but both the progressiv mode and the evanescent
mode are bigger than expected.
Table 6.5: Results using changing phase method, case 2. ∆aprog = aprog,mes −aprog,theo and ∆aevan =
aevan,mes − aevan,theo .
Ch. no. aprog,mes [m] aevan.mes ∆aprog [m] ∆aevan [m]
1
0.0287
0.0098
0.0107
0.0042
2
0.0235
0.0050
0.0055
0.0010
3
0.0182
0.0027
0.0002
0.0000
4
0.0159
0.0011
-0.0021
-0.0006
5
0.0132
0.0036
-0.0048
0.0027
6
0.0166
0.0006
-0.0014
0.0000
7
0.0175
0.0012
-0.0005
0.0009
8
0.0159
0.0002
-0.0021
0.0001
9
0.0169
0.0006
-0.0011
0.0005
10
0.0136
0.0027
-0.0044
0.0027
The results for case 2 using changing phase method can be seen in table 6.5. In this case the
result from the last wave gauge is bad because of a low determinant, see table 6.1, so wave gauge
number 9 is used instead. The wave height in table 6.5 is Hmes,F F T = 0.0338 m and the result from
the time series (wave gauge 9) is Hmes,zero = 0.033 m, in other words approximately 1 mm difference,
but if we take the mean of wage gauge 8 and 9 is the difference 0, see section 5.2.
The evanescent modes are not going to exact zero, but nearly.
The results from case 3 are showed in table 6.6. The wave height at the last wave gauge respectively Hmes,F F T = 0.0324 m from the FFT analysis and Hmes,zero = 0.032 m from the surface
CHAPTER 6. THE RESULT OF THE EVANESCENT MODES ANALYSIS
39
Table 6.6: Results using changing phase method, case 3. ∆aprog = aprog,mes −aprog,theo and ∆aevan =
aevan,mes − aevan,theo .
Ch. no. aprog,mes [m] aevan.mes ∆aprog [m] ∆aevan [m]
1
0.0173
0.0007
-0.0002
0.0001
2
0.0169
0.0002
-0.0006
-0.0002
3
0.0171
0.0003
-0.0004
0.0001
4
0.0171
0.0003
-0.0004
0.0002
5
0.0184
0.0017
0.0009
0.0017
6
0.0102
0.0065
-0.0073
0.0064
7
0.0169
0.0003
-0.0006
0.0003
8
0.0163
0.0002
-0.0012
0.0002
9
0.0165
0.0001
-0.0010
0.0001
10
0.0162
0.0001
-0.0013
0.0001
elevation time serie. The evanescent modes are going slowly to zero, the wave length is L = 2.86 m
so this is nearly as expected.
Chapter 7
Statistical results
The figures 7.1 to 7.6 show the normalized progressiv amplitudes and normalized resulting amplitudes.
The progressive amplitudes are normalized with the theoretical linear progressive amplitude and the
evanescent modes amplitudes are normalized with the theoretical linear evanescent modes amplitudes.
The figures show a weighted mean and a weighted standard deviation too. The weight is the modulus
of the determinant in the power of 2:
W =
Ã
mod
³
¸ !2
0 cos k0 x ´
det
1 sin k0 x
·
(7.1)
W become small when the progressive phase and the evanescent mode phase are nindentical or
nearly identical. Because both depend on ωt in the same way, will the determinat be small when the
kx ≈ nπ/2; n ∈ Z.
The normalized evanescent modes are not calculated because they go to zero and the normalization will not make sense. The normalized resulting amplitudes are calculated instead.
7.1
Case 1
The figures 7.1 and 7.2 show the results of case 1. In table 7.1 it is seen that the ratio between
measured amplitudes and linear theory is the same for the resulting wave and the progressive wave.
It indicates that the weighting is working good, because the errors caused by low determinants are
eliminated. The mean value of the weighted normalized amplitudes are:
ares = 0.89
(7.2)
The same ratio found from the surface elevation time series, see figure 5.2, is approximately 0.82,
measured 1.73 m from the flap. This indicate that the FFT give a higher amplitude than measured
or that the weighting is not good enough.
Table 7.1: Normalized mean amplitude and standard deviation.
Case no. Mean ares Mean aprog Deviation σres Deviation σprog
1
0.89
0.89
0.0013
0.0013
2
0.96
0.96
0.0068
0.0068
3
0.96
0.96
0.0058
0.0058
40
41
CHAPTER 7. STATISTICAL RESULTS
a
/a
prog,mes
prog,theo
Weighted mean
Standard deviation
1.6
Relative amplitude [m]
1.4
1.2
1
0.8
0.6
0.2
0.4
0.6
0.8
1
1.2
Distance to wavermaker [m]
1.4
1.6
1.8
Figure 7.1: The amplitude of the progressive wave normalized with the theoretical, specified progressive wave, linear theory (standard deviation is weighted too), case 1.
1.2
H
/H
res,mes
res,theo
Weighted mean
Standard deviation
1.1
Relative amplitude
1
0.9
0.8
0.7
0.6
0
0.2
0.4
0.6
0.8
1
1.2
Distance to wavermaker [m]
1.4
1.6
1.8
Figure 7.2: The amplitude of the resulting wave normalized with the theoretical, specified resulting
wave, linear theory (standard deviation is weighted too), case 1.
42
CHAPTER 7. STATISTICAL RESULTS
7.2
Case 2
1.6
a
/a
prog,mes
prog,theo
Weighted mean
Standard deviation
1.5
1.4
Relative amplitude [m]
1.3
1.2
1.1
1
0.9
0.8
0.7
0
0.2
0.4
0.6
0.8
1
1.2
Distance to wavermaker [m]
1.4
1.6
1.8
Figure 7.3: The amplitude of the progressive wave normalized with the theoretical progressive wave,
linear theory (standard deviation is weighted too), case 2.
The figures 7.3 and 7.4 show the result for case 2. In table 7.1 is showed that the ratio between
measured amplitudes and linear theory is the same for the resulting wave and the progressive wave.
The weigthed normalized amplitudes are:
ares = 0.96
(7.3)
The same ratio at the surface elevation time series, see figure 5.5, is 0.92, measured 1.73 m from
the flap. This indicate again that the FFT give a higher amplitude than measured or that the
weighting is not good enough, but the difference is smaller this time.
7.3
Case 3
The figures 7.5 and 7.6 show the result. In table 7.1 is seen that the ratio between measured
amplitudes and linear theory is the same for the resulting wave and the progressive wave. The
normalized amplitudes are:
ares = 0.96
(7.4)
The ratio at the time series figure 5.8 is approximately 0.91, measured 1.73 m from the flap.
The weighted standard deviations are small. The waves a around 7% smaller than the specified
waves, see table 7.1.
The results are better if the Changing phase method is used than if the constant phase method
is used. This indicate that the linear theory is useful for finding the amplitudes, but not the phases.
43
CHAPTER 7. STATISTICAL RESULTS
1.1
H
/H
res,mes
res,theo
Weighted mean
Standard deviation
Relative amplitude
1.05
1
0.95
0.9
0.85
0
0.2
0.4
0.6
0.8
1
1.2
Distance to wavermaker [m]
1.4
1.6
1.8
Figure 7.4: The amplitude of the resulting wave normalized with the theoretical resulting wave, linear
theory (standard deviation is weighted too), case 2.
1.2
a
/a
prog,mes
prog,theo
Weighted mean
Standard deviation
1.1
Relative amplitude [m]
1
0.9
0.8
0.7
0.6
0.5
0
0.2
0.4
0.6
0.8
1
1.2
Distance to wavermaker [m]
1.4
1.6
1.8
Figure 7.5: The amplitude of the progressive wave normalized with the theoretical progressive wave,
linear theory (standard deviation is weighted too), case 3.
44
CHAPTER 7. STATISTICAL RESULTS
0.99
H
/H
res,mes
res,theo
Weighted mean
Standard deviation
0.98
Relative amplitude
0.97
0.96
0.95
0.94
0.93
0
0.2
0.4
0.6
0.8
1
1.2
Distance to wavermaker [m]
1.4
1.6
1.8
Figure 7.6: The amplitude of the resulting wave normalized with the theoretical resulting wave, linear
theory (standard deviation is weighted too), case 3.
Part III
Surfzone.
Analysis of MIKE 21’s Boussinesq
Wave Module Handling of Breaking
Waves and Runup.
45
Chapter 8
Analysis of laboratory experiments in
the surfzone
In the following the results of the laboratory experiments of the 6 cases presented in section 2.4.2
and 2.4.2 are analysed with the purpose of explaining the phenomenons as the waves approach the
coastline.
8.1
Analysis of time series of the regular waves in different distances from the wavemaker
To get an overview of the form of the regular, monochromatic waves measured in different distances
from the wavemaker, and what is happening as the waves shoal towards the coast, time series are
analyzed. The time series are analysed in 6 different distances from the wavemaker.
Case 1, monochromatic wave, T = 1.0 s, H = 0.1 m
The wavemaker movement is in this part purely sinusoidal. Because the wave steepness (H/L) is
0.065 the shape of the surface elevation is expected to be relatively sinusoidal on the first couples of
meters, then changing from sinusoidal at the wavemaker into sharper and sharper crests and longer
throughs as the wave moves away from the wavemaker. In theory it is not very accurate to apply
sinusoidal wave theory to a wave of this steepness and we should therefore expect some nonlinear
effects as sharpened crests and longer throughs.
One effect that is happening as we apply sinusoidal waves to a nonlinear system is the release
of parasitic higher harmonic. These waves are free waves with the same frequencies as the bound
waves [Madsen, P.A. and Sørensen, O.R. (1993)]. When we apply a linear boundary condition to a
nonlinear system the system will counteract. As we force the boundary to be sinusoidal the system
reacts by ”making” waves in counter phase to the higher harmonics. The wave in counter phase will
”turn off” the higher harmonics which is needed to satisfy the nonlinear system. But as the free
parasitic waves do not have the same celerity as the bound higher harmonic, they will only cancel
each other on the wavemaker. The wave forced to be sinusoidal on the wavemaker will become a non
sinusoidal wave further from the wavemaker.
As seen from figure 8.2 the wave is close to being sinusoidal at the first wave gauge (on the figure
marked as channel 1) , which is placed 2.3 m from the wavemaker. The wave height is approximately
11 cm. This is 1 cm higher than the specified wave height. This increment in wave height is not due
to shoaling. As seen on the bathymetry in the bottom of the figure there is hardly any change in
water depth (< 10%) on the first 2.3 m and shoaling is therefore out of the question. The difference
between the wave height specified at the wavemaker and the measured wave height is more likely
due to a problem in the wavemaker.
46
CHAPTER 8. ANALYSIS OF LABORATORY EXPERIMENTS IN THE SURFZONE
47
Going to wave gauge number 2 the wave height has decreased by approximately 1 cm. This is
due to the set-down. We can find our shoaling gradient to be less that one until the water depth is
less than h = 0.1 m [Svendsen, I. A and Jonsson, I. G. (1996)]. A rule of thumb says that the wave
will break when the wave height exceeds 80% of the depth. Therefore according to linear theory the
wave will start breaking before it begins having shoaling coefficient larger than 1.
If we look at the lower graph in figure 8.2, where all the wave heights are inserted in connection with the bathymetry we see a very small shoaling. This is because the previously mentioned calculation of the shoaling gradient is based on linear theory which obviously is not be
valid in this domain. Linear theory is not valid in that domain because the increase of the wave
height is a combination of linear shoaling characteristics and non-linear transfer to higher harmonics
[Madsen, P. A. et al., (1997)]. This will be commented more thoroughly in section 12.1.1
At wave gauge number 2 the crests of the waves have sharpened very little compared to the
waves measured at wave gauge number 1 which is expected. The energy has been transferred from
the sinusoidal wave that was specified at the wavemaker to its bounded higher harmonics. This is
what results in the sharper crests and flatter throughs.
At wave gauge number 5 the crests are getting even more sharp. A small increment in wave
height is also observed.
Moving on to wave gauge number 10 we see that breaking occurs. The wave height decreases
dramatically and the peak of the waves move further back in time compared to not-breaking waves,
where the peak is approximately half way between two wave throughs. The shape of the wave becomes
more triangular. Comparing the wave heights from period to period the wave heights do not look
very constant. In the three periods shown in figure 8.2 at wave gauge number 10 the wave height
changes a couple of centimeters. The surf similarity parameter, ζ, [Madsen, P. A. et al., (1997)] is
calculated to be 0.245 which means that the wave is a spilling breaker. On a spilling breaker most
of the ”foam” is on the top of the wave. This could cause some air bobbles which again leads to
instable results.
At wave gauge number 12 we observe that the variation in wave height has stabilized. This can
be caused by the wave breaking is becoming less intense as the waves move from the point of breaking
to the coastline. By visual observing the breaking of a wave, one is convinced that the breaking is
more intense just after breaking compared to some time after the start of the breaking.
Case 2, monochromatic wave , T = 1.5 s, H = 0.08 m
The next case contains the same phenomenons as the previous and will therefore not be commented
as thoroughly. This wave is not as steep (H/L = 0.027) and therefore the first wave looks more
like a sinusoidal than on the previous case. A noticeable difference is the shape of the wave at wave
gauge number 8, which is positioned a short distance just before breaking: A slowly varying through
is followed by fast decrement in surface elevation. If the wave is considered in the spatial domain
the opposite situation will be seen: A pronounced increment in surface elevation followed by a slow
decrement. I. e. the wave is ”leaning forward”. By using linear theory this phenomenon can be
explained. As the waves approach shallower water the waves become less frequency dispersive and
the wave celerity will depend only on the depth. As the wave height becomes a considerable fragment
of the depth the wave celerity will also depend on how large the wave height is (amplitude dispersive).
At the top of the crest there is more water underneath and therefore the crest will move faster. The
through, which has less water underneath, will move slower. Thus the crest will move faster than the
through and the wave will ’lean’ forward [Svendsen, I. A and Jonsson, I. G. (1996)], see figure 8.1.
We also notice that in this case 2 we have a much more pronounced shoaling. According to linear
theory and [Svendsen, I. A and Jonsson, I. G. (1996)] the shoaling coefficient starts being larger than
one (H > H0 ) on a water depth of approximately 20 cm. This agree with the lowest graph in figure
8.3 where we see that shoaling starts approximately at wave gauge 6.
CHAPTER 8. ANALYSIS OF LABORATORY EXPERIMENTS IN THE SURFZONE
Figure 8.1: Sketch of the wave profile as the wave moves to shallower waver.
[Svendsen, I. A and Jonsson, I. G. (1996)]
48
From
Case 3, monochromatic wave, T = 2.0 s, H = 0.13 m
This wave train, shown in figure 8.4, has a wave steepness of 0.031. As seen in figure 8.4, the waves
does not look sinusoidal at any of the wave gauges.
49
CHAPTER 8. ANALYSIS OF LABORATORY EXPERIMENTS IN THE SURFZONE
Channel 1, x =2.30 m
0.10
0.05
0.05
0.00
0.00
-0.05
20:01:30
2002-03-03
20:01:31
20:01:32
-0.05
20:01:30
2002-03-03
20:01:33
Channel 5, x = 15.8 m
0.10
0.05
0.00
0.00
20:01:31
20:01:32
-0.05
20:01:30
2002-03-03
20:01:33
Channel 10, x = 17.83 m
0.10
0.05
0.00
0.00
-0.05
20:01:30
2002-03-03
20:01:31
20:01:32
-0.05
20:01:30
2002-03-03
20:01:33
20:01:32
20:01:33
20:01:31
20:01:32
20:01:33
Channel 12, x = 18.60 m
0.10
0.05
20:01:31
Channel 8, x = 17.02 m
0.10
0.05
-0.05
20:01:30
2002-03-03
Channel 2, x = 9.68 m
0.10
20:01:31
20:01:32
20:01:33
Bathymetry
[m]
Wave gauges
[-]
Avg. measured waveheight [-]
0.00
-0.20
-0.40
0
2
4
6
8
10
12
14
16
18
20
22
03/03/02 20:00:00:000
Figure 8.2: Time series of regular monochromatic waves, T = 1.0 s, H = 0.1 m, in different distances
from the wavemaker. The lower figure shows the bathymetry where the circles mark where the wave
gauges are placed. x’s mark the averaged wave height.
50
CHAPTER 8. ANALYSIS OF LABORATORY EXPERIMENTS IN THE SURFZONE
Channel 1, x =2.30 m
0.10
0.05
0.05
0.00
0.00
-0.05
20:01:30
2002-03-03
20:01:31
20:01:32
-0.05
20:01:30
2002-03-03
20:01:33
Channel 5, x = 15.8 m
0.10
0.05
0.00
0.00
20:01:31
20:01:32
-0.05
20:01:30
2002-03-03
20:01:33
Channel 10, x = 17.83 m
0.10
0.05
0.00
0.00
-0.05
20:01:30
2002-03-03
20:01:31
20:01:32
-0.05
20:01:30
2002-03-03
20:01:33
20:01:32
20:01:33
20:01:31
20:01:32
20:01:33
Channel 12, x = 18.60 m
0.10
0.05
20:01:31
Channel 8, x = 17.02 m
0.10
0.05
-0.05
20:01:30
2002-03-03
Channel 2, x = 9.68 m
0.10
20:01:31
20:01:32
20:01:33
Bathymetry
[m]
Wave gauges
[-]
Avg. measured waveheight [-]
0.20
0.00
-0.20
-0.40
0
2
4
6
8
10
12
14
16
18
20
22
03/03/02 20:00:00:000
Figure 8.3: Time series of regular monochromatic waves, T = 1.5 s, H = 0.08 m, in different distances
from the wavemaker. The lower figure shows the bathymetry where the circles mark where the wave
gauges are placed. x’s mark the averaged wave height.
CHAPTER 8. ANALYSIS OF LABORATORY EXPERIMENTS IN THE SURFZONE
Channel 1, x =2.30 m
Channel 2, x = 9.68 m
0.10
0.10
0.05
0.05
0.00
0.00
-0.05
20:01:30
2002-03-03
20:01:32
-0.05
20:01:30
2002-03-03
20:01:34
Channel 5, x = 15.8 m
0.10
0.05
0.05
0.00
0.00
20:01:32
-0.05
20:01:30
2002-03-03
20:01:34
Channel 10, x = 17.83 m
0.10
0.05
0.05
0.00
0.00
20:01:32
20:01:34
20:01:32
20:01:34
Channel 12, x = 18.60 m
0.10
-0.05
20:01:30
2002-03-03
20:01:32
Channel 8, x = 17.02 m
0.10
-0.05
20:01:30
2002-03-03
51
-0.05
20:01:30
2002-03-03
20:01:34
20:01:32
20:01:34
Bathymetry
[m]
Wave gauges
[-]
Avg. measured waveheight [-]
0.20
0.00
-0.20
-0.40
0
2
4
6
8
10
12
14
16
18
20
22
03/03/02 20:00:00:000
Figure 8.4: Time series of regular monochromatic waves, T = 2.0 s, H = 0.13 m, in different distances
from the wavemaker. The lower figure shows the bathymetry where the circles mark where the wave
gauges are placed. x’s mark the averaged wave height.
CHAPTER 8. ANALYSIS OF LABORATORY EXPERIMENTS IN THE SURFZONE
8.2
52
Analysis of time series measured on the runup gauge in the
cases of regular monochromatic waves
On figure 8.5 we see the time series and the FFT analysis of the measurement from the runup gauge
in case 1. If we only pay attention to the FFT analysis it is seen that the most energetic frequency
is approximately 0.04 Hz. This corresponds to a period of about 25 seconds. On the plot of the time
series we recognize this period as the long wave; the long wave has minimum in the time 20:02:20
and the next minimum is in the time 20:02:42. We can see that the amplitude of this long wave
exceeds the amplitude of the specified frequency (f = 1Hz) for this wave. The specified frequency is
the very short fluctuations on the plot of time series. On the plot of the FFT analysis this is the the
small peak in f = 1 Hz.
The 2% exceedence value, R2% , is defined as the 2% exceedence of the vertical runup maximum
[Ruggiero et al, (2001)]. The total runup exceedence value, R2%,total are define as R2% only not
vertical but total (following the slope of the beach. From the time series, figure 8.5 we can estimate
the value of R2%,total . The total, R2%,total is estimated to 0.3 m. On the slope of 0.063 this gives a
value of R2% of 0.02 m.
An implicit expression of R2% is gives as [Ruggiero et al, (2001)]:
R2% = 0.27
p
aHs L0
(8.1)
Where a is the slope of the beach, Hs is the significant wave height and L0 is the deep water
wave length. By calculating this for case 1, Hs = 0.10 m, L0 = 1.56 m and a = 0.063 we obtain a
value of 0.027 m which is surely in the same order as the one we measured. √
Whether the two are comparable is not for sure as the smallest value of aHs L0 investigated in
the paper is approximately 2 m and ours is 0.1 m. As the
√ implicit expression, equation 8.1 is derived
on basis of investigation in a hole other range of the aHs L0 parameter the results might not be
comparable.
But the value of R2% does not tell anything about the validity of the fluctuation of the larger
wave (T ≈ 25 s) only that the order of magnitude of the measurements are quite good.
The time series of the next case 2, T = 1.5 s, H = 0.08 m is seen on figure 8.6. Here it is
interesting to look at the FFT analysis. Apart from the previous analysis this has almost equally
much energy on the frequency of the main wave (T = 1.5 s) as on the long waves which we also
encountered in the previous analysis.
The last of the three time series shows case 3, T = 2.0 s and H = 0.13 m. In this we see something
in the middle of the to cases above. The long wave is very dominating but not as dominating as in
case 1, figure 8.11.
CHAPTER 8. ANALYSIS OF LABORATORY EXPERIMENTS IN THE SURFZONE
53
G:\skole\midtvejsprojekt\MIKE\bolge_T10_H01\SA_measuredlog1SUB.dfs0
0.38
0.37
0.0012
0.36
0.0010
0.0008
0.35
0.0006
0.0004
0.34
0.0002
0.33
0.0
0.5
1.0
1.5
2.0
0.32
0.31
0.30
0.29
0.28
0.27
0.26
20:02:00
2002-03-03
20:02:05
20:02:10
20:02:15
20:02:20
20:02:25
20:02:30
20:02:35
20:02:40
20:02:45
20:02:50
Figure 8.5: Time series of regular monochromatic waves, T = 1.0 s, H = 0.1 m measured on the
runup gauge. The small plot in the top right corner is the FFT analysis of the same time series. The
FFT analysis express the energy as a function of the frequency.
8.3
Analysis of FFT analysis in different distances from the wavemaker
It is difficult to tell much about the development of an irregular wave just by looking on a small range
of its time series. In the following we have done an FFT analysis of time series from the different
wave gauges. The FFT plots a spectrum of energies. On the axis of abscissa we have the frequency
in Hz ( T1 ) on the axis of ordinate the energy corresponding to that particular frequency in m 2 /Hz.
In the first plot, figure 8.8, we see FFT analysis of the irregular case IR 1. This wave train has a
peak period of 1.5 seconds, a significant wave height of 0.03 m and a γ value of 1. The peak period,
Tp is the period of the wave component containing the highest energy [DHI Lecture Note]. As seen
on the top left plot in figure 8.8 (Channel 1) a frequency around 0.66 Hz (corresponding to T p = 1.5
s) has the highest energy. This is as we expected as we specified a peak period of 1.5 seconds on the
wavemaker. On the wavemaker we also specified the significant wave height, H s to be 0.03 m. The
√
significant wave height can be estimated as 4 m0 where m0 is the integral under the FFT curve.
This quantity is difficult to determine from the visual inspection, but as comparison it is very useful;
by comparing a visual estimate of the areas under the curves we can have an idea of how the wave
height change.
At wave gauge number 1, 2 and 5 there is hardly any change in the form of the spectrum. As
we move to wave gauge number 8 we observe a coming of some low frequency energy. The energy of
these lower frequencies is probably the sub-harmonics of the most energetic waves (i.e. frequencies)
interacting. But also interaction between other frequencies contribute to these sub-harmonics. At
wave gauges 10 and 12 we see the development of some super harmonic energy. This is the small
CHAPTER 8. ANALYSIS OF LABORATORY EXPERIMENTS IN THE SURFZONE
54
G:\skole\midtvejsprojekt\MIKE\bolge_T15_H008\SA_measuredlog1SUB.dfs0
0.54
0.0050
0.52
0.0040
0.50
0.0030
0.48
0.0020
0.46
0.0010
0.44
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.42
0.40
0.38
0.36
0.34
0.32
0.30
20:02:00
2002-03-03
20:02:10
20:02:20
20:02:30
20:02:40
20:02:50
20:03:00
Figure 8.6: Time series of regular monochromatic waves, T = 1.5 s, H = 0.08 m measured on the
runup gauge. The small plot in the top right corner is the FFT analysis of the same time series. The
FFT analysis express the energy as a function of the frequency.
peak with a frequency approximately twice the frequency of the main peak. The development of
these super harmonics is an expected phenomenon of shoaling; the super harmonics will, because of
the smaller wave length, have a larger shoaling coefficient [Svendsen, I. A and Jonsson, I. G. (1996)].
Going from wave gauge number 10 to 12 we see the area under the curve is significantly smaller at
wave gauge number 12 compared to wave gauge number 10. This is because the wave enters the
breaking zone where the waves loose their energy and therefore the wave height decreases.
Figure 8.9, case IR 2, differs from the previous by having γ = 3.3. The previous had a γ of 1. The
increase of γ makes the spectrum less wide. The energy is distributed to frequencies concentrating
near the main frequency. It is noticed that the super harmonics are more pronounced. This is because
energy of the frequencies from where the super harmonics come from, the main frequency, is larger.
The energy loss between wave gauge number 10 and 12 is also more pronounced.
In the last case IR 3 we see an even slimmer energy spectrum as γ has now been increased to 10
(notice the scale of the y-axis). Here we also see the very early development of the super harmonics.
Again this is because of the very energetic peak frequency around the specified peak frequency.
CHAPTER 8. ANALYSIS OF LABORATORY EXPERIMENTS IN THE SURFZONE
55
0.92
G:\skole\midtvejsprojekt\MIKE\bolge_T20_H013\SA_measuredlog1SUB.dfs0
0.94
0.012
0.90
0.010
0.88
0.008
0.86
0.006
0.84
0.004
0.82
0.002
0.80
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0.78
0.76
0.74
0.72
0.70
0.68
0.66
0.64
0.62
0.60
20:02:00
2002-03-03
20:02:05
20:02:10
20:02:15
20:02:20
20:02:25
20:02:30
20:02:35
20:02:40
20:02:45
20:02:50
20:02:55
20:03:00
Figure 8.7: Time series of regular monochromatic waves, T = 2.0 s, H = 0.13 m measured on the
runup gauge. The small plot in the top right corner is the FFT analysis of the same time series. The
FFT analysis express the energy as a function of the frequency.
8.4
Analysis of time series measured on the runup gauge in the
cases of irregular waves
In figure 8.11, 8.12 and 8.13 we see the FFT analysis of the time series measured on the runup gauge
(described in chapter 2).
In all figures we hardly see any energy remaining on the peak frequency. The peak frequency in
all three irregular cases is fp = 1/Tp = 0.67 Hz, (which is also supported by the FFT analysis in
the time series measured in regular wave gauges, see previous section). What is observed through
these FFT analysisses is that all the energy has been transferred into some low frequency energy. In
all cases the measured peak period on the runup gauge is approximately fp = 0.15 Hz and a large
amount of energy in the corresponding to the frequency, f = 0.05 Hz. In the first two cases, figures
8.11 and 8.12 some higher frequency energy is observed near the domain of f ≈ 0.55 Hz.
The reason why all the energy is in the low frequency range is that the high frequency waves
break and thus loose the energy. Figure 8.8 shows how the breaking dissipates the higher frequency
energy. Between wave gauge number 10 and wave gauge number 12 we see that the size of the main
peak decreases, whereas the low energy, f < 0.4 Hz actually seems to grow. Apparently the low
frequency energy keeps growing from wave gauge number 12 where the peak energy is ≈ 2 · 10 −5
m2 /Hz until the runup gauge where the energy is ≈ 7.5 · 10−5 m2 /Hz.
56
CHAPTER 8. ANALYSIS OF LABORATORY EXPERIMENTS IN THE SURFZONE
Channel 1, x =2.30 m
0.00020
Channel 2, x = 9.68 m
0.00020
0.00015
0.00015
0.00010
0.00010
0.00005
0.00005
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
Channel 5, x = 15.8 m
0.00020
0.00015
0.00010
0.00010
0.00005
0.00005
0.5
1.0
1.5
2.0
2.5
3.0
0.0
Channel 10, x = 17.83 m
0.00020
0.00015
0.00010
0.00010
0.00005
0.00005
0.0
0.5
1.0
1.5
2.0
2.5
3.0
1.5
2.0
2.5
3.0
0.5
1.0
1.5
2.0
2.5
3.0
Channel 12, x = 18.60 m
0.00020
0.00015
1.0
Channel 8, x = 17.02 m
0.00020
0.00015
0.0
0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Bathymetry
[m]
Wave gauges
[-]
Significant Wave height [-]
Bottom profile
0.20
depth, [m]
0.00
-0.20
-0.40
0
2
4
6
8
10
12
14
16
18
20
22
x, [m]
03/03/02 20:00:00:000
Figure 8.8: Irregular wave, H = 0.03 m, T = 1.5 s, γ = 1. Plot of FFT analysis of time series in
6 different distances from the wavemaker. The lower figure shows the bathymetry, where the circles
mark where the wave gauges are placed.
57
CHAPTER 8. ANALYSIS OF LABORATORY EXPERIMENTS IN THE SURFZONE
Channel 1, x =2.30 m
0.00040
0.00030
0.00030
0.00020
0.00020
0.00010
0.00010
0.0
1.0
2.0
3.0
0.0
Channel 5, x = 15.8 m
0.00040
0.00030
0.00020
0.00020
0.00010
0.00010
1.0
2.0
3.0
0.0
Channel 10, x = 17.83 m
0.00040
0.00030
0.00020
0.00020
0.00010
0.00010
1.0
2.0
3.0
2.0
3.0
1.0
2.0
3.0
Channel 12, x = 18.60 m
0.00040
0.00030
0.0
1.0
Channel 8, x = 17.02 m
0.00040
0.00030
0.0
Channel 2, x = 9.68 m
0.00040
0.0
1.0
2.0
3.0
Bathymetry
[m]
Wave gauges
[-]
Significant Wave height [-]
Bottom profile
depth, [m]
0.20
0.00
-0.20
-0.40
0
2
4
6
8
10
12
14
16
18
20
22
x, [m]
03/03/02 20:00:00:000
Figure 8.9: Irregular wave, H = 0.03 m, T = 1.5 s, γ = 3.3. Plot of FFT analysis of time series in
6 different distances from the wavemaker. The lower figure shows the bathymetry, where the circles
mark where the wave gauges are placed.
58
CHAPTER 8. ANALYSIS OF LABORATORY EXPERIMENTS IN THE SURFZONE
Channel 1, x =2.30 m
0.0040
0.0030
0.0030
0.0020
0.0020
0.0010
0.0010
0.0
1.0
2.0
3.0
0.0
Channel 5, x = 15.8 m
0.0040
0.0030
0.0020
0.0020
0.0010
0.0010
1.0
2.0
3.0
0.0
Channel 10, x = 17.83 m
0.0040
0.0030
0.0020
0.0020
0.0010
0.0010
1.0
2.0
3.0
2.0
3.0
1.0
2.0
3.0
Channel 12, x = 18.60 m
0.0040
0.0030
0.0
1.0
Channel 8, x = 17.02 m
0.0040
0.0030
0.0
Channel 2, x = 9.68 m
0.0040
0.0
1.0
2.0
3.0
Bathymetry
[m]
Wave gauges
[-]
Significant Wave height [-]
Bottom profile
depth, [m]
0.20
0.00
-0.20
-0.40
0
2
4
6
8
10
12
14
16
18
20
22
x, [m]
03/03/02 20:00:00:000
Figure 8.10: Irregular wave, H = 0.03 m, T = 1.5 s, γ = 10. Plot of FFT analysis of time series in
6 different distances from the wavemaker. The lower figure shows the bathymetry, where the circles
mark where the wave gauges are placed.
59
CHAPTER 8. ANALYSIS OF LABORATORY EXPERIMENTS IN THE SURFZONE
0.000075
0.000070
0.000065
0.000060
0.000055
0.000050
0.000045
0.000040
0.000035
0.000030
0.000025
0.000020
0.000015
0.000010
0.000005
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
Figure 8.11: Irregular wave, H = 0.03 m, T = 1.5 s, γ = 1. Plot of FFT analysis of the vertical
runup time series sampled from the runup gauge.
60
CHAPTER 8. ANALYSIS OF LABORATORY EXPERIMENTS IN THE SURFZONE
0.000075
0.000070
0.000065
0.000060
0.000055
0.000050
0.000045
0.000040
0.000035
0.000030
0.000025
0.000020
0.000015
0.000010
0.000005
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
Figure 8.12: Irregular wave, H = 0.03 m, T = 1.5 s, γ = 3.3. Plot of FFT analysis of the vertical
runup time series sampled from the runup gauge.
61
CHAPTER 8. ANALYSIS OF LABORATORY EXPERIMENTS IN THE SURFZONE
0.000075
0.000070
0.000065
0.000060
0.000055
0.000050
0.000045
0.000040
0.000035
0.000030
0.000025
0.000020
0.000015
0.000010
0.000005
0.000000
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
Figure 8.13: Irregular wave, H = 0.08 m, T = 1.5 s, γ = 10. Plot of FFT analysis of the vertical
runup time series sampled from the runup gauge.
Chapter 9
Introduction to MIKE 21 Boussinesq
wave module
A brief description of MIKE 21 Boussinesq Wave Module 2001 (M21BW) is presented in the following.
For details please refer to [MIKE 21 BW, Scientific Documentation], [Madsen, P. A. et al., (1997)].
M21BW, 1D, is based on Boussinesq formulation of the continuity and momentum equations.
The equations include frequency dispersion and nonlinearity and are solved numerically in the
time domain using a flux formulation. Improved dispersion characteristics are obtained using Padé
[Madsen, P. A. et al (1991)]. Because of the improvements of the Boussinesq equations made compared to the classical Boussinesq formulation M21BW are capable of handling relative wave numbers
(kh) up to a value of kh ≈ 3.1.
M21BW handles: shoaling, bottom friction, partial reflection, nonlinear wave-wave interaction,
wave breaking and moving shoreline.
During this paper we will investigate M21BW handing of wave breaking and moving shoreline by
comparing experiments with a M21BW modelling of the same situation. In the following a description
of how M21BW handles breaking and moving shoreline will be given.
9.1
Handling breaking using surface roller concept
M21BW handles wave breaking by using the surface roller concept. The principles of the surface
roller concept is to consider the breaking part of the wave (the foam) as a mass being transported
with the wave, and therefore with the wave speed. It is assumed that the velocity of the roller is
constant in the vertical direction. Also the velocity distribution in the horizontal (u) direction bellow
the surface roller is assumed to be constant [Madsen, P. A. et al., (1997)].
Figure 9.1: Sketch showing the principles of the surface roller concept. Left part shows a cross section
of the breaking wave. The part of the wave with the height of δ is the roller or ”the foam”. Right
side shows the assumed velocity profile. Figure from [Madsen, P. A. et al., (1997)]
62
CHAPTER 9. INTRODUCTION TO MIKE 21 BOUSSINESQ WAVE MODULE
63
The wave celerity is determined from a roller celerity factor multiplied by the shallow water wave
celerity. By default this factor is set to be the most optimal, 1.3 [Madsen, P. A. et al., (1997)]. Alternatively this could be determined from the slope of the wave and the time derivative [Madsen, P. A. et al., (1997)],
but in practice this is not a stabile solution.
When making the analytical theory into the numerical model several parameters need to be
specified. Among that the factor of velocity of the roller is, as described above, set to 1.3 by default
but the user can specify another value if desired. Also the shape of the roller and the initial and
ending breaking angle can be specified.
9.2
Handling moving shoreline
The concept of modelling this parameter is to, so to speak, ”cut a furrow/slot” in the beach in the
direction of the wave propagation, and the water level in the slot is used to model the wave. The
slot must have such a small width that the wave outside the slot will not be affected from it.
The slot will work as if the beach was very little permeable and the pressure level inside the
permeable beach will represent a wave passing through the beach. Where this surface of this passing
wave intersects the beach we define the runup. See figure 9.2
Figure 9.2: Sketch showing the principle of modelling the moving shoreline. Vertical cut through the
beach bathymetry. Figure from [Madsen, P. A. et al., (1997)]
Chapter 10
Setup of MIKE 21 Boussinesq wave
module
10.1
Physical setup
MIKE 21 Boussinesq Waves module has been set up to simulate the situation of the above situations.
Different physical environments and calibration parameters has to be set in the best way to make a
good simulation.
The physical environment is set up as shown of figure 10.1.
The bathymetry has a discretization, ∆x = 0.05 m i.e. the bathymetry of the experimental
wave flume is split up in nodes with 0.05 m between. In each end the bathymetry is extended
(extrapolated) by 10 m compared to the bathymetry of the laboratory model. This is done to make
room for the sponge layers. By the extension the bathymetry becomes 41 m long. The sponge layers
are applied to absorb the waves and thereby avoid reflection. Behind the wavemaker and also on the
opposite side of the coastline sponge layers are set up. The reason for setting up sponge layers on
the opposite side of the coastline (in x = 36 − 41m) is to absorb the waves passing through the slot
used for modelling of moving shoreline.
In order to make an effective absorbtion the size of the sponge layers must exceed one wavelength.
In this model setup is used sponge layers of a length of 5 m where the sponge layer coefficients are
(n−1) )
exponentially increasing: sponge coefficient = 10(0.92
, where n is the node number from where
the sponge layer coefficient is maximum.
During the modelling of rush-up and rush-down numerical instabilities occur. To remove these
instabilities a low pass filter is applied. The filter is shown as the lower plot in figure 10.1.
The time descretization was set to ∆t = 0.0125 s. Using this time step the model was stabile
and the time consumption was not too big. In the cases of regular monochromatic waves (1-3) the
simulation period was set to 12001 time steps. In the cases of irregular waves (IR 1-3) 69501 time
steps was used. The reason that this simulation period is much larger than the cases with the regular
waves is that to analyse the irregular waves a large amount of data is needed to do the statistical
analysisses. A warm up period of 400 time steps is used to avoid blow up in the beginning of the
simulation.
As previously mentioned (section 9) the dispersion relation is improved. The linear dispersion relation is found from the enhanced Boussinesq equations in which unknowns appear. These unknowns
are collected in one unknown, the linear dispersion factor. By fitting the linear dispersion factor
with in a way that the linear dispersion from the Boussinesq equation match a Padé[2,2] expansion
of Stokes’ linear dispersion relation a very good value of the linear dispersion factor is found. The
optimal value of this is 1/15 ≈ 0.0666667 [Madsen, P. A. and Schäffer, H. A. (1999)]. This is also
the default value in M21BW and the value is also applied during this analysis.
To obtain a good model of the physical experiments we can adjust different parameters influencing
64
65
CHAPTER 10. SETUP OF MIKE 21 BOUSSINESQ WAVE MODULE
Figure 10.1: Set up of physical environment in the MIKE 21 Boussinesq Waves module. Top shows
the bathymetry, middle sponge layer coefficients and bottom the low pass filter.
the breaking and the moving shoreline. The calibration of these parameters are commented more
thoroughly in section 11. The values used in this model are shown i table 10.2, wave breaking and
table 10.1, moving shoreline.
Slot
Slot
Slot
Slot
10.2
Table 10.1: Calibration constants for moving shoreline.
Case 1 Case 2 Case 3 Case IR 1 Case IR 2
depth
-0.3
-0.3
-0.3
-0.3
-0.3
Width
0.001 0.0019 0.008
0.005
0.005
Smoothing
100
100
100
100
100
friction
0
0
0
0.01
0
Case IR 3
-0.3
0.01
100
0
Boundary conditions
Sinusoidal waves, cases 1-3
In order to make the best approximation to the test done in the physical model, the wave generation
was placed in the exact position relative to the bathymetry. Referring to figure 10.1 this corresponds
to placing the wavemaker in x = 10 m.
The wave signal was made using MIKE 21 toolbox. The wave height in the input signal was
specified in a way that the measured wave height in channel 1 was as close to the modelled wave
CHAPTER 10. SETUP OF MIKE 21 BOUSSINESQ WAVE MODULE
66
Table 10.2: Calibration constants for wave breaking.
cases 1-3 case IR 1-3
Roller form factor
1.5
1.5
Roller celerity factor
1.3
1.3
Initial breaking angle
22◦
20◦
Final breaking angle
10◦
20◦
Half-time for cut off roller
T /5
Tp /5
height in the same point as possible. The reason why we did not just used the wave height specified
in the actual physical wavemaking, was that there at the time of testing were problems with the
wavemaker not making the exact same wave height as specified.
Irregular waves, cases IR 1-3
Due to the accuracy problem with making of the specified waves the *spec.dsf0 files could not be
used as boundary conditions. The *spec.dsf0 files contain the signal sent to the wavemaker and the
expected elevation at the wavemaker. To recreate the situation exactly as it was physically modelled
it would be natural to use the signal sent to the wavemaker, but unfortunately this is not satisfactorily
accurate.
Instead the measured signal measured in wave gauge number 1 was used as input signal. The
measured surface elevation was converted into flux and surface slope using MIKE 21 toolbox. This
wavemaker boundary condition was of course set in the position of wave gauge number 1. In figure 10.1 this is in x = 12.3 m.
Chapter 11
Calibrating Breaker and Moving
Shoreline Parameters
11.1
Calibrating parameters used in wave breaking
In this part case 2, T = 1.5 s, H = 0.08 m has been calibrated thoroughly by testing different
wave breaking parameters. This includes the roller form factor, roller celerity factor, initial breaking
angle and half-time for cut-off roller. The analysis area is concentrated to be the range including the
breaking of the wave.
The default parameters are:
Table 11.1: Default calibration parameters used for modelling wave breaking.
Roller form factor
1.5
Roller celerity factor
1.3
Initial breaking angle
20
Final breaking angle
10
Half-time for cut-off roller 0.3
M21BW model of the breaking zone compared to the measured data is seen in figure 11.1. The
range of the plotting has been limited to 23 m from the wavemaker to 30 m from the wavemaker.
The measured average wave heights are drawn on all the figures as circles. In x = 25 m we have
wave gauge number 3 and x = 25.4 m wave gauge number 4 and so fourth. The phase averaged wave
height is drawn as the solid line. The phase averaging was done over 5 periods. It is noticed that the
modelled wave height (the solid line) drop to zero in x = 28.35 m. This is because M21BW can not
phase average near the coastline. The disturbances often observed after the coastline are also caused
by numerical phenomenons, but as these are outside the physical domain they have no significance.
As seen on figure 11.1 the modelling of the shoaling is successful: the wave heights at the first
four displayed wave gauges are modelled quite well. The next three, where the wave height increase
significantly, the model has trouble modelling. This is a well known limitation of the model. The
shoaling is a combination of linear shoaling characteristics and nonlinear energy transfer to higher
harmonics.
The deep water limit of M21BW is kh = 3.1. Just before breaking the higher harmonics (higher
than second) play a significant role, and as these are underestimated due to their high kh value, the
wave height just before breaking will be underestimated [Madsen, P. A. et al., (1997)].
On figure 11.5, top plot, we see the FFT analysis of the time series measured in wave gauge
number 9 (just before breaking). As seen here the higher harmonics has quite large energy up to 4 th
to 5th harmonic. Comparing with the FFT analysis of the modelled time series at wave gauge number
67
CHAPTER 11. CALIBRATING BREAKER AND MOVING SHORELINE PARAMETERS
68
0.15
0.14
0.13
0.12
0.11
0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
-0.01
-0.02
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
03/03/02 20:02:30:000
Figure 11.1: Plot of the wave height variation as a function of distance to the wavemaker. The
circles mark the measurements and the line is the modelled wave height variation with the breaking
parameters listed in table 11.1
9 (figure 11.6 top plot) we see how M21BW does not model more than up to second harmonic.
Judging from the graph the model estimates the point of breaking a little too early compared to
the measurements. By increasing the initial breaking angle to 22◦ the wave will break a little closer
to the shoreline. The initial breaking angle controls at which steepness the breaking will start. By
increasing this parameter the model allows the wave to become more steep before breaking. The
wave becomes steeper as the coastline is approached and thus the point of breaking will move closer
to the shoreline as the initial breaking angle is increased and visa versa.
On figure 11.2 we see the result of change the initial breaking angle to 22 ◦ . The breaking point is
now fairly accurate comparing with the measurements. But the energy loss is not strong enough as
the wave heights in the breaking zone are overestimated by the M21BW model. By decreasing the
half-time for cut off roller to 0.05 (minimum value) we should obtain a more pronounced decrement
in wave heights in the breaking zone [Madsen, P. A. et al., (1997)].
As we see the wave height variation in space has become more pronounced. But the model still
overestimates the wave heights in the breaking zone. As the model estimates the wave heights just
before breaking rather poorly it can be argued that the point of breaking is irrelevant. Decreasing
the breaking angle to 17◦ and having the other parameters as default (table 11.1) we obtain the
result plotted in figure 11.4.
By visually extrapolation of the wave height variation we can imagine that the modelled wave
heights fit the measured quite well in the surf zone using these parameters. It is noticed that the
model results are very poor in the range 26.25 m < x < 27.75 m, but as the wave heights in this
range can not be modelled satisfactorily using present theory, the calibrating parameters are chosen
from comparing data outside this range. Thus the parameters shown in table 11.2 with the result
shown in figure 11.4 are used for the MIKE 21 modelling of the regular cases.
In figure 11.5 the energy spectrum of a wave just before breaking (top) and the energy spectrum
of a wave inside the breaking zone (bottom) is plotted. From these two graphs we can conclude that
the wave has a lot of energy on the waves third harmonic and higher harmonics just before breaking.
After the breaking has started the energy is mostly concentrated on the first and second harmonics.
CHAPTER 11. CALIBRATING BREAKER AND MOVING SHORELINE PARAMETERS
69
0.15
0.14
0.13
0.12
0.11
0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
-0.01
-0.02
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
03/03/02 20:02:30:000
Figure 11.2: Plot of the wave height variation as a function of distance to the wavemaker. The black
dots mark the measurements and the line is the modelled wave height. The breaking parameters
used are listed in table 11.1 but the initial breaking angle has been changed to 22 ◦ .
0.15
0.14
0.13
0.12
0.11
0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
-0.01
-0.02
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
03/03/02 20:02:30:000
Figure 11.3: Plot of the wave height variation as a function of distance to the wavemaker. The black
dots mark the measurements and the line is the modelled wave height. The breaking parameters
differs from table 11.1 by having initial breaking angle on 22◦ and half-time for cut off roller on 0.05.
In the interval between wave gauge number 9 and 12 it seems like all the energy of the harmonics
higher than 2 is dissipated almost totally. As MIKE 21 BW models the shoaling of the higher than
second harmonics poorly near wave gauge number 9 (see figure 11.6) it will also have difficulties in
modelling the first part of the breaking.
CHAPTER 11. CALIBRATING BREAKER AND MOVING SHORELINE PARAMETERS
70
0.15
0.14
0.13
0.12
0.11
0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
-0.01
-0.02
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
03/03/02 20:02:30:000
Figure 11.4: Plot of the wave height variation as a function of distance to the wavemaker. The black
dots mark the measurements and the line is the modelled wave height. In this figure the parameters
differs from table 11.1 by having initial breaking angle on 17◦
Table 11.2: Calibration parameters used for modelling wave breaking.
Roller form factor
1.5
Roller celerity factor
1.3
Initial breaking angle
17
Final breaking angle
10
Half-time for cut-off roller 0.3
When the first part of the breaking is observed visually in the physical experiments this part is
intuitively hard to model. The wave tips and create a lot of stir in the water bellow. It seems like
the first part of the breaking is much more intense compared to breaking closer to the coastline. This
is supported by e.g. figure 11.4 where it is seen that the decrease in wave height from wave gauge
number 9 to 10 is much larger than the decrease going from wave gauge number 10 to 11.
11.2
Calibrating parameters used in moving shoreline
Calibrating the moving shoreline is easier. The calibration is done with the objective of maximizing
or minimizing the parameters but avoiding blow up.
The value of the slot depth is not varied but has the value of -0.3 as this value is only used to
ensure that there is also water in the slot during down rush. Slot width has to be as small as possible
to avoid disturbances in the mass balance in the flow outside the slot. If this value is set too low the
model will blow up and the calibration of this parameter is therefore to keep it as low as possible.
Slot smoothing parameter has to be maximized to avoid the slot influencing on the flow outside the
slot. The slot friction coefficient is usually set to 0, but can be set to non zero values to damp high
frequency noise.
The parameters used for modelling of moving shoreline are shown in table 11.3
71
G:\skole\midtvejsprojekt\MIKE\bolge_T15_H008\SA_measuredlog1.dfs0
CHAPTER 11. CALIBRATING BREAKER AND MOVING SHORELINE PARAMETERS
0.030
0.025
0.020
0.015
0.010
0.005
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
G:\skole\midtvejsprojekt\MIKE\bolge_T15_H008\SA_measuredlog1.dfs0
0.0
0.0050
0.0040
0.0030
0.0020
0.0010
0.0
0.5
Figure 11.5: FFT analysis of the time series measured in wave gauge number 9, just before breaking
(top), and in wave gauge number 12 in the breaking zone (bottom)
Slot
Slot
Slot
Slot
Table 11.3: Calibration constants for moving shoreline.
Case 1 Case 2 Case 3 Case IR 1 Case IR 2
depth
-0.3
-0.3
-0.3
-0.3
-0.3
Width
0.001 0.0019 0.008
0.005
0.005
Smoothing
100
100
100
100
100
friction
0
0
0
0.01
0
Case IR 3
-0.3
0.01
100
0
CHAPTER 11. CALIBRATING BREAKER AND MOVING SHORELINE PARAMETERS
72
0.030
0.025
0.020
0.015
0.010
0.005
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
0.025
0.020
0.015
0.010
0.005
Figure 11.6: FFT analysis of the model output time series at wave gauge number 9, just before
breaking (top), and at wave gauge number 12 in the breaking zone (bottom)
Chapter 12
Results from the MIKE 21 Boussinesq
wave model and comparison with
measurements
In the following the results obtained by the M21BW model will be presented. All the results will
be compared to the measurements from the physical model. Generally one case from the regular
monochromatic cases and one from the irregular cases will be commented thoroughly and the 2 other
cases will be used as comparison.
12.1
Analysis of time series in different distances from the wavemaker, sinusoidal wavemaker signal
12.1.1
Case 1, T = 1.0 s, H = 0.1 m
Time series from case 1 in different distances from the wavemaker are shown in figure 12.1. The dashed
lines mark the time series modelled by M21BW. It is noticed that the modelled wave is approximately
0.05 seconds behind the physical measured model. In the previous part about evanescent modes we
obtained a reason to believe that a synchronizing error between the wave gauges and the wavemaker
could occur. This probably explains the phase difference at wave gauge number 1. Comparing the
phase difference for all the wave gauges in the figure, one is led to believe that the modelled wave
is a little faster than the physically measured. But this case also has a rather high kh value of
2.26. In the limit of the validity range the difference in the wave celerity becomes more pronounced,
[Madsen, P. A. et al (1991)].
The shape of the waves seem to match pretty good at the first two wave gauges. In this range the
energy of the higher harmonics are not significant large, which is seen by the non-sharp crests. As
we proceed to wave gauge number 5 we see the peaks becoming sharper, i.e. more energetic higher
harmonics. At wave gauge number 8 it is terrible. The wave modelled wave height is almost half the
measured. By looking at figure 12.2 we see that the explanation of this large difference is M21BW
not being able to model the measurements just before breaking. As we concluded previously the
higher harmonics of the wave becomes very energetic just before breaking. This case has a large kh
value and therefore M21BW can not model even the second harmonic. When none of the higher
harmonics are modelled the wave heights just before breaking will be very underestimated.
In figure 12.2 the previously discussed effect of shoaling coefficient being less than one is emphasized.
73
74
CHAPTER 12. ANALYSIS OF MIKE 21 BW RESULTS
Channel 1, x =2.30 m
0.10
0.05
0.05
0.00
0.00
-0.05
20:01:30
2002-03-03
-0.05
20:01:30
2002-03-03
20:01:32
Channel 5, x = 15.8 m
0.10
0.05
0.00
0.00
-0.05
20:01:30
2002-03-03
20:01:32
Channel 10, x = 17.83 m
0.10
0.05
0.00
0.00
20:01:32
Channel 12, x = 18.60 m
0.10
0.05
-0.05
20:01:30
2002-03-03
20:01:32
Channel 8, x = 17.02 m
0.10
0.05
-0.05
20:01:30
2002-03-03
Channel 2, x = 9.68 m
0.10
-0.05
20:01:30
2002-03-03
20:01:32
20:01:32
Bathymetry
[m]
Wave gauges
[-]
Avg. measured waveheight [-]
0.00
-0.20
-0.40
0
2
4
6
8
10
12
14
16
18
20
22
03/03/02 20:00:00:000
Figure 12.1: Time series in different distances from the wavemaker. Dashed lines are M21BW
modelled time series and the solid lines are the measured. The lower graph shows the placing of the
wavemaker (marked by circles) and the measured wave heights (marked with x), case 1
75
CHAPTER 12. ANALYSIS OF MIKE 21 BW RESULTS
0.13
0.12
0.11
0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
-0.01
-0.02
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
03/03/02 20:01:00:000
Figure 12.2: Wave heights as a function of the distance to the wavemaker. The wavemaker is placed
in x = 10 m. The solid line represents the M21BW results and the ’x’s the measurements in the
physical model. Case 1
12.1.2
Case 2, T = 1.5 s, H = 0.08 m
The possible synchronizing error is still seen in figure 12.3. What is also noticed in this figure is
that the wave celerity of the measured wave and the modelled wave as more alike compared to the
previous case. Theoretically the error in the wave celerity decreases as we approach shallower water
[Madsen, P. A. et al (1991)]. The form of the waves in the different distances match very well. And
according to figure 12.4 the wave heights are modelled fairly well good. Once more except for the
heights just before breaking.
12.1.3
Case 3, T = 2.0 s, H = 0.13 m
Figure 12.5 indicates that the shoaling starts much sooner than in the previous two cases. Wave
gauge number 2 and especially number 5 have very sharp crests in the measurements. Looking at
figure 12.6 the early shoaling is confirmed. Again the model fits the shoaling rather poorly. On
figure 12.5 we see that the wave throughs at wave gauge number 2 and 5 also are poorly modelled.
Again the reason is the problem of modelling the area just before breaking. The poor modelling of
the waves just before breaking is, as mentioned, caused by the poorly modelled higher harmonics.
The higher harmonics also have a great influence on the wave throughs thus the difference in the
modelled and measured wave throughs.
76
CHAPTER 12. ANALYSIS OF MIKE 21 BW RESULTS
Channel 1, x =2.30 m
0.10
0.05
0.05
0.00
0.00
-0.05
20:01:30
2002-03-03
20:01:31
20:01:32
-0.05
20:01:30
2002-03-03
20:01:33
Channel 5, x = 15.8 m
0.10
0.05
0.00
0.00
20:01:31
20:01:32
-0.05
20:01:30
2002-03-03
20:01:33
Channel 10, x = 17.83 m
0.10
0.05
0.00
0.00
-0.05
20:01:30
2002-03-03
20:01:31
20:01:32
-0.05
20:01:30
2002-03-03
20:01:33
20:01:32
20:01:33
20:01:31
20:01:32
20:01:33
Channel 12, x = 18.60 m
0.10
0.05
20:01:31
Channel 8, x = 17.02 m
0.10
0.05
-0.05
20:01:30
2002-03-03
Channel 2, x = 9.68 m
0.10
20:01:31
20:01:32
20:01:33
Bathymetry
[m]
Wave gauges
[-]
Avg. measured waveheight [-]
0.10
0.00
-0.10
-0.20
-0.30
-0.40
-0.50
0
2
4
6
8
10
12
14
16
18
20
22
03/03/02 20:00:00:000
Figure 12.3: Time series in different distances from the wavemaker. Dashed lines are M21BW
modelled time series and the solid lines are the measured. The lower graph shows the placing of the
wavemaker (marked by circles) and the measured wave heights (marked with x), case 2
77
CHAPTER 12. ANALYSIS OF MIKE 21 BW RESULTS
0.13
0.12
0.11
0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
-0.01
-0.02
0
5
10
15
20
25
30
35
40
03/03/02 20:01:00:000
Figure 12.4: Wave heights as a function of the distance to the wavemaker. The wavemaker is placed
in x = 10 m. The line represents the MIKE BW results and the ’x’s the measurements in the physical
model. Case 2
12.2
Analysis of time series measured on the runup gauge, sinusoidal wavemaker signal
In the following we will take a look at the time series measured on the runup gauge. Both the time
series and the FFT analysis will be analysed.
On figure 12.8 we see the time series of the runup gauge. As we see the measured runup reaches
much higher levels compared to the modelled. The mean value of the measured runup is 0.019 m
and the modelled has a mean value of 0.013 m (calculated by MIKE 21 statistical tool). These two
values distinct quite a lot relatively, but it is noticed that the absolute difference is only 6 mm. But
still this is larger than the accuracy of 1 mm in the vertical runup.
The reason is that it is very difficult to model the wave set-up accurately. The wave set-up is
a consequence of the rapid decrement in wave height, and because the decrement is not modelled
accurately as discussed in chapter 11 the wave-setup will be influenced by this and thus the mean
value of the vertical runup height.
From the plot of the time series we see that the main frequency, the one specified on the wavemaker, is modelled rather well. This is no surprise though as the modelling of the runup is not more
than the wave from the flume evolving into the slot, see section 9.2. The amplitude of the main
frequencies of the modelled and measured height are very different. By a coarse visual inspection of
the time series the modelled amplitude is approximately 3-4 times larger than the measured. In our
laboratory we encountered a damping in the system. This was because of our rather mild sloping
bathymetry: During rush up we observed a rather sharp and well defined edge of the water caused by
the bottom friction acting opposite of the particle velocity, trying to ”hold back” the water. During
78
CHAPTER 12. ANALYSIS OF MIKE 21 BW RESULTS
Channel 1, x =2.30 m
Channel 2, x = 9.68 m
0.10
0.10
0.05
0.05
0.00
0.00
-0.05
-0.05
20:01:30
2002-03-03
20:01:32
20:01:34
20:01:30
2002-03-03
Channel 5, x = 15.8 m
20:01:32
20:01:34
Channel 8, x = 17.02 m
0.10
0.10
0.05
0.05
0.00
0.00
-0.05
-0.05
20:01:30
2002-03-03
20:01:32
20:01:34
20:01:30
2002-03-03
Channel 10, x = 17.83 m
20:01:32
20:01:34
Channel 12, x = 18.60 m
0.10
0.10
0.05
0.05
0.00
0.00
-0.05
-0.05
20:01:30
2002-03-03
20:01:32
20:01:34
20:01:30
2002-03-03
20:01:32
20:01:34
Bathymetry
[m]
Wave gauges
[-]
Avg. measured waveheight [-]
0.20
0.00
-0.20
-0.40
0
2
4
6
8
10
12
14
16
18
20
22
03/03/02 20:00:00:000
Figure 12.5: Time series in different distances from the wavemaker. Dashed lines are M21BW
modelled time series and the solid lines are the measured. The lower graph shows the placing of the
wavemaker (marked by circles) and the measured wave heights (marked with x), case 3
79
CHAPTER 12. ANALYSIS OF MIKE 21 BW RESULTS
0.20
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
-0.02
-0.04
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
03/03/02 20:01:19:000
Figure 12.6: Wave heights as a function of the distance to the wavemaker. The wavemaker is placed
in x = 10 m. The line represents the MIKE BW results and the ’x’s the measurements in the physical
model. Case 3
rush down the opposite was observed. Here the bottom friction again was opposite of the particle
particle velocity. This caused a thin layer of water to remain on the runup gauge. See figure 12.7.
We tried to avoid this phenomenon as described in chapter 2, but some damping remained.
If we look at the FFT analysis of the time series, figure 12.9, it can be confirmed that the
amplitude of the main frequency, f = 1 Hz, is highly overestimated by MIKE 21. But as discussed
the damping due to bottom friction can also cause an error in the measuring of the results, which
makes a smaller amplitude but we estimate from visual observation of the experiments that the
damping can not have such a great influence as the difference observed. Therefore the difference in
amplitudes must be caused by MIKE 21 overestimation. An important result of the FFT analysis
is that the low frequency waves in the measurements are not modelled by MIKE 21. These low
frequencies are rather important as the energy in the low frequency waves are approximately 3-4
times the energy of the main frequency as discussed in section 8.2.
By looking at figure 12.10, showing the FFT analysis of the time series at wave gauge number
12, we see that also here the low frequency waves are not modelled. The low frequency energy
seen on the measurement at wave gauge 12 are sub harmonics. The sub harmonics originates from
two wave components with frequencies very close to each other. All sorts of disturbances, that we
have neglected in theory, might cause that the waves have a finite variance in the frequency, i.e. a
monochromatic wave will become not ¡¡entirely monochromatic and thus create higher harmonics.
In theory we have neglected all disturbances and the waves will therefore be entirely monochromatic
and thus not create sub harmonics on these frequencies. In waves in the flume these low frequency
waves are, as seen in figure 12.10 negligible. But on the runup where all the higher frequency waves
are dissipated due to breaking what is left is the low frequencies, that because of their low frequencies
80
CHAPTER 12. ANALYSIS OF MIKE 21 BW RESULTS
do not break and thus dissipate.
Figure 12.7: Damping of runup signal. Left side is rush up and right side is rush down. The dotted
line is the system how it is modelled by MIKE 21 and the solid line is what we observed in the
laboratory
G:\skole\midtvejsprojekt\MIKE\bolge_T10_H01\measuredlog1.dfs0
0.021
0.020
0.019
0.018
0.017
0.016
0.015
0.014
0.013
0.012
0.011
20:02:00
2002-03-03
20:02:02
20:02:04
20:02:06
20:02:08
20:02:10
20:02:12
20:02:14
20:02:16
20:02:18
20:02:20
Figure 12.8: Comparison of vertical runup. dashed line is MIKE21 BW modelled and the solid line
is the measured. Case 1, T = 1.0 s, H = 0.10 m
On figure 12.11 we see the time series of the runup gauge in case 2. As in case 1 we see that
the mean value of the runup measured in laboratory is larger than the mean value of the modelled
runup.
As in the previous the amplitude of the main frequency wave is overestimated by MIKE 21. This
is again confirmed by 12.12. On the plot of the FFT analysis it can be seen how, as in case 1, the
low frequency wave are not modelled at all.
In case 3 seen on figure 12.13 the difference in the mean value is only 1 mm, which is fairly well.
81
CHAPTER 12. ANALYSIS OF MIKE 21 BW RESULTS
0.000028
0.000026
0.000024
0.000022
0.000020
0.000018
0.000016
0.000014
0.000012
0.000010
0.000008
0.000006
0.000004
0.000002
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Figure 12.9: Comparison of FFT analysis of vertical runup from measurements (solid line) and
M21BW results (dashed line). Case 1, T = 1.0 s, H = 0.10 m
12.3
FFT analysis of time series in different distances from the
wavemaker, irregular waves
In the following we have made FFT analysisses of time series in different distances from the wavemaker. We have made the FFT analysis of the measured time series and the M21BW modelled
time series. The results are plotted as the energy versus logarithm of the frequency. This way the
differences between the model and the measurements in the high frequencies domain are emphasized.
This is necessary as the higher frequency energy is lower than the energy of the main frequency. The
high frequencies corresponds to high values of kh and thus we are emphasizing the differences in the
limits of M21BWs validity range.
We start out by commenting case IR 2, where Tp is 1.5 s, Hs = 0.03 m and γ = 3.3. FFT analysis
of time series in different distances are shown in figure 12.17. As seen the measurements and the
modelled results match rather well in wave gauge number 1, 2 and 5. At wave gauge number 8 we
observe a pronounced difference starting at a frequency around f = 1.2 Hz. This corresponds to
waves with a period of T = 0.83 s. The difference in the modelled and the measured must be caused
by M21BW difficulties in modelling of kh values over 3.1. If we look at a wave at the wavemaker
(h = 0.55 m) with the kh in the limit of what M21BW is able to model, kh = 3.1, we can calculate
from linear theory the period of this wave to be T = 0.85 s. This is very close to where we observe
the difficulties in wave gauge number 8. In wave gauge 10 and 12 we observe differences between
measured and modelled start at frequencies around 0.8 Hz (T = 1.25 s). This corresponds to a kh
value of the wavemaker of kh = 1.55 which is exactly half the M21MW capability range.
The significant wave heights as a function of distances to the wavemaker is seen on figure 12.15.
As in the cases of the regular waves we see problems just before the point of breaking. The variation
82
CHAPTER 12. ANALYSIS OF MIKE 21 BW RESULTS
0.000030
0.000028
0.000026
0.000024
0.000022
0.000020
0.000018
0.000016
0.000014
0.000012
0.000010
0.000008
0.000006
0.000004
0.000002
0.000000
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Figure 12.10: Comparison of FFT analysis time series measured on wave gauge number 12 (solid
line) and M21BW results at wave gauge number 12 (dashed line). Drawn in same scale as figure 12.9.
Case 1, T = 1.0 s, H = 0.10 m
of the wave heights is as seen on the figure more rounded as the point of breaking is approached, this
is because the waves have different heights and lengths and thus starts breaking in different distances.
The same limitation is recognized in case IR 1 on figure 12.16 where the differences between
measured and modelled start at a frequency of approximately 1.3 Hz. But in this case we can also
observe that in wave gauge number 10 and 12 the difference start at lower frequencies.
In case IR 3, figure 12.18 the same is observed. What is also observed there is that in the FFT
analysis of the measured is a discontinuity at f = 2.5 Hz, where the energy goes to zero. This
discontinuity is caused by the wave generation on the boundary being created from the measured
signal. When creating the surface slope and the flux under the wave the smallest wave period 0.4 s
which explains the discontinuity.
12.4
Analysis of time series measured on the runup gauge, irregular
waves
In figure 12.20 we see the FFT analysis of the vertical runup in case IR 2. The area under the two
curves are very different. As mentioned the significant wave height is proportional to the area, and
thus we see M21BW overestimating the significant wave height as also seen in cases of monochromatic
waves.
Unlike the monochromatic cases we do not see the problem of modelling the low frequency components. This is because in these cases of irregular waves we have a frequency spectrum, i.e. many
opportunities of creating the sub harmonics which are not completely dissipated during the breaking.
83
CHAPTER 12. ANALYSIS OF MIKE 21 BW RESULTS
G:\skole\midtvejsprojekt\MIKE\bolge_T15_H008\measuredlog1.dfs0
0.030
0.029
0.028
0.027
0.026
0.025
0.024
0.023
0.022
0.021
0.020
0.019
0.018
0.017
0.016
0.015
0.014
20:02:00
2002-03-03
20:02:05
20:02:10
20:02:15
20:02:20
20:02:25
20:02:30
Figure 12.11: Comparison of vertical runup. dashed line is MIKE21 BW modelled and the solid line
is the measured. Case 2, T = 1.5 s, H = 0.08 m
It is observed that the model predicts an energetic wave with the frequency of approximately 0.3
Hz. The same peak is not seen in the measurements. This might be caused by not describing the
breaking of the waves sufficiently accurate, and the energy on the frequency 0.3 Hz is dissipated in
laboratory experiment but in the model this frequency is not dissipated.
The mean value of the vertical runup is 0.54 cm according to the experimental measurements and
for the computer model the mean value was 0.34 cm. This result is acceptable.
84
CHAPTER 12. ANALYSIS OF MIKE 21 BW RESULTS
0.00044
0.00042
0.00040
0.00038
0.00036
0.00034
0.00032
0.00030
0.00028
0.00026
0.00024
0.00022
0.00020
0.00018
0.00016
0.00014
0.00012
0.00010
0.00008
0.00006
0.00004
0.00002
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
Figure 12.12: Comparison of FFT analysis of vertical runup from measurements (solid line) and
M21BW results (dashed line). Case 2, T = 1.5 s, H = 0.08 m
85
CHAPTER 12. ANALYSIS OF MIKE 21 BW RESULTS
0.060
0.058
0.056
0.054
0.052
0.050
0.048
0.046
0.044
0.042
0.040
0.038
0.036
0.034
0.032
0.030
0.028
0.026
0.024
20:02:00
2002-03-03
20:02:05
20:02:10
20:02:15
20:02:20
20:02:25
20:02:30
Figure 12.13: Comparison of vertical runup. dashed line is MIKE21 BW modelled and the solid line
is the measured. Case 3, T = 2.0 s, H = 0.13 m
86
CHAPTER 12. ANALYSIS OF MIKE 21 BW RESULTS
0.0032
0.0030
0.0028
0.0026
0.0024
0.0022
0.0020
0.0018
0.0016
0.0014
0.0012
0.0010
0.0008
0.0006
0.0004
0.0002
0.0000
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Figure 12.14: Comparison of FFT analysis of vertical runup from measurements (solid line) and
M21BW results (dashed line). Case 3, T = 2.0 s, H = 0.13 m
0.060
0.055
0.050
0.045
0.040
0.035
0.030
0.025
0.020
0.015
0.010
0.005
0.000
-0.005
-0.010
-0.015
-0.020
0
2
4
03/03/02 20:02:26:000
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
Figure 12.15: Significant wave heights as a function of the distance to the wavemaker. ’x’s mark the
measurement and the solid line is the modelled significant wave height. Case IR 2
87
CHAPTER 12. ANALYSIS OF MIKE 21 BW RESULTS
Channel 1, x =2.30 m
Channel 2, x = 9.68 m
0.00020000
0.00020000
0.00005000
0.00005000
0.00001000
0.00001000
0.00000200
0.00000200
0.00000050
0.00000050
0.00000010
0.00000010
0.00000002
0.00000002
0.0
1.0
2.0
3.0
0.0
Channel 5, x = 15.8 m
0.00020000
0.00005000
0.00005000
0.00001000
0.00001000
0.00000200
0.00000200
0.00000050
0.00000050
0.00000010
0.00000010
0.00000002
0.00000002
1.0
2.0
3.0
Channel 8, x = 17.02 m
0.00020000
0.0
1.0
2.0
3.0
0.0
Channel 10, x = 17.83 m
1.0
2.0
3.0
Channel 12, x = 18.60 m
0.00020000
0.00020000
0.00005000
0.00005000
0.00001000
0.00001000
0.00000200
0.00000200
0.00000050
0.00000050
0.00000010
0.00000010
0.00000002
0.00000002
0.0
1.0
2.0
3.0
0.0
1.0
2.0
3.0
Bathymetry [m]
Wave gauges [-]
Bottom profile
0.20
depth, [m]
0.00
-0.20
-0.40
0
2
4
6
8
10
12
14
16
18
20
22
x, [m]
03/03/02 20:00:00:000
Figure 12.16: FFT analysis of time series in different distances from the wavemaker. The logarithm
of the energy is plotted as a function of the frequency. Dashed line is M21BW modelled time series
and the solid line is the measured. The lower graph shows the placing of the wavemaker (marked by
circles). Case IR 1
88
CHAPTER 12. ANALYSIS OF MIKE 21 BW RESULTS
Channel 1, x =2.30 m
0.00050000
Channel 2, x = 9.68 m
0.00050000
0.00005000
0.00005000
0.00000500
0.00000500
0.00000050
0.00000050
0.00000005
0.00000005
0.00000001
0.00000001
0.0
1.0
2.0
3.0
0.0
Channel 5, x = 15.8 m
0.00050000
0.00005000
0.00000500
0.00000500
0.00000050
0.00000050
0.00000005
0.00000005
0.00000001
2.0
3.0
Channel 8, x = 17.02 m
0.00050000
0.00005000
1.0
0.00000001
0.0
1.0
2.0
3.0
0.0
Channel 10, x = 17.83 m
0.00050000
0.00005000
0.00000500
0.00000500
0.00000050
0.00000050
0.00000005
0.00000005
0.00000001
2.0
3.0
Channel 12, x = 18.60 m
0.00050000
0.00005000
1.0
0.00000001
0.0
1.0
2.0
3.0
0.0
1.0
2.0
3.0
Bathymetry [m]
Wave gauges [-]
Bottom profile
0.20
depth, [m]
0.00
-0.20
-0.40
0
2
4
6
8
10
12
14
16
18
20
22
x, [m]
03/03/02 20:00:00:000
Figure 12.17: FFT analysis of time series in different distances from the wavemaker. The logarithm
of the energy is plotted as a function of the frequency. Dashed line is M21BW modelled time series
and the solid line is the measured. The lower graph shows the placing of the wavemaker (marked by
circles). Case IR 2
89
CHAPTER 12. ANALYSIS OF MIKE 21 BW RESULTS
Channel 1, x =2.30 m
Channel 2, x = 9.68 m
0.00020000
0.00020000
0.00005000
0.00005000
0.00001000
0.00001000
0.00000200
0.00000200
0.00000050
0.00000050
0.00000010
0.00000010
0.00000002
0.00000002
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
Channel 5, x = 15.8 m
0.00020000
0.00005000
0.00005000
0.00001000
0.00001000
0.00000200
0.00000200
0.00000050
0.00000050
0.00000010
0.00000010
0.00000002
0.00000002
0.5
1.0
1.5
2.0
1.0
1.5
2.0
2.5
3.0
Channel 8, x = 17.02 m
0.00020000
0.0
0.5
2.5
3.0
0.0
Channel 10, x = 17.83 m
0.5
1.0
1.5
2.0
2.5
3.0
Channel 12, x = 18.60 m
0.00020000
0.00020000
0.00005000
0.00005000
0.00001000
0.00001000
0.00000200
0.00000200
0.00000050
0.00000050
0.00000010
0.00000010
0.00000002
0.00000002
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Bathymetry [m]
Wave gauges [-]
Bottom profile
0.20
depth, [m]
0.00
-0.20
-0.40
0
2
4
6
8
10
12
14
16
18
20
22
x, [m]
03/03/02 20:00:00:000
Figure 12.18: FFT analysis of time series in different distances from the wavemaker. The logarithm
of the energy is plotted as a function of the frequency. Dashed line is M21BW modelled time series
and the solid line is the measured. The lower graph shows the placing of the wavemaker (marked by
circles). Case IR 3
90
CHAPTER 12. ANALYSIS OF MIKE 21 BW RESULTS
0.000085
0.000080
0.000075
0.020
0.000070
0.015
0.000065
0.010
0.000060
0.005
0.000
0.000055
-0.005
0.000050
-0.010
0.000045
20:05:00
2002-03-03
0.000040
20:05:10
20:05:20
20:05:30
20:05:40
0.000035
0.000030
0.000025
0.000020
0.000015
0.000010
0.000005
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
Figure 12.19: FFT analysis of time series from the runup gauge. The plot in the upper right corner
is the time series in a selected time range. Dashed line marks the M21BW modelled vertical runup
and the solid line is the laboratory measurements. Case 1, T = 1.5 s, H = 0.03 m, γ = 1
91
CHAPTER 12. ANALYSIS OF MIKE 21 BW RESULTS
0.000080
0.000075
0.020
0.000070
0.015
0.000065
0.010
0.000060
0.005
0.000055
0.000
0.000050
-0.005
0.000045
20:05:00
2002-03-03
0.000040
20:05:10
20:05:20
20:05:30
20:05:40
0.000035
0.000030
0.000025
0.000020
0.000015
0.000010
0.000005
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Figure 12.20: FFT analysis of time series from the runup gauge. The plot in the upper right corner
is the time series in a selected time range. Dashed line marks the M21BW modelled vertical runup
and the solid line is the laboratory measurements. Case 2, T = 1.5 s, H = 0.03 m, γ = 3.3
92
CHAPTER 12. ANALYSIS OF MIKE 21 BW RESULTS
0.0010
0.0009
0.060
0.0008
0.040
0.0007
0.020
0.000
0.0006
-0.020
0.0005
20:05:00
2002-03-03
20:05:10
20:05:20
20:05:30
20:05:40
0.0004
0.0003
0.0002
0.0001
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Figure 12.21: FFT analysis of time series from the runup gauge. The plot in the upper right corner
is the time series in a selected time range. Dashed line marks the M21BW modelled vertical runup
and the solid line is the laboratory measurements. Case 3, T = 1.5 s, H = 0.08 m, γ = 10
Chapter 13
Conclusions
13.1
Conclusion of the investigation of the near field in front of a
wavemaker
Experimentally we measured times series in different distances from the wavemaker. By analysing
the time series using the linear theory, described in chapter 3 we are able to estimate the validity of
this theory.
Adjusting the phase of the evanescent modes and the progressive to the phase of the theoretical
linear wave the analysis is limited to including estimation of the amplitudes. It turns out that
the amplitudes are determined rather well. Difficulties were encountered but we estimate that the
amplitudes are described fairly well using the linear theory.
We also tried not adjusting the phase of the measured evanescent modes and the progressive
wave to the phase of the theoretical wave, calculated by linear theory. By this method we introduced
new unknown, namely the phases of the progressive wave and the evanescent modes. We found that
the measurements not was comparable to the theoretical results using this procedure. By use of
the conclusion that the amplitudes were estimated sufficiently using linear theory, it leads to the
conclusion that the phases are not determined sufficiently accurate using linear theory.
13.2
Conclusion of the investigation of MIKE 21 BW handling of
the surf zone
The investigation of MIKE 21 Boussinesq Wave Module (M21BW)’s handling of the surf zone was
done by comparison of experiments and the computer model results. It was found that M21BW
described the breaking waves very well; both the shape of the waves and the heights matched the
measured. M21BW has a number of calibration parameters. It can be discussed wether the calibration parameters should be excluded from the model. The purpose of a numerical model is to be able
to model a given situation without the use of measurements.
Large differences between the experimental measurements and the M21BW were found just before
the point of breaking. It is very difficult to extend the current theory to include the higher order
effect that causes these large wave amplitudes near the point of breaking. We think that M21BW
should be able to make an empirical estimate of the wave heights just before breaking. Because the
differences are so pronounced the underestimate of the wave height can lead to underestimation of
forces on structures.
The moving shoreline is not modelled accurately in the case of the sinusoidal wavemaker signal.
But this case is only of theoretical interest as the monochromatic wave will not be encountered in
nature. In cases of irregular waves we found the M21BW computer model to produce satisfying
results.
93
Bibliography
[Asmar, N. (2000)] , Partial Differential Equations and Boundary Value Problems, Prentice Hall
[Bingham, H. (2002)] , Wave Body Interaction, lecture note from course 41223, DTU, Lyngby
[DHI Lecture Note] , Coastal and River Hydraulics, Lecture Notes , Edition: 1st (Preliminary),
August 1989, DHI
[Madsen, P. A. and Schäffer, H. A. (1999)] , A Review of Boussinesq-type Equations for Surface
Gravity Waves, Advances in Coastal and Ocean Engineering, vol 5
[Madsen, P. A. et al., (1997)] , Surf Zone Dynamics Simulated by Boussinesq type Model. Part I.
Model description and cross-shore motion of regular waves, Coastal Engineering 32.
[Madsen, P.A. and Sørensen, O.R. (1993)] , Bound Waves and Triad Interactions in Shallow Water,
Ocean Engineering, Vol 20
[Madsen, P. A. et al (1991)] , A New Form of the Boussinesq Equations with Improved Linear Dispersion Charateristics, Part I, Coastal Engineering, vol 15
[MIKE 21 BW, Scientific Documentation] , DHI.
[MIKE 21 BW Help] MIKE 21 BW Help, From MIKE 21 2001.0 build 4-302, DHI Software 2001.
[Ruggiero et al, (2001)] , Wave Runup, Extreme Water Levels and the Erosion of Properties, Backing
Beaches, Journal of Coastal Research, Vol. 17 No. 2
[Schäffer, H. A. (1994)] , Second-order Wavemaker Theory for Irregular Waves, Ocean Engng vol. 23
[Skourup, J. (1999)] , Wavemaker, Lecture note to ISVA course: 57234 Wave Hydrodynamics, DTU,
Lyngby
[Svendsen, I. A and Jonsson, I. G. (1996)] , Hydrodynamics of Coastal Regions, Den Private Ingeniørfond, DTU, Lyngby
[WS User Manual] , Wave synthesizer for Window NT/2000, version 2.10., DHI
94
Appendix A
Preliminary calculations
A.1
Assuming no shoaling in the near field near the wavemaker
Table A.1: Shoaling of the evanescent waves from the wavemaker to the wave gauge 10.
50
Case no. H52 [m] H50 [m] H52H−H
52
1
0.011
0.011
0.00
2
0.035
0.035
0.00
3
0.035
0.035
0.00
A.2
Changing of wave length from wavemaker to wave gauge 10
Table A.2: Changing of wave length from wavemaker to wave gauge 10.
50
Case no. L52 [m] L50 [m] L52L−L
52
1
0.39
0.39
0.00
2
0.76
0.76
0.00
3
2.86
2.83
0.01
95
Appendix B
FFT results
The results are calculated using changing phase method (only the complex amplitude is different).
Wave gauge no.
1
2
3
4
5
6
7
8
9
10
Table B.1: FFT results case 1.
Distance from flap [m] Period T Complex amplitude ac
0.14
0.50
-0.0052-0.0045i
0.22
0.50
0.0015-0.0042i
0.32
0.50
0.0018+0.0036i
0.44
0.50
-0.0050+0.0016i
0.59
0.50
-0.0007-0.0049i
0.73
0.50
0.0035+0.0029i
0.92
0.50
-0.0042-0.0023i
1.20
0.50
0.0044+0.0012i
1.56
0.50
0.0045-0.0007i
1.73
0.50
-0.0041+0.0020i
Wave gauge no.
1
2
3
4
5
6
7
8
9
10
Table B.2: FFT results case 2.
Distance from flap [m] Period T Complex amplitude ac
0.14
0.70
-0.0170+0.0105i
0.22
0.70
-0.0186-0.0014i
0.32
0.70
-0.0105-0.0136i
0.44
0.70
0.0051-0.0159i
0.59
0.70
0.0163-0.0044i
0.73
0.70
0.0106+0.0128i
0.92
0.70
-0.0119+0.0110i
1.20
0.70
-0.0113-0.0111i
1.56
0.70
0.0144+0.0087i
1.73
0.70
-0.0061+0.0150i
96
APPENDIX B. FFT RESULTS
Wave gauge no.
1
2
3
4
5
6
7
8
9
10
Table B.3: FFT results case 3.
Distance from flap [m] Period T Complex amplitude ac
0.14
1.50
0.0093+0.0143i
0.22
1.50
0.0064+0.0154i
0.32
1.50
0.0023+0.0167i
0.44
1.50
-0.0019+0.0167i
0.59
1.50
-0.0060+0.0156i
0.73
1.50
-0.0105+0.0130i
0.92
1.50
-0.0147+0.0078i
1.20
1.50
-0.0163+0.0015i
1.56
1.50
-0.0125-0.0107i
1.73
1.50
-0.0073-0.0145i
97
Appendix C
Program testing
Case H = 0.011 m and T = 0.5 s
0.01
Measured values
Values from linear theory
0.008
0.006
0.004
n = 3,xn==0.320
6,x = 0.730
0.002
n = 2,x = 0.220
n = 5,x = 0.590
0
−0.002
n = 9,x = 1.560
n = 10,x = 1.730
−0.004
n = 8,x = 1.200
n = 7,x = 0.920
n = 4,x = 0.440
−0.006
n = 1,x = 0.140
−0.008
−0.01
−0.01
−0.008
−0.006
−0.004
−0.002
0
0.002
0.004
0.006
0.008
0.01
Figure C.1: The complex amplitude. H = 0.011 m and T = 0.5 s.
The figure C.4 show the result of case 2. The result looked as expected.
The ratio between the evanescent modes and the progressive waves at the flap can found for a
second order solution to approximatly 1.7 [Schäffer, H. A. (1994)], this linear solution is around 1.6.
98
99
APPENDIX C. PROGRAM TESTING
−3
6
x 10
5
Evan theo
Prog. theo
Evan fft
Prog fft
Amplitude [m]
4
3
2
1
0
−1
0
0.2
0.4
0.6
0.8
1
1.2
Distance from wavemaker [m]
1.4
1.6
1.8
Figure C.2: Progressive and evanescent modes, theo means directly from linear theory, FFT means
linear theory via FFT analysis. H = 0.011 m and T = 0.5 s.
Table C.1: Determinats in different distances from the flap, case 1
Distance from flap [m] determinat
0.00
-1.0000i
0.14
0.6310i
0.22
0.9211i
0.32
-0.4248i
0.44
-0.6970i
0.59
0.9974i
0.73
-0.6856i
0.92
0.6228i
1.20
-0.8929i
1.56
-0.9998i
1.73
0.9106i
100
APPENDIX C. PROGRAM TESTING
Case H = 0.035 m and T = 0.7 s
Measured values
Values from linear theory
0.02
n = 6,x = 0.590
0.015
0.01
n = 9,x = 1.200
n = 5,x = 0.440
0.005
n = 7,x = 0.730
0
n = 10,x = 1.560
−0.005
−0.01
n = 1,x = 0.000
n = 4,x = 0.320
−0.015
n = 8,x = 0.920
n = 11,x = 1.730
−0.02
n = 3,x = 0.220
−0.02
−0.015
−0.01
−0.005
0
n = 2,x = 0.140
0.005
0.01
0.015
0.02
Figure C.3: The complex amplitude. H = 0.035 m and T = 0.7 s.
0.02
Evanescent mode (linear theory)
Progressive mode (linear theory)
Measured evanescent mode
Measured progressive mode
0.018
0.016
Amplitude [m]
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
0
0.2
0.4
0.6
0.8
1
1.2
Distance from wavemaker [m]
1.4
1.6
1.8
Figure C.4: Progressive and evanescent modes, theo means directly from linear theory, FFT means
linear theory via FFT analysis. H = 0.035 m and T = 0.7 s.
The figure C.4 show the result of case 2. The result looked as expected.
The ratio between the evanescent modes and the progressive waves at the flap can found for a
second order solution to approximatly 1.1 [Schäffer, H. A. (1994)], this linear solution is around 1.1.
101
APPENDIX C. PROGRAM TESTING
Case H = 0.035 m and T = 1.5 s
Measured values
Values from linear theory
0.02
0.015
n = 10,x = 1.730
0.01
0.005
n = 9,x = 1.560
0
−0.005
n = 1,x = 0.140
n = 2,x = 0.220
n = 8,x = 1.200
−0.01
n = 3,x = 0.320
n = 4,x = 0.440
−0.015
n = 7,x = 0.920
n = 5,x = 0.590
n = 6,x = 0.730
−0.02
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
Figure C.5: The complex amplitude. H = 0.035 m and T = 1.5 s.
−3
20
x 10
Evan theo
Prog. theo
Evan fft
Prog fft
Amplitude [m]
15
10
5
0
−5
0
0.2
0.4
0.6
0.8
1
1.2
Distance from wavemaker [m]
1.4
1.6
1.8
Figure C.6: Progressive and evanescent modes, theo means directly from linear theory, FFT means
linear theory via FFT analysis. H = 0.035 m and T = 1.5 s.
The figure C.6 show the result of case 3. The result looked as expected. The fail of the result
calculated via FFT in point 7 (0.73 m) is caused by a small determinat, see table C.3.
The ratio between the evanescent modes and the progressive waves at the flap can found for a
second order solution to approximatly 11.0 [Schäffer, H. A. (1994)], this linear solution is around
10.9.
APPENDIX C. PROGRAM TESTING
Table C.2: Determinats in different distances from the flap, case 2
Distance from flap [m] determinat
0.00
-1.0000i
0.14
-0.4083i
0.22
0.2345i
0.32
0.8715i
0.44
0.8900i
0.59
-0.1347i
0.73
-0.9595i
0.92
-0.2909i
1.20
0.9071i
1.56
-0.9687i
1.73
0.0765i
Table C.3: Determinats in different distances from the flap, case 3
Distance from flap [m] determinat
0.00
1.0000i
0.1400
-0.9532i
0.2200
-0.8857i
0.3200
-0.7633i
0.4400
-0.5688i
0.5900
-0.2723i
0.7300
0.0315i
0.9200
0.4337i
1.2000
0.8739i
1.5600
0.9604i
1.7300
0.7927i
102
Appendix D
The code for the evanescent analysis
program
D.1
Program to analysis evanescent modes (constant phase method)
% =======================================================
%%%%%%%%%%%%%%%%% INITIALISERING %%%%%%%%%%%%%%%%%%%%%%%%%
% =======================================================
clear
disp(sprintf(’\n\n===================================================
\n#############
ANALYSE PROGRAM
#############\n==============
=====================================’))
maintic = cputime;
global g
g=9.81;
%%%%%%%%%%%%%%%%% FIL TIL ANALYSE %%%%%%%%%%%%%%%%%%%%%%%%
filNr=12;
faseforskyd=0.2818;%forsoeg 12 case 2
%0.1173;%forsg 3 case 3
%0.5027;%forsoeg 4 case 1
%
[dataArr, H, T, h, dscArr] =readfile(filNr);
afstand=[0 0.14 0.22 0.32 0.44 0.59 0.73 0.92 1.20 1.56 1.73];
distFarest = 1.73;
L = bolgelaengde(h,T);
%%%%%%%%%%%%%%%%%% INDLEDENDE BEREGNINGER %%%%%%%%%%%%%%%
deltaTime = timeWindow(T, h, distFarest);
% opløsning til brug i spline.
% Test viser, at jo lavere den sættes jo mindre fejl fås ved determinant
% minimum.
103
APPENDIX D. THE CODE FOR THE EVANESCENT ANALYSIS PROGRAM
splinedt=0.001;
startInt = ceil(deltaTime(1)/T)
endInt = ceil(deltaTime(2)/T)*T*1000;
startInt = round((ceil(5/T)+startInt)*T*1000);
[AE, AU, cEvan] = bolgehojdeEvan(afstand, T, H, h);
% OBS
% AE er amplituden af evanescent modes
% AU er amplituden på den progressive og
% cEvan er den complekse amplitude af de to lagt sammen. Dvs fase
% indkluderet.
% ============================================================
%%%%%%%%%%%%%%% DE ENKELTE KANALER GENNEMLØBES %%%%%%%%%%%%%%%%
% ============================================================
figure(1)
clf
disp(’.............for l~
A¸kke start...................’)
for n=2:length(afstand)+1
%%%%%%%%%%%%%%%%%%%%%%%%%%
% START
%%
disp(sprintf(’-------------\n--|
dist no = %d
|--’,n-1))
x=afstand(n-1);
disp(sprintf(’x = %3.3f’,x))
%%%%%%%%%%%%%%%%%%%%%%%%%%
% TEST KONSTRUERET EVAN %%
tK = dataArr(:,1);
etaK=AU(n-1).*cos(2*pi/T*tK - 2*pi/L*x) - AE(n-1).*sin(2*pi/T*tK);
etaK=etaK(1:length(etaK));
tK=tK(1:length(tK));
%%%%%%%%%%%%%%%%%%%%%%%%%%
% KANALEN PILLES UD
%%
eta = dataArr(:,n);
t=dataArr(:,1);
dsc=dscArr(:,3);
dsct=dscArr(:,1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SPLINE FOR HØJERE OPLØSNING %%
% KAN EVT SPLINE MED 2^N PUNKTER
104
APPENDIX D. THE CODE FOR THE EVANESCENT ANALYSIS PROGRAM
tic;
tsp = 0:splinedt:t(end);
etaSp = spline(t,eta, tsp);
dscetaSp = spline(dsct,dsc, tsp);
tspK = 0:splinedt:tK(end);
etaSpK = spline(tK,etaK, tspK);
disp(sprintf(’spline done: %3.3f sec’,toc))
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% tidsvinduet uden forstyrrelser + 1 bølgelængde (fjernes senere)
etaSp = etaSp(startInt:endInt+2*T/0.001);
tsp = tsp(startInt+1:endInt+1+2*T/0.001);
tsp=tsp-tsp(1);
dscetaSp = dscetaSp(startInt+1+1000*5:endInt+1+1000*5+2*T/0.001);
% Test vindue
etaSpK = etaSpK(startInt+1:endInt+1);
tspK = tspK(startInt+1:endInt+1);
tspK=tspK-tspK(1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Finder zerocross
j=1;
for p=1:length(dscetaSp)-1
if(dscetaSp(p)*dscetaSp(p+1)<=0 & dscetaSp(p)<=0)
zeroidsc(j) = p;
j=j+1;
end
end
meanDsc = mean(dscetaSp(zeroidsc(1):zeroidsc(end)));
dscetaSpNul = dscetaSp-meanDsc;
j=1;
for p=1:length(dscetaSp)-1
if(dscetaSpNul(p)*dscetaSpNul(p+1)<=0 & dscetaSpNul(p)<=0)
zeroidsc(j) = p;
j=j+1;
end
end
faseInt=(T/splinedt/(2*pi));
forskyd=zeroidsc(1)-floor(faseforskyd*faseInt);
etaSpFase = etaSp(forskyd+1:length(etaSp)-2*T/0.001+forskyd);
105
APPENDIX D. THE CODE FOR THE EVANESCENT ANALYSIS PROGRAM
tspFase = tsp(forskyd+1:length(etaSp)-2*T/0.001+forskyd);
tspFase = tspFase-tspFase(1);
etaSpFase(1)
etaSpK(1)
% PLOT AF TIDSSERIER
figure(6+n)
clf
hold on
plot(tK,etaK,’g:’)
plot(tspFase,etaSpFase,’b’)
legend(’Theoretical phase’,’Measured before FFT’)
xlabel(’Time [s]’)
ylabel(’Elevation [m]’)
grid
hold off
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FFT’EN KØRES
%
tic;
[amp_c(n-1), frekvensVedPeak(n-1)] = runFft(etaSpFase, tspFase);
[amp_cK(n-1), frekvensVedPeakK(n-1)] = runFft(etaSpK, tspFase);
disp(sprintf(’fft done: %3.3f’,toc))
absAmp(n-1) = abs(amp_c(n-1));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% KOMPLEKSE AMPLITUDER PLOTTES %
figure(1)
hold on
% Calculated from HEvan2
xx_c = real(amp_c(n-1));
yy_c = imag(amp_c(n-1));
plot([0 xx_c] , [0, yy_c], ’:ro’)
text(xx_c+0.001, yy_c, sprintf(’n = %d,x = %3.3f’,n-1,x))
xxK = real(amp_cK(n-1));
yyK = imag(amp_cK(n-1));
plot([0 xxK] , [0, yyK], ’:b*’)
text(xxK+0.001, yyK, sprintf(’n = %d,x = %3.3f’,n-1,x))
axis((0.005+absAmp(n-1))*[-1 1 -1 1])
legend(’Measured values’, ’Values from linear theory’)
hold off
grid
dt(n-1)=(atan(yy_c/xx_c)/(2*pi/T)-atan(yyK/xxK)/(2*pi/T));
fasefejl(n-1)=(atan(yy_c/xx_c)-atan(yyK/xxK));
end
106
APPENDIX D. THE CODE FOR THE EVANESCENT ANALYSIS PROGRAM
107
disp(’.............for løkke slut...................’)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% AMPLITUDERNE TRÆKKES UD
%%
tic;
[Ve, Vp, determ]=Hevan2(amp_c, afstand, h, T, H);
[VeK, VpK, determK]=Hevan2(amp_cK, afstand, h, T, H);
[VeD, VpD, determD]=Hevan2(cEvan, afstand, h, T, H);
disp(sprintf(’HEvan2 done: %3.3f’,toc))
% PLOT I SAMME KOORDINATSYSTEM
figure(gcf+1)
clf
hold on
% teoretisk
plot(afstand,-AE,’or-’, afstand, abs(AU), ’r-*’)
% modelforsøg
plot(afstand, Ve,’ob:’, afstand, Vp, ’b*:’ )
xlabel(’Distance from wavemaker [m]’)
ylabel(’Amplitude [m]’)
legend(’Evanescent mode (linear theory)’, ’Progressive mode
(linear theory)’, ’Measured evanescent mode’, ’Measured
progressive mode’)
grid
hold off
xx_c = real(amp_c);
yy_c = imag(amp_c);
xxK = real(amp_cK);
yyK = imag(amp_cK);
fasefejl=(atan(yy_c./xx_c)-atan(yyK./xxK))’
disp(sprintf(’\n\n====================\nAnalys done in %3.3f\
\n====================’, cputime - maintic))
D.2
Program to analysis evanescent modes (changing phase method)
% =======================================================
%%%%%%%%%%%%%%%%% INITIALISERING %%%%%%%%%%%%%%%%%%%%%%%%%
% =======================================================
clear
disp(sprintf(’\n\n===================================================\
\n#############
ANALYSE PROGRAM
#############\n==============\
=====================================’))
maintic = cputime;
APPENDIX D. THE CODE FOR THE EVANESCENT ANALYSIS PROGRAM
global g
g=9.81;
%%%%%%%%%%%%%%%%% FIL TIL ANALYSE %%%%%%%%%%%%%%%%%%%%%%%%
filNr=3;
[dataArr, H, T, h] =readfile(filNr);
afstand=[0.14 0.22 0.32 0.44 0.59 0.73 0.92 1.20 1.56 1.73];
distFarest = 1.73;
L = bolgelaengde(h,T)
%%%%%%%%%%%%%%%%%% INDLEDENDE BEREGNINGER %%%%%%%%%%%%%%%
deltaTime = timeWindow(T, h, distFarest)
% opløsning til brug i spline.
% Test viser, at jo lavere den sættes jo mindre fejl fås ved
% determinant minimum.
splinedt=0.001;
startPer = ceil(deltaTime(1)/T)
endPer = ceil(deltaTime(2)/T)
StartPer = ceil(5/T)+startPer
[AE, AU, cEvan] = bolgehojdeEvan(afstand, T, H, h);
AE=-AE;
% OBS
% AE er amplituden af evanescent modesm
% AU er amplituden på den progressive og
% cEvan er den complekse amplitude af de to lagt sammen. Dvs
% fase indkluderet.
% ============================================================
%%%%%%%%%%%%%%% DE ENKELTE KANALER GENNEMLØBES %%%%%%%%%%%%%%%%
% ============================================================
figure(1)
clf
disp(’.............for l~
A¸kke start...................’)
for n=2:length(afstand)+1
%%%%%%%%%%%%%%%%%%%%%%%%%%
% START
%%
disp(sprintf(’-------------\n--|
x=afstand(n-1);
dist no = %d
|--’,n-1))
108
APPENDIX D. THE CODE FOR THE EVANESCENT ANALYSIS PROGRAM
disp(sprintf(’x = %3.3f’,x))
%%%%%%%%%%%%%%%%%%%%%%%%%%
% KANALEN PILLES UD
%%
eta=dataArr(:,n);t=dataArr(:,1);
% tidsvinduet uden forstyrrelser + 1 bølgelængde (fjernes senere)
eta = eta(startPer*40*T+1:(endPer+1)*40*T+1);
t= t(startPer*40*T+1:(endPer+1)*40*T+1);
t=t-t(1);
%%%%%%%%%%%%%%%%%%%%%%%%%%
% TEST KONSTRUERET EVAN %%
tK = 0:0.025:53.65;% %
etaK=AU(n-1).*cos(2*pi/T*tK - 2*pi/L*x) - AE(n-1).*sin(2*pi/T*tK);
etaK = -etaK(startPer*40*T+1:endPer*40*T+1);
tK = tK(startPer*40*T+1:endPer*40*T+1);
tK=tK-tK(1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SPLINE FOR HØJERE OPLØSNING %%
% KAN EVT SPLINE MED 2^N PUNKTER
tic;
tsp = 0:splinedt:t(end);
etaSp = spline(t,eta, tsp);
tsp = tsp-tsp(1);
tspK = 0:splinedt:tK(end);
etaSpK = spline(tK,etaK, tspK);
tspK = tspK-tspK(1);
disp(sprintf(’spline done: %3.3f sec’,toc))
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FASEN NULSTILLES IFHT KONST %%
j=1;
for p=1:length(tspK)-1
if(etaSpK(p)*etaSpK(p+1)<=0 )
zeroiK(j) = p;
j=j+1;
end
end
j=1;
for p=1:length(tsp)-1
if(etaSp(p)*etaSp(p+1)<=0 )
zeroi(j) = p;
109
APPENDIX D. THE CODE FOR THE EVANESCENT ANALYSIS PROGRAM
j=j+1;
end
end
splinePer = T/splinedt;
forskyd = round(mean(zeroi(2:length(zeroiK)-1)-zeroiK(1:length
(zeroi(2:length(zeroiK)-1)))));
forskydArr(n-1) = splinedt*forskyd;
etaSpFase = etaSp(forskyd+1:length(etaSp) - 2*splinePer + forskyd+1);
tspFase = tsp(forskyd+1:length(etaSp) - 2*splinePer + forskyd+1);
tspFase = tspFase-tspFase(1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FFT’EN KØRES
%
tic;
[amp_c(n-1), frekvensVedPeak(n-1)] = runFft(etaSpFase, tspFase);
[amp_cK(n-1), frekvensVedPeakK(n-1)] = runFft(etaSpK, tspK);
disp(sprintf(’fft done: %3.3f’,toc))
absAmp(n-1) = abs(amp_c(n-1));
absAmpK(n-1) = abs(amp_cK(n-1));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% KOMPLEKSE AMPLITUDER PLOTTES %
figure(1)
hold on
% Calculated from HEvan2
xx = real(amp_c(n-1));
yy = imag(amp_c(n-1));
plot([0 xx] , [0, yy], ’:ro’)
text(xx+0.001, yy, sprintf(’n = %d,x = %3.3f’,n-1,x))
xx = real(amp_cK(n-1));
yy = imag(amp_cK(n-1));
plot([0 xx] , [0, yy], ’:b*’)
xx = real(cEvan);
yy = imag(cEvan);
plot([0 xx] , [0, yy], ’:go’)
title(sprintf(’Kompleks amplitude x = %3.3f’,x))
axis((0.001+absAmp(n-1))*[-1 1 -1 1])
legend(’Fysik model test’, ’Konstrueret eksempel’, ’teoretisk
UDEN fase oplysning’)
hold off
grid
end
disp(’.............for løkke slut...................’)
110
APPENDIX D. THE CODE FOR THE EVANESCENT ANALYSIS PROGRAM
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% AMPLITUDERNE TRÆKKES UD
%%
tic;
[Ve, Vp, determ]=Hevan2(amp_c, afstand, h, T, H);
[VeK, VpK, determK]=Hevan2(amp_cK, afstand, h, T, H);
disp(sprintf(’HEvan2 done: %3.3f’,toc))
Ve=abs(Ve);
Vp=abs(Vp);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% STATISTIK PAA RESULTATERNE %%
vaegt=(abs(determ)).^2;
[resMean, progMean, resMeanVaegt, progMeanVaegt, resdev, progdev,
resdevvaegt, progdevvaegt]=stat(Ve, Vp, AE, AU, vaegt)
% PLOT AF NORMALISEREDE RESULTERENDE BOELGEHOEJDER
figure(gcf+1)
plot(afstand,abs(amp_c)./(sqrt(AU.^2+AE.^2)),’*b-’,[afstand(1)
afstand(end)],[resMeanVaegt resMeanVaegt],’k-’,[afstand(1)
afstand(end)],[resMeanVaegt-resdevvaegt resMeanVaegt-resdevvaegt],
’m:’,[afstand(1) afstand(end)],[resMeanVaegt+resdevvaegt resMeanVaegt
+resdevvaegt],’m:’)
xlabel(’Distance to wavermaker [m]’)
ylabel(’Relative amplitude’)
legend(’H_{res,teo}/H_{res,fft}’,’Weighted mean’,’Standard deviation’)
grid
% PLOT AF NORMALISEREDE PROGRESSIVE BOELGEHOEJDER
figure(gcf+1)
plot(afstand, Vp./AU, ’r-*’,[afstand(1) afstand(end)],[progMeanVaegt
progMeanVaegt],’k-’,[afstand(1) afstand(end)],[progMeanVaegt
-progdevvaegt progMeanVaegt-progdevvaegt],’m:’,[afstand(1)
afstand(end)],[progMeanVaegt+progdevvaegt progMeanVaegt+progdevvaegt],’m:’)
xlabel(’Distance to wavermaker [m]’)
ylabel(’Relative amplitude [m]’)
legend(’a_{prog,FFT}/a_{prog,theo}’,’Weighted mean’,’Standard deviation’)
grid
% PLOT I SAMME KOORDINATSYSTEM
figure(gcf+1)
plot(afstand,AE,’or-’, afstand, AU, ’r-*’, afstand, Ve,’ob-’,
afstand, Vp, ’b*-’ )
xlabel(’Distance to wavermaker [m]’)
ylabel(’Amplitude [m]’)
legend(’Evanescent mode (linear theory)’,’Progressive mode
(linear theory)’,’measured evanescent mode’, ’Measured progressive mode’)
grid
111
APPENDIX D. THE CODE FOR THE EVANESCENT ANALYSIS PROGRAM
disp(sprintf(’\n\n====================\nAnalys done in %3.3f
\n====================’, cputime - maintic))
D.3
readfile.m: open the file
function [dataArr, H_0, T_0, h_0, dscArr] =readfile(filNr)
%%%%%%%%%%%%%%%%%%%%%%%%%
% INDLÆSNING AF FILER
%
%%%%%%%%%%%%%%%%%%%%%%%%%
if length( int2str(filNr) ) == 1
filNrStr = [’0’ int2str(filNr)];
else
filNrStr = int2str(filNr);
end
filNavn = [’00’ filNrStr ’log1.daf’];
logFilNavn = [’00’ filNrStr ’log1.log’];
dscFilNavn = [’00’ filNrStr ’dsc.txt’];
% Load data expect ch 9 and 13
dataArr = dlmread(filNavn,’;’,[3 0 2500 8]);
dataArr(:,10:12) = dlmread(filNavn,’;’,[3 10 2500 12]);
dscArr = dlmread(dscFilNavn, ’\t’, 3, 0);
logFile = textread(logFilNavn, ’%s’, 6, ’headerlines’, 15, ’delimiter’, ’:’);
[h_0 S] = strread(char(logFile(2)), ’%n%s’,’delimiter’,’(’ );
[H_0 S] = strread(char(logFile(4)), ’%n%s’,’delimiter’,’(’ );
[T_0 S] = strread(char(logFile(6)), ’%n%s’,’delimiter’,’(’ );
D.4
bolgelaengde.m: calculating the wave length
function L = bolgelaengde(h, T)
global g
% Første gæt er fladvandstilnærmelsen
% Syntax: L = bolgelaengde(depth, period)
L_old = 2*sqrt(g*h);
L_new = sqrt(g*h);
while any( abs(L_old - L_new) > .0001 )
L_old = L_new;
L_new = (g*T^2/(2*pi)).*tanh(2*pi*h./L_old);
end
112
APPENDIX D. THE CODE FOR THE EVANESCENT ANALYSIS PROGRAM
L = L_new;
D.5
timeWindow.m: calculating the time window
function deltaTime = timeWindow(T, h_0, distFarest)
global g
%
%
%
%
%
%
Calculates the time in which the measurements are undisturbed
from reflektion and
piston gain-up
Syntax: [startTime, endTime] = timeWindow(T, h0, distFarest)
where h0 indicates the waterdepth at the wavemaker and distFarest
is the distance to the farest wavegauge.
% The distance where the wave is being reflected.
reflectionDist = 1.5; % [m]
% Warm-up time
warmUp = 5; % [s]
deltax = 0.1;
[D channelLn] = dybde(0, h_0);
x = 0:deltax:distFarest;
tFarest = timeCalc(x, h_0, T);
startTime = warmUp + tFarest;
x = distFarest:deltax:(channelLn-reflectionDist);
tChannel = 2*timeCalc(x, h_0, T);
endTime = tChannel + tFarest;
% For at regne "skud fra hoften" ud er der brugt fladvandsteori.
skudStart = warmUp + distFarest/sqrt(g*0.55)
skudSlut = distFarest*sqrt(g*0.55) + (channelLn - reflectionDist)
/sqrt(g*2*0.55/3)
deltaTime=[startTime, endTime];
%------------------------function t = timeCalc(x, h_0, T)
deltax = diff(x);
deltax = deltax(1);
h = dybde(x, h_0);
L = bolgelaengde(h, T);
c = L/T;
t = deltax./c;
t = cumsum(t);
t = t(end);
113
APPENDIX D. THE CODE FOR THE EVANESCENT ANALYSIS PROGRAM
D.6
bolgehojdeEvan.m: calculating the evanescent modes amplitude and the progressive modes amplitude
function [AEvan, AUdenEvan, cEvan] = bolgehojdeEvan(x, T, H, h)
%
%
%
%
114
Calculating waveheight including Evanescent modes.
Syntax: [H_with_Evan H_without_Evan] = bolgehojdeEvan(distance
_from_piston, T, H);
x must be one column and a number of rows.
global g
% Wavelength is calculated using the linear dispersion relation,
% assuming constant depth
L = bolgelaengde(h,T);
k0 = 2*pi./L;
omega = 2*pi/T;
% The complex solutions to the linear dispersion relation is found.
kn = ksiteration(T, h);
kn=-i*kn;
kn = [k0; kn];
% The strokes is calculated:
S = H*(sinh(k0*h).*cosh(k0*h) + k0*h)/(2*(sinh(k0*h))^2);
Gn = 2*kn.*h./sinh(2*kn.*h);
% Converting into matrix form
for n = 1:length(kn); xMatrix(n,:) = x; end
for n = 1:length(x); knMatrix(:,n) = kn; GnMatrix(:,n) = Gn; end
% The exponential decrease of the evanescent modes:
pn = exp(-i*knMatrix.*xMatrix);
pn(1,:) = 1;
D = (S*omega^2)/g;
eta_n = D.*pn./(knMatrix.*(1 + GnMatrix));
ampVar = cumsum(eta_n,1);
% The complex for of the evanescent mode:
cEvan = ampVar(end,:);
% The waveheight without Evanescent modes is calculated.
% progressive blge
AUdenEvan = ampVar(1,:);
%evanescent modes
AEvan = imag(ampVar(end,:)-ampVar(1,:));
APPENDIX D. THE CODE FOR THE EVANESCENT ANALYSIS PROGRAM
D.7
ksiteration.m: calculating the wave numbers
function ks = ksiteration(T, h)
global g
medtagneLed = 50;
sigma = 2*pi/T;
konst = h.*(sigma^2)/g;
x = 0.1:.1:4*pi;
for i = 1:medtagneLed
x_gaet = i*pi;
y_gaet = 1;
kontr = 0;
while all( abs( konst./x_gaet + tan(x_gaet) )
y_gaet = konst./x_gaet;
x_gaet = i*pi + atan(-y_gaet);
> 0.0001)
kontr = kontr +1;
if kontr > 500
warning(’blev stoppet af nødstoppen’)
break
end
end
ks(i,:) = x_gaet./h;
end
D.8
runfft.m: doing the FFT analysis
function [amp_c, frek2,amp2, omega, amp, frek, inter] = runFft(eta, t)
deltat = t(2)-t(1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Tjek mm for at g~
A¸re det nemmere for brugeren %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if size(eta, 1) < size(eta, 2)
eta = eta’;
end
Y = fft(eta);%,nfft);
nup = ceil(length(Y)/2);
nfft = 2*nup-1;
115
APPENDIX D. THE CODE FOR THE EVANESCENT ANALYSIS PROGRAM
116
Y2=2*Y(1:nup,1)/(length(eta));
Tmax=deltat*(nfft-1);
df=1/Tmax;
frek=df*[0:1:nup-1]’;
% Amplituden korrigeres
amp=abs(Y2);
[ampmax nampmax]=max(amp);
% Interpolation af modulus
modint=polyfit(frek(nampmax-1:nampmax+1), Y2(nampmax-1:nampmax+1), 2);
frek2=abs(modint(2)/(2*modint(1)));
T2=1/frek2;
amp_c = modint(1)*frek2^2+modint(2)*frek2+modint(3);
amp_c = amp_c;
amp2 = abs(modint(1)*frek2^2+modint(2)*frek2+modint(3));
H2=2*amp2;
% Til plotning
inter(1,:) = frek(nampmax-1):0.001:frek(nampmax+1);
inter(2,:) = modint(1)*inter(1,:).^2 + modint(2)*inter(1,:) + modint(3);
D.9
Hevan2.m: calculating the evanescent mode amplitudes and
the progressive modes amplitude from the complex result from
FFT analysis
function [Ve, Vp, determ]=Hevan2(amp_c, x, h_0, T, H)
global g
L = bolgelaengde(h_0, T);
k=2*pi/L;
for n=1:length(x)
V=[0
cos(k*x(n));
-i
-i*sin(k*x(n))];
determ(n) = det(V);
a_c=[real(amp_c(n)); i*imag(amp_c(n))];
amp=(V\a_c);
APPENDIX D. THE CODE FOR THE EVANESCENT ANALYSIS PROGRAM
Vp(n)=amp(2);
Ve(n)=amp(1);
end
D.10
stat.m: calculating the statistic results
This function is only used for the changing phase method.
function [resMean, progMean, resMeanVaegt, progMeanVaegt, resdev,
progdev, resdevvaegt, progdevvaegt]=stat(Ve, Vp, HE, HU, vaegt)
resMeanVaegt= sum((sqrt(Vp.^2+Ve.^2))./(sqrt(HU.^2+HE.^2)).*vaegt)
/sum(vaegt);
resMean=mean((sqrt(Vp.^2+Ve.^2))./(sqrt(HU.^2+HE.^2)));
progMeanVaegt=sum(Vp./HU.*vaegt)/sum(vaegt);
progMean=mean(Vp./HU);
resdevvaegt=std((sqrt(Vp.^2+Ve.^2)).*vaegt);
resdev=std((sqrt(Vp.^2+Ve.^2))./sqrt(HU.^2+HE.^2));
progdevvaegt=std(Vp.*vaegt);
progdev=std(Vp./HU);
117
Appendix E
Suggestions to improvements of
MIKEZero
I det følgende er en optegnelse over de fejl jeg stødte på under mit arbejde med MIKE 21 BW version
2001.0 build no. 4-302.
Når jeg stødte på en fejl prøvede jeg at genskabe samme situation før jeg nedskrev fejlen. Jeg
genstartede dog ikke programmet eller mit styresystem før genskabelse af situationen.
Jeg har ligeledes nedskrevet ting jeg synes kunne gøre MIKEZero et bedre og mere brugervenligt
program.
Da fejlene og forslagene til forbedringer er nedskrevet sideløbende med arbejdet beder jeg om
læserens forståelse for det sproglige output.
Min computer er en Athlon Thunderbird 1333 MHz med 256 MB DDR RAM
Styresystem er windows 2000 version 5.00.2195
E.1
Fejl
1. Når en model køre (trykkes på run) og man suspenderer optræder alle outputfilerne som
skrivebeskyttede.
2. I vinduet, der kommer efter tryk på ”New” virker hjælp-knappen ikke.
3. Ændres ”equation type” til at inkludere deepwater terms, efter at der er valgt en intern wave
generation kommer der en fejlmeddelelse hver gang der klikkes med musen.
4. Benyttes muligheden konstant filter kommer en fejlmeddelse, at filen ikke kan findes. (Der skal
ikke specificeres nogen fil)
5. I toolbox: æ, ø, å kan ikke benyttes ved navngivning (den første indtastning i wizarden.
6. Æ, ø, å og blanktegn kan ikke benyttes i bibliotekets navn, hvis wizarden fra toolboxen skal
bruges.
7. Toolbox, FFT filter: kan ikke lave default værdier, som fungerer til den valgte tidsserie.
8. Plot Composer: Hvis man trykker på lav video (under et plot af profil) og cancler laver den en
video alligevel.
9. Generate Porosity Map: Den laver en underlig sponge, hvis man forsøge at lave en sponge
både i positiv og negativ bathymetry. Eksempelvis bruger eksemplets (bw-sloaping beach)
bathymetry.
118
APPENDIX E. SUGGESTIONS TO IMPROVEMENTS OF MIKEZERO
119
10. Under BW -¿ Simulation period kan man ikke indtaste 00 sekunder. Man kan godt indtaste
01, men ikke 00.
11. Ændres sponge og bathymetry opløsningen uden at filter opløsningen ændres fanges denne fejl
først når modellen forsøges kørt. Der kommer heller ikke rødt kryds, hvis man sletter filter
filen.
12. Undo kan ikke undo en interpolation.
13. Hvis man reducerer MIKEZero’s vinduesstørrelsen vertikalt kan man ikke nå de nederste
liggende elementer og der kommer ingen scrool.
14. Det virker som om programmet ikke tømmer hukommelsen før man afslutter hele MIKE 21.
15. Når man browser til en licens fil promter den a drevet og at der ikke er nogen disk i a drevet.
16. Installeres licens filen og datoen derefter rettes fortæller programmet, at licensen er udløbet.
Det virker også den anden vej, altså at man blot kan stille uret tilbage, hvorved licensen virker
igen.
17. Det er muligt under output at lave en dfs1 fil, hvor det skulle være en dsf0 fil. Filen med det
”forkerte” kan ikke åbnes men kommer blot med en fejl om at der er ”out of memory”.
18. Moving shoreline output er ”ligeglad” med om samplingsperioden overstiger simalations period.
I modtætning til deterministic output.
19. Når man har en toolbox åben kan man ikk tvinge den i baggrunden ved at klikke på et andet
vindue, men skal klikke oppe i baren for oven.
20. Under proporties tilføjes en ny række. Når man udfylder typen: Den skifter automatisk til
store bostaver (upper case), når den finder en match. Det er irriterende, hvis man vil skrive
noget der ligner, blot med små bogstaver.
21. Da jeg lavede et output fra alle punkter (0-600) med alle tidskridt og med elementerne valgt:
”water level”, ”still water depth” og ”surface elevation” og jeg åbnede filen kom flg problem:
legend viste, at jeg havde ”still water level” som vist item, mens jeg under view item kun
kunne vælge ”surface elevation” (som så i øvrigt ikke skiftede til det item når jeg trykkede).
Jeg havde tidligere haft samme fil åben, hvor der kun var ét item, nemlig ”surface elevation”.
Da jeg lukkede programmet og startede det igen forsvandt problemet.
22. Det ser ud til, at hvis man lave en bathymetri, hvor der er et stykke med konstant kote, som
ligger over swl laver det en underlig/forkert total water depth.
23. Under MIKEZero toolbox, TxStat: Når man specificerer et output-filnavn ved blot at skrive
ændringerne i tekstfeltet, og dernæst trykker på enter (i stedet for next) ændres filnavnet ikke.
24. Under MIKE 21 toolbox, regular wavegeneration: Når man skal specifiere outputfil viser den
ikke de filer, som der står i ”save as type”. Pr. default står der dfs0 filer, hvis man vil have
dem vist skal man skifte til all files.
25. Under WS Spectral Analisys: Har man specificeret input og ændrer spectrum kommer der ikke
”et rødt kryds” i boksen med item selection hvsi det er forkert (det røde kryds kommer hvis
man klikker lidt frem og tilbage).
26. Det ser umiddelbart ud til, at man ikke kan nøjes med at sample hvert fjerde tidsskridt ved
phase avarage. (hvis man skriver ”0-19200, 4” bliver filen også meget større end programmet
fortæller i boksen lige over.)
APPENDIX E. SUGGESTIONS TO IMPROVEMENTS OF MIKEZERO
120
27. I WS spectral analisys er decimaltals seperatoren ”,”, hvor det på engelsk (og i resten af M21
er ”.”)
28. Hvis man includerer moving shoreline (både i calibration og output): Hvis man derefter exkluderer moving shoreline i kalibreringen forsvinder punktet moving shoreline under output ikke.
Hvis man så trykker på exkluder under moving shoreline, output forsvinder punktet. Fejlen
er så, at nu kan man ikke få moving shoreline frem igen. Heller ikke ved inkludering under
kalibrering. Kun hvis man inkluderer i kalibreringen og dernæst justerer på number of output
areas under phase averaged output...
29. M21BW: Under Calibration, moving shoreline: trykker man F1 for at få hjælpevinduet frem,
kan man ikke længere redigere i talene. Det kan man igen, når hjælp-vinduet lukkes.
30. Hvis man har et andet program åbent ved siden af og man ønsker, at gøre MIKE21 aktiv kan
man ikke gøre det ved at klikke der hvor man indtaster parametre som ”type of equation”,
”bottom friction mm.” (gråt område til højre). Det gælder dog ikke for det grå område under
”Deterministic Output”, ”Phase-averaged Output”, ”Internal wave generation” og ”Boundary”
31. MIKE21: Phase avaraged output: Det er muligt at bruge et sidste tidskrift, der ligger efter
simulationsperioden er afsluttet til den kumulerede statistik
32. Under plot composer. Hvis man åbner et plot, der inkluderer filer, som man ”kommer til” at
slette kommer der en masse fejl. Tager man så egenskaber/properties for et plot, som ikke er
der og klikker ind på fanen y axes kommer der fejl. Man skal indtaste et tal, men det er stort
set umuligt at komme væk fra den fane. Også umuligt at cancle.
33. Det er under deterministic output muligt at skrive ud til to filer af samme navn. Det sker uden
der kommer fejlmeddelelse.
34. Plot composer: Hvis man opereret med flere plotvinduer i samme plot ”sejler” de rundt, når
man scroller op og ned og forsøger at markere dem.
35. Plot Composer: Mutate plot virker ikke, hvis man skal konvertere ADCP 2D Plot og man har
flere plot i samme plot fil. Jeg har ikke prøvet at konvertere andre plot.
36. Hvis man har plot composer åben, hvori der er en tidsserie fra det aktuelle modelsetup, samtidig
med model setup’en kommer der fejl i model setup’en. Når man så lukker plot composer
forsvinder fejlen ikke før man gemmer den fejlbehæftede outputfil igen.
37. Der er en trykfejl i hjælpefilen under wavebreaking -¿ roller form factor: ”...tangent of the slope
an dthe resulting....”.
38. Trykfejl i hjælpen under hjælp til moving shoreline -¿ smoothing parameter: ”In principle the
slot smooting....”
39. Har man et model setups åben med fokus/aktiv phase average og man trækker en ny/anden
model setup over, ændres feltet ”time” i alle output items’ne i den først åbne fil til den samme
værdi, som indtastet i det vindue der netop er hevet over.
40. Hvis man trækker en fil (drag and drop) og lader den falde på et vindue hvori der står en
hjælpe/introduktions tekst promptes det, om man vil gemme eller åbne filen.
41. Plot composer kan ikke opdatere periode informationen. Hvis man inkluderer en fil der, eksempelvis, varer 2 min. og man så ændrer i filen, således, at dsen varer 10 min, skal man først
slette filen fra plot composer og tilføje den igen før de sidste 8 min kommer med.
APPENDIX E. SUGGESTIONS TO IMPROVEMENTS OF MIKEZERO
121
42. WS bruger henviser til den komplette sti (C:\...\fil.dsf0) og ikke til det aktuelle bibliotek
(\fil.dsf0) som resten af M21.
43. WS: Hvis man laver et fft setup af en fil og senere ændrer antallet af punkter i den analyserede
fil kan man ikke køre WS analyse, men skal importere filen igen.
44. Plot Composer: Hvis man under properties skifter til/fra logaritmisk y akse kan man ikke bruge
”apply” knappen, men skal klikke OK for at se ændringerne.
45. Plot Composer: Hvis man laver et plot af en tidsserie og ved et uheld kommer til at tilføje
en fil, som på x-aksen har større udbredelse bliver det oprindelige plot skaleret. Man kan ikke
nøjes med at plotte et lille område af x aksen, ej heller kan man ved at fjerne filen igen komme
tilbage til den ”gamle” skalering.
46. Plot Composer: når man redigerer tidsranget i en tidsserie, kan man ikke skrive ”0” i felterne
med klokken (vha. tastaturet, men skal skrive ”1” og så bruge piletasterne eller musen for at
komme ned på tiden 0.
47. Der er stavefejl under BW, ”wave breaking”, første box: ”roler form factor” mangler et ”l” i
roller.
E.2
Forslag
1. Zoom ind skal ikke blive fravalgt når man har zoomet én gang.
2. Det skal være muligt at putte flere snit i én fil under output i en bw model.
3. Når der vælges properties under profile series står number of grid point og number of time
steps ikke på samme række, hvilket er en smulle ulogisk.
4. Still water level er negativt. Det synes jeg er ulogisk.
5. I toolboxen burde den gule box have samme funktion som +’et. Dvs folde dir’et ud.
6. Create porosity map: Det ville være smart hvis der, under parameter settings, var default
indstillinger, der svarede til den bathymetry der var valgt.
7. Under profile editor kan man ved en vis størrelse opløsning kun klikke på nogen af punkterne.
Det ville være rart, hvis curseren blev flyttet til ca. det område, hvor der er klikket, som det
sker ved en lav opløsning.
8. Under tollbox kan man ikke dobbeltklikke på et ”gult” element for at folde underelementerne
ud.
9. Det ville være rart med en mere beskrivende fejl ved blow up. Der står blot fatal error. Man
skal kigge i logfilen for at finde ud af hvad der er galt.
10. Ved porosity map generation er hjælpe vinduet forrest hele tiden. Det er en smule irriterende.
11. Når man har valgt at vise enkelte items og laver en beregning vha. calculator vises alle items’ene
der ellers var fravalgt.
12. Under toolbox ville det være lækkert at kunne kopiere et element. Eksempelvis hvis man vil
have 3 forskellige størrelser bølger skal man blot kopiere, ændre filnavn og H.
13. Der burde være mulighed for at gemme sine outputfiler som ASCII.
APPENDIX E. SUGGESTIONS TO IMPROVEMENTS OF MIKEZERO
122
14. Der burde være muligehed for at vælge en bestemt række/celle i calculatoren. Eksempelvis
i3(n)-i3(n-1) ville give en tilnærmet surface slope.
15. Der mangler noget om import af datafiler (.dsf0 filer) fra andre programmer som eksempelvis
matlab.
16. Den mørkeblå baggrund i hjælpen gør det lidt svært at læse linksene.
17. I eksempelvis en bathymetri fil kan kun nogle af prikkerne vælges klik med musen. Det kan
godt virke en smule mærkeligt og irriterende.
18. Hvis man bruger ”tryk-F1-hjælpen” er det en smule irriterende, at man ikke kan få hjælpen
om bag MIKEZero vinduet ved blot at klikke på MIKEZero vinduet.
19. MIKE 21 1D: phase avaraged output: Når man bruger tab for at komme fra tekstboks til
tekstboks kommer man ned i input filerne, hvor den kører i ring. Det er irriterende, hvis man
vil tilbae og specificere noget.
20. Det er en smule irriterende, at man, når man redigerer i et tool (ex skal lave en bølge til input
i modellen), at man så ikke kan benytte resten af M21.
21. Når Plot composer: Når man ved et plot går ind og tager properties, sletter det gamle plot og
tilføjer et andet kommer der en ny farve på kurven. Det ville være rart, hvis den kunne vælge
den samme farve som der var til det netop slettede plot.
22. Det ville være rart, hvis man kunne slå sit output fra og så indkludere dem igen uden at skulle
indtaste dem alle forfra.
23. Det er en smule irriterende, at hvis man har MIKE 21 åben og dobbeltklikker på en tilfældig
dfs eller anden M21 fil åbner en ny MIKE21. Den nye fil benytter ikke blot den allerede åbne
MIKE21.
24. Det er ikke så pænt, at når man har maksimeret vinduerne (eksempelvis setuppet) og trækker
en gemt toolbox over bliver den også maksimeret, hvilket ikke ser så pænt ud.
25. Jeg synes plot composer mangler et godt zoom værktøj. Magnifikation er ikke så god at arbejde
med.
26. Plot composer: Det ville være rart, hvis man kunne rette i stien på de objekter man indsætter.
Således vill plotcomposer holde på information om linie tykkelser, markør osv.
27. Det ville være lækkert, hvis WS spectral an. kunne udregne og udskrive den signifikante
bølgehøjde.
28. M21BW: Det ville være en god idé, at sammen med .log filen at lave en fil, som indeholder alle
parametrene brug, eks. slot width, smotthing factor mm.
29. Det ville være rart med en knap til at opdatere plot composer. Hvis man ændrer i filer mens
man har et plot åbent skal man først lukke plot filen og åbne den igen før man kan se øndringen.
30. Af pladshensyn ville det være godt at indføre eksponentiel notation til akserne i plotcomposer.
Især hvis man skifter til logaritme akser kommer y aksens værdier til at fuylde ret meget.
31. Under plot composer: Man burde kunne ”trække” et tomt plot vindue blot ved at ”trække” et
tilfældigt sted. Som det er nu skal man have fat i kanten for at flytte plottet (også selv om det
er tomt)
APPENDIX E. SUGGESTIONS TO IMPROVEMENTS OF MIKEZERO
123
32. Det ville være en god idé, at have mulighed for at have vinduet splittet op i to. I den ene skulle
det normale MIKEZero være og i det andet filerne i det bibliotek man arbejder i. I stil med
MatLab løsning, med ’command window’ til højre og ’current directory’ til venstre.
Er der spørgsmål til ovenstående kan jeg kontaktes på:
Hans Jacob Simonsen
39 27 26 93 - 26 22 26 93
[email protected]
Appendix F
Experiments in the near field in front
of a piston wavemaker and in the surf
zone
General for all: The files containing the time series are in the format .dfs0 which can be opened by
MIKEZero software. The files named *log1.dfs0 contain time series sampled from the wave gauges
placed according to the respectable tables. The sample rate is 40 Hz. The files named *dsc.dfs0 are
the time series sampled on the wavemaker. In the dsc files ”SP” is the specified value and ”PV” is
the processed value. The sample rate of the dsc measurements are 20 Hz. Not every log1 file has a
corresponding dsc file.
To every file there is a *.log file containing different specified parameters of the run e.g. specified
wave height, period among others. The files ”calibration.cal” contain the calibration constants used
in the sampling of the time series.
The experiments were split up in experiments investigating the near field near field near the
piston wavemaker and experiments investigating the surf zone.
Near field near the piston wavemaker: Two run with different positions of the wave gauges were
carried out. The sampling of the time series from the wave gauges (the *log1.dfs0 files) start when
the wavemaker is fully gained. The sampling of the time series from the wave gauge mounted on
the wavemaker (the *dsc.dfs0 files) start 5 seconds before sampling of the time series from the wave
gauges and are therefore including the warmup period. The wavemaker signal are in all cases of
evanescent modes experiments sinusoidal.
Surf zone Two run with different positions of the wave gauges were carried out. The sampling
of the time series from the wave gauges (the *log1.dfs0 files) start 30 seconds after the wavemaker is
fully gained. The sampling of the time series from the wave gauge mounted on the wavemaker (the
*dsc.dfs0 files) start 5 seconds before sampling of the time series from the wave gauges.
Description of the experiments please refer to chapter 2.
124
APPENDIX F. EXPERIMENTS PERFORMED DURING THE PROJECT
125
Table F.1: Position of wave gauges in the first run of experiments near the wavemaker.
Channel Distances from the wavemaker, [m]
1
0.498
2
0.661
3
0.855
4
1.27
5
1.73
6
2.33
7
4.02
8
Runup gauges
Table F.2: Experiments investigating the near field near the wavemaker using sinusoidal wavemaker
signal, run 1. Depth at the wavemaker is h = 0.525 m. The wave gauges was places according to
table F.1
Filename
T [s] H [m]
kh
0001log1.dsf0 0.900 0.012 2.633
0002log1.dsf0 1.200 0.022 1.592
0003log1.dsf0 1.500 0.035 1.148
0004log1.dsf0 0.500 0.011 8.442
0005log1.dsf0 0.600 0.015 5.863
0006log1.dsf0 0.700 0.021 4.309
0007log1.dsf0 0.900 0.035 2.633
0008log1.dsf0 1.200 0.060 1.592
0009log1.dsf0 1.500 0.100 1.148
0010log1.dsf0 0.500 0.018 8.442
0011log1.dsf0 0.600 0.026 5.863
0012log1.dsf0 0.700 0.035 4.309
0013log1.dsf0 0.900 0.060 2.633
0014log1.dsf0 1.200 0.105 1.592
0015log1.dsf0 1.500 0.168 1.148
0016log1.dsf0 0.500 0.036 8.442
0017log1.dsf0 0.600 0.054 5.863
0018log1.dsf0 0.700 0.073 4.309
0019log1.dsf0 0.900 0.122 2.633
0020log1.dsf0 1.200 0.210 1.592
APPENDIX F. EXPERIMENTS PERFORMED DURING THE PROJECT
126
Table F.3: Position of wave gauges in the second run of experiments near the wavemaker.
Channel Distances from the wavemaker, [m]
1
0.14
2
0.22
3
0.32
4
0.44
5
0.59
6
0.73
7
0.92
8
1.20
9
1.37
10
1.56
11
1.73
12
2.56
14
runup
Table F.4: Experiments investigating the near field near the wavemaker using sinusoidal wavemaker
signal, run 2. Depth at the wavemaker is h = 0.52 m. The wave gauges was places according to table
F.3
Filename
T [s] H [m]
kh
0001log1.dsf0 0.900 0.012 2.609
0002log1.dsf0 1.200 0.022 1.580
0003log1.dsf0 1.500 0.035 1.141
0004log1.dsf0 0.500 0.011 8.362
0005log1.dsf0 0.600 0.015 5.807
0006log1.dsf0 0.700 0.021 4.268
0007log1.dsf0 0.900 0.035 2.609
0008log1.dsf0 1.200 0.060 1.580
0009log1.dsf0 1.500 0.100 1.141
0010log1.dsf0 0.500 0.018 8.362
0011log1.dsf0 0.600 0.026 5.807
0012log1.dsf0 0.700 0.036 4.268
0013log1.dsf0 0.900 0.060 2.609
0014log1.dsf0 1.200 0.105 1.580
0015log1.dsf0 1.500 0.168 1.141
0016log1.dsf0 0.500 0.036 8.362
0017log1.dsf0 0.600 0.054 5.807
0018log1.dsf0 0.700 0.073 4.268
0019log1.dsf0 0.900 0.122 2.609
APPENDIX F. EXPERIMENTS PERFORMED DURING THE PROJECT
Table F.5: Position of wave
Channel
1
2
3
4
5
6
7
8
9
10
11
12
13
14
127
gauges in the first run of experiments in the surf zone.
Distances from the wavemaker, [m]
02.30
09.68
14.99
15.40
15.80
16.20
16.60
17.02
17.40
17.83
18.21
18.60
19.05
runup
Table F.6: Experiments investigating the surf zone using sinusoidal wavemaker signal, run 1. Depth
at the wavemaker is h = 0.55 m. The wave gauges was places according to table F.5.
Filename
T [s] H [m]
kh
0001log1.dsf0 1.500 0.080 1.185
0002log1.dsf0 1.500 0.080 1.185
0003log1.dsf0 2.000 0.080 0.819
0004log1.dsf0 1.000 0.100 2.260
0005log1.dsf0 2.000 0.100 0.819
0006log1.dsf0 1.500 0.120 1.185
0007log1.dsf0 2.000 0.130 0.819
0008log1.dsf0 1.500 0.150 1.185
0009log1.dsf0 1.500 0.170 1.185
Table F.7: Experiments investigating the surf zone using irregular JONSWAP wavemaker signal, run
1. Depth at the wavemaker is h = 0.55 m. The wave gauges was places according to table F.5. Water
depth at the experiment marked with ’*’ is h = 0.535 m.
Filename
T [s] H [m] γ
kh
1001log1.dsf0
1.5
0.03
1 1.185
1002log1.dsf0
1.5
0.03 3.3 1.185
1003log1.dsf0
1.5
0.03
10 1.185
1004log1.dsf0
1.5
0.08
1 1.185
1005log1.dsf0
1.5
0.08 3.3 1.185
1006log1.dsf0
1.5
0.08
10 1.185
1007log1.dsf0
1.5
0.12
1 1.185
1008log1.dsf0
1.5
0.12 3.3 1.185
1009log1.dsf0
1.5
0.12
10 1.185
1010log1.dsf0* 2.0
0.10
10 0.806
APPENDIX F. EXPERIMENTS PERFORMED DURING THE PROJECT
128
Table F.8: Experiments investigating the surf zone using cnoidal wavemaker signal, run 1. Depth at
the wavemaker is h = 0.55 m. The wave gauges was places according to table F.5.
Filename
T [s] H [m]
kh
2001log1.dsf0 2.500 0.150 0.632
2002log1.dsf0 2.100 0.050 0.773
2003log1.dsf0 2.100 0.100 0.773
2004log1.dsf0 3.000 0.150 0.517
2005log1.dsf0 5.000 0.050 0.302
Table F.9: Experiments investigating the surf zone using solitary wavemaker signal, run 1. Depth at
the wavemaker is h = 0.535 m. The wave gauges was places according to table F.5. Notice that the
experiment with the specified H of 0.05 m the wave spilled over the top of the wave flume.
Filename
H [m]
3001log1.dsf0 0.01
3002log1.dsf0 0.03
3003log1.dsf0 0.05
3004log1.dsf0 0.04
Table F.10: Position of wave gauges in the second run of experiments in the surf zone.
Channel Distances to the wavemaker, [m]
1
02.22
2
09.68
3
17.06
4
17.29
5
17.51
6
17.70
7
17.93
8
18.09
9
18.30
10
18.45
11
18.65
12
18.85
13
18.98
14
runup
Table F.11: Experiments investigating the surf zone using sinusoidal wavemaker signal, run 2. Depth
at the wavemaker is h = 0.55 m. The wave gauges was places according to table F.10.
Filename
T [s] H [m]
kh
0001log1.dfs0 1.5
0.030 1.185
0002log1.dfs0 1.0
0.040 2.260
0003log1.dfs0 3.0
0.050 0.517
0004log1.dfs0 1.0
0.070 2.260
0005log1.dfs0 2.0
0.050 0.819
0006log1.dfs0 3.0
0.080 0.517
0008log1.dfs0 1.5
0.080 1.185
0009log1.dfs0 2.0
0.080 0.819
APPENDIX F. EXPERIMENTS PERFORMED DURING THE PROJECT
129
Table F.12: Experiments investigating the surf zone using cnoidal wavemaker signal, run 2. Depth
at the wavemaker is h = 0.55 m. The wave gauges was places according to table F.10.
Filename
T [s] H [m]
kh
2001log1.dfs0 1.5
0.03 1.185
2002log1.dfs0
5
0.03 0.302
F.1
All experiments performed
Table F.13 shows all experiments performed by Anders Wedel Nielsen and Hans Jacob Simonsen in
the spring of 2002. The wave heights (for irregular this corresponds to significant wave height) and
periods are listed. To find the filename, please go to previous section
In
Si
Ir
Cn
So
l
s
1
2
table F.13 following notation has been used:
: Sinusoidal wavemaker signal.
: Irregular wavemaker signal.
: Cnoidal wavemaker signal.
: Solitary wavemaker signal.
: Wave gauges placed in the local field near the wavemaker
: Wave gauges placed in the surf zone
: Run 1.
: Run 2.
0.5
0.6
0.7
0.9
1.0
1.2
1.5
2.0
2.1
2.5
3.0
5.0
0.01
0.011
Si,l,1&2
0.012
Si,l,1&2
Si,l,1&2
Si,l,1&2
0.021
Si,l,1&2
0.022
Si,l,1&2
0.026
Si,l,1&2
0.03
Si,s,2+Ir,s,1+Cn,s,2
0.035
0.036
Si,l,1
Si,l,1&2
Si,l,1&2
So,s,1
Si,s,2
Cn,s,1
Si,s,2
Si,l,1&2
Si,l,1&2
0.07
Si,l,1&2
Si,s,2
Si,l,1&2
0.08
0.10
Si,s,1
0.105
Si,s,1&2 + Ir,s,1
Si,s,1&2
Si,l,1&2
Si,s,1+Ir,s,1
Si,s,2
Cn,s,1
Si,l,1&2
0.120
Si,s,1+Ir,s,1
Si,l,1&2
0.130
Si,s,1
0.150
Si,s,1
0.168
Si,l,1&2
0.170
0.210
So,s,1
Si,s,2
0.06
0.122
Cn,s,1
Si,l,1&2
0.05
0.073
So,s,1
Si,l,2
0.04
0.054
Cn,s,2
Cn,s,1
Table F.13: Overview of all experiments carried out.
0.015
0.018
∞
So,s,1
APPENDIX F. EXPERIMENTS PERFORMED DURING THE PROJECT
H/T
Cn,s,1
Si,s,1
Si,l,1
130
APPENDIX F. EXPERIMENTS PERFORMED DURING THE PROJECT
F.2
131
CD-ROM containing all the timeseries
The CD-ROM attached contains all the time series as described in previous section. Requisition of
CD-ROM can be done by sending email to [email protected].