Numerical validation of the mild-slope wave propagation model
MILDwave, using test cases from literature
Jonas Fahy
Promotor: prof. dr. ir. Peter Troch
Supervisor: ir. Vicky Stratigaki
Masterproef ingediend tot het behalen van de academische graad van Master in de
ingenieurswetenschappen: bouwkunde
Vakgroep Civiele Techniek
Voorzitter: prof. dr. ir. Peter Troch
Faculteit ingenieurswetenschappen
Academiejaar 2012-2013
De auteur geeft de toelating deze masterproef voor consultatie beschikbaar te stellen en
delen van de masterproef te kopiëren voor persoonlijk gebruik. Elk ander gebruik valt
onder de beperkingen van auteursrecht, in het bijzonder met betrekking tot de
verplichting de bron uitdrukkelijk te vermelden bij het aanhalen van resultaten uit deze
masterproef.
The author grants permission to make this master dissertation available for consultation
and to copy parts of this master dissertation for personal use. Any other use is subject to
the limitations of the copyright, in particular with regard to the obligation of referencing
explicitly to this thesis when quoting obtained results.
Jonas Fahy Ghent, June, 2013
Voorwoord
Laat dit afstudeerwerk niet zozeer de sluitsteen van een opleiding zijn, maar eerder een
hoeksteen van de kathedraal aan kennis die ik nog wens te vergaren. Het fundament van
mijn opleiding werd gegoten in de lagere school Paus Johannes College in Merelbeke, zij
werd vervolgd op het Sint-Lievenscollege in Gent en afgesloten bij de vakgroep Weg- en
Waterbouwkunde aan de Universiteit Gent. Ik wens dan ook mijn dank uit te drukken
aan alle onderwijzend personeel van deze instellingen.
In het bijzonder ook dank aan ir. Vicky Stratigaki die de begeleiding van dit werk op zich
nam. Zij stond steeds open voor vragen en voorzag de auteur van waardevolle feedback
en bruikbare informatie, zowel op wetenschappelijk als tekstueel vlak.
Dank ook aan mijn promotor, prof. dr. ir. Peter Troch, om mij de kans te geven inzicht te
verwerven in golftransformatieprocessen en mij gebruik te laten maken van het in huis
ontwikkelde numerieke golfvoortplantingsmodel MILDwave. Zijn positieve en
stimulerende attitude waren van grote waarde bij de totstandkoming van dit werk.
Tenslotte dank ik ook de mensen uit mijn omgeving, familie en vrienden, voor hun niet
aflatende steun en vertrouwen.
Summary
In the present work, the numerical wave propagation model MILDwave is validated
using four different test cases.
In Chapter 1, an overview of the different numerical models for wave propagation is
provided. In this context, the numerical wave propagation model MILDwave is situated.
The underlying mathematics are explained and an overview of the user interfaces for the
MILDwave Preprocessor and MILDwave Calculator is provided.
In Chapter 2, wave propagation over a submerged, spherical island is considered. The
simulation is performed three times to learn weather grid cell size has an influence on the
resulting values of the distrubrance coefficient Kd. Resulting values of the disturbance
coefficient Kd are analyzed along three sections in the vicinity of the submerged island. A
comparison is made between the results of the MILDwave simulations and the
experimental data provided by Ito (1972) by using two sets of parameters for evaluation
of model performance, as suggested by Dingemans (1997). Evaluation of these sets of
parameters is in accordance with graphical representations of resulting Kd values and
show a good accordance between the numerical results of MILDwave simulations and
the experimental data provided by Ito (1972). Moreover, it is observed that the grid cell
size has only very small influence on the numerical results provided by MILDwave.
Small grid cell size result in slightly better agreement with experimental data by Ito
(1972) but also requires a far greater computation time. In conclusion, MILDwave is able
to simulate wave propagation over a submerged island very accurately.
In Chapter 3, a second test case involving the phenomenon of resonance in a rectangular
harbour is studied. This resonance can occur when waves with a wave period equal to an
eigen period of an enclosed basin enter this basin. A rectangular harbour is implemented
in MILDwave and 63 simulations for different periods are performed. A measure for the
resonance, the amplification factor R, is defined at the center of the rear wall of the
harbour and R is plotted as a function of the parameter kl, with k the wave number and l
the length of the harbour. As predicted by Raichlen (1966), two resonance peaks are
observed in the range of wave periods that is considered. Also, the numerical results of
MILDwave show good accordance with experimental data of Ippen and Goda (1963) and
the theoretical solution and experimental data by Lee (1969). Parameters for model
performance are calculated and confirm that MILDwave is able to accurately describe
harbour resonance in a rectangular harbour.
In Chapter 4, wave deformation in the surf zone is studied. Three types of beaches are
considered: a uniform slope beach, a step type beach and a bar type beach. Results of
wave deformation and wave breaking obtained with MILDwave simulations are
compared to experimental data points by Nagayama (1983) and a numerical simulation
by Watanabe (1988). For the uniform slope beach, relatively good agreement is achieved,
for the step and bar type beaches, non linear effects are observed and agreement is less
satisfactory. As an attempt to achieve better results with MILDwave, in a next step the
influence of wave breaking parameters for the MILDwave wave breaking module are
studied. A relationship between two parameters and physical characteristics is observed.
Using this knowledge the three beach types are revisited with different wave breaking
parameters and the results are compared to a theoretical result by Goda (2010) and to a
SwanOne simulation.
In Chapter 5, the influence of different parameters on transmission of waves through
partially reflecting structures is studied. Some recommendations are formulated. Using
the obtained insights, diffraction diagrams for waves passing through partially reflective,
semi-infinite breakwaters are created for waves propagating perpendicular to the
breakwater. The diffraction diagram obtained for an impermeable semi-infinite
breakwater are compared to the theoretical solution provided by Wiegel (1962).
Table of Contents
CHAPTER 1 - Numerical models for simulation of water wave propagation
1.1 Introduction 1-2
1.2 Phase-averaged models 1-2
1.3 Phase-resolving models 1-3
1.3.1 Boundary Integral models 1-3
1.3.2 Mild Slope Equation models 1-3
1.3.3 Boussinesq Equation models 1-4
1.4 The Mild Slope Equation model MILDwave 1-4
1.4.1 The solution scheme of MILDwave 1-5
1.4.2 Wave generation in MILDwave 1-7
1.4.2.1 Regular wave generation 1-7
1.4.2.2 Irregular wave generation 1-8
1.4.3 Numerical domain boundaries 1-8
1.4.4 Wave breaking in MILDwave 1-9
1.4.5 The MILDwave user interface 1-10
1.4.5.1 The MILDwave preprocessor 1-10
1.4.5.2 The MILDwave Calculator 1-15
1.4.5.3 The MILDwave calculator output files 1-15
CHAPTER 2 - Test case: waves travelling over a submerged island
2.1 General description of the experiment 2-2
2.2 Aim of the experiment 2-2
2.3 Simulation process and input 2-2
2.3.1 Bathymetry and grids 2-2
2.3.2 Wave conditions and MILDwave parameters for the wave propagation
experiment over a submerged island 2-4
2.4 Results of the Kd value throughout the domain, provided by MILDwave 2-5
2.4.1 Plan view plots of the Kd value throughout the domain 2-5
2.4.2 Section plots of the Kd value through three sections near the submerged
island 2-6
2.5 Quantitative analysis of the results using parameters for model performance
2-9
2.6 Conclusions 2-11
CHAPTER 3 - Test case: resonance in a rectangular harbour
3.1 Introduction and aim of the experiment 3-2
3.2 General description of the experiment 3-2
3.3 Simulation process and input 3-2
3.3.1 Bathymetry and grids 3-2
3.3.2 Wave conditions and MILDwave parameters 3-4
3.4 MILDwave results of the Kd-value throughout the domain and the
amplification factor R at the centre of the harbour rear wall 3-6
3.4.1 Plan view plots of the Kd-value throughout the effective domain 3-6
3.4.2 Section plots of the R value along the vertical symmetry axis of the domain
3-8
3.5 Analysis of the results and conclusions 3-9
CHAPTER 4 - Test case: wave deformation in the surf zone
4.1 General description of the experiment 4-2
4.1.1 Aim of the experiment 4-2
4.2 Simulation process and numerical input data 4-2
4.2.1 Bathymetry and numerical domains (I will add an additional figure of the
entire layout in the simulation, including sponge layers) 4-2
4.2.2 Wave conditions and MILDwave input parameters 4-4
4.3 Results and discussion 4-5
4.3.1 Potential energy density 4-5
4.3.2 Uniform slope beach 4-6
4.3.2.1 Section plots of the wave height and potential energy density for the
uniform slope beach 4-6
4.3.2.2 Analysis of the results for the uniform slope beach 4-6
4.3.3 Step-type beach 4-7
4.3.3.1 Section plots of the wave height and potential energy density for the
step-type beach 4-7
4.3.3.2 Analysis of the results for the step-type beach 4-8
4.3.4 Bar-type beach 4-9
4.3.4.1 Section plots of the wave height and potential energy density for the
bar-type beach 4-9
4.3.4.2 Analysis of the results for the bar-type beach 4-10
4.3.5 Conclusions 4-11
4.4 Influence of breaking coefficient K1, K2, K3 and K4 4-12
4.5 Comparison between MILDwave simulations and other models 4-15
4.5.1 MILDwave numerical results, Goda (2010) theoretical results and SwanOne
numerical results for the uniform slope beach 4-15
4.5.2 MILDwave numerical results, Goda (2010) theoretical results and SwanOne
numerical results for the step type beach 4-16
4.5.3 MILDwave numerical results, Goda (2010) theoretical results and SwanOne
numerical results for the bar type beach 4-17
CHAPTER 5 - Wave transmission and diffraction through a semi-infinite breakwater
5.1 General description of the experiment 5-2
5.2 Study on the influence of numerical basin width, sponge layers, wave period T,
time step Δt, MILDwave transmission coefficient S and width of the breakwater on the
transmission in MILDwave 5-2
5.2.1 Influence of the numerical basin width and sponge layers on the wave
transmission in MILDwave 5-2
5.2.1.1 Setup of the experiments and calculation of MILDwave input
parameters 5-2
5.2.1.2 Results: influence of the basin width and side sponge layers on the Kd
value. 5-4
5.2.2 Influence of the number of cells in the permeable structure on the Kd value
5-5
5.2.2.1 Setup of the experiment and calculation of MILDwave input parameters
5-6
5.2.2.2 Results: influence of the number of cells inside the permeable structure
on the Kd value 5-6
5.2.3 Influence of the wave period T and time step Δt on the transmission of a
permeable structure with fixed absorption coefficient S 5-7
5.2.3.1 Setup of the experiment and calculation of MILDwave input parameters
5-7
5.2.3.2 Results: influence of the time step and wave period on the wave
transmission behind the breakwater 5-8
5.2.4 Concluding recommendations for the implementation of permeable
structures in MILDwave 5-11
5.3 Diffraction diagrams for semi-infinite breakwaters with partial transmission of
wave energy 5-11
5.3.1 Bathymetry setup and MILDwave parameters 5-11
5.3.2 Results: graphical representation of Kd value behind a semi-infinite
breakwater with partial transmission of wave energy 5-13
A.2 MATLAB code for preparation of the .bmp file of a spherical island that is used
in the test case of waves propagating over a submerged island 5-11
APPENDIX
List of symbols
a [m] wave amplitude
Bs [grid cells] the number of grid cells along the length of the sponge layer
C [m/s] phase velocity
Cg [m/s] group velocity
Cθ [m/s] wave propagation velocity in spectral (σ, θ) space
Cσ [m/s] wave propagation velocity in spectral (σ, θ) space
DB [J/s] dissipation term for depth-induced wave breaking in hyperbolic
mild-slope equation
d [m] water depth
dmax [m] maximum water depth in a basin with variable water depth
d(2) [-] index of agreement
d(1) [-] modified index of agreement
E [J] total energy
E(f, θ) [m²s/rad] wave density spectrum
g [m/s²] gravitational acceleration (=9.81 m/s²)
Hb [m] maximum wave height that can exist at a given depth before wave
breaking occurs
Hr [m] the mean resulting wave height as calculated by MILDwave
Hg [m] the generated wave height in numerical simulations
Hrms [m] the root mean square value of the wave height
Hrms0 [m] the root mean square value of the wave height in deep water
Hs [m] significant wave height
HSGB [m] the incident significant wave height at the wave generation
boundary
Hw [m] wave height of regular wave
K1 [-] parameter in the expression of Hb
K2 [-] parameter in the expression of Hb
K3 [-] parameter in the expression of Hb
K4 [-] parameter in the expression of Hb
Kd [-] disturbance coefficient
k [rad/m] wave number
= 2π/L
L [m] wave length
L0p [m] the wave length in deep water for a wave with period Tp
N [m²s²/rad²] action density
Nas [grid cells] number of absorbing grid cells along the length of the partially
permeable structure
Nx [-] number of cells in x-direction
Ny [-] number of cells in y-direction
S [-] the MILDwave absorption coefficient
Stot [m²/rad] total input term for all dissipation phenomena in action-balance
equation
Tp [s] peak wave period
Ts [s] significant wave period
T [s] wave period of regular wave
tb [s] warm-up time to establish the wave conditions in the basin and
start calculation
twfin [s] total simulation time
te [s] the end of the calculation time in a simulation
u [m/s] the x component of the velocity vector
[m/s] the depth averaged horizontal velocity
v [m/s] the y component of the velocity vector
w [m/s] the z component of the velocity vector
x [m] direction perpendicular to the wave propagation direction
y [m] wave propagation direction
Δt [s] time step
Δx [m] grid cell size in x-direction
Δy [m] grid cell size in y-direction
α [-] proportionality factor in the Baldock formula for DB
η [m] surface elevation
η* [m] additional surface elevation
ϕ [m²/s] velocity potential at SWL
ρw [kg/m³] water density
σ [rad/s] relative radian frequency of wave
θ [°] angle between the direction of the waves and the wave generation
line
ω [rad/s] absolute radian frequency of wave
= 2πf
[-] horizontal gradient operator
1 Numerical models for simulation of
water wave propagation
CHAPTER 1 - Numerical models for simulation of water wave propagation 1-2
1.1 Introduction
These days, scientists and engineers can use the computational power to predict and
describe waves in the open ocean, or to obtain the distribution of waves in harbor and
coastal areas. There are many numerical models available on the market, and this makes
it beneficial to know which model is suited for which task. The right choice of model can
save both time and energy, and avoid unnecessary computing time.
Fundamentally, the way in classifying these models is based on the rate of spatial
evolution of the wave field (Boshek, 2009). Two families of models exist; the phase-
resolving and the phase-averaged models. The difference between these models will be
explained below.
1.2 Phase-averaged models
Phase-averaged models deal with slowly changing waves (Boshek, 2009). Variations in
wave characteristics and bathymetry vary weakly over the scale of a wavelength L. These
models deal with averaged properties over a spectrum. The wave spectrum at a certain
location describes the average sea state in a finite area around that location. The models
predicts averaged properties such as the significant wave height Hs, group velocity Cg,
wave number k, the average wave period T and the spectrum at a certain location in the
ocean. Therefore, phase-averaged models take into account effects of wave generation by
wind, wave-wave interaction and dissipation. The basic equation is the spectral action
balance equation provided by Mei (1983) (1.1):
(1.1)
Where
, the action density with the spectrum which distributes the
wave energy over frequencies f and directions θ and the relative radian frequency. The
second term denotes the propagation of wave energy in two-dimensional geographical -
space, with the group velocity
following from the dispersion relation
, where is the wave number vector and d the water depth. The third term
represents the effect of shifting of the radian frequency due to variations in depth and
mean currents. The fourth term represents depth-induced and current-induced refraction.
The quantities and are the propagation velocities in spectral space (σ,θ). The right
hand side of Equation (1.1) contains the source term , which represents all physical
processes which generate, dissipate or redistribute wave energy, such as wave generation
by wind energy input, non-linear interaction between different wave frequencies in a
CHAPTER 1 - Numerical models for simulation of water wave propagation 1-3
spectrum resulting in energy transfer to another component and energy loss in
dissipation processes like breaking and white-capping. (SWAN (2006)
The phase-averaged model is very well applicable to wind driven waves with a large
directional spreading. However, this model cannot accurately represent regular or
unidirectional waves since is based on spectral wave action.
1.3 Phase-resolving models
Contrary to phase-averaged models, phase-resolving models are able to accurately model
local properties which very strongly within a small scale of wavelength L. These models
are used when average waves properties and bathymetry change rapidly. The models
consist of equations describing the immediate state of motion, either the time domain or
in the frequency domain. Phase-resolving models are much more computational
intensive than phase-averaged models and thus are only considered necessary in the near
field of wave-structure interaction. The phase resolving models can be further classified
by their foundation equations: boundary integral, mild-slope equations and Boussinesq.
1.3.1 Boundary Integral models
The boundary integral models do not involve any assumptions for wave conditions or
site conditions. They solve the Laplace equation (1.2):
(1.2)
With is the scalar velocity potential, which is defined such that
, with u, v and w respectively the x, y and z component of the
velocity vector.
This type of model is good for irrotationality dominance cases such as breaking, but bad
for viscosity dominance situations, as well as for wave-structure interaction.
1.3.2 Mild Slope Equation models
In the Mild Slope Equation models, the assumption is that the sea bad slope is very much
less than kd, with k the wave number and d the water depth. Also, it is assumed that the
waves are only weakly nonlinear. The underlying equation of these models was
originally presented by Berkhoff (1972) as Eq. (1.3):
(1.3)
with the gradient operator, C the phase speed of the waves, the group velocity of the
waves, k the wavenumber and the velocity potential.
CHAPTER 1 - Numerical models for simulation of water wave propagation 1-4
This formulation can cater to effects of wave shoaling, wave refraction, wave diffraction
and wave reflection. For computational efficiency, a parabolic version of the original
elliptic equation is often used. Over the full numerical domain, boundary conditions are
required. The parabolic equation can be extended to include current, non-linear
dispersion, dissipation and wind input.
The numerical model MILDwave, which will be validated in the present work, is a model
of the Mild Slope Equation type, and uses the parabolic equation extended with a wave
breaking module. Other software using the Mild Slope Equation are ARTEMIS
(Aelbraecht, 1997) and RDE (Maa, 1998)
1.3.3 Boussinesq Equation models
These models assume that the bed slope and are very small compared to unity and
that the waves are weakly non-linear. This model includes refraction, diffraction,
shoaling, reflection, wave-current interaction effects, dissipation and wind input.
The underlying formulation was given by Peregrine (1967) as equations (1.4) and (1.5):
(1.4)
(1.5)
With the water elevation, d the water depth, the depth averaged horizontal velocity
and g the gravitational acceleration.
1.4 The Mild Slope Equation model MILDwave
As mentioned above, the numerical model MILDwave is a Mild Slope wave propagation
model which uses a parabolic version of the Mild Slope Equation (1.3). These parabolic
equations are developed by Radder and Dingemans (1985) and describe the
transformation of linear irregular waves with a narrow frequency band over a mildly
varying bathymetry with bed steepness up to 1/3. These equations are (1.6) and (1.7)
(1.6)
(1.7)
With
(1.8)
(1.9)
CHAPTER 1 - Numerical models for simulation of water wave propagation 1-5
The overbar denotes that the wave characteristic is calculated for the carrier
frequency.
MILDwave is able to simulate linear water waves by calculating instantaneous surface
elevations throughout the domain. Wave transformation processes such as refraction,
shoaling, reflection, transmission and diffraction are simulated intrinsically. MILDwave
can generate regular and irregular long- and short crested waves, as well as radiated
waves. Furthermore, wind effects and wave breaking can be introduced in MILDwave
simulations.
Since MILDwave is a phase-resolving model, typical applications are wave penetration in
harbours, the behaviour of wave energy converters or wave transformation studies along
coastlines. The model is used in the coastal engineering research group at Ghent
University for research and educational purposes, and is the official wave propagation
model used by the Flemish government for calculating and modelling wave penetration
into several Belgian coastal harbours, such as Zeebrugge and Ostend.
MILDwave is developed by Troch (1998) and is written in C++. It is easily operated using
two executable files in a user friendly interface. The MILDwave Preprocessor is used for
the preparation of the input files, and the MILDwave Calculator performs the actual
calculations and provides several types of output files. Within the numerical domain,
wave gauges can be introduced to measure the surface elevation on predefined location.
The surface elevations in the numerical domain can be saved in multiple time instants
and the disturbance coefficient Kd and the vector field of the wave power p can be
calculated in the numerical domain and is saved in an output file. This disturbance
coefficient Kd is given by Eq. (1.10):
(1.10)
where is the local significant wave height and is the incident significant wave
height at the wave generation boundary.
1.4.1 The solution scheme of MILDwave
Equations (1.6) are (1.7) solved on a finite difference scheme, as shown in Figure 1. The
numerical domain is divided in grid cells with dimensions and and central
differences are used for spatial and time derivates. The water elevation and velocity
potential are calculated in the centre of each grid cell at different time levels,
and using the discredited Equations (1.11) and (1.12):
CHAPTER 1 - Numerical models for simulation of water wave propagation 1-6
(1.11)
(1.12)
A en B are computed using Equations (1.8) and (1.9). Lower index i,j defines the spatial
grid cell at position and , upper index n signifies the time step .
Figure 1: Finite difference scheme (computational space-centred, time-staggered grid) used by MILDwave.
(Beels, 2009).
Wave generation starts from quiescent water conditions at t=0. Each time step, first
and then is calculated in the centre of each grid cell.
CHAPTER 1 - Numerical models for simulation of water wave propagation 1-7
1.4.2 Wave generation in MILDwave
In MILDwave, waves are generated at an offshore boundary and subsequently propagate
further into the simulation domain. In order to avoid undesirable reflection from the
domain boundaries, which may disturb of influence the wave patter in the simulation
domain, outgoing waves must be absorbed at these open boundaries. Also, a reflected
wave may not be re-reflected at the wave generation boundary. To obtain this, internal
wave generation techniques in combination with absorbing layers, called sponge layers,
at the open boundaries are used.
1.4.2.1 Regular wave generation
The method used to generated the waves at an offshore boundary is called the source
term addition method. For each time step, an additional surface elevation is added to
the calculated value on a wave generation line. Figure 2 shows a set-up to generate a
regular wave with phase velocity C and wave elevation that propagates at an angle θ
from the x-axis.
Figure 2: Definition sketch of wave generation on two wave generation lines. Beels (2009).
The wave elevation is calculated using Equation (1.13)
(1.13)
with a the wave amplitude and the phase shift. The phase velocity along the x-axis and
y-axis can be calculated as and (Larsen and Dancy (1983)). Each time step
, the volume flux across the wave generation line parallel to the y-axis equals
. Since this flux occurs in two directions (in both the positive, and negative
x-direction), and since a wave generation cells covers an area , the additional surface
CHAPTER 1 - Numerical models for simulation of water wave propagation 1-8
elevation on the wave generation line parallel to the y-axis, can be calculated with
Equation (1.14):
(1.14)
and analogous, the additional surface elevation on the wave generation line parallel to
the y-axis can be calculated using Equation (1.15)
(1.15)
1.4.2.2 Irregular wave generation
Lee (1998) shows that the model of Radder and Dingemans (1985) can be used to simulate
transformation of uni- and directional random waves, or irregular long-crested and
short-crested waves. For the generation of random waves, the peak frequency is used as a
carrier frequency in Equations (1.8) and (1.9). For generation of irregular waves, the same
set-up as in Figure 2 is used. For uni-directional irregular waves, the wave elevation is
given in Equation (1.16):
(1.16)
with , the angular frequency,
the
frequency interval and the random phase. A parameterized JONSWAP spectrm is
used as an input frequency spectrum .
Short-crested wave generation has also been implemented with a single summation
model. Here, each wave component has a unique frequency while several wave
components are travelling in the same direction.
Implementation and validation of the source code for irregular wave generation in
MILDwave is performed by Caspeele (2006).
1.4.3 Numerical domain boundaries
As mentioned before, it is necessary to insert absorbing sponge layers at boundaries of
the domain, in order to avoid wave reflection on these boundaries, which may disturb
the wave pattern in the domain. Typically, these sponge layers with length are placed
against the edges of the wave basin. Numerical absorption is obtained by multiplying the
calculated surface elevation on each time step with an absorption function that
continuously and smoothly decreases from 1 at the start to 0 at the end of the sponge
layer. Three different absorption functions are available, they are defined by there
CHAPTER 1 - Numerical models for simulation of water wave propagation 1-9
equations as given in (1.17), (1.18) and (1.19) for S1(b), S2(b) and S3(b) respectively (Beels,
2009):
(1.17)
(1.18)
(1.19)
These functions are shown in Figure 3:
Figure 3: sponge layer function S1, S2 and S3 (Beels, 2009).
The best choice of sponge layer and the sponge layer length depend on the type of the
generated waves (regular or irregular) and the wave period. A good criterion for these
parameters is given by Finco (2011).
1.4.4 Wave breaking in MILDwave
In MILDwave, wave breaking is implemented by Gruwez (2008) based on the Battjes -
Janssens model (1978) by adding an energy dissipation term to Equation (1.6), giving
Equation (1.20):
(1.20)
CHAPTER 1 - Numerical models for simulation of water wave propagation 1-10
When working with regular waves, is given by Deigaard (1991) as Equation (1.21)
(1.21)
For irregular waves the dissipation term proposed by Baldock (1998) is used:
(1.22)
with a proportionality factor according to the intensity of the wave breaking, the
wave height at breaking, this is related to the local water depth d, the root mean
square value of the wave height, the water density and the peak wave period. is
calculated using Equation (1.23):
(1.23)
With the wave number for the wave with the peak wave period , the root
mean square value of the water wave in deep water and , the wave length in deep
water for a wave with period . Parameters and have to be calibrated and
can be changed in the MILDwave preprocessor.
1.4.5 The MILDwave user interface
1.4.5.1 The MILDwave preprocessor
The MILDwave preprocessor consists of 7 tab windows in which various parameters for
the simulation have to be implemented. In Figure 4, the first tab, called 'Grid' is shown.
This tab specifies the size and the grid cell size of the numerical domain.
CHAPTER 1 - Numerical models for simulation of water wave propagation 1-11
Figure 4: MILDwave preprocessor, tab 'Grid'.
Also, it is possible to implement sponge layers at the four sides of the numerical domain,
both the sponge layer function and length have to be specified. It is also possible to
perform a one dimensional simulation. In this case, the number of horizontal cells Nx is
set to 3.
The second tab window is called 'Wave' (Figure 5). Here it is specified whether the
generated waves are regular or irregular, the wave height and wave period
are specified. Also, the time instant when wave generation is starting and
stopping is specified, as well as the (mean) angle of wave propagation θ. When
generating irregular waves, it is possible to choose for a Pierson Moskovictch or a
JONSWAP frequency spectrum and parameters have to be specified. Also, the number of
generation lines and their location are specified.
CHAPTER 1 - Numerical models for simulation of water wave propagation 1-12
Figure 5: MILDwave preprocessor, tab 'Wave'.
The 'Model options' tab (Figure 6) enables or disables the depth-induced wave-breaking
module. When enabled, it is possible to change the wave breaking parameters
and , default parameters values are given.
Figure 6: MILDwave preprocessor, tab 'Model options'.
The next tab window is called 'Time step' (Figure 7). Here the time step and the time
instant when the simulation stops are specified.
CHAPTER 1 - Numerical models for simulation of water wave propagation 1-13
Figure 7: MILDwave preprocessor, tab 'Time step'.
In the tab 'Bathymetry' (Figure 8), different options are available to implement the
desired bathymetry into the preprocessor. First, the water level and minimum water
depth are specified. For constant water depth throughout the domain, a numerical value
for the bottom level can be inserted. Otherwise, a bathymetry can be inserted by using a
color-coded bitmap file with size Nx, Ny. Each different color represents a different
bottom level. It is also possible to use a text file of size Nx, Ny, with each cell containing
the numerical value of the bottom level.
Figure 8: MILDwave preprocessor, tab 'Bathymetry'.
CHAPTER 1 - Numerical models for simulation of water wave propagation 1-14
The tab 'Cell Type' enables the user to implement a specific geometry and specify its
properties. This is done using a colour coded bitmap image of size Nx, Ny with each
colour representing a different 'Cell type value'. This makes it possible to make structures
partially reflective.
Figure 9: MILDwve preprocessor, tab 'Cell Type'.
The 'Output' tab (Figure 10) specifies whether wave gauges are used, as well as there
position. The output parameters are chosen, and the time instants when the calculation of
these parameters start and stop are specified.
Figure 10: MILDwave preprocessor, tab 'Output'.
CHAPTER 1 - Numerical models for simulation of water wave propagation 1-15
Finally, it is necessary to press the button 'Creat ct&d file', this causes MILDwave to
generate some additional input files for the MILDwave Calculator.
1.4.5.2 The MILDwave Calculator
To start the computation, it is necessary to open the MILDwave Calculator (Figure 11),
this is an executable program. Using the first button 'Select project directory', the location
of the input files is inserted. The button 'Load-allocate-prepare' loads these input files,
allocates memory and prepares for calculation. When continuing an already started but
interrupted calculation, the button' Load snapshot date' is used. The buttons 'Start
calculation' and 'Stop calculation' start and stop the calculation. During calculation, it is
possible to view and save the water elevation throughout the numerical domain and at
the position of the wave gauges.
Figure 11: MILDwave Calculator.
1.4.5.3 The MILDwave calculator output files
Depending on the options selected in the MILDwave preprocessor, the MILDwave
Calculator creates different output files, as shown in Table 1:
Table 1: list of MILDwave Calculator output files
CTdata.out Cell type values along the domain
D3dataTime.out Instant wave elevations along the grid at t=time
Edata.out Wave energy along the domain
gradFItxdate.out Wave power in x-direction along the domain
gradFItydate.out Wave power in y-direction along the domain
GRIDdaa.out Depth values along the domain
CHAPTER 1 - Numerical models for simulation of water wave propagation 1-16
Pdata.out Distrubrance coefficient along the domain
WGdataNr Instant wave elevations at the location of wave gauges Nr
along the simulation
In the present study, most results are analyzed using MATLAB 2011 (The MathWorks,
Inc. 1984-1997).
2 Test case: waves travelling over a
submerged island
In this chapter, the test case of waves travelling over a submerged, spherical island is used to test
validate the numerical model for wave propagation MILDwave.
CHAPTER 2 - Test case : waves travelling over a submerged island 2-2
2.1 General description of the experiment
In 1972 Ito and Tanimoto present a numerical method to obtain wave patterns in regions
of arbitrary shape. Their method is validated by comparing several numeric results with
hydraulic model experiments. One of these tests consists of a submerged shoal with a
concentric, circular shape where a cusped caustics is formed. The calculated wave height
distribution around the shoal is compared with that obtained from model tests.
2.2 Aim of the experiment
The experimental data provided by Ito (1972) will now be used to validate the wave
propagation in MILDwave above a circular, submerged island.
First, the basin layout and the geometry of the submerged island is implemented in
MILDwave. Calculation of the disturbance coefficient is performed using the
MILDwave Calculator. This value is defined in Eq. (2.1):
(2.1)
is the mean resulting wave height between two time instants, which are specified in
the MILDwave Preprocessor.
The resulting distribution of -values is analysed using MATLAB and compared with
experimental data using Microsoft EXCEL.
2.3 Simulation process and input
2.3.1 Bathymetry and grids
The hydraulic model tests are conducted in a basin with maximum water depth,
minimum water depth and wave length respectively:
The wave length can be found by using Eq. (2.2)
(2.2)
with
, the gravitational constant, T the wave period and
, the wave
number. The wave period is an important parameter for the MILDwave preprocessor,
this period can be found by rearranging Eq. (2.2):
CHAPTER 2 - Test case : waves travelling over a submerged island 2-3
(2.3)
The wave height H=0.0064 m, in accordance with Ito (1972).
Numerical modelling for the same situation was conducted in MILDwave. The width and
length of the numerical basin are x , where L is the wave length. The layout of the
basin used for the MILDwave simulations is shown in Figure 12. The hatched area at the
top and bottom boundaries represents the sponge layers ,the circular area represents the
submerged island. Cross sections across the centre point of the island are displayed on
the bottom and right hand side of the figure.
Figure 12: wave basin layout.
Two numerical sponge layers of type S1 as defined in chapter 1 and width are
used along the upwave and downwave boundaries (hatched area in Figure 12). The wave
generation line is located four grid cells after the sponge layer. To prevent energy
absorption and undesirable effects from the basin side walls there are no sponge layers
on the side boundaries. This is possible because the wave propagation is perpendicular to
the wave generation line and no reflection on the side boundaries is to be expected.
Dashed lines in Figure 12 represent sections on which the disturbance coefficient is
studied.
CHAPTER 2 - Test case : waves travelling over a submerged island 2-4
Throughout the simulation domain the water depth remains constant, except at the
location of the island, where the water depth d decreases from h1=15 cm to h2=5 cm, as
seen in Figure 12. The bathymetry was implemented using a bitmap file which was
generated with MATLAB, the code to do this is added in the appendix. As Figure 13
indicates, 14 colour shades represent different heights on the island. The submerged
island has the form of a part of a sphere.
Figure 13:spherical shaped, submerged island modelled with 14 shades of red
2.3.2 Wave conditions and MILDwave parameters for the wave propagation
experiment over a submerged island
Three numerical simulations were performed, all with the same basin configuration but
different grid spacing .
Simulation 1 was performed with the wave breaking module active. To avoid
discretisation errors Gruwez (2011) suggested that for the wave breaking module, grid
sizes , and time step should be chosen as follows:
(2.4)
(2.5)
with
, the phase velocity of the wave. Therefore and
was chosen as grid size and time step.
Simulation 2 and 3 where performed without the wave breaking module and with grid
size and time step of respectively and
these values lay between boundaries
proposed by De Doncker (2002)
CHAPTER 2 - Test case : waves travelling over a submerged island 2-5
based on empirical practice. Table 2 summarizes grid size, time step, basin dimensions
and sponge layer thickness.
Table 2: MILDwave parameters for 3 simulations of wave propagation over a spherical shaped submerged
island
Simulation 1 2 3
grid cell size Δx=Δy [m] 0.01 0.025 0.02
time step Δt [s] 0.01 0.025 0.02
number of grid cells in horizontal direction Nx 1200 480 600
Number of grid cells in vertical direction Ny 920 368 460
number of grid cells in sponge layer Ns 140 56 70 Because of the constant wavelength in this experiment only regular waves were
generated. As mentioned before, the direction of wave propagation is perpendicular to
the wave generation line. In all three simulations the wave height is .
During the simulation values were computed during 200 wave periods, starting from
2.4 Results of the Kd value throughout the domain, provided by MILDwave
2.4.1 Plan view plots of the Kd value throughout the domain
During computation MILDwave generates a Nx x Ny matrix, where Nx is the number of
horizontal grid cells and Ny the number of vertical grid cells, wich contains in every grid
cell a value of the disturbance coefficient
Analysis and graphical presentation
where performed using MATLAB. A qualitative view of the values throughout the
domain is given in Figure 14. This result was generated for Simulation 1, Simulations 2
and 3 give very similar results.
CHAPTER 2 - Test case : waves travelling over a submerged island 2-6
Figure 14: Kd values throughout the basin [simulation. 1], T=0.51 s, d=0.15 m.
Maximum and minimum values of are 2.11 and 0.00 respectively.
2.4.2 Section plots of the Kd value through three sections near the submerged island
To compare results of Kd-values between different simulations and to compare the
simulations with experimental data provided by Ito (1972), the values along three
different sections are studied. As shown in Figure 12 and Figure 14. Section 1 has a length
of 8L, is located in the middle of the basin and is parallel to the wave propagation
direction. Sections 2 and 3 have lengths 6L, are oriented perpendicular to the wave
propagation direction and are located respectively at 0L and 1L after the submerged
island.
As seen in Figure 15 there are 29 measured values of in section 1, ranging from
to
, where x is the distance starting from zero at the bottom end of the section. There
are no experimental data for
, thus results of numerical simulations are not plotted
in this region.
CHAPTER 2 - Test case : waves travelling over a submerged island 2-7
Figure 15: Kd-values along Section 1. 'x' = MILDwave results for Δx=0.010 m, 'Δ'= MILDwave results for
Δx=0.020 m, 'О'= MILDwave results for Δx=0.025 m, '●' = experimental data by Ito (1972)
In Sections 2 and 3 there are 25 experimental data points provided by Ito (1972). For each
simulation the corresponding MILDwave results are shown in Figure 17 and Figure 17.
0.0
0.5
1.0
1.5
2.0
1 2 3 4 5 6 7 8 9
Kd-v
alu
e
x/L
Kd along Section 1 exp. results
Δx = 0.01
Δx = 0.025
Δx = 0.02
CHAPTER 2 - Test case : waves travelling over a submerged island 2-8
Figure 16: Kd-values along Section 2'x' = MILDwave results for Δx=0.010 m, 'Δ'= MILDwave results for
Δx=0.020 m, 'О'= MILDwave results for Δx=0.025 m, '●' = experimental data by Ito (1972)
Figure 17: Kd-values along Section 3, 'x' = MILDwave results for Δx=0.010 m, 'Δ'= MILDwave results for
Δx=0.020 m, 'О'= MILDwave results for Δx=0.025 m, '●' = experimental data by Ito (1972)
0.0
0.5
1.0
1.5
2.0
0 1 2 3 4 5 6
Kd-v
alu
e
y/L
Kd along Section 2 exp. results
Δx = 0.01
Δx = 0.025
Δx = 0.02
0.0
0.5
1.0
1.5
2.0
0 1 2 3 4 5 6
Kd-v
alu
e
y/L
Kd along Section 3 exp. results
Δx = 0.01
Δx = 0.025
Δx = 0.02
CHAPTER 2 - Test case : waves travelling over a submerged island 2-9
Full lines are added to provide better insight in the results. They consist of a value for
every grid cell in Simulation 1.
2.5 Quantitative analysis of the results using parameters for model performance
As suggested in Dingemans (1997), for comparing results of model testing and numerical
simulation, several parameters are chosen in order to give a quantitative measure for the
performance of the mathematical model. These parameters are divided into a primary
and a secondary set of parameters. The primary set consists of:
Table 3: primary set of statistical parameters used for numerical model validation
: mean value of measured values
: mean value of computed values
: standard deviation of measured values
: standard deviation of computed values
: parameters of ordinary least square regression of
y over x:
: mean absolute deviation between y and x
: root mean square deviation between y and x
: systematic part of rmse
: unsystematic part of rmse
: index of agreement, for perfect
agreement is reached
: modified index of agreement, more sensitive then
and nearly always
The secondary set of parameters is derived from the primary set and consists of
CHAPTER 2 - Test case : waves travelling over a submerged island 2-10
Table 4: secondary set of statistical parameters used for numerical model validation
All parameters, except s(d) are expressed relative to and as a percentage. For each
section in each simulation, these parameters where computed. Also, since there are 29
points in section 1 and 25 in section 2 and 3, there is a total of 79 points in every
simulation. For these 79 values, the parameters where computed as well, providing a
more global view of the model performance. These values are summarized in Table 5.
Moreover, the graphs of the linear regression analysis for these values are shown in
Figure 18.
CHAPTER 2 - Test case : waves travelling over a submerged island 2-11
Table 5: primary and secondary parameters of 3 simulations
Simulation 1 2 3
n 79 79 79
y ̅ 1.1215 1.1495 1.1382
x ̅ 1.0459 1.0459 1.0459
s(y) 0.4840 0.5208 0.5062
s(x) 0.4656 0.4656 0.4656
b 1.0046 1.0765 1.0478
a 0.0707 0.0236 0.0423
mae 0.1196 0.1464 0.1355
rmse 0.1448 0.1782 0.1645
rmses 0.0818 0.1129 0.0994
rmseu 0.1195 0.1379 0.1311
d(2) 0.9667 0.9534 0.9591
d(1)0.8088 0.7732 0.7875
bias 7.22% 9.91% 8.82%
mae 11.44% 14.00% 12.95%
rmse 13.85% 17.04% 15.73%
rmses 7.82% 10.80% 9.50%
rmseu 11.43% 13.18% 12.53%
pes 31.89% 38.38% 34.91%
peu 68.11% 57.23% 60.70%
sd 1.48% 2.04% 1.80%
pri
mar
y p
ar.
seco
nd
ary
par
.
2.6 Conclusions
Firstly, it is clear that MILDwave produces consistent results for different grid sizes
. For every simulation good agreement with experimental results is obtained. From
Figure 15 one can learn that further away from the submerged island the difference
between measured and computed values becomes larger.
Still, the model performs very well and gives accurate values in all three sections, as
shown in Fout! Verwijzingsbron niet gevonden.. For all simulations the index of
agreement is high, the root mean square errors, and most importantly, the systematic part
of the latter are small and the regression analysis shows very good agreement between
measured and computed values.
Even though no wave breaking occurs, Simulation 1 provides better results than the two
other simulations. Also, Simulation 3 provides slightly better results than Simulation 2. It
CHAPTER 2 - Test case : waves travelling over a submerged island 2-12
seems that a smaller grid size and time step is beneficial for the model performance.
However, the difference is small and computation time for Simulation 1 is far greater
than for Simulation 2.
Finally, a linear regression analysis of the results is plotted in Figure 18.
Figure 18: linear regression analysis of results for Simulations 1-3.'x' = MILDwave results for Simulation 1,
'О'= MILDwave results for Simulation 2, 'Δ' = MILDwave results for Simulation 3
0
0.5
1
1.5
2
0 0.5 1 1.5 2
y
x
y = a+bx
Simulation 1
simulation 2
Simulation 3
1:1
Lineair (Simulation 1)
Lineair (simulation 2)
Lineair (Simulation 3)
3 Test case: resonance in a rectangular
harbour
In this chapter the phenomenon of resonance in a rectangular harbour is studied using the
numerical wave propagation model MILDwave. Results are compared with experimental studies
performed by Ippen and Goda (1963) and a theoretical solution by Lee (1969)
CHAPTER 3 – Test case: resonance in a rectangular harbour 3-2
3.1 Introduction and aim of the experiment
In coastal regions, there are several structures such as harbours and fjords that can be
regarded as (partially) enclosed basins. These structures all have resonant or eigen
periods that are determined by the basin geometry and the water depth d. When water
waves with the eigen period of a harbour enters this harbour, resonance occurs and large
water oscillations can produce damaging surging and yawing and swaying of ships. It is
thus important to be able to predict these eigen periods. In the present chapter, it is
examined whether MILDwave can accurately predict the harbour eigen periods and the
amplitude of the resulting oscillating waves.
3.2 General description of the experiment
In 1963 the problem of resonance of a rectangular harbour connected to the open sea is
studied both theoretically and experimentally by Ippen and Goda (1963). The theoretical
model showed good agreement with experiments conducted in a wave basin. Lee (1969)
developed a theory to describe wave induced oscillations in harbours of arbitrary shape.
Here also, theoretical results shows good agreement with the experimental results from
Ippen and Goda (1963), and Lee (1969). In the present chapter, the experimental results
and the theoretical results of the Rectangular Harbour Theory are used to validate
MILDwave. Results are plotted on a chart where the abscissa is the wave number
parameter , with
L=wave length and l=the harbour length. The ordinate is the
amplication factor, R, defined as the wave amplitude at the center of the rear wall of the
harbour divided by the average standing wave amplitude at the harbour entrance when
the entrance is closed.
3.3 Simulation process and input
3.3.1 Bathymetry and grids
The width and length of the harbour in het numerical model is:
Where is the harbour width, the harbour length. The water depth d is constant
throughout the entire basin:
CHAPTER 3 – Test case: resonance in a rectangular harbour 3-3
Using the geometric information of the harbour, a prediction about the eigen period
can be made. The first mode or fundamental mode for a rectangular basin with
constant water depth d is given by Raichlen (1966) as Eq. 3).
(3.1)
The second mode can be found as:
(3.2)
It is thus expected to find two resonance peaks around these specific eigen periods and
.
In MILDwave, the numerical domain has width w’ and length l’:
(3.3)
(3.4)
where L is the wave length and the grid size in y-direction. The additional term of
10m forms the harbour rear wall.
As seen in Figure 19 the numerical basin contains sponge layers of width 3L at the side
boundaries. Sponge layers are represented by area containing a hatch. The sponge layers
are implemented because waves radiated from the harbour can reflect from these
boundaries of the numerical domain and cause a disturbed wave pattern. For the same
reason, another sponge layers is added behind the wave generation line. The sponge
layer type is for every experiment S1, which is defined in chapter 1. On the north side of
the simulation domain, the harbour of width is implemented. The distance
between the harbour and the wave generation line is 7L, where L is the wave length.
CHAPTER 3 – Test case: resonance in a rectangular harbour 3-4
Figure 19: basin layout, the bottom and side boundaries of the numerical domain contain a sponge layer
(hatch), the top boundary contains the harbour and fully reflective walls (black filled area).
To implement the harbour, a bitmap file was generated with MATLAB. An example of
this can be seen in Figure 20:
Figure 20: harbour detail, the black area represents the land side with fully reflecting walls. The gap in the
middle of the black area represents the opening and length of the harbour.
Black grid cells are given an absorption coefficient of 1, so no absorption occurs and total
reflection is achieved.
3.3.2 Wave conditions and MILDwave parameters
Since the amplification factor R is a function of the parameter kl with k the wave number
and l the length of the harbour, it is necessary to perform several simulations for different
values of kl and thus wave length L. Lee (1969) and Ippen & Goda (1963) provide
experimental results for the kl range . Since it is not possible to generate waves
with infinite wave length (kl=0 thus and since the wave basin size and simulation
time increases with increasing wave length, numerical simulations in MILDwave are
CHAPTER 3 – Test case: resonance in a rectangular harbour 3-5
performed in the kl range , which corresponds to
Since the water depth d=6.0 m throughout the numerical domain the wave
period T varies between 5.78 s and 25.55 s.
As mentioned before, grid sizes and should be selected between L/20 and L/10.
However, the minimum grid size would then be
This makes it difficult to
implement a harbour of width w=6m. Moreover, the amplification factor is measured at
the center of the rear wall in the harbour and thus the harbour should have a width of at
least multiple grids. To reduce calculation time but still have sufficient accuracy of the
measured amplification factor, the harbour consists of 5 grid cells and thus in all
simulations
and The maximum time step
with
, the phase velocity of the wave varies between . In order
to achieve sufficient accuracy and avoid numerical errors, which result from a too long
time step, for all simulations is chosen.
A total of 63 simulations for different values of kl were performed. The values of kl that
are considered are:
(3.5)
Smaller intervals between kl values were chosen where it was seen that resonance
occurred.
The wave-breaking module was set off, because no wave dissipation processes are
considered in this test case. Since MILDwave is a linear model, the wave height H in
these experiments is not an important factor. It is chosen to be H=0.25 m in every
experiment. Calculation of the amplification factor R starts when the first reflected wave
hits the generation line ( ) and stops at .
Because of the large number of simulations that have to be carried out, a MATLAB-file
was created to automatically prepare the necessary maps and files. Basically the script
generates the MILDwave.ini file, which is otherwise created by the MLDwave pre-
processor, and the bitmap file shown above. The code of this file is provided in the
appendix.
CHAPTER 3 – Test case: resonance in a rectangular harbour 3-6
3.4 MILDwave results of the Kd-value throughout the domain and the amplification
factor R at the centre of the harbour rear wall
3.4.1 Plan view plots of the Kd-value throughout the effective domain
Analysis and graphical presentation where performed using MATLAB. In order to give a
qualitative view of the different reflection patterns, Figure 21 shows plan view plots of
the amplification factor R for kl=1, 1.35, 3, 4.3, 5. In these plots, the sponge layers are not
displayed. Hence, in every simulation, the width of the basin w' without sponge layers is
300 m, the length l' varies with the wavelength.
CHAPTER 3 – Test case: resonance in a rectangular harbour 3-7
Figure 21: R values throughout the effective domain in MILDwave for kl=1.00, 1.35, 3.00, 4.30, 5.00 and
d=6.0 m.
For kl=1.35 a first resonance peak is observed, for kl=4.30 a second but smaller peak is
observed. This will be made more clearly visible below.
CHAPTER 3 – Test case: resonance in a rectangular harbour 3-8
3.4.2 Section plots of the R value along the vertical symmetry axis of the domain
A better quantitative comparison is possible when plotting R-values along the central
section. This is shown in Figure 22 for the same kl values as in Figure 21. The abscissa is
zero meter at the harbour rear wall and 31m at the harbour mouth.
Figure 22: R-values along the central section for 5 different values of kl, d=6.0 m. Continuous line for
kl=1.00, dotted line for kl=1.35, medium dashed line for kl=3.00, short dashed line for kl=4.3, long dashed line
for kl=5.00.
A very clear resonance peak for kl=1.35 is observed (dotted line). At kl=1.00 (continuous
line) and kl=4.3 (short dashed line) resonance is less strong. For kl=3.00 (short dashed
line) and kl=5.00 (long dashed line) no resonance is observed. Outside the harbour, the
maximum amplification factor R quickly reduces to 1 and a stable standing wave pattern
is observed, as is also shown in Figure 21.
0
1
2
3
4
5
6
7
0 20 40 60 80 100 120 140
R [
-]
distance from rear wall harbor [m]
R-values along the central section kl=1
kl=1.35
kl=3
kl=4.3
kl=5
CHAPTER 3 – Test case: resonance in a rectangular harbour 3-9
3.5 Analysis of the results and conclusions
Resulting R-values at the center of the harbour rear wall are plotted in function of kl.
Also, experimental results of Ippen & Goda (1963) and Lee (1969), as well as a theoretical
solution of Lee (1969) are plotted in Figure 23:
Figure 23: R-values in function of kl, d=6.00 m. ● = MILDwave Results, + = Ippen & Goda (1963)
experimental data, О = Lee(1969) experimental data, ─ = Lee (1969) theoretical solution.
Ippen & Goda (+), Lee (О) and Lee (theoretical) (─) provide respectively 104, 101 and 256
data points.
As predicted above, two resonance peaks are observed. The fist resonance peak for small
kl=1.35 results in very large amplification factor R. This peak corresponds with a period
of and is known as the Helmholtz (1885) mode of the basin. Eq. 3) it was
expected to find this mode at , which turned out to be a relatively good
approximation. Another eigen period was expected to exist at and indeed, the
0
1
2
3
4
5
6
7
8
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
R
kl
R-values in function of kl MILDwave
Ippen & Goda
Lee
Lee (theoretical)
CHAPTER 3 – Test case: resonance in a rectangular harbour 3-10
second peak in Figure 23 is observed at kl=4.3 or at The amlification factor R
of this eigen mode is however much smaller than at the Helmholtz mode.
It is clear that very good resemblance is achieved with measured and theoretical results.
Especially the first resonance peak (for lowest value of kl) is simulated accurately. In the
second resonance peak, R-values obtained with MILDwave are lower than theoretical
and measured results.
This close agreement to experimental data is also observed when looking at the primary
and secondary parameters for model evaluation. Index of agreement d2 is smaller for
than for . In Table 6, primary and secondary parameters are given.
These parameters are comparing MILDwave results for R (y) with respectively
experimental data from Ippen & Goda (1963), Lee and the theoretical solution of Lee
(1969). Model evaluation can only occur for the 63 values of kl where a MILDwave
simulation was performed. Therefore, to achieve data points for specific kl-values in the
three data sets linear interpolation between the MILDwave results was performed. The
secondary set of parameters is expressed as a percentage relative to , the mean value of
the measured values.
CHAPTER 3 – Test case: resonance in a rectangular harbour 3-11
Table 6: primary and secondary parameters of MILDwave simulations compared to three data sets.
data set Ippen & Goda (1963) Lee (1969) Lee theoretical (1969)
n 63 63 63
y ̅ 2.0927 2.0927 2.0927
x ̅ 2.0736 2.1641 2.3331
s(y) 1.4890 1.4890 1.4890
s(x) 0.9376 1.2520 1.6362
b 1.3278 1.1325 0.8831
a -0.6606 -0.3581 0.0324
mae 0.5920 0.3746 0.3666
rmse 0.8660 0.4857 0.4704
rmses 0.3055 0.1793 0.3062
rmseu 0.8104 0.4513 0.3570
d(2)0.8608 0.9674 0.9769
d(1)0.6515 0.8113 0.8397
bias 0.92% -3.30% -10.30%
mae 28.55% 17.31% 15.71%
rmse 41.77% 22.44% 20.16%
rmses 14.73% 8.29% 13.13%
rmseu 39.08% 20.86% 15.30%
pes 6.00% 6.30% 18.17%
peu 42.22% 39.91% 24.70%
sd 36.74% 10.84% 7.12%
prim
ary
para
met
ers
seco
ndar
y pa
ram
eter
s
Especially compared with results provided by Lee, low root mean square errors are
observed. Also the index of agreement is much higher for these data sets. Compared to
results provided by Ippen & Goda (1963), the model still performs well, but larger errors
are observed. This is probably due to a significant difference between these data sets,
especially in the region of the first resonance peak.
MILDwave gives very good results for the first resonance peak. Similarity between
values by MILDwave and the theoretical solution by Lee (1969) is very high. However,
the second resonance peak is described less accurately, notably the peak is located
slightly towards higher values of kl when compared to other data sets. Also, the R-values
are smaller.
Finally, graphs of linear regression analysis for the amplification factor provided by
MILDwave and for the date sets by Ippen & Goda (1963) and Lee (1969) are shown in
Figure 24:
CHAPTER 3 – Test case: resonance in a rectangular harbour 3-12
Figure 24: regression analysis of MLIDwave results compared to three data sets. + = Ippen & Goda (1963), О
= experimental date by Lee (1969), ● = theoretical data by Lee (1969).
Here it is very clear to see the good performance of MILDwave compared to the
theoretical solution by Lee for high values of R.
The overall conclusion is that MILDwave is capable to simulate resonance in a
rectangular harbour relatively accurate.
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6
y
x
y=a+bx Ippen & Goda
Lee
Lee theoretical
1:1
Lineair (Ippen & Goda)
Lineair (Lee)
Lineair (Lee theoretical)
4 Test case: wave deformation in the
surf zone
In this chapter the numerical simulation of wave deformation and wave breaking by MILDwave is
studied and compared to experimental results. Three different types of beach are considered: a
uniform slope beach, a step-type beach and a bar-type beach.
CHAPTER 4 – Test case: wave deformation in the surf zone 4-2
4.1 General description of the experiment
4.1.1 Aim of the experiment
Watanabe and Dibajnia (1988) studied near shore wave deformation due to shoaling and
wave breaking, and due to wave height decay and recovery in the surf zone. A model
based on a set of time-dependent mild slope equations is developed including a term of
wave energy dissipation caused by wave breaking. These time-dependent mild-slope
equations are then used to compute cross-shore change of wave height and wave energy
in a one-dimensional wave field. Three different nearshore geometric layouts are studied
by comparing the numerical and experimental results (wave-height and potential energy)
from a hydraulic model study performed by Nagayama (1983). The numerical model is
able to reproduce very well cross-shore wave transformation due to wave shoaling,
breaking, decay and recovery. In the present study both the numerical and experimental
results are used for validation of MILDwave in the present study.
4.2 Simulation process and numerical input data
4.2.1 Bathymetry and numerical domains (I will add an additional figure of the
entire layout in the simulation, including sponge layers)
Three different geometric beach layouts are considered, (i) a uniform slope (1:20), (ii) a
step-type beach and (iii) a bar-type beach. A graphical representation of these types is
given in Figure 25. In the numerical wave propagation model MILDwave (1998), every
beach is preceded by a 6 m long run-up zone. In MILDwave, the water depth does not
take the zero value, as no surface-piercing bottom slopes are modeled; therefore,
downwave the beach slope the water depth has a constant value of 0.005 m. The
numerical domain is enclosed along its length by two sponge-layers of type and width
according to Finco (2011). No side sponge layers are used. In Table 7, the wave period T,
the generated wave height H, the maximum water depth dmax, the maximum wavelength
Lmax, the size of the grid cells Δx=Δy and the time step Δt are given for the three different
beach layout types. Apart from dmax and Lmax, these values are input data for the
MILDwave preprocessor.
CHAPTER 4 – Test case: wave deformation in the surf zone 4-3
Table 7: period, wave height, maximum depth, maximum wavelength, grid size and time step for each
experiment
beach layout T (s) H (m) dmax (m) Lmax (m) ∆x=∆y (m) ∆t (s)
Uniform 1.19 0.06 0.30 1.75 0.04377 0.030
Step 1.18 0.07 0.25 1.62 0.04062 0.029
Bar 0.94 0.07 0.25 1.19 0.02985 0.024
Figure 25: layout of the beaches used in the MILDwave simulations.
The values of Table 7 and the beach layout result in three wave flumes with 2252, 2352
and 2342 grid cells for the uniform (1:20), the step-type and the bar-type beach
respectively. In this case, the one dimensional wave flume of MILDwave is used. This
wave flume has a width of three cells.
The bathymetries of the three beach types are generated using Microsoft Excel. For every
position along the length of the beach, a value for the depth is calculated. A file with a .txt
format containing this information is then used as input for the MILDwave pre-
processor. Since the width of the numerical domain is three grid cells, the text file
contains every depth value three times.
6, -0.3
1.6, -0.08
7, -0.25
3.6, -0.08
2.6, -0.13
0, 0
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
water d
epth
d (m
)
position (m)
uniform
step
bar
CHAPTER 4 – Test case: wave deformation in the surf zone 4-4
4.2.2 Wave conditions and MILDwave input parameters
All simulations are performed with the wave breaking module active in MILDwave.
Since the water depth d decreases to zero, the wavelength L becomes zero, as seen in Eq.
(4.1):
(4.1)
Where g = 9.81 m/s², the gravitational acceleration and T is the wave period. This creates
a conflict since the criteria for grid cell size when using the wave breaking model
proposed by Gruwez (2008) are given in Eq. (4.2) and the Courant-Friedrichs-Lewy (1928)
condition for time step Δt is given in Eq. (4.1):
(4.2)
(4.3)
Where L is the wavelength, and the grid size, the time step and
the phase
velocity of the wave. Which leads to . This of course is not possible. The size
of the grid cell is chosen to be
, with L the maximum wavelength. Criteria
(4.2) and (4.3) then lead to Eq. (4.4) and Eq. (4.5):
(4.4)
(4.5)
These criteria result in the grid cell size and time step given in Table 7.
The wave height H of the generated waves is 0.060 m for the uniform slope of 1:20 and
0.070 m for the step-type and bar-type beaches.
The moment when calculation of the starts and ends are given in Eq. (4.6) and (4.7):
(4.6)
(4.7)
CHAPTER 4 – Test case: wave deformation in the surf zone 4-5
4.3 Results and discussion
4.3.1 Potential energy density
To compare results from MILDwave simulations with the experimental results from
Nagayama (1983) and the computed results from Watanabe (1988), the values are
multiplied by the generated wave height H as denoted in Table 7. The mean square value
of the potential energy density E is given by Eq. (4.8):
(4.8)
Dividing Eq. (4.8) by the water density results in Eq. (4.9):
(4.9)
For each test case, the results of both the wave height and the potential energy density
along the beach layout are plotted and an analysis is performed. Next, the primary and
secondary parameters for model performance are presented. For the calculation of these
parameters only the values for the resulting wave height are used. Values for the
potential energy density along the beach layout are plotted in the graphs below, but are
not used in the calculation of the parameters for model performance.
CHAPTER 4 – Test case: wave deformation in the surf zone 4-6
4.3.2 Uniform slope beach
4.3.2.1 Section plots of the wave height and potential energy density for the uniform
slope beach
In Figure 26, the wave height H and the potential energy density are plotted along
the uniform beach of slope 1:20. Hollow dots represent the measured data by Nagayama
(1983), black dots are the numerical results of Watanabe (1988) and the continuous black
line are the MILDwave results.
0.00
0.02
0.04
0.06
0.08
Wave
he
ight H
(m)
computed
experimental
MILDwave
0
500
1,000
1,500
2,000
2,500
Po
ten
tial en
ergy
de
nsity E
p /ρ(cm
³/s²)
0
-0.3
-0.2
-0.1
0
0123456
wate
r de
pth
d
(m)
position along the effective domain x (m)
Figure 26: Calculated Kd for the uniform slope beach (1:20) experiment. Wave period T=1.19 s,wave height
H=0.060 m.'∙'= numerical data points by Watanabe (1988), '◦'=experimental data points by Nagayama (1983), '-'MILDwave simulations results.
4.3.2.2 Analysis of the results for the uniform slope beach
As in previous test cases, quantitative analysis of the results is performed using several
parameters suggested by Dingemans (1997). Table 7Table 8 summarizes both the primary
and secondary parameters for model performance analysis. For comparison of the
numerical MILDwave results n with the experimental results e, a total of 19 points is
used. The location of these points is determined by the data given by Nagayama (1983)
and is depicted in Figure 26. For comparison of the numerical MILDwave results n with
CHAPTER 4 – Test case: wave deformation in the surf zone 4-7
the computed results c, a total of 181 points is used. These points are represented by the
black dots in Figure 26. In the first column, the results of the MILDwave simulation are
compared with the experimental values, in the second column, the results of the
MILDwave simulation are compared with the computed values.
Table 8: MILDwave performance parameters for waveheight H on uniform slope beach (y=MILDwave results,
x=measured or computed results)primary experimental (e) computed (c) secondary experimental (e) computed (c)
number of data points (e or c) 19 181 bias (%) 8.13% 15.98%mean(n) (m) 0.060 0.056 mae (%) 12.23% 15.67%
mean(e or c) (m) 0.055 0.049 rmse (%) 13.54% 16.89%s (n) (m) 0.009 0.014 rmses (%) 12.41% 15.80%
s (e or c) (m) 0.014 0.015 rmseu (%) 5.41% 5.95%b (-) 0.618 0.933 pes (%) 84.04% 87.58%
a (m) 0.026 0.011 peu (%) 15.96% 12.42%mae (m) 0.007 0.008
rmse (m) 0.007 0.008rmses (m) 0.007 0.008rmseu (m) 0.003 0.003
d(2) (-) 0.891 0.925d(1) (-) 0.623 0.711
4.3.3 Step-type beach
4.3.3.1 Section plots of the wave height and potential energy density for the step-
type beach
In Figure 27, computed results for the step-type beach by Watanabe (1988), experimental
results by Nagayama (1983) and MILDwave results are plotted along the beach profile.
CHAPTER 4 – Test case: wave deformation in the surf zone 4-8
0.00
0.02
0.04
0.06
0.08
wave
he
ight H
[m]
computed
experimental
MILDwave
0
500
1,000
1,500
2,000
2,500
3,000
Po
ten
tial en
ergy
de
nsity E
p /ρ[cm
³/s²]l
0
-0.3
-0.2
-0.1
0
01234567
wate
r de
pth
d
[m]
position along the effective domain x (m)
Figure 27: Calculated Kd for the step-type beach experiment. Wave period T=1.18 s, wave height H=0.070 m. '∙'=
numerical data points by Watanabe (1988), '◦'=experimental data points by Nagayama (1983), '-'MILDwave simulations results.
4.3.3.2 Analysis of the results for the step-type beach
Quantitative analysis is performed using the primary and secondary parameters for
model performance analysis, as suggested by Dingemans (1997). As previous, the
experimental and computed points, respectively 25 and 129 are depicted in Figure 27.
Table 9 summarizes the parameters for both the experimental and the computed wave
heights.
CHAPTER 4 – Test case: wave deformation in the surf zone 4-9
Table 9: model performance parameters for waveheigth on step-type beach (y=MILDwave results, x=measured
or computed results).
primary experimental (e) computed (c) secondary experimental (e) computed (c) # (e or c) 25 129 bias (%) 30.06% 57.41%
mean(n) (m) 0.060 0.059 mae (%) 31.30% 45.46%
mean(e or c) (m) 0.046 0.040 rmse (%) 34.12% 47.57%
s(n) (m) 0.013 0.016 rmses (%) 31.79% 45.46%
s(e or c) (m) 0.015 0.015 rmseu (%) 12.40% 14.03%
b (-) 0.824 0.994 pes (%) 86.79% 91.31%
a (m) 0.022 0.019 peu (%) 13.21% 8.69%
mae (m) 0.014 0.018
rmse (m) 0.016 0.019
rmses (m) 0.015 0.018
rmseu (m) 0.006 0.006
d(2) (-) 0.768 0.751
d(1)
(-) 0.524 0.496
4.3.4 Bar-type beach
4.3.4.1 Section plots of the wave height and potential energy density for the bar-type
beach
In Figure 28, the computed, experimental and MILDwave results for the bar-type beach
are plotted along the beach profile.
CHAPTER 4 – Test case: wave deformation in the surf zone 4-10
0.00
0.02
0.04
0.06
0.08
wave
he
ight H
(m)
comp
meas
MILDwave
0
500
1,000
1,500
2,000
2,500
3,000
po
ten
tial en
ergy d
en
sity E
p /ρ(cm
³/s²)l
0
-0.3
-0.2
-0.1
0
01234567
wate
r de
pth
d
(m)
position along the effective domain x (m)
Figure 28: Calculated Kd for the bar-type beach experiment.Wave period T=0.94 s, wave height H=0.070 m. '∙'=
numerical data points by Watanabe (1988), '◦'=experimental data points by Nagayama (1983), '-'MILDwave
simulations results.
4.3.4.2 Analysis of the results for the bar-type beach
Table 10 summarizes the primary and secondary parameters for model performance
analysis. Again, the experimental points (23) and computed points (136) used in the
analysis are depicted in Figure 28.
CHAPTER 4 – Test case: wave deformation in the surf zone 4-11
Table 10: model performance parameters for wave height on uniform slope beach (y=MILDwave results,
x=measured or computed results)
primary experimental (e) computed (c) secondary experimental (e) computed (c) # (e or c) 23 136 bias (%) 17.17% 48.58%
mean(n) (m) 0.067 0.061 mae (%) 16.91% 25.63%
mean(e or c) (m) 0.057 0.048 rmse (%) 20.66% 29.79%
s(n) (m) 0.007 0.017 rmses (%) 19.28% 25.85%
s(e or c) (m) 0.012 0.017 rmseu (%) 7.42% 14.81%
b (-) 0.519 0.903 pes (%) 87.10% 75.29%
a (m) 0.037 0.017 peu (%) 12.90% 24.71%
mae (m) 0.010 0.012
rmse (m) 0.012 0.014
rmses (m) 0.011 0.012
rmseu (m) 0.004 0.007
d(2)
(-) 0.723 0.852
d(1)
(-) 0.551 0.651
4.3.5 Conclusions
For the uniform slope beach (1:20), the resulting wave height H is modeled well using
MILDwave. The wave height H at breaking is slightly underestimated, as is the case for
the model developed by Watanabe and Dibajnia (1988). This underestimation is to be
expected whenever linear theories are used. Also, wave heights are slightly
overestimated after breaking. Here, excessive shoaling combined with nonlinear effects
are the probable cause.
Before wave breaking, a standing wave pattern is observed. This is due to reflection on
the slope and is a natural occurring phenomenon. This is also observed by Gruwez (2011)
while validating the implemented wave-breaking module. However, for high wave
periods T and small water depths d (T=1.75 s and d=0.420 m to d=0.00 m on a uniform
slope), the numerical model MILDwave generates a reflected wave of very large wave
amplitude. This is not observed for shorter wave periods (T=0.75 s). It is reasonable to
assume that the same phenomenon also occurs in the experiments conducted in this case,
but it is clear that the wave amplitude a of the standing wave pattern is not dramatically
large compared to the generated wave height H.
In conclusion, MILDwave performs very well in the case of a uniform slope, with small
errors and a high index of agreement.
CHAPTER 4 – Test case: wave deformation in the surf zone 4-12
The other two test cases suffer much more of the non-linear effect described above.
Particularly in the downwave region after wave breaking occurs, the wave height H
predicted by MILDwave is much higher than the measured value during the
experimental study. Especially in the step-type beach experiment, the errors appear to be
larger. The difference in performance of the model between these two test cases is rather
great, but is most probably due to the difference in the values of the wave period T. This
causes sharper non-linear effects in the step-type beach case.
4.4 Influence of breaking coefficient K1, K2, K3 and K4
As mentioned in Paragraph 1.4.4, the value of the wave breaking coefficients K1, K2, K3
and K4 can be altered in the MILDwave Preprocessor. In the present paragraph, the
influence of these parameters on the numeric results is studied. The default values of the
coefficients are . These values where used in
the preceding paragraphs. In the present paragraph, 50%, 75%, 100% and 150% of these
values are used in a total of 12 experiments for each beach type, as shown in Table 11:
Table 11: 12 different simulation for varying breaking coefficient
K1 K2 K3 K4
Simulation 1 0.44 0.50 0.40 33.00
Simulation 2 0.66 0.50 0.40 33.00
Simulation 3 1.32 0.50 0.40 33.00
Simulation 4 0.88 0.25 0.40 33.00
Simulation 5 0.88 0.38 0.40 33.00
Simulation 6 0.88 0.75 0.40 33.00
Simulation 7 0.88 0.50 0.20 33.00
Simulation 8 0.88 0.50 0.30 33.00
Simulation 9 0.88 0.50 0.60 33.00
Simulation 10 0.88 0.50 0.40 16.50
Simulation 11 0.88 0.50 0.40 24.75
Simulation 12 0.88 0.50 0.40 49.50
Simulation 13 0.88 0.50 0.40 33.00
When analyzing influence of these wave breaking parameters, it might be possible to
correlate a physical process to each parameter. To do this, the simulation with the
uniform slope beach is considered. This beach has a slope of 1:20 from water depth
d=0.30 m to d=0.00 m and the toe of the beach is located at the position 6.00 m. Figure 29
shows the wave height along the uniform slope beach for varying K1 and constant values
of K2, K3 and K4.
CHAPTER 4 – Test case: wave deformation in the surf zone 4-13
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0123456
Wav
e h
eig
ht
[m]
position along the effective domain [m]
K1
Simulation 1
Simulation 2
Simulation 13
Simulation 3
Figure 29: wave height along the uniform slope beach for varying values of K1. Dotted line = Simulation 1,
medium dashed line = Simulation 2, long dashed line = Simulation 13, solid line = Simulation 3.
Both the location and the wave height at wave breaking are varying with K1. For higher
K1, wave breaking occurs at lower water depth, and waves are higher when they break.
Figure 30 shows the wave height along the uniform slope beach for varying K2 and
constant values of K1, K3 and K4.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0123456
Wav
e h
eig
ht
[m]
position along the effective domain [m]
K2
Simulation 4
Simulation 5
Simulation 13
Simulation 6
Figure 30: wave height along the uniform slope beach for varying values of K2. Dotted line = Simulation 4,
medium dashed line = simulation 5, long dashed line = simulation 13, solid line = simulation 6.
K2 has a similar influence on the location of wave breaking and the wave height at wave
breaking as K1. However, for varying K2, differences between the simulations are smaller
CHAPTER 4 – Test case: wave deformation in the surf zone 4-14
compared to differences for varying K1. Figure 31 shows the wave height along the
uniform slope beach for varying K3 and constant values of K1, K2 and K4.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0123456
Wav
e h
eig
ht
[m]
position along the effective domain [m]
K3
Simulation 7
Simulation 8
Simulation 13
Simulation 9
Figure 31: wave height along the uniform slope beach for varying values of K3. Dotted line = Simulation 7,
medium dashed line = simulation 8, long dashed line = simulation 13, solid line = simulation 9.
Varying values of K3 seem to have a small influence on physical processes. The four
simulations give comparable results. For lower K3, wave breaking starts at slightly larger
water depth. Figure 32 shows the wave height along the uniform slope beach for varying
K4 and constant values of K1, K2 and K3.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0123456
Wav
e h
eig
ht
[m]
position along the effective domain [m]
K4
Simulation 10
Simulation 11
Simulation 13
Simulation 12
Figure 32: wave height along the uniform slope beach for varying values of K3. Dotted line = Simulation 10,
medium dashed line = simulation 11, long dashed line = simulation 13, solid line = simulation 12.
Results are almost identical to the results obtained for varying K3 and constant values of
K1, K2 and K4.
CHAPTER 4 – Test case: wave deformation in the surf zone 4-15
In conclusion only K1 and K2 have a significant (and similar) influence on the location of
the wave breaking and the wave height at breaking. K3 and K4 have only limited
influence in the considered range.
4.5 Comparison between MILDwave simulations and other models
In the context of his phd thesis, Dogan K. (2012) developped an Excel sheet based on
Goda's (2010) theoretical approach. As input, this Excel sheet uses the still water level
TAW, the equivalent offshore wave height H0', the wave period T, the slope of the beach,
the level of the toe of the beach in m TAW, and the bathymetry of the beach. After
calculation, the significant wave height Hs is provided along the beach.
Also the phase-averaged model SwanOne is used to calculate Hs along the profile of the
beach. SwanOne is the same model as SWAN (2006) but uses the program in 1D mode.
However, results obtained using SwanOne are only indicative since the minimum grid
cell size of this model is . Therefore, all length dimensions are scaled with a
factor 100.
4.5.1 MILDwave numerical results, Goda (2010) theoretical results and SwanOne
numerical results for the uniform slope beach
As mentioned in Paragraph 4.4, 13 simulations of the uniform slope beach are performed
using MILDwave, in addition the phase-averaged model SwanOne and the Excel sheet
provided by Dogan (2012) based on the theoretical approach of Goda (2010) are used to
simulate wave breaking on a uniform slope beach.
In Figure 33, resulting wave heights along the uniform slope beach from thee MILDwave
simulations, the results of the SwanOne simulation and the results from the Excel sheet
by Dogan (2012) based in Goda (2010). MILDwave Simulation 10 gave best results,
compared to the theoretical solution by Goda (2010) and the SwanOne simulation. Other
MILDwave simulations results are to be found within the extremes of Simulation 1 and 3,
but are not displayed for the sake of clearness of the figure.
CHAPTER 4 – Test case: wave deformation in the surf zone 4-16
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.001.002.003.004.005.006.00
Wav
e h
eig
ht
[m]
Position along the effective domain [m]
theoretical by Goda (2010)SwanOne numerical modelMILDwave Simulation 10MILDwave Simulation 1MILDwave Simulation 3
Figure 33: Wave height along the uniform slope beach. Dotted line = MILDwave Simulation 1, short dashed
line = MILDwave Simulation 3, long dashed line line = MILDwave Simulation 10, Solid line = theoretical
result by Goda (2010) using an Excel sheet by Dogan (2012), line-point line = SwanOne numerical results.
Relatively good agreement is observed between the MILDwave Simulation 10 results and
both the theoretical solution by Goda (2010) and the SwanOne numerical model. In
MILDwave the wave height when wave breaking initiates is slightly larger than in Goda
and SwanOne, but when dissipation of energy initiates, the wave height decreases
towards the results of the SwanOne model.
4.5.2 MILDwave numerical results, Goda (2010) theoretical results and SwanOne
numerical results for the step type beach
As in Paragraph 4.5.1, in the case of the step type beach, 13 MILDwave simulations for
varying values of K1, K2, K3 and K4 are performed. Also one SwanOne simulation and one
solution according to Goda (2010) using the Excel file by Dogan (2012) are considered.
The bathymetry of the beach is given in Figure 25 and the names of the MILDwave
simulations for varying breaking coefficients are given in Table 11. SwanOne results are
obtained by scaling the length with a factor 100, since the maximal grid cell size in
SwanOne is
CHAPTER 4 – Test case: wave deformation in the surf zone 4-17
Figure 34 shows the resulting wave height along the step type beach for three MILDwave
simulations, the theoretical solution by Goda (2010) and the SwanOne simulation.
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.001.002.003.004.005.006.007.00
Wav
e h
eig
ht
[m]
Position along the effective domain [m]
theoretical by Goda (2010)SwanOne numerical modelMILDwave Simulation 11MILDwave Simulation 1MILDwave Simulation 3
Figure 34: Wave height along the step type beach. Dotted line = MILDwave Simulation 1, short dashed line =
MILDwave Simulation 3, long dashed line line = MILDwave Simulation 11, Solid line = theoretical result by
Goda (2010) using an Excel sheet by Dogan (2012), line-point line = SwanOne numerical results.
All MILDwave simulations are to be found between Simulation 1 and 3, but for the sake
of clearness only these two extremes are plotted. Simulation 11 provides the best results
compared to the theoretical solution by Goda (2010) and the SwanOne simulation. Again,
in MILDwave, wave breaking occurs at a higher wave height compared to the other
solutions.
4.5.3 MILDwave numerical results, Goda (2010) theoretical results and SwanOne
numerical results for the bar type beach
Again 13 MILDwave simulations are performed for varying breaking coefficients. Also
one SwanOne simulation and one solution according to Goda (2010) using the Excel file
by Dogan (2012) are considered. The bathymetry of the beach is given in Figure 25 and
the names of the MILDwave simulations for varying breaking coefficients are given in
CHAPTER 4 – Test case: wave deformation in the surf zone 4-18
Table 11. SwanOne results are obtained by scaling the length with a factor 100, since the
maximal grid cell size in SwanOne is
In Figure 35, the results are plotted. The Excel file provided by Dogan (2012) cannot
handle negative slopes so the results between x=3.60 m and x=1.60 m are not reliable
(solid line). Not all MILDwave Simulations are plotted, but only the two extreme
Simulations 1 and 3, together with the best result compared with the Goda (2010)
solution and SwanOne simulation, i.e. Simulation 10.
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.001.002.003.004.005.006.007.00
Wav
e h
eig
ht
[m]
Position along the effective domain [m]
theoretical by Goda (2010)SwanOne numerical modelMILDwave Simulation 10MILDwave Simulation 1MILDwave Simulation 3
Figure 35: Wave height along the bar type beach. Dotted line = MILDwave Simulation 1, short dashed line =
MILDwave Simulation 3, long dashed line line = MILDwave Simulation 11, Solid line = theoretical result by
Goda (2010) using an Excel sheet by Dogan (2012), line-point line = SwanOne numerical results.
A very close agreement between the MILDwave Simulation 10 and the SwanOne
simulation is observed. Again, wave breaking in MILDwave starts at the higher wave
height compared to the SwanOne and Goda (2010) solutions.
5 Wave transmission and diffraction
through a semi-infinite breakwater
In this chapter, partial wave absorption and reflection are studied in MILDwave. Different
MILDwave parameters that can influence the magnitude of the absorption are studied. Finally,
diffraction diagrams for waves passing through a partially absorbing semi-infinite breakwater are
generated using MILDwave.
CHAPTER 5 – Wave transmission and diffraction through a semi-infinite breakwater 5-2
5.1 General description of the experiment
Wave diffraction is a phenomenon in which energy is transferred laterally along a wave
crest. If this lateral transfer of wave energy along a wave crest and across orthogonals
would not occur, straight, long-crested waves passing the tip of a structure like a
breakwater would leave a region of perfect calm in the lee of the barrier, while beyond
the edge of the structure the waves would pass unchanged in form and height. The line
separating two regions would be a discontinuity.
It is important to be able to calculate diffraction effects because the wave height
distribution around natural or manmade structures in a harbour or sheltered bay is to
some degree determined by the diffraction characteristics of these structures. Wiegel
(1962) developed a theoretical approach to study wave diffraction around a semi-infinite
breakwater.
In this test case, the solution of Wiegel (1962) compared with the diffraction pattern
obtained using MILDwave. Also, diffraction patterns are presented for breakwaters with
some transmission. First, the influence of specific MILDwave parameters on the
transmission of structures are studied.
5.2 Study on the influence of numerical basin width, sponge layers, wave period T,
time step Δt, MILDwave transmission coefficient S and width of the breakwater
on the transmission in MILDwave
5.2.1 Influence of the numerical basin width and sponge layers on the wave
transmission in MILDwave
5.2.1.1 Setup of the experiments and calculation of MILDwave input parameters
In this section, the influence the numerical basin with and side sponge layer is studied.
Eight experiments for different wave periods ranging from 1 s to 4.5 s (
) were conducted in the 1D wave flume and in the 2D wave basin of width 3
cells. These experiments were also conducted in the 2D wave basin with an effective
basin width of 22 wavelengths, once with and once without side sponge layers. In all
these experiments, the length of the numerical basin t equals:
(5.1)
CHAPTER 5 – Wave transmission and diffraction through a semi-infinite breakwater 5-3
with the width of the sponge layer according to Finco (2011)1, the wave
length, For wave periods T<6, Finco advises to use sponge layer
function S1 so this sponge layer function is used in all the experiments. Since the wave
period is a variable and the water depth is kept constant at d=50m, the wave length
and thus the length of the numerical basin is a variable.
Further, the cell width
, with meaning that the second
decimal digit of the real number x is rounded upward, the time step
and
the moments when the calculation of Kd starts and stops are dependent on the wave
period. Furthermore, the wave height of the generated waves is H=0.10 m. In MILDwave,
a permeable breakwater or any porous structure can be modelled using the sponge layer
technique. Each grid cell of the breakwater contains an absorption coefficient S which
results in a certain degree of reflection and transmission by the structure. In this
experiment the breakwater has a width of , and each cell has an
absorption coefficient of S=0.25. An overview of the MILDwave input parameters is given
in Table 12.
Table 12: overview of MILDwave input parameters
T (s) 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
water depth d (m) 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00
wave length L (m) 1.56 3.51 6.25 9.76 14.05 19.13 24.98 31.62
wave height H (m) 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
grid cell size Δx=Δy (m) 0.04 0.08 0.13 0.20 0.29 0.39 0.50 0.64
time step Δt (s) 0.026 0.034 0.042 0.051 0.062 0.071 0.080 0.091
start calculation Ts (s) 65 97 130 161 194 226 258 291
end calculation Te (s) 265 397 530 661 794 926 1058 1191
numerical domain height ti (cells) 631 707 775 785 781 791 803 797
sponge layer width Bs (cells) 118 132 145 147 146 148 150 149
Sponge layer function S1 S1 S1 S1 S1 S1 S1 S1
MILDwave absorption coefficient S (-) 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
One exception is made, when using these parameters for the experiments with the 2D
module and an effective numerical domain of 22 wave lengths with side sponge layers,
CHAPTER 5 – Wave transmission and diffraction through a semi-infinite breakwater 5-4
an ‘overflow error’ is encountered. This is avoided by choosing a time step that is a
little less than the value from Table 12, as can be seen in Table 13:
Table 13: time step Δt for MILDwave simulations in 2D basin
T (s) 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
Δt (s) 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090
5.2.1.2 Results: influence of the basin width and side sponge layers on the Kd value.
In Figure 36, the results of experiments are plotted, performed for 8 values of wave
period T. Each wave period T is tested in four different layout modes in MILDwave: a) a
1D wave flume with a flume length of ti, b) a 2D wave basin of width 3 grid cells and
length ti, c) a 2D wave basin with an effective domain of 22L, with L the wave length but
without side sponge layers and length ti and d) a 2D wave basin with an effective domain
of 22L with side sponge layers and length ti . Experiments performed with the 1D module
of MILDwave are denoted with ‘1D’, experiments performed with the 2D module with a
basin width of 3 cells are denoted ‘2D3’, and experiments performed with the 2D module
with a effective numerical basin width of 22 wave lengths are denoted 2D(22L). As
mentioned above, the latter experiments are performed twice, once with and once
without side sponge layers.
0.04
0.045
0.05
0.055
0.06
0.065
0.07
0.075
0.08
1 1.5 2 2.5 3 3.5 4 4.5
Kd
(-)
T (s)
Kd - value in function of the period T
1D & 2D3 & 2D(22L) without sponge layer
2D(22L) with sponge layer
Figure 36: influence of wave period and numerical test area width on Kd testing 8 values of wave periods and
for four different layout modes in MILDwave. 'Δ' = results of the 1D, 2D3 and D2(22L) without sponge layer simulations (identical), '●'=results of the 2D(22L) with sponge layer simulations.
As expected, Kd-values from 1D, 2D3 and 2D(22L) without sponge layer experiments are
identical, meaning that basin width has no influence on the numerical permeability of the
CHAPTER 5 – Wave transmission and diffraction through a semi-infinite breakwater 5-5
breakwater. The Kd-values of 2D(22L) with sponge layer differ only marginally from the
1D Kd-values for higher wave periods T. The difference between 2D(22L) with sponge
layer and 1D results increases with decreasing waveperiod T. However, this difference
may not be the result of the decreasing wave period T, but rather of the increasing
difference in , as will be explained later in the Chapter.
Furthermore, there seems to be a small but noticeable dependence of the Kd-value of the
wave period T, especially for longer wave periods T. However, this dependency is
probably the cause of an increasing difference between
and
,
which would be the value of if is not rounded upward. Plotting the ratios of
and
, where represents the Kd values in the
corresponding experiment and where represents the Kd values in the 1D, 2D3 and
2D(22L) without sponge layer experiments, in function of the wave period results in
Figure 37, enhancing the assumption that there is a dependency between the time step
and the permeability of a structure with a fixed absorption coefficient S.
0.650.750.850.951.05
1 1.5 2 2.5 3 3.5 4 4.5
(-)
T (s)
ratios of Δt and Kd in function of T
Δti,a/Δti Kd2D,22,with sponge/Kd1D
Figure 37: ratios of Δt and Kd in function of the wave period T. '●' =
, '◊'=
5.2.2 Influence of the number of cells in the permeable structure on the Kd value
As mentioned above, a permeable structure is implemented by assigning a specific
absorption coefficient to each cell in the structure. It is therefore to be expected that when
the number of cells along the width of the structure increases, the total amount of
absorbed energy also increases. It is expected that if the number of cells in the structure
increases, the wave height of the waves behind the structure decreases. This will be
examined in the present paragraph.
CHAPTER 5 – Wave transmission and diffraction through a semi-infinite breakwater 5-6
5.2.2.1 Setup of the experiment and calculation of MILDwave input parameters
A total of 10 simulations are executed, each time with a different number of cells in the
permeable structure. In all simulations the wave period T, water depth d and thus
wavelength L are kept constant. The layout of the 1D domain is shown in Figure 38:
Figure 38: numerical basin layout for variable number of cells in breakwater in MILDwave.
In these simulations, the grid size
and time step
. With d=50,00
m and T=4.50 s this results in , and . The number of
cells varies from 3 to 30 with an increment of 3 cells resulting in a total of 10 simulations.
Each time, every cell in the permeable breakwater is assigned the same absorption
coefficient of S=0.90. The sponge layer width is 3L and the sponge layer function is S1,
according to Finco (2011).
5.2.2.2 Results: influence of the number of cells inside the permeable structure on
the Kd value
In Figure 39, black dots are the calculated value of Kd, plotted in function of the number
of cells in the permeable structure. This Kd value is the mean value in the region of
length 5L behind the structure. The diamonds in Figure 39 represent the function
(5.2)
With , the number of absorbing cells It is to be expected that the calculated values of
Kd follow this function closely since every extra cell in the permeable structure is just
another multiplication by S.
CHAPTER 5 – Wave transmission and diffraction through a semi-infinite breakwater 5-7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
3 6 9 12 15 18 21 24 27 30
Kd
thickness of the breakwater (# cells)
Kd in function of the number of absorbing cells
calculated Kd
y=0.9^N[as]
Figure 39: Kd in function of the number of absorbing cells, '●'=the calculated Kd by MILDwave, '◊'=expected
Kd basin on Equation (5.2).
Thus, to model a permeable structure with a number of cells and a permeability P, each
cell in the structure can be given a transmission coefficient equal given by Eq. (9.1).
(9.1)
5.2.3 Influence of the wave period T and time step Δt on the transmission of a
permeable structure with fixed absorption coefficient S
As mentioned in Paragraph 5.2.1, it is assumed that the time step Δt of the numerical
simulation has an important influence on the transmission of a permeable structure. The
present Paragraph focuses more closely on this assumption. Also, since the time step Δt is
dependent on the wave period T via the Courant-Friedrichs-Lewy (1928) criterion (Eq.
3.3), the influence of the wave period T on the permeability is studied.
5.2.3.1 Setup of the experiment and calculation of MILDwave input parameters
Three wave periods are considered: T1=3.0 s, T2=4.5 s and T3=6.0 s. The water depth d is 50
m. For each experiment, the grid cell size
and the maximum time step according
to the Courant-Friedrichs-Lewy criterion is
. Table 14 summarizes grid cell size
and maximum time step for the different wave periods T1, T2 and T3.
CHAPTER 5 – Wave transmission and diffraction through a semi-infinite breakwater 5-8
Table 14: grid cell size and maximum time step for 3 different wave periods.
T (s) 3.0 4.5 6.0
Δx (m) 0.35 0.75 1.40
Δtmax (s) 0.0747 0.1067 0.1495
The geometry of the simulation layout is the same as in Paragraph 5.2.2 and is shown in
Figure 38, the only exception being that for T=6.0 s, the sponge layer function is now S3,
according to Finco (2011). The absorption coefficient S=0.98 for each cell in every
experiment. For each wave period, 25 simulations are performed for five different values
of the time step Δt. These values are and .
For each time step, five experiments are performed with varying number of cells in the
breakwater. The number of cells in the permeable breakwater are 3, 5, 10, 20 and 30.
5.2.3.2 Results: influence of the time step and wave period on the wave transmission
behind the breakwater
MILDwave calculates a Kd value for every cell of the numerical domain. A mean value
for all the Kd values behind the breakwater is used in the present section. This value is
plotted in Figure 40, which contains 15 series of simulations with different wave periods
T and time steps Δt. Every series contains five Kd-values for different numbers of cells in
the breakwater. Also, the function is plotted.
As in paragraph 5.2.2, the Kd-value decreases with increasing number of cells in the
breakwater. However, the experiments conducted with follow very closely
the function , whereas the experiments with lower give increasingly lower
values of Kd. It is thus clear that the amount of energy absorbed by the cells in the
breakwater is dependent on the time step . Also, the difference between these Kd
values becomes greater with increasing number of cells in the breakwater. Vermeeren
(2011) describes how this phenomenon occurs. In MILDwave, absorption is simulated by
multiplying the water elevation η with the absorption coefficient S. Since S is not
dependent on the time step , this gives the following implication:
Consider one grid cell on position (i,j) with absorption coefficient within a numerical
domain in MILDwave. The elevations η' after a time step within this domain are
multiplied with the absorption coefficient, in this case S. The elevation in this
cell after multiplication with its absorption coefficient is given in Eq. (9.2)
(9.2)
In Eq. (9.2) is the elevation calculated for a time step on time t + . If the
time step is now reduced to a smaller value, say , then the elevation η' for has to
be calculated and then multiplied by the absorption coefficient of the cell :
CHAPTER 5 – Wave transmission and diffraction through a semi-infinite breakwater 5-9
The elevation at is calculated by repeating this procedure, giving Eq. (9.3)
(9.3)
In Eq. (9.3), is the elevation calculated for a time step at ,
this value is directly dependent on . The elevation η in a cell with absorption
coefficient at , calculated with a time step is thus dependent on and
more absorption occurs than when calculated with a time step . Vermeeren (2011)
solves this problem by using an absorption coefficient that is time step dependent.
For a small number of absorbing cells in the breakwater , some difference is observed
between Kd-values of experiments with different wave period T but the same time step
. These simulations have the same colour, but a different symbol in Figure 40. This
difference decreases with increasing number of absorbing cells in the breakwater . For
, this difference is negligible. Therefore, for a low number of absorbing cells
in any permeable structure, the calculated Kd value is dependent on the wave period
T. To remove this dependence, it is wise to use a higher number of absorbing cells in
the structure.
CHAPTER 5 – Wave transmission and diffraction through a semi-infinite breakwater 5-10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
3 5 10 20 30
Kd
# cells in breakwater
Kd as a function of T, Δt and N[as]
y=0.98^N[as]
T=3.0 s & Δt=0.0747 s
T=4.5 s & Δt=0.1067 s
T=6.0 s & Δt=0.1495 s
T=3.0 s & Δt=0.0598 s
T=4.5 s & Δt=0.0854 s
T=6.0 s & Δt=0.1196 s
T=3.0 s & Δt=0.0448 s
T=4.5 s & Δt=0.064 s
T=6.0 s & Δt=0.0897 s
T=3.0 s & Δt=0.0299 s
T=4.5 s & Δt=0.0427 s
T=6.0 s & Δt=0.0598 s
T=3.0 s & Δt=0.0149 s
T=4.5 s & Δt=0.0213 s
T=6.0 s & Δt=0.0299 s
Figure 40: Kd value in function of #cells in breakwater, T and Δt
CHAPTER 5 – Wave transmission and diffraction through a semi-infinite breakwater 5-11
5.2.4 Concluding recommendations for the implementation of permeable structures
in MILDwave
If possible, it is recommended to use a high number of absorbing cells in the
structure. This removes the dependency between the absorbed energy and the wave
period T.
For
, the Kd-value behind the permeable structure can be very well
predicted by the relation or, if Kd is given, the value of S can be found by using
relation (9.1). If , then the value of Kd will be lower than predicted. If
, the Courant-Friedrichs-Lewy (1928) criterion is not fulfilled.
5.3 Diffraction diagrams for semi-infinite breakwaters with partial transmission of
wave energy
In this section, diffraction diagrams will be presented for semi-infinite breakwaters with a
partial transmission of 0%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80% and 90%. Wiegel
(1962) created the first diffraction diagrams for semi-infinite breakwaters for incident
angles between 0° and 90°. In the present paragraph, the incident angle is always 90°.
5.3.1 Bathymetry setup and MILDwave parameters
Figure 41 shows the numerical domain in MILDwave. Waves are generated in front of
the bottom sponge layer towards the breakwater, represented by the black area. Behind
the breakwater, a region with a length of 11L=347.82 m is implemented to allow the
development of the diffraction pattern. At the top of Figure 41, another sponge layer is
implemented to absorb the diffracted waves due to the presence of the breakwater.
Sponge layer width (3L) and type (S1) are in accordance with Finco (2011). The numerical
width of the domain is 22L=695.64 m, no side sponge layers are added. All simulations
are performed for wave height H= 1.00 m, wave period T=4.50 s and water depth d=50.00
m, resulting in a wave length L=31.62 m. The width of the breakwater is kept constant at
30 grid cells (=24 m) as to ensure a minimal dependency of the results on the wave period
T. The grid cell size
and
. The calculation starts
after 64 waves lengths have travelled throughout the domain and stops when 264 wave
lengths are reached.
CHAPTER 5 – Wave transmission and diffraction through a semi-infinite breakwater 5-12
Figure 41: bathymetry layout used in MILDwave to generate diffraction diagramsbehind a semi-infinite and
partially permeable breakwater.
The same absorption coefficient S is assigned to each cell in the breakwater.. The S value
found by using Eq. (9.1) gives a slightly higher transmission than desired, especially for
experiments with 10% and 20% transmission. This problem is resolved in the following
way: if the found transmission is y times greater than desired, the value P in Eq. (9.1) has
to be divided by y:
(9.4)
The value S' gives much better results when implemented in MILDwave, as illustrated in
Table 15.
CHAPTER 5 – Wave transmission and diffraction through a semi-infinite breakwater 5-13
Table 15: Iterative calculation of the MILDwave absorption coefficient S, for the semi-infinite and partially
absorbing breakwater.
# cells 30
Desired transmission P 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
MILDwave absorption coefficent S 0.93 0.95 0.96 0.97 0.98 0.98 0.99 0.99 1.00
Resulting transmission with S 0.12 0.22 0.32 0.41 0.51 0.61 0.71 0.81 0.91
y 1.24 1.10 1.05 1.03 1.02 1.01 1.01 1.01 1.01
Corrected MILDwave absorption coefficient S' 0.92 0.94 0.96 0.97 0.98 0.98 0.99 0.99 1.00
Resulting transmission with S' 0.11 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
5.3.2 Results: graphical representation of Kd value behind a semi-infinite
breakwater with partial transmission of wave energy
Using MATLAB, the Kd value behind the semi-infinite breakwater with a partial
transmission of wave energy can be displayed. However, the initial results are rather
irregular and difficult to interpret as illustrated in Figure 42 for a permeability of 50%.
Figure 42: initial MILDwave results for P=50%, t=4.5 s, d=50.0 m
These results, as presented in Figure 42 are not very useful. In order to make the results
more easy to use, the Kd value in every cell has been replaced by the mean value of the
Kd values of all the cells within a square with a side of 30 cells around this cell. This post-
CHAPTER 5 – Wave transmission and diffraction through a semi-infinite breakwater 5-14
processing results to more useful result representation. Below, Figure 43 to Figure 52 are
results for a breakwater permeability ranging from 0 to 90%.
Figure 43: Wiegel diagram for a semi-infinite and non-permeable breakwtaer in MILDwave. Permeability
P=0%, wave period T=4.5 s, water depth d=50.0 m. Thicker lines are theoretical results by Wiegel (1962).
Very good resemblance is achieved between the post-processed MILDwave simulation
data and the theoretical solution by Wiegel (1962). This was also observed by Beels (2009)
Since MILDwave is clearly capable to simulate wave diffraction on a semi-infinite
impermeable breakwater, it is assumed that results of the simulations for semi-infinite
permeable breakwaters are reliable.
CHAPTER 5 – Wave transmission and diffraction through a semi-infinite breakwater 5-15
Figure 44: Wiegel diagram for a semi-infinite and partially permeable breakwater. Permeability P=10%, time
step T=4.5 s, water depth d=50.0 m.
Figure 45: Wiegel diagram for a semi-infinite and partially permeable breakwater. Permeability P=20%, time
step T=4.5 s, water depth d=50.0 m.
CHAPTER 5 – Wave transmission and diffraction through a semi-infinite breakwater 5-16
Figure 46: Wiegel diagram for a semi-infinite and partially permeable breakwater. Permeability P=30%, time
step T=4.5 s, water depth d=50.0 m.
Figure 47: Wiegel diagram for a semi-infinite and partially permeable breakwater. Permeability P=40%, time
step T=4.5 s, water depth d=50.0 m.
CHAPTER 5 – Wave transmission and diffraction through a semi-infinite breakwater 5-17
Figure 48: Wiegel diagram for a semi-infinite and partially permeable breakwater. Permeability P=50%, time step
T=4.5 s, water depth d=50.0 m.
Figure 49: Wiegel diagram for a semi-infinite and partially permeable breakwater. Permeability P=60%, time
step T=4.5 s, water depth d=50.0 m.
CHAPTER 5 – Wave transmission and diffraction through a semi-infinite breakwater 5-18
Figure 50: Wiegel diagram for a semi-infinite and partially permeable breakwater. Permeability P=70%, time
step T=4.5 s, water depth d=50.0 m.
Figure 51: Wiegel diagram for a semi-infinite and partially permeable breakwater. Permeability P=80%, time
step T=4.5 s, water depth d=50.0 m.
CHAPTER 5 – Wave transmission and diffraction through a semi-infinite breakwater 5-19
Figure 52: Wiegel diagram for a semi-infinite and partially permeable breakwater. Permeability P=90%, time
step T=4.5 s, water depth d=50.0 m.
Conclusions
Considering the test case of wave propagation over a submerged spherical shaped island
(Chapter 2), MILDwave produces results that are consistent with experimental data. To
quantitatively compare the results of the MILDwave simulation with experimental data
points, a numerical analysis is performed using parameters for model performance. This
analysis shows indeed very good agreement between the MILDwave simulation and
experimental data. This test case shows that MILDwave is very well able to simulate
wave propagation over a submerged island.
In Chapter 3, the test case of resonance in a rectangular harbour is studied. MILDwave
results are compared to analytical solutions, experimental data points and another
numerical model. Again, very good agreement between MILDwave results and these
data is observed. Leading to the conclusion that MILDwave can simulate the
phenomenon of harbour resonance very accurately. Both the eigen periods of the harbour
and the maximum amplitude of the standing wave pattern that is formed are simulated
very well.
Chapter 4 studies wave deformation in the surf zone. Three different types of beaches are
considered. This test case is primarily chosen to study the wave breaking module in
MILDwave. MILDwave simulations are compared to experimental data points, a
theoretical solution and another wave propagation model. Good agreement is observed,
however, since MILDwave is a linear wave propagation model, nonlinear effects are not
simulated leading to less accurate results than both previous test cases.
In Chapter 5, wave propagation through partially permeable structures is considered.
The relationship between the time step of the MILDwave simulation and the
absorption coefficient S assigned to a partially permeable grid cell is studied, and some
recommendations for simulations with partially permeable structures are formulated.
Diffraction diagrams for partially permeable, semi-infinite breakwaters are developed.
For a semi-infinite and impermeable breakwater, a comparison is made between the
MILDwave simulation and the theoretical solution by Wiegel (1962), showing very good
comparison, as was also observed in Beels (2009).
Instruments
For the research and compilation of the thesis the following software is used:
Adobe ® Photoshop ® CS6 v. 13.0
AutoCAD ® 2012
GetData Graph Digitizer 2.24
MATLAB ® 2011
Microsoft ® Office Excel® 2007
Microsoft ® Office Word 2007
MILDwave preprocessor v.3.02
MILDwave calculator v.3.02
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Appendix: MATLAB code for preparation and analysis of
MILDwave simulations
A.1 MATLAB Code for preparation of directories and files for the harbor
resonance test case
Since 63 simulations were conducted, it has proven to be meaningful to use a MATLAB
script to automatically generate files that are otherwise manually created using the
MILDwave preprocessor. This makes it easy and quick to change specific parameters as
the numerical basin width w', grid cell size Δx, Δy or time step Δt and sponge layer width
and type and to examine the influence of these parameters on the resulting standing
wave pattern. Moreover, errors that can occur by manually entering parameters in the
MILDwave preprocessor are avoided. Below, this MATLAB code is presented.
A.1.1 STEP 1: creating a text file which contains important information for the
simulation
First, a text file 'data.txt' is generated using Microsoft Excel. The 'data.txt' file contains
necessary input data for the MILDwave preprocessor. 'data.txt' contains 63 rows, one for
each simulation and 9 columns, for variable parameters. Table 16 shows the structure of
'data.txt':
Table 16: structure of the text file 'data.txt' used for input in the MILDwave Preprocessor
kl [-] L [m] T [s ] h width sponge y-position wg1 y-position wg2 start calculation [s ] timesteps/wave length
1.05 186 24.3 1897 557 558 1886 352 487
… … … … … … … … …
h stands for the length of the numerical basin, this value is dependent on the wave length
L and thus the wave period T, as mentioned in the text.
A.1.2 STEP 2: importing 'data.txt' into MATLAB
The text file 'data.txt' is now imported in MATLAB:
cd('Directory Location')
da=importdata('data.txt'); b=300; %the width of the effective domain
A.1.3 STEP 3: generating input files for the MILDwave Preprocessor and MILDwave
Calculator using the imported data from 'data.txt'
The MILDwave preprocessor generates input files for the MILDwave Calculator.
Extensive use of the preprocessor is avoided by automatic generation of these input files
using MATLAB, saving time and avoiding errors. These input files are 'Pos_WG.txt',
which contains information about the number and the location of inserted wave gauges.
'RGB.txt', to assign a cell type to a cell with a certain RGB color. 'CTOBST.txt', containing
absorption coefficients for different cell types. 'DEPTH_T', to assign different water
depths for different cell types.
for i=1:length(da(:,1)) %This for loop repeats the same operations
%for every experiment in the data.txt file da(i,5)=round(da(i,5)/2)*2; %make directory mkdir(strcat('Directory Location\subdirectory',int2str(i))); cd(strcat('Directory Location\subdirectory',int2str(i)));
%Pos_WG.txt setup %2 wave gauges are inserted in the MILDwave
%file, their location varies with the wave length. fileID=fopen('Pos_WG.txt','w'); fprintf(fileID,'%s\r\n','[WG settings]'); fprintf(fileID,'%s\r\n','WG_series=1'); fprintf(fileID,'%s\r\n','Num_WG=2'); fprintf(fileID,'%s\r\n','[WG_series_1]');
fprintf(fileID,'%s\r\n',strcat('xcoordinaat_WG1=',int2str(da(i,5)/
0.4+b/0.4/2)));
fprintf(fileID,'%s\r\n',strcat('ycoordinaat_WG1=',int2str(da(i,6)+
2)));
fprintf(fileID,'%s\r\n',strcat('xcoordinaat_WG2=',int2str(da(i,5)/
0.4+b/0.4/2)));
fprintf(fileID,'%s\r\n',strcat('ycoordinaat_WG2=',int2str(da(i,7))
)); fprintf(fileID,'%s\r\n','number of WG in series_1=2'); fclose(fileID);
%CTOBST.txt setup (copy of default file which is located in
%'Directory Location\subdirectory') copyfile('Directory Location\subdirectory\CTOBST.txt');
%RGB.txt setup (copy of default file which is located in
%'Directory Location\subdirectory') copyfile('Directory Location\subdirectory\RGB.txt');
%MILDwave.ini fileID2=fopen('MILDwave.ini','w'); fprintf(fileID2,'%s\r\n','[grid]');
fprintf(fileID2,'%s\r\n',strcat('gridsize_x=',int2str(2*da(i,5)/0.
4+b/0.4)));
fprintf(fileID2,'%s\r\n',strcat('gridsize_y=',int2str(da(i,4)))); fprintf(fileID2,'%s\r\n',strcat('gridfile=Directory
Location\subdirectory\,int2str(i),'\grid')); fprintf(fileID2,'%s\r\n','Deltax=0,4'); fprintf(fileID2,'%s\r\n','Deltay=1'); fprintf(fileID2,'%s\r\n',strcat('ixsL=',int2str(da(i,5)))); fprintf(fileID2,'%s\r\n',strcat('ixsR=',int2str(da(i,5)))); fprintf(fileID2,'%s\r\n',strcat('jysB=',int2str(da(i,5)))); fprintf(fileID2,'%s\r\n','jysT=0'); fprintf(fileID2,'%s\r\n','No of intervals for frequency=50'); fprintf(fileID2,'%s\r\n','Type Spongelayer=1'); fprintf(fileID2,'%s\r\n',strcat('Wave generation j-
line=',int2str(da(i,5)+3))); fprintf(fileID2,'%s\r\n','Wave generation i-line=0'); fprintf(fileID2,'%s\r\n','LenD_mu=1.04'); fprintf(fileID2,'%s\r\n','LenD_aaa=60'); fprintf(fileID2,'%s\r\n','[Timestep]'); fprintf(fileID2,'%s\r\n','delt=0.05');
fprintf(fileID2,'%s\r\n',strcat('twfin=',int2str(round(da(i,3)*24+
2)))); fprintf(fileID2,'%s\r\n','[Bathymetry]'); fprintf(fileID2,'%s\r\n','dw=6'); fprintf(fileID2,'%s\r\n','dmin=5'); fprintf(fileID2,'%s\r\n','dmax=0'); fprintf(fileID2,'%s\r\n','usedtxt=0'); fprintf(fileID2,'%s\r\n','usedbmp=0'); fprintf(fileID2,'%s\r\n',strcat('text depth Filename=Directory
Location\subdirectory\',int2str(i),'\dtxt')); fprintf(fileID2,'%s\r\n','type depth file=2'); fprintf(fileID2,'%s\r\n','set cell type k4=0'); fprintf(fileID2,'%s\r\n','set elevation k4=0'); fprintf(fileID2,'%s\r\n','set cell type k3=0'); fprintf(fileID2,'%s\r\n','set elevation k3=0'); fprintf(fileID2,'%s\r\n','dct1=1'); fprintf(fileID2,'%s\r\n','dct2=-0.0001'); fprintf(fileID2,'%s\r\n','dct3=6.5'); fprintf(fileID2,'%s\r\n','dk1=-5'); fprintf(fileID2,'%s\r\n','dk4=-4'); fprintf(fileID2,'%s\r\n','cell type k4=2'); fprintf(fileID2,'%s\r\n','cell type k1=3'); fprintf(fileID2,'%s\r\n','[Wave_char]'); fprintf(fileID2,'%s\r\n','Hw=0,25');
fprintf(fileID2,'%s\r\n',strcat('Tw=',
strrep(num2str(da(i,3)),'.',','))); fprintf(fileID2,'%s\r\n','Rw°=90'); fprintf(fileID2,'%s\r\n','[Wave_gen]'); fprintf(fileID2,'%s\r\n','startWGEN=0');
fprintf(fileID2,'%s\r\n',strcat('stopWGEN=',int2str(round(da(i,3)*
24+2)))); fprintf(fileID2,'%s\r\n','IRRwave=0'); fprintf(fileID2,'%s\r\n','D1D2gen=2'); fprintf(fileID2,'%s\r\n','Generation Type=1'); fprintf(fileID2,'%s\r\n','v_iline=0'); fprintf(fileID2,'%s\r\n','v_jline=1'); fprintf(fileID2,'%s\r\n','[IRRWave_gen]'); fprintf(fileID2,'%s\r\n','Stype=1'); fprintf(fileID2,'%s\r\n','theta_o°=90'); fprintf(fileID2,'%s\r\n','FStype=2'); fprintf(fileID2,'%s\r\n','gamma=3.3'); fprintf(fileID2,'%s\r\n','[Wave_breaking]'); fprintf(fileID2,'%s\r\n','Wave breaking=0'); fprintf(fileID2,'%s\r\n','K1=0.150000005960464'); fprintf(fileID2,'%s\r\n','K2=0.589999973773956'); fprintf(fileID2,'%s\r\n','K3=0.509999990463257'); fprintf(fileID2,'%s\r\n','K4=25'); fprintf(fileID2,'%s\r\n','K1_reg=0.879999995231628'); fprintf(fileID2,'%s\r\n','K2_reg=0.5'); fprintf(fileID2,'%s\r\n','K3_reg=0.400000005960464'); fprintf(fileID2,'%s\r\n','K4_reg=33'); fprintf(fileID2,'%s\r\n','[Variance_calc]'); fprintf(fileID2,'%s\r\n','useVAR=1'); fprintf(fileID2,'%s\r\n',strcat('tv1=',int2str(da(i,3)*8))); fprintf(fileID2,'%s\r\n',strcat('tv2=',int2str(da(i,3)*24))); fprintf(fileID2,'%s\r\n','rho=1026'); fprintf(fileID2,'%s\r\n','[Data_output]'); fprintf(fileID2,'%s\r\n','useWG=1'); fprintf(fileID2,'%s\r\n',strcat('Wavegauges filename=Directory
Location\subdirectory\',int2str(i),'\Pos_WG.txt')); fprintf(fileID2,'%s\r\n','number of WG series=1'); fprintf(fileID2,'%s\r\n','number of WG per series=2'); fprintf(fileID2,'%s\r\n','use3DMeshfile=0'); fprintf(fileID2,'%s\r\n','Num3D=1'); fprintf(fileID2,'%s\r\n','Power vector field=0'); fprintf(fileID2,'%s\r\n','[Bitmap]'); fprintf(fileID2,'%s\r\n','UseBitmapCT=1'); fprintf(fileID2,'%s\r\n',strcat('Bitmap CT Filename=Directory
Location\subdirectory\',int2str(i),'\Harb.bmp')); fprintf(fileID2,'%s\r\n',strcat('Bitmap depth Filename=
Directory Location\subdirectory\',int2str(i),'\dbitmap'));
%make bitmap of harbor ('Harb.bmp') B=zeros(da(i,4),(b+da(i,5)*2)/0.4);
for a=1:41 if a<11 for j=1:(b+2*da(i,5))/0.4 B(a,j)=1; end; else for j=1:((2*da(i,5)+b)/0.4-15)/2 B(a,j)=1; B(a,((b+da(i,5)*2)/0.4)-j+1)=1; end; end; end; matr=repmat((~B)*255,[1,1,3]); imwrite(matr,'Harb.bmp','bmp');
%copy of preprocessor (optional)
%copyfile('DirectoryLocation\subdirectory\Mildwave_PREPRO.exe') end;
When preparation of all the directories and files has finished, it is necessary to open the
MILDwave.ini file in the MILDwave Preprocessor in order to generate the 'ct.dat' and
'd.dat' files. These files cannot be generated using MATLAB but are indispensible for the
MILDwave Calculator.
Computation starts by importing the MILDwave.ini file into the MILDwave Calculator.
A.2 MATLAB code for preparation of the .bmp file of a spherical island that is
used in the test case of waves propagating over a submerged island
In chapter 2, the test case of wave propagation over a submerged island is studied. In
order to create a spherical shaped submerged island, a MATLAB code is developed. This
code creates the bitmap file that is used as an input file in MILDwave to implement the
submerged island. This bitmap file has the dimensions NxxNy and contains the island,
which is modeled with 15 different colour shades. Each of these colour shades can be
assigned to a specific water height in the MILDwave preprocessor.
cd('Directory');%define the directory where the bmp file will be
saved l=0.40;%the wave length deltal=0.01;%the grid size dx sp1=3.5*l;%the top and bottom sponge layer width
sp2=0*l;%the left and right sponge layer width aanloop=2*l;%distance between wave generation line and the most
southern point of the island afloop=10*l;%distance between the most northern point of the
island and the top sponge layer breedte=30*l;%width of the effective domain M=zeros(2*sp1/deltal+afloop/deltal+4*l/deltal+aanloop/deltal,2*sp2
/deltal+breedte/deltal,3);%creating a matrix with the dimensions
of the numerical domain, the third dimension defines the RGB
colour matr=zeros(2*l/deltal,2*l/deltal);%creating a matrix in which a
quarter of the island will be created, this matrix will be
mirrored to create the island matr1=zeros(4*l/deltal,4*l/deltal);%the diameter of the island is
4x the wave length L, creating a 4Lx4L matrix which circumscribes
the island %matr2=zeros(4*l/deltal,4*l/deltal,3); %a=zeros(2*l/deltal,2*l/deltal); matr(:,:,1)=0; y=((2*l/deltal)^2+100)/20; for i=1:2*l/deltal for j=1:2*l/deltal a=-(y-10)+sqrt(y^2-(j-1)^2-(i-1)^2); if a>0 matr(i,j)=round(a/10*15)/15; else j=2*l/deltal; end; end; end;%creation of one quarter of the island where the height of
each grid cell is defined by 15 rational numbers between 0 and 1 matr1(1:2*l/deltal,1:2*l/deltal)=flipdim(flipdim(matr,1),2);%mirro
r of matr is copied at the right place matr1(1:2*l/deltal,2*l/deltal+1:4*l/deltal)=flipdim(matr,1);%idem matr1(2*l/deltal+1:4*l/deltal,1:2*l/deltal)=flipdim(matr,2);%idem matr1(2*l/deltal+1:4*l/deltal,2*l/deltal+1:4*l/deltal)=matr;%idem for i=1:3 M(sp1/deltal+afloop/deltal+1:sp1/deltal+afloop/deltal+4*l/deltal,s
p2/deltal+(breedte/l/2-2)*l/deltal+1:sp2/deltal+(breedte/l/2-
2)*l/deltal+4*l/deltal,i)=matr1;%the island matr1 is copied at the
right place within the numerical domain end; imwrite(M,'eilandbw.bmp','bmp');%write the bmp file to the
specified Directory.
When the bmp image is created in the desired Directory, the MILDwave preprocessor can
be executed and in the tab 'Bathymetry', each colour of the bmp file can be assigned to a
desired water height. To assign the correct water height to the colours, the continuous
sphere has to be discretized, as is shown in Figure 53:
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0 20 40 60 80
he
igh
t [m
], 0
=se
a b
ott
om
leve
l
Number of cells 0=top the island, 81=the bottom of the island
Discretized sphere
Continuous sphere
Figure 53: continuous and discretizes distribution of the water height along the radius of the island
A.3 MATLAB code to prepare MILDwave input files for the MILDwave Calculator
and to analyze and display the Kd value in the effective domain after the MILDwave
simulation
Since a large number of MILDwave simulations is conducted, it is necessary to be
able to automatically create the input files for the MILDwave Calculator. When
the calculation is executed, a MATLAB code can analyze the data of multiple
simulations by using the output files of the MILDwave Calculator. A convenient
way to provide a qualitative view on the results is by plotting the value of the
disturbance coefficient Kd along the effective domain.
A.3.1 Preparation of the MILDwave Calculator input files using MATLAB
Below, the MATLAB code used to generate the MILDwave input files for the creation of
diffraction diagrams is presented. In 'Directory', 9 subdirectories are created, each
corresponding with a transmission of the breakwater from 10% to 90%.
jbegin=1; T=[4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5];%a list of wave periods
for each simulation d=[50 50 50 50 50 50 50 50 50];%a list of water depths H=[1 1 1 1 1 1 1 1 1];%a list of wave heights Trans=[0.9196 0.9446 0.959 0.969 0.9765 0.9827 0.9879
0.9924 0.9963];%a list of MILDwave transmission coefficients S doel=[10 20 30 40 50 60 70 80 90];%the expected transmission
behind the breakwater
g=9.81;%gravitational acceleration L=zeros(size(T,2),1);% a list of wavelengths (to be filled) C=zeros(size(T,2),1);% a list of phase velocities (to be filled) dx=zeros(size(T,2),1);% a list of grid cell sizes (to be filled) dt=zeros(size(T,2),1);% a list of time steps (to be filled) sp=zeros(size(T,2),1);% a list of sponge layer widths (to be
filled) z=zeros(size(T,2),1);% a list of distances (measured in a number
of grid cells) between the breakwater and the wave genaration line
(to be filled) t=zeros(size(T,2),1);% a list of the total length of the numerical
basin (to be filled) b=zeros(size(T,2),1);% a list of the total width of the numerical
basin (to be filled) tb=zeros(size(T,2),1);% a list of time instances when the
calculation of Kd starts (to be filled) te=zeros(size(T,2),1);% a list of time instances when the
calculation of Kd stops (to be filled) L0=zeros(10,1); % a list of wave lengths (to be filled) info=zeros(12,size(T,2)); % this matrix will contain important
parameters for the MILDwave simulations (to be filled)
for j=jbegin:1:size(T,2);
loc=strcat(Directory\Trans',num2str(doel(j)));
L0(1)=g*T(j)^2/(2*pi); mkdir(loc); cd(loc); for i=2:10 %the wavelenght L is calculated using the Newton-
Raphson convergence method for faster convergence L0(i)=1/2*L0(i-1)*g*T(j)^2*(2*pi*d(j)+L0(i-
1)*sinh(2*pi*d(j)/L0(i-1))*cosh(2*pi*d(j)/L0(i-1)))/(pi*(L0(i-
1)^2*cosh(2*pi*d(j)/L0(i-1))^2+g*T(j)^2*d(j))); end L(j)=L0(10); C(j)=L(j)/T(j); dx(j)=round(L(j))/40; %defining the grid cell size as 1/40th
of the wave length dt(j)=floor(dx(j)/C(j)*100)/100; %defining the time step
according to the Courant-Friedrichs-Lewy (1928) criterion if T(j)>9 %assigning the right with to the sponge layers,
according to Finco (2011) sp(j)=ceil(3.5*L(j)/dx(j)); else sp(j)=ceil(3*L(j)/dx(j)); end z(j)=ceil(5*L(j)/dx(j)); t(j)=2*sp(j)+1*z(j)+30+ceil(11*L(j)/dx(j)); b(j)=0*2*sp(j)+2*ceil(11*L(j)/dx(j)); tb(j)=ceil(4*t(j)*dx(j)/C(j));
te(j)=ceil(tb(j)+200*T(j)); M=zeros(t(j),b(j),3)+255;
M(sp(j)+ceil(11*L(j)/dx(j))+1:sp(j)+ceil(11*L(j)/dx(j))+30,b(j)/2+
1:end,2:3)=0; imwrite(M,'golfbreker.bmp','bmp'); info(1,j)=T(j); info(2,j)=d(j); info(3,j)=L(j); info(4,j)=H(j); info(5,j)=dx(j); info(6,j)=dt(j); info(7,j)=tb(j); info(8,j)=te(j); info(9,j)=sp(j); info(10,j)=z(j); info(11,j)=t(j);
%CTOBST.txt setup (copy of default file) %copyfile(strcat(loc,'\CTOBST.txt')); fileID1=fopen('CTOBST.txt','w'); fprintf(fileID1,'%s\r\n','[Obstacle]'); fprintf(fileID1,'%s\r\n','celltype 0=1'); fprintf(fileID1,'%s\r\n','celltype 1=0'); fprintf(fileID1,'%s\r\n',strcat('celltype
2=',num2str(Trans(j)))); for e=3:19 fprintf(fileID1,'%s\r\n',strcat('celltype
',int2str(e),'=1')); end
%RGB.txt setup (copy of default file) copyfile(aDirectory\RGB.txt');
%MILDwave.ini fileID2=fopen('MILDwave.ini','w'); fprintf(fileID2,'%s\r\n','[grid]'); fprintf(fileID2,'%s\r\n',strcat('gridsize_x=',int2str(b(j)))); fprintf(fileID2,'%s\r\n',strcat('gridsize_y=',int2str(t(j)))); fprintf(fileID2,'%s\r\n',strcat('gridfile=',loc,'\grid')); fprintf(fileID2,'%s\r\n',strcat('Deltax=',num2str(dx(j)))); fprintf(fileID2,'%s\r\n',strcat('Deltay=',num2str(dx(j)))); fprintf(fileID2,'%s\r\n','ixsL0='); %fprintf(fileID2,'%s\r\n',strcat('ixsL=',int2str(sp(j)))); fprintf(fileID2,'%s\r\n','ixsR0='); %fprintf(fileID2,'%s\r\n',strcat('ixsR=',int2str(sp(j)))); fprintf(fileID2,'%s\r\n',strcat('jysB=',int2str(sp(j)))); fprintf(fileID2,'%s\r\n',strcat('jysT=',int2str(sp(j)))); fprintf(fileID2,'%s\r\n','No of intervals for frequency=50'); if T(j)>5 && T(j)<10 fprintf(fileID2,'%s\r\n','Type Spongelayer=3');
else fprintf(fileID2,'%s\r\n','Type Spongelayer=1'); end fprintf(fileID2,'%s\r\n',strcat('Wave generation j-
line=',int2str(sp(j)+1))); fprintf(fileID2,'%s\r\n','Wave generation i-line=0'); fprintf(fileID2,'%s\r\n','LenD_mu=1.04'); fprintf(fileID2,'%s\r\n','LenD_aaa=60'); fprintf(fileID2,'%s\r\n','[Timestep]'); fprintf(fileID2,'%s\r\n',strcat('delt=',num2str(dt(j)))); fprintf(fileID2,'%s\r\n',strcat('twfin=',int2str(te(j)+1))); fprintf(fileID2,'%s\r\n','[Bathymetry]'); fprintf(fileID2,'%s\r\n','dw=0'); fprintf(fileID2,'%s\r\n','dmin=5'); fprintf(fileID2,'%s\r\n',strcat('dmax=-',int2str(d(j)))); fprintf(fileID2,'%s\r\n','usedtxt=0'); fprintf(fileID2,'%s\r\n','usedbmp=0'); fprintf(fileID2,'%s\r\n',strcat('text depth
Filename=',loc,'golfbreker.bmp')); fprintf(fileID2,'%s\r\n','type depth file=2'); fprintf(fileID2,'%s\r\n','set cell type k4=0'); fprintf(fileID2,'%s\r\n','set elevation k4=0'); fprintf(fileID2,'%s\r\n','set cell type k3=0'); fprintf(fileID2,'%s\r\n','set elevation k3=0'); fprintf(fileID2,'%s\r\n','dct1=1'); fprintf(fileID2,'%s\r\n','dct2=-0.0001'); fprintf(fileID2,'%s\r\n','dct3=6.5'); fprintf(fileID2,'%s\r\n','dk1=-5'); fprintf(fileID2,'%s\r\n','dk4=-5'); fprintf(fileID2,'%s\r\n','cell type k4=2'); fprintf(fileID2,'%s\r\n','cell type k1=3'); fprintf(fileID2,'%s\r\n','[Wave_char]'); fprintf(fileID2,'%s\r\n',strcat('Hw=',num2str(H(j)))); fprintf(fileID2,'%s\r\n',strcat('Tw=', num2str(T(j)))); fprintf(fileID2,'%s\r\n','Rw°=90'); fprintf(fileID2,'%s\r\n','[Wave_gen]'); fprintf(fileID2,'%s\r\n','startWGEN=0');
fprintf(fileID2,'%s\r\n',strcat('stopWGEN=',int2str(te(j)+1))); fprintf(fileID2,'%s\r\n','IRRwave=0'); fprintf(fileID2,'%s\r\n','D1D2gen=2'); fprintf(fileID2,'%s\r\n','Generation Type=1'); fprintf(fileID2,'%s\r\n','v_iline=0'); fprintf(fileID2,'%s\r\n','v_jline=1'); fprintf(fileID2,'%s\r\n','[IRRWave_gen]'); fprintf(fileID2,'%s\r\n','Stype=1'); fprintf(fileID2,'%s\r\n','theta_o°=90'); fprintf(fileID2,'%s\r\n','FStype=2'); fprintf(fileID2,'%s\r\n','gamma=3.3'); fprintf(fileID2,'%s\r\n','[Wave_breaking]'); fprintf(fileID2,'%s\r\n','Wave breaking=0');
fprintf(fileID2,'%s\r\n','K1=0.150000005960464'); fprintf(fileID2,'%s\r\n','K2=0.589999973773956'); fprintf(fileID2,'%s\r\n','K3=0.509999990463257'); fprintf(fileID2,'%s\r\n','K4=25'); fprintf(fileID2,'%s\r\n','K1_reg=0.879999995231628'); fprintf(fileID2,'%s\r\n','K2_reg=0.5'); fprintf(fileID2,'%s\r\n','K3_reg=0.400000005960464'); fprintf(fileID2,'%s\r\n','K4_reg=33'); fprintf(fileID2,'%s\r\n','[Variance_calc]'); fprintf(fileID2,'%s\r\n','useVAR=1'); fprintf(fileID2,'%s\r\n',strcat('tv1=',int2str(tb(j)))); fprintf(fileID2,'%s\r\n',strcat('tv2=',int2str(te(j)))); fprintf(fileID2,'%s\r\n','rho=1026'); fprintf(fileID2,'%s\r\n','[Data_output]'); fprintf(fileID2,'%s\r\n','useWG=0'); fprintf(fileID2,'%s\r\n',strcat('Wavegauges
filename=',loc,'\Pos_WG.txt')); fprintf(fileID2,'%s\r\n','number of WG series=1'); fprintf(fileID2,'%s\r\n','number of WG per series=1'); fprintf(fileID2,'%s\r\n','use3DMeshfile=0'); fprintf(fileID2,'%s\r\n','Num3D=1'); fprintf(fileID2,'%s\r\n','Power vector field=0'); fprintf(fileID2,'%s\r\n','[Bitmap]'); fprintf(fileID2,'%s\r\n','UseBitmapCT=1'); fprintf(fileID2,'%s\r\n',strcat('Bitmap CT
Filename=',loc,'\golfbreker.bmp')); fprintf(fileID2,'%s\r\n',strcat('Bitmap depth
Filename=',loc,'\dbitmap')); fclose(fileID2); %preprocessor kopieren
%copyfile('C:\Users\Student\Documents\Jonas\TC4_1\Mildwave_PREPRO.
exe') end fclose('all');
A.3.2 Analysis of the results: creation of a contour plot of Kd values throughout the
effective domain
When the MILDwave Calculator has finished the calculations of all the simulations (9 in
this case), another MATLAB code can be used to generate contour plots of the Kd value
throughout the effective domain. The code presented below loads the relevant
MILDwave output files, processes them, and finally creates the contour plots of the Kd
values for all simulation. These plots are saved as jpeg files to a specified Directory. It is
necessary to run this code after the code presented in Paragraph 0, since variables defined
in this code are used again in the code for the analysis of the results.
jbegin=1;
for j=jbegin:1:5%size(T,2) loc=strcat('Directory\Trans',num2str(doel(j))); cd(strcat(loc,'\data')); VARdata=load('VARdata.out'); eta = transpose(VARdata); S = size(eta); Ny = S(1,1); Nx = S(1,2); Beginx = sp(j);%da(i,5); %start value on the x-axis (!!!it has
to be =< compared to the dx of the basin) Beginy = sp(j)+ceil(L(j)/dx(j));%da(i,5); %start value on the y-
axis (!!!it has to be =< compared to the dy(=dx) of the basin) Stapx = 1; %step on the x-axis (it has to be changed according
to the used dx) Stapy = 1; %step on the y-axis (it has to be changed according
to the used dy(=dx)) cd('Directory'); figure colormap('gray'); a=61;%this pice of code is used to replace the value in each
cell by the mean value of all the cells in a square of side 61
grid cells surrounding this cell, thus creating a smoother and
more easely to interpret result A=(1/a^2)*ones(a); B=(1/a)*ones(1,a); eta2=conv2(eta,A,'same'); for b=0:floor(30)
eta2(sp(j)+ceil(5*L(j)/dx(j))+30+b,:)=mean(conv2(eta(sp(j)+ceil(5*
L(j)/dx(j))+30:sp(j)+ceil(5*L(j)/dx(j))+30+b*2,:),B,'same'),1); end ind1 = Beginx+ceil(5*L(j)/dx(j))+30:Stapx:Ny-sp(j)-
ceil(L(j)/dx(j)); % for the length of the basin or flume (it has
to be changed according to the used dx) ind2 = Beginy:Stapy:Nx-sp(j)-ceil(L(j)/dx(j))+1; % for the
width of the basin or flume (it has to be changed according to the
used dx) v=[0.01 0.05 0.10 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
0.65 0.7 0.75 0.8 0.85 0.9 0.95 1.0 1.05 1.10]; [C,h]=contour((ind2-Beginy)/(max(ind2)-min(ind2))*20-10,(ind1-
Beginx-ceil(5*L(j)/dx(j))-30)/(max(ind1)-
min(ind1))*10,eta(sp(j)+ceil(5*L(j)/dx(j))+30:Ny-sp(j)-
ceil(L(j)/dx(j)),Beginy:Nx-sp(j)-ceil(L(j)/dx(j))+1),v,'k'); clabel(C,h,v);%The greater this value is, the further the
labels are from each other xlabel('Width of domain [xL]','Fontsize',16); %The size of the
legend is modified automatic with the adjustment of the label of
the axes ylabel('Length of domain [xL]','Fontsize',16); axis equal; title('Calculated Kd [-]','Fontsize',16);
view([0 90]) % plan view axis ([-10 10 0 10]) saveas (h,strcat('basin_Kd_ ',num2str(doel(j))),'jpg');
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