Numerical validation of the mild-slope wave propagation...

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


3.3 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


4.1.1 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


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


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


numerical results for the step type beach 4-16


numerical results for the bar type beach 4-17

CHAPTER 5 - Wave transmission and diffraction through a semi-infinite breakwater


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


5-7

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




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


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.


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)


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):


(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.


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


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


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)


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.


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.


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.


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'.


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'.


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


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):


(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.


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)


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.


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.


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


Figure 16: Kd-values along Section 2'x' = MILDwave results for Δx=0.010 m, 'Δ'= MILDwave results for


Figure 17: Kd-values along Section 3, 'x' = MILDwave results for Δx=0.010 m, 'Δ'= MILDwave results for


0.0

0.5

1.0

1.5

2.0

0 1 2 3 4 5 6

Kd-v

alu

e

y/L


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


Δx = 0.01

Δx = 0.025

Δx = 0.02


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


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.


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


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.


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:


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.


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


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.


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.


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.


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


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)


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.


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:


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


4.1.1 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.


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


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)


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.


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


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.


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]


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.


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.


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)


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.


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.


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.


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]


K2

Simulation 4

Simulation 5

Simulation 13

Simulation 6


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


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]


K3

Simulation 7

Simulation 8

Simulation 13

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]


K4

Simulation 10

Simulation 11

Simulation 13

Simulation 12



Results are almost identical to the results obtained for varying K3 and constant values of

K1, K2 and K4.


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.


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.


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.


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


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]



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.


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


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]



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


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)


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,


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


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.



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.


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.


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.


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 :


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.


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


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.


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.


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-


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.


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 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.



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

References

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coastal areas and harbours including dissipation. Computer Modelling of Seas and Coastal

Regions III. International Conference. Southampton, United Kingdom: WIT Press, pp.

343-352

Baldock, T.E., Holmes, P., Bunker, S., Van Weert, P. (1998). Cross-shore hydrodynamics

within an unsaturated surf zone. Coastal Engineering, Vol. 34, pp. 173-196.

Battjes, J.A., Janssens, P.P.F.M. (1978). Energy loss and set-up due to breaking of random

waves. Proceedings of 16th Conference on Coastal Engineering, ASCE, New York, pp.

569-587.

Beels, C. (2009). Optimization of the lay-out of a farm of wave energy converters in the North

Sea. Ph.D. Thesis, Ghent University.

Berkhoff (1972). Computation of combined refraction-diffraction. Proceedings 13th

International Conference on Coastal Engineering, Vancouver, pp. 471-490.

Boshek, M. R. (2009). Reflection and Diffraction Around Breakwaters. Master's thesis, Delft

University.

Caspeele, R. (2006). Generatie van onregelmatige lang- en kortkruinige golven in een numeriek

model voor golfvoorplanting: implementatie, validatie en toepassing. Master's thesis, Ghent

University.

Courant, R., Friedrichs, K. Lewy, H. (1928). Über die partiellen Differenzengleichungen der

mathematischen Physik, Mathematische Annalen 100 (1) : 32-74.

Deigaard, R., Justesen, P., Fredsoe, J. (1991). Modelling of undertow by a one-equation

terbulance model. Coastal Engineering, Vol. 15, pp. 431-458.

Dingemans, M.W. (1997). Water wave propagation over uneven bottoms - Part I: Linear wave

propagation. Advanced Series on Ocean Engineering, Volume 13, World Scientific.

Finco, V. (2011). Validation test cases for the numerical wave propagation model “MILDwave”.

Master Dissertation for the degree of Civil Engineering, Ghent University.

Goda, Y. (2010). Random seas and design of maritime structures. World Scientific Publishing

Company.

Gruwez, V. (2008). Implementatie en validatie van golfbreking in het numeriek golfvoort-

plantingsmodel MILDwave (in Dutch). Master’s Thesis, Ghent University.

Helmholtz, H. von. (1885). On the sensations of tone as a physiological basis for the theory of

music. London: Longmans, Green, and Co.

Ito, Y., Tanimoto, K. (1972). A method of numerical analysis of wave propagation. Proceedings

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Larsen, J., Dancy, H. (1983). Open boundaries in short wave simulations - a new apprach.

Coastel Engineering, 7:285-297.

Lee, C., Suh, K.D. (1998). Internal generation of waves for time-dependent mild-slope equations.

Coastal Engineering, 34:35-57.

Maa, J.P.-Y., HSu, T.-W., Hwung, H.-H. (1998). RDE Model: A Program for Simulating Water

Wave Transformation for Harbor Planning. Special Scientific Report, No. 136.

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Nagayama, S. (1983). Study on the change of wave height and energy in the surf zone, Bachelor

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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');

Numerical validation of the mild-slope wave propagation...

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