Update on the use of parapets at storm walls for wave

Michiel Vanhalewyn, Arno Soetaert

overtopping: 2D model testsUpdate on the use of parapets at storm walls for wave

Academic year 2015-2016Faculty of Engineering and ArchitectureChair: Prof. dr. ir. Peter TrochDepartment of Civil Engineering

Master of Science in Civil EngineeringMaster's dissertation submitted in order to obtain the academic degree of

Counsellors: Maximilian Streicher, David Gallach SanchezSupervisor: Prof. dr. ir. Andreas Kortenhaus

Michiel Vanhalewyn, Arno Soetaert

overtopping: 2D model testsUpdate on the use of parapets at storm walls for wave

Academic year 2015-2016Faculty of Engineering and ArchitectureChair: Prof. dr. ir. Peter TrochDepartment of Civil Engineering

Master of Science in Civil EngineeringMaster's dissertation submitted in order to obtain the academic degree of

Counsellors: Maximilian Streicher, David Gallach SanchezSupervisor: Prof. dr. ir. Andreas Kortenhaus

i

PREFACE

As finalization of 5 years academic career as student Civil Engineering, major Dredging and

Offshore Engineering at Ghent University, we present our master’s dissertation. Over the last

six months we have intensively researched wave overtopping at storm walls and the effect of

parapets in further reducing this wave overtopping. We hope we brought up some new

knowledge about this topic and that the thesis can form a basis for further research.

In the process of making this master’s dissertation we relied upon the knowledge of the

members of Civil Engineering, unit Coastal Engineering, Bridges and Roads. Therefore our

sincere thanks go to the people who made this dissertation possible and helped us creating our

work.

First of all we want to thank our supervisor Prof. dr. ir. Andreas Kortenhaus who introduced

us in the interesting domain of wave overtopping and the coastal measure techniques present

to mitigate it. Also we’d like to thank him for the organizing several meetings, being

accessible for all our questions and his expertise in wave overtopping. In the same line we

would like to thank our counsellors Maximilian Streicher and David Gallach Sanchez to guide

us through the execution of the model tests itself and their guidance in respectively wave

forces and wave overtopping expertise. For all our technical questions and the preparation of

our experiments in the wave flume, we would like to thank the technical staff; Herman, Sam,

Dave and Tom.

Last thanking words go to our family, friends to support us through our academic career and

especially during the preparation of our dissertation.

Deze pagina is niet beschikbaar omdat ze persoonsgegevens bevat.Universiteitsbibliotheek Gent, 2021.

This page is not available because it contains personal information.Ghent University, Library, 2021.

v

ABSTRACT

Update on the use of parapets at storm walls for wave overtopping: 2D

model tests

By Arno Soetaert and Michiel Vanhalewyn

Master’s dissertation submitted in order to obtain the academic degree of Master of Science in

Civil Engineering

Supervisor: Prof. dr. ir. Andreas Kortenhaus

Counsellors: Maximilian Streicher, David Gallach Sanchez

Academic year 2015-2016

University Ghent – Faculty of Engineering and Architecture

Department of Civil Engineering

Chairman: Prof. dr. ir. Peter Troch

The average overtopping has been tested and analysed for many different kinds of

configurations, but many of those studies only included one single reduction factor. By

performing hydraulic models tests in the large wave flume of Ghent University, the effect of

combined measures is tested. In these tests a slope is combined with a berm and a wall or a

parapet. When performing the tests also the individual amounts of overtopping are measured

as well as the forces acting on the storm wall. By analysing the data, the current prediction

formula for the average overtopping is updated with new reduction factors for a wall, a

parapet and a berm. For the individual overtopping and wave forces, new prediction formulas

are drafted and an introduction to their relationship is given.

Keywords

wave overtopping, individual overtopping, wave forces, hydraulic model tests, storm walls,

parapets, smooth impermeable slope, berm

Update on the use of parapets at storm walls for

wave overtopping: 2D model tests

Arno Soetaert and Michiel Vanhalewyn

Supervisors: Prof. dr. ir. Andreas Kortenhaus, ir. Maximilian Streicher, ir. David Gallach Sanchez

Abstract – The average overtopping has been tested

and analysed for many different kinds of configurations,

but many of those studies only included one single

reduction factor. By performing hydraulic models tests

in the large wave flume of Ghent University, the effect of

combined measures is tested. In these tests a slope is

combined with a berm and a wall or a parapet. When

performing the tests also the individual amounts of

overtopping are measured as well as the forces acting on

the storm wall. By analysing the data, the current

prediction formula for the average overtopping is

updated with new reduction factors for a wall, a parapet

and a berm. For the individual overtopping and wave

forces, new prediction formulas are drafted and an

introduction to their relationship is given.

Keywords - wave overtopping, individual overtopping,

wave forces, hydraulic model tests, parapet.

I. INTRODUCTION

In literature much is already known about the

prediction of overtopping characteristics for a simple

slope, for non-breaking conditions. The average

overtopping q [l/s/m], individual overtopping

characteristics Vmax [kg/m] and V1/250 [kg/m] and the

probability of overtopping Pow can already been

predicted quite accurately [1,2].

Placing a vertical wall or a parapet on top of the

slope, including a berm between the top of the slope

and the vertical wall or parapet, will change the

geometry and influence the prediction formulas for

the overtopping characteristics. Hydraulic model tests

are performed to investigate what the influence is of

modifying the simple slope.

II. THEORETICAL BACKGROUND

A. Average overtopping q

For non-breaking wave conditions, the new

prediction formula proposed by van der Meer &

Bruce [2] for the dimensionless overtopping for a

simple slope, is given by Eq.(1).

A. Soetaert and M. Vanhalewyn are graduating students at the

Civil Engineering Department, Ghent University (UGent), Gent,

Belgium. E-mail: [email protected],

[email protected].

Eq.(1)

Eq.(2)

Eq.(3)

Eq.(4)

The dimensionless overtopping will be further

notated as Q. The formula takes the form of a Weibull

distribution. The factors a, b are dependent on the

angle α of the slope. The formula is valid for all

relative freeboards Rc/Hm0.

B. Probability of overtopping Pow

The probability of overtopping Pow is defined as the

ratio of the number of overtopping waves Now and the

total amount of waves Nw reaching the construction.

Eq.(5)

In literature [1], Pow is mostly predicted by a

Rayleigh type prediction formula in function of the

relative freeboard Rc/Hm0. (Eq.(6))

Eq.(6)

is a factor dependent on the structure.

C. Individual overtopping characteristics Vmax and

V1/250

Besides average overtopping quantities, also the

individual overtopping volumes are of importance.

Vmax [kg/m] is the maximum volume encountered

during a certain time frame, V1/250 [kg/m] the average

of 0.4% largest volumes of the same time frame.

III. HYDRAULIC MODEL TESTS

A. Geometrical configurations

Six different configurations, modifying the

reference case, will be tested. The reference case is a

simple slope with slope 1:2 (V:H). The six

configurations can be divided into two main groups:

configurations without berm and configurations with a

berm.

For the configurations without berm either a

vertical wall or parapet is installed on the top of the

slope. Two heights for the walls and parapets are

used: 5 and 10 cm.

For the configurations with a berm, two berm

lengths are used: 20 and 40 cm. At the end of the

berm either a vertical wall or parapet is installed.

Again two heights for the walls and parapets are used:

5 and 10 cm.

B. Test conditions

The hydraulic model tests were performed in the

Large Wave Flume of the Coastal Engineering

Section of the Civil Engineering Department at Ghent

University. All tests were carried out with irregular

waves (JONSWAP3.3 spectrum). A summary of the

significant wave heights Hs, wave steepnesses sp and

water depths d at the wave paddle used for the tests

can be found in Table 1:

Table 1: Wave conditions for the hydraulic model tests

Wave height Hs [cm] 5, 10 and 15

Wave steepness sp [-] 0.02 and 0.05

Water depth d [cm] 68, 70, 73 and 76

Not all water depths were used for each geometrical

configuration. In total 166 tests were performed (150

unique and useful tests and 16 repetition tests).

IV. RESULTS OVERTOPPING

A. Reference case: simple slope

The reference case is drawn in Fig. 1. The slope has

a height of 53 cm and rest in reality on a 20 cm thick

return channel.

Fig. 1: Reference case, simple slope

1) Average overtopping q

The average overtopping q for the 12 tests

performed was made dimensionless. The existing

prediction formula Eq.(1) with the correct coefficients

(cotα = 2) is given in Eq.(7) and plotted against the

data.

Eq.(7)

A good fitting of the dimensionless overtopping Q

was found for the test results, such that Eq.(7) will be

used as the reference equation for a simple slope.

Eq.(7) will be named the ‘updated vdM formula’.

2) Probability of overtopping Pow

By fitting Eq.(6) against the 12 test results, a

corresponding χ-factor can be determined. A χ-factor

of 0.917 is found. In literature already equations are

present for the determination of χ. One of them is

proposed by Victor [3] and given in Eq.(8).

Eq.(8)

With cotα = 2 a value for χ equal to 1.25 is found.

As Eq.(8) is based on much more test results the χ-

factor corresponding with Eq.(8) will be used for

comparison with other equations.

3) Vmax and V1/250

To investigate Vmax and V1/250 these quantities are

made dimensionless. In Eq.(9) and Eq.(10) the

dimensionless quantities considered are given. Due to

a lack of number of tests, no prediction formulas are

created for the configuration of a simple slope.

Eq.(9)

Eq.(10)

B. No berm and vertical wall

The configuration without berm and a vertical wall

(5 or 10 cm) on top of the slope is represented in Fig.

2.

Fig. 2: No berm with wall on top of slope


The average overtopping q is made dimensionless

and plotted against the relative freeboard Rc/Hm0.

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5

Q [-]

0,0001

0,001

0,01

0,1

no berm walltest results wallupdated vdM

Fig. 3: Data points no berm wall and the pred. form.

The data points are compared with the updated vdM

formula for a simple slope in Fig. 3. The data points

lie clearly below the updated vdM formula. The

updated vdM formula needs to be modified such that

it can be applied for the configuration with a wall on

top of the slope. A reduction factor is used to take

into account the influence of the wall. The adapted

equation is given in Eq.(11).

Eq.(11)

The -factors are determined by rewriting Eq.(11)

to and to fill in all the required quantities of the

considered test results. The resulting equation is given

in Eq.(12).

Eq.(12)

The values resulting out of Eq.(12) are plotted

against Rc/Hm0 (Fig. 4). For these values a prediction

formula is created, only function of Rc/Hm0. This

prediction formula is given in Eq.(13).

Rc/H

m0 [-]

0 1 2 3 4

v [-]

0,0

0,2

0,4

0,6

0,8

1,0

1,2

reduced data set

prediction formula v

v = 0.1441x + 0.5692

R² = 0.5876

Fig. 4: Reduction factor in function of Rc/Hm0

Eq.(13)

Eq.(13) is really valid for Rc/Hm0 in [0.3;2.2]. An

extrapolation is made for Rc/Hm0 > 2.2. Remarkable

for Eq.(13) is that for Rc/Hm0 > 2.99, the values for

is 1. This means that no extra reduction will be found

in comparison with a simple slope. The prediction

formulas for the dimensionless overtopping Q for the

configuration of a wall on top of the slope (Eq.(11)) is

plotted in Fig. 3. Eq.(11) is also strictly valid for

Rc/Hm0 in [0.3; 2.2].


Fitting Eq.(6) against the test results, the

corresponding χ-factor can be determined. A χ factor

of 1.083 is found.

3) Vmax and V1/250

The dimensionless quantities represented in Eq.(9)

and Eq.(10) are plotted in function of Rc/Hm0. An

exponential and power law relationship described best

the dependency on Rc/Hm0. For smaller relative

freeboards the power law relationship gave better

results and is used for the prediction formulas.

Eq.(14) and Eq.(15) give the found prediction

formulas. These are valid for Rc/Hm0 in the interval

[0.4; 3].

Eq.(14)

Eq.(15)

C. No berm parapet

The configuration without berm and parapet (5 or

10 cm) on top of the slope is represented in Fig. 5.

Fig. 5: No berm with parapet on top of slope


The data points are compared with the updated vdM

formula for a simple slope in Fig. 3. The data points

lie clearly below the updated vdM formula and also

the prediction formula for a wall on top of the slope.

Further, Q depends on the wave steepness sm-1,0.

Rc/Hm0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Q [-]

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

1e+0

sm-1,0 = 0.03

sm-1,0 = 0.05

no berm par. sm-1,0 = 0.03


updated vdMno berm wall

Fig. 6: Data points no berm parapet and the pred. form.

The updated vdM formula needs to be modified

such that it can be applied for the configuration with a

parapet on top of the slope. A reduction factor is

used to take into account the reduction in Q due to the

presence of a parapet. The reduction is considered

relative to the reduction already present by placing a

wall. The adapted equation is given in Eq.(16).

Eq.(16)

A similar procedure as for the configuration with a

wall is followed to determine with the help of the

test results. Only the resulting fitted equation for is

given.

Eq.(17)

Eq.(17) is valid for Rc/Hm0 in [0.3;1.7]. The prediction

formula for the dimensionless overtopping Q for the

configuration of a parapet on top of the slope

(Eq.(17)) is plotted in Fig. 6. For the two target

steepnesses another formula is plotted. Eq.(17) is also

strictly valid for Rc/Hm0 in [0.3;1.7].



corresponding χ-factor can be determined. A χ-factor

of 0.724 is found.

3) Vmax and V1/250

The dimensionless quantities Vdim,max and Vdim,1/250

are described by a power law. Eq.(18) and Eq.(19)

give the respective equations.

Eq.(18)

Eq.(19)

The equations are valid for Rc/Hm0 in [0.3;1.7]. The

same equations are valid when a berm of 20 or 40 cm

is present in front of the parapet. Therefor it can be

said that the berm width has no influence on the

dimensionless quantities Vdim,max and Vdim,1/250.

D. 20 cm or 40 cm berm and vertical wall

The configuration with berm (20 or 40 cm) and a

vertical wall behind the berm is represented in Fig. 7.

Fig. 7: 20 or 40 cm berm with vertical wall behind berm


In Fig. 8 the data points of the tests with a 20 or 40

cm berm are plotted along with the prediction formula

for the average overtopping in case of a vertical wall

without berm (Eq.(11)). It can be seen that the

formula over predicts the average overtopping,

especially for larger relative freeboards. An adaption

to the formula will have to be made to include the

influence of the berm.

Rc/Hm0 [-]

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Q [-]

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

Data points 20/40 cm berm wall no berm wall

Fig. 8: Data points berm wall and the pred. form.

The influence of the berm is included by adding a

reduction factor γb in the formula. The new prediction

formula is given in Eq.(20).

Eq.(20)

A similar procedure is followed to determine

with the help of the test results and only the equation

is given. The average overtopping with a berm and a

vertical wall was dependent on the steepness and

therefore this factor is included in the formula.

Eq.(21)

Eq.(21) is valid for Rc/Hm0 in the range of

[0.3;2.2]. The prediction formula for the average

overtopping in case of a berm and a vertical wall

(Eq.(20)) is plotted on Fig. 8.



corresponding χ-factors can be determined. A χ factor

of 0.972 and 0.810 are found for respectively a 20

and 40 cm berm.

3) Vmax and V1/250


are described by a power law. Eq.(22), Eq.(23) and

Eq.(24), Eq.(25) give the equations for respectively

20 and 40 cm berm. The equations are valid for

Rc/Hm0 in [0.3; 1.9].

Eq.(22)

Eq.(23)

Eq.(24)

Eq.(25)

E. 20 cm or 40 cm berm and parapet

The configuration with berm (20 or 40 cm) and a

parapet behind the berm is represented in Fig. 9.

Fig. 9: 20 or 40 cm berm with parapet behind berm


Installing a 20 or 40 cm does not influence the

average overtopping when a parapet is present.

Therefore the prediction formula is the same as

without the berm (Eq.(16)).



corresponding χ-factors can be determined. A χ factor

of 0.712 and 0.797 are found for respectively a 20

and 40 cm berm.

3) Vmax and V1/250


are described by a power law. Eq.(18) and Eq.(19)

give the respective equations.

V. RESULTS FORCES

For the analysis of the wave forces, only tests with a

vertical wall are considered. To verify the influence

of the wave height and the relative freeboard, F1/250 is

examined. F1/250 is the average force of the 0.4 %

highest wave impacts. It varies between 13.4 N/m and

330.0 N/m. This variation is the consequence of the

influence that the wave height and relative freeboard

have on F1/250. A larger wave height leads to a higher

wave force and an increasing relative freeboard leads

to a decreasing wave force. To check the influence of

the water depth, steepness and berm width and to

come up with a prediction formula, a dimensionless

force needs to be introduced. The dimensionless force

P is equal to:

Eq.(26)

In this formula F1/250 is the force as mentioned

before, ρ is the water density, g is the gravitational

acceleration and Rc is the freeboard. The

dimensionless force P of every test is plotted in

function of the relative freeboard. This plot is shown

in Fig. 10. On this plot, all data points are closely

gathered and therefore the influence of the water

depth, steepness and berm width can be assumed zero.

With the help of Excel, a prediction formula can be

determined. The only parameter that will have to be

included in this formula is the relative freeboard since

the other parameters do not influence the wave force.

The prediction formula is equal to:

Eq.(27)

On Fig. 10, this prediction formula is plotted

alongside the test results. The plot shows that the

prediction line follows the test data quite nicely and

the scatter is limited. The correlation coefficient R² is

equal to 0.91.

Rc/Hm0 [-]

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

P [-]

0.01

0.1

1

10

100

Prediction formula

P = 10.7*exp(-1.67x)

R² = 0.91

Fig. 10: Prediction formula plotted along with the test

results

To compare the wave forces acting on a simple

vertical wall to the ones acting on a parapet, 18 tests

are selected. When comparing the dimensionless

wave forces, the analysis shows that the forces acting

on the parapet are about 22 % higher than the ones

acting on the vertical wall. The reason for this rise in

horizontal forces is probably due to the fact that the

vertical forces that act on the parapet nose cause a

bending of the wall and this bending causes an

additional horizontal force onto the load cell.

However the exact reason for this phenomenon cannot

be determined since only horizontal forces were

measured. No prediction formula for the forces acting

on a parapet is determined because not enough data

was used for the analysis.

As a last part of the analysis of the forces, the

relationship between the wave forces and overtopping

was examined. To investigate whether there is a

relationship between both, the first 100 seconds of

three tests are studied. The three selected tests have a

water depth of 70 cm because the impacts and

overtopping events can be seen more clearly in this

case. The berm width is either 0, 20 or 40 cm. For the

three tests, a quadratic relationship between the wave

forces and overtopping can be seen. On Fig. 11 the

result for the test with a 20 cm berm is displayed. It

can also be seen that there needs to be a minimum

wave force before overtopping events occur. This

threshold is dependent on the berm width. For the test

without a berm the minimum force is equal to 15.2

N/m, for the 20 cm berm the minimum force is 19.6

N/m and for the 40 cm berm, the minimum force is

25.6 N/m. The influence of the water depth and

steepness on this threshold have not been determined

and therefore no prediction formula has been drafted.

Wave force [N/m]

0 20 40 60 80 100 120 140 160

Wave

ove

rto

pp

ing

[kg

/m]

-2

0

2

4

6

8

10

12

14

16

18

Fig. 11: Relationship between wave force and wave

overtopping

REFERENCES

[1] Platteeuw, J. (2015). Analysis of individual wave

overtopping volumes for steep low crested coastal structures

in deep water conditions.

[2] van der Meer, J., & Bruce, T. (2014). New Physical Insights

and Design Formulas on Wave Overtopping at Sloping and

Vertical Structures.

[3] Victor, L. (2012). Probability distribution of individual

wave overtopping volumes for smooth impermeable steep

slopes with low crest freeboard. Coastal Eng., pp. 87-101.

xiii

TABLE OF CONTENTS

Preface .............................................................................................................................................................. i

Abstract ........................................................................................................................................................... v

Extended Abstract .......................................................................................................................................... vii

Table of contents .......................................................................................................................................... xiii

List of Figures ................................................................................................................................................ xvii

List of Tables .................................................................................................................................................. xxi

List of Symbols and Abbreviations ................................................................................................................ xxiii

Chapter 1 Introduction .............................................................................................................................. 1

1.1 Objectives ............................................................................................................................................. 2 1.2 Thesis outline ....................................................................................................................................... 2

Chapter 2 Literature study ......................................................................................................................... 5

2.1 Wave parameters ................................................................................................................................. 5 2.2 Overtopping parameters ...................................................................................................................... 6 2.3 Single overtopping reduction measures ............................................................................................... 7

2.3.1 Chamfered and overhanging vertical structures ........................................................................ 7 2.3.2 Other types of parapets.............................................................................................................. 9 2.3.3 Simple slope ............................................................................................................................. 11

2.4 Wave overtopping (EurOtop Manual) ................................................................................................ 11 2.5 Combined overtopping reduction measures ...................................................................................... 12

2.5.1 Slope and vertical wall .............................................................................................................. 13 2.5.2 Slope and parapet .................................................................................................................... 14 2.5.3 Conclusion ................................................................................................................................ 15

2.6 Wave forces on storm walls ............................................................................................................... 16 2.6.1 Conclusion ................................................................................................................................ 17

Chapter 3 Hydraulic model tests .............................................................................................................. 19

3.1 The wave flume at Ghent University .................................................................................................. 19 3.2 Model geometry ................................................................................................................................. 20 3.3 Test matrix ......................................................................................................................................... 23

3.3.1 Time series and test names ...................................................................................................... 25 3.3.2 Repetition tests ........................................................................................................................ 26

3.4 Calibration .......................................................................................................................................... 27 3.4.1 Pump ........................................................................................................................................ 27 3.4.2 Weigh cell ................................................................................................................................. 29 3.4.3 Wave gauges............................................................................................................................. 30 3.4.4 Force load cell ........................................................................................................................... 32

Chapter 4 Data processing ....................................................................................................................... 35

4.1 Wave measurements ......................................................................................................................... 35 4.1.1 Difference in generated and theoretical wave characteristics ................................................. 38

Table of contents

xiv

4.2 Average overtopping measurements ................................................................................................. 40 4.2.1 Scatter on the average overtopping ......................................................................................... 43

4.3 Individual overtopping measurements............................................................................................... 44 4.3.1 Scatter on the individual overtopping measures ..................................................................... 45

4.4 Force measurements .......................................................................................................................... 46 4.4.1 Scatter on the force measurements ......................................................................................... 52 4.4.2 Influence of the installation of a cover ..................................................................................... 53

Chapter 5 Average overtopping analysis .................................................................................................. 55

5.1 Influence of the water depth .............................................................................................................. 56 5.1.1 No berm .................................................................................................................................... 56 5.1.2 With berm ................................................................................................................................ 57

5.2 Influence of the wave steepness ........................................................................................................ 59 5.2.1 Reference case: slope no protection ........................................................................................ 60 5.2.2 No berm .................................................................................................................................... 62 5.2.3 With berm ................................................................................................................................ 65

5.3 Influence vertical wall/parapet .......................................................................................................... 66 5.3.1 No berm .................................................................................................................................... 67 5.3.2 20 cm berm ............................................................................................................................... 69 5.3.3 40 cm berm ............................................................................................................................... 70

5.4 Influence of the berm width ............................................................................................................... 71 5.4.1 Vertical wall .............................................................................................................................. 71 5.4.2 Parapet ..................................................................................................................................... 73

5.5 Comparison with literature ................................................................................................................ 74 5.5.1 No berm .................................................................................................................................... 74 5.5.2 With berm ................................................................................................................................ 78

5.6 Reduction factors ............................................................................................................................... 80 5.6.1 No berm .................................................................................................................................... 80 5.6.2 With berm ................................................................................................................................ 88

5.7 Conclusion .......................................................................................................................................... 94

Chapter 6 Individual overtopping analysis ............................................................................................... 95

6.1 Probability of overtopping ................................................................................................................. 95 6.1.1 Factor 𝝌 determining the probability of overtopping Pow ........................................................ 96

6.2 Dimensionless individual overtopping quantities ............................................................................... 99 6.3 Influence of the water depth ............................................................................................................ 100 6.4 Influence of the wave steepness ...................................................................................................... 101 6.5 Influence of the berm width ............................................................................................................. 103 6.6 Prediction formulas for the dimensionless parameters ................................................................... 104

6.6.1 Vertical wall ............................................................................................................................ 104 6.6.2 Parapet ................................................................................................................................... 107 6.6.3 Comparison vertical wall/parapet .......................................................................................... 108 6.6.4 Prediction formulas Vdim,1/250 .................................................................................................. 109

Chapter 7 Force analysis ........................................................................................................................ 111

7.1 Influence of the wave height ............................................................................................................ 111 7.2 Influence of the relative freeboard ................................................................................................... 112 7.3 Dimensionless wave force ................................................................................................................ 112 7.4 Influence of the water depth, steepness and berm width ................................................................ 112 7.5 Prediction formula............................................................................................................................ 113 7.6 Wave force when a parapet is installed ........................................................................................... 114 7.7 Relationship between wave forces and individual overtopping ....................................................... 115

Chapter 8 Conclusion ............................................................................................................................. 117

xv

8.1 Average overtopping ........................................................................................................................ 117 8.2 Individual overtopping ..................................................................................................................... 117 8.3 Wave forces ...................................................................................................................................... 118 8.4 Further research ............................................................................................................................... 118

References ................................................................................................................................................... 121

Annex A: Average overtopping analysis ....................................................................................................... 123

Annex B: Individual overtopping .................................................................................................................. 133

Annex C: Wave forces .................................................................................................................................. 137

Annex D: Test Matrix .................................................................................................................................... 139

xvii

LIST OF FIGURES

Figure 1-1: Maximum average overtopping rates for different activities ................................................................ 1 Figure 2-1: Overtopping parameters displayed on geometrical configuration ........................................................ 7 Figure 2-2: Range of chamfered and overhanging wall geometries and water levels ............................................. 8 Figure 2-3: FSS in comparison to upright seawall ................................................................................................ 10 Figure 2-4: Small recurve (left) and medium and large recurve (right); dimensions in m at prototype scale ....... 10 Figure 2-5: Principle of the vertical wall built in the dike .................................................................................... 13 Figure 2-6:Smooth slope with parapet, right case increased walking space ......................................................... 14 Figure 2-7: Schematically representation of parameters β and λ .......................................................................... 14 Figure 2-8: Generalised cross section of SSP walls .............................................................................................. 16 Figure 3-1: Large Wave Flume at Ghent University ............................................................................................. 19 Figure 3-2: Wave paddle ....................................................................................................................................... 20 Figure 3-3: Drawing of the whole flume; all dimensions are in cm ...................................................................... 21 Figure 3-4: The reference case .............................................................................................................................. 21 Figure 3-5: No berm and a parapet of 10 cm ........................................................................................................ 22 Figure 3-6: 20 cm berm and a parapet of 10 cm.................................................................................................... 22 Figure 3-7: 40 cm berm and a parapet of 10 cm.................................................................................................... 22 Figure 3-8: Drawing of the used parapets ............................................................................................................. 23 Figure 3-9: Generation of a time series in LabVIEW ........................................................................................... 25 Figure 3-10: Comparison between tests with the same time series (0091A: blue, 0091B: green, 0091C: red) .... 27 Figure 3-11: Individual pumping curves ............................................................................................................... 28 Figure 3-12: Average pumping curve ................................................................................................................... 29 Figure 3-13: Calibration curve weigh cell ............................................................................................................. 30 Figure 3-14: Set of 3 wave gauges near the toe of the structure ........................................................................... 31 Figure 3-15: Three hammer blows to the load cell................................................................................................ 32 Figure 3-16: Resonance frequency of the 10 cm wall/parapet .............................................................................. 32 Figure 3-17: Resonance frequency of the 5 cm wall/parapet ................................................................................ 33 Figure 3-18: Load cell without cover (left) and with cover (right) ....................................................................... 33 Figure 4-1: Input required to perform the reflection analysis ............................................................................... 35 Figure 4-2: Example of the output in the frequency domain in table form ........................................................... 37 Figure 4-3: Example of the output in the time domain in table form .................................................................... 37 Figure 4-4: Wave spectrum test 0011A ................................................................................................................. 37 Figure 4-5: Wave height distribution test 0011A .................................................................................................. 38 Figure 4-6: Generated significant wave heights plotted against the target significant wave heights (at the toe) .. 38 Figure 4-7: Generated peak periods plotted against the target peak periods (at the toe) ....................................... 40 Figure 4-8: Pumped absolute mass as function of the time ................................................................................... 41 Figure 4-9: Absolute mass in the reservoir as function of the time ....................................................................... 42 Figure 4-10: Cumulative absolute mass in the reservoir ....................................................................................... 42 Figure 4-11: Comparison 𝑞-values calculated with average overtopping script and individual overtopping script

.............................................................................................................................................................................. 44 Figure 4-12: Comparison between 𝑉𝑚𝑎𝑥 and 𝑉1/250 ........................................................................................ 45 Figure 4-13: Comparison between the measured, removed and filtered signal (low pass filter of 40 Hz) ............ 47 Figure 4-14: Comparison between the measured (black), removed (red) and filtered (blue) signal (low pass

filter of 85 Hz)....................................................................................................................................................... 48 Figure 4-15: Wave impact corresponding to the smallest wave is around 0.12 N ................................................ 49 Figure 4-16: Multiple peaks for one wave impact................................................................................................. 49 Figure 4-17: L-Davis only selects the highest peak per impact by applying a time domain search equal to the

peak period ............................................................................................................................................................ 49 Figure 4-18: Number of impacts according to the water depth ............................................................................. 50 Figure 4-19: Number of impacts according to the configuration .......................................................................... 51 Figure 4-20: Comparison between Fmax and F1/250 ................................................................................................. 51 Figure 4-21: Comparison between Fmax and F1/10 .................................................................................................. 52 Figure 4-22: Scatter analysis for test number 119 ................................................................................................. 53 Figure 5-1: Dimensionless overtopping discharge Q versus relative freeboard Rc/Hm0 for all the tests ............... 55 Figure 5-2: Influence of the water depth when there’s no berm and a wall (wave steepness 0.05) ...................... 56 Figure 5-3: Influence of the water depth when there’s no berm and a parapet (wave steepness 0.05) ................. 57 Figure 5-4: Influence of water depth when there's a 40 cm berm and a vertical wall (steepness 0.05) ................. 58 Figure 5-5: Influence of water depth when there's a 40 cm berm and a parapet (steepness 0.05) ......................... 59

List of Figures

xviii

Figure 5-6: Influence of the wave steepness when there's a slope without protection .......................................... 60 Figure 5-7: 𝑘𝑠-factors when there’s a slope without protection ............................................................................ 62 Figure 5-8: Influence of the wave steepness when there's no berm and a wall ..................................................... 63 Figure 5-9: ks-factors when there’s no berm and a wall ........................................................................................ 63 Figure 5-10: Influence of the steepness when there’s no berm and a parapet ....................................................... 64 Figure 5-11: 𝑘𝑠-factors when there’s no berm and a parapet ................................................................................ 65 Figure 5-12: 𝑘𝑠-factors configurations with wall .................................................................................................. 65 Figure 5-13: 𝑘𝑠-factors configurations with parapet ............................................................................................. 66 Figure 5-14: Comparison between the overtopping of a wall (left) and a parapet (right) ..................................... 67 Figure 5-15: Influence of parapet when there’s no berm (steepness 0.03) ............................................................ 68 Figure 5-16: 𝑘𝑝-factors no berm (steepness 0.03) ................................................................................................ 68 Figure 5-17: Influence of parapet when there’s a 20 cm berm (steepness 0.03) ................................................... 69 Figure 5-18: 𝑘𝑝-factors when there’s a 20 cm berm (steepness 0.03) .................................................................. 69 Figure 5-19: Influence of parapet when there’s a 40 cm berm (steepness 0.03) ................................................... 70 Figure 5-20: 𝑘𝑝-factors when there’s a 40 cm berm (steepness 0.03 left and 0.05 right) ..................................... 71 Figure 5-21: Influence of the berm width when there’s a vertical wall (steepness 0.05) ...................................... 72 Figure 5-22: 𝑘𝑏- factors 20/40 cm berm (with a vertical wall and steepness 0.05) .............................................. 73 Figure 5-23: Influence of the berm width when there’s a parapet (steepness 0.05) .............................................. 73 Figure 5-24: Comparison of test results and EurOtop Manual in case there's a slope with no protection ............ 75 Figure 5-25: Comparison of test results and updated vdM formula in case there's a slope with no protection..... 76 Figure 5-26: Comparison of test results and updated vdM formula in case there's no berm and wall .................. 77 Figure 5-27: Comparison of test results and updated vdM formula in case there's no berm and a parapet........... 78 Figure 5-28: Comparison between prediction formula of van Doorslaer and test results ..................................... 79 Figure 5-29: 𝛾𝑣-factors split up in steepness. ....................................................................................................... 81 Figure 5-30: 𝛾𝑣-factors without the values above 1 and linear trendline .............................................................. 81 Figure 5-31: Prediction formula 𝛾𝑣 based on the reduced data set ....................................................................... 82 Figure 5-32: Relationship between the predicted γv and the γv based on the test results ..................................... 83 Figure 5-33: Prediction formula Q no berm wall plotted through the test results ................................................. 84 Figure 5-34: γp-factors split up in wave steepness .............................................................................................. 85 Figure 5-35: Reduced test results with their linear trendline................................................................................. 86 Figure 5-36: Relationship between the predicted γp and the γp based on the test results .................................... 87 Figure 5-37: Prediction formulas Q parapet and no berm plotted through the test results .................................... 87 Figure 5-38: Test results plotted against prediction formula Q for a wall without berm ..................................... 88 Figure 5-39: Reduction factors relative to berm width B ...................................................................................... 89 Figure 5-40: Determination of the prediction formula for γb with a simple vertical wall .................................... 90 Figure 5-41: γb real relative to γb predicted ........................................................................................................ 91 Figure 5-42: Comparison between prediction formulas Q for wall without berm and wall with berm ................. 91 Figure 5-43: γb factor relative to the actual berm width ...................................................................................... 92 Figure 5-44: Determination of prediction formula for γb with parapet ................................................................ 93 Figure 5-45: Comparison between predicted overtopping and test results ............................................................ 94 Figure 5-46: Flowchart prediction formulas Q ...................................................................................................... 94 Figure 6-1: Probability of overtopping in function of the relative freeboard for all the tests ................................ 96 Figure 6-2: Prediction formula Pow for configurations without a berm ............................................................... 97 Figure 6-3: Prediction formula Pow for configurations with a 20 cm berm ......................................................... 98 Figure 6-4: Prediction formula Pow for configurations with 40 cm berm ............................................................ 99 Figure 6-5: Maximal volume in function of the relative freeboard ....................................................................... 99 Figure 6-6: Influence of the water depth on the maximum volumes for the cases with a wall ........................... 100 Figure 6-7: Influence of the water depth on the maximum volumes for the cases with parapet ......................... 101 Figure 6-8: Influence of the steepness on the maximum volume for the cases with wall ................................... 102 Figure 6-9: Influence of the steepness on the maximum volume for the cases with parapet .............................. 102 Figure 6-10: Influence of the berm width on the maximum volume for the cases with wall .............................. 103 Figure 6-11: Influence of the berm width on the maximum volume for the cases with parapet ......................... 104 Figure 6-12: Dimensionless maximum volume in function of the relative freeboard ......................................... 105 Figure 6-13: Two different forms of prediction formulas for the dimensionless maximum volume .................. 105 Figure 6-14: Two different forms of prediction formulas for the dimensionless maximum volume (linear) ..... 106 Figure 6-15: Graphical summary of the prediction formulae used for the configurations with a wall ............... 107 Figure 6-16: Graphical summary of the prediction formulae used for the configurations with a parapet ........... 108 Figure 6-17: Comparison between pred. formulas wall and parapet ................................................................... 109 Figure 7-1: F1/250 plotted against Hm0................................................................................................................... 111 Figure 7-2: Wave force F1/250 plotted against the relative freeboard Rc/Hm0 ....................................................... 112

xix

Figure 7-3: Dimensionless wave force P in function of relative freeboard Rc/Hm0 ............................................. 113 Figure 7-4: Prediction formula plotted along with the test results ...................................................................... 114 Figure 7-5: Comparison between wave forces with a vertical wall and a parapet .............................................. 115 Figure 7-6: Relationship between wave force and wave overtopping ................................................................. 116 Figure A-1: Influence of water depth when there's no berm and a wall (steepness 0.03) ................................... 123 Figure A-2: Influence of the water depth when there’s no berm and a parapet (steepness 0.03) ........................ 123 Figure A-3: Influence of water depth when there's a 40 cm berm and a wall (steepness 0.03) ........................... 124 Figure A-4: Influence of water depth when there's a 40 cm berm and a parapet (steepness 0.03) ...................... 124 Figure A-5: Influence of the wave steepness when there’s a 20 cm berm and a wall ......................................... 125 Figure A-6: Influence of the wave steepness when there’s a 40 cm berm and wall ............................................ 125 Figure A-7: Influence of the wave steepness when there’s a 20 cm berm and parapet ....................................... 126 Figure A-8: Influence of the wave steepness when there’s a 40 cm berm and parapet ....................................... 126 Figure A-9: Influence of parapet when there’s no berm (steepness 0.05) ........................................................... 127 Figure A-10 : 𝑘𝑝-factors no berm (steepness 0.05) ............................................................................................. 127 Figure A-11: Influence of parapet when there’s a 20 cm berm (steepness 0.05) ................................................ 128 Figure A-12: 𝑘𝑝-factors 20 cm Berm (steepness 0.05) ....................................................................................... 128 Figure A-13: Influence of parapet when there’s a 40 cm berm (steepness 0.05) ................................................ 129 Figure A-14: 𝑘𝑝-factors 40 cm berm (steepness 0.05) ........................................................................................ 129 Figure A-15: Influence of the berm width when there’s a wall (steepness 0.03) ................................................ 130 Figure A-16: kb-factors when there’s a 20/40 cm berm and a wall (steepness 0.03) .......................................... 130 Figure A-17: Influence of the berm width when there’s a parapet (steepness 0.03) ........................................... 131 Figure A-18: γv-factors without the values above 1 and quadratic trendline ..................................................... 131 Figure A-19: γv-factors without the values above 1 and power trendline .......................................................... 132 Figure B-1: Influence of the water depth on the 0.4% volumes for the cases with wall ..................................... 133 Figure B-2: Influence of the water depth on the 0.4% volumes for the cases with parapet ................................ 133 Figure B-3: Influence of the wave steepness on the 0.4% volumes for the cases with wall ............................... 134 Figure B-4: Influence of the water depth on the 0.4% volumes for the cases with parapet ................................ 134 Figure B-5: Influence of the berm width on the 0.4% volumes for the cases with wall ...................................... 135 Figure B-6: Influence of the berm width on the 0.4% volumes for the cases with parapet ................................. 135 Figure B-7: Comparison between pred. formulas wall and parapet .................................................................... 136 Figure C-1: Influence of the water depth on the wave force ............................................................................... 137 Figure C-2: Influence of the steepness on the wave force ................................................................................... 137 Figure C-3: Influence of the berm width on the wave force ............................................................................... 138

xxi

LIST OF TABLES

Table 2-1: Overtopping for different configurations with a relative freeboard of 1.67 ........................................... 8 Table 3-1: Chosen peak periods Tp with their corresponding wave height Hs .................................................... 23 Table 3-2: Summary of the test matrix .................................................................................................................. 24 Table 3-3: average overtopping for test 0091A and its repetition tests 0091B, 0091C and 0091D ...................... 26 Table 4-1: summary of reduction in wave height at the toe .................................................................................. 39 Table 4-2: Modifications performed in the MATLAB -script to predict the average overtopping 𝑞 ................... 41 Table 4-3: Scatter present on the average overtopping ......................................................................................... 43 Table 4-4: Comparison of the individual volume characteristics repetition tests 0091A, 0091B, 0091C and

0091D .................................................................................................................................................................... 45 Table 4-5: Comparison of the force measurements between a covered and uncovered test ................................. 53 Table 5-1: Target wave steepnesses and the range of the experimental reached values ....................................... 60 Table 5-2: Summary of tests with water depth 68 cm at the wave paddle ............................................................ 61 Table 6-1: Factor 𝜒 and its 95% confidence intervals determined with SPSS ...................................................... 96 Table 6-2: Prediction formulas Vdim,1/250 and applicability range ........................................................................ 109 Table A-1: Different formulas for γp with their unknown parameters and coefficient of determination R² ...... 132 Table D-1: Test matrix of the conducted tests .................................................................................................... 139

xxiii

LIST OF SYMBOLS AND ABBREVIATIONS

Symbols

a [-] coefficient used in the new form (Weibull distribution) of prediction formula for Q

A [-] parameter used in SPSS to fit prediction formulas

B [m] berm width of the structure

b [-] specific factor describing the behaviour of wave overtopping for a certain structure

b [-] coefficient used in the new form (Weibull distribution) of prediction formula for Q

B [-] parameter used in SPSS to fit prediction formulas

𝐵𝑏 [m] berm width of the structure defined by Kortenhaus et al.

c [-] coefficient used in the new form (Weibull distribution) of prediction formula for Q

C [-] parameter used in SPSS to fit prediction formulas

𝐶𝑟𝑒𝑓𝑙 [-] reflection coefficient determined with WaveLab at the slope of the structure

d [m] water depth at the specified location of the structure

D [-] parameter used in SPSS to to fit prediction formulas

E [-] parameter used in SPSS to to fit prediction formulas

𝐹1/10 [N/m] average horizontal force of the 10% highest force impacts

𝐹1/250 [N/m] average horizontal force of the 0,4% highest force impacts

𝐹ℎ [N/m] horizontal force defined by Kortenhaus et al.

𝐹𝑚𝑎𝑥 [N/m] maximal horizontal force found on the load cell for each test

g [m/s²] acceleration due to gravity, taken equal to 9.81 m/s²

ℎ𝑏 [m] foreland height defined by Kortenhaus et al.

𝐻𝑖 [m] wave height individual waves

𝐻𝑚0 [m] significant wave height in the frequency domain

𝐻𝑚0,𝑡𝑎𝑟𝑔𝑒𝑡 [m] target significant wave height in the frequency domain at the toe of the structure

𝐻𝑚0,𝑡𝑜𝑒 [m] significant wave height in the frequency domain measured at the toe of the structure

ℎ𝑛 [m] height of the inclined part of the parapet

𝐻𝑠 [m] significant wave height in the time domain

ℎ𝑡 [m] total parapet height

ℎ𝑤𝑎𝑙𝑙 [m] height of the storm wall

k [-] Ratio of two wave overtopping quantities

𝑘𝑏 [-] ratio of two dimensionless overtopping quantities Q; with berm or no berm, but all the

other test conditions the same

List of Symbols and Abbreviations

xxiv

𝑘𝑝 [-] ratio of two dimensionless overtopping quantities Q; with wall or parapet, but all the


𝑘𝑠 [-] ratio of two dimensionless overtopping quantities Q; with different steepness, but all the


𝐿𝑝 [m] wave length determined with peak period 𝑇𝑝

𝐿𝑚−1,0 [m] wave length determined with spectral wave period 𝑇𝑚−1,0

𝐿𝑝𝑎𝑑𝑑𝑙𝑒 [m] wave length determined with the peak period 𝑇𝑝 at the wave paddle

𝑚0 [m²] zero moment of the incident wave spectrum

𝑚−1 [m²s] first negative moment of the incident wave spectrum

𝑀𝑐.𝑎𝑏𝑠.,𝑒𝑛𝑑 [kg] end mass of the cumulative absolute mass curve

𝑀𝑐.𝑎𝑏𝑠.,𝑠𝑡𝑎𝑟𝑡 [kg] start mass of the cumulative absolute mass curve

𝑁𝑜𝑤 [-] number of overtopping waves

𝑁𝑤 [-] number of waves

P [-] Dimensionless maximum force

𝑃𝑜𝑤 [-] probability of wave overtopping

Q [-] dimensionless overtopping quantity

q [m³/s/m] average overtopping discharge

𝑄0 [-] dimensionless overtopping quantity when the relative freeboard is zero

R² [-] Coefficient of determination: defined as 𝑅² = 1 −𝑟𝑟𝑒𝑠

𝑟𝑡𝑜𝑡 and 𝑟𝑟𝑒𝑠 the residual sum of

squares and 𝑟𝑡𝑜𝑡 the total sum of squares

𝑅𝑐 [m] crest freeboard

𝑅𝑐 𝐻𝑚0⁄ [-] relative crest freeboard defined with the significant wave height of the frequency

domain, also equal to R*

𝑅𝑐 𝐻𝑠⁄ [-] relative crest freeboard defined with the significant wave height of the time domain

𝑅𝑢2% [m] 2 % run-up height: On average 2% of the waves run-up to that specified height

𝑠𝑝 [-] wave steepness calculated with peak period 𝑇𝑝

𝑠𝑚−1,0 [-] wave steepness calculated with spectral period 𝑇𝑚−1,0

𝑠0 [-] fictitious wave steepness calculated with spectral period 𝑇𝑚−1,0

𝑡𝑒𝑛𝑑 [s] end time of a test, excluding the extra outrun time

𝑇𝑖 [s] wave period individual wave

𝑇𝑚 [s] average wave period

𝑇𝑚−1,0 [s] spectral wave period

𝑇𝑝 [s] peak wave period

xxv

𝑇𝑝,𝑡𝑎𝑟𝑔𝑒𝑡 [s] target peak wave period at the toe of the structure

𝑇𝑝,𝑡𝑜𝑒 [s] peak wave period measured at the toe of the structure

𝑡𝑠𝑡𝑎𝑟𝑡 [s] start time of a test, excluding the extra time at the start of the test

𝑉1/250 [kg/m] average volume of the 0.4% individual overtopping volumes 𝑉𝑖

𝑉𝑖 [kg/m] individual overtopping volumes per meter

𝑉𝑑𝑖𝑚,𝑚𝑎𝑥 [-] dimensionless maximum overtopping volume

𝑉𝑑𝑖𝑚,1/250 [-] Dimensionless 0.4% overtopping volume

𝑉𝑚𝑎𝑥 [kg/m] maximal volume of the individual overtopping volumes 𝑉𝑖

𝑤𝑠 [-] ratio of the significant wave heights Hm0 with different steepness, but all the other test

conditions the same

𝑊𝑡𝑟𝑎𝑦 [m] width of the tray used to guide the overtopped water volume

𝑥1,2 [m] distance between wave gauge 1 and 2 of set of wave gauges

𝑥1,3 [m] distance between wave gauge 1 and 3 of set of wave gauges

α [° or

rad]

angle of the sloping structure

β [°] the angle the inclined part of the parapet makes with the vertical part

𝛾𝑏 [-] reduction factor dimensionless overtopping Q due to the presence of a berm

𝛾𝑏𝑣 [-] reduction factor dimensionless overtopping Q due to the presence of a wall and a berm

𝛾𝑓 [-] reduction factor dimensionless overtopping Q due to the roughness of the bottom

𝛾𝑓,𝑖 [-] empirical factor for impacting waves used in the horizontal force Fh calculation defined

by Kortenhaus et al.

𝛾𝑓,𝑛𝑖 [-] empirical factor for non-impacting waves used in the horizontal force Fh calculation

defined by Kortenhaus et al.

𝛾𝑝 [-] reduction factor dimensionless overtopping Q due to the presence of a parapet

𝛾𝑝𝑎𝑟 [-] reduction factor dimensionless overtopping Q due to the presence of a parapet,

according to Van Doorslaer et al.

𝛾𝑣 [-] reduction factor dimensionless overtopping Q due to the presence of a vertical wall

𝛾𝛽 [-] reduction factor dimensionless overtopping Q due to oblique waves

𝛾𝛽,𝑝𝑎𝑟 [-] reduction factor dependent on the parapet angle 𝛽, present in the formula for 𝛾par

𝛾λ,par [-] reduction factor dependent on height ratio λ of the parapet, present in the formula for

λ [-] ratio of the height of the inclined part of the parapet on the total height of the parapet

𝛾par

𝜉0 [-] breaker parameter to separate breaking and non-breaking wave conditions at the toe of

the structure

List of Symbols and Abbreviations

xxvi

𝜌𝑤 [kg/m³] mass density of pure water, assumed to have a value of 1000 kg/m³

χ [-] factor used in the formula for Pow, taking into account the geometrical configuration of

the structure

Abbreviations and Acronyms

AWASYS

Active Wave Absorption SyStem

BAL

balans

FFT

Fast Fourier Transform

FSS

Flearing Shaped Seawall

ghm

golfhoogtemeter

JONSWAP

Joint North Sea Wave Project

SSP

Storm Surge Protection

SWB

Stilling Wave Basin

vdM

van der Meer

WG

wave gauge

1

CHAPTER 1 INTRODUCTION

More and more people are keen on living by the sea side because of the beautiful view. The

downside of living near the coastline is the unpredictability of the behaviour of the waves.

When a large storm approaches the coastline, water will overtop the defense structures and

can cause damage to buildings and even people. To minimize the risk for damage, better

defense structures need to be build, but of course these structures can’t be too high because

then the aesthetics of the coast line view would be lost.

The last couple of years, there were many investigations to reduce the amount of overtopping

water without obstructing the view too much. Some of these measures are a slope (or dike

structure), a berm, a wall, a parapet and of course combinations are possible. Overtopping can

be quantified in different terms. The average overtopping q is measured as the volume water

(m³ or l) that overtops the structure per meter per second. Another way to measure the

overtopping is the individual overtopping; this is the volume of water that overtops the

structure per meter for a certain wave impact. The individual overtopping is more critical to

cause damage to structures or people. In practice already more research has been conducted

on average overtopping. Therefore the last parameter is mostly used for the design of coastal

defense structures.

FIGURE 1-1: MAXIMUM AVERAGE OVERTOPPING RATES FOR DIFFERENT ACTIVITIES

Chapter 1 Introduction 1.1 Objectives

2

In the Coastal Engineering Manual, the average overtopping corresponding with a certain

safety level is given for some types of traffic and structures. To give the reader an idea about

these values of average overtopping, they are shown in Figure 1-1.When designing coastal

defense structures, the aim is to minimize the average overtopping until they are lower than

the allowable average overtopping rates (corresponding with a certain safety level) presented

in Figure 1-1. A difficult task in this design is the prediction of the average overtopping,

especially when multiple reduction measures are combined together.

1.1 Objectives

The objectives for this study are threefold:

- The influence of different parameters such as water depth, wave height and steepness

on the average overtopping are tested and analysed. Different measures such as a

vertical wall, a parapet and a berm on top of a slope are tested (individually and

combined) to reduce the overtopping and to come up with new reduction factors which

can be implemented in the current prediction formula for a better prognosis of the

average overtopping.

- For individual overtopping, the probability of overtopping and two other

characteristics are investigated. Existing or new prediction formulas are fitted against

the test results.

- The wave forces on the storm wall accompanying the wave overtopping are also

measured and analysed with the intention of composing a prediction formula for the

wave forces. The influence of several parameters on the wave force are also tested and

analysed. A possible relationship between wave overtopping and wave forces is

examined.

In order to fulfill these objectives hydraulic model tests are conducted in the Big Wave Flume

of Ghent University. The test matrix is drafted in such a way that the influence of the different

parameters (water depth, wave height, steepness, berm width, wall or parapet) can be

analysed. The test matrix is added in Annex D.

1.2 Thesis outline

In Chapter 2 (the first one being the introduction), the current literature on wave overtopping

and wave forces is reviewed. Based upon this literature study, the configuration that will be

built in the wave flume is designed to fill up some existing gaps in the literature.

Chapter 3 gives more information about the wave flume in which the tests are conducted. The

used model geometries are also explained more detailed in this Chapter. The measurement

Chapter 1 Introduction 1.2 Thesis outline

3

devices and instrumentation for obtaining the wave parameters, wave overtopping and wave

forces are also described.

The data processing is explained in Chapter 4. The acquired data has to be processed in order

to be able to compare different tests with each other. For the wave data specifically, the

measured wave parameters are compared with the target wave parameters to see if the tests

were conducted in a good way.

The actual analysis of the data happens in Chapter 5, Chapter 6 and Chapter 7. First the

average overtopping is examined. A study on the influence of water depth, steepness, vertical

wall/parapet and berm width on the overtopping is conducted. For the parameters that have an

influence, a reduction factor is created that can be implemented in the current prediction

formula to improve its accuracy. In Chapter 6, the individual overtopping analysis is

performed. Also the influencing parameters are investigated and prediction formulas are

proposed for different characteristics of individual overtopping. For the force analysis in

Chapter 7, a similar procedure is followed. First the influence of the different parameters on

the wave force is checked and afterwards a prediction formula is created. A first initiation is

given to relate measured individual overtopping volumes to the measured forces.

In the final chapter, a general conclusion of the performed study is given. Also guidance is

given in which domains further research can be performed.

5

CHAPTER 2 LITERATURE STUDY

In the literature study, first a short summary will be given of the wave parameters used in the

test matrix and the main overtopping parameters used in literature for structures. After the

discussion of these parameters, the literature about overtopping is looked in more into detail.

The different measures for reducing wave overtopping are looked at individually. The

calculation methods of overtopping for a single measure are already fairly accurate known but

for the combinations of these different measures, there are still some gaps in the EurOtop

Manual (2007) and other literature. The objective of this literature study is to find one of these

gaps so it can be tested during the hydraulic model tests. The research that exists will be

shortly summarized and the possible gaps in the research domains will be investigated.

Overtopping measures standing alone will be discussed first; afterwards the combined

measures will be examined.

Combining different measures also leads to a variation in the force acting on the wall and this

variation found in literature is discussed shortly in paragraph 2.6.

2.1 Wave parameters

The wave parameters needed to create the test matrix of this thesis and some target values will

be shortly discussed below. The wave characteristics are roughly determined by the

significant wave height Hs, the peak period Tp and the wave steepness sp. Also the wave

breaking parameter 𝜉0 will be shortly discussed.

The significant wave height Hs is a characteristic of a wave signal in the time domain. Once

all the wave heights are determined out of a signal, representing the surface elevation at a

certain location, the significant wave height Hs can be determined. Sorting all the wave

heights from high to low and taking the average of the 1/3th largest waves leads to the

definition of the significant wave height Hs. This is represented in equation (1) with Nw the

number of waves and Hi the individual wave heights found out of the time signal.

𝐻𝑠 =1

𝑁𝑤 3⁄∑ 𝐻𝑖

𝑁𝑤 3⁄

1

(1)

Once the individual time periods Ti of the individual waves are known, the average wave

period Tm can be determined. Out of this average wave period the peak wave period Tp can be

calculated. The definition for the peak wave period is given in equation (2):

𝑇𝑝 = 1.1 ∙ 𝑇𝑚 (2)

Chapter 2 Literature study 2.2 Overtopping parameters

6

With the peak period the wave length can be determined. The general formula following the

linear wave theory is used to calculate the wave length. In equation (3) the formula is given:

𝐿𝑝 =

𝑔𝑇𝑝²

2𝜋tanh(

2𝜋𝑑

𝐿𝑝) (3)

In the formula above, d is the water depth at the specified location and g the gravity constant

(9.81 m²/s). The formula needs to be solved iteratively as the wave length is found as well left

as right in the formula.

With the knowledge of the significant wave height and wave length, the wave steepness can

be determined. The wave steepness is given by the following formula:

𝑠𝑝 =

𝐻𝑠𝐿𝑝

(4)

The wave breaker parameter 𝜉0 for waves encountering a slope with slope angle α is defined

in equation (5). 𝑠0 is a fictitious wave steepness. It is the wave steepness calculated in deep

water conditions, independently what the real depth conditions are. Equation (6) represents

the formula for 𝑠0.

𝜉0 =

tan(α)

√𝑠0 (5)

𝑠0 =

2𝜋 ∙ 𝐻𝑚0

𝑔 ∙ 𝑇𝑚−1,02 (6)

In equation (6) Hm0 is the significant wave height in the frequency domain, Tm-1,0 the spectral

wave period (more about those two parameters in section 4.1) and g the gravity constant (9.81

m/s²

Roughly it can be stated that for 𝜉0-values larger than 2 non-breaking waves can be assumed,

where for 𝜉0-values smaller than 2 breaking wave conditions are assumed.

2.2 Overtopping parameters

Besides wave parameters, also parameters that characterise the overtopping behaviour of a

structure are considered. The freeboard height Rc with respect to the water level d, the relative

freeboard Rc/Hs and the berm width B. All these parameters are sketched in Figure 2-1 to

make their definition clearer.

Chapter 2 Literature study 2.3 Single overtopping reduction measures

7

FIGURE 2-1: OVERTOPPING PARAMETERS DISPLAYED ON GEOMETRICAL CONFIGURATION

In overtopping calculations and prediction formulas, the dimensionless overtopping quantity

Q is used frequently. The definition of Q is given in equation (7). In equation (7) q is the

average overtopping rate in m³ per meter per second, g the gravity constant (9.81 m/s²) and

Hm0 the significant wave height determined in the frequency domain.

𝑄 =𝑞

√𝑔 ∙ 𝐻𝑚03

(7)

In what follows the overtopping reduction measures used in literature will be discussed. First

the single overtopping reduction measures, thereafter the combined reduction measures.

2.3 Single overtopping reduction measures

The simplest way to reduce overtopping is to increase the crest freeboard of the sea defence

structure. This can be done by installing a simple vertical wall as protection. The downside of

this measure is that a high vertical wall blocks the sight and can’t be integrated well in the

environment. Therefore other single measures are used to reduce overtopping and they are

discussed in the following sections. Also a short summary of the tests and analysis of the

structure considered is given.

2.3.1 Chamfered and overhanging vertical structures

Cornett

In a study performed by Cornett (1999) the influence of the wall geometry on the overtopping

was investigated, both chamfered and overhanging upper sections were considered.

Overhanging structures are sometimes also called parapets. The overtopping was examined

and quantified relative to the overtopping at a simple vertical wall. The study was executed in

the Canadian Hydraulics Centre in Ottawa in a wave flume with an active wave absorption

system. In total over 450 experiments were conducted with regular and irregular waves. The


8

global set up of every test was the same: there was a fixed vertical wall of 60 cm and on top of

that was a hinged upper section of 15 cm. With this hinged upper section a large range of

chamfered and overhanging geometries can be reached. The base of the wall was sheltered by

a 15 cm high gravel berm. Each configuration was tested with three different water depths.

FIGURE 2-2: RANGE OF CHAMFERED AND OVERHANGING WALL GEOMETRIES AND WATER LEVELS

In general it can be said that in comparison with a simple vertical wall, chamfered upper

sections allow larger overtopping whereas overhanging structures reduce the overtopping

significantly. In Table 2-1 the results of the experiments can be seen with a relative freeboard

Rc/Hs of 1.67. In this table α is the angle as shown in Figure 2-2. The overtopping for

chamfered walls can be 5 times higher than a simple vertical wall, whereas for overhanging

walls, the overtopping can be reduced to less than 3 % of the original overtopping.

TABLE 2-1: OVERTOPPING FOR DIFFERENT CONFIGURATIONS WITH A RELATIVE FREEBOARD OF 1.67

α 30° 45° 60° 90° 120° 135° 150°

Mean Q/Qvertical

[%] 511 334 149 100 12 5.2 2.5

Kortenhaus

Kortenhaus (2003) performed in total 300 hydraulic model tests. In these model tests, three

wall heights, nine parapet types, six water levels, four wave heights and three wave periods

have been used throughout the tests. In this study the average overtopping with a wall or with

a parapet was considered in function of the relative freeboard. Only tests with the same

overall crest level are compared to each other. To compare these tests, a reduction factor k

was introduced:


9

𝑘 =𝑞𝑝𝑎𝑟𝑎𝑝𝑒𝑡

𝑞𝑤𝑎𝑙𝑙 (8)

When the relative freeboard is low (Rc/Hs < 0.3), k is almost equal to 1. This means that the

parapet doesn’t really influence the overtopping because the waves flow right over the wall.

When the relative freeboard increases, the effectiveness of the parapet increases. The

effectiveness keeps increasing until the relative freeboard reaches a value of 1.4. From this

point, the overtopping is already 0 for the vertical wall so the parapet doesn’t affect the

overtopping anymore.

2.3.2 Other types of parapets

Kamikubo

There are not only straight parapets; there are also curved parapets to reduce the wave

overtopping. An example of this kind of parapet is the Flaring Shaped Seawall (FSS), as

shown in Figure 2-3. This parapet has a deep circular cross section. In a study performed by

Kamikubo, the effect of the FSS on wave overtopping and spray was tested. The purpose of

this study wasn’t to lower the overtopping, but to find the lowest crest elevation which

resulted in acceptable overtopping.

To check the effect of the FSS on wave overtopping, model tests were performed. The wave

steepness 𝑠𝑚−1,0 was fixed at 0.036 or 0.012 and the incident wave height Hs was changed in

the range of 6 to 14 cm. The water depth was also varied. The results of these tests show that

the average overtopping is larger for the smallest wave steepness (0.012). When the results of

a conventional upright seawall are compared to the FSS, it can be seen that the overtopping is

reduced much for the FSS. This means that the crown height for a FSS can become smaller

than for a vertical wall, under the same allowable overtopping quantity.

There are still some variations possible on this FSS, but as they aren’t of interest to this

master thesis, they aren’t discussed further.


10

FIGURE 2-3: FSS IN COMPARISON TO UPRIGHT SEAWALL

Pearson

Pearson (2004) investigated the influence of certain types of recurve walls (see Figure 2-4) on

wave overtopping. The tests are performed in the wave channel at the University of

Edinburgh. This channel is 20 m long, 0.4 m wide with a working depth of 0.7 m. During

these tests also the induced forces were measured.. The test matrix included both uni- and bi-

modal seas with nominal steepnesses around 0.04 and 0.02. When looking at the results, it

immediately stands out that the overtopping is more dependent of the steepness than of the

parapet. The overtopping response of these structures is very sensitive to the wave period.

Shorter period waves are generally more affected by the presence of the parapet than the

longer waves.

FIGURE 2-4: SMALL RECURVE (LEFT) AND MEDIUM AND LARGE RECURVE (RIGHT); DIMENSIONS IN M AT

PROTOTYPE SCALE

Chapter 2 Literature study 2.4 Wave overtopping (EurOtop Manual)

11

2.3.3 Simple slope

The case of a simple slope is a typical form of coastal protection found at the Belgian coast. It

is a very basic and old form of coastal protection. In order to protect the Belgian coastal

regions, a Coastal Safety Plan is worked out to protect these cities against storms with a return

period of a 1000 years. Based on theoretical and physical models it can be stated that the

safety of one third of the Belgian coast is insufficient. New solutions have to be searched for

such that the safety is guaranteed. A restriction is that the attraction of the open sea cannot be

lost and the apartments and promenades near the coastline have to remain untouched.

Van Doorslaer et al., 2010 performed 51 tests with non-breaking waves (𝜉0 > ~2) on a

smooth slope of 1:2. They received a similar prediction formula for the dimensionless

overtopping Q as van der Meer (see section 2.4) but with a slightly higher trend:

𝑞

√𝑔 ∙ 𝐻𝑚03= 0.2 ∙ exp (−2.335 ∙

𝑅𝑐𝐻𝑚0

) (9)

These tests and the resulting formula for the dimensionless overtopping will be used in

section 2.5. In section 2.4, the prediction formula proposed by van der Meer, found in the

EurOtop Manual (2007), for the dimensionless overtopping Q for a simple slope will be

discussed.

2.4 Wave overtopping (EurOtop Manual)

The general expression for the calculation of the dimensionless overtopping quantity Q (non-

breaking waves) according to the EurOtop Manual (Pullen et al, 2007) is equal to:

𝑞

√𝑔 ∙ 𝐻𝑚03= 𝑄0 ∙ exp (−𝑏 ∙ 𝑅∗) (10)

The left hand side of equation (10) is equal to Q, 𝑄0 is the dimensionless overtopping quantity

when the relative freeboard is zero, 𝑅∗ is equal to dimensionless freeboard and b is a

coefficient which describes the specific behaviour of wave overtopping for a certain structure.

In the EurOtop Manual, values for b and the resulting formula as in equation (10) can be

found for simple slopes, wave walls and other structures. The resulting formula for simple

slopes with non-breaking waves is equal to:

𝑞

√𝑔 ∙ 𝐻𝑚03= 0.2 ∙ exp (−2.6 ∙

𝑅𝑐𝐻𝑚0 ∙ 𝛾𝑓 ∙ 𝛾𝛽

) (11)

Chapter 2 Literature study 2.5 Combined overtopping reduction measures

12

In this formula, proposed by van der Meer, 𝛾𝑓 and 𝛾𝛽 are reduction factors related to the

roughness of the slope and the angle of the incident waves. Further it must be remarked that

this formula is only valid for configurations with a relative freeboard Rc/Hm0 larger than 0.5.

When the relative freeboard is lower, this formula significantly over predicts the overtopping.

This gap in the EurOtop Manual has been filled up by van der Meer & Bruce (2014). With the

help of data of Victor (2012), a new prediction formula could be fitted which takes into

account the slope angle of the sloping structure and which is valid for a relative freeboard

Rc/Hm0 upward of zero. In equation (12) the new prediction formula is presented. It takes the

form of a Weibull distribution:

𝑞

√𝑔 ∙ 𝐻𝑚03= 𝑎 ∙ exp (− (𝑏

𝑅𝑐𝐻𝑚0

)𝑐

) (12)

In prediction formula (12) the coefficients a, b and c are defined in equations (13), (14) and

(15).

𝑎 = {

0.09 − 0.01(2 − cot 𝛼)2.1 𝑓𝑜𝑟 cot 𝛼 ≤ 2 0.09 𝑓𝑜𝑟 cot 𝛼 ≥ 2

(13)

𝑏 = {

1.5 − 0.42(2 − cot 𝛼)1.5 𝑓𝑜𝑟 cot 𝛼 ≤ 2 1.5 𝑓𝑜𝑟 cot 𝛼 ≥ 2

(14)

𝑐 = 1.3 (15)

Some other gaps in the EurOtop Manual are also filled up by Van der Meer et al (2014), but

as they are not related to this master thesis, these cases aren’t described here. The interested

reader can always check the references for further information.

2.5 Combined overtopping reduction measures

Up to now, only single measures have been discussed. To further reduce the wave

overtopping, the measures can be combined. The effect of the combined reduction measures is

not equal to the sum of the single measures, therefore extra research is needed to analyse the

influence of the combined reduction measures on the wave overtopping. In this paragraph two

specific combinations are further treated: a slope combined with a vertical wall and a slope

combined with a parapet. The effect of these measures on prediction formula (9) will also be

discussed shortly.


13

2.5.1 Slope and vertical wall

Van Doorslaer et al. (2010) performed a study to measure the reduction in overtopping when

a vertical wall is built into a dike without changing the crest freeboard, as can be seen on

Figure 2-5. By doing so, the zone behind the wall can be filled up and a wider promenade is

possible.

FIGURE 2-5: PRINCIPLE OF THE VERTICAL WALL BUILT IN THE DIKE

The study exists of 88 tests that have been carried out in the wave flume of Ghent University

(more detailed information in Chapter 3). All tests were performed on the same slope as in

2.3.3 with a slope of 1:2. The overtopping was compared with respect to the reference case (a

slope without vertical wall). Different wall heights were tested along with non-breaking

waves. Adding a wall to the dike leads to a reduction in overtopping. Therefore the general

formula for the dimensionless overtopping in (9) must be adapted by adding a reduction factor

𝛾𝑣 taken into account the reduction in overtopping. This leads to the following formula:

𝑞

√𝑔 ∙ 𝐻𝑚03= 0.2 ∙ 𝑒𝑥𝑝 (−2.335 ∙

𝑅𝑐𝐻𝑚0

∙1

𝛾𝑣) (16)

For breaking waves, the reduction factor proposed by EurOtop (2007) is independent of the

wall height and equal to 0.648 for vertical walls. In the study performed by Van Doorslaer et

al. (2010), the data clearly show a dependency of the wall height. Larger walls lead to a

smaller reduction because the wall is closer to the still water line. The reduction factor for

non-breaking waves found by Van Doorslaer et al. can be written as:

𝛾𝑣,𝑛𝑜𝑛−𝑏𝑟𝑒𝑎𝑘𝑖𝑛𝑔 = exp ( −0.57 ∙

ℎ𝑤𝑎𝑙𝑙𝑅𝑐

) (17)

When this reduction factor is introduced in the prediction formula for Q, all obtained data

points are now located in the 90% confidence interval whereas this wasn’t the case for the

formula without the reduction factor 𝛾𝑣. This means that the adapted formula gives a much

better prediction of the overtopping over a smooth dike with a vertical wall.


14

2.5.2 Slope and parapet

To continue on the case of a slope (1:2) and a vertical wall, (Van Doorslaer et al., 2010)

investigated two innovative crest designs with a significant reduction of wave overtopping

without increasing the height of the crest level. One is the use of a parapet on the existing dike

structures, and the other one is the Stilling Wave Basin (SWB). Only the solution using a

parapet will be further discussed, because the concept of the SWB is less relevant for the

purpose of the thesis. In Figure 2-6 the solution is represented schematically. The solution

with the wave parapet can make the walking space of the dike wider, as represented at the

right of Figure 2-6. The space between the parapet and the dike is filled up with material.

FIGURE 2-6:SMOOTH SLOPE WITH PARAPET, RIGHT CASE INCREASED WALKING SPACE

In the research performed by Van Doorslaer et al. (2010) the optimal geometry of the parapet

is searched for such that for the same crest freeboard, the overtopping is reduced the most.

Smoothly curved parapets have not been investigated, because these types of coastal

protection don’t allow simple adaption of the existing vertical crown walls.

The tests were also performed in the large wave flume of the Department of Civil Engineering

at Ghent University. For dimensions and other features see Chapter 3. Irregular waves were

generated using mainly the JONSWAP spectrum with peak enhancement factor 𝛾 = 3.3.

Some tests were repeated with a standard Pierson-Moskowitz spectrum, but no significant

difference was found regarding overtopping.

To define the geometry of the parapet the factor 𝜆 and the angle 𝛽 are used. The factor 𝜆 is the

ratio of the height of the inclined part on the total height of the parapet. The angle 𝛽 is the

angle the inclined part makes with the vertical. The two parameters are represented in Figure

2-7.

FIGURE 2-7: SCHEMATICALLY REPRESENTATION OF PARAMETERS Β AND Λ


15

In total 92 tests were performed with the following different geometrical combinations: total

parapet height (ℎ𝑡) was 2, 5 and 8 cm, the angle 𝛽 was 15°, 30°, 45° and 60° and 𝜆 varied

between 1/8 and 1. A major reduction of the overtopping was found. In order to use the

formula stated by van der Meer to predict the overtopping, a reduction factor 𝛾𝑝𝑎𝑟 is

introduced in the exponential part of formula (16). This factor relates the reduction of

overtopping with the parapet nose in comparison to the vertical wall. The adapted equation

can be found in equation (18). The expression for 𝛾𝑝𝑎𝑟 can be found in equation (19).

𝑞

√𝑔 ∙ 𝐻𝑚03= 0.2 ∙ 𝑒𝑥𝑝 (−2.335 ∙

𝑅𝑐𝐻𝑚0

∙1

𝛾𝑣 ∙ 𝛾𝑝𝑎𝑟) (18)

𝛾𝑝𝑎𝑟 =(𝛾𝛽 ∙ 𝛾𝜆 − 0.0317)

0.541 (19)

The factor 𝛾𝑝𝑎𝑟 is a function of the parapet parameters 𝛽 and 𝜆 which are translated into

reduction factors 𝛾𝛽 and 𝛾𝜆 respectively. General remarks about these two parameters can be

stated as following: when only the angle 𝛽 is increased (with all the other parameters the

same), the reduction factor 𝛾𝑝𝑎𝑟 and the overtopping decrease. The reduction in 𝛾𝑝𝑎𝑟 is

significant until an angle of 𝛽 = 50°. From that angle on, no reduction in overtopping can be

realized by increasing the angle of the parapets nose angle. When the height ratio 𝜆 is

increased 𝛾𝑝𝑎𝑟 is reduced and so also the overtopping.

As conclusion Van Doorslaer et al. (2010) found that for ease of construction and to limit

wave impacts on the parapet nose, it is advised to take a nose angle 𝛽 ≤ 45° and a height ratio

𝜆 of 1/3. With these values for the parapet geometry the minimal value of 𝛾𝑝𝑎𝑟 will be

reached.

2.5.3 Conclusion

There are many different ways to protect the hinterland from overtopping. A lot of these

measures are already described in literature and accurate formulas have been drafted to

predict the amount of overtopping for those structures. However there are still some gaps in

the current literature. One of those gaps is the combination of a slope with a wall or parapet

on top the slope. Further also the configurations existing of a slope followed by a berm and a

wall or parapet at the end of the berm are investigated. The influence of placing a wall or

parapet on top of your slope and the influence of placing a berm between the top of the slope

and the wall/parapet will form the points of interest for this master thesis. Both the influence

on the overtopping and forces will be checked. For more detailed drawings of the different

test configurations, reference is made to section 3.2.

Chapter 2 Literature study 2.6 Wave forces on storm walls

16

2.6 Wave forces on storm walls

Kortenhaus et al. (2001) performed more than 900 hydraulic model tests in the wave flume of

the Leichtweiss Institute (90 m long, 2 m wide and 1.2 m deep) in order to study the loading

and overtopping on storm surge protection (SSP) walls. The purpose of the paper was to

propose very simple and effective methods for the prediction of the breaker type at the

structure, the average overtopping rate over the SSP walls and the magnitude of forces and

pressures at the wall taking into account the dynamic aspect of the loading. Furthermore

overtopping reduction elements and under water barriers were designed to reduce overtopping

and wave loading.

Figure 2-8 displays the generalized cross section of SSP walls. A slope of 1:2 with a certain

berm width is found in front of the SSP wall. Three different foreland heights ℎ𝑏, five

different berm widths 𝐵𝑏, three wall heights, three water depths, four wave heights, three

wave periods and two different overtopping reducers have been varied through the tests. The

scale of the model was 1:6.

FIGURE 2-8: GENERALISED CROSS SECTION OF SSP WALLS

Via video analysis the different breaker types on the foreland have been determined. One

could distinct standing waves (non-breaking on the slope or edge of slope), broken waves

(breaking on the slope or edge of the slope) and plunging or impact breaker (a broken wave

that breaks directly at the wall, inducing severe dynamic loading at the wall). The load cases

resulting from these breaker types are simplified to two cases: non-impact loading and impact

loading. A semi-empirical formula based on the wave height in front of the slope was

proposed for determining the maximum horizontal force in case of non-impact waves:

𝐹ℎ = 𝛾𝑓,𝑛𝑖𝜌𝑤𝑔𝐻𝑠2 (20)

𝛾𝑓,𝑛𝑖 is the empirical factor which can be set equal to 8.0 for the conditions investigated in the

paper. The validity of the equation is limited to wave heights 𝐻𝑠 up to 80 cm in prototype

conditions.

Chapter 2 Literature study 2.6 Wave forces on storm walls

17

The prediction formula for the maximum horizontal force caused by impact waves has a

similar form as eq. (20) in which the factor 𝛾𝑓,𝑖 was found to be 11.0 and the wave heights 𝐻𝑠

is limited to 1.2 m:

𝐹ℎ = 𝛾𝑓,𝑖𝜌𝑤𝑔𝐻𝑠2 (21)

Besides the magnitude of the resulting horizontal force, also the duration of the force must be

considered. For an impact force the duration is rather short and the behavior highly dynamic.

For a non-impacting wave the duration is longer and the behavior more static.

2.6.1 Conclusion

For the wave forces, there are also some gaps left in the literature. But as mentioned before,

the main focus of this master thesis is to investigate combinations of wave overtopping

reducing measures, therefore no extra tests were carried out to try and fill up the gaps of the

wave forces. Nevertheless, the influence of different wave and model parameters on the wave

forces will be examined and a general prediction will be drafted based upon the performed

tests.

19

CHAPTER 3 HYDRAULIC MODEL TESTS

In order to investigate the chosen structures and see their influence on overtopping and forces,

hydraulic model tests will be performed. These model tests are necessary to acquire data that

can be analysed and will give a better understanding of how the chosen structures deal with

the incident waves. In this chapter the test facility at Ghent University will be described, the

different model geometries and also the performed tests are discussed.

3.1 The wave flume at Ghent University

The hydraulic model tests are performed in the Big Wave Flume of Ghent University, located

in the AWW department in Zwijnaarde. The wave flume has a width of 1 m, a height of 1.2 m

and the total length is equal to 30 m. The wave paddle is 3.15 m long and is integrated in the

total length. Hence, an effective wave flume length of approximately 27 m can be used for

testing. The maximum water depth in the flume is around 80 cm and the maximum wave

height is 35 cm.

The wave flume is equipped with an active wave absorption system. This system that is

implemented in the flume software is derived from AWASYS. For more details on this

absorption system, the references can be checked (Troch, 2000).

FIGURE 3-1: LARGE WAVE FLUME AT GHENT UNIVERSITY

The wave paddle used in the flume is a piston type wave paddle. The wave paddle is fixed to

a moving open framework and moves on linear bearings. The maximum stroke length is 1.50

m. The displacement of the paddle is controlled by LabVIEW, a software developed at Ghent

University.

Chapter 3 Hydraulic model tests 3.2 Model geometry

20

FIGURE 3-2: WAVE PADDLE

To measure the overtopping, a tray is placed behind the structure. The width of the tray is

dependent on the model. For this thesis most tests were carried out with a width of 20 cm. The

tray leads the overtopping water to a container which is placed on a balance. By continuously

weighing the container and its content, the amount of overtopping water can be determined.

To determine this amount correctly, the scale needs to be calibrated, but more on this in

section 3.4.2. In the effort to try and measure individual overtopping, a wave gauge is placed

in front of the tray. By doing so, a signal is received every time there is water overtopping and

so individual amounts of overtopping can be related to certain waves (Troch, 2000).

The horizontal forces that act on the wall are measured by a 5 kg load cell. This load cell is

placed in such way that there is a very small gap between the load cell and the wooden wall.

Without this gap, there would be friction between the load cell and the wooden wall and this

would lead to incorrect results. If this gap should be constructed too big, the water flowing

through it would cause unwanted forces onto the load cell. Therefore, the gap should be

present but it has to be constructed as small as possible. For the force measurement a

sampling frequency of 1000 Hz is used (Hohls, 2015).

3.2 Model geometry

Before starting the model tests, the model geometry needs to be described accurately. In the

flume there is a fixed return channel in smooth concrete. This return channel has a height of

20 cm and a length of approximately 18 m. At the exit of the return channel, a wooden plate is

placed with a slope to avoid turbulent flow. On top of this concrete layer, at the end of the

flume, the test geometry is installed. The test geometry consists of a smooth slope of 1:2 and

has a height of 53 cm and is present for every conducted test. The tests can be split up into

three stages: first there’s no berm, secondly there’s a berm of 20 cm and finally a berm of 40


21

cm is installed. The whole flume is shown on Figure 3-3. On further Figures only detail A is

shown because only this part of the geometry changes.

FIGURE 3-3: DRAWING OF THE WHOLE FLUME; ALL DIMENSIONS ARE IN CM

The test geometries are not a scaled version of a real structure, but if needed scaling up of the

structure and test results can be done according to the Froude scaling laws. For the first stage,

when there’s no berm present, the reference case is tested. The reference case is the

configuration without any measures to reduce the overtopping (only a slope). This

configuration is shown on Figure 3-4. On this Figure only detail A as depicted in Figure 3-4 is

shown.

FIGURE 3-4: THE REFERENCE CASE

For the next tests without the berm, measures were taken to reduce the overtopping. First a

simple vertical wall of 5 cm was mounted on top of the slope. For the second series of tests, a

parapet of 5 cm was installed. More information on the used parapet can be found further in

this section. The next tested measure is a simple vertical wall of 10 cm. For the last series of

tests, a parapet of the same height was installed. On Figure 3-5, the geometry with a parapet

of 10 cm is shown.


22

FIGURE 3-5: NO BERM AND A PARAPET OF 10 CM

For the second stage, the wall, with or without parapet, is moved 20 cm behind the slope. The

berm that is created in this way is given a 1% slope to guarantee water drainage. To keep the

freeboard equal to the tests without the berm, the slope is 0.2 cm lower than before. A

drawing of this geometry can be seen on Figure 3-6.

FIGURE 3-6: 20 CM BERM AND A PARAPET OF 10 CM

For the third stage, the wall, with or without parapet, is moved 40 cm behind the slope. The

same principles are taken into account as with the 20 cm berm so now the slope is 0.4 cm

lower in comparison to the tests without any berm. A drawing of this configuration can be

found on Figure 3-7.

FIGURE 3-7: 40 CM BERM AND A PARAPET OF 10 CM

The parapet used in the model tests is very easy to install. A triangle shaped wooden piece is

fixed to the vertical wall by some screws. This makes it easy to change between different

configurations. The dimensions of the parapet are chosen based upon the literature study

(section 2.5.2). The angle between the parapet and the vertical was chosen to be 45° and the

Chapter 3 Hydraulic model tests 3.3 Test matrix

23

ratio of the height of the parapet to the total height of the wall was chosen equal to 0.3.

According to literature this would result in the most efficient parapet. Both parapets used are

shown on Figure 3-8.

FIGURE 3-8: DRAWING OF THE USED PARAPETS

3.3 Test matrix

In section 3.2 the different model geometries were determined. Besides the model geometry

also the test conditions must be defined. It is chosen to generate irregular waves with a

JONSWAP spectrum and peak enhancement factor 𝛾 = 3.3. Also the waves need to be non-

breaking at the toe of the slope. Three different significant wave heights Hs are chosen: 5 cm,

10 cm and 15 cm. The peak periods Tp of the waves are chosen in such a way that the wave

steepness 𝑠𝑝 lie close to two target values: 0.02 and 0.05. The peak periods chosen per

significant wave height are summarized in Table 3-1. The corresponding wave steepness with

these wave periods are not shown in Table 3-1 because they depend on the water depths used

in the tests.

TABLE 3-1: CHOSEN PEAK PERIODS TP WITH THEIR CORRESPONDING WAVE HEIGHT HS

Wave height 𝐻𝑠 [cm]

peak period 𝑇𝑝

[s] 5 0.8

1.2 10 1.1

1.8 15 1.4

1.8

At last, each configuration is tested with two different water depths d (more about it further

in this section). Table 3-2 gives a summary of test conditions and geometrical configurations

that will be varied. For each geometrical configuration the combination of the different test


24

conditions results in 12 unique tests. In total 156 different tests were planned to be performed.

To check the validity of the tests some configurations will be tested more than once, either

with the same time series or either with a different one. In total 166 tests were performed with

150 useful individual tests and 16 repetition tests.

TABLE 3-2: SUMMARY OF THE TEST MATRIX

Geometrical configuration Reference case, no berm, 20 cm berm, 40 cm berm

Type of wall/ parapet 5 cm/10 cm wall and 5 cm/10 cm parapet

Water depth d Two water depths, depends on geometrical conf.

Wave height 𝐻𝑠 5 cm, 10 cm, 15 cm

Wave steepness 𝑠𝑝 0.02 and 0.05

In Table 3-2 it is stated that the two water depths are chosen and that this depends on the

geometrical configuration. One water depth of 70 cm (50 cm at the toe of the slope) at the

paddle is chosen as the reference water depth, wherefore the other one depends on the

configuration. For the reference case a lower water level is chosen, otherwise the water would

flow into the overtopping box with the smallest change in surface water level. For the cases

with a wall/parapet of 5 cm, the other water level is set on 73 cm (53 cm at the toe). For the

cases with a wall/parapet of 10 cm in combination with a berm, the water level of 70 cm is

increased to 76 cm (56 cm at the toe), because otherwise too low overtopping quantities are

expected.

To check if the individual tests will be useful (to make up the test matrix before testing) some

rules of thumb are checked. The first one is that the relative freeboard needs to be in the

following range:

0.5 ≤

𝑅𝑐𝐻𝑠≤ 2 (22)

The lower limit makes sure that there is not too much overtopping, where the upper limit

prevents tests being performed without much overtopping.

The second one is to check that the waves are non-breaking. Therefore the ratio of the

significant wave height over the water depth at the toe of the slope is considered and needs to

be below the indicated limit:

𝐻𝑠𝑑< 0.5 (23)

If the individual tests don’t lie in the boundaries given by (22) and (23), the tests will be

performed once in the wave flume. If it is clear no useful results are found (for example no


25

overtopping due to too high relative freeboard or too wide promenade), tests with similar test

conditions will not be executed. In total 6 tests were not performed due to too low

overtopping quantities. The different tests performed together with their test conditions can be

found in the test matrix, put in Annex D.

3.3.1 Time series and test names

For each test, a specific time series needs to be created in the program LabVIEW. The time

series forms the surface elevation that the wave paddle tries to attain. Figure 3-9 gives an

example screen how the creation of a time series in LabVIEW is done. The time series is

defined by the water height at the paddle, significant wave height 𝐻𝑠, peak period 𝑇𝑝, the

duration time of the test, extra time which needs to be added before and after the duration

time and the spectrum which needs to be generated.

FIGURE 3-9: GENERATION OF A TIME SERIES IN LABVIEW

The pre time is fixed to 5 s where the post time of the test is fixed to 120 s. If the post time is

longer, it allows the active wave absorption system still to be active and damping out the

reflected waves still present in the flume. The result is that a still water level is much faster

attained. The duration time of each test is defined by stating that each test should contain

about 1000 waves. Therefore the average period 𝑇𝑚 of a wave should be calculated. With this

average period the required duration time can be defined by multiplying 𝑇𝑚 with 1000. On

average the test will comprise 1000 waves. The average period of the waves is defined by

(24).

𝑇𝑚 =

𝑇𝑝

1.1 (24)


26

Each different time series is numbered with a number and a letter. A time series with other

hydraulic parameters is given another number, but a different time series with the same

hydraulic parameters is given the same number, but a different letter. For example, 0016A for

the sixteenth tests with hydraulic parameter combination that wasn’t used before, but 0016B

for a new time series with the same hydraulic parameters. In total 24 unique time series were

made.

The different tests are numbered on a similar way as for the time series; a number followed by

a letter. Just as for the time series, the number stands for uniqueness of the test, where the

letter stands for possible repetition. For example, repeating test 003A twice with the same

time series results in two tests with names 003B and 003C. Repeating a test with another time

series is given the name 003D (see Annex D).

3.3.2 Repetition tests

Mentioned earlier in the section about the test matrix, some repetition tests were performed to

check the consistency of both the overtopping and force measurements. For each berm width,

at least one test was repeated twice with the same time series and once with a new time series.

For the repetition tests performed with the same time series the average overtopping volume

and the force measurements are approximately the same. The average overtopping rates

should be close to each other, and when the force measurements are put over each other, the

different peaks should occur at the same time with approximately the same magnitude. For the

repetition test with another time series the overtopping quantity should be in the same range

and the time series of the forces will be different, but the magnitudes of the forces should be

in the same order.

In Table 3-3 the average overtopping is compared for test 0091A and its repetition tests

0091B, 0091C and 0091D (different time series). It can be seen that for 0091A, 0091B and

0091C the overtopping quantity is almost the same. The overtopping quantity for 0091D

differs a little bit more due to the use of another time series, but lays in the same ranges as for

0091A.

TABLE 3-3: AVERAGE OVERTOPPING FOR TEST 0091A AND ITS REPETITION TESTS 0091B, 0091C AND 0091D

test q [l/s/m]

0091A_d70_Hs10_T1.1 0.168 0091B_d70_Hs10_T1.1 0.162 0091C_d70_Hs10_T1.1 0.169 0091D_d70_Hs10_T1.1 0.137

The force measurements of 0091A, 0091B and 0091C are put over each other with the help of

the program LDavis. This is represented in Figure 3-10. It can be seen that the peaks are

Chapter 3 Hydraulic model tests 3.4 Calibration

27

reached at the same time and lay in the same order. Plotting 0091D over the other plots would

be useless as a different time series is used.

FIGURE 3-10: COMPARISON BETWEEN TESTS WITH THE SAME TIME SERIES (0091A: BLUE, 0091B: GREEN, 0091C:

RED)

3.4 Calibration

Once the first test setup is installed into the wave flume, some measuring equipment needs to

be calibrated before the model tests can be performed; namely the pump which empties the

overtopping box, the weigh cell, the wave gauges and the force load cell. In the following

sections the calibration of each of them will be discussed.

3.4.1 Pump

The pump starts working automatically when a mass of 80 kg is measured by the weigh cell.

The pump removes water for 4.5 s, leaving enough water in the reservoir such that the pump

stays submerged. Calibration of the pump is required to correct the weight measurements by

the weigh cell and find the actual amount of overtopped water during a pump cycle. A

possible calibration method is to determine the average pumping curve.

The pumping curve is determined by filling the reservoir manually till the weigh cell

measures almost 80 kg. The measurements are started and the reservoir is filled carefully to

exactly 80 kg. The change in weight of the reservoir and the pumping time of 4.5 s define the

pumping curve. To have an accurate pumping curve, the above defined procedure is

performed 10 times and the average pumping curve is calculated. The 10 established pumping

curves are displayed in Figure 3-11.


28

FIGURE 3-11: INDIVIDUAL PUMPING CURVES

From Figure 3-11 it can be seen that the 10 pumping curves don’t differ much from each other

and that the average pumping curve will approximate the real pumping curve close enough.

The average pumping curve is represented in Figure 3-12. At the start, a small delay of 0.5 s

can be remarked. During the actual pumping a linear decrease of the water mass can be seen

until the minimum value of 49.533 kg is reached. Remarkable is that a certain amount of

water is pumped out the reservoir when the pump is shut down; it is after the pumping period

of 4.5 s. A total amount of 30.468 kg of is pumped out on average.

time [s]

0 2 4 6 8

am

ount of

wate

r [k

g]

45

50

55

60

65

70

75

80

85

pumping curve 1pumping curve 2pumping curve 3pumping curve 4pumping curve 5pumping curve 6pumping curve 7pumping curve 8pumping curve 9pumping curve 10


29

FIGURE 3-12: AVERAGE PUMPING CURVE

3.4.2 Weigh cell

The scale or weigh cell is used to measure the amount of overtopped water collected in a

reservoir above the weigh cell. Due to the inconsistency between the mass indicated by the

weigh cell and the total sum of the reservoir mass and the effective water mass in the

reservoir, a calibration of the weigh cell is necessary (Vroman and Pintelon, 2013). A larger

value is registered by the weigh cell due to the appearance of Archimedes forces, induced by

the presence of the submerged pump. The calibration curve gives the relation between the

mass registered by the weigh cell and the actual mass in the reservoir. The following

procedure is followed to come up with the calibration curve of the calibration curve for the

weigh cell.

The reservoir is filled in several steps, such that each step corresponds with approximately 1

kg. The exact mass of water is measured on a calibrated scale, where after it is poured into the

reservoir. The reference weight on the weigh cell is measured and corresponds with a value of

41.235 kg. This value corresponds with an absolute value of 0 kg on the calibrated scale. By

repeating the above procedure and measuring the weight of the individual steps on the

calibrated scale and the weight on of the reservoir, the calibration curve can be established.

Figure 3-13 corresponds with this calibration curve. A clear linear relationship can be found.

The filling procedure is followed until the weigh cell measures 85.155 kg. This value is higher

than the value of 80 kg where the pump starts pumping. The reason for this is that when the

scale is close to 80 kg and a large overtopping event takes place, the scale will measure a

time [s]

0 2 4 6 8

am

ount of

wate

r [k

g]

45

50

55

60

65

70

75

80

85

average pumping curve


30

quantity larger than the 80 kg. To correlate also values above the 80 kg, the scale must be

calibrated for these larger values.

FIGURE 3-13: CALIBRATION CURVE WEIGH CELL

The calibration is performed in gentle conditions. During the models tests, large overtopping

quantities can happen which cause a serious impact on the submerged pump. Therefore the

submerged pump needs to be fixed to prevent possible movement under this water impact.

The calibration curve presented in Figure 3-13 will be used together with the average

pumping curve (Figure 3-12) to calculate the average and individual overtopping quantities.

More about that can be found in sections 4.2 and 4.3.

3.4.3 Wave gauges

For the chosen test setup three sets of wave gauges are used. One set of two wave gauges is

used to guide the active wave absorption system (named AWA1 and AWA2), a set of three

wave gauges (named WG1, WG2 and WG3) is used near the wave paddle to measure the

deep water wave conditions and the last set of three wave gauges (named WG4, WG5, WG6)

is used to determine the surface elevation a distance away from the toe of the structure. The

guidelines of Mansard and Funke (1980) are used to determine the intermediate distances

between the different wave gauges and for the sets with three wave gauges also the minimum

distance from the toe of the structure and the wave paddle respectively.

According to Mansard and Funke (1980) there is a certain minimum distance that needs to be

present between the wave gauges. The results are based on experimental investigations. The

relative mass weigh cell [kg]

30 40 50 60 70 80 90

absolu

te m

ass c

alib

rate

d s

cale

[kg]

0

10

20

30

40

50

calibration curve weigh celltrendline


31

distances need to be calculated as presented in equations (25) and (26). The distances between

the gauges only depend on the wave length at the paddle 𝐿𝑝𝑎𝑑𝑑𝑙𝑒 determined with the peak

period 𝑇𝑝. In these equations, 𝑥1,2 refers to the distance between the first and second wave

gauge of the set of 3 wave gauges closest to the wave paddle and 𝑥1,3 refers to the distance

between the first and third wave gauge of this set of wave gauges. Similar formulas are found

for the set of wave gauges near the toe of the slope, only the indices change (wave gauges 4, 5

and 6). The actual used distances can be found in the test matrix (Annex D).

𝑥1,2 =

𝐿𝑝𝑎𝑑𝑑𝑙𝑒

10 (25)


6< 𝑥1,3 <


3 𝑎𝑛𝑑 𝑥1,3 ≠


5 𝑎𝑛𝑑 𝑥1,3 ≠

3𝐿𝑝𝑎𝑑𝑑𝑙𝑒

10 (26)

FIGURE 3-14: SET OF 3 WAVE GAUGES NEAR THE TOE OF THE STRUCTURE

For irregular waves, the minimal distance between the wave gauge (number 6) closest to the

toe of the slope and the slope should be 0.4𝐿𝑝𝑎𝑑𝑑𝑙𝑒 . As the peak period 𝑇𝑝 changes for every

test it is more convenient to use a fixed value. If the maximum peak period 𝑇𝑝= 2.2 s is used

to calculate the fixed distance, all the distances for the other peak periods will also fulfil. A

fixed distance of 3 m is chosen between the toe of the slope and the first wave gauge

encountered. Similar a fixed distance should be present between the first wave gauge and the

wave paddle.

To make sure the wave gauges measure the real surface elevation, they need to be attained a

certain zero level. By calibrating them before each test, this is realised. The calibration is

defined by the linear relationship between the measured voltage and the depth the wave

gauges are found in the water. This relationship is determined by recording the voltages of the

wave gauges at two positions, low and high. First the wave gauges are put in high position

and the voltage is measured. The wave gauges can be pneumatically operated, which makes it


32

easier to change their position in the vertical direction. Thereafter the wave gauges are

lowered 10 cm and again the voltage is measured. The knowledge of the two voltages at

different levels and the relative distance the wave gauges is lowered in low position (10 cm),

the calibration can be performed.

3.4.4 Force load cell

The load cell used to measure the forces is 5 kg load cell. This load cell needs to be calibrated

daily to check whether the output is still acceptable. The acceptability of the output is tested

through a very precise calibrated pressure meter. This pressure meter is pushed against the

load cell and the outcome of the force read on the pressure meter and the one registered with

the load cell are compared. When the two forces are the same, the load cell is still accurately

calibrated.

For the analysis of the forces, the resonance frequency of the load cell also needs to be

determined. This is done by inducing three hammer blows (see Figure 3-15) to the load cell

for each configuration: 5 cm wall/parapet and 10 cm wall/parapet. For the 10 cm wall/parapet

the resonance frequency is around 90 Hz (see Figure 3-16) and for the 5 cm wall/parapet the

resonance frequency is around 103 Hz (see Figure 3-17). When applying the data filter in

Chapter 4, these frequencies will have to be filtered out to become good results.

FIGURE 3-15: THREE HAMMER BLOWS TO THE LOAD CELL

FIGURE 3-16: RESONANCE FREQUENCY OF THE 10 CM WALL/PARAPET


33

FIGURE 3-17: RESONANCE FREQUENCY OF THE 5 CM WALL/PARAPET

At the start of the model tests, the load cell was installed as seen on Figure 3-18 (left). During

testing it was seen that the load cell was also hit by water overtopping the wall. Since the aim

of the model tests was to measure the forces on the storm wall, this extra force impact due to

the overtopping water needed to be filtered out from the test results. The solution to this

problem was to install a cover (see Figure 3-18 (right)) in front of the load cell above the

storm wall so overtopping water could not hit the load cell anymore. The influence of this

cover on the wave force will be discussed in section 4.4.2.

FIGURE 3-18: LOAD CELL WITHOUT COVER (LEFT) AND WITH COVER (RIGHT)

35

CHAPTER 4 DATA PROCESSING

Once the model tests are performed, some initial data processing must be done such that the

data collected by the model tests can be analysed. A reflection analysis is performed to

determine the generated wave characteristics, the forces measured are filtered and the average

and individual overtopping parameters are calculated with the help of existing MATLAB

scripts. The following sections describe these different data processing steps and the

programs/scripts that were used.

4.1 Wave measurements

In the program LabVIEW a significant wave height and peak wave period can be given as

input which form the target values for the time signal which is sent to the wave paddle and

translated into the movement of the wave paddle. By measuring the surface elevation with the

two sets of three wave gauges, the real generated wave characteristics can be determined. For

the data processing of these wave characteristics the software package WaveLab, developed at

the Aalborg University, is used. By performing a reflection analysis in WaveLab, some wave

characteristics can be determined in both the time domain as in the frequency domain.

WaveLab is user friendly and the method to perform the reflection analysis will be explained

shortly below.

FIGURE 4-1: INPUT REQUIRED TO PERFORM THE REFLECTION ANALYSIS

Chapter 4 Data processing 4.1 Wave measurements

36

The input information requested for the wave analysis is summarized in Figure 4-1. The first

step is to load all the ghm-files created by LabVIEW that represent the surface elevation in

time measured by the gauges. These ghm-files are attained with a sample frequency of 40 Hz.

After loading the required files in it and applying the right sample frequency, the

corresponding water depth for the set of wave gauges considered, needs to be filled in. This

data needs to be filled in for each ghm-file separately. Length of the scale Prototype/Model is

kept on one, because we want to analyse the wave characteristics generated in the wave flume

itself.

Also it is requested to define the number of gauges that were used to measure the surface

elevation and also the distance between these gauges. The number of gauges is always three,

but the distance between them is defined by the wave characteristics that were given as input

to the wave paddle (see section 3.4.3). To distinct between the different wave gauges, they are

given a certain channel number. The three wave gauges measuring the offshore conditions at

the wave paddle have the numbers 1,2 and 3, where 1 is the wave gauge nearest to the wave

paddle and 3 the wave gauge furthest from the paddle. The wave gauges near the toe of the

slope are also numbered. They have the numbers 4,5 and 6, where 6 is the gauge nearest to the

toe and 4 furthest from the toe. The distance between the gauges needs to be filled in again for

each file separately.

Because there is some time defined in the beginning (5 seconds) before the wave paddle starts

generating waves and some time at the end of the generated series (120 seconds) were the

wave paddle only moves to compensate the reflected waves in the flume, some data points

can be skipped. The last step before the reflection analysis can be performed is to define this

amount of data points that need to be skipped at the start and end of the signal. This is done to

have a time signal which forms a representative signal to calculate the wave characteristics.

There is chosen to skip 2000 data points at the start and 6000 data points at the end. With the

knowledge of the sample frequency of 40 Hz, this corresponds with shortening the time signal

with 50 s at the start and 150 s at the end. The reason for shortening the signal with 50

seconds at the start is because it takes also some time before the first generated waves at the

wave paddle reach the toe and the wave gauges at the toe.

When everything is set as described above, the calculation can be executed and a summary of

the wave parameters in the time and frequency domain will be displayed as represented in

Figure 4-2 and Figure 4-3. In the frequency domain the wave parameters that will be used for

the analysis are the significant wave height Hm0, the reflection coefficient Crefl, the peak

period Tp and the wave period Tm-1,0. The spectral wave period Tm-1,0 is defined in equation

(27). In this equation 𝑚−1 and 𝑚0 are the spectral moments of order -1 and 0 of the found

wave spectrum.


37

𝑇𝑚−1,0 =𝑚−1

𝑚0 (27)

The only wave parameter that will be used from the time domain analysis is the number of

waves found in the time series.

FIGURE 4-2: EXAMPLE OF THE OUTPUT IN THE FREQUENCY DOMAIN IN TABLE FORM

FIGURE 4-3: EXAMPLE OF THE OUTPUT IN THE TIME DOMAIN IN TABLE FORM

The significant wave height in the frequency domain does not differ much from the value

found in the time domain and in most of the overtopping calculations the significant wave

height in the frequency domain is used, therefore Hm0 will also be used in the thesis. Besides

an output of the results in table form, it is also possible to give the results in graphical form. In

the frequency domain the generated wave spectrum can be displayed as also the theoretical

JONSWAP spectrum (Figure 4-4). By changing the amount of data used in a Fast Fourier

Transform (FFT) block, the generated spectrum can be modified such that it approximates the

theoretical spectrum as optimal as possible. For most of the generated spectra 1024 data

points in a FFT block was enough to approximate the theoretical spectrum.

FIGURE 4-4: WAVE SPECTRUM TEST 0011A


38

In the time domain it can be graphically checked if the measured wave heights are Rayleigh

distributed. Figure 4-5 displays an example of such a graphical output. When the measured

wave heights lie close to the line for the Rayleigh distribution, it can be said that the wave

heights are Rayleigh distributed. In the graph below the wave heights are not Rayleigh

distributed. The check if the wave heights are Rayleigh distributed is needed to run the script

for the calculation of the individual overtopping.

FIGURE 4-5: WAVE HEIGHT DISTRIBUTION TEST 0011A

4.1.1 Difference in generated and theoretical wave characteristics

Performing the reflection analysis for all the 166 tests, some conclusions can be made. For

each test the wave height is compared with the target wave height and the difference in

percentage of that target wave height is calculated. Figure 4-6 gives the target wave height

against the generated wave height measured with the wave gauges at the toe of the structure.

FIGURE 4-6: GENERATED SIGNIFICANT WAVE HEIGHTS PLOTTED AGAINST THE TARGET SIGNIFICANT WAVE

HEIGHTS (AT THE TOE)

Hm0, target

0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14 0,16 0,18

Hm

0,t

oe

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

0,18

perfect generationHm0,toe

regression line Hm0,toe


39

The red line displays the relation when the target wave height is exactly the same as the

generated one. The black dots represent the real relationship between them. By plotting the

resulting trendline it can be seen that the generated wave heights are predominantly smaller

than the target wave heights.

In general a reduction of the significant wave height can be seen which is larger for the wave

heights measured with the set of wave gauges at the toe than for the set of wave gauges at the

wave paddle. In Table 4-1 a summary is given of the different configurations with the tests

which result in the largest wave height reduction at the toe. Also the reflection coefficient

measured at the toe of the structure is included in Table 4-1.

TABLE 4-1: SUMMARY OF REDUCTION IN WAVE HEIGHT AT THE TOE

test configuration Hm0

target Hm0 123

123 % from

target Hm0 456

456 % from

target Crefl

0012A no wall no promenade 0,1 0,09558 4,42 0,08221 17,79 0,456

0019A 5 cm wall no promenade 0,05 0,04988 0,24 0,04277 14,46 0,665

0029A 5 cm parapet no promenade 0,1 0,09704 2,96 0,08652 13,48 0,576

0038A 10 cm wall no promenade 0,05 0,04953 0,94 0,04371 12,58 0,688

0053A 10 cm parapet no promenade 0,1 0,09796 2,04 0,08685 13,15 0,597

0067A 10 cm parapet 20 cm promenade 0,15 0,1376 8,27 0,1359 9,40 0,357

0077A 10 cm wall 20 cm promenade 0,1 0,09642 3,58 0,0839 16,10 0,361







The largest reduction is up to 29% of the target significant wave height. The other reductions

don’t exceed 18% of the target value. The reason for the smaller wave height at the toe is due

to wave transformations and energy dissipation. The exact reasons are unknown.

A similar analysis can be done for the peak wave period 𝑇𝑝. Again the target periods are

displayed against the generated periods. This is illustrated in Figure 4-7. Now it can be seen

that the trendline for the generated periods lies close to perfect relation between theoretical

and generated peak period. The same can be found for peak periods generated at the wave

paddle.

Chapter 4 Data processing 4.2 Average overtopping measurements

40

FIGURE 4-7: GENERATED PEAK PERIODS PLOTTED AGAINST THE TARGET PEAK PERIODS (AT THE TOE)

4.2 Average overtopping measurements

With the help of an existing MATLAB-script (Goormachtigh & Harchay, 2010), the average

overtopping discharge q [m³/s/m] can be calculated. The script requires as input data the

average pumping curve from the pump and the calibration curve of the weigh cell (determined

in section 3.4). During each test the values the weigh cell measures are continuously

transferred to a text file. These text files are attained the same name as the test name and are

given the extension .BAL (abbreviation of ‘balance’) and must be put in the directory

prescribed in the MATLAB script. Further the script requires also an input Excel-file

indicating the name of the test, the freeboard Rc and the seconds to skip at the start (50 s) and

the end of the test (120 s), each given in a separate column. The seconds which are skipped in

front and at the end of the test are the same as for the wave measurements.

By using an Excel input file, it is possible to analyse multiple tests in a single MATLAB

script execution. Before the script could be used small modifications were needed, such that it

becomes applicable for the test configurations considered. Table 4-2 lists up these

modifications and refers also to the line in the MATLAB-script. Besides these modifications

all path names are altered to work with the correct directories.

Tp, target

0,0 0,5 1,0 1,5 2,0 2,5 3,0

Tp

, to

e

0,0

0,5

1,0

1,5

2,0

2,5

3,0

perfect generationTp,toe

regression line Tp,toe


41

TABLE 4-2: MODIFICATIONS PERFORMED IN THE MATLAB -SCRIPT TO PREDICT THE AVERAGE OVERTOPPING 𝑞

Line number Modification

19 Reference to the correct calibration file of the weigh cell

32 Reference to the correct calibration file of the pump

161 Adjust the reservoir’s capacity (80 kg)

166 Adjust the correct pumping time (4.5 s)

302 Fill in the suitable width of the overtopping area (20 cm for normal

overtopping behaviour, 10 cm for extreme overtopping behaviour)

The script will produce four graphs and an output file containing the average overtopping

discharge q in m³/s/m for each test. These four Figures are discussed for test 0091A (for the

test conditions see Annex D). In Figure 4-8 the pumped absolute mass is plotted as function of

time. The jump around 400 s indicates a pumping event corresponding with an increase of the

pumped mass. For test 0091A the pump was only activated once.

FIGURE 4-8: PUMPED ABSOLUTE MASS AS FUNCTION OF THE TIME

In Figure 4-9 the absolute mass in the reservoir is shown as function of the time. From the

start the reservoir is filled with overtopped water.


42

FIGURE 4-9: ABSOLUTE MASS IN THE RESERVOIR AS FUNCTION OF THE TIME

The absolute mass in the reservoir increases until a value around 37 kg is reached. The

absolute mass of 37 kg corresponds with a relative mass of 80 kg measured on the weigh cell.

The pump is calibrated such that when the weigh cell indicates a value equal or larger than 80

kg, the pump starts working (see section 3.4.1). This is visible by the sudden decrease of the

absolute mass in the reservoir around 400 s. For test 0091A this occurred only once, but in

other tests the cycle described above repeats itself more often.

Figure 4-10 displays the cumulative absolute mass in the reservoir in function of time.

FIGURE 4-10: CUMULATIVE ABSOLUTE MASS IN THE RESERVOIR


43

The steepness of the slope of this curve indicates how fast the reservoir is filled. By

calculating the derivative of the curve in Figure 4-10, the rate at which absolute mass is

entering the reservoir can be seen over time. This last Figure, displaying the derivative, is not

shown in this thesis, because it is of less importance.

The average overtopping q itself is determined by the start and end value of the cumulative

absolute mass curve and the time interval of the test. The time interval is determined by

skipping seconds in front and at the end of the test as was discussed in section 4.1 about the

wave measurements. These are given in in the input Excel file as mentioned before. The

formula to determine the average overtopping discharge in m³/s/m is given in equation (28):

𝑞 =

𝑀𝑐.𝑎𝑏𝑠.,𝑒𝑛𝑑 −𝑀𝑐.𝑎𝑏𝑠.,𝑠𝑡𝑎𝑟𝑡

𝑡𝑒𝑛𝑑 − 𝑡𝑠𝑡𝑎𝑟𝑡∙

1

𝜌𝑤𝑊𝑡𝑟𝑎𝑦 (28)

In formula (28) 𝜌𝑤 is mass density of water (1000 kg/m³) and 𝑊𝑡𝑟𝑎𝑦 the width of the tray used

(mostly 20 cm, special cases 10 cm).

4.2.1 Scatter on the average overtopping

When looking at the repetition tests, there is some scatter present when comparing the average

overtopping. When different configurations will be compared to each other, it is important to

know if the difference in average overtopping is due to the change in configuration or due to

the scatter. Therefore the amount of scatter needs to be determined. Test 0091A (Annex D)

has been repeated four times, three times with the same time series (0091B and 0091C) and

once with another time series (0091D). The average overtopping of the first three repetitions

are almost equal but the last one is a bit different (see Table 4-3). Due to the different time

series used, the scatter can be up to 19 %. For the two other tests (test numbers 119 and 143)

that have been repeated four times, the scatter is respectively 1 and 5 %. On average the

scatter on the average overtopping is equal to 8 %. This means that if the difference in

average overtopping between two configurations is less than 8 %, it cannot be said that the

one configuration is better or worse than the other.

TABLE 4-3: SCATTER PRESENT ON THE AVERAGE OVERTOPPING

test q [l/s/m]

0091A 0.168 0091B 0.162 0091C 0.169 0091D 0.137

maximum scatter [%] 19

Chapter 4 Data processing 4.3 Individual overtopping measurements

44

4.3 Individual overtopping measurements

The individual overtopping volumes are also determined with the help of an existing script

(Goormachtigh & Harchay, 2010). As the script has a serious length only the output elements

used for this thesis will be shortly discussed.

A first parameter that can be retrieved is the average overtopping quantity q but calculated on

a different way than represented in equation (28). The difference with equation (28) is that not

the difference of end and start value of the cumulative mass curve is taken but the sum of the

individual overtopping quantities. The q-values calculated with the individual overtopping

script can be compared with the values found for the average overtopping script.

FIGURE 4-11: COMPARISON 𝑞-VALUES CALCULATED WITH AVERAGE OVERTOPPING SCRIPT AND INDIVIDUAL

OVERTOPPING SCRIPT

In Figure 4-11 the q-values calculated with the average overtopping script are displayed

against the values calculated with the individual script. The black line drawn shows when the

two q-values would be exactly the same for the two scripts. The red line gives the real relation

between the two calculated q-values. As the red line almost coincides with the black line, it

forms an extra confirmation that the q-values determined with the average script are trustable.

Other parameters determined with the individual overtopping script are the amount of

overtopping events Now and the individual overtopping volumes Vi [kg/m]. More about the

use of Now for the analysis can be found in Chapter 6.

With the individual overtopping volumes Vi some characteristics can be determined. Vmax is

the maximum volume of all the individual overtopping volumes and V1/250 is the average

q [l/s/m] with avg. script

0 2 4 6 8 10 12 14 16

q [l/s/m

] w

ith ind. script

0

2

4

6

8

10

12

14

16

data pointsreal relationperfect relation

Chapter 4 Data processing 4.3 Individual overtopping measurements

45

overtopping volume of the 0.4% largest overtopping volumes. On average 1000 waves are

generated per test. But not all these waves will lead to an individual overtopping event. The

consequence of this is that for tests which have less than 250 overtopping events, Vmax and

V1/250 will be equal to each other. In Figure 4-12 the maximum volume Vmax is set against the

volume V1/250. The scatter of the results increases with the magnitude of the volumes. The

values for Vmax lie between 0 and 124.92 kg/m. The values for V1/250 lie between 0 and 120.86

kg. On average V1/250 is equal to 94% of Vmax.

FIGURE 4-12: COMPARISON BETWEEN 𝑉𝑚𝑎𝑥 AND 𝑉1/250

4.3.1 Scatter on the individual overtopping measures

To have an idea of the amount of scatter on the measurement results, the repetition tests are

investigated. In Table 4-4, the characteristic volume quantities for test 0091A and its

repetition tests 0091B, 0091C and 0091D are displayed (see Annex D for test conditions).

TABLE 4-4: COMPARISON OF THE INDIVIDUAL VOLUME CHARACTERISTICS REPETITION TESTS 0091A, 0091B,

0091C AND 0091D

test Vmax [kg/m] V1/250 [kg/m]

0091A 7.18 7.18

0091B 7.36 7.36

0091C 9.70 9.70

0091D 8.95 8.95

Vmax and V1/250 are the same, because only one overtopping volume was considered for the

0.4% threshold. For the tests with the same time series, a maximum difference is found

between test 0091A and 0091C. The difference is up to 24% of the largest value. The reason

Vmax

[kg/m]

0 20 40 60 80 100 120 140

V1/2

50 [kg/m

]

0

20

40

60

80

100

120

140

data pointsreal relationshipperfect relationship

Chapter 4 Data processing 4.4 Force measurements

46

that even for tests with the same time series, such a scatter is attained, is because Vmax is based

on one quantity, which has a certain statistical variability and also the measuring equipment

has also its limitations.

4.4 Force measurements

The forces induced by the overtopped waves are measured at the wall as described in section

Chapter 1. The analysis of the force measurement is done with L-Davis, a data analysis and

visualization software developed at LWI Braunschweig. The data file generated by the load

cell acquisition system needs to be converted from a tdms-file into an L-Davis readable

format. This conversion can be done with a software tool developed in LabVIEW.

In L-Davis a new project that contains these files needs to be created and a certain folder

structure obeyed. First the right timeframe for the force measurements needs to be

implemented. The total duration of the timeframe depends on the duration of the test. The

same pre (50 s) and post time (120 s), as in the data processing for waves (section 4.1) and

overtopping (section 4.2), needs to be excluded from the total timeframe to receive

comparable results. For some reason, this function did not work properly in L-Davis so the

setting of the right timeframe will be done later on in Excel (see further in this section).

A second step in the processing of the force measurements is to find a suitable filter in the

frequency domain to reduce the noise from the signal. By applying a filter, also the artificial

forces in the range of the resonance frequency of the wall can be filtered out (section 3.4.4).

Finding the right filter is often trial and error but there are some restraints. First of all, no high

pass filter is used because this filter shifts the data to a lower position and this is not

acceptable. Second of all, a band stop filter is introduced from 49 to 51 Hz because the

electricity runs on 50 Hz and introduces noise to the signal. Thirdly, a low pass filter is

implemented. If this filter is set too low, the signal will be very smooth but the peak forces in

the signal will be reduced too much (Figure 4-13). The measured (black), removed (red) and

filtered (blue) signals are shown for one wave impact when a low pass filter of 40 Hz is

applied. The filtered signal is nicely smoothened but the peak force is reduced from around

8.5 to 4 N. This reduction about 50% is considered too large. This peak force represents the

maximum impact that a structure will endure during a certain wave impact. Therefore it’s

important that this peak isn’t reduced too much by filtering, otherwise the maximum impact is

underestimated. To limit the peak force reduction, a higher low pass filter needs to be chosen.


47

FIGURE 4-13: COMPARISON BETWEEN THE MEASURED, REMOVED AND FILTERED SIGNAL (LOW PASS FILTER OF 40

HZ)

The resonance frequency of the wall also has to be filtered out. In section 3.4.4, the resonance

frequency of each configuration was determined. The lowest resonance frequency found was

around 90 Hz, therefore the low pass filter is set on 85 Hz. This ensures that the resonance

frequency of each wall configuration is filtered out and the peak values are not significantly

reduced (Figure 4-14). The same wave impact as in Figure 4-13 is shown. The reduction of

the peak force is very small (from around 8.5 to 7.5 N). The signal isn’t as smooth as before

but since the peak force is more important than a smooth signal, the low pass filter of 85 Hz is

chosen.


48

FIGURE 4-14: COMPARISON BETWEEN THE MEASURED (BLACK), REMOVED (RED) AND FILTERED (BLUE) SIGNAL

(LOW PASS FILTER OF 85 HZ)

After the filter is applied, the processing of the force measurements continues with the ‘Event

Analysis’ tool in L-Davis. With this tool, the peak values are traced. The problem with this

analysis is that some detected peaks aren’t caused by wave impact but by noise. To avoid that

these peaks enter the final data selection, a certain threshold is implemented below which all

peaks are considered noise. Finding the value for this threshold needs to be done by combined

signal and video analysis. First a test with very few impacts is chosen. The smallest impacts

will occur in the tests with the smallest wave heights (5 cm) and the largest berm

configuration (40 cm). Therefore test 00109A is selected for the video analysis (see Annex D

for test conditions). After 433 seconds, a very small wave reaches the structure and causes an

impact on the wall. The force value of this impact needs to be checked in the force time

series. Hence, video and force measurement need to be synchronized. To find the correct

impact, the first wave that hits the wall is viewed in the video, this happens after 36 seconds.

The first impact occurs in the time series after 49 seconds, this means that the video time is 13

seconds in front of the data time. With this delay, it is now known that the impact of the

smallest wave will happen 13 seconds after the video time, so around 446 seconds. In the

force time series, indeed a small impact is found around 446 seconds. To be certain that this

impact relates to the specified smallest wave, the waves and impacts just before and after are

checked as well. This impact is equal to approximately 0.12 N (Figure 4-15). With 0.12 N

defined as high pass threshold, the peak values can be collected.


49

FIGURE 4-15: WAVE IMPACT CORRESPONDING TO THE SMALLEST WAVE IS AROUND 0.12 N

However, in the ‘Event Analysis’ another issue came up. Some wave impacts caused more

than one peak value (Figure 4-16). Since only the largest peak value per impact is required a

time domain search equal to the peak period is set. With this extra criterion, L-Davis only

selects the largest value within the timeframe (Figure 4-17). The selected data points are

exported to an Excel file for further analysis.

FIGURE 4-16: MULTIPLE PEAKS FOR ONE WAVE IMPACT

FIGURE 4-17: L-DAVIS ONLY SELECTS THE HIGHEST PEAK PER IMPACT BY APPLYING A TIME DOMAIN SEARCH

EQUAL TO THE PEAK PERIOD

In Excel, the timeframe that couldn’t be applied in L-Davis is implemented. The data points

that appear in the first 50 or last 150 seconds are deleted from the data set. When looking at

the data points, it can be seen that L-Davis still selected some points with a peak values below

the threshold of 0.12 N. These data points are also filtered out of the data set. Now that the

right force measurements have been collected, they have to be divided by the width of the


50

load cell to find the force per unit length. In this case, the load cell was 0.1 m wide so all

forces need to be divided by 0.1 m.

For the force analysis, only the tests with a wall are considered. This selection is introduced to

limit the number of tests and afterwards a few tests are examined to investigate the difference

in wave force when a parapet is installed (see section 7.6). For each test, the number of

impacts, Fmax, F1/250 and F1/10 were determined in Excel. F1/250 and F1/10 are respectively the

averages of the 0.4 % and 10 % highest force impacts. During testing, it was tried to achieve

around 1000 waves so the maximum amount of impacts will also be around 1000. In that case

F1/250 will be the average of the 4 highest wave impacts. To come up with a statistically more

constant value, also F1/10 is determined. Because more, and smaller, impacts are used to

determine F1/10, this value will be a lot smaller then Fmax, but more on this later in this section.

For some tests, the number of impacts was less than the proposed 1000 so first the number of

impacts of each test needs to be determined in order to determine F1/250 and F1/10.

The number of waves that cause an impact on the wall is most dependent on the water depth

(Figure 4-18).

FIGURE 4-18: NUMBER OF IMPACTS ACCORDING TO THE WATER DEPTH

When the water is below the promenade (water depth of 70 cm) the number of impacts varies

between 135 and 1013 whereas when water is present on top of the berm (water depth of 76

cm), the number of impacts varies between 798 and 1089. For the tests where the water

reaches the promenade (water depth of 73 cm), the number of impacts varies between 407 and

1060. This means that for the lowest water depth, only the largest wave height will cause an

impact on the wall and for the highest water depth, almost every wave causes an impact. Note

that in case of number of impacts, the water depths plays a greater role then the freeboard Rc.

Water depth [cm]

69 70 71 72 73 74 75 76 77

Num

be

r o

f im

pa

cts

[-]

0

200

400

600

800

1000

1200


51

This is because for the number of impacts, the distance between the still water level and the

underside of the wall is most important and not the distance between the still water level and

the upside of the wall which is defined as the freeboard Rc. This principle is shown in Figure

4-19. In this Figure both the left and right configuration have the same freeboard Rc, but there

will be a lot more impacts on the right configuration because the water depth is larger and

therefore the distance between the underside of the wall and still water line is smaller.

FIGURE 4-19: NUMBER OF IMPACTS ACCORDING TO THE CONFIGURATION

To see how F1/250 and F1/10 compare to Fmax, Figure 4-20 and Figure 4-21 are shown. On

Figure 4-20, it can be seen that F1/250 is in general not much lower than Fmax and scatter is

limited, especially for the lowest force measurements. For the lowest force measurements,

often there were no more than 250 impacts so in those cases, F1/250 is equal to Fmax. The range

of the maximum measured forces is between 13.4 N/m and 363.2 N/m. On Figure 4-21, the

relationship between F1/10 and Fmax is shown. The scatter is large for the higher force

measurements. The F1/10 value is also significantly smaller than Fmax. On average F1/10 is 44 %

of Fmax whereas F1/250 is 88 %.

FIGURE 4-20: COMPARISON BETWEEN FMAX AND F1/250

Fmax

[N/m]

0 100 200 300 400

F1

/25

0 [N

/m]

0

100

200

300

400

Regression line

45° line


52

FIGURE 4-21: COMPARISON BETWEEN FMAX AND F1/10

4.4.1 Scatter on the force measurements

As mentioned previously, some scatter is present on the results. To determine the amount of

scatter, the repetition tests are looked at. Three tests have been executed multiple times, three

times with the same time series and once with another. The force signals were very similar for

the tests with the same time series (see Figure 3-10), but does this also lead to a similar force

measurement? On Figure 4-22, the test results of test number 119 (see Annex D for the test

conditions) are displayed. It can be seen that the most scatter is found for Fmax which makes

sense since this result is only dependent on one impact. The maximum impact ranges from

222.5 N/m to 278.8 N/m, this means that for the same test configuration the scatter can be as

much as 20 %. For F1/250 and F1/10, the maximum difference due to scatter is around 17 % and

8 % respectively. For these tests the overtopping was measured as well and the scatter on the

average overtopping was only 1% so the scatter on the forces is a lot higher than the one on

average overtopping (see section 3.3.2). This higher scatter is probably because for the forces

only the highest forces are looked at whereas for the overtopping, all events are taken into

account. Another possible reason is that the load cell is more sensitive for small differences

than the weigh cell which measures larger quantities.

Fmax

[N/m]

0 100 200 300 400

F1

/10 [N

/m]

0

100

200

300

400

Regression line

45° line


53

FIGURE 4-22: SCATTER ANALYSIS FOR TEST NUMBER 119

4.4.2 Influence of the installation of a cover

As discussed in section 3.4.4, there was a cover installed after some tests to exclude the

influence of the impact of the overtopping water on the force measurement. After this cover

was installed, test 0067A (Annex D) was repeated (test 0067B) to see if there was a difference

in force measurement with or without cover. The test results are displayed in Table 4-5. The

difference between both tests is very minimal (maximum 1.37 %) and probably more caused

by scatter than by an actual difference in force measurement. This means that the water

overtopping the storm wall has almost no influence on the force measurement and so the tests

that were conducted before the cover was installed were not repeated with cover.

TABLE 4-5: COMPARISON OF THE FORCE MEASUREMENTS BETWEEN A COVERED AND UNCOVERED TEST

Test F1/250 [N/m] F1/10 [N/m] Fmax [N/m]

0067A Uncovered 326.6 248.4 330.1

0067B Covered 325.3 249.3 334.7

Difference [%] 0.40 0.36 1.37

F1/250 F1/10 Fmax

Fo

rce

me

asu

rem

ent

s [N

/m]

0

50

100

150

200

250

300

119A

119B

119C

119D

55

CHAPTER 5 AVERAGE OVERTOPPING ANALYSIS

With the help of the average overtopping script the average overtopping discharge per test is

calculated (section 4.2). Non breaking wave conditions were found for all the tests (𝜉0>2) As

found in literature (section 2.4) it is common to plot (semi-logarithmic) the dimensionless

overtopping Q (𝑞 √𝑔 ∙ 𝐻𝑚03⁄ ) in function of the relative freeboard Rc/Hm0 (Figure 5-1). The

resulting data for all tests performed is plotted.

FIGURE 5-1: DIMENSIONLESS OVERTOPPING DISCHARGE Q VERSUS RELATIVE FREEBOARD RC/HM0 FOR ALL THE

TESTS

During the model tests, different parameters were varied (see Annex D). First the influence of

the water depth will be analysed, especially when the water depth is 76 cm and water is

present on the berm (section 5.1). Secondly the influence of the steepness will be checked

(section 5.2). Thereafter the influence of the parapet in comparison to a simple vertical wall

will be analysed (section 5.3) and finally the influence of the berm width will be verified

(section 5.4). To investigate the influence of the different parameters, the tests are split up into

different groups. Within these groups, tests with a similar geometrical configuration and some

of the wave parameters the same, are combined. By doing so, different groups that only have

one parameter that differs in value can be compared with each other to verify the influence of

that parameter on the average overtopping. To verify the influence of one parameter, the Q

versus Rc/Hm0 plot is made.

In section 5.5 a short comparison is made between the existing prediction formula for the

average overtopping in literature and the test data obtained by the hydraulic model tests. Gaps

found in the prediction formula for the different model geometries are tried to be filled up in

section 5.6.

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Q [

-]

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

1e+0

All data points

Chapter 5 Average overtopping analysis 5.1 Influence of the water depth

56

5.1 Influence of the water depth

In the hydraulic model tests, different water depths have been tested. To verify whether this

parameter influences the overtopping, different test results are plotted. In total four different

water depths have been tested: 68, 70, 73 and 76 cm (measured at the wave paddle). First the

influence of the water depth is checked when there’s no berm installed, afterwards the effect

on the overtopping is inspected when a berm is present. When analysing the influence of the

water depth, a distinction is made between the tests with a vertical wall and a parapet because

this parameter will have an influence on the overtopping (discussed in section 5.3). In this

analysis only tests with a steepness of 0.05 are taken into account because also the steepness

plays a role in the overtopping (section 5.2). For the other tests, similar results are found and

therefore these aren’t discussed elaborately but the Figures can be found in Annex A.

5.1.1 No berm

Vertical wall

When there was no berm and a vertical wall present, two different water depths have been

used for testing: 70 and 73 cm. First the tests with a vertical wall will be reviewed. In Figure

5-2 the results of these tests are shown.

FIGURE 5-2: INFLUENCE OF THE WATER DEPTH WHEN THERE’S NO BERM AND A WALL (WAVE STEEPNESS 0.05)

In this Figure, the relative overtopping Q is plotted against the relative freeboard Rc/Hm0 on a

semi logarithmic scale. The different tests results are plotted as data points. When comparing

the different water depths, it can be seen that the data points all follow the same trend. This

Rc/H

m0 [-]

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Q [

-]

0.0001

0.001

0.01

0.1

depth 70 cm depth 73 cm


57

means that the influence of the water depth is very minimal. This result is logic since the

freeboard Rc, which is affected by the water depth, is included in the x-axis of the graph.

Parapet

Now the influence of the water depth is evaluated when there’s a parapet installed. For the

case without a berm, two different water depths are used: 70 and 73 cm. In Figure 5-3, the

data is plotted. It can be seen that the difference between the two water depths is a bit larger

than for the tests with a vertical wall but they still follow the same trend. This difference is

mainly caused by the scatter on the results (section 4.2.1). In general, the overtopping is

smaller when a parapet is installed and therefore the results are more susceptible to small

effects that affect the overtopping and cause scatter on the results. These small effects include

measuring errors, water entering the tray through small splits in the set-up and water getting

under the scale which leads to an incorrect weighing of the container with water. The

difference however between the two water depths is rather small and therefore it can be said

that the influence of the water depth is again negligible.

FIGURE 5-3: INFLUENCE OF THE WATER DEPTH WHEN THERE’S NO BERM AND A PARAPET (WAVE STEEPNESS 0.05)

5.1.2 With berm

Vertical wall

For the different tests with a berm, three different water depths have been used for testing: 70,

73 and 76 cm. For the case with a water depth of 76 cm, there’s water present on top of the

berm. Now it has to be verified if this water has an influence on the overtopping or if the

presence of Rc in the x-axis is still sufficient to balance this change in water depth. The data

of the tests with a berm width of 40 cm is plotted on Figure 5-4. On Figure 5-4, all data points

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5

Q [

-]

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1



58

are closely gathered and follow the same trend for the different cases (70-73-76 cm).

Therefore it can be concluded that even when there’s water present on top of the berm, the

presence of Rc in the x-axis is still sufficient to balance the change in water depth. From this it

can be concluded that the water depth has a very minimal influence on the overtopping and

therefore it can be neglected as a parameter which influences the overtopping. For the tests

with a berm width of 20 cm, similar results were found and therefore these aren’t discussed

elaborately.

FIGURE 5-4: INFLUENCE OF WATER DEPTH WHEN THERE'S A 40 CM BERM AND A VERTICAL WALL (STEEPNESS 0.05)

Parapet

For the configurations with a berm and with a parapet, again the test results of the set-ups

with a berm width of 40 cm are examined. In Figure 5-5. Small differences between the two

cases are again caused by scatter of the results. Since the overtopping is lower with the berm

of 40 cm, the results are even more susceptible to small errors and so the scatter is a bit larger

than before. However the data points generally follow the same trend so it can be concluded

that here again, the water depth can be neglected as a parameter which influences the

overtopping.

Rc/H

m0 [-]

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Q [

-]

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

depth 70 cm depth 73 cm depth 76 cm

Chapter 5 Average overtopping analysis 5.2 Influence of the wave steepness

59

FIGURE 5-5: INFLUENCE OF WATER DEPTH WHEN THERE'S A 40 CM BERM AND A PARAPET (STEEPNESS 0.05)

5.2 Influence of the wave steepness

In section 2.1 the wave steepness sp was calculated according to equation (4) with use of the

peak period to define the wave length. This in order to attain the target steepness values 0.03

and 0.05. In literature about average overtopping calculations it is more convenient to use the

wave steepness 𝑠𝑚−1,0, which is defined with the help of the significant wave height Hm0 in

the frequency domain and the wave period Tm-1,0. Hm0 and Tm-1,0 are determined with

WaveLab (see section 4.1). The wave steepness 𝑠𝑚−1,0 is than defined by:

𝑠𝑚−1,0 =

𝐻𝑚0𝐿𝑚−1,0

(29)

In which 𝐿𝑚−1,0 is defined in Equation (30) as the wavelength corresponding with the wave

period 𝑇𝑚−1,0 :

𝐿𝑚−1,0 =

𝑔𝑇𝑚−1,02

2𝜋tanh(

2𝜋𝑑

𝐿𝑚−1,0) (30)

d is the water depth and g the gravity constant (9.81 m²/s). The wave steepness 𝑠𝑚−1,0 is

calculated for each test and compared with the target values of 0.03 and 0.05. The intervals in

which these wave steepnesses are found around the two target values are displayed in Table

5-1.

Rc/H

m0 [-]

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Q [

-]

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1



60

TABLE 5-1: TARGET WAVE STEEPNESSES AND THE RANGE OF THE EXPERIMENTAL REACHED VALUES

target value 𝑠𝑚−1,0

interval

0.03 [0.025-0.036] 0.05 [0.043-0.058]

To check what the influence is of the wave steepness on the average overtopping quantity, the

data for the average overtopping is divided in two groups. One group contains the data which

has a target steepness of 0.03, where the other group has the target value 0.05. These two

groups are further divided in the different geometrical configurations (section 3.2). Plots are

made which show the two different groups of steepnesses against each other for the different

geometrical configurations.

5.2.1 Reference case: slope no protection

First case that will be discussed is the configuration of only a slope which is not protected at

its end. Figure 5-6 displays the dimensionless overtopping quantity Q in function of the

dimensionless freeboard Rc/Hm0 for the two target steepnesses.

FIGURE 5-6: INFLUENCE OF THE WAVE STEEPNESS WHEN THERE'S A SLOPE WITHOUT PROTECTION

The plot seems to indicate that the data points for the steeper waves result in slightly smaller

Q-values than for less steep waves. In order to check how much the Q-values of the steeper

waves are reduced with respect to the tests with steepness 0.03, the Q-values for the tests with

steepness 0.03 are divided by the Q-values with steepness 0.05 which have the same test

conditions apart from the wave steepness (and resulting, the wave periods).

Rc/H

m0 [-]

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

Q [

-]

0,0001

0,001

0,01

0,1

1

sm-1,0 = 0.03

sm-1,0 = 0.05


61

This division of two dimensionless overtopping quantities will be represented by the factor ks:

𝑘𝑠 =

𝑄0.03𝑄0.05

(31)

Besides the k-factor, it is also checked what the ratio is of the incoming significant wave

heights. This ratio is given the name 𝑤𝑠 and is defined as:

𝑤𝑠 =

𝐻𝑚0;0.03𝐻𝑚0;0.05

(32)

This ratio is important as two tests with the same test conditions (except for the steepness) are

compared with each other, but due certain effects in the wave flume, the significant wave

height generated for these tests will not be exactly the same. The difference in significant

wave height Hm0 results also in a difference in relative freeboard Rc/Hm0. Too large

differences in Hm0 would also mean that the tests cannot be compared with each other as the

relative freeboard Rc/Hm0 differs too much. In Figure 5-7 the 𝑘𝑠–values are displayed in

function of the relative freeboard. Table 5-2 gives a summary of the tests compared with a

water depth of 68 cm at the wave paddle for the configuration of a slope without protection.

Parameters that are given are the freeboard Rc, the significant wave height Hm0, the

overtopping amount q, the dimensionless overtopping Q and the ratio 𝑤𝑠.

TABLE 5-2: SUMMARY OF TESTS WITH WATER DEPTH 68 CM AT THE WAVE PADDLE

Test Rc

[m] Hm0

[m] q

[l/s/m] Q [/]

ks [/]

ws [/]

001A_d68_Hs5_T1.2 0.05 0.04618 0.348 0.011 1.385 1.065

004A_d68_Hs5_T0.8 0.05 0.04337 0.228 0.008

005A_d68_Hs10_T1.8 0.05 0.09358 3.326 0.037 1.278 1.138 0012A_d68_Hs10_T1.1 0.05 0.08221 2.144 0.029

008A_d68_Hs15_T2.2 0.05 0.1468 10.090 0.057 1.217 1.145 009A_d68_Hs15_T1.4 0.05 0.1282 6.764 0.047


62

FIGURE 5-7: 𝑘𝑠-FACTORS WHEN THERE’S A SLOPE WITHOUT PROTECTION

Some remarks can be given when looking at Table 5-2 and Figure 5-7. When the wave height

ratio 𝑤𝑠 is considered, it can be seen that the values are larger than one. This is also the case

for the tests performed with a water depth of 70 cm at the wave paddle. This means that

waves with steepness 0.05 are flattened out more in the wave flume than the waves with a

steepness of 0.03. The factors 𝑤𝑠 stay below 1.15 such that the difference in significant wave

height Hm0 does not influence the relative freeboard a lot. It can be seen that all the 𝑘𝑠–values

lay above 1.2 and the tests with a steepness of 0.03 result in a higher dimensionless

overtopping Q and resulting, higher average overtopping q. As the difference is rather limited,

the influence of the wave steepness for the geometrical configuration of a slope without

protection is neglected.

5.2.2 No berm

Vertical wall

A same procedure can be followed for the geometrical case where a vertical wall is placed on

top of the slope (without berm). The tests with 5 and 10 cm wall are taken together because

the effect of the wall height is already present in the (relative) freeboard and the test with

walls are compared with each other. This means that a test with a wall of 5 cm is compared

with a test also with a wall of 5 cm, wherefore only the steepness is different. The same

reasoning is valid for the tests with a 10 cm wall. In Figure 5-8 the dimensionless overtopping

Q is plotted against the dimensionless freeboard Rc/Hm0. Again it is better to plot the ks–

values to make a conclusion about the influence of the wave steepness. Figure 5-9 displays

these ks–values. From this Figure it can be seen that the ks–values are mostly higher than 1.5

Rc/H

m0 [-]

0,0 0,2 0,4 0,6 0,8 1,0 1,2

ks [-]

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

no berm no wall


63

which indicates again that the dimensionless overtopping for the steeper waves is smaller and

so is the average overtopping q. The wave height ratio 𝑤𝑠 is again always larger than 1, which

indicates smaller Hm0 for the steeper waves, but limited to 1.15. It can be concluded that the

wave steepness plays a role for the dimensionless overtopping, but not large enough to

separate the data according to wave steepness for a slope with a wall on top of it.

FIGURE 5-8: INFLUENCE OF THE WAVE STEEPNESS WHEN THERE'S NO BERM AND A WALL

FIGURE 5-9: ks-FACTORS WHEN THERE’S NO BERM AND A WALL

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Q [

-]

0,0001

0,001

0,01

0,1

sm-1,0 = 0.03

sm-1,0 = 0.05

Rc/H

m0 [-]

0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2

ks [

-]

0,0

0,5

1,0

1,5

2,0

2,5

3,0

no berm 5 cm/10 cm wall


64

Parapet

Same plots can be made for the cases with a parapet without a berm. Figure 5-10 displays the

Q versus Rc/Hm0 plot and Figure 5-11 the 𝑘𝑠 versus Rc/Hm0 plot. Looking to the plot of the ks-

values directly shows three points which results in a high value. The reason for this is that the

values for Q for the different steepnesses have very small values and differ from each other

some decimals. In section 5.2.3 these values are withdrawn from the plot so it is more clear

the 𝑘𝑠-values are found above the value 2. It can be concluded that steeper waves result in

smaller Q values and a separation of the tests according to steepness, with a parapet on top of

the slope, for further analysis will be necessary.

FIGURE 5-10: INFLUENCE OF THE STEEPNESS WHEN THERE’S NO BERM AND A PARAPET

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0

Q [-]

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

sm-1,0 = 0.03

sm-1,0 = 0.05


65

FIGURE 5-11: 𝑘𝑠-FACTORS WHEN THERE’S NO BERM AND A PARAPET

5.2.3 With berm

Vertical wall

To limit the amount of overflow on graphs, the graphs of Q in function of Rc/Hm0 for the cases

with a vertical wall and a berm of 20 or 40 cm could be found in Annex A. The graphs of the

𝑘𝑠-values of the different berm widths are plotted on the same plot in Figure 5-12.

FIGURE 5-12: 𝑘𝑠-FACTORS CONFIGURATIONS WITH WALL

Rc/H

m0 [-]

0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2

ks [-]

0

50

100

150

200

250

300

no berm 5 cm/10 cm parapet

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0

ks [-]

0

2

4

6

8

10

no berm 20 cm berm 40 cm berm

Chapter 5 Average overtopping analysis 5.3 Influence vertical wall/parapet

66

The ratio ws is again higher than 1 and bounded by the value 1.16. The ks-values take up

larger values, which mean that the difference in Q for the two steepnesses is more pronounced

than for the cases without a berm. Separation of the tests with a 20 or 40 cm berm in

combination with a vertical wall on base of steepness will become necessary.

Parapet

At last, the ks-values are plotted in Figure 5-13, graphs of Q versus Rc/Hm0 can be found in

Annex A. Same conclusion can be made as for the walls in combination with a 20 or 40 cm

berm. Separation of the tests according to steepness will become necessary.

FIGURE 5-13: 𝑘𝑠-FACTORS CONFIGURATIONS WITH PARAPET

5.3 Influence vertical wall/parapet

In this section the effect of a parapet with reference to the tests performed with a vertical wall

is examined. As the reference case (only slope) has no wall or parapet in its configuration, this

case will not be discussed. The three geometrical configurations considered are the

configurations with 0, 20 and 40 cm berm in combination with 5 or 10 cm wall or parapet. As

it follows from section 5.2 the tests should be divided on basis of their steepness. Again plots

are made of the dimensionless overtopping Q in function of the relative freeboard Rc/Hm0.

These plots compare the tests with a wall with those with a parapet. The concept of k-values

is also used again but now with a different meaning. The 𝑘𝑝-value is defined as:

𝑘𝑝 =

𝑄𝑝𝑎𝑟𝑎𝑝𝑒𝑡

𝑄𝑤𝑎𝑙𝑙 (33)

Rc/H

m0 [-]

0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2

ks [

-]

0

5

10

15

20

25



67

From literature it is known that a parapet is more efficient in reducing overtopping in

comparison with a vertical wall. The kp-factor will show how much this reduction is in

function of the relative freeboard Rc/Hm0. Figure 5-14 (left) displays how the water is guided

upwards due to the presence of a vertical wall. Figure 5-14 (right) displays how the water is

not only deflected upwards, but also backwards due to the presence of a parapet.

FIGURE 5-14: COMPARISON BETWEEN THE OVERTOPPING OF A WALL (LEFT) AND A PARAPET (RIGHT)

5.3.1 No berm

Figure 5-15 and Figure 5-16 give respectively the Q versus Rc/Hm0 plot and the 𝑘𝑝-values in

function of R for the tests with steepness 𝑠𝑚−1,0 = 0.03. To reduce the amount of overflow of

Figures, the Figures for the tests with steepness 𝑠𝑚−1,0 = 0.05 can be found in Annex A.

From Figure 5-15 it is clear that when the relative freeboard Rc/Hm0 increases, the reduction in

dimensionless overtopping Q also increases. This means that the parapet becomes more

efficient for the cases with a larger relative freeboard and more specifically for relative

freeboards Rc/Hm0 larger than 1. The reduction is larger for the steeper waves than for the

waves with a steepness of 0.03.

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0

Q [-]

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

wall sm-1,0 = 0.03

parapet sm-1,0 = 0.03


68

FIGURE 5-15: INFLUENCE OF PARAPET WHEN THERE’S NO BERM (STEEPNESS 0.03)

In Figure 5-16 the same conclusions can be made. The kp-values decrease with increasing

relative freeboard, which means that the tests with parapet result in lower Q-values compared

to tests performed with walls and the efficiency increases with increasing relative freeboard

Rc/Hm0.

FIGURE 5-16: 𝑘𝑝-FACTORS NO BERM (STEEPNESS 0.03)

Rc/H

m0 [-]

0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2

kp [-]

0,0

0,2

0,4

0,6

0,8

1,0

no berm sm-1,0 = 0.03

Rc/H

m0 [-]

0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

kp [-]

0,0

0,2

0,4

0,6

0,8

1,0

1,2

no berm sm-1,0 = 0.03


69

5.3.2 20 cm berm

Figure 5-17 and Figure 5-18 display the Q versus Rc/Hm0 plot and the kp-values in function of

Rc/Hm0 for the tests with 20 cm berm with steepness sm-1,0 = 0.03. For the tests with steepness

sm-1,0 = 0.03 the presence of the 20 cm berm seems not to be changing much about the effect

that with increasing relative freeboard Rc/Hm0, the efficiency of the parapet with respect to

walls is higher.

FIGURE 5-17: INFLUENCE OF PARAPET WHEN THERE’S A 20 CM BERM (STEEPNESS 0.03)

FIGURE 5-18: 𝑘𝑝-FACTORS WHEN THERE’S A 20 CM BERM (STEEPNESS 0.03)

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0

Q [-]

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

wall sm-1,0 = 0.03


Rc/H

m0 [-]

0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

kp [-]

0,0

0,2

0,4

0,6

0,8

1,0

1,2

no berm sm-1,0 = 0.03


70

For the tests with a wave steepness sm-1,0 = 0.05 some other conclusions can be made (see

Annex A for the Figures). Due to the presence of the 20 cm berm the effect of the parapet

(reduction of Q) seems to decrease in comparison when there is no berm present. The

reduction in overtopping seems to be stabilizing once the relative freeboard Rc/Hm0 becomes

equal to 1.

5.3.3 40 cm berm

Figure 5-19 and Figure 5-20 display the Q versus Rc/Hm0 plot and the kp-values in function of

Rc/Hm0 for the tests with 40 cm berm with steepness sm-1,0 = 0.03. Again the tests with

parapets show a reduction in Q compared to the tests with wall as protection. For both

steepnesses the 𝑘𝑝-values are more spread but show an increasing efficiency of the parapet

with respect to the tests performed with a wall.

FIGURE 5-19: INFLUENCE OF PARAPET WHEN THERE’S A 40 CM BERM (STEEPNESS 0.03)

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0

Q [

-]

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

wall sm-1,0 = 0.03


Chapter 5 Average overtopping analysis 5.4 Influence of the berm width

71

FIGURE 5-20: 𝑘𝑝-FACTORS WHEN THERE’S A 40 CM BERM (STEEPNESS 0.03 LEFT AND 0.05 RIGHT)

The findings above show that for further analysis, the configurations with vertical wall and

parapet should dealt with separately.

5.4 Influence of the berm width

In this section the influence of the berm width on the overtopping Q will be tested. To test this

influence, first, a distinction is made between the tests with a vertical wall and a parapet and

with a different steepness because these parameters influence the overtopping. No distinction

is made between different wall heights or water depths because they don’t have an effect on

the overtopping as discussed in section 5.1. The data points are plotted on a semi logarithmic

scale of dimensionless overtopping Q in function of relative freeboard Rc/Hm0. Once the

influence is established, a reduction factor 𝑘𝑏 is determined according to equation (34).

𝑘𝑏 =

𝑄𝑏𝑒𝑟𝑚𝑄𝑛𝑜 𝑏𝑒𝑟𝑚

(34)

5.4.1 Vertical wall

First of all, all tests with a vertical wall and a steepness of 0.05 are considered. These tests are

divided into three cases: no berm, 20 cm berm and 40 cm berm. Each case is plotted on a semi

logarithmic scale and the different trendlines are drawn. The result of this plot can be seen on

Figure 5-21. On this Figure, it can immediately be seen that Q decreases with increasing berm

width. The reduction in Q increases with increasing relative freeboard. When the relative

freeboard would be equal to zero, the overtopping should theoretically be the same for every

Rc/H

m0 [-]

0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6

k p [

-]

0,0

0,2

0,4

0,6

0,8

1,0

no berm sm-1,0 = 0.03


72

berm width. This can be verified by extrapolating the trendlines until the y-axis and there they

would intersect the axis at approximately the same point.

FIGURE 5-21: INFLUENCE OF THE BERM WIDTH WHEN THERE’S A VERTICAL WALL (STEEPNESS 0.05)

Reduction factors kb

Now that the influence of the berm has been established, the amount of reduction can be

determined. This reduction is determined by a kb-factor as seen in equation (34). The kb -

values are shown on Figure 5-22. On this Figure, similar observations as for Figure 5-21 can

be made. The reduction in overtopping is bigger for the 40 cm berm and for larger relative

freeboards. When the relative freeboard is close to zero, the reduction is almost negligible (kb

≈ 1).

The analysis of the tests with a vertical wall and a steepness of 0.03 shows very similar results

and therefore aren’t discussed again. The Figures can be found in Annex A.

Rc/H

m0 [-]

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Q [

-]

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

1e+0



73

FIGURE 5-22: 𝑘𝑏- FACTORS 20/40 CM BERM (WITH A VERTICAL WALL AND STEEPNESS 0.05)

5.4.2 Parapet

For the tests with parapet, the same steps are repeated. The three different cases: no berm, 20

cm berm and 40 cm berm are plotted on a semi logarithmic scale. The results can be seen on

Figure 5-23.

FIGURE 5-23: INFLUENCE OF THE BERM WIDTH WHEN THERE’S A PARAPET (STEEPNESS 0.05)

On this Figure, it can be seen that there’s no clear distinction between the three berm widths

as it was the case with a vertical wall. Through the results of these few tests, it can be

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

kb [

-]

0,0

0,2

0,4

0,6

0,8

1,0

20 cm berm 40 cm berm

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Q [

-]

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

no berm 20 cm berm40 cm berm

Chapter 5 Average overtopping analysis 5.5 Comparison with literature

74

assumed that when a parapet is installed, the berm width does not decrease the overtopping

much anymore. The effect of the parapet is larger than the one of the berm width and

therefore the berm width can be neglected as a parameter when a parapet is installed.

However, if one wants to make a general conclusion about the influence of the berm width

when a parapet is installed, more tests should be carried out (with longer, shorter or equal

berms).

5.5 Comparison with literature

The goal of this section is to compare the performed tests with prediction formulas for the

dimensionless overtopping already available in literature. When literature is not present for a

certain geometrical configuration, the strategy is to come up with a modified prediction

formula or reduction factor. The actual execution of the strategy is explained more thoroughly

in 5.6.

5.5.1 No berm

Slope no protection

For the case of a smooth slope without protection a prediction formula for the dimensionless

average overtopping quantity Q is available in the EurOtop Manual (2007). This formula is

already discussed in section 2.4 and represented by equation (11). In this formula the

roughness factor 𝛾𝑓 and the factor for the angle of wave incidence 𝛾𝛽 are taken equal to 1. The

test results together with the prediction formula and its 90% confidence interval are plotted in

Figure 5-24.


75

FIGURE 5-24: COMPARISON OF TEST RESULTS AND EUROTOP MANUAL IN CASE THERE'S A SLOPE WITH NO

PROTECTION

It can be remarked that the prediction formula given by (11) is only valid for relative

freeboards Rc/Hm0 larger than 0.5. For 7 of the 18 tests results this prediction formula does not

provide a useful result. Recently a new prediction formula was found which take into account

the slope angle and is also valid for relative freeboards Rc/Hm0 below 0.5 (van der Meer &

Bruce, 2014). The general form of this formula (Weibull-distribution) and its coefficients are

given in section 2.4 by equations (12), (13), (14) and (15).

The slope has a 1:2 (V:H) ratio such that cot 𝛼 = 2 and the coefficients and equation (12)

become:

{𝑎 = 0.09𝑏 = 1.5𝑐 = 1.3

(35)

𝑞

√𝑔 ∙ 𝐻𝑚03= 0.09 ∙ exp (−(1.5

𝑅𝑐𝐻𝑚0

)1.3

) (36)

The formula presented in equation (36) will be named the updated van der Meer (vdM)

formula and is plotted in Figure 5-25 together with test results and the older formula of the

EurOtop Manual with its 90% confidence interval. For larger relative freeboards the updated

vdM formula gets more closely to the old prediction formula stated in the EurOtop Manual.

The test results for relative freeboards below 0.5 fit closely to the updated vdM formula as

also the test results with Rc/Hm0 larger than 1. The test results with Rc/Hm0 around 0.7 seem to

lie below the updated vdM formula. As the updated vdM formula predicts the test results

Rc/H

m0 [-]

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

Q [-]

0,001

0,01

0,1

slope no prot.Eurotop 2007upper 5% bound.lower 5% bound.


76

quite well and based on lot more test results, it is chosen to work further with equation (36) to

describe the overtopping of the configuration consisting of only a slope.

FIGURE 5-25: COMPARISON OF TEST RESULTS AND UPDATED VDM FORMULA IN CASE THERE'S A SLOPE WITH NO

PROTECTION

Vertical wall

Placing a wall on top of the slope increase the freeboard Rc and should result in smaller

average overtopping quantities than without a wall. The water is directed parallel upwards the

wall, which results in less overtopping over the wall. In Figure 5-26 the tests performed with a

wall without a berm are plotted. No separation of the data according to wave steepness is

made (see section 5.2.2). Also included in Figure 5-26 is the updated vdM formula.

Rc/H

m0 [-]

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

Q [

-]

0,001

0,01

0,1

1

slope no prot.Eurotop 2007upper 5% bound.lower 5% bound.updated vdM


77

FIGURE 5-26: COMPARISON OF TEST RESULTS AND UPDATED VDM FORMULA IN CASE THERE'S NO BERM AND WALL

The test results for the configuration with a wall protecting the top of the slope lie clearly

below the updated prediction formula of vdM for relative freeboards between 0 and 2. The

two data points with a larger relative freeboard can be found above the updated vdM formula.

A possible explanation is that for relative freeboards Rc/Hm0 larger than 2 the effect of the wall

diminishes in comparison to the case of only a 1:2 slope with the same relative freeboard.

Another fact is that the predictions of the average overtopping are more sensitive for larger

relative freeboards. Small amounts of water ingress in the reservoir will have an influence on

the prediction of the average overtopping q.

As in general the data lies below the updated prediction formula for vdM, either a new

formula must be created, or a correction factor must be applied on the prediction formula for

only a slope. The last option is chosen and will be discussed in section 5.6.

parapet

Attaching a parapet nose to the vertical wall does result in an overtopping reduction with

respect to a wall. The water is curved back leading to the reduction in overtopping (see 5.3). It

was also found in 5.2.2 that the steepness plays a role on the quantity of overtopping.

Therefore test results are plotted (Figure 5-27) in two groups with their respective target wave

steepness. The updated vdM formula and the test results for a wall are also included in Figure

5-27.

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Q [-]

1e-5

1e-4

1e-3

1e-2

1e-1

1e+0



78

FIGURE 5-27: COMPARISON OF TEST RESULTS AND UPDATED VDM FORMULA IN CASE THERE'S NO BERM AND A

PARAPET

Clear is that the reduction of overtopping with a parapet, compared to the case with a wall

really starts around Rc/Hm0 equal to 1. Further the reduction in overtopping stabilizes for

values of Rc/Hm0 larger than 2. But it is difficult to say at which specific Rc/Hm0 value it

stabilizes, due to the limited data points for relative freeboards larger than 2. To take the

reducing effect of a parapet into account with respect to the prediction formula with a wall, an

extra reduction factor will be determined. More about the determination of this reduction

factor can be found in section 5.6.

5.5.2 With berm

When a berm is installed, the updated vdM formula also over predicts the dimensionless

overtopping. Van Doorslaer et al. (2010) came up with a new formula to predict the wave

overtopping when a berm and vertical wall were installed. His formula had the same form as

the one in the EurOtop Manual but the coefficients were a bit different. The formula is

presented in equation (37).

𝑞

√𝑔 ∙ 𝐻𝑚03= 0.2 ∙ 𝛾𝑏 ∙ exp (−2.335 ∙

𝑅𝑐𝐻𝑚0 ∙ 𝛾𝑏𝑣

) (37)

This formula looks like the one presented in section 2.5.1 but now 𝛾𝑏𝑣 represents a reduction

factor for when a vertical wall and berm are installed. 𝛾𝑏𝑣 can be determined in the following

way:

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Q [

-]

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

1e+0

updated vdMno berm parapet sm-1,0 = 0.03

no berm parapet sm-1,0 = 0.05

no berm wall


79

𝛾𝑏𝑣 = 1.039 ∙ 𝛾𝑏 ∙ 𝛾𝑣 − 0.11 (38)

𝛾𝑣 (reduction factor when only a wall is present) has been determined in section 2.5.1 and 𝛾𝑏

(reduction factor when only a berm is present) is equal to:

𝛾𝑏 = 0.923 − 0.055 ∙

𝐵ℎ2

𝐿0 ∙ 𝑅𝑐 (39)

The attentive reader can immediately see that this formula isn’t valid for small relative berm

widths because 𝛾𝑏 does not become one when the berm width is zero. This formula is only

valid for relative berm widths in the range of 0.05 to 4.0.

To be able to plot the prediction formula presented by Van Doorslaer et al. (2010), the mean

values for 𝛾𝑏 and 𝛾𝑣 have been determined based upon the test results. With this knowledge,

the reduction factor 𝛾𝑏𝑣 can be determined and so the prediction formula (37) can be plotted.

The result of this plot can be seen on Figure 5-28. In this Figure also the test results are

displayed. Only tests with a vertical wall are plotted because the formula is only valid for

those configurations. On Figure 5-28, it can be seen that the prediction formula slightly under

predicts the overtopping, especially for larger relative freeboards.

In section 5.6, a similar strategy as applied by Van Doorslaer et al.(2010) is used to come up

with reduction factors for a combination of wall or parapet with a berm. Only equation (36) is

used as the reference formula.

FIGURE 5-28: COMPARISON BETWEEN PREDICTION FORMULA OF VAN DOORSLAER AND TEST RESULTS

Rc/H

m0 [-]

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Q [-]

1e-9

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

1e+0

Test results

Prediction formula Koen van Doorslaer

Chapter 5 Average overtopping analysis 5.6 Reduction factors

80

5.6 Reduction factors

As it is the purpose to adapt the chosen reference equation for other geometrical

configurations, reduction factors need to be determined. In this section the methods to

determine the reduction factors are described and final reduction factors will be displayed.

5.6.1 No berm

Reduction factor vertical wall: 𝛄𝐯

In order to adapt the updated prediction formula of van der Meer for the geometrical

configuration with a vertical wall on top of the slope, a reduction factor γv is introduced in the

right part of equation (36) such that it has the form as described by equation (40).The idea of

introducing a reduction factor into the formula is based on similar reduction factors

determined by Van Doorslaer et al. (2010) as described in section 2.5.1.

𝑞

√𝑔 ∙ 𝐻𝑚03= 0.09 ∙ exp (−(1.5

𝑅𝑐𝐻𝑚0 ∙ 𝛾𝑣

)1.3

) (40)

Equation (40) is rewritten such that an expression is found which describes γv. This

expression is shown in equation (41).

γv = 1.5 ∙

𝑅𝑐𝐻𝑚0

1

[− ln (𝑄0.09

)]1 1.3⁄

(41)

By filling in the dimensionless overtopping Q and the relative freeboard Rc/Hm0 for the 23

tests performed with a wall on top of the slope, γv can be estimated. To be a real reduction

factor, the value of γv needs to be smaller than 1 such that in equation (40) a smaller

dimensionless overtopping value Q is found in comparison for the case without a wall on top

of the slope. Figure 5-29 displays the calculated values of γv as described above. The 23 tests

are split up in their respective target steepness to show again that the wave steepness does not

influence the determination of γv a lot (as was seen in 5.2.2).


81

FIGURE 5-29: 𝛾𝑣-FACTORS SPLIT UP IN STEEPNESS.

It must be remarked that for the largest relative freeboards tested two values are found with a

γv-value larger than 1. This would mean that for the case without protection on the top of the

slope is more effective than the case with a wall. This is according to common sense not logic.

To take this phenomenon into account, the two tests resulting in a γv-value larger than 1 are

withdrawn from the Figure and the optimal trendline to describe the remaining γv-values in

function of the relative freeboard is determined.

FIGURE 5-30: 𝛾𝑣-FACTORS WITHOUT THE VALUES ABOVE 1 AND LINEAR TRENDLINE

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

v [-]

0,0

0,2

0,4

0,6

0,8

1,0

1,2

sm-1,0 = 0.03

sm-1,0 = 0.05

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5

v [

-]

0,0

0,2

0,4

0,6

0,8

1,0

all datalinear trendline

v = 0.1441x + 0.5692

R² = 0.5876


82

In Figure 5-30 the trendline is plotted together with its equation and coefficient of

determination R². Other trendlines fitted with a nonlinear relationship between γv-value and

the relative freeboard can be found in Annex A. A linear trendline seemed to describe the

relationship between γv-value and the relative freeboard the best and it is also to most simple

relationship. To deal with the phenomenon of γv-values larger than 1 for larger relative

freeboards, the following procedure is worked out. With the formula for the linear trendline

the relative freeboard is calculated which corresponds with a γv-value equal to 1. This

relative freeboard Rc/Hm0 corresponds with a value of 2.99. For freeboards larger than 2.99 a

γv-value equal to 1 is assumed. This last measure takes into account that a reduction factor

larger than 1 for large Rc/Hm0 is not physical correct. This is represented in Figure 5-31.

FIGURE 5-31: PREDICTION FORMULA 𝛾𝑣 BASED ON THE REDUCED DATA SET

The linear relationship between the γv-value and the relative freeboard Rc/Hm0 can then

described by equation (42).

𝛾𝑣 =

{

0.1441 ∙𝑅𝑐𝐻𝑚0

+ 0.5692 0.3 ≤𝑅𝑐𝐻𝑚0

≤ 2.99

1 𝑅𝑐𝐻𝑚0

≥ 2.99

(42)

The upper part in equation (42) is strictly only valid for Rc/Hm0-values between 0.3 and 2.2.

An extrapolation is done for Rc/Hm0-values larger than 2.2, but the uncertainty of this

extrapolation is large because not enough test results were present to check the validity of

equation (42) for Rc/Hm0 larger than 2.2. The real transition Rc/Hm0-value could be further

Rc/H

m0 [-]

0 1 2 3 4

v [

-]

0,0

0,2

0,4

0,6

0,8

1,0

1,2

reduced data set

prediction formula v

v = 0.1441x + 0.5692

R² = 0.5876


83

investigated by performing model tests with the same configuration but relative freeboards

larger than 2.2.

Another plot can be made to check how the prediction formula (42) corresponds with the

reduction factors γv found by using equation (41). On the horizontal axis the reduction

factors γv are plotted calculated with the prediction formula and on the vertical axis γv-values

found with equation (41). Figure 5-32 represents the perfect relationship between the two and

also the real found relationship. From Figure 5-32 it can be seen that the prediction formula

slightly under predicts the γv-values found with the test results, as the real relationship

between the two is found always above the perfect relation line.

FIGURE 5-32: RELATIONSHIP BETWEEN THE PREDICTED γv AND THE γv BASED ON THE TEST RESULTS

Ultimately with the knowledge of the formula for the reduction factor γv, the prediction

formula for the dimensionless overtopping represented in equation (40) is plotted through the

test results in Figure 5-33. Equation (40) is only plotted up to relative freeboards of 2.2,

because the prediction formula for γv is predicted best up to that Rc/Hm0.

v predicted [-]

0,0 0,2 0,4 0,6 0,8 1,0

v re

al [-

]

0,0

0,2

0,4

0,6

0,8

1,0

1,2



84

FIGURE 5-33: PREDICTION FORMULA Q NO BERM WALL PLOTTED THROUGH THE TEST RESULTS

Reduction factor parapet: 𝛄𝐩

To take into account the extra reducing effect in overtopping by adding a parapet nose to the

wall, in comparison with the case of a simple vertical wall, a reduction factor γp is

determined. Equation (40) is modified by adding the reduction factor γp in at the same

location as for the reduction factor of the wall. The prediction formula for the case of a

parapet on top of the slope becomes:

𝑞

√𝑔 ∙ 𝐻𝑚03= 0.09 ∙ exp (−(1.5

𝑅𝑐𝐻𝑚0

1

γv ∙ γp)

1.3

) (43)

Equation (43) is rewritten such that an expression for γp can be determined. This expression

is shown in equation (44).

γp = 1.5 ∙𝑅𝑐𝐻𝑚0

∙1

𝛾𝑣

1

[− ln (𝑄0.09)]

1 1.3⁄

(44)

By the knowledge of the dimensionless overtopping Q, relative freeboard Rc/Hm0 for the 23

tests performed with a parapet on top of the slope and calculating γv with equation (42), γp

can be calculated. To be a real reduction factor, the values need to be smaller than 1. For very

large relative freeboards the case of a parapet would result in the same average overtopping

quantities as a vertical wall, as only very small volumes of water will be able to get over the

wall or parapet.

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5

Q [

-]

0,0001

0,001

0,01

0,1

no berm walltest results wallupdated vdM


85

FIGURE 5-34: γp-FACTORS SPLIT UP IN WAVE STEEPNESS

Figure 5-34 represents the γp-values as calculated with equation (44). The results are split up

in their target wave steepness as was found necessary in section 5.3.1. In Figure 5-34 it is

clear that the wave steepness do play a role for the value of the reduction factor γp. The

steeper waves result in smaller γp-values. Further it can be remarked that the γp-values

decrease until a relative freeboards of 1.7, where after they start increasing again for larger

freeboards.

With the test results presented in Figure 5-34 it is not known what the transition relative

freeboard is between increasing and decreasing γp-values and what Rc/Hm0 results in a γp-

value equal to 1 for large relative freeboards. Because of these uncertainties, the

determination of a prediction formula will be based on the data with a relative freeboard

smaller than 1.7 and so only on the test results leading to a decreasing γp-value with

increasing Rc/Hm0. Figure 5-35 represents the test results with a relative freeboard below 1.7.

For the two wave steepnesses the linear trendline is drawn. The trendlines can be found

almost parallel to each other. To come up with a formula to describe the γp-value, two

parameters are of importance; the relative freeboard Rc/Hm0 and the wave steepness sm-1,0.

With the help of the statistical program SPSS, a nonlinear regression analysis is performed on

the test results with relative freeboard smaller than 1.7.

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0

p [-]

0,0

0,2

0,4

0,6

0,8

1,0

1,2

sm-1,0 = 0.03

sm-1,0 = 0.05


86

FIGURE 5-35: REDUCED TEST RESULTS WITH THEIR LINEAR TRENDLINE

Different forms of formulas are tried to check which one would result into the best coefficient

of determination R². The method to do this in SPSS is by establishing a formula which is a

function unknown parameters (A,B,C,D and E) and the known parameters Rc/Hm0 and sm-1,0.

A nonlinear regression analysis is started and the unknown parameters are estimated. All the

tested formulas and their coefficients of determination can be found in Table A-1 in Annex A.

The formula with the largest coefficient of determination R² (0.866) has the simplest form

possible with two variables and this form will be used to describe the γp-factor. This formula

is linear dependent on the relative freeboard and the relative freeboard. In the 3-dimensional

space it would represent a plane. Equation (45) represents this formula for γp with the

unknown parameters filled in. This formula is explicitly valid for relative freeboards between

0.3 and 1.7.

γp = −11.611 ∙ (𝑠𝑚−1,0) − 0.324 ∙ (𝑅𝑐𝐻𝑚0

) + 1.446 (45)

To check how the formula in (45) relates to the γp-values found based on the test results,

Figure 5-36 is made. A perfect relationship between the two axes is represented by the red

line, the found relation by the black line. Base on the relative small difference of the real

relation and the perfect relation it can be stated that the prediction formula predicts quite good

the γp-values for relative freeboards between 0.3 and 1.7.

Rc/H

m0 [-]

0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8

p [-]

0,0

0,2

0,4

0,6

0,8

1,0

1,2

sm-1,0 = 0.03

sm-1,0 = 0.05

linear trendlines


87

FIGURE 5-36: RELATIONSHIP BETWEEN THE PREDICTED γp AND THE γp BASED ON THE TEST RESULTS

Ultimately with the knowledge of the formula for the reduction factor γp, the prediction

formula for the dimensionless overtopping represented in equation (43) is plotted through the

test results in Figure 5-37 for each wave steepness separately. To do so, the target wave

steepness is filled in equation (45), because otherwise no γp-value can be determined for

relative freeboards that were not encountered in the test results. The prediction formula for

the case of a parapet on top of the slope is only plotted for relative freeboards up to 1.7,

because the prediction formula for γp is only valid for Rc/Hm0-values lower than 1.7.

FIGURE 5-37: PREDICTION FORMULAS Q PARAPET AND NO BERM PLOTTED THROUGH THE TEST RESULTS

p

predicted [-]

0,0 0,2 0,4 0,6 0,8 1,0 1,2

p real [-

]

0,0

0,2

0,4

0,6

0,8

1,0

1,2


Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Q [-]

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

1e+0

sm-1,0 = 0.03

sm-1,0 = 0.05





88

5.6.2 With berm

Now that the influence of the wall and parapet on the overtopping and the related reduction

factors have been established, the influence of the berm width can be looked at more closely.

First the influence will be examined when there’s a simple wall installed. Afterwards, the tests

results with parapet will be investigated.

Reduction factor berm: 𝛄𝐛 (vertical wall)

On Figure 5-38, the tests with a vertical wall and 20 or 40 cm berm are depicted along with

equation (40) which predicts the overtopping when there’s a wall and no berm. In this Figure,

a distinction is made between the two different target steepnesses because they influence the

overtopping as described in section 5.2.3. On Figure 5-38 it can be seen that the berm width

still has an influence on the overtopping, especially for higher relative freeboards. To include

the effect of the berm width, an extra reduction factor 𝛾𝑏 needs to be added to equation (40).

FIGURE 5-38: TEST RESULTS PLOTTED AGAINST PREDICTION FORMULA Q FOR A WALL WITHOUT BERM

The reduction factor 𝛾𝑏 is determined in the same way as the reduction factors before. First

the new reduction factor is added to equation (40) and so the new formula for predicting the

overtopping becomes:

𝑞

√𝑔 ∙ 𝐻𝑚03= 0.09 ∙ exp (− (1.5

𝑅𝑐𝐻𝑚0 ∙ 𝛾𝑣 ∙ 𝛾𝑏

)1.3

) (46)

This equation can be rewritten to determine the value of the reduction factor for each test:

γb = 1.5 ∙

𝑅𝑐𝛾𝑣 ∙ 𝐻𝑚0

1

[− ln (𝑄0.09)]

1 1.3⁄

(47)

Rc/H

m0 [-]

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Q [-]

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

1e+0

sm-1,0 = 0.03

sm-1,0 = 0.05

No berm wall


89

By definition, the 𝛾𝑏 factor is equal to 1 when there’s no berm present. For all 47 available

tests with a berm (20 and 40 cm), the found overtopping, relative freeboard and 𝛾𝑣 according

to equation (42) were filled in and a value of 𝛾𝑏 could be determined. On Figure 5-39, all

found 𝛾𝑏 are displayed relative to the actual berm width. The tests are split up according to

their target steepness. On Figure 5-39, it can be seen that the reduction is larger for the tests

with a steepness of 0.05. For the tests with a target steepness of 0.03, some 𝛾𝑏 factors are

larger than 1 and this would lead to an increase in overtopping according to the formula. The

reason that these values are sometimes larger than 1 is that already use is made of a prediction

formula for γv. Due to this unrealistic values for γb can be encountered. On Figure 5-39 it can

also be seen that it won’t be possible to come up with a formula for 𝛾𝑏 using the real berm

width, a relative berm width will have to be introduced. Also the steepness will have to be

included into the formula since there’s a clear difference between the two steepnesses.

FIGURE 5-39: REDUCTION FACTORS RELATIVE TO BERM WIDTH B

In accordance with the relative freeboard, the actual berm width B is divided by the wave

height Hm0 to find a relative, dimensionless berm width. This relative berm width is then

multiplied by the reached steepness and a plot of 𝛾𝑏 against 𝐵 𝐻𝑚0⁄ ∙ 𝑠𝑚−1,0 was made. A

linear trendline was added to try and find a formula to describe 𝛾𝑏 but it was found that a

linear trendline did not reach a large enough coefficient of determination and so a more

complex trendline was searched with the help of SPSS. With SPSS, it was found that the

highest correlation was found when the relative berm width was raised to the power of 0.65.

On Figure 5-40, the found 𝛾𝑏 factors are displayed along with the trendline formula and

coefficient of determination R². In this formula it is made sure that 𝛾𝑏 is equal to 1 when the

relative berm width is zero because this is true by definition.

Berm width B [cm]

0 10 20 30 40 50

b [

-]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

sm-1,0 = 0.03

sm-1,0 = 0.05


90

FIGURE 5-40: DETERMINATION OF THE PREDICTION FORMULA FOR γb WITH A SIMPLE VERTICAL WALL

A formula for 𝛾𝑏 is now found and is equal to:

𝛾𝑏 = 1 − 2.2112 ∙ (

𝐵

𝐻𝑚0)0.65 ∙ 𝑠𝑚−1,0 (48)

To verify if this formula (48) gives a good representation of the actual 𝛾𝑏 found by equation

(47), the real 𝛾𝑏 factors are plotted against the predicted values. The result of this plot can be

seen on Figure 5-41. On Figure 5-41, it can be seen that all predicted values are fairly close to

the perfect 45° line and the trendline of all found values almost coincides with this perfect 45°

line. This means that the prediction formula gives a good representation of the actual 𝛾𝑏

values.

(B/Hm0

)0.65

* sm-1,0

[-]

0.00 0.05 0.10 0.15 0.20 0.25

b [

-]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

b according to data set

Prediction formula b

b = 1 - 2.112 x

R2 = 0.77


91

FIGURE 5-41: γb REAL RELATIVE TO γb PREDICTED

To plot the test results in comparison to the prediction formula as presented by equation (48),

some difficulties arise. To determine 𝛾𝑏, the relative berm width has to be known. Since this

value is different for every test, an average over all tests is taken to represent this value. On

Figure 5-42 the two prediction formulas, with and without the influence of 𝛾𝑏, are plotted

alongside the test results. The formula which includes the influence of the berm, gives a better

prediction for the overtopping, especially for larger relative freeboards. Both formulas are

valid for relative freeboards between 0.3 and 2.2 because γv is only valid in that range.

FIGURE 5-42: COMPARISON BETWEEN PREDICTION FORMULAS Q FOR WALL WITHOUT BERM AND WALL WITH BERM

b predicted [-]

0.0 0.2 0.4 0.6 0.8 1.0 1.2

b r

ea

l [-]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

data points

real relation

perfect realation

Rc/H

m0 [-]

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Q [-]

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

Data points

20/40 cm berm wall

no berm wall


92

Reduction factor berm: 𝛄𝐛 (parapet)

For the determination of the 𝛾𝑏 factor when a parapet is installed, the same procedure of

before is repeated, only now the 𝛾𝑏 factor is added in equation (43). This leads to the

following prediction formula for wave overtopping when a berm and parapet are present:

𝑞

√𝑔 ∙ 𝐻𝑚03= 0.09 ∙ exp (−(1.5

𝑅𝑐𝐻𝑚0

1

γv ∙ γp ∙ 𝛾𝑏)

1.3

) (49)

Here again, equation (49) can be rewritten to determine the 𝛾𝑏 values:

γb = 1.5 ∙

𝑅𝑐𝛾𝑣 ∙ 𝛾𝑝 ∙ 𝐻𝑚0

1

[− ln (𝑄0.09)]

1 1.3⁄

(50)

In equation (50), the relative freeboard and dimensionless overtopping can be filled in

directly. 𝛾𝑣 and 𝛾𝑝 can be determined using equations (42) and (45) respectively. Since

equation (45) is only valid for a relative freeboard up to 1.7, the γb factors are also only

determined up to that point. All determined 𝛾𝑏 factors are shown in Figure 5-43. There are a

lot of 𝛾𝑏 factors that are bigger then 1, and this isn’t a logical result since 𝛾𝑏 should be a

reduction factor. It can also be seen that all 𝛾𝑏 values are spread around. This is a logic result

since in section 5.4.2, it was assumed that the berm width does not play a major role when a

parapet is installed.

FIGURE 5-43: γb FACTOR RELATIVE TO THE ACTUAL BERM WIDTH

To find a formula for 𝛾𝑏 when a parapet is installed, the same procedure as before is repeated.

The 𝛾𝑏 values are plotted against (𝐵 𝐻𝑚0⁄ )0.65 ∙ 𝑠𝑚−1,0 and the result can be seen on Figure

5-44.

Berm width B [cm]

0 10 20 30 40 50

b [

-]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

sm-1,0 = 0.03

sm-1,0 = 0.05


93

FIGURE 5-44: DETERMINATION OF PREDICTION FORMULA FOR γb WITH PARAPET

It can immediately be seen on Figure 5-44 that the coefficient of determination is very low

and therefore it can be concluded that all 𝛾𝑏 values are just scatter around the mean value of 1.

Hereby it is concluded that the berm width has no influence on the overtopping when a

parapet is already installed, which means that 𝛾𝑏 is always equal to 1 in that case. Note that

this generalization is only valid for relative freeboards between 0.3 and 1.7 and for the berm

widths tested in this thesis, for tests out of this range, no assumption can be made on the value

of 𝛾𝑏 due to the fact that not enough tests were carried out.

Now that the reduction factor 𝛾𝑏 has been determined, the test results can be plotted alongside

the prediction formula. The prediction formula presented on Figure 5-45 is the one presented

in equation (49) with 𝛾𝑏 = 1 and 𝛾𝑣 and 𝛾𝑝 equal to equation (42) and (45) respectively. In

Figure 5-45 the test results and prediction formulas are split up according to their target wave

steepness because 𝛾𝑝 is dependent of 𝑠𝑚−1,0. On Figure 5-45 it can be seen that the prediction

formula follows the trend of the test results quite nicely but of course some scatter is always

present.

0.0 0.1 0.2 0.3 0.4 0.5 0.6

b [-]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Data points

Prediction formula b

b =1

(B/Hm0

)0.65

* sm-1,0

[-]

Chapter 5 Average overtopping analysis 5.7 Conclusion

94

FIGURE 5-45: COMPARISON BETWEEN PREDICTED OVERTOPPING AND TEST RESULTS

5.7 Conclusion

To conclude this section about the average overtopping, a general overview is given on which

prediction formula should be used for the determination of the average overtopping. To

decide which formula should be used, the flowchart presented on Figure 5-46 can be used.

This flowchart is valid for configurations which have a slope of 1:2.

FIGURE 5-46: FLOWCHART PREDICTION FORMULAS Q

Rc/H

m0 [-]

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Q [

-]

1e-10

1e-9

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

1e+0

sm-1,0 = 0.03

sm-1,0 = 0.05

20/40 cm berm par. s = 0.0320/40 cm berm par. s = 0.05

Wall?

no

Use Eq. (36)

yes

Parapet?

no

Berm?

no

Use Eq. (40) with 𝛾𝑣 (42)

yes

Use Eq. (46) with 𝛾𝑣 (42) and 𝛾𝑏 (48)

yes

Use Eq. (43) with 𝛾𝑣 (42) and 𝛾𝑝 (45)

95

CHAPTER 6 INDIVIDUAL OVERTOPPING ANALYSIS

Besides an average overtopping analysis, an individual overtopping analysis is performed

with the help of the individual overtopping script discussed in section 4.3. Before a part of the

analysis is started, a short theoretical background is given about the probability of

overtopping (section 6.1). The background is based on a master thesis (Platteeuw, 2015) also

performed at Ghent University. Further in section 6.1 a specific factor is determined, used in

the prediction formula for the probability of overtopping, for each configuration separately. In

section 6.2 till 6.5 the different influences on the individual overtopping quantities Vmax and

V1/250 are investigated and dimensionless quantities are introduced. With the knowledge of the

influencing parameters, prediction formulas will be proposed for these dimensionless

quantities and this for the different geometrical configurations (see section 6.6).

6.1 Probability of overtopping

With the help of a wave gauge placed in front of the overtopping tray, a signal is received

every time there is an overtopping event (see Chapter 3). The amount of overtopping waves

Now can be determined with this signal and it is done automatically by the individual

overtopping script. With the knowledge of the number of incident waves Nw determined with

the help of Wavelab, the probability of overtopping Pow can be calculated. It is defined as the

probability for which a certain volume of water will flow over the defense structure and it is

quantified as the ratio of the number of overtopping waves and the number of incident waves:

𝑃𝑜𝑤 =

𝑁𝑜𝑤𝑁𝑤

(51)

In literature there is already some research performed to search for prediction formulas of Pow

for different structures. The most common form of prediction formula is of the Rayleigh type

and function of the relative freeboard Rc/Hm0. Out of this formula it is also clear that zero

relative freeboard results in a value of 1 for Pow. This means that for zero relative freeboards

all the waves will result in overtopping.

𝑃𝑜𝑤 = exp(− (

1

𝜒

𝑅𝑐𝐻𝑚0

)2

) (52)

The factor 𝜒 depends on the structure and is in some cases related to the 2% run-up height

Ru2%/Hm0. In this thesis the factor 𝜒 is determined for the different geometrical configurations

as also its 95% confidence interval values. For simplicity it is chosen to determine the 𝜒-

factors as constant values depending only on the configuration tested.

Chapter 6 Individual overtopping analysis 6.1 Probability of overtopping

96

6.1.1 Factor 𝝌 determining the probability of overtopping Pow

For each test the corresponding probability of overtopping Pow is calculated and plotted

against the relative freeboard Rc/Hm0 for its corresponding geometrical configuration. In

Figure 6-1 Pow is plotted against the relative freeboard Rc/Hm0 for all the tests. Formula (52) is

fitted against the data points with the help of SPSS for each configuration and the factor 𝜒 is

determined. A summary of the found 𝜒 factors, the 95% confidence interval values and the

coefficient of determination R² per configuration can be found in Table 6-1. The range of

Rc/Hm0 for which 𝜒 is valid is also summarized in Table 6-1.

FIGURE 6-1: PROBABILITY OF OVERTOPPING IN FUNCTION OF THE RELATIVE FREEBOARD FOR ALL THE TESTS

TABLE 6-1: FACTOR 𝜒 AND ITS 95% CONFIDENCE INTERVALS DETERMINED WITH SPSS

Configuration 𝜒 𝜒95% 𝑙𝑜𝑤𝑒𝑟 𝜒95% 𝑢𝑝𝑝𝑒𝑟 R² Rc/Hm0

no berm no wall 0.917 0.850 0.996 0.914 [0.2; 1.2]

no berm wall 1.083 1.000 1.182 0,863 [0.4; 3]

no berm parapet 0.724 0.667 0.792 0,894 [0.4; 2.7]

20 cm berm wall 0.972 0.883 1.080 0,858 [0.4; 3]

20 cm berm parapet 0.712 0.643 0.797 0,804 [0.4; 1.7]

40 cm berm wall 0.810 0.754 0.875 0,907 [0.4; 2.7]

40 cm berm parapet 0.797 0.945 0.727 0,820 [0.4; 3]

For the configuration without a berm or wall, the factor 𝜒 found with SPSS gives a lower

value found than the values already available in literature. The reason for this is that only 12

tests were available with relative freeboards Rc/Hm0 not larger than 1.15. The existing formula

for the factor 𝜒 is based on a much larger data set and depends on the slope angle (cot 𝛼 = 2)

of the slope (Victor, 2012):

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Pow [

-]

0,0

0,2

0,4

0,6

0,8

1,0

1,2

all data


97

𝜒 =

1

(1.4 − 0.3 ∙ cot 𝛼)= 1.25 (53)

This formula is valid for relative freeboards Rc/Hm0 larger than 0.4. The value of 1.25 for 𝜒

will be used to plot the prediction formula for Pow for the case without a berm or wall.

Another indication that a value of 0.917 is not realistic for 𝜒 is that the value would be lower

than for the case without a berm and a wall. Common sense state that placing a wall on top of

the slope should not only reduce the average overtopping but also reduce the amount of

overtopping waves. In the Figures that follow the prediction formula for Pow presented in

equation (52) are plotted. The configurations with the same berm width are compared with

each other.

FIGURE 6-2: PREDICTION FORMULA POW FOR CONFIGURATIONS WITHOUT A BERM

No berm

For the configurations without a berm represented in Figure 6-2 it can be seen that the

prediction curve for Pow for the case with a parapet lies below the one for a wall and the one

for a wall lies below the one without protection. This means that from the waves that reach

the top of the slope, in general the least wave overtop is found when a parapet is present. The

construction with a wall results in more overtopping waves in comparison with the

construction with a parapet and the most waves overtop is found for an unprotected slope. The

difference between the three configurations is present for relative freeboards Rc/Hm0 up to 2.5.

Coming back to the results for average overtopping, similar results were found. The lowest

average overtopping volumes q were found for the case with a parapet, then the case with a

wall and the largest values were found for the case without protection.

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Pow [

-]

0,0

0,2

0,4

0,6

0,8

1,0

1,2

no wallwallparapetpred. formula Pow no wall

pred. formula Pow wall

pred. formula Pow parapet


98

20 cm berm

The prediction formulas for the configurations with a 20 cm berm are plotted in Figure 6-3.

Same conclusions can be made as for the configurations without a berm, but the prediction

curves for Pow lie closer to each other and almost no difference is found of the curves for

relative freeboards larger than 2.

FIGURE 6-3: PREDICTION FORMULA POW FOR CONFIGURATIONS WITH A 20 CM BERM

40 cm berm

The prediction formulas for the configurations with a 40 cm berm are plotted in Figure 6-4.

Now it can be remarked that the prediction curves for Pow lie almost on one line. It means that

for the probability of overtopping it doesn’t make a difference if a vertical wall or parapet is

present in combination with a 40 cm berm.

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Pow [

-]

0,0

0,2

0,4

0,6

0,8

1,0

1,2

wallparapetpred. formula Pow wall


Chapter 6 Individual overtopping analysis 6.2 Dimensionless individual overtopping quantities

99

FIGURE 6-4: PREDICTION FORMULA POW FOR CONFIGURATIONS WITH 40 CM BERM

6.2 Dimensionless individual overtopping quantities

In the following sections the individual overtopping quantities Vmax and V1/250 are further

investigated. The influence of different parameters on these quantities is checked.

FIGURE 6-5: MAXIMAL VOLUME IN FUNCTION OF THE RELATIVE FREEBOARD

The reference case with a simple slope is not considered due to the lack of data points with

𝑅𝑐/𝐻𝑚0 >1.2. In Figure 6-5 the maximum volumes are plotted against the relative freeboard

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Pow [

-]

0,0

0,2

0,4

0,6

0,8

1,0

1,2

wallparapetpred. formula Pow wall


Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Vm

ax [

kg/m

]

0

20

40

60

80

100

120

140

Chapter 6 Individual overtopping analysis 6.3 Influence of the water depth

100

for all the test results. For the volumes V1/250 a similar plot could be made, only most values

would be smaller. In the further sections, the use of dimensionless individual overtopping

quantities will be made. The dimensionless parameters for Vmax and V1/250 are given by

equations (54) and (55) respectively. 𝜌𝑤 is the water density (1000 kg/m³), Hm0 the significant

wave height and Rc the freeboard. The dimensionless parameters are the result of trial and

error and the equations below gave the best result to describe Vmax and V1/250. About the role

of the wave steepness 𝑠𝑚−1,0 in the dimensionless parameters, reference is made to section

6.4.

𝑉𝑑𝑖𝑚,𝑚𝑎𝑥 =

𝑉𝑚𝑎𝑥 ∙ 𝐻𝑚0

𝜌𝑤𝑅𝑐3 ∙ √𝑠𝑚−1,0

(54)

𝑉𝑑𝑖𝑚,1/250 =

𝑉1/250 ∙ 𝐻𝑚0

𝜌𝑤𝑅𝑐3 ∙ √𝑠𝑚−1,0

(55)

6.3 Influence of the water depth

To investigate the influence of the different parameters, a separation is always made between

configurations with a wall and configurations with a parapet. Later in section 6.6, it will be

checked if this separation is necessary for the prediction formulas.

FIGURE 6-6: INFLUENCE OF THE WATER DEPTH ON THE MAXIMUM VOLUMES FOR THE CASES WITH A WALL

In Figure 6-6 the maximum volumes with a wall (and different berm widths) are divided by

their water depth at the paddle. No specific influence of the water depth can be seen on this

Figure for relative freeboards larger than 0.5. One reason for this is that the influence of the

water depth is already taken up in the x-axes (dimensionless freeboard). A same plot can be

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Vm

ax [kg/m

]

0

20

40

60

80

100

120

140

water depth 70 cmwater depth 73 cmwater depth 76 cm

Chapter 6 Individual overtopping analysis 6.4 Influence of the wave steepness

101

made for the cases with parapet (Figure 6-7). A similar conclusion is valid for the cases with

parapet.

FIGURE 6-7: INFLUENCE OF THE WATER DEPTH ON THE MAXIMUM VOLUMES FOR THE CASES WITH PARAPET

Similar plots can be made to investigate the influence of the water depth on V1/250. To limit the

amount of Figures, they can be found in Annex B. The two Figures lead to similar conclusions

as for Vmax.

6.4 Influence of the wave steepness

In Figure 6-8 and Figure 6-9 the tests are separated in their two target wave steepnesses 0.03

and 0.05 for the configurations with wall and parapet respectively. For both Figures the

maximum volumes for the waves with target steepness equal to 0.05 lie below the maximum

volumes for the waves with steepness equal to 0.03. For relative freeboards larger than 1.5 the

difference in maximum volumes for the two target steepnesses is rather limited. Due to the

difference for the two target steepnesses it is required to take up the wave steepness either in

the dimensionless parameters or in the prediction formulas itself for the dimensionless

parameters. In the thesis the first option is chosen.

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Vm

ax [kg/m

]

0

20

40

60

80

100

120

140


Chapter 6 Individual overtopping analysis 6.4 Influence of the wave steepness

102

FIGURE 6-8: INFLUENCE OF THE STEEPNESS ON THE MAXIMUM VOLUME FOR THE CASES WITH WALL

FIGURE 6-9: INFLUENCE OF THE STEEPNESS ON THE MAXIMUM VOLUME FOR THE CASES WITH PARAPET

A similar influence of the wave steepness was found for average overtopping. The waves with

a wave steepness of 0.03 lead to smaller values for the dimensionless overtopping Q waves

with a target steepness of 0.05 (section 5.2).The same conclusions about wave steepness are

also valid for the Figures with V1/250. Again these can be found in Annex B.

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Vm

ax [

kg/m

]

0

20

40

60

80

100

120

140

sm-1,0 = 0.03

sm-1,0 = 0.05

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Vm

ax [kg/m

]

0

20

40

60

80

100

120

140

sm-1,0 = 0.03

sm-1,0 = 0.05

Chapter 6 Individual overtopping analysis 6.5 Influence of the berm width

103

6.5 Influence of the berm width

In Figure 6-10 and Figure 6-11 the tests are separated by their berm width for the

configurations with wall and parapet respectively. With these Figures it is difficult to state if

the presence of a berm has an effect on the maximum volumes. In section 6.6, prediction

formulas for the dimensionless individual overtopping parameters will be proposed,

considering different berm widths. Once it is clear that these prediction formulae for certain

berm widths lie close to each other, a global prediction formula will be proposed for multiple

berm widths.

FIGURE 6-10: INFLUENCE OF THE BERM WIDTH ON THE MAXIMUM VOLUME FOR THE CASES WITH WALL

Similar Figures for V1/250 can be made and the same conclusions as for Vmax are valid. These

Figures can be found in Annex B.

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Vm

ax [

kg/m

]

0

20

40

60

80

100

120

140

no berm20 cm berm40 cm berm

Chapter 6 Individual overtopping analysis 6.6 Prediction formulas for the dimensionless

parameters

104

FIGURE 6-11: INFLUENCE OF THE BERM WIDTH ON THE MAXIMUM VOLUME FOR THE CASES WITH PARAPET

6.6 Prediction formulas for the dimensionless parameters

The goal of this section is to come up with formulas that describe the dimensionless

parameters represented in (54) and (55). The dimensionless parameters do take into account

the difference in results due to wave steepness (section 6.4). As mentioned before, a

separation is made between configurations with wall/parapet and also the tests are split up in

different berm widths. A detailed analysis is performed for Vdim,max. A similar analysis with

Figures can be done for Vdim,1/250. To limit the amount of Figures, the Figures for Vdim,1/250 can

be found in Annex B. The prediction formulas for Vdim,1/250 will be summarized in at the end

of this section.

6.6.1 Vertical wall

The analysis is started by plotting Vdim,max against the relative freeboard Rc/Hm0. If its

required, Vdim,max is plotted on a log scale. The procedure to come up with a prediction

formula for Vdim,max is done extensively for the case without a berm. For the configurations

with a 20 cm and 40 cm berm a similar procedure is followed, but only the results will be

discussed.

No berm

In Figure 6-12 the dimensionless maximum volume Vdim,max is plotted against the relative

freeboard. An exponential or power law decrease can be seen for increasing relative

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Vm

ax [

kg/m

]

0

20

40

60

80

100

120

140



parameters

105

freeboard. Therefore prediction formulas that are proposed have an exponential form or obey

a power law.

FIGURE 6-12: DIMENSIONLESS MAXIMUM VOLUME IN FUNCTION OF THE RELATIVE FREEBOARD

With the help of Excel two forms of prediction formulas are plotted in Figure 6-13, together

with their coefficient of determination R². The dimensionless maximum volume is plotted on

a logarithmic scale such that also a difference is seen in the very small values.

FIGURE 6-13: TWO DIFFERENT FORMS OF PREDICTION FORMULAS FOR THE DIMENSIONLESS MAXIMUM VOLUME

The power law results in a larger coefficient of determination R² than the exponential

prediction formula. The difference is rather small such that on another basis the right

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Vd

im,

ma

x [

-]

0

5

10

15

20

25

no berm wall

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Vd

im,

ma

x [-]

0,0001

0,001

0,01

0,1

1

10

100

1000

no berm wallexp. pred. formulapow. pred. formula

R² = 0.928

Vdim,max = 0.321*x-4.345

R² = 0.914Vdim,max = 25.053*exp(-3.719x)


parameters

106

prediction formula is chosen. The power law gives better predictions for the lower relative

freeboards with larger values of Vdim,max. This can be seen in Figure 6-14, in which the y-axes

is set back on linear.

FIGURE 6-14: TWO DIFFERENT FORMS OF PREDICTION FORMULAS FOR THE DIMENSIONLESS MAXIMUM VOLUME

(LINEAR)

Due to this observation and the better results for configurations with another berm length, the

power law is chosen to be the general form of the prediction formula. In equation (56), the

prediction formula valid for Vdim,max in the case of a wall without berm is given. The formula

is valid for Rc/Hm0 between 0.4 and 3.

𝑉𝑑𝑖𝑚,𝑚𝑎𝑥 = 0.321 ∙ (𝑅𝑐𝐻𝑚0

)−4.345

(56)

20 cm berm

A same fitting procedure is done for the tests with a 20 cm berm. The prediction formula for

Vdim,max is given in equation (57). With this formula a coefficient of determination R² of 0.939

is found.


)−4.872

(57)

Figure 6-15 forms a graphical representation of all the prediction formulas used for the

configurations with a wall and different berm widths. The red line represents the prediction

formula for the case with a 20 cm berm. The prediction formula lays below the formula the

case without a berm. The difference in dimensionless maximum volume increases with

relative freeboard. Formula (57) is valid for Rc/Hm0 between 0.4 and 1.8.

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Vdim

, m

ax [-]

0

10

20

30

40

50

60

70

no berm wallexp. pred. formulapow. pred. formula

Vdim,max = 25.053*exp(-3.719x)

Vdim,max = 0.321*x-4.345

R² = 0.928

R² = 0.914


parameters

107

FIGURE 6-15: GRAPHICAL SUMMARY OF THE PREDICTION FORMULAE USED FOR THE CONFIGURATIONS WITH A

WALL

40 cm berm

For the tests with a 40 cm berm the prediction formula is plotted in blue in Figure 6-15. A

coefficient of determination R² of 0.934 is attained. The formula itself is given in equation

(58). In Figure 6-15 it can be remarked that the reduction in Vdim,max is smaller between a 20

cm berm and 40 cm berm than the reduction between no berm and 20 cm berm.


)−4.891

(58)

6.6.2 Parapet

The prediction formulas for Vdim,max, for the configurations with a parapet, are determined on

a similar way as in 6.6.1. In Figure 6-16 they are graphically represented. The formula for no

berm seems almost to coincide with the prediction formula for a 20 cm berm. Also the

formula for a 40 cm berm lies close to the two others. Therefore it is decided to determine 1

design formula without making difference in berm width for the configurations with parapet.

The global formula is given in equation (59). A coefficient of determination R² of 0.936 is

found.


)−5.083

(59)

Equation (59) will be plotted in Figure 6-17 when the comparison between the prediction

formulas of the configurations with a wall and the formula for the configurations with parapet.

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Vdim

, m

ax [-]

0,0001

0,001

0,01

0,1

1

10

100

1000

no berm20 cm berm40 cm bermpred. formula no bermpred. formula 20 cm bermpred. formula 40 cm berm


parameters

108

FIGURE 6-16: GRAPHICAL SUMMARY OF THE PREDICTION FORMULAE USED FOR THE CONFIGURATIONS WITH A

PARAPET

6.6.3 Comparison vertical wall/parapet

To compare all the prediction formulas, which were created in the previous sections, they are

plotted on the same Figure (Figure 6-17). It can be seen that the general prediction formula for

the parapets forms the lower curve of the formulas. This means that according to the formulas,

the configurations with parapet result in lower Vdim,max-values than the configurations with

walls, for relative freeboards higher than 0.5. For relative freeboards below 0.5 the difference

in Vdim,max-values is rather limited for the different configurations.

Further it is remarkable that the global prediction formula for the parapets almost coincide

with the prediction formula for a 40 cm berm with wall. This means that the different

configurations with a parapet result in almost the same Vdim,max-values as for the configuration

with a wall and 40 cm berm.

Rc/H

m0 [-]

0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

Vd

im ,

ma

x [-]

0,001

0,01

0,1

1

10

100

1000

no berm20 cm berm40 cm bermpred. formula no bermpred. formula 20 cm bermpred. formula 40 cm berm


parameters

109

FIGURE 6-17: COMPARISON BETWEEN PRED. FORMULAS WALL AND PARAPET

6.6.4 Prediction formulas Vdim,1/250

As mentioned before the Figures for the analysis of the dimensionless parameter Vdim,1/250 can

be found in Annex B. The prediction formulas for Vdim,1/250 for the different geometrical

configurations are summarized in Table 6-2. In this table also the range of applicable relative

freeboard is mentioned and the corresponding coefficient of determination R².

TABLE 6-2: PREDICTION FORMULAS VDIM,1/250 AND APPLICABILITY RANGE

Configuration R² Rc/Hm0 range Pred. formula Vdim,1/250

No berm wall 0.926 [0.3; 3] 0.301 ∙ (𝑅𝑐𝐻𝑚0

)−4.27

20 cm berm wall 0.938 [0.3;1.9] 0.191 ∙ (𝑅𝑐𝐻𝑚0

)−4.732

40 cm berm wall 0.940 [0.3; 1.9] 0.128 ∙ (𝑅𝑐𝐻𝑚0

)−4.777

No/20 /40 cm berm parapet 0.931 [0.3; 1.7] 0.142 ∙ (𝑅𝑐𝐻𝑚0

)−4.946

Also no difference is made in berm width for the configurations with parapet, because again

these different formulas lie close together.

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Vdim

, m

ax [

-]

0,0001

0,001

0,01

0,1

1

10

100

1000

no berm wall20 cm berm wall40 cm berm wallall data parapetpred. formula no berm wallpred. formula 20 cm berm wallpred. formula 40 cm berm wallglob. pred. formula parapet

111

CHAPTER 7 FORCE ANALYSIS

The aim of this section is to come up with a prediction formula for the wave loads on the

storm wall. To do so first the influence of different parameters such as wave height (section

7.1), relative freeboard (section 7.2), water depth, steepness and berm width (section 7.4) is

checked. These influences are first checked with the absolute force values. To be able to make

a prediction formula, it’s better to work with a dimensionless force and finding the right

dimensionless force is the next step in the analysis. The last step of the analysis is then

combining al the parameters that have an influence on the force impact to one general

prediction formula (section 7.5). For the general analysis of the forces, only tests with a wall

are considered and in section 7.6 the influence of installing a parapet on the wave forces is

examined. In the final section of this chapter, an introduction to the relationship between

wave forces and individual overtopping is given.

7.1 Influence of the wave height

In Figure 7-1 the absolute wave force F1/250 is shown in function of the incident wave height

Hm0 at toe of the dike. It can immediately be seen that the incident wave height has a great

influence on the wave force, a larger incident wave height leads to a higher wave force.

Because in Figure 7-1 no distinction is made between tests with different configurations

(berm width, relative freeboard), a lot of scatter is present and therefore no trendline is added.

FIGURE 7-1: F1/250 PLOTTED AGAINST HM0

Hm0

[m]

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18

F1

/25

0 [N

/m]

0

50

100

150

200

250

300

350

Chapter 7 Force analysis 7.2 Influence of the relative freeboard

112

7.2 Influence of the relative freeboard

On Figure 7-2 the relationship between the absolute wave force F1/250 and relative freeboard

Rc/Hm0 is shown. The relative freeboard is, just like with the overtopping, an important

parameter. It combines the water depth, wave height and the crest level. According to the plot,

the wave force increases with decreasing relative freeboard.

FIGURE 7-2: WAVE FORCE F1/250 PLOTTED AGAINST THE RELATIVE FREEBOARD RC/HM0

7.3 Dimensionless wave force

Up to now only absolute values of the wave force have been used. To be able to come up with

a general prediction formula, it is required to use a dimensionless wave force. To make the

force dimensionless, it is divided by the water density ρ, the gravitational acceleration g and

the square of the freeboard Rc.

𝑃 =

𝐹1/250

𝜌 ∙ 𝑔 ∙ 𝑅𝑐2 (60)

The chosen form of the dimensionless force P is based on the master thesis of (Hohls, 2015).

By introducing the dimensionless wave force it is easier to check the influence of the water

depth, wave steepness and berm width.

7.4 Influence of the water depth, steepness and berm width

To investigate the influence of the different parameters on to the dimensionless wave force P,

Figure 7-3 is shown. The data points shown in this Figure are closely gathered and the

Rc/H

m0 [-]

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

F1

/25

0 [N

/m]

0

50

100

150

200

250

300

350

Chapter 7 Force analysis 7.5 Prediction formula

113

difference between them is smaller than the possible scatter that is present and therefore it can

be concluded that the water depth, steepness and berm width have no influence on the

determination of the wave force. In Annex C, the data points are split up according to water

depth, steepness and berm width and there it can be seen that all these parameters do not

influence the wave force.

The water depths play a role in the number of waves that hit the wall as was described in

section 4.4 but it has no influence on the wave force because the freeboard is present in both

axes. For the influence of the steepness and berm width, it has to be mentioned that this

conclusion is only valid for steepnesses between 3 and 5 % and for the berm widths presented

in this thesis. For other steepnesses and berm widths, no conclusion can be given.

FIGURE 7-3: DIMENSIONLESS WAVE FORCE P IN FUNCTION OF RELATIVE FREEBOARD RC/HM0

7.5 Prediction formula

Now that the parameters which influence the wave force are determined, an empirical

prediction formula can be developed with the use of Excel. This formula will be dependent on

the relative freeboard only since the other parameters have less influence on the wave force.

The prediction formula is plotted in Figure 7-4 along with the test results. The prediction

formula is shown in equation (61).

𝐹1/250

𝜌 ∙ 𝑔 ∙ 𝑅𝑐2= 10.7 ∙ exp (−1.67 ∙

𝑅𝑐𝐻𝑚0

) (61)

In this formula F1/250 is the average force of the highest 0.4 % impacts, ρ is the water density,

g is the gravitational acceleration, Rc is the freeboard and Hm0 is the incident wave height at

Rc/H

m0 [-]

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

P [

-]

0.01

0.1

1

10

100

Chapter 7 Force analysis 7.6 Wave force when a parapet is installed

114

the toe of the structure. In Figure 7-4, it can be seen that the prediction formula follows the

test results quite nicely and this leads to a coefficient of determination equal to 0.91.

FIGURE 7-4: PREDICTION FORMULA PLOTTED ALONG WITH THE TEST RESULTS

7.6 Wave force when a parapet is installed

Up to now, only test configurations with a vertical wall were considered. In this paragraph a

comparison will be made between tests with a wall and a parapet but with every other

parameter the same. A selection of tests is made to compare the wave forces. The selection

contains nine tests in total with three different wave heights and three different berm widths.

The dimensionless force is used to compare the wave forces. In Figure 7-5 the dimensionless

wave force is plotted against the relative freeboard Rc/Hm0. The 18 data points (9 with a wall

and 9 with a parapet) are plotted alongside with the prediction formula presented in equation

(61). In Figure 7-5, the trendline of the data points of the tests with a vertical wall is fairly

close to the prediction formula. The trendline of the data points of the tests with parapet is

further off. Based upon this small selection of tests, it can be concluded that the wave forces

induced on a storm wall with a parapet are higher than the ones on a simple vertical storm

wall. The difference between both configurations is about 22 %. The reason for this rise in

horizontal forces is probably due to the fact that the vertical forces that act on the parapet

cause a bending of the wall and this bending causes an additional horizontal force onto the

load cell. However the exact reason for this phenomenon cannot be determined since only

horizontal forces were measured. To come up with a prediction formula for the wave forces

on a storm wall with a parapet, more tests should be used for the analysis.

Rc/H

m0 [-]

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

P [-]

0.01

0.1

1

10

100

Prediction formula

P = 10.7*exp(-1.67x)

R² = 0.91

Chapter 7 Force analysis 7.7 Relationship between wave forces and individual overtopping

115

FIGURE 7-5: COMPARISON BETWEEN WAVE FORCES WITH A VERTICAL WALL AND A PARAPET

7.7 Relationship between wave forces and individual overtopping

During testing, the wave forces and overtopping have been measured. In literature not much

research has been done yet to the relationship between wave forces and individual

overtopping amounts. In this section a small introduction is given to this subject to see if there

is a clear relationship between them and if this is the case, further research might be useful.

To investigate whether there is a relationship, the first 100 seconds of three tests are looked at

more closely. The wave force and individual overtopping of each wave impact during this

time period are related to each other. The three selected tests are 0048A, 0073A and 00145A

(see Annex D). The chosen tests have a water depth of 70 cm and a berm width of 0 cm, 20

cm and 40 cm respectively. The chosen water depth is 70 cm because in this case, the

overtopping events are more limited and the wave impacts can be seen more clearly. To

correlate the wave impact to the right overtopping event, the videos are examined. In the

video analysis, every impact and potential overtopping can be seen. This impact can then be

correlated to the right wave force with the help of the excel files that were created in Chapter

4. When the impact lead to an overtopping event, the overtopping volume could be linked to

the impact with the help of the excel files created by the individual overtopping script. This

course of action was repeated for all the wave impacts in the first 100 seconds of the three

videos and so every overtopping event was linked to a certain wave force.

For test number 0073A the relationship between the wave force and overtopping is shown in

Figure 7-6. The lowest force for which an overtopping event occurs is 19.6 N/m and the

overtopping then is very limited (0.14 kg/m). Beyond this threshold the wave overtopping

Rc/H

m0 [-]

0.0 0.5 1.0 1.5 2.0 2.5

P [

-]

0.1

1

10

With parapet Trendline with parapetWithout parapetTrendline without parapetPrediction formula without parapet

Chapter 7 Force analysis 7.7 Relationship between wave forces and individual overtopping

116

shows a quadratic relationship compared to the wave force. The two other selected tests (test

number 0048A and 00145A) show a similar relationship. The only difference between the

three different tests is that the quadratic trendline is shifted to the left when there is no berm

or to the right with a 40 cm berm. The threshold for overtopping without a berm is 15.2 N/m

and for the 40 cm berm the threshold equals 25.6 N/m. This indicates that the berm width has

an influence on the relationship between the wave force and overtopping. The influences of

the other parameters (steepness and water depth) on this relationship were not determined

because not enough data was checked and therefore no formula based relationship was

established as well. However this small introduction into this domain shows that there might

be a fixed relationship between the wave force and overtopping and therefore further research

would be useful.

FIGURE 7-6: RELATIONSHIP BETWEEN WAVE FORCE AND WAVE OVERTOPPING

Wave force [N/m]

0 20 40 60 80 100 120 140 160

Wa

ve o

vert

op

pin

g [kg

/m]

-2

0

2

4

6

8

10

12

14

16

18

117

CHAPTER 8 CONCLUSION

8.1 Average overtopping

The analysis of the average overtopping happened in Chapter 5. To start the analysis, the

influence of different parameters on the overtopping was verified. For the verification of the

influences, the dimensionless overtopping Q was used. The water depth was the first

parameter to be checked and it was found that it has no influence on the dimensionless

overtopping because the freeboard, which is influenced by the water depth, is present in the x-

axis. Secondly the influence of steepness was examined. For the tests with a parapet, an

influence of the steepness was found whereas for the tests with a vertical wall, an influence of

the steepness was only present when there was a berm installed. When there was no berm and

a vertical wall, the steepness did not influence the overtopping. The third tested parameter

was the influence of a parapet in comparison to a simple vertical wall. It was found that

installing a parapet instead of a wall, greatly reduced the overtopping, especially for a larger

relative freeboard. The fourth and last investigated parameter was the berm width. For this

parameter it was found that it has an influence on the overtopping when a vertical wall was

installed. When a parapet was installed, the influence was much less and even negligible.

Once the influence of the different parameters was determined, the results were compared to

the existing literature. Here it was established that the updated van der Meer formula is the

most accurate prediction formula for the tests without any protection (wall or berm).

Therefore this formula was chosen as the basis for new prediction formulas. To adapt the

updated vdM formula, reduction factors were added when a wall, parapet or berm was

installed. These reductions are dependent on the previous discussed parameters (steepness and

berm width). In total three new prediction formulas, dependent on one or two reduction

factors, were drafted.

8.2 Individual overtopping

In the individual overtopping analysis (Chapter 6), different parameters were investigated that

characterize the individual overtopping behaviour of a structure. In a first part the probability

of overtopping Pow was investigated per structure. For each configuration separately,

prediction formulas of the Rayleigh type are used to predict Pow in function of the relative

freeboard Rc/Hm0. The factor χ determined the shape of the prediction formula.

In section 6.2 till 6.6, the characteristics Vmax and V1/250 were looked into. The different

parameters influencing Vmax and V1/250 were investigated. With the knowledge of the

influencing parameters (wave steepness), Vmax and V1/250 were made dimensionless. The

Chapter 8 Conclusion 8.3 Wave forces

118

reason for making those characteristics dimensionless is that the fitting procedure of

prediction formulas is facilitated. For the different geometrical configurations, prediction

formulas for Vdim,max and Vdim,1/250 were proposed. The prediction formulas follow a power

law and are again function of the dimensionless freeboard Rc/Hm0. In the last part, the

different formulas are compared to each other and some observations were explained.

8.3 Wave forces

In Chapter 7 an analysis of the horizontal wave forces was performed. For this analysis only

tests with a simple vertical wall were considered. To start the analysis the influence of

different parameters was checked. To verify the influence of the wave height, the average of

the 0.4 % highest wave forces was examined. It could be noticed that the wave force increases

for increasing wave height. The relative freeboard has on opposite effect on the absolute wave

force. The wave forces decreases with increasing relative freeboard. To verify the influence of

the water depth, steepness and berm width, a dimensionless wave force was created. With the

help of this dimensionless wave force it was found that none of the previous mentioned

parameters has an influence on the wave force. The only parameter that is taken up in the

prediction formula is therefore the relative freeboard Rc/Hm0. With this prediction formula all

the test data can be predicted quite well and this leads to a coefficient of determination of

0.91.

After the prediction formula was drafted, a comparison was made between the wave forces

acting on a storm wall and the ones acting on a parapet. To compare these two configurations,

a few tests were selected. Based upon the selected tests, it could be concluded that the forces

on a parapet are about 22 % higher than on a simple vertical wall. For the wave forces acting

on a parapet, no prediction formula was drafted because not enough data was examined to be

able to come up with an accurate formula. In the last part of Chapter 7, the relationship

between wave forces and individual overtopping was examined. Through the analysis of three

tests, a quadratic relationship was found. For every test, a certain threshold for overtopping

was found. Below a minimum force, there were no overtopping events. The analysis of the

three selected tests shows that there might be a fixed relationship between the wave force and

overtopping and therefore further research would be useful.

8.4 Further research

Regarding the overtopping there are still some influences that can be further examined. First

of all, the influence of a larger berm width needs to be verified. In this thesis it was often

found that the berm width does not have a clear influence but this is probably because the

tested berm widths are too small in comparison to the wave lengths of the used waves.

Chapter 8 Conclusion 8.4 Further research

119

Secondly, more tests with different steepnesses should be conducted to further understand the

effect that the steepness has on the overtopping. A third possibility for further research is the

influence of curved parapets. Coming up with reduction factors for different types of parapets

can be useful in the future. At last, the formulas created can be checked and possibly updated

with tests, performed with other relative freeboards or wave characteristics.

For the individual overtopping volumes also some further research can be performed. One

interesting research domain is to look into the distribution function of the individual

overtopping volumes Vi. Some literature is already present for the case of a simple slope. An

extension of this literature is to consider also the configurations described in the master thesis.

When looking at the forces, there are also still some things that can be investigated more

closely. The horizontal forces that act on a parapet are only briefly discussed in this thesis.

For these horizontal forces acting on a parapet, it can be investigated why they are larger in

comparison to the forces acting on a vertical wall and also a prediction formula can be

drafted. Besides horizontal forces also vertical forces will be present on the parapet.

Measuring and investigating these forces can be interesting. The relationship between wave

forces and individual overtopping has also been introduced in this thesis and here further

research can be conducted to be able to predict the overtopping when the forces are known or

vice versa.

121

REFERENCES

Altomare. (2014). Characterization of Wave Impacts on Curve Faced Storm Return Walls

within a Stilling Wave Basin Concept.

Cornett, A. (1999). Wave Overtopping at Chamfered and Overhanging Vertical Structures.

Daemrich, K. (2006). Overtopping at Vertical Walls and Parapets - Regular Wave Tests for

Irregular Simulation.

Goormachtigh, J., & Harchay, W. (2010). Experimentele studie van golfoploop en overslag

ter optimalisatie van golfenergieconvertoren gebaseerd op golfoverslag. Ghent,

Belgium.

Hohls, C. (2015). Wave-Induced Loading of a Storm Walls at the Belgian Coast.

Kamikubo, Y. (sd). Reduction of Wave Overtopping and Water Spray with Using Flaring

Shaped Seawall.

Kisacik, D. (2011). Description of Loading Conditions due to Violent Wave Impacts on a

Vertical Structure with an Overhanging Horizontal Cantilever Slab.

Kortenhaus et al. (2008). Storm Surge Protection Walls in Germany.

Kortenhaus et al. (2001). Design Aspects of Verticall Walls with Steep Foreland Slopes.

Kortenhaus, A. (2003). Influence of Parapets and Recurves on Wave Overtopping and Wave

Loading of Complex Vertical Walls.

Mansard and Funke. (1980). The Measurement of Incident and Reflected Spectra Using a

Least Squares Method.

Pearson. (2004). Effectiveness of Recurve Walls in Reducing Wave Overtopping.

Platteeuw, J. (2015). Analysis of individual wave overtopping volumes for steep low crested

coastal structures in deep water conditions.

Pullen et al. (2007). EurOtop.

Troch, P. (2000). Wave Flume Manuel.

van der Meer, J., & Bruce, T. (2014). New Physical Insights and Design Formulas on Wave

Overtopping at Sloping and Vertical Structures.

Van Doorslaer et al. (2015). Crest Modifications to Reduce Wave Overtopping of Non-

Breaking Waves over a Smooth Dike Slope.

References

122

Van Doorslaer, K. (2010). Reduction of Wave Overtopping on a smooth Dike by means of a

Parapet.

Van Doorslaer, K. (2010). The Influence of a Berm and a Vertical Wall above SWL on the

Reduction of Wave Overtopping.

Van Doorslaer, K. (2011). Measures to Control Wave Overtopping Inside the Harbor of

Oostende.

Van Doorslaer, K. (sd). Reduction of Wave Overtopping: from Research to Practice.

Victor, L. (2012). Probability Distribution of Individual Wave Overtopping Volumes for

Smooth Impermeable Steep Slopes with Low Crest Freeboard. Coastal Eng., pp. 87-

101.

Vroman and Pintelon, T. L. (2013). Experimental Study of the Overtopping Performance of

Steep Slopes with Small Freeboards in Shallow Water Conditions. Gent.

123

ANNEX A: AVERAGE OVERTOPPING ANALYSIS

Influence of water depth, wall/parapet and berm width

FIGURE A-1: INFLUENCE OF WATER DEPTH WHEN THERE'S NO BERM AND A WALL (STEEPNESS 0.03)

FIGURE A-2: INFLUENCE OF THE WATER DEPTH WHEN THERE’S NO BERM AND A PARAPET (STEEPNESS 0.03)

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5

Q [

-]

0,0001

0,001

0,01

0,1


Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0

Q [

-]

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1


Annex A: Average overtopping analysis

124

FIGURE A-3: INFLUENCE OF WATER DEPTH WHEN THERE'S A 40 CM BERM AND A WALL (STEEPNESS 0.03)

FIGURE A-4: INFLUENCE OF WATER DEPTH WHEN THERE'S A 40 CM BERM AND A PARAPET (STEEPNESS 0.03)

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0

Q [

-]

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

depth 70/73 cm depth 76 cm

Rc/H

m0 [-]

0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8

Q [

-]

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

depth 70/73 cm depth 76 cm


125

FIGURE A-5: INFLUENCE OF THE WAVE STEEPNESS WHEN THERE’S A 20 CM BERM AND A WALL

FIGURE A-6: INFLUENCE OF THE WAVE STEEPNESS WHEN THERE’S A 40 CM BERM AND WALL

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Q [

-]

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

sm-1,0 = 0.03

sm-1,0 = 0.05

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0

Q [

-]

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

sm-1,0 = 0.03

sm-1,0 = 0.05


126

FIGURE A-7: INFLUENCE OF THE WAVE STEEPNESS WHEN THERE’S A 20 CM BERM AND PARAPET

FIGURE A-8: INFLUENCE OF THE WAVE STEEPNESS WHEN THERE’S A 40 CM BERM AND PARAPET

Rc/H

m0 [-]

0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8

Q [

-]

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

sm-1,0 = 0.03

sm-1,0 = 0.05

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Q [

-]

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

sm-1,0 = 0.03

sm-1,0 = 0.05


127

FIGURE A-9: INFLUENCE OF PARAPET WHEN THERE’S NO BERM (STEEPNESS 0.05)

FIGURE A-10 : 𝑘𝑝-FACTORS NO BERM (STEEPNESS 0.05)

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Q [-]

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

wall sm-1,0 = 0.05


Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5

kp [

-]

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

no berm sm-1,0 = 0.05


128

FIGURE A-11: INFLUENCE OF PARAPET WHEN THERE’S A 20 CM BERM (STEEPNESS 0.05)

FIGURE A-12: 𝑘𝑝-FACTORS 20 CM BERM (STEEPNESS 0.05)

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Q [

-]

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

wall sm-1,0 = 0.05


Rc/H

m0 [-]

0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

kp [-]

0,0

0,2

0,4

0,6

0,8

1,0

no berm sm-1,0 = 0.05


129

FIGURE A-13: INFLUENCE OF PARAPET WHEN THERE’S A 40 CM BERM (STEEPNESS 0.05)

FIGURE A-14: 𝑘𝑝-FACTORS 40 CM BERM (STEEPNESS 0.05)

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Q [-]

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

wall sm-1,0 = 0.05


Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0

kp [-]

0,0

0,2

0,4

0,6

0,8

1,0

no berm sm-1,0 = 0.05


130

FIGURE A-15: INFLUENCE OF THE BERM WIDTH WHEN THERE’S A WALL (STEEPNESS 0.03)

FIGURE A-16: kb-FACTORS WHEN THERE’S A 20/40 CM BERM AND A WALL (STEEPNESS 0.03)

Rc/H

m0

0,0 0,5 1,0 1,5 2,0 2,5 3,0

Q [

-]

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

1e+0

no berm20 cm berm 40 cm berm

Rc/H

m0 [-]

0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

kb [

-]

0,0

0,2

0,4

0,6

0,8

1,0

1,2

20 cm berm 40 cm berm


131

FIGURE A-17: INFLUENCE OF THE BERM WIDTH WHEN THERE’S A PARAPET (STEEPNESS 0.03)

Reduction factors

FIGURE A-18: γv-FACTORS WITHOUT THE VALUES ABOVE 1 AND QUADRATIC TRENDLINE

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0

Q [

-]

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

no berm 20 cm berm40 cm berm

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5

v [

-]

0,0

0,2

0,4

0,6

0,8

1,0

all dataquadr. trendline

v = 0.0061x² + 0.1301x+0.5757

R² = 0.5878


132

FIGURE A-19: γv-FACTORS WITHOUT THE VALUES ABOVE 1 AND POWER TRENDLINE

TABLE A-1: DIFFERENT FORMULAS FOR γp WITH THEIR UNKNOWN PARAMETERS AND COEFFICIENT OF

DETERMINATION R²

Formula R² A B C D E

𝐴 ∙ (𝑠𝑚−1,0)𝐵∙ (𝑅𝑐𝐻𝑚0

)𝐶

0.813 0.078 -0.641 -0.337 - -

𝐴 ∙ (𝑠𝑚−1,0)𝐵∙ (𝑅𝑐𝐻𝑚0

)𝐶

+ 𝐷 0.838 -4.281 0.210 0.127 2.801 -

𝐴 ∙ (𝑠𝑚−1,0) + 𝐵 ∙ (𝑅𝑐𝐻𝑚0

) + 𝐶 0.866 -11.611 -0.324 1.446 - -

𝐴 ∙ (𝑠𝑚−1,0)𝐵+ 𝐶 ∙ (

𝑅𝑐𝐻𝑚0

)𝐷

0.836 -41.126 0.11 40.287 -0.006 -

𝐴 ∙ (𝑠𝑚−1,0)𝐵+ 𝐶 ∙ (

𝑅𝑐𝐻𝑚0

)𝐷

+ 𝐸 0.839 -50.725 1.62 971.309 0.000 -970.38

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5

v [

-]

0,0

0,2

0,4

0,6

0,8

1,0

all datapower trendline

v = 0.137x1.05

+ 0.576

R² = 0.588

133

ANNEX B: INDIVIDUAL OVERTOPPING

Analysis F1/250

FIGURE B-1: INFLUENCE OF THE WATER DEPTH ON THE 0.4% VOLUMES FOR THE CASES WITH WALL

FIGURE B-2: INFLUENCE OF THE WATER DEPTH ON THE 0.4% VOLUMES FOR THE CASES WITH PARAPET

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

V1

/25

0 [kg/m

]

0

20

40

60

80

100

120

140


Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

V1

/25

0 [kg/m

]

0

20

40

60

80

100

120

140


Annex B: Individual overtopping

134

FIGURE B-3: INFLUENCE OF THE WAVE STEEPNESS ON THE 0.4% VOLUMES FOR THE CASES WITH WALL

FIGURE B-4: INFLUENCE OF THE WATER DEPTH ON THE 0.4% VOLUMES FOR THE CASES WITH PARAPET

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

V1/2

50 [

kg/m

]

0

20

40

60

80

100

120

140

sm-1,0 = 0.03

sm-1,0 = 0.05

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

V1/2

50 [

kg/m

]

0

20

40

60

80

100

120

140

sm-1,0 = 0.03

sm-1,0 = 0.05


135

FIGURE B-5: INFLUENCE OF THE BERM WIDTH ON THE 0.4% VOLUMES FOR THE CASES WITH WALL

FIGURE B-6: INFLUENCE OF THE BERM WIDTH ON THE 0.4% VOLUMES FOR THE CASES WITH PARAPET

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

V1/2

50 [kg/m

]

0

20

40

60

80

100

120

140


Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

V1

/25

0 [kg/m

]

0

20

40

60

80

100

120

140



136

FIGURE B-7: COMPARISON BETWEEN PRED. FORMULAS WALL AND PARAPET

Rc/H

m0 [-]

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

Vd

im ,

1/2

50 [-]

0,0001

0,001

0,01

0,1

1

10

100

no berm wall20 cm berm wall40 cm berm wallall data parapetpred. formula no berm wallpred. formula 20 cm berm wallpred. formula 40 cm berm wallglob. pred. formula parapet

137

ANNEX C: WAVE FORCES

FIGURE C-1: INFLUENCE OF THE WATER DEPTH ON THE WAVE FORCE

FIGURE C-2: INFLUENCE OF THE STEEPNESS ON THE WAVE FORCE

Rc/H

m0 [-]

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

P [-]

0.01

0.1

1

10

100

water depth 70 cm

water depth 73 cm

water depth 76 cm

Rc/H

m0 [-]

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

P [-]

0.01

0.1

1

10

100

sm-1,0=0.03

sm-1,0=0.05

Annex C: Wave forces

138

FIGURE C-3: INFLUENCE OF THE BERM WIDTH ON THE WAVE FORCE

Rc/H

m0 [-]

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

P [-]

0.01

0.1

1

10

100

no berm

20 cm berm

40 cm berm

139

ANNEX D: TEST MATRIX

TABLE D-1: TEST MATRIX OF THE CONDUCTED TESTS

Configuration RC

[m]

dpaddle

[m]

Hs

[m]

TP

[s]

Lpaddle

[m]

Ltoe

[m]

Spaddle

[-]

Stoe

[-]

XAWA1-2

[m]

XGHM6-toe

[m]

XGHM1-2

[m]

XGHM2-3

[m]

XGHM1-3

[m] Test # Timeseries

no wall, no promenade 0.05 0.68 0.05 1.2 2.16 2.03 0.02 0.02 3.38 3.0 0.22 0.28 0.50 001A 001A












no wall, no promenade 0.03 0.7 0.05 0.8 1.00 1.00 0.05 0.05 3.38 3.0 0.16 0.16 0.32 003B 003A

no wall, no promenade 0.03 0.7 0.05 0.8 1.00 1.00 0.05 0.05 3.38 3.0 0.16 0.16 0.32 003C 003A

no wall, no promenade 0.03 0.7 0.05 0.8 1.00 1.00 0.05 0.05 3.38 3.0 0.16 0.16 0.32 003D 003B

no wall, no promenade 0.03 0.7 0.15 2.2 5.20 4.53 0.03 0.03 3.50 3.0 0.51 0.39 0.90 007B 007A

no wall, no promenade 0.03 0.7 0.15 2.2 5.20 4.53 0.03 0.03 3.50 3.0 0.51 0.39 0.90 007C 007A

no wall, no promenade 0.03 0.7 0.15 2.2 5.20 4.53 0.03 0.03 3.50 3.0 0.51 0.39 0.90 007D 007B

no berm, vertical wall 5 cm 0.08 0.7 0.15 1.4 2.81 2.57 0.05 0.06 3.38 3.0 0.28 0.32 0.60 0013A 0010A





Annex D: Test Matrix

140

Configuration RC

[m]

dpaddle

[m]

Hs

[m]

TP

[s]

Lpaddle

[m]

Ltoe

[m]

Spaddle

[-]

Stoe

[-]

XAWA1-2

[m]

XGHM6-toe

[m]

XGHM1-2

[m]

XGHM2-3

[m]

XGHM1-3









no berm, parapet 5 cm 0.08 0.7 0.15 2.2 5.20 4.53 0.03 0.03 3.50 3.0 0.51 0.39 0.90 0025A 007A




















141

Configuration RC

[m]

dpaddle

[m]

Hs

[m]

TP

[s]

Lpaddle

[m]

Ltoe

[m]

Spaddle

[-]

Stoe

[-]

XAWA1-2

[m]

XGHM6-toe

[m]

XGHM1-2

[m]

XGHM2-3

[m]

XGHM1-3



















berm 20 cm, parapet 10 cm 0.13 0.7 0.05 0.8 1.00 1.00 0.05 0.05 3.38 2.8 0.19 0.41 0.60 0061A 003A










142

Configuration RC

[m]

dpaddle

[m]

Hs

[m]

TP

[s]

Lpaddle

[m]

Ltoe

[m]

Spaddle

[-]

Stoe

[-]

XAWA1-2

[m]

XGHM6-toe

[m]

XGHM1-2

[m]

XGHM2-3

[m]

XGHM1-3





berm 20 cm, parapet 10 cm 0.07 0.76 0.15 1.4 2.85 2.66 0.05 0.06 3.38 2.8 0.28 0.32 0.60 0067B 0022A

berm 20 cm, vertical wall 10 cm 0.13 0.7 0.15 2.2 5.20 4.53 0.03 0.03 3.50 3.0 0.51 0.39 0.90 0073A 007A




















berm 20 cm, parapet 5 cm 0.08 0.7 0.1 1.1 1.86 1.78 0.05 0.06 3.38 3.0 0.19 0.41 0.60 0091C 0011A

berm 20 cm, parapet 5 cm 0.08 0.7 0.1 1.1 1.86 1.78 0.05 0.06 3.38 3.0 0.19 0.41 0.60 0091D 0011B


143

Configuration RC

[m]

dpaddle

[m]

Hs

[m]

TP

[s]

Lpaddle

[m]

Ltoe

[m]

Spaddle

[-]

Stoe

[-]

XAWA1-2

[m]

XGHM6-toe

[m]

XGHM1-2

[m]

XGHM2-3

[m]

XGHM1-3





























144

Configuration RC

[m]

dpaddle

[m]

Hs

[m]

TP

[s]

Lpaddle

[m]

Ltoe

[m]

Spaddle

[-]

Stoe

[-]

XAWA1-2

[m]

XGHM6-toe

[m]

XGHM1-2

[m]

XGHM2-3

[m]

XGHM1-3




berm 40 cm, vertical wall 5 cm 0.05 0.73 0.15 2.2 5.29 4.65 0.03 0.03 3.50 3.0 0.51 0.39 0.90 00119B 0018A

berm 40 cm, vertical wall 5 cm 0.05 0.73 0.15 2.2 5.29 4.65 0.03 0.03 3.50 3.0 0.51 0.39 0.90 00119C 0018A

berm 40 cm, vertical wall 5 cm 0.05 0.73 0.15 2.2 5.29 4.65 0.03 0.03 3.50 3.0 0.51 0.39 0.90 00119D 0018B























145

Configuration RC

[m]

dpaddle

[m]

Hs

[m]

TP

[s]

Lpaddle

[m]

Ltoe

[m]

Spaddle

[-]

Stoe

[-]

XAWA1-2

[m]

XGHM6-toe

[m]

XGHM1-2

[m]

XGHM2-3

[m]

XGHM1-3






berm 40 cm, parapet 10 cm 0.07 0.76 0.15 2.2 5.37 4.75 0.03 0.03 3.50 3.0 0.51 0.39 0.90 00143C 0024A

berm 40 cm, parapet 10 cm 0.07 0.76 0.15 2.2 5.37 4.75 0.03 0.03 3.50 3.0 0.51 0.39 0.90 00143D 0024B














Documents

Update on the use of parapets at storm walls for wave