Wave overtopping of coastal structures in case studies ...lib.ugent.be/fulltxt/RUG01/002/224/511/RUG01... · Wave overtopping of coastal structures in case studies using the EurOtop

Jan Axters

using the EurOtop Design ManualWave overtopping of coastal structures in case studies

Academic year 2014-2015Faculty of Engineering and ArchitectureChairman: 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

Supervisors: Prof. dr. ir. Andreas Kortenhaus, Prof. dr. ir. Peter Troch

De auteur(s) geeft (geven) de toelating deze masterproef voor consultatie beschikbaar te stellen

en delen van de masterproef te kopiëren voor persoonlijk gebruik. Elk ander gebruik valt onder

de bepalingen van het auteursrecht, in het bijzonder met betrekking tot de verplichting de bron

uitdrukkelijk te vermelden bij het aanhalen van resultaten uit deze masterproef.

The author(s) gives (give) permission to make this master dissertation available for consultation

and to copy parts of this master dissertation for personal use. In the case of any other use, the

copyright terms have to be respected, in particular with regard to the obligation to state expressly

the source when quoting results from this master dissertation.

Gent, 19/01/2015

Jan Axters

Preface After five years of hard work, it has finally come to an end with this master dissertation. I had

a very enjoyable student period and I am sad that it will be over soon. I’ve had my ups and downs

but when submitting this report, the only things I am going to remember are the good things. One

of these good things was doing this report. I enjoyed making it and I was very motivated by the

subject.

I would also like to thank my mother for supporting me during my years in college, both financially

and mentally. Without you, this would not have been possible for. When I did not believe in myself,

you did.

Special gratitude goes to professor Troch for allowing me to do this thesis subject and giving me

this chance to finish it. This also true for professor Kortenhaus who was also my guidance during

the thesis and always available for questions and a quick response.

My friends, and my girlfriend, I would also like to give credit as they were my much needed

entertainment during this report. It were moments with them that always refreshed my

motivation when it was low.

Wave overtopping of coastal structures in case

studies using the EurOtop design manual

Jan Axters

Counsellors: Prof. dr. ir. Andreas Kortenhaus, Prof. dr. ir. Peter Troch

Abstract - This article is about using the EurOtop design manual

in order to calculate overtopping at several case studies. The

results are compared with other calculations methods such as PC-

Overtopping and the Neural Network.

Keywords - EurOtop, waves, overtopping, coastal structures,

PC-Overtopping, Neural Network, Wenduine

I. INTRODUCTION

Overtopping is a very important topic nowadays. Recent

cases of lethal dike breaches and unaware pedestrians being

swept away by overtopping waves, together with the threat of

global warming causing higher water levels and more severe

storms, it is very important to have capable defense structures.

Structures can be considered safe when they limit the

overtopping so that no damage can happen to the surroundings

and structures itself, as well as prevent loss of life for people in

the surroundings. To create such a safe structure, the

overtopping must be evaluated. This can be done with several

methods. One of these methods is the use of the EurOtop design

manual.

In this report, case studies mentioned in EurOtop are

calculated with the equations from this manual and compared

with the results from other calculation methods. This includes

results from the updated formulae from Van der Meer et al [1],

the software package PC-Overtopping (PCO), and the Neural

Network (NN).

The calculations details are not given in this report as the

length would be unnecessarily large. Instead, each case is

briefly explained in chapter 2, what kind of structure is

assumed, if there were any difficulties and so on. In chapter 3

the results are then compared and the report ends in chapter 4

with some conclusions.

II. CASE STUDIES

A. Case A

Case A [1] is a simple slope, consisting of 2 slopes with

different slope angles, both covered in a rip-rap cover. It is clear

that this structure is thus calculated according to a composite

slope structure, which is done by finding the composite slope

angle. The roughness factor 𝛾𝑓 is found to be 0.6. The waves

are considered breaking in all wave conditions. No difficulties

or special considerations are found for this case.

Figure 1 Cross section case A

B. Case B

The structure in case B [1] is somewhat complex. It is a beach

with slope angle of 1:10 and has a vertical wall with a bullnose

on the end. The beach has a healthy and an eroded condition.

Some problems are found here as the simple slope equation

from EurOtop cannot be applied to the cases with a water level

below the toe of the vertical structure. This is because the beach

has a too mild slope angle (<1:8) and thus outside the

application range of this equation. Instead, the equation for

emergent vertical structures is chosen. Some boundary

conditions for this equation are also not met, but they are only

mildly exceeded so that the use is accepted. For water levels

which do reach the vertical structure toe, the equations for a

vertical structure are applied.

Figure 2 Cross section Case B

C. Case C

The structure type of case C [1] depends again on the water

level. The structure consists of a simple slope, ending in a

vertical wall. The slope is rather short and steep, having an

angle of around 1:2. The vertical wall has a recurved form.

When the water levels are below the toe of the vertical

structure, the structure is considered a simple slope with a wave

wall on top. The slope is covered in steps and a roughness factor

of 0.6 is picked [2][3]. The wave conditions for these are these

water levels are assumed to be non-breaking according to the

breaker parameter.

For water levels reaching above the vertical structure, it is

seen as a composite structure with d the water height above the

toe and h the water height before the structure. The

impulsiveness parameter 𝑑∗ lead to non-impulsive conditions.

Figure 3 Cross section Case C

D. Case D

Case D [1] is again a simple slope with just one slope angle.

It is covered in grass so that no roughness is considered. The

wave conditions are all have breaker parameters below the

wave breaking threshold. No difficulties are found here for this

case.

Figure 4 Cross section case D

E. Case E

The following case, case E [1], is a composite sloping dike

consisting of several slopes with different angles. 2 slopes have

a slope angle which cannot be considered a berm (<1:15) or a

slope (>1:8). This thus requires interpolation. However, as

EurOtop is not clear on how to interpolate in case of multiple

of these intermediate slopes, only one slope is interpolated, the

first promenade (1:10). The other intermediate slope, the

second promenade (1:11), is just considered a slope.

The structure is thus first calculated entirely as a slope,

without the influence factor for a berm. After this, the structure

is calculated according to the presence of a berm. This has an

effect on the influence factors for total roughness and berm.

Then the results are interpolated to achieve the final result.

Wave heights are relatively high for this case with long wave

periods. Together with the mild slope, the wave conditions are

just beneath wave breaking threshold.

Figure 5 Cross section case E

F. Case F

Case F [1] is a clear case of a composite vertical structure.

The toe structure is well defined and all wave conditions have

a SWL above the toe. The relevant wave conditions are

considered impulsive. This leads to the use for impulsive waves

at composite structures in EurOtop. Only one wave condition,

does not meet the application range for this equation but is not

relevant as this wave conditions has the lowest SWL and wave

height. No other difficulties are present for this case.

Figure 6 Cross section case F

G. Case M

The last case in EurOtop, case M [1] is a steep armoured

slope. It is comprised out of 2 parts which have the same steep

angle of 1:2. Between these 2 parts, a berm with a walkway is

present. Overtopping is calculated at this walkway as well as at

the crest at the end of the 2nd part. The armour consists of two

layers on top of a permeable core, leading to a roughness factor

of 0.4. Due to the steep slope, wave conditions are considered

non-breaking.

Figure 7 Cross section case M

H. Case Wenduine

Another case outside those of EurOtop is picked. This case is

the dike of Wenduine in Belgium [4][5]. There are two

structures discussed: a current structure and a future one. The

future structure is being built right now in order of the Belgian

government in order to strengthen the Belgian coast.

The current structure is a long beach run-up with a steep

slope at the end. At the end of this dike, a wide crest is present.

The future structure has the promenade at the crest extended

and a vertical wall built instead of a steep dike. A parapet wall

is place on top of this vertical structure, increasing the

freeboard further.

The wave conditions are very extreme for this case as

1:17000 year storm is considered.

The current structure is calculated as a simple slope, but due

to the slope angles of the beach and promenade, both <<1:8,

and the high breaker parameter, no berm influence factors can

be included. This means that EurOtop does not hold into

account the beach in front of the slope, and the promenade after

the dike.

The future structure is considered a vertical structure and not

a composite one.

Figure 8 Cross section current situation Wenduine

Figure 9 Cross section future situation Wenduine

III. RESULTS

The results are summarized in Table 1Table 1 Overtopping

results for worst wave conditions and Figure 10. Only the worst

wave conditions are considered in order to have accessible

comparisons.

Table 1 Overtopping results for worst wave conditions

Case 𝑅𝑐 𝐻𝑚0[𝑚] 𝐿𝑚−1,0[𝑚] 𝑞[𝑙/𝑠/𝑚] 𝑞𝑉𝑑𝑀[𝑙/𝑠/𝑚] 𝑞𝑃𝐶𝑂[𝑙/𝑠/𝑚] 𝑞𝑁𝑁[𝑙/𝑠/𝑚]

A 0.56 0.72 15.81 0.515 0.505 0.93 0.855

B 1.48 0.95 23.75 0.504 15 - 0.678

C 2.9 2.00 120.91 3.310 3.84 - 3.97

D 1.38 1.93 26.13 0.454 0.395 0.913 15

E 5 3.5 290.33 37.4 39 90.8 39.5

F 5.69 2.51 44.5 6.8 4.31 - 2.86

M 2.65 1.29 45.53 0.039 0.0077 0.311 0.174

W. current 0.44 4.97 126.47 5514 2717 - -

W. future 1.14 4.97 126.47 3442 2167 - -

Figure 10 Overtopping results for worst wave conditions

It is clear that from table 1 that the results for Wenduine are

too large and they would skew Figure 10 if they are put in the

same graph. Therefore, they are left out of this figure.

It is clear from Figure 10 and Table 1 that the overtopping

results are mostly limited to around 0.5 to 7 l/s/m and thus fall

into a safe regime for structure damage and pedestrians, except

for case E and for Wenduine. Case E has rather larger

overtopping but could still be considered viable when

precautions are taken to protect the structure and pedestrians.

However, for Wenduine, both the current and future structure,

an abnormal high value is found for the overtopping. As

mentioned before, the beach and promenade are not included in

the calculation process albeit skewing the results. Model tests

where these items are included, show a much smaller

overtopping, around 500 l/s/m [4][5]. This is still high so that

this location can be considered dangerous.

Another thing the results show is that the different methods

often produce a varying overtopping result. This is especially

true for the cases B and Wenduine future, where a beach is

present with a vertical wall at the end. An explanation for these

cases might be that the use of the impulsiveness parameter ℎ∗ in the equations are prone to mistakes.

IV. CONCLUSIONS

Overtopping is difficult to describe in an analytical manner.

Therefore it’s hard to predict the overtopping correctly. The use

of EurOtop in the case studies showed this. The results are often

very different compared to other methods which rely on other

calculation principles. This unreliability of the results often

increases when the complexity of the structure increases.

Structure elements such as a second wave wall, or a wide crest

cannot be modelled with EurOtop together with a number of

other structure types. For this reasons, it is advised that further

research time is spent on the improvement of the equations

from EurOtop. The papers from Van der Meer et al are a first

step towards this, but these papers still lack in some

departments, such as influence factors or emergent vertical

structures. EurOtop is thus not a trustable source when

designing and fine tuning defense structures. It does however

excel in predicting simple structures and can be used to quickly

calculate and compare different kinds of structures.

REFERENCES,

[1] Pullen, T., Allsop, N.W.H., Bruce, T., Kortenhaus, A., Schüttrumpf, H., van der Meer, J.W. 2007. EurOtop, Wave Overtopping of Sea Defences

and Related Structures: Assessment Manual, [online] available at

www.overtopping-manual.com [2] Xiaomin, W., Liehong, JU., Treuel, F.M. 2013. The study on wave run-

up roughness and permeability coefficient of stepped slope dike,

Proceedings of the 7th International Conference on Asian and Pacific

Coasts, Bali, Indonesia.

[3] Hughes, S. 2005. Estimating irregular wave run-up on rough,

impermeable slopes, ERDC/CHL CHETN-III-70, U.S. Army Engineer Research and Development Center, Vicksburg, MS.

[4] Veale, W., Suzuki, T., Verwaest, T., Trouw, K., Mertens, T. 2012.

Integrated design of coastal protection works for Wenduine, Belgium. Coastal Engineering Proceedings, 1(33), structures.70.

[5] Baelus, L. & De Rouck, J. 2012. Fysische schaalmodelproeven

overtopping reducerende maatregelen promenade Wenduine, Rapportnummer OPW320/005, Vakgroep Civiele Techniek, UGent &

Afdeling kust, Vlaamse overheid.

Table of contents

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

1.1 Background ........................................................................................................................................................ 1

1.2 Goal & scope of this report .......................................................................................................................... 2

1.3 Structure .............................................................................................................................................................. 3

Chapter 2 Literature ................................................................................................................................................... 4

2.1 Hydrodynamics ................................................................................................................................................ 4

2.1.1 Wave Height ....................................................................................................................................... 4

2.1.2 Wave period ....................................................................................................................................... 4

2.1.3 Wavelength ......................................................................................................................................... 4

2.1.4 The Iribarren number .................................................................................................................... 5

2.1.5 Water level, tides, and surges ...................................................................................................... 5

2.1.6 Wave run-up ....................................................................................................................................... 6

2.1.7 Wave overtopping ............................................................................................................................ 7

2.1.8 Statistical data ................................................................................................................................... 8

2.2 Calculation tools ............................................................................................................................................... 9

2.2.1 EurOtop ................................................................................................................................................ 9

2.2.2 van der Meer et al: Updated EurOtop formulae................................................................ 22

2.2.3 PC-Overtopping .............................................................................................................................. 29

2.2.4 Neural Network .............................................................................................................................. 30

2.2.5 CLASH Database ............................................................................................................................. 31

2.3 Methodology ................................................................................................................................................... 32

Chapter 3 Results ...................................................................................................................................................... 33

3.1 Case A ................................................................................................................................................................ 33

3.1.1 Cross section and info.................................................................................................................. 33

3.1.2 EurOtop calculations .................................................................................................................... 34

3.1.3 van der Meer et al: updated EurOtop .................................................................................... 36

3.1.4 PC-Overtopping .............................................................................................................................. 37

3.1.5 Neural network .............................................................................................................................. 38

3.1.6 CLASH-database ............................................................................................................................. 38

3.1.7 Summary ........................................................................................................................................... 39

3.2 Case B................................................................................................................................................................. 41


3.2.2 EurOtop ............................................................................................................................................. 42

3.2.3 van der Meer et al: updated EurOtop .................................................................................... 48

3.2.4 PC-Overtopping .............................................................................................................................. 49

3.2.5 Neural Network .............................................................................................................................. 49

3.2.6 Clash Database................................................................................................................................ 50

3.2.7 Summary ........................................................................................................................................... 50

3.3 Case C ................................................................................................................................................................. 53


3.3.2 EurOtop ............................................................................................................................................. 55

3.3.3 van der Meer et al: Updated EurOtop ................................................................................... 59

3.3.4 PC-Overtopping .............................................................................................................................. 60

3.3.5 Neural network .............................................................................................................................. 60

3.3.6 CLASH Database ............................................................................................................................. 60

3.3.7 Summary ........................................................................................................................................... 61

3.4 Case D ................................................................................................................................................................ 63


3.4.2 EurOtop ............................................................................................................................................. 64


3.4.4 PC-Overtopping .............................................................................................................................. 65

3.4.5 The Neural Network ..................................................................................................................... 66

3.4.6 CLASH ................................................................................................................................................. 66

3.4.7 Summary ........................................................................................................................................... 66

3.5 Case E ................................................................................................................................................................. 68


3.5.2 EurOtop ............................................................................................................................................. 69


3.5.4 PC-Overtopping .............................................................................................................................. 74

3.5.5 Neural Network .............................................................................................................................. 76

3.5.6 CLASH ................................................................................................................................................. 77

3.5.7 Summary ........................................................................................................................................... 77

3.6 Case F ................................................................................................................................................................. 79


3.6.2 EurOtop ............................................................................................................................................. 81


3.6.4 PC-Overtopping .............................................................................................................................. 83

3.6.5 Neural Network .............................................................................................................................. 83

3.6.6 CLASH database ............................................................................................................................. 84

3.6.7 Summary ........................................................................................................................................... 84

3.7 Case M................................................................................................................................................................ 86


3.7.2 EurOtop ............................................................................................................................................. 87


3.7.4 PC-Overtopping .............................................................................................................................. 90

3.7.5 Neural Network .............................................................................................................................. 92

3.7.6 CLASH database ............................................................................................................................. 92

3.7.7 Summary ........................................................................................................................................... 93

3.8 Case study Wenduine .................................................................................................................................. 95


3.8.2 EurOtop ............................................................................................................................................. 97


3.8.4 PC-Overtopping ........................................................................................................................... 101

3.8.5 Neural Network ........................................................................................................................... 101

3.8.6 CLASH .............................................................................................................................................. 101

3.8.7 Summary ........................................................................................................................................ 102

Chapter 4 Discussion ............................................................................................................................................ 105

4.1 Comparison of the results ...................................................................................................................... 105

4.2 Safety of the structures ........................................................................................................................... 108

4.3 Use of the EurOtop manual and van der Meer et al ..................................................................... 109

Chapter 5 Summary .............................................................................................................................................. 111

Chapter 6 References ........................................................................................................................................... 113

Chapter 7 Annexes ................................................................................................................................................. 116

List of Figures

Figure 1 Waves hitting the beach in Wales (Jem Rowland) ....................................................................... 1

Figure 2 Wave wall failure and the subsequent damage at Dawlish, February 2014 ..................... 2

Figure 3 Types of wave breaking (FHWA) ........................................................................................................ 5

Figure 4 Wave run-up on a simple slope............................................................................................................ 6

Figure 5 Overtopping on a slope ........................................................................................................................... 7

Figure 6 Wave run-up for smooth and straight slopes (EurOtop 2007) ........................................... 10

Figure 7 Defining a composite slope for structure with different slope angles .............................. 10

Figure 8 Transforming an angled berm to a horizontal berm ................................................................ 12

Figure 9 Definition of wave attack angle relative to the structure ....................................................... 12

Figure 10 Wave overtopping data for breaking waves and overtopping equation 8 with 5%

lower and 95% upper exceedance limit (EurOtop 2007) .............................................................................. 13

Figure 11 Run-up level and location for overtopping differ (EurOtop 2007) ................................. 15

Figure 12 Non-impulsive waves above, impulsive waves below (EurOtop 2007) ....................... 15

Figure 13 Definition of vertical structure parameters .............................................................................. 16

Figure 14 Overtopping at a plain vertical wall under non-impulsive conditions (EurOtop 2007)

................................................................................................................................................................................................ 17

Figure 15 Overtopping at a plain vertical wall under impulsive conditions, equation 18.a

(EurOtop 2007) ............................................................................................................................................................... 18

Figure 16 Overtopping at a plain vertical wall under impulsive conditions and broken waves

for ℎ ∗ 𝑅𝑐𝐻𝑚0 < 0.02, equation 18.b; the area between ℎ ∗ 𝑅𝑐𝐻𝑚0 = 0.02 and ℎ ∗ 𝑅𝑐𝐻𝑚0 = 0.03

is vague and both extrapolated equations 18.a and 18.b are shown (EurOtop 2007) .................... 18

Figure 17 Overtopping for emergent vertical structures (EurOtop 2007) ...................................... 19

Figure 18 Impulsive overtopping on composite structures (EurOtop 2007) ................................. 20

Figure 19 Definition of parameters for recurve/parapet wall and a decision chart on how to

calculate the reducing factor 𝑘 for these walls (EurOtop 2007) ................................................................. 21

Figure 20 Overtopping equation 3 from EurOtop plotted together with the proposed formula

from Battjes/TAW (van der Meer et al 2013) ..................................................................................................... 22

Figure 21 A fitted Weibull distribution for vertical walls based on experimental data, equation

24 , under non-breaking wave conditions (van der Meer et al 2013) ...................................................... 23

Figure 22 Fitted Weibull distributions for overtopping on steep slopes with different angles ,

under non-breaking wave conditions (van der Meer et al 2013) ............................................................... 24

Figure 23 The values of parameters a and b in the Weibull distribution for overtopping on steep

slopes with different angles, under non-breaking wave conditions and c=1.3, equation 25 (van

der Meer et al 2013) ...................................................................................................................................................... 24

Figure 24 Representation of equation 28 and Allsop et al (1995) and Franco et al (1994) ‘s

interpretation of equation 29; Left CLASH dataset 802 is plotted: a vertical structure with

foreshore; Right CLASH dataset 914 is plotted: a vertical structure without a foreshore in relative

deep water (Bruce et al 2013) ................................................................................................................................... 26

Figure 25 Vertical structures on relatively deep water, no sloping foreshore (Bruce et al 2013)

................................................................................................................................................................................................ 26

Figure 26 All data of seawalls on sloping foreshore for impulsive waves, and with optimim

values of 𝑎 = 0.5 and 𝑏 = −0.5. The 5% and 95% exceedance lines are plotted as dotted lines. For

lower freeboards, the exponential equation 32.a is valid while for larger freeboards, the power

curve is valid, equation 32.b (Bruce et al 2013) ................................................................................................ 27

Figure 27 Comparison of overtopping at composite and plain vertical structures for non-

impulsive waves as well as for impulsive waves (Bruce et al 2013) ........................................................ 28

Figure 28 Decision chart showing new schemes; vertical to left; composite to right (Bruce et al

2013) .................................................................................................................................................................................... 29

Figure 29 Possible structures which can be modelled with the Neural Network (EurOtop 2007)

................................................................................................................................................................................................ 31

Figure 30 Simplified cross section of case A .................................................................................................. 33

Figure 31 Wave directions for case A, with the dotted line as the normal to the structure ...... 34

Figure 32 Composite slope in red, with the original structures in black, and the still water line

in blue .................................................................................................................................................................................. 35

Figure 33 PCO input for case A, wave condition 4 ...................................................................................... 37

Figure 34 PCO output for case A, wave condition 4.................................................................................... 37

Figure 35 Different overtopping results for case A .................................................................................... 39

Figure 36 Wave run-up for smooth and straight slopes: breaking waves, case A ......................... 40

Figure 37 Overtopping for smooth and straight slopes: breaking waves, case A .......................... 40

Figure 38 Simplified cross section case B ....................................................................................................... 41

Figure 39 Extrapolation of the dike in case of eroded conditions and when the water level

reaches the vertical wall .............................................................................................................................................. 42

Figure 40 Results of different type of structures for case B, not considering the parapet and

wave obliquity .................................................................................................................................................................. 45

Figure 41 Overtopping according to different structure interpretations with wave height

𝐻𝑚0 = 0.95𝑚, wave period 𝑇𝑚 − 1,0 = 0.93𝑠, slope 1:12.5, vertical wall at 4.2mODN. Grey is

overtopping for a simple slope with a vertical wall factor; Blue is assuming a vertical wall with ℎ ∗;

Green is the emergent wall with foreshore slope 1:12.5 ................................................................................ 46

Figure 42 Recurve modelling in Neural Network ........................................................................................ 49

Figure 43 Overtopping results for case B ....................................................................................................... 51

Figure 44 Overtopping for emergent vertical wall structures for Case B ......................................... 51

Figure 45 Overtopping for vertical structures, impulsive condition, Case B ................................... 52

Figure 46 Cross section Case C ............................................................................................................................ 53

Figure 47 Simplified cross section Case C....................................................................................................... 54

Figure 48 Comparison between a composite structure or a simple slope, for case C, calculated

with EurOtop, without recurve factor. ................................................................................................................... 58

Figure 49 Overtopping results for case C ........................................................................................................ 61

Figure 50 Overtopping for smooth and straight slopes, non-breaking waves, with results from

case C .................................................................................................................................................................................... 62

Figure 51 Overtopping for vertical walls, non-impulsive conditions, with results from case C

................................................................................................................................................................................................ 62

Figure 52 Cross section for case D ..................................................................................................................... 63

Figure 53 Input example for PCO for case D wave condition 1 ............................................................. 65

Figure 54 Output PCO for case D wave condition 1 ................................................................................... 65

Figure 55 Overtopping results for case D ....................................................................................................... 67

Figure 56 Run-up for smooth and sloping dikes with results from case D ....................................... 67

Figure 57 Overtopping for smooth and sloping dikes, breaking waves, with results from case D

................................................................................................................................................................................................ 68

Figure 58 Cross section of case E ....................................................................................................................... 69

Figure 59 Making the first promenade a horizontal berm for calculation purposes .................... 71

Figure 60 The structure with berm (solid black), without berm ( dashed black), and the

composite slope (red) ................................................................................................................................................... 71

Figure 61 PCO input example for case E, wave condition 1 .................................................................... 75

Figure 62 Preparing the cross section for NN input ................................................................................... 77

Figure 63 Overtopping results for case E........................................................................................................ 78

Figure 64 Run –up for smooth and sloping dikes with results from case E ..................................... 78

Figure 65 Overtopping for smooth and sloping dikes, breaking waves, with results from case E

................................................................................................................................................................................................ 79

Figure 66 Cross section case F ............................................................................................................................. 80

Figure 67 Simplified cross section of case F .................................................................................................. 80

Figure 68 Overtopping results for case F ........................................................................................................ 85

Figure 69 Overtopping for composite vertical structures, impulsive breaking conditions with

results from case F .......................................................................................................................................................... 85

Figure 70 Cross section for case M .................................................................................................................... 86

Figure 71 PCO input for case M, wave condition 1 at first overtopping location ........................... 90

Figure 72 PCO input for case M, wave condition 1 at second overtopping location ..................... 91

Figure 73 Overtopping results for case M ...................................................................................................... 93

Figure 74 Run-up for smooth and simple slopes with results from case M ..................................... 94

Figure 75 Overtopping for smooth and sloping dikes, non-breaking waves, with results from

case M ................................................................................................................................................................................... 94

Figure 76 Current situation Wenduine cross section ................................................................................ 95

Figure 77 Cross section of future dike of Wenduine .................................................................................. 97

Figure 78 Overtopping results for case Wenduine .................................................................................. 102

Figure 79 Overtopping for smooth and simple slopes, non-breaking waves, with results from

case Wenduine, current situation ......................................................................................................................... 103

Figure 80 Overtopping for vertical structures, impulsive conditions, breaking waves, with

results from case Wenduine, future situation .................................................................................................. 103

Figure 81 graphical comparison of overtopping results for the worst wave conditions ......... 105

List of tables

Table 1 Reference water levels .............................................................................................................................. 6

Table 2 Consequences of overtopping discharge rates on defence structures .................................. 7

Table 3 Input parameters for the Neural Network ..................................................................................... 30

Table 4 Wave conditions for case A, 1:1000 year joint probability ..................................................... 33

Table 5 Influence factors for case A, calculated with EurOtop .............................................................. 34

Table 6 Run-up and overtopping for case A, calculated with EurOtop .............................................. 36

Table 7 Run-up and overtopping for case A, calculated with van der Meer et al (2013) ........... 36

Table 8 Run-up and overtopping for case A, calculated with PCO ....................................................... 38

Table 9 Overtopping for case A, calculated with Neural Network, and the input parameters . 38

Table 10 Case A wave condition 3 different methods ............................................................................... 40

Table 11 Wave conditions case B for a 1:200 year joint probability function ................................ 41

Table 12 Boundary conditions for case B, equation 18 ............................................................................ 44

Table 13 Boundary conditions for emergent wall structure for case B ............................................. 44

Table 14 Reduction factor for bullnose wall for case B ............................................................................ 47

Table 15 Results for case B using EurOtop, Emergent structures and Vertical wall structure 47

Table 16 Wave breaking according to van der Meer et al for Case B .................................................. 48

Table 17 Overtopping results for case B calculated with van der Meer et al updated equations

for vertical structures .................................................................................................................................................... 49

Table 18 Neural Network Case B ........................................................................................................................ 49

Table 19 Overtopping for Case B ........................................................................................................................ 50

Table 20 future climate change conditions for the worst current conditions ................................. 50

Table 21 Overtopping results for climate change conditions applied to the worst current

calculated cases................................................................................................................................................................ 50

Table 22 Wave conditions case C ....................................................................................................................... 54

Table 23 Boundary conditions on the use of a vertical wall factor ...................................................... 55

Table 24 Results for 𝑅𝑢2% and 𝑅𝑢2%, 𝑚𝑎𝑥 for case C ............................................................................ 56

Table 25 EurOtop results for case B assuming a simple slope ............................................................... 56

Table 26 Final overtopping results using EurOtop for case C, assuming composite wall for water

levels reaching the vertical wall, calculated at the recurved wall .............................................................. 58

Table 27 Overtopping results using the updated formulae from van der Meer et al for case C,

calculated at the recurved wall ................................................................................................................................. 59

Table 28 Overtopping results using PCO for case C, calculated at recurved wall .......................... 60

Table 29 Overtopping results for worst case scenario for case C ......................................................... 63

Table 30 Wave conditions for case D ................................................................................................................ 63

Table 31 Overtopping results using EurOtop for case D .......................................................................... 64

Table 32 Overtopping results using updated formulae from van der Meer et al for case D ...... 64

Table 33 Overtopping results using PCO for case D ................................................................................... 66

Table 34 Overtopping results using NN for case D ..................................................................................... 66

Table 35 Worst case conditions for case D, overtopping ......................................................................... 68

Table 36 Wave conditions for case E, return period unknown ............................................................. 69

Table 37 Values influence factors for case E, assuming no berms, after 6 iterations ................... 70

Table 38 Overtopping results for case E, using EurOtop, assuming no berms ............................... 70

Table 39 Influence factors for case E, assuming presence of berm, after 6 iterations ................. 72

Table 40 Overtopping results for case E, using EurOtop, assuming presence of berm ............... 73

Table 41 Overtopping results for case E, using EurOtop, interpolated values ................................ 73

Table 42 Overtopping results for case E, using updated formulae from van der Meer et al,

assuming no berm .......................................................................................................................................................... 74


assuming presence of a berm ..................................................................................................................................... 74


assuming the interpolated values ............................................................................................................................ 74

Table 45 Influence factors for case E, assuming no berms ...................................................................... 75

Table 46 Influence factors for case E, assuming presence of berm ..................................................... 75

Table 47 Overtopping results for case E, using PCO, assuming no berm. ......................................... 75

Table 48 Overtopping results for case E, using PCO, assuming presence of berm ........................ 75

Table 49 Overtopping results for case E, using PCO, assuming interpolated values .................... 76

Table 50 Overtopping results for case E, using NN .................................................................................... 77

Table 51 Worst case conditions for case E, overtopping .......................................................................... 79

Table 52 Wave conditions for case F, unknown return period.............................................................. 80

Table 53 Composite impulsiveness parameter and equation 20 boundary conditions for case F

wave conditions ............................................................................................................................................................... 81

Table 54 Overtopping results for case F, using EurOtop .......................................................................... 82

Table 55 Overtopping results for case F, using updated formulae from van der Meer et al ..... 83

Table 56 Overtopping results for case F, using NN ..................................................................................... 83

Table 57 Worst case conditions for overtopping for case F .................................................................... 85

Table 58 Wave conditions for case M ............................................................................................................... 86

Table 59 Influence factors for overtopping at the first location for case M according to EurOtop

................................................................................................................................................................................................ 88

Table 60 Overtopping results for case M, using EurOtop, at the first overtopping location ..... 88

Table 61 Influence factors for overtopping at the second location for case M according to

EurOtop ............................................................................................................................................................................... 89

Table 62 Overtopping results for case M, using EurOtop, at the second overtopping location89

Table 63 Overtopping results for case M, using updated formulae from van der Meer et al, at

the first overtopping location .................................................................................................................................... 89

Table 64 Overtopping results for case M, using updated formulae from van der Meer et al, at

the second overtopping location .............................................................................................................................. 90

Table 65 Influence factors for overtopping at the first location for case M according to PCO . 91

Table 66 Influence factors for overtopping at the second location for case M according to PCO

................................................................................................................................................................................................ 91

Table 67 Overtopping results for case M, using PCO, at the first overtopping location .............. 91

Table 68 Overtopping results for case M, PCO, at the second overtopping location .................... 92

Table 69 Overtopping results for case M, using NN, at the first overtopping location ................ 92

Table 70 Overtopping results for case M, using NN, at the second overtopping location .......... 92

Table 71 Worst case conditions for case M at overtopping location 1 ............................................... 94

Table 72 Wave conditions for case Wenduine .............................................................................................. 95

Table 73 Current existing overtopping values, calculated at Flanders Hydraulic Lab and

UGent(2011) *: extrapolated value ........................................................................................................................ 96

Table 74 Overtopping values found with model tests on the future dike in Flanders Hydraulic

Lab and UGent *: measured with a higher wave height of 5.25m instead of 4.97m ........................... 97

Table 75 Overtopping results for Wenduine current situation, using EurOtop ............................. 98

Table 76 Reduction factors for the parapet wall in the future dike at Wenduine ......................... 99

Table 77 Overtopping results for future dike at Wenduine, using EurOtop .................................... 99

Table 78 Overtopping results for current dike at Wenduine, using updated formulae from van

der Meer et al .................................................................................................................................................................... 99

Table 79 Overtopping results for the future dike at Wenduine, using the updated formulae from

van der Meer et al ........................................................................................................................................................ 100

Table 80 Overtopping results for the future dike at Wenduine, using the updated formulae from

van der Meer et al, considering a composite slope ........................................................................................ 100

Table 81 Overtopping results for current situation Wenduine, using PCO ................................... 101

Table 82 Worst case wave conditions for each case ............................................................................... 105

Table 83 Comparing the EurOtop overtopping results with the overtopping limits, taken from

EurOtop ............................................................................................................................................................................ 108

List of symbols

𝐴𝑐 [𝑚] Armour crest freeboard of structure

𝐵 [𝑚] Berm width, measured horizontally

𝐵𝑡 [𝑚] Width of toe of structure

𝐵ℎ [𝑚] Width of horizontally schematised berm

𝐵𝑟 [𝑚] Width (seaward extension) in front of main vertical wall of recurve/

parapet/ wave return wall section

𝑔 [𝑚/𝑠²] Acceleration due to gravity (=9.81)

𝐺𝑐 [𝑚] Width of structure crest

ℎ [𝑚] Water depth at toe of structure

ℎ𝑏 [𝑚] Water depth on berm

ℎ𝑟 [𝑚] Height of recurve/ parapet/ wave return wall section

ℎ𝑡 [𝑚] Water depth on toe of structure

𝐻 [𝑚] Wave height

𝐻𝑠 [𝑚] Significant wave height

𝐻𝑚0 [𝑚] Estimate of significant wave height from spectral analysis

𝑘 [−] Multiplier for mean discharge giving effect of recurve wall

𝑘′, 𝑘23 [−] Dimensionless parameters used (only) in intermediate stage of

calculation of reduction factor for recurve walls

𝐿𝐵𝑒𝑟𝑚 [𝑚] Horizontal length between two points on slope, 1.0𝐻𝑚0 above and

1.0𝐻𝑚0 below middle of the berm

𝐿𝑆𝑙𝑜𝑝𝑒 [𝑚] Horizontal length between two points on slope, 𝑅𝑢2% above and 1.5𝐻𝑚0

below S.W.L.

𝐿 [𝑚] Wave length measured in direction of wave propagation

𝐿𝑚−1,0 [𝑚] Deep water wave length based on 𝑇𝑚−1,0

𝑚 [−] Slope of the foreshore, 1unit vertical corresponds to m units horizontal

𝑚∗, 𝑚 [−] Dimensionless parameters used (only) in intermediate stage of


𝑃𝑐 [𝑚] Height of vertical wall from SWL to bottom of recurve/ parapet/ wave

return wall section

𝑞 [𝑚3/𝑠/𝑚] Mean overtopping discharge per meter structure width

𝑅𝑐 [𝑚] Crest freeboard of structure

𝑅0∗ [−] Dimensionless length parameter used (only) in intermediate stage of


𝑅𝑢 [𝑚] Run-up level, vertical measured with respect to the SWL

𝑅𝑢2% [𝑚] Run-up level exceeded by 2% of incident waves

𝑠0 [−] Wave steepness, based on 𝑇𝑚−1,0 and 𝐿𝑚−1,0

𝑇 [𝑠] Wave period

𝑇𝑚−1,0 [𝑠] Wave period based on spectral analysis

𝑇𝑚 [𝑠] Average wave period

𝑇𝑝 [𝑠] Peak wave period

𝑉 [𝑚3/𝑚] Volume of overtopping wave per unit crest width

𝑋 [𝑚] Landward distance of falling overtopping jet from rear edge of wall

𝛼 [°] Angle between overall structure slope and horizontal

𝛼 [°] Angle of parapet/ recurve/ wave return wall above seaward horizontal

𝛼𝐵 [°] Angle that sloping berm makes with horizontal

𝛼𝑢 [°] Angle between structure slope upward berm and horizontal

𝛼𝑑 [°] Angle between structure slope downward berm and horizontal

𝛽 [°] Angle of wave attack relative to normal on structure

𝛾𝑏 [−] Correction factor for a berm

𝛾𝑓 [−] Correction factor for permeability and roughness of or on the slope

𝛾𝛽 [−] Correction factor for oblique wave attack

𝛾𝑣 [−] Correction factor for a vertical wall on the slope

𝜉𝑚−1,0 [−] Iribarren number or wave breaker parameter based on 𝑠0

𝜇(𝑥) [−] Mean of measured parameter X with normal distribution

𝜎(𝑥) [−] Standard deviation of measured parameter X with normal distribution

Chapter 1 -Background

1

Chapter 1 Introduction

1.1 Background

Wave overtopping has been a problem as early as people were living in coastal areas and it will

remain a problem as long as they stay there. Situations when a high tide is combined with large

waves, chances are, that some waves will overtop the structure protecting the mainland. This can

create serious risks for the environment, structures and people living there as can be clearly seen

in Figure 1. Unsuspecting people can be dragged away when a large wave hits the beach and a

large amount of overtopping crashes on the walkway.

Figure 1 Waves hitting the beach in Wales (Jem Rowland)

It is then obvious that the overtopping is an important topic in scientific research.

Hydrodynamic research can predict and analyse the amount of overtopping which occur. But not

only is this volume important, the structural engineering plays an important role as well. When

waves crash into a vertical wall, the structure is subjected to high loading in a very short time

burst, resulting in very large momentum. This can cause structural failure and loss of life as a

result. Not only the immediate surroundings are in danger, but entire provinces or even countries

can be at risk. When a dike breaks, flooding can occur which can result into high amount of

casualties. Such as was the case in 1953 when a storm hit the Low Lands and in 2005 in New

Orleans, USA, where hurricane Katrina struck. Both of these cases have a casualty number around

1800 and millions of euros in damage due to failure of structures and flooding. A more recent case,

is February 2014, when a storm hit Dawlish in the UK and destroyed the defence wall and

simultaneously swept away the railroad and buildings lying just behind the structure, Figure 2.

Chapter 1 -Goal & scope of this report

2

Figure 2 Wave wall failure and the subsequent damage at Dawlish, February 2014

With rising sea levels due to global warming, the risks are only getting bigger. Preventing these

risks cannot be 100% done but reducing them, however, is possible. Building defence structures

such as dikes or vertical walls, can reduce the overtopping by a considerable amount. Installing

specific elements, such as artificial roughness, berms, recurved walls, parapet wave walls… can

reduce the run-up and overtopping as well. How much they reduce this, and what the optimal

dimensions are, depends on a lot of factors and is the subject of many research topics today.

Several countries already have a manual which describes guidelines on how to dimension these

structures as well as predict the amount of overtopping. For example, the Netherlands used to

have the ‘Technical Report on wave run-up and wave overtopping at dike’ (TAW) edited by van

der Meer (2002), Germany had ‘Die Küste ‘(EAK) edited by Erchinger (2002), the UK the

‘Environment Agency Manual’ (EA) edited by Besley (1999), and the US had the ‘Coastal

Engineering Manual’ (CEM) edited by Pope (2012). The prediction methods described in these

manuals are based on several solution methods, such as empirical data, numerical models, and

neural networks.

In 2007, the EurOtop manual was released. This is a manual based on the TAW, EAK, and EA

from the Netherlands, Germany and the UK and the result of a cooperation of different European

countries. It is this manual that is used as main guidelines when calculating overtopping and

designing dikes nowadays in Europe and replaces the previously mentioned manuals.

1.2 Goal & scope of this report

The use of EurOtop is been spreading all over Europe since its release in 2007. It is a convenient

tool which has accurate results due to the background of several studies originating in different

countries. However, since 2007, new insights and research lead to new and updated formulas. In

this thesis, these new formulas are examined by evaluating them in different case studies. These

case studies are the original ones included in the EurOtop (2007). These case studies are not yet

elaborated or worked out in the manual released in 2007. For this reason, the first goal of the

Chapter 1 -Structure

3

thesis is to work out these case studies, so that they might be included in a next version of EurOtop.

The second goal is to compare the results with the new updated formulae and draw conclusions

on the use of the old and new formulae. Comparisons with the other methods such as PC-

Overtopping, Neural Network, and CLASH, are also included where they are applicable. The results

are only presented on wave run-up and overtopping. No information is given on the subjects of

velocities or wave forces. A third objective is to formulate advice on specific subjects on how to

improve EurOtop and its guidelines or where important information is missing.

1.3 Structure

The first part of the report consists out of brief explanation on wave hydrodynamics. This

happens in chapter 2. Sufficient background information is given on waves, so that an

unexperienced user can fully understand the actions taken further along the report. In this

chapter, an overview of the different methods used in this report, such as the Neural Network or

PCO, is given as well. The next chapter, chapter 3, contains the eight evaluated case studies plus

an extra case study which was not given in EurOtop. Only the results are given here. The

discussion of the results and conclusions are for chapter 4. The report finishes with chapter 5, a

summary of the report, and chapter 6, the references.

Chapter 2 -Hydrodynamics

4

Chapter 2 Literature

2.1 Hydrodynamics

In this chapter, a few hydrodynamic properties of waves are explained. This is necessary in

order to fully understand the formulae and calculation processes and make correct assumptions

and conclusions.

2.1.1 Wave Height

The wave height is one of the most important parameters in wave run-up and overtopping as

well as in every other hydrodynamical application. The height is measured from the toe of the

wave until the crest. The comprehension of the term ‘height’ appears to be that it is simple, but in

fact, it is a very difficult parameter to measure. The irregularity of the sea does not allow for a

clear definition of wave height. This is why a lot of different definitions exist for parameter and is

the topic of a lot of research.

In an irregular sea state, the individual wave heights can be accurately approximated by a

Rayleigh distribution. In this statistical distribution, a significant wave height 𝐻𝑠 can be defined as

the average of the 1/3 largest waves. Another definition is the spectral wave height 𝐻𝑚0 which is

the preferred use of wave height in EurOtop and thus this report. This definition of wave height is

based on the wave spectrum.

The wave heights mentioned in the case studies are of course those measured at the toe of the

structure. In most practical cases, there is no information on wave heights at the desired location.

Only knowledge of wave properties offshore are available. This requires a transformation of the

wave height. How 𝐻𝑚0 is calculated and what the transformations process is, can be found in other

more specific literature such as the CEM.

2.1.2 Wave period

The wave period is the time between two wave crests at a certain point. Again, for irregular

waves, this is hard to measure as the waves comprise of an entire spectrum of wave height with

different wave periods. Multiple definitions exist such as the mean period 𝑇𝑚 derived from wave

spectra like JONSWAP. The spectral mean period 𝑇𝑚−1,0 gives more weight to longer period

waves. The peak period 𝑇𝑝 is the period belonging to the peak of a spectrum. In case there is only

one peak, the relation 𝑇𝑝/ 𝑇𝑚−1,0 is fixed at around 1.1. In other cases, this ratio lies between 1.1

and 1.25. The formulae in EurOtop make use of the spectral period 𝑇𝑚−1,0 and the ratio of 1.1 is

used.

2.1.3 Wavelength

The wavelength is a property which is very difficult to measure in reality and varies a lot. There

is also a difference between deep sea wave length and near-shore wave length. Because the

wavelength depends on the slope of the shore and local water depth, the wave length is calculated


5

at deep water conditions with the spectral mean wave period where it has no influence of the sea

floor and depth:

𝐿𝑚−1,0 =

𝑔 ∙ 𝑇𝑚−1,02

2𝜋 (1)

2.1.4 The Iribarren number

The wave steepness 𝑠0 is the ratio between the wave height and length 𝐻𝑚0/𝐿𝑚−1,0 . This

steepness combined with the slope of the sea bed determines whether the wave will break or not.

The Iribarren number 𝜉𝑚−1,0 or Wave Breaker parameter, is the ratio between the tangent of the

slope and the square root of the steepness:

𝜉𝑚−1,0 =

tan(𝛼)

√𝑠0

(2)

Small values for 𝜉𝑚−1,0 mean that the wave is breaking. For the same slope, steeper waves

break easier and for the same steepness, more gentle slopes induce wave breaking. Different type

of wave breaking exist according to a range of 𝜉𝑚−1,0. This can be seen in Figure 3.

Figure 3 Types of wave breaking (FHWA)

The Iribarren number is the prime parameter affecting the run-up and overtopping value.

Depending on the range, different formulae exists, which becomes clear in the following chapters.

2.1.5 Water level, tides, and surges

Another important parameter is the depth or water height. The still water line (SWL) is always

expressed as height relative to a standardized basis level derived from a mean sea level at a

specific time. This is called an ordnance datum (OD). Different countries use different OD. The

different reference heights which are being used in this thesis can be found in Table 1.

Knowing the correct water level allows the user to determine water heights near the

structures. If the area where the structure is built is subject to a tidal window, this water level can

vary a lot. The normal tide variation is a consequence of the attraction force of the sun and moon.

Predicting this tide can be done with empirical tools or formulae based on the astronomical cycle.

This is however not the only cause for a rise in water level as storm surges can push the water

higher as well. These surges can be superimposed on the normal tidal range to get exteme water


6

level heights. Predicting these extreme heights is done based on statistical data and is the base for

designing defence structures.

Table 1 Reference water levels

Country OD abbrevitation

The United Kingdom Ordance Datum Newlyn ODN

Germany Normalnull NN

Ireland Main Ordance Datum MHD

Belgium Tweede Algemene Waterpassing TAW

A last factor influencing the water height is global warming. It is a well proven fact that this

global warming causes a general increase of the sea water levels. In some case studies, this effect

is included as a rise of 30cm of the water level. It is clearly stated where this effect is hold into

account and where it is not.

2.1.6 Wave run-up

The definition that EurOtop gives is very clear and concise:

The wave run-up height is defined as the vertical difference between the

highest point of wave run-up and the still water level (EurOtop 2007)

Figure 4 gives more clarity on this definition:

Figure 4 Wave run-up on a simple slope

The run-up can thus be seen as the highest point the wave will reach on a slope. When the run-up

becomes higher, the slope of the dike is extrapolated with the same slope in order to find the

theoretical run-up value.

The incoming waves however are of stochastic nature so that waves with the same wave height

can have different run-up values. This lead to the definition of the parameter 𝑅𝑢2% which is the

run-up value of which 2% of the incoming waves exceed this value. This definition was used to

design dikes in the early days to decrease the chance of overtopping. This is however a

conservative design process, and nowadays, some overtopping is allowed and 𝑅𝑢2% finds less use

in design procedures. Structures are designed now on an allowable overtopping instead of the

run-up height.


7

2.1.7 Wave overtopping

Wave overtopping is defined by the amount of water going over the defence structure when

the wave run-up reaches the crest, Figure 5. Overtopping thus happens when the maximum run-

up is higher than the crest level. This overtopping can be expressed in 𝑚3/𝑠 per m width or 𝑙/𝑠

per m width and is mostly expressed in mean values.

Figure 5 Overtopping on a slope

The overtopping formulae from EurOtop (see later) are always mean results. As waves have an

irregular characteristic, the overtopping is thus not constant in time. Therefore, a maximum

discharge can also be calculated by multiplying the mean discharge with a certain factor. This

maximum can be significantly higher than the mean discharge.

The effects of overtopping can be put into 4 general categories, taken from EurOtop:

Direct hazard of injury or death to people immediately behind the fence;

Damage to property, operation and or infrastructure in the area defended, including

loss of economic, environmental or other resource, or disruption to an economic

activity or process;

Damage to defence structures(s), either short-term or longer-term, with the possibility

of breaching and flooding;

Low depth flooding

It is thus clear that the overtopping should be limited. Table 2 describes briefly the consequences

of the value of the discharge on the structure itself.

Table 2 Consequences of overtopping discharge rates on defence structures

q[l/s/m] Consequences

<0.1 Insignificant with respect to strength of crest and rear of structure

1 Erosion on crest and inner slope made from grass/clay

10 Significant overtopping for dikes/embankments. Some overtopping for rubble mound breakwaters

100 Crest and inner slopes should be protected

Tolerable discharges depend on the location of the structure and the area it is defending. When

pedestrians are present, other discharge limits are valid than when only vehicles are present or


8

only buildings. For pedestrians who are aware, a limit of 0.1 𝑙/𝑠/𝑚 is advised while for trained

staff, protected and low risk of falling from the walkway, this limit is increased to 1 − 10 𝑙/𝑠/𝑚.

For vehicles limits range from 50 to 0.01 𝑙/𝑠/𝑚 depending on the vehicle velocity. Equipment can

be damaged at discharges as low as 0.4 𝑙/𝑠/𝑚 and even yachts sinking at 50 𝑙/𝑠/𝑚. Careful

attention has to be given thus when selecting overtopping limits.

This overtopping is heavily influenced by the structure parameters as well as the wave

conditions. It is clear that higher water levels increase the run-up and thus overtopping. The same

is true for the wave height. The wave period also plays an important role: a large wave period

means large wave lengths. Wave heights with a large wave length also lead to higher overtopping.

Other parameters such as the wind direction and spray also play a role in the amount of

overtopping. These parameters are however difficult to measure.

2.1.8 Statistical data

Statistics play an important role in the hydrodynamical sector. The complex processes involved

in this science are hard or sometimes near-impossible to describe with analytical formulae. This

is why a lot of the research is based on empirical data. These data can then be formed with

statistics into a fitting equation with a degree of uncertainty. For example, extreme wave heights

are described by a Rayleigh distribution in order to predict the largest wave heights on a certain

location. The parameters of this Rayleigh distribution can be assembled by measuring the wave

heights at this location for an extended period of time. When a certain probability threshold is

picked, an extreme wave height can be derived. The same method goes for a water level, but not

necessarily with a Rayleigh distribution. Combining the probability functions for wave height and

water level creates a joint probability function. It is this function which is of importance when

picking overtopping criteria. Important in this process is picking a return period for a storm, based

on the design life time on a structure. This return period, for example 1 in 200 years gives a certain

probability chance after which the desired wave height and water level can be derived from the

joint probability function. In this report, the wave heights and water levels are already calculated

so that no further information is given on this and the reader is referred to more specific reports

on this topic.

Another important part when calculating overtopping, is the confidence range of the used

equations. The equations used in this report are assumed to be normally distributed. This is done

by using the fitted parameters in the equations as a normally distributed variable with mean 𝜇

and standard deviation 𝜎. A confidence interval can then be created by defining that the interval

should contain for example 90% of the results. This means a lower limit solution is found by using

𝜇′ = 𝜇 − 1.64 ∙ 𝜎 and an upper solution by using 𝜇′ = 𝜇 + 1.64 ∙ 𝜎. In some formulas, it is not the

parameters which are normally distributed but the result of the equation itself, using a variation

coefficient 𝜎′ = 𝜎/𝜇.

Chapter 2 -Calculation tools

9

2.2 Calculation tools

2.2.1 EurOtop

The EurOtop manual is the basis on which this thesis is built. This manual is basically a guide

on how to predict wave run-up and overtopping for different structures. It intended use is to

provide coastal engineers the tools to design wave defence structures. It was published in 2007

and is the result of a cooperation between several authors spread over Europe. It is based on the

previous works of national manuals from Germany, the Netherlands, and the UK: EAK, TAW, and

EA respectively. New results from the European CLASH database provided new insights and new

tools. These new results combined with the national manuals is the basis for EurOtop. The

formulae defined in EurOtop are mostly based on empirical data. This is due to the complex

physical process and stochastic nature of overtopping which makes it very difficult to create

analytical formulae. As this thesis does not revolve around designing, the probabilistic formulae

are given each time. In this paragraph a short summary is given on the relevant structures and

formulae for overtopping and run-up.

Coastal dikes and embankment seawalls

A first kind of structures are the simple slopes. These slopes are easy to explain with

straightforward empirical formulae, based on a large international dataset. This dataset includes

many different slopes. A structure is considered a slope when the angle of the slope, tan (𝛼), is

between 1:1 and 1:8. When the slope is milder than 1:15, it is considered a berm. A berm requires

a different approach and is explained later. Slopes in between those values, need to be calculated

as slope and berm and then interpolated. Slopes steeper than 1:1 are considered battered or

vertical walls.

Run-up

For the wave run-up on slopes, Equation 3 can be used. This equation is also shown in figure 6

as the solid line. The statistical distribution around the curve of Equation 3, can be described by a

normal distribution with a variation coefficient 𝜎′ = 0.07. The exceedance limit of 5% and 95%

are used.

𝑅𝑢2%

𝐻𝑚0= 1.65 ∙ 𝛾𝑏 ∙ 𝛾𝑓 ∙ 𝛾𝛽,𝑟 ∙ 𝜉𝑚−1,0

𝑤𝑖𝑡ℎ 𝑎 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑜𝑓 𝑅𝑢2%

𝐻𝑚0= 𝛾𝑏 ∙ 𝛾𝑓 ∙ 𝛾𝛽,𝑟 ∙ (4.0 −

1.5

√𝜉𝑚−1,0

)

𝑅𝑢2% = wave run-up height exceeded by 2% of the incoming waves[𝑚]

𝛾𝑏 = influence factor for a berm [−]

𝛾𝑓 = influence factor for the roughness [−]

𝛾𝛽,𝑟 = influence factor for the wave obliqueness [−]

𝜉𝑚−1,0 = wave breaker parameter [−]

(3)


10

Figure 6 Wave run-up for smooth and straight slopes (EurOtop 2007)

Equation 3 is influenced by several parameters, in which the most important is the breaker

parameter 𝜉𝑚−1,0 and the wave height 𝐻𝑚0. From equation 2 the reader can see that 𝜉𝑚−1,0 is

dependent on the slope angle 𝛼. For a structure with a single slope, this is easy to calculate.

However, when the structure has several parts with different slopes, a composite slope angle

𝛼𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒 is defined, as seen in figure 7. This is done by removing any possible berms and

calculating the average slope between 1.50 ∙ 𝐻𝑚0 below SWL, and 1.0 ∙ 𝑅𝑢2% above SWL. Parts of

the structure which are not in this interval are said not to have any influence on the overtopping.

𝑅𝑢2% is however not known at first, so an iterative calculation process is necessary. An initial value

and upper limit for run-up is taken to be 1.50 ∙ 𝐻𝑚0. When this is higher than the crest, the crest

is taken as limit and initial value. The composite slope can then be calculated according to

Equation 4.

Figure 7 Defining a composite slope for structure with different slope angles

tan(𝛼𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒) =

1.50 ∙ 𝐻𝑚0 + 𝑅𝑢2%

𝐿 − 𝐵 (4)


11

The roughness factor 𝜸𝒇 is a value to measure the roughness and permeability for structures.

A smooth and impermeable structure allows the wave to run up the structure unobstructed

leading to a higher run-up 𝑅𝑢2%. The values for these kind of slopes is then 𝛾𝑓 = 1. More roughness

and permeability reduces the run-up and thus the roughness influence factor. This increased

roughness has however less effect when the parameter 𝛾𝑏 ∙ 𝜉𝑚−1,0 goes over 1.8. and is increased

linearly to 1 for 𝛾𝑏 ∙ 𝜉𝑚−1,0 = 10. When the structure has parts with different roughness factors, a

weighted average is taken:

𝛾𝑓 =

∑ 𝛾𝑓,𝑖 ∙ 𝐿𝑖

∑ 𝐿𝑖 (5)

When calculating this weighted average, the reader should know that roughness elements lying

lower than 0.25 ∙ 𝑅𝑢2% beneath SWL, and higher than 0.50 ∙ 𝑅𝑢2% above, do not have any effect on

the run-up. This requires an iterative calculation comparable with a composite slope. The initial

run-up is calculated assuming the structure is smooth and impermeable with roughness factor

𝛾𝑓 = 1.

When a berm, slope milder than 1:15, is present, the run-up and overtopping is reduced,

especially when the still water line is close to the berm level. This is translated in a berm factor 𝜸𝒃.

The width of the berm is limited to 0.25 ∙ 𝐿𝑚−1,0, with 𝐿𝑚−1,0 the wave length. The factor can be

calculated with the following equation:

𝛾𝑏 = 1 − 𝑟𝑏(1 − 𝑟𝑑𝑏)

𝑟𝑏 =𝐵

𝐿𝑏𝑒𝑟𝑚

𝑟𝑑𝑏 = 0.5 − 0.5 cos (𝜋𝑑𝑏

𝑅𝑢2%) for 𝑑𝑏 < 0

𝑟𝑑𝑏 = 0.5 − 0.5 cos (𝜋𝑑𝑏

2∙𝐻𝑚0) for 𝑑𝑏 ≥ 0

(6)

Important is that 𝐿𝑏𝑒𝑟𝑚 is the length of the structure between 1.00 ∙ 𝐻𝑚0 below and above still

water line. This is different from 𝐿 shown in figure x where the interval is 1.50 ∙ 𝐻𝑚0 below and

1.00 ∙ 𝑅𝑢2% above. 𝑑𝑏 is the height of the berm above or below still water line, and is negative

when the berm is located above water level. For a berm lying outside the influence area (𝑑𝑏

𝑅𝑢2%<

−1 or 𝑑𝑏

2∙𝐻𝑚0> 1), 𝑟𝑑𝑏 = 1. For berms lying above SWL this requires again an iterative solution.

Run-up is first calculated as no berm is present and then the factor is calculated and applied. One

last thing to note is when a non-horizontal berm is present, it should be transformed to a

horizontal berm as proposed in Figure 8. After this process, 𝛾𝑏 can be calculated according to

equation 6. Last thing about berms is that 𝛾𝑏 is limited to 0.6.


12

Figure 8 Transforming an angled berm to a horizontal berm

A last factor in Equation 3 for run-up is the oblique wave attack factor 𝜸𝜷,𝒓. When incoming

waves are not perpendicular, a reduction of run-up and overtopping occurs. The angle of wave

attack is the angle of the wave direction relative to the perpendicular of the structure as proposed

in Figure 9. Equation 7 gives the value for the influence factor.

Figure 9 Definition of wave attack angle relative to the structure

𝛾𝛽,𝑟 = 1 − 0.0022|𝛽| for 0° ≤ 𝛽 ≤ 80°

𝛾𝛽,𝑟 = 0.824 for 80° < 𝛽 (7)

Overtopping

Unlike for run-up, the formula for overtopping cannot be put into one single equation. There

are different formulae depending on the wave breaking. For breaking waves and non-breaking

waves with 𝜉𝑚−1,0<5 the probabilistic formula is seen in equation 8 and Figure 10.

𝑞

√𝑔 ∙ 𝐻𝑚03

=0.067

√tan(𝛼)∙ 𝛾𝑏 ∙ 𝜉𝑚−1,0 ∙ exp (−4.75 ∙

𝑅𝐶

𝜉𝑚−1,0 ∙ 𝐻𝑚0 ∙ 𝛾𝑏 ∙ 𝛾𝑓 ∙ 𝛾𝛽,𝑞 ∙ 𝛾𝑣)

with a maximum of 𝑞

√𝑔∙𝐻𝑚03

= 0.2 ∙ exp (−2.6 ∙𝑅𝐶

𝐻𝑚0∙𝛾𝑓∙𝛾𝛽)

𝑞 = overtopping discharge [𝑚3

𝑚∙𝑠]

𝑅𝑐 = Freeboard [𝑚]

𝛾𝑣 = influence factor for a vertical wave wall

𝛾𝛽,𝑞= influence factor for oblique wave attack

(8)


13

Figure 10 Wave overtopping data for breaking waves and overtopping equation 8 with 5% lower and 95% upper exceedance limit (EurOtop 2007)

As mentioned before, this is the probabilistic equation of the data. The scatter in the data is

incorporated through factors 4.75 and 2.6 being normally distributed with standard deviations 𝜎

of respectively 0.5 and 0.35. This equation also uses a lot of the same parameters from the run-up

equation. There are however some new ones.

One of these, is the influence factor for the presence of a wave wall 𝜸𝒗. This is a vertical wall or

very steep slope on top of the structure. This can reduce the overtopping and is presented in the

factor 𝛾𝑣 . There is however limited information on this and has several strict conditions when it

can be used. These conditions, as quoted from EurOtop, are:

The composite slope taken from 1.50 ∙ 𝐻𝑚0 to the foot of the wall, excluding berms,

should be between 1:2.5 and 1:3.5. These are relatively steep slopes.

The width of the berms must not exceed 3.00 ∙ 𝐻𝑚0

The foot of the wall must lie between 1.20 ∙ 𝐻𝑚0 below and above of the still water line.

The minimum height of the wall is about 0.50 ∙ 𝐻𝑚0, while the maximum is 3.00 ∙ 𝐻𝑚0

Outside of these boundary conditions, the reliability of the factor is lower. Equation 9 calculates

this factor using 𝛼, the slope of the wall, with 𝛼 = 90° for a vertical wall.

𝛾𝑣 = 1.35 − 0.0078 ∙ 𝛼 for 45° ≤ 𝛼 ≤ 90 (9)

𝛾𝛽,𝑞 is the influence factor for wave oblique attack for overtopping. It is calculated on the same

manner as 𝛾𝛽,𝑟 but with different coefficients:

𝛾𝛽,𝑞 = 1 − 0.0033|𝛽| for 0° ≤ 𝛽 ≤ 80°

𝛾𝛽,𝑞 = 0.736 for 80° < 𝛽 (10)

For large 𝜉𝑚−1,0 > 7 another equation is used in order to avoid underestimation:


14

𝑞

√𝑔 ∙ 𝐻𝑚03

= 10𝑐 ∙ exp (−𝑅𝐶

𝐻𝑚0 ∙ 𝛾𝑓 ∙ 𝛾𝛽,𝑞 ∙ (0.33 − 0.022𝜉𝑚−1,0))

(11)

Armoured rubble slopes and mounds

When dikes are covered by rubble or armour units, the structure can be seen as an armoured

rubble slope. As the structure is similar to smooth impermeable slopes, the equations used for

overtopping and run-up are also very similar. In fact, for run-up the same equations are used:

Equation 3. For wave overtopping, Equation 8 is used which is also the same. Armoured slopes

mostly have a steep slope which leads to the use of equation 8.b for most cases.

The only difference are the influence factors. 𝛾𝑓 is replaced by 𝛾𝑓,𝑠𝑢𝑟𝑔𝑖𝑛𝑔, which holds into

account the fact that surging waves slowly run up and down the slope, storing water in the armour

layer. Following waves do not ‘feel’ the roughness due to this effect. For 𝜉𝑚−1,0 ≤ 1.8 the normal

roughness factor 𝛾𝑓 can be used. For increasing 𝜉𝑚−1,0, 𝛾𝑓,𝑠𝑢𝑟𝑔𝑖𝑛𝑔 linearly increases to 1 until

𝜉𝑚−1,0 = 10.

𝛾𝑓,𝑠𝑢𝑟𝑔𝑖𝑛𝑔 = 𝛾𝑓 for 𝜉𝑚−1,0 ≤ 1.8

𝛾𝑓,𝑠𝑢𝑟𝑔𝑖𝑛𝑔 = 𝛾𝑓 + (𝜉𝑚−1,0 − 1.8)1−𝛾𝑓

(10−1.8) for 1.8 < 𝜉𝑚−1,0 < 10

𝛾𝑓,𝑠𝑢𝑟𝑔𝑖𝑛𝑔 = 1 for 10 ≤ 𝜉𝑚−1,0

(12)

Another important note is that when a permeable core is present, the relative run-up 𝑅𝑢2%/𝐻𝑚0

is limited to a constant value of 1.97 as water can be stored into this core as well.

𝑅𝑢2%,𝑚𝑎𝑥

𝐻𝑚0= 1.97 (13)

The oblique wave factor 𝛾𝛽 is the same as for simple slopes but with different values:

𝛾𝛽 = 1 − 0.0063|𝛽| for 0 ≤ 𝛽 ≤ 80°

𝛾𝛽 = 0.504 for 80° < 𝛽 (14)

For the berm influence, the same factor can used as for simple slopes. Due to the high slope in

armoured slopes, equation 8.b is used in which there is no berm factor present. This leads to the

conclusion that the overtopping for armoured slopes is mostly the same with and without a berm.

When a berm is present at the crest existing out of armour layers, this is defined as 𝐺𝑐: the width

of the armour layer at crest. The presence of this 𝐺𝑐 reduces the overtopping with a factor 𝐶𝑟.

𝐶𝑟 = 3.03exp (−1.5𝐺𝑐

𝐻𝑚0) with maximum 𝐶𝑟 = 1 (15)

The height of the armour layer crest 𝐴𝑐 can be either lower or higher than the freeboard 𝑅𝑐,

where overtopping is measured. As information on wave walls on armoured slopes is also limited,

wave wall extending above the armour crest 𝐴𝑐 , are calculated into 𝑅𝑐 and 𝛾𝑣 = 1, as seen in Figure

11. Another handy definition for 𝑅𝑐 considering 𝐴𝑐 is that 𝑅𝑐 is the height of the dike where waves

cannot run back into the sea anymore.


15

Figure 11 Run-up level and location for overtopping differ (EurOtop 2007)

Vertical and steep seawalls

The last kind of structures are the vertical walls or the battered walls which have a slope

steeper than 1:1. Similar to simple slopes, there are different equations depending on the wave

conditions. For sloping structures, this is the separation between plunging and surging waves. For

vertical structures, this is impulsive and non-impulsive conditions. Non-impulsive conditions

happen when the wave steepness is relatively low and the water depth is high near the vertical

wall. Overtopping in these conditions are not influenced a lot by the structure toe or approach

slope. Loads on the wall are smooth varying and overtopping is sort of ‘calm’, called ‘green water

overtopping’, Figure 12. Unlike for impulsive conditions, where loads are very high in a short

window of time. Overtopping is also highly aerated and violent. The conditions are impulsive

when the wave steepness is high and there is a low water depth. This automatically leads to the

overtopping having a large effect of local bathymetry.

Figure 12 Non-impulsive waves above, impulsive waves below (EurOtop 2007)

The separation between impulsive conditions is defined by the impulsiveness parameter. This

parameter changes with the type of structure as well: a plain vertical wall, ℎ∗, or a composite wall,

𝑑∗. The difference is the presence of a toe structure before the wall. The equation for ℎ∗ or 𝑑∗ can


16

be seen in Equation 16 and the parameters used in this equation are found in Figure 13. Impulsive

conditions dominate when its respectively ℎ∗ and 𝑑∗ are smaller than 0.02. Non-impulsive

conditions when they are larger than 0.03

ℎ∗ = 1.35 ∙ℎ𝑠

𝐻𝑚0

ℎ𝑠

𝐿𝑚−1,0 for plain vertical walls

𝑑∗ = 1.35 ∙𝑑

𝐻𝑚0

ℎ𝑠

𝐿𝑚−1,0 for composite vertical walls

(16)

Figure 13 Definition of vertical structure parameters

Plain vertical walls:

For simple vertical breakwaters, the overtopping in non-impulsive conditions (ℎ∗ > 0.3) is

presented in the following equation and by the solid dark blue line in Figure 14.

𝑞


= 0.04exp (−2.6

𝛾𝛽

𝑅𝑐

𝐻𝑚0) valid for 0.1 < 𝑅𝑐/𝐻𝑚0 < 3.5

(17)


17

Figure 14 Overtopping at a plain vertical wall under non-impulsive conditions (EurOtop 2007)

This equation has the same structures as for surging waves for simple slopes. The scatter around

this curve can be represented by assuming that the factor 2.6 has a normal distribution with

standard deviation 𝜎 = 0.8.

In impulsive conditions (ℎ∗ ≤ 0.2), this equation changes due to the fact that the local

bathymetry plays an important role. Figure 15 and 16 represents this equation in a graph.

𝑞

ℎ∗2√ℎ𝑠

3= 1.5 ∙ 10−4 (ℎ∗

𝑅𝑐

𝐻𝑚0)

−3.1 valid for 0.03 ≤ ℎ∗

𝑅𝑐

𝐻𝑚0≤ 1.0

𝑞

ℎ∗2√𝑔ℎ𝑠

3= 2.7 ∙ 10−4 (ℎ∗

𝑅𝑐

𝐻𝑚0)

−2.7 valid for ℎ∗

𝑅𝑐

𝐻𝑚0≤ 0.02, broken waves

(18)


18

Figure 15 Overtopping at a plain vertical wall under impulsive conditions, equation 18.a (EurOtop 2007)

Figure 16 Overtopping at a plain vertical wall under impulsive conditions and broken waves for ℎ∗𝑅𝑐

𝐻𝑚0< 0.02,

equation 18.b; the area between ℎ∗𝑅𝑐

𝐻𝑚0= 0.02 and ℎ∗

𝑅𝑐

𝐻𝑚0= 0.03 is vague and both extrapolated equations 18.a and

18.b are shown (EurOtop 2007)


19

For values ℎ∗𝑅𝑐

𝐻𝑚0 that are between 0.02 and 0.03, there is a transition zone. It is recommended to

use the first formula for values up to 0.02 unless there is certainty that waves will break. In that

case, the equation for broken waves is used until a value of 0.03.

A last equation is given on emergent vertical walls where ℎ𝑠 < 0. This a common structure

where at the end of a smooth slope or beach a vertical wall is placed. Despite the wide spread use

of this kind of structure, not much studies have been done on these structures. Available studies

point out that the use of plunging waves on simple slopes is valid in most cases, equation 8. In

specific conditions an equation has been developed which describes these kinds of structures

better. Equation 19 is presented in Figure 17.

𝑞

√𝑔 ∙ 𝐻𝑚03

∙ √cot(𝛼𝑓𝑠) 𝑠𝑚−1,0 = 0.043 exp (−2.16 𝑐𝑜𝑡(𝛼𝑓𝑠) 𝑠𝑚−1,00.33

𝑅𝑐

𝐻𝑚0)

Valid for:

2.0 < 𝑐𝑜𝑡(𝛼𝑓𝑠) 𝑠𝑚−1,00.33 < 5.0

0.55 ≤𝑅𝑐

𝐻𝑚0≤ 1.6

𝑠𝑚−1,0 ≥ 0.025

cot(𝛼𝑓𝑠) = 10 (slope foreshore)

(19)

Figure 17 Overtopping for emergent vertical structures (EurOtop 2007)

Composite walls

The process for composite vertical walls is equivalent as for plain vertical walls. When 𝑑∗ >

0.03 non-impulsive conditions are assumed and the same equation as for a plain vertical walls can


20

be used, equation 17. This corresponds with the statement that in non-impulsive conditions, the

overtopping is not affected by the local bathymetry.

For impulsive conditions, 𝑑∗ < 0.02, a modified version of equation 18 is used. This new

equation is presented in Figure 18.

𝑞

𝑑∗2√𝑔ℎ𝑠

3= 4.1 ∙ 10−4 (𝑑∗

𝑅𝑐

𝐻𝑚0)

−2.9 valid for 0.05 ≤ 𝑑∗

𝑅𝑐

𝐻𝑚0≤ 1.0 (20)

Figure 18 Impulsive overtopping on composite structures (EurOtop 2007)

Influence factors

The roughness is not relevant in these kind of structures and is therefore neglected in the

equations. Wave obliquity does still play a role however but is not as simple to apply anymore.

Only for the non-impulsive conditions a simple influence factor 𝛾𝛽 is introduced again.

𝛾𝛽 = 1 − 0.0062|𝛽| for 0° ≤ 𝛽 ≤ 45°

𝛾𝛽 = 0.72 for 45° < 𝛽 (21)

For impulsive conditions, the process is somewhat more complicated and again not a lot of

information is known. Overtopping decreases again with increasing wave obliquity as well as

impulsiveness. When 𝛽 > 60°, what should be impulsive waves at 𝛽 = 0°, are becoming non-

impulsive. That’s why EurOtop proposed an adjustment of the formula for discrete wave attack

angles.

𝛽 = 15°; ℎ∗

𝑅𝑐

𝐻𝑚0≥ 0.02 =>

𝑞


3= 5.8 ∙ 10−5 (ℎ∗

𝑅𝑐

𝐻𝑚0)

−3.7

𝛽 = 15°; ℎ∗𝑅𝑐

𝑚0< 0.02 => equation 18.a: impulsive 𝛽 = 0°

(22)


21

𝛽 = 30°; ℎ∗𝑅𝑐

𝐻𝑚0≥ 0.07 =>

𝑞


3= 8.0 ∙ 10−6 (ℎ∗

𝑅𝑐

𝐻𝑚0)

−4.2

𝛽 = 60°; ℎ∗𝑅𝑐

𝐻𝑚0≥ 0.07 => eq 17: non impulsive 𝛽 = 0°

A last reduction factor can come from a part of the wall hanging seaward: a recurve/ parapet/

wave return wall/ bullnose. This recurve factor 𝑘 can be calculated with the following diagram:

Figure 19 Definition of parameters for recurve/parapet wall and a decision chart on how to calculate the reducing factor 𝑘 for these walls (EurOtop 2007)


22

2.2.2 van der Meer et al: Updated EurOtop formulae

Since the release of EurOtop in 2007, a lot of extra research has been done. In 2013, two papers

were released containing suggestions on improvements for the EurOtop. The first paper (van der

Meer et al, 2013) revisits the topic on the sloping structure. The second paper (Bruce et al, 2013)

discusses vertical walls, both plain and composites. In this paragraph, a brief explanation and

summary is given on both papers.

Sloping structures: van der Meer et al (2013)

The first problem of the formulae for sloping structures is that they significantly overestimate

the overtopping at low relative freeboards. A solution is proposed in the paper, based on previous

data from Battjes (1974) and the Dutch guidelines TAW (1989). Based on a bivariate Rayleigh

distribution for wave heights and periods, Battjes derived an expression for wave overtopping.

This expression was also a function of the bivariate Rayleigh distribution with parameter 𝜅

causing a curved line in a log-lineair graph. This is in contrast with the straight line produced from

Equation 8, the exponential equation for overtopping at simple slopes. In Figure 20, both formulae

are plotted together with available experimental data.

Figure 20 Overtopping equation 3 from EurOtop plotted together with the proposed formula from Battjes/TAW (van der Meer et al 2013)

The results are quite extraordinary because not only does the expression from Battjes match the

experimental data at positive freeboards, it also has a good match at zero freeboard. From these

results, it became clear that the theory from Battjes (1974) not only gave the correct curve but

also an analytical background. van der Meer et al concluded that a new fit through the data would

be the next step. Because the current EurOtop uses an exponential curve, they proposed to use a

Weibull curve in order to receive the best fit. The result after fitting the coefficients was:


23

𝑞

√𝑔 ∙ 𝐻𝑚03

=0.023

√tan(𝛼)∙ 𝛾𝑏 ∙ 𝜉𝑚−1,0 ∙ exp (− (2.7 ∙

𝑅𝐶

𝜉𝑚−1,0 ∙ 𝐻𝑚0 ∙ 𝛾𝑏 ∙ 𝛾𝑓 ∙ 𝛾𝛽,𝑞 ∙ 𝛾𝑣

)

1.3

)

with a maximum of 𝑞


= 0.09 ∙ exp (− (1.5 ∙𝑅𝐶


1.3

)

(23)

This is the probabilistic version of the proposed new overtopping formula on sloping structures

which gives a significant better result at lower relative freeboard and zero freeboard. The

reliability can be given by 𝜎(0.023) = 0.003 and 𝜎(2.7) = 0.2 for equation 23.a. For eq 23.b the

reliability is 𝜎(0.09) = 0.013 and 𝜎(1.5) = 0.15. This formulae give similar results for

overtopping as the original EurOtop formulae, but give better results for 𝑅𝑐/𝐻𝑚0 < 0.8.

The second problem for sloping structures is the lack of transition from steep slopes to vertical

walls. To solve this, the researchers marked an upper and lower bound for non-breaking waves.

The upper bound is for steep smooth slopes, equation 23.b, and the lower bound is an expression

for vertical walls. In order to have a good transition between both cases, the formulae should have

the same structure. This is why the new expression for vertical walls is a Weibull equation as well,

fitted through the available data on vertical walls, Equation 24 and Figure 21. The expressions for

the lower and upper bound have now the same shape and the only difference are the coefficients

and exponent.

𝑞

√𝑔 ∙ 𝐻𝑚03

= 0.047 ∙ exp [− (2.35 ∙𝑅𝐶

𝐻𝑚0 ∙ 𝛾𝑓 ∙ 𝛾𝛽)

1.3

] (24)

Figure 21 A fitted Weibull distribution for vertical walls based on experimental data, equation 24 , under non-breaking wave conditions (van der Meer et al 2013)

The connecting parameter between the expressions is the slope angle cot(𝛼). Victor (2012)

analysed data on overtopping for several steep slopes Figure 22, mostly non-breaking waves and

derived a relation for the coefficients in the expression, based on the slope angle, Figure 23 and

Equation 25.

23.b

24


24

Figure 22 Fitted Weibull distributions for overtopping on steep slopes with different angles , under non-breaking wave conditions (van der Meer et al 2013)

Figure 23 The values of parameters a and b in the Weibull distribution for overtopping on steep slopes with different angles, under non-breaking wave conditions and c=1.3, equation 25 (van der Meer et al 2013)

𝑞

√𝑔 ∙ 𝐻𝑚03

= 𝑎 ∙ exp [− (𝑏 ∙𝑅𝐶

𝐻𝑚0 ∙ 𝛾𝑓 ∙ 𝛾𝛽)

1.3

]

𝑎 = 0.09 − 0.01(2 − cot(𝛼))2.1

𝑏 = 1.5 + 0.42(2 − cot(𝛼))1.5

(25)

The new equation for smooth slopes is now:

𝑞

√𝑔 ∙ 𝐻𝑚03

=0.023

√tan(𝛼)∙ 𝛾𝑏 ∙ 𝜉−1,0 ∙ exp (− (2.7 ∙

𝑅𝐶

𝑚−1,0 ∙ 𝐻𝑚0 ∙ 𝛾𝑏 ∙ 𝛾𝑓 ∙ 𝛾𝛽,𝑞 ∙ 𝛾𝑣

)

1.3

)

with a maximum 𝑞


= 𝑎 ∙ exp [− (𝑏 ∙𝑅𝐶


1.3

]

(26)


25

𝑎 = 0.09 − 0.01(2 − cot(𝛼))2.1

𝑏 = 1.5 + 0.42(2 − cot(𝛼))1.5

Vertical structures: Bruce et al (2013)

Bruce et al (2013) raised the issue that in EurOtop, the formulae for impulsive and non-

impulsive waves do not have an equal structure, making a comparison or a transition regime very

difficult. For non-impulsive waves (ℎ∗ > 0.3) and impulsive, the equations used in EurOtop are:

𝑞


= 𝑎𝑛𝑖 exp (−𝑏𝑛𝑖𝑅𝑐

𝐻𝑚0) Non-impulsive

𝑞


3= 𝑎𝑖 (ℎ∗

𝑅𝑐

𝐻𝑚0)

−𝑏𝑖

Impulsive waves (27)

where the coefficients a and b are different depending on the wave and structure properties. The

reader can see that the dimensionless overtopping 𝑞∗ and relative freeboard 𝑅𝑐∗ are different for

each equation making a direct comparison difficult. This is why van der Meer et al (2008) made

an algebraic expression for the impulsive waves with the standard dimensionless overtopping

𝑞∗ =𝑞


and 𝑅𝑐∗ =

𝑅𝑐

𝐻𝑚0:

𝑞

√𝑔 ∙ 𝐻𝑚03

= 𝑎√𝐻𝑚0

ℎ𝑠

1

𝑠𝑚−1,0(

𝑅𝑐

𝐻𝑚0)

−3

(28)

This new equation is compared with CLASH datasets and the base equation on which the equation

for non-impulsive overtopping for vertical walls in EurOtop is derived:

𝑞

√𝑔 ∙ 𝐻𝑚03

= 𝑎 ∙ exp (−𝑏 ∙𝑅𝑐

𝐻𝑚0)

(29)

Both Franco et al (1994) and Allsop et al (1995) derived their own values for the parameters 𝑎

and 𝑏:

Franco et al (1994): 𝑎 = 0.2 and 𝑏 = 4.3

Allsop et al (1995): 𝑎 = 0.05 and 𝑏 = 2.78

Figure 24 shows both these equations, as well as equation 17 from EurOtop for non-impulsive

overtopping at vertical walls, in two different plots with CLASH data in different conditions. One

plot shows the CLASH data for vertical structures with a foreshore present, while the other gives

CLASH datasets of a vertical structure without a foreshore, in relative deep water. In both these

plots, an example of Equation 28 is plotted as well with a=0.000192, ℎ/𝐻𝑚0 = 0.9, and 𝑠𝑚−1,0 =

0.03. Assessing these two plots, the conclusion came that a separation for structures with and

without foreshore was necessary. As can be seen from Figure 24, the new proposed Equation 28

does fit the data well when a foreshore is present. For structures without the foreshore, the

equations from Franco et al, and Allsop et al are a better match. Therefore, a distinction is made

between structures with and without a foreshore.


26

Figure 24 Representation of equation 28 and Allsop et al (1995) and Franco et al (1994) ‘s interpretation of equation 29; Left CLASH dataset 802 is plotted: a vertical structure with foreshore; Right CLASH dataset 914 is

plotted: a vertical structure without a foreshore in relative deep water (Bruce et al 2013)

When considering all the CLASH datasets without a foreshore mentioned in Bruce et al(2013),

the equations from Franco et all (1994) and from Allsop et al (1995) each had their separate

region where their equations fits best. Based on this, Equation 30 is proposed for vertical

structures without a foreshore, Figure 25. These are for example vertical flood walls in harbours,

caissons, gates of locks, and etc…

Figure 25 Vertical structures on relatively deep water, no sloping foreshore (Bruce et al 2013)

𝑞


= 0.05 exp (−2.78𝑅𝑐

𝐻𝑚0) Franco et al (1994) for 𝑅𝑐/𝐻𝑚0 < 0.91

𝑞


= 0.2 exp (−4.3𝑅𝑐

𝐻𝑚0) Allsop et al (1995) for

𝑅𝑐

𝐻𝑚0≥ 0.91

(30)

The same is done for the datasets from CLASH where a foreshore is present. For these cases, a

distinction is made between non-impulsive and impulsive conditions. This discriminator is

ℎ2/(𝐻𝑚0𝐿𝑚−1,0). When presenting the data, a generalized formula from the manipulation by van

28 28


27

der Meer et al (2008) and the non-dimensional 𝑞 is presented by equation 31. The parameters 𝑎

and 𝑏 are also sought for the best fit.

𝑞

√𝑔 ∙ 𝐻𝑚03

/((𝐻𝑚0

ℎ𝑠)

𝑎

𝑠𝑚−1,0b )

(31)

The conclusion was that for 𝑎 and 𝑏 the best value is 0.5 and -0.5 respectively and that the

discriminator ℎ2/(𝐻𝑚0𝐿𝑚−1,0) = 0.23 for separating impulsive and non-impulsive conditions.

This value responds with the general ℎ∗ = 0.3 as used in EurOtop.

When separating the data according to this discriminator, it is found that for non-impulsive

waves the equation from Allsop et al (1995), equation 29, is the best fit. For impulsive conditions,

the remaining cases are given in Figure 26. Here it can be seen that for larger freeboard, a power

function would be good a fit. When the relative freeboard decreases, such a power function has

the vertical axis as asymptote so for this region, an exponential equation is picked. The resulting

formulae are presented in equation 32.

Figure 26 All data of seawalls on sloping foreshore for impulsive waves, and with optimim values of 𝑎 = 0.5 and 𝑏 = −0.5. The 5% and 95% exceedance lines are plotted as dotted lines. For lower freeboards, the exponential

equation 32.a is valid while for larger freeboards, the power curve is valid, equation 32.b (Bruce et al 2013)

𝑞


= 0.011√𝐻𝑚0

ℎ𝑠𝑠𝑚−1,0 exp ( −2.2

𝑅𝑐

𝐻𝑚0) for 𝑅𝑐/𝐻𝑚0 < 1.35

𝑞


= 0.0014√𝐻𝑚0

ℎ𝑠𝑠𝑚−1,0 (

𝑅𝑐

𝐻𝑚0)

−3for 𝑅𝑐/𝐻𝑚0 ≥ 1.35

(32)

Composite structures

The last topic of the EurOtop revisions is about composite vertical structures. For these type of

structures the same problem exists as with plain vertical walls, that it is very hard to compare


28

impulsive and non-impulsive conditions. The solution is analogue to the plain walls with a

generalized formula with the standard non-dimensional 𝑞 and freeboard:

𝑞

√𝑔 ∙ 𝐻𝑚03

= 𝑏 (𝑑

ℎ)

0.5

(𝐻𝑚0

ℎ𝑠𝑚−1,0)

0.5

(𝑅𝑐

𝐻𝑚0)

−3

(33)

By using this, the formulae for composite and plain structures becomes the same besides the

factor 𝑏 and 𝑑/ℎ, which becomes 1 when there is no berm, 𝑑 = ℎ. Separating impulsive and non-

impulsive conditions is done by the parameter 𝑑∗. The best solution is when 𝑑∗ = 0.85 is used

opposed to EurOtop’s 𝑑∗ = 0.3. Results after analysing the CLASH data after this separation,

Figure 27, are the following equations for impulsive conditions:

𝑞


= 1.3√𝑑

ℎ0.011√

𝐻𝑚0

ℎ𝑠𝑠𝑚−1,0 exp ( −2.2

𝑅𝑐

𝐻𝑚0) for 𝑅𝑐/𝐻𝑚0 < 1.35

𝑞


= 1.3√𝑑

ℎ0.0014√

𝐻𝑚0

ℎ𝑠𝑠𝑚−1,0 (

𝑅𝑐

𝐻𝑚0)

−3for 𝑅𝑐/𝐻𝑚0 ≥ 1.35

(34)

These equations become the same for plain vertical walls when 𝑑/ℎ = 0.6. Physically, this seems

rational as a berm has no influence anymore when it’s depth 𝑑 > 0.6ℎ. For non-impulsive

conditions, it was shown that the structure could be handled as a plain vertical wall without

foreshore, using Allsop et al, or Franco et al depending on the freeboard, Equation 29.

The following decision chart, Figure 28, presents a summary on the results for vertical

structures.

Figure 27 Comparison of overtopping at composite and plain vertical structures for non-impulsive waves as well as for impulsive waves (Bruce et al 2013)


29

Figure 28 Decision chart showing new schemes; vertical to left; composite to right (Bruce et al 2013)

When using the equations from both these papers further along this report, it is referred as van

der Meer et al (2013) for both papers.

2.2.3 PC-Overtopping

Another tool for calculating overtopping is the software package PC-Overtopping. This is an

easy to use tool in which simple slopes can be defined. It is based on the results from the Technical

Report on Wave Run-Up and Wave overtopping at Dikes, TAW (2002).

Slope angles should be between 1:1 and 1:8 and a small vertical wall can be included into the

structure. The slopes can be divided into different sections by assigning x and y coordinates. Each

section has to be assigned a roughness factor as well. A permeable crest existing out of rough rock

or armour units cannot be modelled. The lengths of each section are limited to 100m each. Berms

with a slope angle milder than 1:8 can be modelled but not as first or last slope and are limited to

a length of 0.25 ∙ 𝐿𝑚−1,0

The input of the program is the wave height 𝐻𝑚0, the wave period 𝑇𝑝 or 𝑇𝑚−1,0, the water level

ℎ, and wave obliquity 𝛽. Optional input parameters are the time length of a storm and the mean

period.

Eq. 32.a Eq. 32.b Eq. 30.b Eq. 30.a

Eq. 34.a Eq. 34.b


30

The output is the 2% run-up 𝑅𝑢2%, the mean wave overtopping 𝑞 and for the optional input

parameters, the number of overtopping waves and volume. The additional parameters such as the

breaker parameter 𝜉𝑚−1,0 is also shown. Just like for the EurOtop, if the run-up is higher than the

dike crest, 𝑅𝑢2% is calculated by extending the last slope.

Advantages , taken from EurOtop, are:

Each slope can be modelled with each slope its own independent roughness factor

Calculation of multiple overtopping parameters, not only the mean discharge

Disadvantages:

It cannot calculate vertical structures or rough/permeable crests.

2.2.4 Neural Network

Overtopping can also be calculated by a neural network. This is a tool which tries to describe a

complex process with a large amount of input parameters. When there is a lot of data available on

this process, the tool can be programmed to predict solutions based on the available data by a

regression model. This tool can actually be compared to the human brain and its structure, hence

the name ‘neural network’. The tool is divided into several layers and within each layer, there are

processing units called neurons. The first layer is the input layer and has as many neurons as it

has input parameters. The last layer is the output layer with the number of neurons equal to the

number of parameters the network needs to predict. The layers in between the first and last are

hidden layers. Each neuron in these hidden layers are committed to a single standard operation

with an input from the preceding layer and the output going to the next layer. This is done with

connections and each connection has a calibrated weight factor attached to it. It is clear that the

calibration phase, based on the existing data is very important. It also needs to be calibrated so

that when ‘garbage goes in, garbage goes out’.

These neural network tools have a large success in the coastal engineering domain. It is used

to predict armour stability, forces on walls, wave transmission and wave overtopping. From

previous sections, it is seen that calculating overtopping requires different formulae, often

depending on the type of structure. With the Neural Network, the full range of different structures

can be covered, provided that sufficient data is available on which the tool can be ‘trained’, making

it a very useful tool for coastal engineers to quickly calculate a structure.

The Neural Network developed for wave overtopping is based on the large CLASH database.

The input parameters for the program are:

Table 3 Input parameters for the Neural Network

parameter

𝛽 Wave direction

ℎ Water height

𝐻𝑚0 Wave height 𝑇𝑚−1,0 Wave period

ℎ𝑡 Water height on toe


31

𝐵𝑡 Width of toe 𝛾𝑓 Roughness factor

cot (𝛼𝑑) Slope downward of berm

cot (𝛼𝑢) Slope upward of berm

𝐵 Width of berm

ℎ𝑏 Water height on berm

tan (𝛼𝑏) Angle of berm

𝑅𝑐 Freeboard

𝐴𝑐 Armour height

𝐺𝑐 Crest width Figure 29 gives the range of possible structures which can be modelled in the Neural Network.

It is clear that, opposed to PCO, vertical structures are perfectly suited to be calculated with this

tool.

Figure 29 Possible structures which can be modelled with the Neural Network (EurOtop 2007)

2.2.5 CLASH Database

The last described method to predict overtopping is the CLASH database. This was a research

project originating in the EU, where countries all over the world submitted the results from model

tests on wave overtopping. Each model test was described by 31 parameters, both hydraulic and

structural. The result is a database of more than 10,000 results on wave overtopping. This CLASH

database was used in the programming for the Neural Network and it can be used to look up

previously tested structures. When someone wants to find the results on overtopping for a specific

Chapter 2 -Methodology

32

structure, the person can use the Excel file of the database to look up similar structures and

compare results. Sometimes, the structure can already have been tested with the same wave

conditions. In this report, the CLASH database is used to search for additional results for similar

structures compared to the one in the case studies.

2.3 Methodology

In order to find results for the case studies, above tools are used in the next chapter to

calculated the overtopping. This is done systematically for each case, starting with defining the

type of structure. Once the type of structure is defined, as well as other special properties such as

the wave direction and structure roughness, the EurOtop manual is used to calculate the reference

overtopping results for each wave condition mentioned in the case study. For the calculations, the

intermediate steps are shown for one relevant wave conditions or more if it is of any interest.

These results are then put into a table for the EurOtop results. After this, the structure is again

calculated with the van der Meer et al formulae. The results are contained in a similar table from

the EurOtop results, with relevant information. When this is done, the structure is modelled in

PCO if it is possible. When applicable, the necessary changes are noted in order for the model to

be working. The same is done for the Neural Network. Results from both methods are again

summarized in a table with the relevant information. Extra information useful information for the

case, such as the NN input and CLASH data, can be found in it’s respective annex.

After all the results have been calculated, a short summary of the results is given, as well as

figures with the non-dimensional overtopping. A bar graph is also included so that a quick

graphical comparison can be made on which wave condition is the worst. This is supported with

a table containing all the overtopping results for the different methods.

The final results are then shown in an in- and output sheet. This sheet is a short and concise 1-

page item which can be used to quickly validate intermediary results and consult the results. It

contains a figure of the cross section, with some notes on any encountered difficulties. A table and

figure with relevant information and results are also included.

Chapter 3 -Case A

33

Chapter 3 Results

3.1 Case A

3.1.1 Cross section and info

The first case study is a sloping dike, located in the Outer Thames near the Blyth Sands in the

United Kingdom (UK). The south bank has an orientation of 344°N, facing north-west. The dike’s

dimensions are derived from LIDAR, which is an instrument which can measure the distance to

an object with laser pulses. The slope composes of two parts with a different slope, which means

the dike can be considered a composite sloping dike. The first part has a slope of 1:6.667. The

upper slope angle is 1:4.5. The embankment on the seaward face is covered with a rip-rap

roughness. A foreshore is also present with an approximate slope of around 1:100. These details

can be seen in figure 30.

Figure 30 Simplified cross section of case A

The wave conditions are described in Table 4. The values in this table are derived from a

1:1000 year joint probability function. Careful attention has to be given to the wave period as the

peak period is given and not the mean period.

Table 4 Wave conditions for case A, 1:1000 year joint probability

Condition Water level[mODN] Hm0[m] Tp[s] Wave direction N [°]

1 5.17 0.62 2.7 290

2 5.17 0.39 2.1 19

3 5.17 0.72 3.5 55

4 3.39 1.41 4.6 304

5 3.39 0.86 3.1 18

6 3.39 1.16 4.5 52

Another point of attention is the wave direction. The angle given in the table is relative to the

north. An additional calculation is required to translate this in a relative wave direction 𝛽 to the

normal of the dike. This can be explained in Figure 31 where a schematized orientation of the

waves and dike are represented.

Chapter 3 -Case A

34

Figure 31 Wave directions for case A, with the dotted line as the normal to the structure

The resulting relative wave direction to the perpendicular of the dike are respectively 36°, 57°,

19°, 50°, 56° and 22°.

3.1.2 EurOtop calculations

This structure can be seen as a simple slope, described in chapter 2. There are 6 different wave

conditions, but only for one condition, condition 4, the calculation is shown as example. Wave

condition 4 has a water level of 3.39mODN and waves with a wave height 𝐻𝑚0 of 1.41m, a peak

period of 4.6s and relative wave direction 𝛽 of 50°.

First, the run-up is calculated with Equation 3, the probabilistic run-up equation for simple

slopes. No berm is present in this case, which sets the value for the influence factor 𝛾𝑏 equal to 1.

The roughness factor can be acquired from literature. EurOtop (2007) specifies a value of 0.6 for

1-layered rocks on an impermeable core, whereas Hughes (2005) determined a value closer to

0.5 for rip-rap. The safer presumption is to set the value at 0.6. This factor remains the same for

all wave conditions. For oblique attacking waves, a value for the influence factor can be

determined according to Equation 7. Working out equation 7 for a relative wave direction of 50°

gives a value 𝛾𝛽,𝑟 = 0.89. The other influence factors can be found in Table 5 for the other wave

conditions.

𝛾𝛽,𝑟 = 1 − 0.0022|50°| = 0.890

Table 5 Influence factors for case A, calculated with EurOtop

condition γf γβ,r γβ,q γb γv

1 0.600 0.921 0.881 1.000 1.000

2 0.600 0.879 0.819 1.000 1.000

3 0.600 0.958 0.937 1.000 1.000

4 0.600 0.890 0.835 1.000 1.000

5 0.600 0.877 0.815 1.000 1.000

6 0.600 0.952 0.927 1.000 1.000

Chapter 3 -Case A

35

Calculating the breaker parameter requires the knowledge of the wave steepness and thus the

wave length, and the slope. The wave length can be calculated with the spectral mean period

𝑇𝑚−1,0. The ratio between 𝑇𝑝 and 𝑇𝑚−1,0 is around 1.1.

𝐿𝑚−1,0 =𝑔 ∙ (

4.6𝑠1.1 )²

2𝜋= 27.30𝑚

For the slope, careful attention has to be given to the change in slopes at 3.83mODN. As

mentioned in Chapter 2, a composite slope has to be defined. This is initially done by taking an

interval of the dike between 1.5 ∙ 𝐻𝑚0 above and below the still water line, hereby setting the

initial 𝑅𝑢2% = 1.5 ∙ 𝐻𝑚0, as shown in Figure 32. This interval is then between 1.275𝑚𝑂𝐷𝑁 and

5.05𝑚𝑂𝐷𝑁. The calculation process for the composite slope is shown below.

Figure 32 Composite slope in red, with the original structures in black, and the still water line in blue

𝐿𝑠𝑙𝑜𝑝𝑒,𝑑𝑜𝑤𝑛 =3.39𝑚𝑂𝐷𝑁 − 1.275𝑚𝑂𝐷𝑁

0.150= 14.10𝑚

𝐿𝑠𝑙𝑜𝑝𝑒,𝑢𝑝 =5.505𝑚𝑂𝐷𝑁 − 3.39𝑚𝑂𝐷𝑁

0.222= 9.52𝑚

tan (𝛼)𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒 =3 ∙ 𝐻𝑚0

𝐿𝑠𝑙𝑜𝑝𝑒,𝑑𝑜𝑤𝑛 + 𝐿𝑠𝑙𝑜𝑝𝑒,𝑢𝑝= 0.172

𝜉𝑚−1,0 = tan(𝛼) /√𝑠0 = 0.75754

𝑅𝑢2% = (1.65 ∙ 1 ∙ 0.6 ∙ 0.89 ∙ 0.75754) ∙ 1.41𝑚 = 0.941𝑚

𝑅𝑢2%,max = 1.00 ∙ 1 ∙ 0.6 ∙ 0.89 ∙ (4 −1.5

√0.75754) ∙ 1.41𝑚 = 1.714𝑚

An initial run-up of 0.941m is found, as the maximum value is not reached due to the small breaker

parameter. This run-up defines the new interval [1.275mODN; 4.331mODN] and leads to a new

composite slope of 0.158. The run-up value then decreases to 0.866m. 2 Additional iterations give

a final composite slope of tan(𝛼) = 0.157, a breaker parameter of 𝜉𝑚−1,0 = 0.692 and run-up

value of 𝑅𝑢2% = 0.859𝑚. The absolute run-up height for this wave condition is then 4.249𝑚𝑂𝐷𝑁.

The results for the other wave conditions are found in Table 6.

Next, the overtopping is calculated. The formula in EurOtop for simple slopes are given by

equation 8: wave overtopping for simple slopes with wave breaking (𝜉𝑚−1,0 < 5). Besides the

influence factor for wave obliqueness, the other influence factors remain the same as they are in

the run-up formula. For the obliquity, Equation 10 is used for the overtopping influence factor.

𝛾𝛽,𝑞 = 1 − 0.0033|50°| = 0.835

Chapter 3 -Case A

36

For condition 4, 𝛾𝛽,𝑞 = 0.835. There is no vertical wave wall present in this case, which makes this

influence factor 𝛾𝑣 = 1. Using the composite slope calculated in the previous run-up section, the

overtopping for this condition using Equation 8 is 𝑞 = 8.12 ∙ 10−11 𝑚³

𝑚∙𝑠.

𝑞 = [0.067

√0.157∙ 1 ∙ 0.692 ∙ exp (−4.75 ∙

2.34𝑚

0.692 ∙ 1.41𝑚 ∙ 1 ∙ 0.6 ∙ 0.835 ∙ 1)] √𝑔 ∙ 1.41𝑚3 = 8.12 ∙ 10−11 𝑚3/𝑠/𝑚

𝑞𝑚𝑎𝑥 = [0.2 ∙ exp (−2.6 ∙2.34𝑚

1.41𝑚 ∙ 0.6 ∙ 0.835)] √𝑔 ∙ 1.41𝑚3 = 0.1 ∙ 10−3 𝑚3/𝑠/𝑚

For the other wave conditions, the reader is referred to Table 6. Important to note is that for every

condition, the validity of the equations used above is correct as the breaker parameters are well

below the value of 5 and wave breaking occurs.

Table 6 Run-up and overtopping for case A, calculated with EurOtop

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,0[−] Ru2%[m] q[m3/s/m]

1 0.560 0.62 9.41 0.865 0.489 1.60E-05

2 0.560 0.39 5.69 0.848 0.288 7.21E-09

3 0.560 0.72 15.81 1.040 0.711 5.15E-04

4 2.340 1.41 27.30 0.696 0.859 8.13E-11

5 2.340 0.86 12.40 0.571 0.424 1.52E-21

6 2.340 1.16 26.13 0.753 0.817 4.93E-11

3.1.3 van der Meer et al: updated EurOtop

Continuing from the previous section, the new updated formulae for overtopping are used to

calculate condition 4. Using van der Meer et al’s equation 26.a for simple slopes, the overtopping

for wave condition 4 can be calculated as followed.

𝑞𝑉𝑑𝑀 = [0.023

√0.157∙ 1 ∙ 0.692 ∙ exp [− (2.7 ∙

2.34𝑚

0.692 ∙ 1.41𝑚 ∙ 1 ∙ 0.6 ∙ 0.835 ∙ 1)

1.3

]] √𝑔 ∙ 1.41𝑚3 = 1.67 ∙ 10−13 𝑚3/𝑠/𝑚

𝑞𝑉𝑑𝑀,𝑚𝑎𝑥 = [0.09 ∙ exp [− (1.5 ∙2.34𝑚

1.41𝑚 ∙ 0.6 ∙ 0.835)

1.3

]] √𝑔 ∙ 1.41𝑚3 = 1.525 ∙ 10−5 𝑚3/𝑠/𝑚

The influence factors remain the same for this updated formula and the composite slope

calculated in the run-up is still used. The update overtopping discharge is 𝑞𝑉𝑑𝑀 = 1.67 ∙

10−13𝑚3/𝑠/𝑚.

Table 7 Run-up and overtopping for case A, calculated with van der Meer et al (2013)

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,0[−] Ru2%[m] qVdM[m3/s/m]

1 0.560 0.62 9.41 0.865 0.489 9.71E-06

2 0.560 0.39 5.69 0.848 0.288 4.11E-10

3 0.560 0.72 15.81 1.040 0.711 5.05E-04

4 2.340 1.41 27.30 0.696 0.859 1.67E-13

Chapter 3 -Case A

37

5 2.340 0.86 12.40 0.571 0.424 1.75E-32

6 2.340 1.16 26.13 0.753 0.817 8.57E-14


The structure here is suitable for use in PC-Overtopping (PCO) and is straightforward to enter.

A screenshot of the input can be seen in Figure 33. Two slopes are entered with the correct

dimensions. The roughness factor is manually entered as no information on rip-rap is found in

PCO. The peak period is automatically converted to the correct spectral mean period.

Figure 33 PCO input for case A, wave condition 4

Figure 34 PCO output for case A, wave condition 4

Chapter 3 -Case A

38

The program gives a result for run-up 𝑅𝑢2%,𝑉𝑑𝑀 = 0.917𝑚 and the overtopping discharge 𝑞𝑉𝑑𝑀 =

0 𝑚3/𝑠/𝑚 for a composite slope angle of 0.158. The zero overtopping is because the calculated

absolute run-up is lower than the crest.

Table 8 Run-up and overtopping for case A, calculated with PCO

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,0,PCO[−] Ru2%,PCO[m] qPCO[m3/s/m]

1 0.560 0.62 9.41 0.865 0.518 3.80E-05

2 0.560 0.39 5.69 0.848 0.298 -

3 0.560 0.72 15.81 1.040 0.753 9.30E-04

4 2.340 1.41 27.30 0.712 0.917 -

5 2.340 0.86 12.40 0.584 0.452 -

6 2.340 1.16 26.13 0.771 0.872 -

3.1.5 Neural network

The neural network can also be used with a significant simplicity for this structure. There is no

toe structure present so ℎ = ℎ𝑡 and 𝐵𝑡 = 0𝑚 and the berm has a width of 𝐵 = 0𝑚. The armour

width is also 𝐺𝑐 = 0𝑚 and the armour freeboard is equal to the normal freeboard 𝐴𝑐 = 𝑅𝑐. The

values of the down- and upward slope can be derived in a similar method as described above for

the run-up in EurOtop: an interval of water level ± 1.5 ∙ 𝐻𝑚0 is considered again. For this structure,

there is only 1 slope on each side of the berm, which makes it easy to determine the slopes,

cot(𝛼𝑑) = 6.667 and cot(𝛼𝑢) = 4.5. The berm height for condition 4 is -0.44m as the berm is

located above water level. The results are found in Table 9 while the complete input can be found

in Table A-1 in Annex A.

Table 9 Overtopping for case A, calculated with Neural Network, and the input parameters

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,0,PCO[−] Ru2%,PCO[m] qPCO[m3/s/m]

1 0.560 0.62 9.41 0.866 - 4.10E-04

2 0.560 0.39 5.69 0.849 - 2.00E-05

3 0.560 0.72 15.81 1.041 - 8.55E-04

4 2.340 1.41 27.30 0.692 - 1.28E-04

5 2.340 0.86 12.40 0.568 - 0.00E+00

6 2.340 1.16 26.13 0.748 - 1.29E-04

For condition 4, the overtopping is found to be 1.28 ∙ 10−4𝑚3/𝑠/𝑚. The other conditions are found

in Table 9.

3.1.6 CLASH-database

Lastly, the CLASH-database is searched to find similar structures and their overtopping values.

Finding the correct filters to receive similar structures can be quite difficult. A preliminary search,

Chapter 3 -Case A

39

showed that there are no structures with the same slope variation in the database. Therefore, the

composite slope is simplified to a single slope. The filter on the column of cot (𝛼)𝑖 is set on values

between 5.8 and 6.1 as the total slope has an average slope of around 5.91. The width 𝐵 of berm

is 0 which results in 229 remaining cases. The following filter which can be used is to select a

range of wave steepness’s. The smallest steepness for case A is 0.044 and the largest 0.069. The

range is thus picked between 0.04 and 0.08. This leaves 58 cases. An additional filter on the

parameter 𝑅𝑐/𝐻𝑚0 is placed so that similar cases are picked. For the lower water level of 3.39, this

parameter ranges between 1.66 and 2.72 whereas for the higher water level conditions, it ranges

between 0.78 and 1.4. The range of the remaining cases goes from 0 to 1.5 so the limit is set at

greater than 0.7. This results in 7 remaining cases, which can be found in Table A-2 in Annex A.

3.1.7 Summary

The results of the different methods for all the conditions can be seen in the following figures.

Conditions where there is a very low overtopping are left out of the dimensionless figures as they

have nog significant practical meaning besides a mathematical value. A complete summary on

every result and graphic can be found in Annex A. Figure 35 is the graphical comparison of each

method for each wave condition. Figure 36 represents the relative run-up versus the breaker

parameter, while figure 37 is the relative overtopping versus the relative freeboard.

Figure 35 Different overtopping results for case A

0,E+00

1,E-04

2,E-04

3,E-04

4,E-04

5,E-04

6,E-04

7,E-04

8,E-04

9,E-04

1,E-03

1 2 3 4 5 6

Wave condition

EurOtop

VdM

PCO

NN

𝑞[𝑚

3/

𝑠/𝑚

]

Chapter 3 -Case A

40

Figure 36 Wave run-up for smooth and straight slopes: breaking waves, case A

Figure 37 Overtopping for smooth and straight slopes: breaking waves, case A

Only the relevant wave conditions are represented as very low overtopping only has a

mathematical value, and not a practical one. As can be seen on Figure 35 and 37, only 2 wave

conditions remain: wave condition 1 and 3 with the largest overtopping for condition 3, see Table

10. It is clear that for the lower water level conditions, the overtopping is negligible. The smaller

slopes at lower levels causes smaller 𝜉𝑚−1,0 due to the milder slope at the bottom part of the

structure, and thus smaller run-up. When the water level increases, 𝜉𝑚−1,0 increases as well

because of the steeper upper slope. This increases the risks by a twofold as the absolute run up

value is determined by the increase in run-up and the increase in still water line. However, the

wave heights are small at high water levels which limits the overtopping at a safe value.

Looking at the other methods, it is clear that most values fall between the 90% confidence

range of the EurOtop formula. The new updated formulae proposed by van der Meer et al predict

smaller overtopping values. The difference in some cases are very small though. The Neural

Network mostly gives a larger prediction as well as PCO for this case.

Table 10 Case A wave condition 3 different methods

condition qEurOtop[𝑙/𝑠/𝑚] qVdM [𝑙/𝑠/𝑚] qPCO[l/s/m] qNN[l/s/m]

3 0.515 0.505 0.930 0.855

Based on these results and the simplicity of the structure, it is safe to say that the results are fairly

reliable. The overtopping is also limited to a safe value of < 1 𝑙/𝑠/𝑚. The end results can be seen

in Annex J as an in- and output sheet.

0

0,5

1

1,5

2

2,5

3

3,5

4

0 1 2 3 4 5 6

EurOtop(2007)

PC-Overtopping

PCO

5%

95%

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

0 0,5 1 1,5 2

EurOtop(2007)

VdM & B (2013)

PC-Overtopping

Neural Network

CLASH

01,0

*

3

0

/1

[]

tan

9.8

1

mm

bm

HL

qq

H

Eq. 3.a

Eq. 3.b

Eq. 8.a

5

%

9

5%

Chapter 3 -Case B

41

3.2 Case B


The next case is somewhat more complicated. At first glance, the structure appears to be a

simple slope found in the Outher Thames in Southend, UK. There are however a few additional

characteristics which make the structure more complex than a simple sloping dike. At the end of

the sloping beach, a vertical wall is found. It also has a bullnose at the top of the structure. The

beach is covered in shingle with a slope of 1:10 and starts at 0.2mODN and ends at 5.2mODN in

healthy conditions. However, the beach is prone to erosion which can lead to a reduction in height

of the beach to a level of 4.2mODN in extreme conditions. No information is given on the lower

beach level in eroded condition, so it is assumed to stay constant. The vertical wall goes up until

5.4mODN where the bullnose starts. The bullnose has a height of 0.3m, as well as a width of 0.15m,

which makes the crest of the structure at 5.7mODN. The vertical wall thus has a height ranging

from 0.5m in healthy conditions to 1.5m in eroded conditions. The wall has an orientation of 180°

relative to the North. The cross section can be seen in Figure 38, with a simplification of the

bullnose as a parapet wall. The wave conditions are given in Table 11 for a 1:200 year joint

probability function.

Figure 38 Simplified cross section case B Table 11 Wave conditions case B for a 1:200 year joint probability function

Condtion Water level[mODN] Hm0[𝑚] T𝑚−1,0[s] Wave direction N [°]

1 2.40 1.32 5.80 30

2 2.90 1.60 5.40 30

3 3.35 1.55 5.00 30

4 3.75 1.25 4.50 30

5 4.22 0.95 3.90 30

6 4.70 0.50 2.90 30

The cases where the beach is in a healthy state with the toe level at 5.2𝑚𝑂𝐷𝑁 are noted with a

subscript a. The eroded conditions are noted with the subscript b. So condition 3b has a wave

height of 1.55𝑚 and the toe of the vertical structure at 4.2𝑚𝑂𝐷𝑁 in eroded conditions.

All wave conditions are assumed to have waves coming from 120°N so that 𝛽 = 30°. Besides

the conditions given in Table 11, extra information was given on future conditions with effects of

the climate change. These effects are a uniform rise of sea level of 30cm and an increase in wave

Chapter 3 -Case B

42

heights and periods. Only for the severe cases in Table 11, conditions for climate change are

applied.

Depending on the water level, the dike can be seen as different structures. For low water levels,

the structure can be seen as a simple sloping dike, whereas for high water levels combined with

eroded conditions, a vertical wall structure arises. This creates a certain difficulty in the

calculation as each type of structure requires a different approach. To see what the best solution

is, every type of structure is calculated, where applicable.

3.2.2 EurOtop

As mentioned before, there are several types of structures which can be seen in this case. It

resembles most on an emergent vertical wall structure which is discussed in Chapter 2. These

structures however have some strict conditions in which they can be applied. EurOtop mentions

that outside of this applicability range, the normal formula for simple slopes can be used,

independent of water level. Another possibility is when the water level reaches the vertical wall,

is to assume the structure as a vertical wall. Each type is carefully explained in the following text

and compared. For an easier comparison, some factors are ignored such as wave direction and the

recurve. After this comparison, the most appropriate structure is picked and applied to all wave

conditions, including climate change and applying the correct influence factors.

Type 1: A simple slope

First, the dike is considered to be a simple slope, even when the water level reaches the vertical

wall. When the water level reaches the vertical wall, it is assumed that the slope before the wall is

extrapolated until crest level of 5.7mODN. This case thus increases the width of the beach and

removes the wall and recurve as shown in Figure 39.

Condition 5b, with a water level of 4.22m and wave height and period of respectively 0.95m

and 3.90s is used as an example for the calculations.

Figure 39 Extrapolation of the dike in case of eroded conditions and when the water level reaches the vertical wall

The first problem with this assumption is that the slope of the beach, 1:10-12.5 is not

considered as a slope according to the EurOtop. It is also too steep to be considered a berm which

have slope angles of 1:15 or less. There is currently no suggestion to approach this problem other

than the use of the formulae for simple slopes. It is thus assumed that the structure is a simple

slope in this type.

Equation 3 is used again for the run-up on simple slopes to calculate one condition as example.

The roughness factor 𝛾𝑓 for a shingle beach can be assumed 1. The wave obliquity is currently

Chapter 3 -Case B

43

ignored and set at 1. No berm is present in this case so the value is also 1. As no slope change is

present, no iteration is needed and the breaker parameter is:

𝜉𝑚−1,0 =0.08

√0.95𝑚/23.75𝑚= 0.40

The run-up value is then 𝑅𝑢2% = 0.627m with the use of equation 3.a as the breaker parameter is

well in wave breaking range.

For the overtopping, the vertical wall requires the calculation of the factor 𝛾𝑣 . This factor can

be calculated with Equation 9 but has strict conditions.

𝛾𝑓 = 1.35 − 0.0078 ∙ 90° = 0.648

Some of these conditions given in Chapter 2, are not fulfilled however, so the values might be

skewed. In order to have a relevant comparison between the different structures, the factor 𝛾𝑣 is

applied to all conditions, whether the range of application is met or not. This is a second problem

with assuming a simple slope as type of structure. The boundary conditions are:

The average slope of 1.5 ∙ 𝐻𝑚0 below still water line to the foot of the wall (excluding a

berm) should be between 1:2.5 and 1:3.5: Not fulfilled for any condition.

The width of all the berms together should be no more than 3 ∙ 𝐻𝑚0: Fulfilled.

The foot of the wall must be between 1.2 ∙ 𝐻𝑚0 under and above the still water line:

Fulfilled for every condition but 1a, 1b, and 2a.

The minimum height of the wall is about 0.5 ∙ 𝐻𝑚0, the maximum around 3 ∙ 𝐻𝑚0: Not

fulfilled for condition 1a, 2a, 3a, 4a, and 5a.

Knowing all influence factors, the overtopping can be calculated with Equation 8 for condition 5b.

𝑞 = [0.067

√0.08∙ 1 ∙ 0.40 ∙ exp (−4.75 ∙

1.48𝑚

0.40 ∙ 0.95𝑚 ∙ 1 ∙ 1 ∙ 1 ∙ 0.648)] √𝑔 ∙ 0.95𝑚3 = 1.10 ∙ 10−13 𝑚3/𝑠/𝑚

𝑞𝑚𝑎𝑥 = [0.2 ∙ exp (−2.6 ∙1.48𝑚

0.95𝑚 ∙ 1 ∙ 1)] √𝑔 ∙ 0.95𝑚3 = 1.01 ∙ 10−2𝑚3/𝑠/𝑚

With these factors, a freeboard of 𝑅𝑐 = 1.48𝑚 and a breaker parameter of 𝜉𝑚−1,0 = 0.40 well

below the wave-breaking limit, the overtopping becomes 𝑞 = 1.10 ∙ 10−13 𝑚3/𝑚/𝑠 for wave

condition 5B. Annex B contains the results for the other wave conditions in table B-1.

Type 2: A simple slope going into a vertical wall

Another possibility can be to see the structure as a vertical wall when the water level reaches

the toe of the wall. For lower water level, the structure can then be seen again as a simple slope

with a wave wall on top. Besides the problem of the slope which is too small and the applicability

of the parapet wall, another problem is present here as well, namely the transition from one type

of structure to another is not a realistic or a practical statement and should be carefully looked at.

A composite structure is not considered as it not applicable here as the beach is too long to be

considered a toe.

Vertical walls don’t have any run-up, so only the wave overtopping is calculated. The first

procedure in the case of a vertical wall, is to check the impulsiveness parameter ℎ∗ using equation

16.a. For wave condition 5b, the water depth at the toe is ℎ𝑠 = 0.02𝑚.

Chapter 3 -Case B

44

ℎ∗ = 1.35 ∙

0.02𝑚

0.95𝑚∙

2𝜋0.02𝑚

9.81𝑚/𝑠2 ∙ (3.90𝑠)2= 2.39 ∙ 10−5

The wave conditions are considered impulsive when ℎ∗ ≤ 0.2 which they definitely are.

Another parameter, ℎ∗𝑅𝑐

𝐻𝑚0, has to be checked whether waves are breaking or not. If this parameter

is larger than 0.03, waves are not breaking and the use of equation 18.a is proposed. For

parameters lower than 0.02, equation 18.b is used as waves are breaking. In this case however,

due to the beach, it can be assumed that waves between 0.02 and 0.03 are also breaking and

equation 18.b is proposed for these cases.

Table 12 Boundary conditions for case B, equation 18

𝑐𝑜𝑛𝑑. ℎ∗[−] ℎ∗

𝑅𝑐

𝐻𝑚0[−]

Eq.

5b 2.39E-05 3.73E-05 18.b

6b 5.14E-02 1.03E-01 18.a For wave condition 5b waves are definitely breaking and equation 18.b is used, Table 12.

𝑞


3= 2.7 ∙ 10−4 (ℎ∗

𝑅𝑐

𝐻𝑚0)

−2.7

= 1.24 ∙ 10−3𝑚3/𝑚/𝑠

The results when the water level did not reach the vertical wall structure are the same as in

the previous part, for a simple slope with an influence factor for the vertical wall. The wave

conditions where the SWL did reach the vertical wall, the results are found in the Table B-2 in

Annex B.

Third type: Emergent wall going into a vertical wall

The EurOtop manual also mentions a third type of structure which can be applied here, namely

an emergent vertical wall where the toe of the wall lies above the water level (ℎ𝑠 < 0). The

conditions in which this structure type can be applied are very strict as there is limited data

available on these kind of structures, as written in chapter 2. The shore in front of the emergent

wall is in this case 1:10 for non-eroded conditions and about 1:12.5 for eroded. This is in the

application range of Equation 19. The other boundary conditions are tested for wave condition 4b

as condition 5b is considered a vertical wall due to the high water level. For a freeboard 𝑅𝑐 =

1.95𝑚, a wave height 𝐻𝑚0 = 1.25𝑚, and a wave period 𝑇𝑚−1,0 = 4.5𝑠, the boundary conditions

are written here for wave condition 4b while the other wave conditions are found in Table 13.

𝑠𝑚−1,0 = 0.040 ≥ 0.02

0.55 ≤ 𝑅𝑐

𝐻𝑚0= 1.56 ≤ 1.6

2.0 < cot(𝛼𝑓) 𝑠𝑚−1,00.33

𝑅𝑐

𝐻𝑚0= 6.71 ≮ 5

Table 13 Boundary conditions for emergent wall structure for case B

𝑐𝑜𝑛𝑑. 𝑠0[−] 𝑅𝑐/𝐻𝑚0[−] cot(𝛼𝑓) 𝑠00.33𝑅𝑐/𝐻𝑚0[−]

1a 0.025 2.50 7.41

Chapter 3 -Case B

45

2a 0.035 1.75 5.80

3a 0.040 1.52 5.23

4a 0.040 1.56 5.37

5a 0.040 1.56 5.39

6a 0.038 2.00 6.80

1b 0.025 2.50 9.27

2b 0.035 1.75 7.25

3b 0.040 1.52 6.54

4b 0.040 1.56 6.71

5b - - -

6b - - -

It can be seen that several boundary conditions are not met. It is however important to note

that these are not far outside the application range so that despite this issue, the structure is still

seen an emergent wall. The results might however be somewhat skewed by this and have to be

interpreted accordingly. For example, condition 4b is written here as example, using equation 19.

The other results are found in Annex B, Table B-3. The results for wave condition 5b and 6b are the

same as those found in previous section for vertical walls.

𝑞 = (0.043 exp (−2.16 cot(𝑚) 𝑠𝑚−1,0

0.33𝑅𝑐

𝐻𝑚0) √𝑔𝐻𝑚0

3 √cot(𝑚) 𝑠𝑚−1,0 = 1.34 ∙ 10−7𝑚3/𝑚/𝑠

Picking the best type

Each type of structure is calculated with the appropriate formula. The results can be

summarized in a single graph, Figure 40.

Figure 40 Results of different type of structures for case B, not considering the parapet and wave obliquity

The highest values are found for an emergent vertical wall structure at low water levels and a

vertical wall structure at high water level. Another conclusion is that the transition from emergent

to normal vertical wall structure is more realistic as seen in Figure 41. In this graph, the three

different structure interpretations are plotted as overtopping versus water level. The parameters

1,00E-17

1,00E-15

1,00E-13

1,00E-11

1,00E-09

1,00E-07

1,00E-05

1,00E-03

1,00E-01

0 1 2 3 4 5 6 7 8 9 10 11 12

Ove

rto

pp

ing

[m³/

m/s

]

Case

Slope-VerticalWall

Slope

Emergent vertical wall-vertical wall

Chapter 3 -Case B

46

used in the calculations for this graph are the same as for wave condition 4b: eroded conditions

with wave height 0.95𝑚 and period 3.9𝑠.

Figure 41 Overtopping according to different structure interpretations with wave height 𝐻𝑚0 = 0.95𝑚, wave period 𝑇𝑚−1,0 = 0.93𝑠, slope 1:12.5, vertical wall at 4.2mODN. Grey is overtopping for a simple slope with a vertical

wall factor; Blue is assuming a vertical wall with ℎ∗; Green is the emergent wall with foreshore slope 1:12.5

For these practical reasons, the structures is viewed as an emergent vertical wall, going into a

vertical wall when the water level reaches this vertical wall.

The next step is now including the other factors such as wave obliquity. The problem is that

there is also not much data for oblique wave attack on vertical structures. The EurOtop only has

suggestions at discrete values of wave obliquity when the waves are impulsive. Luckily, the waves

are attacking in a 30° angle, which is one of the angles which EurOtop has a suggestion for. This

suggestion is found in chapter 2 under Equation 22.

𝑞


3= 8.0 ∙ 10−6 (ℎ∗

𝑅𝑐

𝐻𝑚0)

−4.2valid for 0.07 ≤ ℎ∗

𝑅𝑐

𝐻𝑚0 (22.c)

For other values of ℎ∗𝑅𝑐

𝐻𝑚0 no further information is given and it is assumed that the standard

formulae are used for impulsive overtopping at vertical structures. From Table 12 it is clear that

only wave condition 6b is in the applicability range of 22.c so that only this wave condition is

‘affected’ by the wave obliquity.

The recurve at the end of the vertical structure also causes a reduction in overtopping. In order

to calculate this reduction factor, the reader can use the chart in Figure 19 in chapter 2. This chart

is not for emergent structures ℎ𝑠 < 0. So the factor can only be calculated for when the water

level reaches the toe and is considered a vertical wall, like for wave conditions 5B and 6B. The

bullnose is considered to have a width 𝐵𝑟 of 0.15m and height ℎ𝑟 of 0.3m. For wave condition 5B,

𝑃𝑐 is 1.18m and thus:

𝑅0∗ = 0.25

ℎ𝑟

𝐵𝑟+ 0.05

𝑃𝑐

𝑅𝑐= 0.540

1,E-19

1,E-17

1,E-15

1,E-13

1,E-11

1,E-09

1,E-07

1,E-05

1,E-03

1,E-01

1,E+01

3,5 4 4,5 5 5,5

slope

vertical

emergent

𝑊𝑎𝑡𝑒𝑟 𝑙𝑒𝑣𝑒𝑙 [𝑚𝑂𝐷𝑁]

𝑞[𝑚

3/

𝑠/𝑚

]

Eq. 8

Eq. 18.b

Eq. 19

Chapter 3 -Case B

47

𝑚 = 1.1√ℎ𝑟

𝐵𝑟+ 0.2

𝑃𝑐

𝑅𝑐= 1.715

𝑚∗ = 𝑚(1 − 𝑘23) = 1.372

0.540 <𝑅𝑐

𝐻𝑚0= 1.56 < 𝑅0

∗ + 𝑚∗ = 1.912

𝑘 = 1 −1

𝑚(

𝑅𝑐

𝐻𝑚0− 𝑅0

∗) = 0.405

For wave condition 6b 𝑅𝑐

𝐻𝑚0 is rather large and causes the reduction factor 𝑘 to drop below 0.05.

Values lower than this are however not reliable so that it is set at 0.05 for this wave condition. The

values for 𝑘 for the other wave conditions are found in Table 14.

Table 14 Reduction factor for bullnose wall for case B

𝑐𝑜𝑛𝑑. 𝑘

1a 1.00

2a 1.00

3a 1.00

4a 1.00

5a 1.00

6a 1.00

1b 1.00

2b 1.00

3b 1.00

4b 1.00

5b 0.41

6b 0.05

Applying the recurve factor 𝑘 to the value from wave condition 5b then means an overtopping

value of

𝑞 = 0.405 ∙ 1.24 ∙ 10−3𝑚3/𝑚/𝑠 = 0.502 ∙ 10−3𝑚3/𝑚/𝑠

The other results are now summarized in Table 15. It’s important to note that for the wave

conditions where an emergent wall is assumed, no wave obliquity and bullnose could be modelled

and that for wave condition 5b no wave obliquity is modelled as well due to the application range.

Table 15 Results for case B using EurOtop, Emergent structures and Vertical wall structure

condition 𝑅𝑐[𝑚] 𝐻𝑚0[𝑚] 𝐿𝑚−1,0[𝑚] ℎ∗[−] 𝑞[𝑚3/𝑠/𝑚]

1a 3.30 1.32 52.52 - 4.53E-08

2a 2.80 1.60 45.53 - 1.68E-06

3a 2.35 1.55 39.03 - 5.14E-06

4a 1.95 1.25 31.62 - 2.73E-06

5a 1.48 0.95 23.75 - 1.75E-06

6a 1.00 0.50 13.13 - 3.21E-08

1b 3.30 1.32 52.52 - 7.39E-10

2b 2.80 1.60 45.53 - 6.56E-08

Chapter 3 -Case B

48

3b 2.35 1.55 39.03 - 2.73E-07

4b 1.95 1.25 31.62 - 1.34E-07

5b 1.48 0.95 23.75 2.39E-05 5.04E-04

6b 1.00 0.50 13.13 5.14E-02 1.65E-05

3.2.3 van der Meer et al: updated EurOtop

Van der Meer et al suggests for vertical structures to follow a decision chart according to

several parameters. This chart is shown in Figure 28. It is however not clear if the paper of van

der Meer et al is valid on negative ℎ𝑠. It is assumed that this is not the case, so that no alternative

solutions are given on those where an emergent structure is applied. This means that only wave

condition 5b and 6b are calculated with the new updated equations.

The first step is to check whether the structure is a composite or vertical wall. Because of the

smooth sloping beach and the large width, it is not realistic to see it as a composite wall, ℎ = 𝑑.

Instead, the beach is considered as a foreshore For all water levels below the toe, the waves will

break before reaching the wall. For higher water levels, the parameter ℎ2/(𝐻𝑚0𝐿𝑚−1,0) should be

smaller than 0.23. Checking for wave condition 5B, this is true as ℎ2/(𝐻𝑚0𝐿𝑚−1,0) = 1.77 ∙ 10−5.

The other wave conditions are calculated in Table 16.

Table 16 Wave breaking according to van der Meer et al for Case B

𝑐𝑜𝑛𝑑. ℎ𝑠²/𝐿𝑚−1,0/𝐻𝑚0

5b 0.00002

6b 0.034

It is then safe to say that wave breaking does occur for every wave condition. Depending on the

ratio 𝑅𝑐/ 𝐻𝑚0, Equation 32 is used. For wave condition 5B, 𝑅𝑐/ 𝐻𝑚0 = 1.56 and the overtopping

becomes 𝑞𝑛𝑒𝑤 = 37.0 ∙ 10−3𝑚3/𝑚/𝑠 with equation 32.b.

There is no further information given on how to handle wave obliquity so an improvised

method is used. When calculating the overtopping according to EurOtop for a certain wave

condition, it is calculated once with and once without wave obliquity. The ratio is then calculated

to have an alternative factor for wave obliquity which can be applied to the value derived from

van der Meer et al. For example, for wave condition 6B, ℎ∗𝑅𝑐

𝐻𝑚0= 0.103

𝑞𝑛𝑜𝛽,𝐸𝑢𝑟𝑂𝑡𝑜𝑝 = 5.07 ∙ 10−4𝑚3/𝑚/𝑠

𝑞𝛽,𝐸𝑢𝑟𝑂𝑡𝑜𝑝 = 3.30 ∙ 10−4𝑚3/𝑚/𝑠

𝑘𝛽 = 0.651

Wave condition 5b is actually the only wave condition where a wave obliquity is applied due to

ℎ∗𝑅𝑐

𝐻𝑚0. Therefore, 𝑘𝛽 = 1 for the other wave conditions.

Chapter 3 -Case B

49

The recurve factor is taken the same as in EurOtop so that for wave condition 5B: 𝑘𝛽 = 1 and

𝑘 = 0.405. This makes the overtopping value according to van der Meer et al for condition 5B 𝑞 =

14.5 ∙ 10−3𝑚3/𝑚/𝑠. Other results are in Table 17.

Table 17 Overtopping results for case B calculated with van der Meer et al updated equations for vertical structures

condition 𝑅𝑐[𝑚] 𝐻𝑚0[𝑚] 𝐿𝑚−1,0[𝑚] ℎ∗[−] 𝑞[𝑚3/𝑠/𝑚]

5b 1.48 0.95 23.75 2.39E-05 1.50E-02

6b 1.00 0.50 13.13 5.14E-02 3.23E-05


PCO is not applicable here as no recurve can be modelled or slopes milder than 1:8 as first or

last slope. This section is thus skipped for this case.


Configuring the details for the neural network for this structure is quite complicated. Modelling

a recurve can be done by connecting the toe of the vertical wall and the upper part of the recurve,

with the same freeboard. This becomes the new wall with an equivalent slope as can be seen in

Figure 42. For the eroded condition, the wall has a total height of 1.5m and a width of 0.3 leaning

seawards. The slope of this wall is then cot (−0.15𝑚

1.5).

Figure 42 Recurve modelling in Neural Network

No toe is present in this case, and the berm is located at the toe of the vertical wall, with 0 width.

For the low water levels, this berm is located higher than the allowed range of ℎ𝑏 given in the

manual of the Neural Network. This manual allows the berm to be located 5 ∙ 𝐻𝑚0 below still water

line, and 1 ∙ 𝐻𝑚0 above. The limit for slopes is 1:10 so for the eroded conditions this has to be

adjustments accordingly as can be seen in Table B-4 in Annex B. Table 18 contains the overtopping

results.

Table 18 Neural Network Case B

Condition 𝑅𝑐[𝑚] 𝐻𝑚0[𝑚] 𝐿𝑚−1,0[𝑚] ℎ∗[−] 𝑞[𝑚3/𝑠/𝑚]

1a 3.30 1.32 52.52 - 1.38E-04

2a 2.80 1.60 45.53 - 6.34E-04

3a 2.35 1.55 39.03 - 9.01E-04

Chapter 3 -Case B

50

4a 1.95 1.25 31.62 - 5.58E-04

5a 1.48 0.95 23.75 - 3.05E-04

6a 1.00 0.50 13.13 - 2.74E-05

1b 3.30 1.32 52.52 - 1.35E-04

2b 2.80 1.60 45.53 - 7.26E-04

3b 2.35 1.55 39.03 - 1.27E-03

4b 1.95 1.25 31.62 - 9.06E-04

5b 1.48 0.95 23.75 2.39E-05 6.78E-04

6b 1.00 0.50 13.13 5.14E-02 1.45E-04

For condition 5B, the results is an overtopping value of 𝑞 = 0.678 ∙ 10−3𝑚3/𝑚/𝑠.

3.2.6 Clash Database

The same process of placing correct filters is applied on the CLASH database. First off, a filter

is set on the upper slope of the berm such that only slopes of cot(𝛼𝑢) ≤ 0 remain. Another filter is

placed on the lower slope, cot(𝛼𝑢) ≥ 5. These filters result into 7 remaining cases from dataset

036. In each of these case, the water level is located near the berm, below and above which is

similar to the desired case B. They do have a berm width which is not present in case B. The ratio

of 𝑅𝑐/𝐻𝑚0 and 𝑠0 are in a similar range like case B. The results can be consulted in Table B-5 in

Annex B.

3.2.7 Summary

The results of the previous calculations can be seen in Appendix B Table B-6. A short summary

is presented by figures 43, 44 and 45. The conditions with the largest overtopping, both for eroded

and healthy conditions, are given in Table 19. These wave conditions are also checked with the

climate change, give in Table 20.

Table 19 Overtopping for Case B

Condition qEurOtop[𝑙/𝑠/𝑚] qVdM [𝑙/𝑠/𝑚] qPCO[l/s/m] qNN[l/s/m]

3a 0.00514 - - 0.901

5b 0.504 15.0 - 0.678

Table 20 future climate change conditions for the worst current conditions

Condtion Water level[mODN] Hm0[𝑚] Tp[s] Wave direction β [°] Toe level[mODN]

3a” 3.65 1.71 5.30 30 5.2

5b” 4.52 1.05 4.10 30 4.2

The results for these climate change conditions are found in Table 21. Nothing changed regarding

the calculation method and these are calculated according to the methods described above.

Table 21 Overtopping results for climate change conditions applied to the worst current calculated cases


3a” 0.067 - - 2.08

Chapter 3 -Case B

51

5b” 3.06 19.10 - 2.38

Regarding the graphs, the results of the emergent structure cannot be put in the same graph as

those of a normal vertical structure because the formula is entirely different. Two different graphs

must then be used to present the results of case B. Figure 43 is again a bar graph to compare the

different methods for each wave condition. In figure 44, the relevant results of the wave conditions

when an emergent structure is used, are shown. In the other graph, figure 45, those of the vertical

wall structure are presented.

Figure 43 Overtopping results for case B

Figure 44 Overtopping for emergent vertical wall structures for Case B

0,E+00

2,E-03

4,E-03

6,E-03

8,E-03

1,E-02

1,E-02

1,E-02

2,E-02

1a 2a 3a 4a 5a 6a 1b 2b 3b 4b 5b 6b

Wave condition

EurOtop

VdM

PCO

NN

𝑞[𝑚

3/𝑠

/𝑚]

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

0 1 2 3 4 5 6

EurOtop(2007)

VdM & B (2014)

Neural Network

CLASH

*

0.3

3

1,0

3

09.

[81

]

m

m

qq

ms

H

* 0.33

1,0

0

[ ]cc m

m

RR ms

H

95%

Eq. 19

Chapter 3 -Case B

52

Figure 45 Overtopping for vertical structures, impulsive condition, Case B

Looking at figure 43, the reader can concluded that one value really springs out: the van der Meer

et al result for wave condition 5b. Considering the low water depth there at the toe of the structure,

and considering a vertical structure, this result might not be reliable. Also seen from this figure is

that EurOtop predicts the highest overtopping for 5b but NN gives the highest prediction at 3b.

When creating the dimensionless graph for the vertical wall, it’s important to divide the

relative overtopping by the parapet reducing factor k and the obliquity factor 𝑘𝛽 in order to have

a correct graph. The CLASH database where a positive ℎ𝑠 could be defined are also plotted in

Figure 45.

When studying the figures with the relative overtopping, it can be concluded that for the

emergent walls, the data of other methods do not lay in the confidence interval of emergent

structures. Despite the fact that the boundary conditions for the formula of emergent wall are

generally well met for case B, the reliability is a bit low. This can be due to the fact that other

methods do not have much information on these kinds of structures.

However, when the structures is viewed as a vertical wall, when the still water line is high, the

calculated data from EurOtop matches the data from the other methods quite well, despite the

result from van der Meer et al for wave condition 5b.. It are these conditions, with high water level,

which are of significance when calculating overtopping. Given the good match in data on the

figures, and based on the relative simple geometry of the structure, the results for high water level

have a good reliability. For lower water levels, including water levels reaching the vertical

structure but with a very low water height above toe, the reliability is a lot less due to the lack of

information on these kind of cases. However, the other considered types of structures at low water

level give a smaller overtopping, have a more realistic transition to vertical walls, and have a

better match in boundary conditions, so that the reliability can be considered moderate.

Annex K is the in- and output sheet.

1,E-031,E-021,E-011,E+001,E+011,E+021,E+031,E+041,E+051,E+061,E+071,E+081,E+091,E+10

0 0,02 0,04 0,06 0,08 0,1 0,12 0,14 0,16 0,18 0,2

EurOtop(2007)

VdM & B (2014)

Neural Network

CLASH

𝑞∗

=𝑞

ℎ∗

2√

𝑔ℎ

𝑠3∙

1

𝑘∙𝑘

𝛽

*

*

0

[ ]cc

m

RR h

H

5%

95%

Eq. 18.b

Eq. 18.a

Chapter 3 -Case C

53

3.3 Case C


In the Dover harbour in the UK, the next case can be found. It’s a dock exit seawall which has

steep slope and at the end a vertical wall with a recurve form, Figure 46. This slope has a stair like

structure also called wave steps. Above the vertical wall, there is a promenade where people can

walk. At the end of this promenade, there is another vertical wall to protect the dock exit road

lying behind it.

Figure 46 Cross section Case C

This is not an ordinary structure and the overtopping cannot simply be calculated with

formulae from EurOtop. Some simplifications have to be done first. The first simplification is to

propose the slope as just a smooth slope with a certain roughness factor. Xiaomin et al

experimented with wave steps in order to find a roughness factor for these kind of wave steps.

The research was based on the relative step height 𝑑ℎ/ℎ. They defined the roughness and

permeability factor as:

𝑅 = 𝐾Δ𝑅1𝐻

where 𝑅1 is the run-up on a smooth impermeable slope. This definition is equivalent to 𝛾𝑓 in

EurOtop which is why the results for 𝐾Δ from this paper can be used. The step height in this case

is around 0.285m while the total height of the slope is 5.25m. The relative step height 𝑑ℎ/ℎ is then

approximately 0.054. They proposed a roughness and permeability coefficient of 0.63 for 𝑑ℎ/ℎ =

0.029 and 0.5 for 𝑑ℎ/ℎ = 0.063. Hughes et al also noted a coefficient between 0.5 and 0.6 for rough

impermeable slopes. From both these sources, the roughness value of 𝛾𝑓 = 0.6 is picked, which is

a little higher than proposed to offset for any uncertainties and the limited other information

available on this.

Chapter 3 -Case C

54

At the end of the steps, a berm is placed with a vertical wall with a recurve. After this vertical

wall, another wave wall is placed on the promenade. Separate vertical walls cannot be calculated

with the EurOtop formula, so some simplifications should be done. These simplifications differ

depending on the place where overtopping is calculated. Initially, the overtopping is calculated at

the end of the recurve, on the promenade, ignoring the second wave wall. Secondarily, the

overtopping is calculated at the wave wall. This is done by removing the recurve and extending

the crest of the first vertical wall to the level of the wave wall. The simplifications can be seen in

Figure 47. The wave conditions are summarized in Table 22. The wave direction is assumed to be

0° as the structure is enclose by two quay walls and waves travel perpendicular in the basin.

Figure 47 Simplified cross section Case C

Table 22 Wave conditions case C

Condtion Water level[mODN] Hm0[m] Tm−1,0[s] 𝑊𝑎𝑣𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝛽[°]

1 3.40 0.50 2.30 0

2 3.60 0.80 5.60 0

3 3.40 1.20 6.80 0

4 3.90 0.80 5.60 0

5 3.40 2.00 8.80 0

6 3.60 2.00 8.80 0

7 3.80 1.60 7.90 0

8 4.20 1.20 6.80 0

9 4.45 0.80 5.60 0

10 5.00 0.50 2.30 0

Chapter 3 -Case C

55

Like the previous case, the structure has different interpretations at different water levels. For

the lower water levels, the simple slope formulae can be used with a vertical wave wall factor and

a berm factor. This berm factor is explained in the next paragraph. For the highest water level,

5mODN, the structure can be analysed as a composite vertical wall structure or also as a simple

slope. An emergent vertical wall structure cannot be used in this case for low water levels as the

boundary conditions are way off and the results would be even more unreliable then they were in

case B, where the boundary conditions were generally well met. For wave condition 1, 𝑅𝑐/𝐻𝑚0 =

6.2 and the ‘foreshore’ has a slope of 2. This is indeed far out of the application range which makes

the emergent structure not a valid option. The other wave conditions have similar conclusions.

3.3.2 EurOtop

A similar method is used as for case B, where a comparison between the different

interpretations is visualized with a graph. Calculating the run-up and overtopping for the simple

slope interpretation is the same as in previous cases. This is done by considering the point of

overtopping at the recurve wall, so not at the second wave wall.

Simple slope

The first possible type of structure the reader can identify is a simple slope with a berm and a

wave wall at the crest. This is similar to the previous cases and Equation 3 is used again, the only

thing new is the presence of a berm. The procedure to calculate 𝛾𝑏 is also described in Chapter 2.

The example is given here for wave condition 6. The procedure involves the value of 𝑅𝑢2%

requiring several iterations. First, the other influence factors are calculated. For the influence

factor for the wave wall, the range of application is tested and can be seen in Table 23. The slope

is slightly out of bounds and only for 3 wave conditions, other criteria are not met. The criteria are

however, exceeded each time by a small margin so the general conclusion is that the wave wall

factor can be applied but some attention has to be given to the results. It is already clear that when

using the second wave wall as overtopping location, the last criteria is exceeded by a significant

margin. This is kept in mind when deciding which structure type is best.

Table 23 Boundary conditions on the use of a vertical wall factor

Wave condition slope width foot height

1 0.29 < 0.51 ≮ 0.4 0.8 < 1.5 2.8 < 4.8 ≮ 4.00 0.25 < 1.7 ≮ 1.50

2 0.29 < 0.515 ≮ 0.4 0.8 < 2.4 2.64 < 4.8 ≮ 4.56 0.40 < 1.7 < 2.40

3 0.29 < 0.515 ≮ 0.4 0.8 < 3.6 1.96 < 4.8 < 4.84 0.60 < 1.7 < 3.60

4 0.29 < 0.515 ≮ 0.4 0.8 < 2.4 2.94 < 4.8 < 4.86 0.40 < 1.7 < 2.40

5 0.29 < 0.515 ≮ 0.4 0.8 < 6.0 1.00 < 4.8 < 5.80 1.00 < 1.7 < 6.00

6 0.29 < 0.515 ≮ 0.4 0.8 < 6.0 1.20 < 4.8 < 6.00 1.00 < 1.7 < 6.00

7 0.29 < 0.515 ≮ 0.4 0.8 < 4.8 1.88 < 4.8 < 5.72 0.80 < 1.7 < 4.80

8 0.29 < 0.515 ≮ 0.4 0.8 < 3.6 2.76 < 4.8 < 5.64 0.60 < 1.7 < 3.60

9 0.29 < 0.515 ≮ 0.4 0.8 < 2.4 3.49 < 4.8 < 5.41 0.40 < 1.7 < 2.40

Chapter 3 -Case C

56

10 0.29 < 0.515 ≮ 0.4 0.8 < 1.5 4.4 < 4.8 < 5.60 0.25 < 1.7 ≮ 1.50

For comparisons reasons, the recurve is neglected in the intial calculations and 𝛾𝑣 becomes 0.65.

No wave obliquity is present and the roughness is already explained so 𝛾𝛽 = 1 and 𝛾𝑓 = 0.6.

Using Equation 3 gives the result for initial run-up 𝑅𝑢2% without berm, applied to wave

condition 6:

𝜉𝑚−1,0 =0.515

√2.00𝑚/121𝑚= 4.00

𝑅𝑢2% = (1.65 ∙ 1 ∙ 0.6 ∙ 1 ∙ 4.00) ∙ 0.80𝑚 = 7.92𝑚

𝑅𝑢2%,𝑚𝑎𝑥 = (1.00 ∙ 0.6 ∙ 1 ∙ (4 −1.5

√4.00)) = 3.90𝑚

Because of the high breaker parameter, waves are surging up the slope instead of plunging or

spilling which leads to the second part of Equation 3. In this equation there is actually no berm

factor present. The influence of the berm is neglected. This makes the further calculations a lot

easier as there is no iteration aspect involved. When checking Table 24, it is clear that this is also

the case for every other wave condition but wave condition 1 and 10 due to the small wave height

and small wave period.

Table 24 Results for 𝑅𝑢2% and 𝑅𝑢2%,𝑚𝑎𝑥 for case C

Condition 𝑅𝑐[𝑚] 𝜉𝑚−1,0[−] 𝑅𝑢2%[𝑚] 𝑅𝑢2%,𝑚𝑎𝑥[𝑚]

1 3.10 2.09E+00 1.04E+00 8.89E-01

2 2.90 4.03E+00 3.19E+00 1.56E+00

3 3.10 3.99E+00 4.74E+00 2.34E+00

4 2.60 4.03E+00 3.19E+00 1.56E+00

5 3.10 4.00E+00 7.92E+00 3.90E+00

6 2.90 4.00E+00 7.92E+00 3.90E+00

7 2.70 4.02E+00 6.36E+00 3.12E+00

8 2.30 3.99E+00 4.74E+00 2.34E+00

9 2.05 4.03E+00 3.19E+00 1.56E+00

10 1.50 2.09E+00 1.04E+00 8.89E-01

For overtopping, Equation 8.a is used for wave condition 1 and 10 while 8.b is used for the

other wave conditions.

Table 25 EurOtop results for case B assuming a simple slope

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,0,PCO[−] Ru2%,PCO[m] 𝑞[𝑚3/𝑚/𝑠]

1 3.10 0.50 8.26 2.092 8.89E-01 4.07E-17

2 2.90 0.80 48.96 4.027 1.56E+00 6.75E-08

3 3.10 1.20 72.19 3.992 2.34E+00 1.13E-05

4 2.60 0.80 48.96 4.027 1.56E+00 3.43E-07

5 3.10 2.00 120.91 4.002 3.90E+00 2.14E-03

6 2.90 2.00 120.91 4.002 3.90E+00 3.31E-03

Chapter 3 -Case C

57

7 2.70 1.60 97.44 4.017 3.12E+00 8.46E-04

8 2.30 1.20 72.19 3.992 2.34E+00 2.04E-04

9 2.05 0.80 48.96 4.027 1.56E+00 6.75E-06

10 1.50 0.50 8.26 2.092 8.89E-01 5.32E-09

Despite wave conditions 1 and 10 being in the spilling regime, where a berm factor should be

applied, this is not done. This is justified as both these wave conditions have very low overtopping

and the berm width is very small and might even be considered in the recurve which is ignored at

first. Calculating a berm factor would be useless in these cases.

Composite vertical wall

A second possibility is to see the structure as a composite vertical wall when water levels reach

the wall. For lower water levels, the structure is still seen as slope as a negative ℎ𝑠 is only possible

in emergent vertical walls, which is explained that is not a valid choice here. How to calculate a

composite structure is explained in chapter 2. The difference with a plain vertical wall is the

impulsiveness parameter 𝑑∗. As only wave condition 10 has a water level which reaches the wall,

it is explained with this condition as example. The water depth at the toe is 𝑑 = 0.2𝑚 and the

water depth before the composite structure starts is ℎ = 7.9𝑚 as the bottom is at −2.9𝑚𝑂𝐷𝑁.

𝑑∗ = 1.350.2𝑚

0.5𝑚

7.9𝑚

8.26𝑚= 0.517 > 0.3

This means that the waves can be considered non-impulsive, leading to the use of Equation 17.

0.1 < 𝑅𝑐/𝐻𝑚0 = 0.3 < 3.5

𝑞 = 0.04exp (−2.6 ∙ 𝑅𝑐/𝐻𝑚0) ∙ √𝑔𝐻𝑚03 = 1.81 ∙ 10−5 ∙ 𝑚3/𝑠/𝑚

The recurve factor 𝑘 has been ignored initially until the correct type of structure is chosen as this

factor cannot be modelled in simple slopes, otherwise leading to a skewed comparison.

Picking the most appropriate structure

When picking the best structure, the choice comes down to whether to pick a composite

vertical wall at high water levels or just assume a simple slope. As mentioned above, the range of

application to apply a vertical wall influence factor for slopes, is a bit off. This increases even more

for large freeboard ( picking the second wave wall as location for overtopping ). This is a first

argument to pick a composite vertical wall. A second argument is that the value for overtopping

is higher for composite structures, which means a more safe approach, as seen in Figure 48. A last

reason is that the structures looks more like a composite structure than a slope, which is more

realistic.

Chapter 3 -Case C

58

Figure 48 Comparison between a composite structure or a simple slope, for case C, calculated with EurOtop, without recurve factor.

Now, the recurve factor can be applied to receive the correct value for the overtopping at the

recurve wall. Using Figure 19, the 𝑘-value is found at 0.036 which is lower than 0.05. This means

that it is somewhat less reliable and therefore set at 0.05. The new overtopping value becomes:

𝑞′ = 𝑞 ∙ 𝑘 = 9.07 ∙ 10−7 ∙ 𝑚3/𝑠/𝑚

The overtopping is also calculated now at the second location, near the second wave wall. This

is done by just increasing the freeboard. It is clear that overtopping is reduced by this action so a

berm factor and recurve factor are neglected. By increasing the crest level, there is however

danger in that the ratio 𝑅𝑐/𝐻𝑚0 exceeds it validity range. This happens for almost all the cases

with low water level or small wave height. As these are mostly not relevant, this criteria is ignored

for these wave conditions but some thoughts should be given on these results and not blindly

accepted. The results for the second overtopping location are given in Annex C in table C-1, while

the final results for the first location, at the recurved wall, are given in Table 26.

Table 26 Final overtopping results using EurOtop for case C, assuming composite wall for water levels reaching the vertical wall, calculated at the recurved wall

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,0[−] Ru2%[m] 𝑞[𝑚3/𝑚/𝑠]

1 3.10 0.50 8.26 2.09E+00 8.89E-01 4.07E-17

2 2.90 0.80 48.96 4.03E+00 1.56E+00 6.75E-08

3 3.10 1.20 72.19 3.99E+00 2.34E+00 1.13E-05

4 2.60 0.80 48.96 4.03E+00 1.56E+00 3.43E-07

5 3.10 2.00 120.91 4.00E+00 3.90E+00 2.14E-03

6 2.90 2.00 120.91 4.00E+00 3.90E+00 3.31E-03

7 2.70 1.60 97.44 4.02E+00 3.12E+00 8.46E-04

8 2.30 1.20 72.19 3.99E+00 2.34E+00 2.04E-04

9 2.05 0.80 48.96 4.03E+00 1.56E+00 6.75E-06 condition Rc[m] Hm0[m] Lm−1,0[m] 𝑑∗[−] q[m3/s/m]

1,E-36

1,E-33

1,E-30

1,E-27

1,E-24

1,E-21

1,E-18

1,E-15

1,E-12

1,E-09

1,E-06

1,E-03

1,E+00

0 1 2 3 4 5 6 7 8 9 10 11

Ove

rto

pp

ing

q[m

³/m

/s]

Cases

Slope+composite

Slope

Chapter 3 -Case C

59

10 1.500 0.500 8.260 0.52 0.000

3.3.3 van der Meer et al: Updated EurOtop

The lower water levels can make use of the equation used in the previous cases as well for

simple slopes. The steepness of the slope is just ‘mild’ enough to make use of the standard equation

26 for mild simple slopes, cot (𝛼) ≈ 2. The coefficients 𝑎 and 𝑏 in equation 26 are then respectively

0.09 and 1.5. Wave condition 6 is calculated again as an example, starting with a berm factor 𝛾𝑏 =

1.

𝑞 =0.023

√0.5151.00 ∙ 4.00 ∙ exp (− (2.7

2.9𝑚

4.00 ∙ 2.00𝑚 ∙ 1.00 ∙ 0.60 ∙ 1.00 ∙ 0.648)

1.3

) ∙ √𝑔 ∙ (2.00𝑚)3 = 4.11 ∙ 10−2 ∙

𝑚3

𝑠𝑚

𝑞𝑚𝑎𝑥 = 0.09 exp (− (1.5 ∙2.9𝑚

2.00𝑚 ∙ 0.60 ∙ 1.00)

1.3

) = 3.84 ∙ 10−3 ∙ 𝑚3/𝑠/𝑚

For condition 10, the chart of Figure 28 has to be followed because it’s a composite structure. At

first it is checked whether the structure can be seen as a plain or composite wall by calculating

𝑑/ℎ = 0.2𝑚/7.9𝑚 = 0.025 < 0.6. This low value means it can indeed be seen as a composite

structure. The structures is in a quay which leads to the absence of a foreshore. 𝑑∗ = 0.517 as

calculated in previous section and 𝑅𝑐/𝐻𝑚0 = 3. This leads to the exponential equation from

Franco et al (1994). This is an important difference with the traditional EurOtop equation: van

der Meer et al suggests impulsive breaking wave conditions, while EurOtop proposed non-

impulsive conditions.

𝑞 = 0.2 exp(−4.3 ∙ 3) √𝑔(0.5𝑚)3 = 5.53 ∙ 10−7 ∙ 𝑚3/𝑠/𝑚

The recurve factor 𝑘 = 0.05 has to be applied as well making the new overtopping value:

𝑞 = 0.05 ∙ 5.53 ∙ 10−7 ∙ 𝑚3/𝑠/𝑚 = 2.77 ∙ 10−8 ∙ 𝑚3/𝑠/𝑚

The results can be summarized in the following table, Table 27, keeping in mind the same notes

as in previous section on the range of 𝑅𝑐/ℎ𝑚0. The results for the overtopping at the second wave

wall is included in Annex C, Table C-2.

Table 27 Overtopping results using the updated formulae from van der Meer et al for case C, calculated at the recurved wall

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,0,VdM[−] Ru2%,VdM[m] 𝑞𝑉𝑑𝑀[𝑚3/𝑚/𝑠]

1 4.52 0.50 8.26 2.09 8.89E-01 5.30E-24

2 4.32 0.80 48.96 4.03 1.56E+00 4.79E-09

3 4.52 1.20 72.19 3.99 2.34E+00 4.58E-06

4 4.02 0.80 48.96 4.03 1.56E+00 4.89E-08

5 4.52 2.00 120.91 4.00 3.90E+00 2.37E-03

6 4.32 2.00 120.91 4.00 3.90E+00 3.84E-03

7 4.12 1.60 97.44 4.02 3.12E+00 8.60E-04

8 3.72 1.20 72.19 3.99 2.34E+00 1.73E-04

9 3.47 0.80 48.96 4.03 1.56E+00 2.80E-06 condition Rc[m] Hm0[m] Lm−1,0[m] 𝑑∗[−] q[m3/s/m]

10 2.92 0.50 8.26 0.52 2.77E-08

Chapter 3 -Case C

60


The structure is not appropriate for PC-Overtopping due to the vertical wall and thus is

neglected in this case.

3.3.5 Neural network

The neural network is the ideal tool for these kind of structures as they can model most details

from this case. The jump from the bottom to the slope can be modelled as a toe and the transition

from the slope to the vertical wall is seen as a berm. For lower water levels, the same problem as

for case B exists that the height of the berm above the water is limited to 𝐻𝑚0. For condition 1, the

freeboard is also reduced because the limit is 5 ∙ 𝐻𝑚0. This influences the results but could still be

realistic. The resulting input table for the Neural Network can be seen in Annex C in Table C-3

while the overtopping results are shown in Table 28. The input and the results for overtopping at

the second wave wall are also included in Annex C.

Condition 1, 2, 4, and 10 have a value lower than 𝑞 = 10−6 𝑚3/𝑠/𝑚. The Neural Network does

not give the overtopping when they are below this threshold. For all conditions a correction factor

is applied because it’s a rough slope with results between 10−4 and 10−6 𝑚3/𝑠/𝑚.

Table 28 Overtopping results using PCO for case C, calculated at recurved wall

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,0,NN[−] Ru2%,NN[m] 𝑞𝑁𝑁[𝑚3/𝑚/𝑠]

1 4.52 0.50 8.26 2.09 8.89E-01 #N/A

2 4.32 0.80 48.96 4.03 1.56E+00 #N/A

3 4.52 1.20 72.19 3.99 2.34E+00 9.83E-05

4 4.02 0.80 48.96 4.03 1.56E+00 #N/A

5 4.52 2.00 120.91 4.00 3.90E+00 3.06E-03

6 4.32 2.00 120.91 4.00 3.90E+00 3.97E-03

7 4.12 1.60 97.44 4.02 3.12E+00 8.81E-04

8 3.72 1.20 72.19 3.99 2.34E+00 2.44E-04

9 3.47 0.80 48.96 4.03 1.56E+00 2.27E-05 condition Rc[m] Hm0[m] Lm−1,0[m] 𝑑∗[−] q[m3/s/m]

10 2.92 0.50 8.26 0.52 #N/A

3.3.6 CLASH Database

The first criteria is the lower slope, which is set between cot(𝛼) = 1.5 and cot(𝛼) = 2.5 and the

vertical wall, or upper slope at cot(𝛼) = 0. The wave conditions in this case are primarily non-

breaking so the berm is neglected and no filter is placed on B. 𝑠𝑚−1,0 is another criteria and set at

lower than 0.02. These actions leave out 10 results, 8 from Dataset 035 from Bradbury et al, and

2 from Dataset 509 from Pullen et al. The relative freeboard 𝑅𝑐/𝐻𝑚0 in these cases range from

1.5 to around 5, which is similar to this case’s conditions. Table C-5 in Annex C contains the

datasets.

Chapter 3 -Case C

61

3.3.7 Summary

The overtopping results can be found in Figure 49. From this figure it is clear that wave

condition 6 gives the highest overtopping, for every method. For the other wave conditions, the

methods give similar results, leading to a high realiability.

For the non-dimensional graphs, despite that the equations for simple slopes and for vertical

wall with non-impulsive conditions, have the same structure of formula, an exponential curve of

𝑅𝑐/𝐻𝑚0, they cannot be put in the same graph. For the simple slope, the roughness factor is

included whereas for vertical walls, this is not the case. EurOtop does not give any guidelines on

how to approach this. For oblique waves at vertical walls, they give the same method as for simple

slopes by including the factor 𝛾𝛽 but not for the roughness factor 𝛾𝑓 . Therefore, a distinction is

made between the dimensionless freeboard for simple slopes and vertical (non-impulsive). These

are respectively 𝑅𝑐/(𝐻𝑚0𝛾𝛽𝛾𝑓) and 𝑅𝑐/(𝐻𝑚0𝛾𝛽). This can be explained by the statement that the

overtopping in non-impulsive conditions at vertical walls is less influenced by the bottom as

mentioned before. The results from above are now summarized in two graphs, Figure 50 for

simple slopes, and figure 51 for vertical walls (non-impulsive conditions). The results from van

der Meer et al, the Neural Network, and clash, are made non-dimensionless such as in the

appropriate equation in EurOtop. This means that for the vertical wall graph, no roughness factor

is included.

Figure 49 Overtopping results for case C

0,E+00

5,E-04

1,E-03

2,E-03

2,E-03

3,E-03

3,E-03

4,E-03

4,E-03

5,E-03

1 2 3 4 5 6 7 8 9 10

Wave condition

EurOtop

VdM

PCO

NN

𝑞[𝑚

3/

𝑠/𝑚

]

Chapter 3 -Case C

62

Figure 50 Overtopping for smooth and straight slopes, non-breaking waves, with results from case C

Figure 51 Overtopping for vertical walls, non-impulsive conditions, with results from case C

For the simple slope, the results of CLASH and neural network confirm with the 90%

confidence band of EurOtop Equation 8. For the vertical wall however, this is less the case,

especially for some CLASH database values. The CLASH results in the vertical wall graph which

lay below the 5% confidence limit, have a roughness factor of 0.4 and the berm, the transition

from slope to vertical wall, is above still water line. There are two apparent reasons as for why the

CLASH data does not fit the graph for vertical walls well. The first is that the roughness is not

included in the representation in the graphs. If the relative freeboard is applied a factor 1/𝛾𝑓 , the

results would shift to the right, causing a better fit. The second reason is that the vertical wall

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5

EurOtop(2007)

VdM & B(2014)

Neural Network

EurOtop(2007) location2

Van der Meer (2014) location 2

Neural Network location 2

CLASH

𝑅𝑐∗ =

𝑅𝑐

𝐻𝑚0 ∙ 𝛾𝛽 ∙ 𝛾𝑓

[−]

𝑞∗

=𝑞

√𝑔

∙𝐻𝑚

03 [−

]

1,E-08

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5

EurOtop(2007)

VdM & B (2014)

CLASH

𝑅𝑐∗ =

𝑅𝑐

𝐻𝑚0 ∙ 𝛾𝛽

[−]

𝑞∗

=𝑞

√𝑔

∙𝐻𝑚

03 [−

] Eq. 17

Eq. 8.b

5%

95%

5%

95%

Chapter 3 -Case D

63

interpretation might be a wrong interpretation. The author however thinks this is not the case

due to resemblance found with the other methods.

The largest overtopping in this case is for wave condition 6, where the still water line is at

3.8𝑚𝑂𝐷𝑁 and with 𝐻𝑚0 = 2.00𝑚. The different overtopping values for the different methods are

summarized in the following table:

Table 29 Overtopping results for worst case scenario for case C


6 3.31 3.84 - 3.97

This is measured at the lower location, just behind the recurve wall. It can be assumed that the

overtopping is limited to a safe region with this structure design. Other information regarding

overtopping at the second location and a complete comparison between the results can be found

in Annex C. The in- and output sheet is found in Annex L

3.4 Case D


In case D, a simple sloping dike is discussed again, located in St. Peter-Ording in Germany.

Figure 52 represents the cross section. No foreshore is present, and the slope has an angle of

around 1:8 with a height difference of 4.38m from the toe until the crest and is covered in grass.

The dike has an orientation of 315°N. The width of the crest is 3.5m but the overtopping is

calculated at the beginning of the crest. The wave conditions are given in Table 30.

Figure 52 Cross section for case D Table 30 Wave conditions for case D

Condtion Water level[mNN] Hm0[m] Tp[s] Wave direction N[°]

1 6 1.93 4.50 225

2 5.5 1.65 4.50 225

3 5 1.38 4.50 225

4 4.7 1.21 4.50 225

The return periods for each wave condition is different with 1:1000 years for wave condition 1

increasing to 1:10 years for wave condition 4.

Chapter 3 -Case D

64

The wave direction 𝛽 relative to the perpendicular of the dike is then 0° as 315° − 225° = 90°

and no oblique wave attacks occur in either condition. Also important is to adjust the peak period

𝑇𝑝 for the mean period 𝑇𝑚−1,0.

𝑇𝑚−1,0 =4.50𝑠

1.1= 4.09𝑠

No other difficulties are found in this case.

3.4.2 EurOtop

This case is a standard example and Equation 3 can be used without any doubt. There is no

composite slope, no oblique wave attack, no roughness factor as the cover is grass, and no berm.

Calculating the breaker parameter suggests that waves are breaking in each condition and the

maximum overtopping value is not reached. Calculated here as an example is wave condition 4,

with the largest waves and highest water level. The other conditions are found in Table 31.

𝐿𝑚−1,0 = 𝑔 ∙(4.09𝑠)2

2𝜋= 26.13𝑚

𝜀𝑚−1,0 =0.125

√1.93𝑚/26.13𝑚= 0.460

𝑅𝑢2% = 1.65 ∙ 1 ∙ 1 ∙ 1 ∙ 0.460 ∙ 1.93𝑚 = 1.46𝑚

𝑞 =0.067

√0.125∙ 1 ∙ 0.460 ∙ exp (−4.75 ∙

1.38𝑚

0.460 ∙ 1.93𝑚 ∙ 1 ∙ 1 ∙ 1) ∙ √𝑔 ∙ (1.93𝑚)3 = 0.454 ∙ 10−3𝑚3/𝑠/𝑚

Table 31 Overtopping results using EurOtop for case D


1 1.38 1.93 26.13 0.460 1.465 4.54E-04

2 1.88 1.65 26.13 0.497 1.354 1.18E-05

3 2.38 1.38 26.13 0.544 1.239 1.51E-07

4 2.68 1.21 26.13 0.581 1.160 6.25E-09


The slope is considered mild which leads to the use of equation 26.a. No difficulties are found

again and the results can be viewed in Table 32, with wave condition 1 presented as a calculation

example.

𝑞 =0.023

√0.125∙ 1 ∙ 0.460 ∙ exp (− (2.7 ∙

1.38𝑚

0.460 ∙ 1.93𝑚 ∙ 1 ∙ 1 ∙ 1)

1.3

) ∙ √𝑔 ∙ (1.93𝑚)3 = 0.395 ∙ 10−3𝑚3/𝑠/𝑚

Table 32 Overtopping results using updated formulae from van der Meer et al for case D


1 1.38 1.93 26.13 0.460 1.465 3.95E-04

2 1.88 1.65 26.13 0.497 1.354 4.93E-06

3 2.38 1.38 26.13 0.544 1.239 1.49E-08

4 2.68 1.21 26.13 0.581 1.160 1.58E-10

Chapter 3 -Case D

65


The case here is also a perfect example for PCO and can be easily implemented in the software.

In figure 53, the input is shown and in Figure 54, the output. Table 33 shows the results of all cases.

Figure 53 Input example for PCO for case D wave condition 1

Figure 54 Output PCO for case D wave condition 1

Chapter 3 -Case D

66

Table 33 Overtopping results using PCO for case D

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,0,PCO[−] Ru2%,PCO[m] 𝑞𝑃𝐶𝑂[𝑚3/𝑚/𝑠]

1 1.38 1.93 26.13 0.460 1.553 9.13E-04

2 1.88 1.65 26.13 0.497 1.436 3.30E-05

3 2.38 1.38 26.13 0.544 1.313 1.00E-06

4 2.68 1.21 26.13 0.581 1.230 -

3.4.5 The Neural Network

In this application, the structure does not create any problems. No toe means ℎ = ℎ𝑡 = 3𝑚 for

condition 1 and 𝐵𝑡 = 0. There is no berm present so the upper and lower slope are equal and berm

width is 0. This gives the following table of overtopping results. For the input, the reader is

referred to Annex D. For wave condition 1, a problem occurred, stating that the wave steepness

is out of range. For this reason, the wave period is increased to 4.21s instead of 4.09s. This causes

the overtopping to be larger.

Table 34 Overtopping results using NN for case D

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,0[−] Ru2%,NN[m] 𝑞𝑁𝑁[𝑚3/𝑚/𝑠]

1 1.38 1.93 26.13 0.460 1.465 1.50E-02

2 1.88 1.65 26.13 0.497 1.354 2.80E-03

3 2.38 1.38 26.13 0.544 1.239 3.82E-04

4 2.68 1.21 26.13 0.581 1.160 1.10E-04

3.4.6 CLASH

The structure of case D is relatively straightforward so a similar structure should be easy to

find in the CLASH database. When setting the filters for the slope, greater than 1:6 and no berm, a

single data set is found: Perdijk et al, 1987. This data set tested a slope with angle of 1:7. This is

the mildest slope which is found in CLASH data base with no berm. The wave steepness and

relative freeboard are approximately in the same range as for the wave conditions of case D. Annex

D contains these datasets.

3.4.7 Summary

Figure 55 is a bar graph again, comparing the different results for the different methods for

each wave condition. The run-up data is only available for PC-Overtopping and EurOtop and is

shown in figure 56. The dimensionless overtopping is shown in figure 57.

Chapter 3 -Case D

67

Figure 55 Overtopping results for case D

Figure 56 Run-up for smooth and sloping dikes with results from case D

0,E+00

2,E-03

4,E-03

6,E-03

8,E-03

1,E-02

1,E-02

1,E-02

2,E-02

1 2 3 4

Wave conditions

EurOtop

VdM

PCO

NN𝑞

[𝑚3

/𝑠/

𝑚]

0

0,5

1

1,5

2

2,5

3

3,5

4

0 1 2 3 4 5 6

EurOtop(2007)

PC-Overtopping

𝜉𝑚 −1,0 = tan(𝛼) /√𝐻𝑚0/𝐿𝑚−1,0[−]

𝑅∗

=𝑅

𝑢2

%

𝐻𝑚

0

1

𝛾 𝑓𝛾 𝑏

𝛾 𝛽[−

]

95%

5%

Eq. 3.a

Chapter 3 -Case E

68

Figure 57 Overtopping for smooth and sloping dikes, breaking waves, with results from case D

Figure 55 makes it clear that the overtopping is largest for wave condition 1, Table 35. It also

shows that the Neural Network gives substantial larger results, especially for the first wave

condition. The other methods seem to give similar results. This is also clear in the dimensionless

graphs. Most of the other results seem to fit the EurOtop equation very well. The CLASH data is in

the 90% interval range as well as the results for PCO. The NN results, however, have significant

higher estimations of the overtopping and is larger than the 95% upper limit of EurOtop. This is

very strange as the complexity of the structure is very low.

Table 35 Worst case conditions for case D, overtopping


1 0.454 0.395 0.913 15

The in- and output sheet can be found in Annex M.

3.5 Case E


In case E, an important revetment is discussed in Norderney, Germany. This revetment has an

important history and protects the city against the large waves found in this area. A picture of

the situation can be seen in Figure 58 and the simplified cross section Figure 59. The toe starts

with a basalt slope with an angle of 1:4.5 and transitions in an s-profile slope of concrete, which is

simplified into a straight slope with an angle of 1:2.4. After this, the first promenade is found which

can be used for pedestrians when the still water level is low. This promenade has a slope of 1:10

and is covered in concrete. A steep slope with an angle of 1:3 and covered with blocs, separates

the promenade with the second promenade. This promenade is very wide and has a slope angle

of 1:11 and ends at the crest at 10𝑚𝑁𝑁. This is where the overtopping is measured. The

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

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

EurOtop(2007)

VdM & B (2014)

PC-Overtopping

Neural Network

CLASH

𝑞∗=

𝑞

√9.

81∙𝐻

𝑚0

3 √

𝐻𝑚

0/𝐿

𝑚−

1,0

tan(

𝛼)

1 𝛾 𝑏

[−]

𝑅𝑐∗ =

𝑅𝑐

𝐻𝑚0

√𝐻𝑚0/𝐿𝑚−1,0

tan(𝛼)

1

𝛾𝑣𝛾𝑏 𝛾𝛽 𝛾𝑓

[−]

95%

5%

Eq. 8.a

Chapter 3 -Case E

69

orientation of the dike is 315°N and the wave conditions are shown in Table 36. The waves

occurring here are relatively large which implies the importance of this revetment.

Figure 58 Cross section of case E Table 36 Wave conditions for case E, return period unknown

Condtion Water level[mNN] Hm0[m] Tp[s] Wave direction N[°]

1 5 3.5 15 315

2 4.14 3.5 15 315

The relative wave direction 𝛽 is 0° and the mean wave period 𝑇𝑚−1,0 = 13.64𝑠. No information

is given on the dimensions of the roughness blocks. As the stroke is relative small, the roughness

factor is assumed to be 𝛾𝑓 = 0.75.

3.5.2 EurOtop

The different parts of the slopes with each their own roughness and slope angle make this case

more complicated than the normal composite slope. Also, there are two slopes which have angles

milder than 1:8 but steeper than 1:15 which makes that these slopes can be considered neither a

slope nor a berm. The solution thus requires an interpolation between both interpretations.

Without berm

First, the entire structure is seen as a slope with a composite slope angle. This leads to the

iterative method of calculating this slope. Careful attention has to be given to the roughness in this

case. As different parts of the structure have different roughness’s, a weighted average should be

taken of the structure above 𝑤𝑎𝑡𝑒𝑟 𝑙𝑒𝑣𝑒𝑙 − 0.25𝑅𝑢2% and below 𝑤𝑎𝑡𝑒𝑟 𝑙𝑒𝑣𝑒𝑙 + 0.5𝑅𝑢2%. Any

roughness outside of this region, does not influence the run-up or overtopping much.

The start of the iteration is thus calculating the composite slope of the dike between

𝑤𝑎𝑡𝑒𝑟 𝑙𝑒𝑣𝑒𝑙 − 1.5𝐻𝑚0 = −0.25𝑚𝑁𝑁 and 𝑤𝑎𝑡𝑒𝑟 𝑙𝑒𝑣𝑒𝑙 + 1.5𝐻𝑚0 = 10.25𝑁𝑁 as explained in

Chapter 2. When this upper limit goes above the crest, as in this case, the crest level is taken as

upper limit. In this iteration, the roughness factor is also set at 𝛾𝑓,1,𝑠𝑙𝑜𝑝𝑒 = 1. The other influence

factors are also 1.

tan(𝛼𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒,1,𝑠𝑙𝑜𝑝𝑒) =10 − (−0.25)

(2.38 − (−0.25)) ∙ 4.5 + (4.41 − 2.38) ∙ 2.4 + (5.13 − 4.41) ∙ 10 + (7 − 5.13) ∙ 3 + (10 − 7) ∙ 11 = 0.164

𝐿𝑚−1,0 =𝑔(13.64𝑠)2

2𝜋= 290.33𝑚

𝜉𝑚−1,0,1,𝑠𝑙𝑜𝑝𝑒 =0.164

√ 3.5𝑚290.33𝑚

= 1.493

Chapter 3 -Case E

70

𝑅𝑢2%,1,𝑠𝑙𝑜𝑝𝑒 = 1.65 ∙ 1 ∙ 1 ∙ 1 ∙ 1.493 ∙ 3.5𝑚 = 8.624𝑚

𝑅𝑢2%,𝑚𝑎𝑥,1,𝑠𝑙𝑜𝑝𝑒 = 1 ∙ 1 ∙ 1 ∙ 1 ∙ (4 −1.5

√1.493) ∙ 3.5𝑚 = 9.703𝑚

The maximum run-up value is not reached despite the high breaker parameter. The run-up is quite

high and reaches higher than the crest level again (5𝑚𝑁𝑁 + 8.624𝑚 = 13.624𝑚𝑁𝑁). This causes

that the upper limit in the next iteration is the crest level again and the composite slope does not

change in this iteration. The roughness influence factor is however included in the next iteration.

𝑊𝑎𝑡𝑒𝑟 𝑙𝑒𝑣𝑒𝑙 − 0.25𝑅𝑢2% = 2.844𝑚𝑁𝑁

𝑊𝑎𝑡𝑒𝑟 𝑙𝑒𝑣𝑒𝑙 + 0.50𝑅𝑢2% = 9.311𝑚𝑁𝑁

𝛾𝑓,2,𝑠𝑙𝑜𝑝𝑒 =(4.41𝑚 − 2.844𝑚) ∙ 2.4 ∙ 1 + (5.13𝑚 − 4.41𝑚) ∙ 10 ∙ 1 + (7𝑚 − 5.13𝑚) ∙ 3 ∙ 0.75 + (9.311𝑚 − 7𝑚) ∙ 11 ∙ 1

(4.41𝑚 − 2.844𝑚) ∙ 2.4 + (5.13𝑚 − 4.41𝑚) ∙ 10 + (7𝑚 − 5.13𝑚) ∙ 3 + (9.311𝑚 − 7𝑚) ∙ 11= 0.967

𝑅𝑢2%,2,𝑠𝑙𝑜𝑝𝑒 = 8.624𝑚 ∙ 0.966 = 8.335𝑚

Table 37 Values influence factors for case E, assuming no berms, after 6 iterations

𝑐𝑜𝑛𝑑 𝛾𝑓[−] 𝛾𝑏[−]

1 0.965 1.000

2 0.954 1.000

It is clear that roughness factor reduces the run-up, but by a very small amount. The run-up is still

higher than the crest but the lower run-up causes the roughness factor to decrease in next

iterations. This is because the length of the last slope, which has no roughness, becomes smaller

and contributes less to the weighted roughness factor, leading to a lower roughness factor. After

6 iterations in total, it becomes clear that the decrease in roughness factor becomes smaller and

smaller after each iteration and that the run-up stabilizes at certain value, still above the crest

level:

𝛾𝑓,𝑠𝑙𝑜𝑝𝑒 = 0.965

𝑅𝑢2%,𝑠𝑙𝑜𝑝𝑒 = 8.322𝑚

The other influence factors are noted in Table 37.

The overtopping can now be calculated with the final values of slopes, roughness, and breaker

parameter. The freeboard 𝑅𝑐 = 5𝑚.

𝑞𝑠𝑙𝑜𝑝𝑒 =0.067

√0.164∙ 1 ∙ 1.493 ∙ exp (−4.75 ∙

5𝑚

1.493 ∙ 3.5𝑚 ∙ 0.965 ∙ 1 ∙ 1) ∙ √𝑔 ∙ (3.5𝑚)3 = 45.7 ∙ 10−3𝑚3/𝑠/𝑚

The result is a large overtopping of 46𝑙/𝑠/𝑚 that could be dangerous when unaware people

are walking at the crest. The width of the crest is not included in this calculation. Wave condition

1 can be found in Table 38.

Table 38 Overtopping results for case E, using EurOtop, assuming no berms

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,0,slope[−] Ru2%,slope[m] 𝑞𝑠𝑙𝑜𝑝𝑒[𝑚3/𝑚/𝑠]

1 5 3.5 290.33 1.493 8.322 45.7E-03

2 5.86 3.5 290.33 1.524 8.399 21.6E-03

Chapter 3 -Case E

71

With berm

In the previous section, the calculation is done based on that the entire structure can be seen

as a slope. In this section however, a part of the dike is seen as a berm. There are actually two parts

of the dike which are eligible to be seen as neither a slope nor a berm: the first promenade between

4.41𝑚𝑁𝑁 and 5.13𝑚𝑁𝑁 and the second promenade between 7𝑚𝑁𝑁 and the crest. The problem

is that EurOtop only gives a method when 1 berm is present. When multiple berms are present

and lay next to each other, they can be simplified as 1 averaged berm. In this case, this cannot be

correctly done without any clear guidance given in EurOtop as the interpolation method for

multiple berms is not clear. As the parts of the dike are not a 100% berm, it is justified to ignore

one part as berm and see it as a slope. This is best done for the second promenade as the influence

of berms is decreased the further away it is from the still water line. The run-up is now calculated

assuming the first promenade is a berm. The first step in this process is making the berm

horizontal. This is made clear in Figure 59. The width of the berm becomes:

𝐵𝑏𝑒𝑟𝑚,ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 = (5.13𝑚 − 4.41𝑚) ∙ 10 − ((4.47 − 4.41𝑚) ∙ 2.4 + (5.13𝑚 − 4.47𝑚) ∙ 3) = 5.26𝑚

Figure 59 Making the first promenade a horizontal berm for calculation purposes

The next step is calculating the composite slope of the structure without the berm. This is again

an iterative process, starting with the crest as upper limit as 𝑤𝑎𝑡𝑒𝑟 𝑙𝑒𝑣𝑒𝑙 + 1.5𝐻𝑚0 > 10𝑚𝑁𝑁.

This is shown in Figure 60. The slope is then calculated as:

tan(𝛼𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒,1) =10 − (−0.25)

(2.38 − (−0.25)) ∙ 4.5 + (4.77 − 2.38) ∙ 2.4 + (7 − 4.77) ∙ 3 + (10 − 7) ∙ 11 = 0.179

Figure 60 The structure with berm (solid black), without berm ( dashed black), and the composite slope (red)

Both the influence factors for roughness and a berm are also dependent on the run-up value so

they are set at 1 in the initial iteration:

Chapter 3 -Case E

72

𝜉𝑚−1,0,1,𝑏𝑒𝑟𝑚 =0.179

√ 3.5𝑚290.33𝑚

= 1.630

𝑅𝑢2%,1,𝑏𝑒𝑟𝑚 = 1.65 ∙ 1 ∙ 1 ∙ 1 ∙ 1.630 ∙ 3.5𝑚 = 9.42𝑚

𝑅𝑢2%,𝑚𝑎𝑥,1,𝑏𝑒𝑟𝑚 = 1 ∙ 1 ∙ 1 ∙ 1 ∙ (4 −1.5

√1.630) ∙ 3.5𝑚 = 9.89𝑚

The run-up is larger than in previous section but the roughness and berm are not included yet. It

is clear that again, the slope won’t change in the next iteration due to this high run-up and the

crest serving as upper limit. The roughness is done in the exact same way as in previous section,

but using the horizontal berm instead of angled berm:

𝛾𝑓,2,𝑏𝑒𝑟𝑚 =(4.77𝑚 − 2.645𝑚) ∙ 2.4 ∙ 1 + 5.26𝑚 ∙ 1 + (7𝑚 − 4.77𝑚) ∙ 3 ∙ 0.75 + (9.71𝑚 − 7𝑚) ∙ 11 ∙ 1

(4.77𝑚 − 2.645𝑚) ∙ 2.4 + 5.26𝑚 + (7𝑚 − 4.77𝑚) ∙ 3 + (9.71𝑚 − 7𝑚) ∙ 11= 0.964

The berm influence factor can be calculated as described in Chapter 2. First off, the calculation

of the characteristic berm length between 𝑏𝑒𝑟𝑚 𝑙𝑒𝑣𝑒𝑙 − 𝐻𝑚0 = 1.27𝑚𝑁𝑁 and 𝑏𝑒𝑟𝑚 𝑙𝑒𝑣𝑒𝑙 +

𝐻𝑚0 = 8.27𝑚𝑁𝑁.

𝐿𝑏𝑒𝑟𝑚 = (2.38𝑚 − 1.27𝑚) ∙ 4.5 + (4.77𝑚 − 2.38𝑚) ∙ 2.4 + 5.26𝑚 + (7𝑚 − 4.77𝑚) ∙ 3 + (8.27𝑚 − 7𝑚) ∙ 11 = 36.651𝑚

𝑑𝑏 = 5𝑚 − 4.77𝑚 = 0.23𝑚

𝑟𝑏 =𝐵𝑏𝑒𝑟𝑚

𝐿𝑏𝑒𝑟𝑚=

5.26𝑚

36.65𝑚= 0.144

𝑟𝑑𝑏 = 0.5 − 0.5 cos (𝜋𝑑𝑏

2 ∙ 3.5𝑚 ) = 0.027

𝛾𝑏 = 1 − 𝑟𝑏(1 − 𝑟𝑑𝑏) = 0.85

Because the berm is actually below the water line, the berm factor is not iterative and is the same

for each iteration. For wave condition 2 this is not the case and should be reminded when

calculating. The run-up now becomes:

𝑅𝑢2%,1 = 9.42𝑚 ∙ 0.85 ∙ 0.964 = 7.71𝑚

The run-up is now reduced more than if only roughness should have been present. The value

is still higher than the crest level, and as the berm is lower than the still water line, only the

roughness changes with further iterations. After 4 more iterations, the following values for the

composite slope, roughness, and run-up are assumed:

𝜉𝑚−1,0,𝑏𝑒𝑚 = 1.630

𝛾𝑓,𝑏𝑒𝑟𝑚 = 0.953

𝛾𝑏,𝑏𝑒𝑟𝑚 = 0.85

𝑅𝑢2%,𝑏𝑒𝑟𝑚 = 7.62𝑚

Table 39 Influence factors for case E, assuming presence of berm, after 6 iterations


1 0.953 0.849

2 0.940 0.851

Chapter 3 -Case E

73

The run-up is less than when assuming the slope was not a berm and the overtopping is thus

also less, as expected. The results can be seen in Table 40.

𝑞𝑏𝑒𝑟𝑚 =0.067

√0.179∙ 0.85 ∙ 1.630 ∙ exp (−4.75 ∙

5𝑚

1.630 ∙ 3.5𝑚 ∙ 0.953 ∙ 0.85 ∙ 1) ∙ √𝑔 ∙ (3.5𝑚)3 = 26.3 ∙ 10−3𝑚3/𝑠/𝑚

Table 40 Overtopping results for case E, using EurOtop, assuming presence of berm

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,0,berm[−] Ru2%,berm[m] 𝑞𝑏𝑒𝑟𝑚[𝑚3/𝑚/𝑠]

1 5 3.5 290.33 1.630 7.620 26.3E-03

2 5.86 3.5 290.33 1.655 7.646 11.2E-03

Correct value

To receive the most correct overtopping value, an interpolation is done between both results,

based on the slope angle of the first promenade. The angle is 1:10 and interpolation can be done

with the following formula derived from EurOtop. The interpolation method for overtopping is

not given in EurOtop, but a similar process should be used as for the run-up interpolation.

𝑅𝑢2% = 𝑅𝑢2%,𝑠𝑙𝑜𝑝𝑒 + (𝑅𝑢2%,𝑏𝑒𝑟𝑚 − 𝑅𝑢2%,𝑠𝑙𝑜𝑝𝑒)(1/8 − tan(𝑎𝑙𝑝ℎ𝑎))

(1/8 − 1/15)= 8.02𝑚

𝑞 = 𝑞𝑠𝑙𝑜𝑝𝑒 + (𝑞𝑏𝑒𝑟𝑚 − 𝑞𝑠𝑙𝑜𝑝𝑒)(1/8 − tan(𝑎𝑙𝑝ℎ𝑎))

(1/8 − 1/15)= 37.4 ∙ 10−3𝑚3/𝑠/𝑚

Table 41 gives a summary of the EurOtop results.

Table 41 Overtopping results for case E, using EurOtop, interpolated values


1 5 3.5 290.33 - 8.02 37.4E-03

2 5.86 3.5 290.33 - 8.08 17.1E-03 No interpolated values for the breaker parameter are given because no information is given in

EurOtop on how to calculate this. If it’s reverse calculated through the equation for run-up, it won’t

fit for the overtopping equation and vice versa. Also, the influence factors are not correct either

for interpolated values so that only the overtopping and run-up values are calculated and given.


The paper presented by van der Meer et al only proposes a new adjusted overtopping value. It

does not give any new guidelines on how to calculate influence factors, or composite slopes. The

composite slope and influence factors are thus the same as in the previous section. The slopes are

also definitely not steep, so that equation 26.a can be used again which has been explained before.

This sections is thus kept short and overtopping values are calculated according to a slope or berm

for promenade 1 and the subsequent interpolation for the correct value.

𝑞𝑠𝑙𝑜𝑝𝑒 =0.023

√0.164∙ 1 ∙ 1.493 ∙ exp (− (2.7 ∙

5𝑚

1.493 ∙ 3.5𝑚 ∙ 0.965 ∙ 1 ∙ 1)

1.3

) ∙ √𝑔 ∙ (3.5𝑚)3 = 47.7 ∙ 10−3 ∙ 𝑚3/𝑠/𝑚

𝑞𝑏𝑒𝑟𝑚 =0.023

√0.179∙ 0.85 ∙ 1.630 ∙ exp (− (2.7 ∙

5𝑚

1.630 ∙ 3.5𝑚 ∙ 0.965 ∙ 1 ∙ 1)

1.3

) ∙ √𝑔 ∙ (3.5𝑚)3 = 27.4 ∙ 10−3 ∙ 𝑚3/𝑠/𝑚

Chapter 3 -Case E

74

𝑞 = 47.7 + (27.4 − 47.7) ∙ (1/8 − 1/10)/(1/8 − 1/15) ∙ 10−3 ∙ 𝑚3/𝑠/𝑚 = 39.0 ∙ 10−3 ∙ 𝑚3/𝑠/𝑚

Table 42 Overtopping results for case E, using updated formulae from van der Meer et al, assuming no berm

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,0,sl,VdM[−] Ru2%,slVdM[m] 𝑞𝑠𝑙𝑉𝑑𝑀[𝑚3/𝑚/𝑠]

1 5 3.5 290.33 1.493 8.322 47.7E-03

2 5.86 3.5 290.33 1.524 8.399 22.3E-03

Table 43 Overtopping results for case E, using updated formulae from van der Meer et al, assuming presence of a berm

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,0,bVdM[−] Ru2%,bVdM[m] 𝑞𝑏𝑉𝑑𝑀[𝑚3/𝑚/𝑠]

1 5 3.5 290.33 1.630 7.620 27.4E-03

2 5.86 3.5 290.33 1.655 7.646 11.2E-03

Table 44 Overtopping results for case E, using updated formulae from van der Meer et al, assuming the interpolated values


1 5 3.5 290.33 - 7.620 39.0E-03

2 5.86 3.5 290.33 - 7.646 17.5E-03

The overtopping results are a bit higher than for the traditional EurOtop formulas.


The many different slopes in this structure is ideal for a program such as PC-Overtopping. All

the slopes can be entered with their respective roughness factor and coordinates. The only issue

for this case is that the last slope cannot be milder than 1:8, the structure cannot end as a berm.

The first promenade, with slope 1:10, does not give any issues. To resolve this, a tiny part at the

end of the second promenade is set as a slope 1:8, the limit slope. The freeboard remains the same

and the dike ends at 10𝑚𝑁𝑁. This is done by ending the second promenade with slope 1:11 at

9𝑚𝑁𝑁 and making a last slope with angle 1:8 until 10𝑚𝑁𝑁. This is presented in Figure 61.

Chapter 3 -Case E

75

Figure 61 PCO input example for case E, wave condition 1

Just as in EurOtop, PCO does an interpolation based on the assumption of either a slope or a

berm. Though the results are somewhat different. The values of the influence factors are found in

both table 45 and 46. The results for overtopping and run-up are presented in Table 47 and 48 for

both wave conditions. Table 49 contains the interpolated values.

Table 45 Influence factors for case E, assuming no berms


1 0.949 1

2 0.937 1

Table 46 Influence factors for case E, assuming presence of berm


1 0.929 0.758

2 0.906 0.775

Table 47 Overtopping results for case E, using PCO, assuming no berm.

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,0,slPCO[−] Ru2%,slPCO[m] 𝑞𝑠𝑙𝑃𝐶𝑂[𝑚3/𝑚/𝑠]

1 5 3.5 290.33 1.744 10.137 129E-03

2 5.86 3.5 290.33 1.763 10.117 67.3E-03

Table 48 Overtopping results for case E, using PCO, assuming presence of berm

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,0,bPCO[−] Ru2%,bPCO[m] 𝑞𝑏𝑃𝐶𝑂[𝑚3/𝑚/𝑠]

1 5 3.5 290.3262 2.020 8.711 59.4E-03

Chapter 3 -Case E

76

2 5.86 3.5 290.3262 2.020 8.689 28.6E-03

Table 49 Overtopping results for case E, using PCO, assuming interpolated values

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,0,PCO[−] Ru2%,PCO[m] 𝑞𝑃𝐶𝑂[𝑚3/𝑚/𝑠]

1 5 3.5 290.3262 - 9.358 90.8E-03

2 5.86 3.5 290.3262 - 9.337 46.1E-03

It can be seen that the results are quite significant larger, for each assumption of type of structure.

Not only is the amount of interpolated overtopping more than double for wave condition 1, the

assumption of a slope as structure is triple the value: 130 − 45 𝑙/𝑠/𝑚. This could signify that PCO

uses an entirely different method than the one proposed in EurOtop on how to incorporate

multiple berms into a structure. Also, the last part of the promenade having a slope of 1:8 instead

of 1:11 contributes to this difference as well. The pattern for the influence factors is the same, both

𝛾𝑓 and 𝛾𝑏 decrease when assuming a berm but they are clearly lower than those found for EurOtop.


In order to be able to input the structure into the neural network, some modifications have to

be done to the cross section. This has an effect on the results which should be kept in mind when

comparing them with other results from other methods.

The Neural Network separates the structures into three different parts: the toe, the slope and

the crest. The slope part can only exist out of 2 slopes with a berm lying between those two slopes.

In order for the structure from case E to fit this, a separation is also done. First, the toe is defined

as the lowest slope. The Neural Network, does not give an option to include sloping toes, which is

why this slope is transformed to a horizontal one, as seen in Figure 62. The height of the toe is at

the average of the toe and crest level of this slope, with the following slope extrapolated to this

level. This following slope is then considered as the down slope in NN, with slope angle 1:2.4. NN

does give an option for angled berms so that nothing has to be changed here compared to the

original structure for the first promenade. The next slope, together with the second promenade is

combined to have the upper slope in the Neural Network. Combined, this slope has an angle of

1:7.93. For the roughness, due to the small areas which do have roughness and previous results

of a high roughness factor, it is set at 1 for calculations in NN. The resulting input table and results

are found in annex E while the results are found in Table 50.

Chapter 3 -Case E

77

Figure 62 Preparing the cross section for NN input Table 50 Overtopping results for case E, using NN

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,0,NN[−] Ru2%,NN[m] 𝑞𝑁𝑁[𝑚3/𝑚/𝑠]

1 5 3.5 290.3262 - 7.620 39.5E-02

2 5.86 3.5 290.3262 - 7.646 1.7E-02

3.5.6 CLASH

Searching for similar results in the CLASH database is somewhat more difficult compared to

other cases due to the many different slopes. As it is harder to find equivalent structures with toe,

down slope, berm and upper slope, the structure is simplified to just a down slope with berm and

then an upper slope. In this case, the slope downward from the berm is the part from the toe of

the structure at −1.27𝑚𝑁𝑁 until the berm at 4.41𝑚𝑁𝑁, with an average slope of cot(𝛼𝑑) = 3.75.

The upper slope is taken from the berm at 5.13𝑚𝑁𝑁 until the crest level at 10𝑚𝑁𝑁. This slope

has an angle of cot(𝛼𝑢) = 7.93. Putting a filter in the CLASH database on these two slopes and

setting a berm width greater than zero, results in 81 results of a case with down slope of cot(𝛼𝑑) =

4, cot(𝛼𝑢) = 8, and berm width 𝐵 = 1.45𝑚. The average slope of the entire structure, with and

without slope for this case is respectively cot(𝛼𝑤𝑖𝑡ℎ𝑏𝑒𝑟𝑚) = 5.95 and cot(𝛼𝑤𝑖𝑡ℎ𝑜𝑢𝑡𝑏𝑒𝑟𝑚) = 5.49.

This is a lot lower than for the cases found in CLASH, therefore, the lowest are picked with

cot(𝛼𝑤𝑖𝑡ℎ𝑏𝑒𝑟𝑚) = 7.08/7.11 and cot(𝛼𝑤𝑖𝑡ℎ𝑜𝑢𝑡𝑏𝑒𝑟𝑚) = 6.61/6.64 leaving out 50 results from

dataset 111. A last filter is set for the breaker parameter, which is set at greater than 1.2. This

results in 7 remaining results, found in Annex E.

3.5.7 Summary

Putting the results into one graphic brings an additional difficulty as for some results, the

breaker parameter is not known. The Neural Network for example, does not give a result for this

and the interpolated results from EurOtop and van der Meer updates don’t give a result for this as

well. For this reason, both interpretations of the structure for EurOtop and van der Meer et al, are

given in figure 64 and 65. This is also done for PCO as this program also gives the breaker

parameter and overtopping for both interpretations. The relative freeboard, run-up and

overtopping are calculated with their respective correct influence factors. Figure 63 also contains

both interpretations in a bar graph. For the Neural Network, the breaker parameter is based on

the entire structure, from the toe at −1.27𝑚𝑁𝑁 until the crest at 10𝑚𝑁𝑁, resulting into 𝜉𝑚−1,0 =

Chapter 3 -Case E

78

1.53. The berm factor for the NN-result is 1 as the total slope is considered for 𝜉𝑚−1,0. Looking at

the CLASH-database results, a similar problem is encountered. The breaker parameter for these

results is calculated with the total angle including berms, given in the CLASH-database.

Figure 63 Overtopping results for case E

Figure 64 Run –up for smooth and sloping dikes with results from case E

0,E+00

2,E-02

4,E-02

6,E-02

8,E-02

1,E-01

1,E-01

1,E-01

1_slope 1_berm 2_slope 2_berm

Wave conditions

EurOtop

VdM

PCO

NN

𝑞[𝑚

3/

𝑠/𝑚

]

0

0,5

1

1,5

2

2,5

3

3,5

4

0 1 2 3 4 5 6

EurOtop_berm

EurOtop_noberm

PCO_noberm

PCO_berm

𝜉𝑚 −1,0 = tan(𝛼) /√𝐻𝑚0/𝐿𝑚−1,0[−]

𝑅∗

=𝑅

𝑢2

%

𝐻𝑚

0

1

𝛾 𝑓𝛾 𝑏

𝛾 𝛽[−

]

Eq. 3

95%

5%

Chapter 3 -Case F

79

Figure 65 Overtopping for smooth and sloping dikes, breaking waves, with results from case E

It’s obvious that the largest overtopping happens for wave condition 1. The following table

contains the final overtopping values for this wave condition.

Table 51 Worst case conditions for case E, overtopping


1 37.4 39.0 90.8 39.5

For every method, the same pattern is valid: the overtopping is larger assuming a slope than

assuming a berm. The different methods give similar results as well. Almost every result falls into

the 90% confidence band, pointing to reliable results. It does stand out that PC-Overtopping does

have a significant higher result than the other methods. Annex N contains the in- and output sheet.

3.6 Case F


The following case is a vertical structure in Samphire Hoe, Dover UK. It was designed so that

the vertical wall and promenade experience frequently overtopping. In Figure 66 is the given cross

section and in Figure 67 is the simplified cross section which is used in the calculation. The

structure has a toe of rubble mound with a width of 2.25m for 8.5m long, with a sloping foreshore

in front of this toe. After this rubble mound, the bottom part of the vertical structure is built from

Larsen piles. At 4.2𝑚𝑂𝐷𝑁, a concrete wall is built above the Larsen piles with a wave wall on top,

from 6.97𝑚𝑂𝐷𝑁 until 8.22𝑚𝑂𝐷𝑁. Behind this wave wall, a horizontal structure is present which

is assumed to be a promenade. At the end of this horizontal structure a recurve wall is present

and ending the structure at 12.35𝑚𝑂𝐷𝑁. The overtopping is however measured at the end of the

1,E-08

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

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

EurOtop_Noberm

EurOtop_berm

VdM_noberm

VdM_berm

PCO_noberm

PCO_berm

NN_noberm

CLASH

𝑅𝑐∗ =

𝑅𝑐

𝐻𝑚0

√𝐻𝑚0/𝐿𝑚−1,0

tan(𝛼)

1


[−]

𝑞∗=

𝑞

√9.

81∙𝐻

𝑚0

3 √

𝐻𝑚

0/𝐿

𝑚−

1,0

tan(

𝛼)

1 𝛾 𝑏

[−]

Eq.8.a

5%

95%

Chapter 3 -Case F

80

first vertical structure, at the wave wall at 8.22𝑚𝑂𝐷𝑁. This is because the promenade is just

behind this part, and including the horizontal part into the structure would complicate the

calculations a lot, not allowing the use of the standard EurOtop equations. The little berms, at each

different part of the vertical wall, are also neglected as these don’t have any major impact on the

overtopping. The wall has a 90°N orientation. The wave conditions are summarized in Table 52.

Figure 66 Cross section case F

Figure 67 Simplified cross section of case F Table 52 Wave conditions for case F, unknown return period

Condtion Water level[mODN] Hm0[m] Tm−1,0[s] Wave direction N[°]

1 1.88 2.37 5.33 180

2 2.28 2.53 5.34 180

3 2.53 2.51 5.34 180

4 2.62 2.47 5.34 180

5 2.55 2.22 5.35 180

6 2.34 2.07 5.46 180

7 1.56 1.75 5.85 180

8 1.01 1.56 5.97 180

9 0.45 1.40 5.86 180

Chapter 3 -Case F

81

10 -0.12 1.26 5.52 180

The wave direction for all waves is coming from 180°N, which means they are perpendicular to

the vertical structure: 𝛽 = 0°. Also, the mean wave period 𝑇𝑚−1,0 is given, so no further

adjustments are done for this period.

The lowest water level is at −0.12𝑚𝑂𝐷𝑁, which is just above the toe, going up to a maximum

at 2.62𝑚𝑂𝐷𝑁. This maximum is however not the absolute maximum which can occur because the

highest astronomical tide goes up to 3.63𝑚𝑂𝐷𝑁. The chances of this water level occurring with

relevant wave heights are however so small, that these are not discussed in the wave conditions.

Lower water levels than those mentioned in Table 52 are also possible, but are not relevant for

overtopping measures and are thus discarded. It is clear that despite the large range of water

levels, the structure can be seen as a composite vertical structure for all wave conditions.

3.6.2 EurOtop

For composite structures, the first thing which needs to be checked is the impulsiveness of the

wave conditions with 𝑑∗, Equation 16.b. Taking wave condition 4 as calculation example, ℎ𝑠, the

water height before the toe, and 𝑑, the water height on the toe, are respectively 5.04m and 2.79m.

𝑑∗ becomes:

𝑑∗ = 1.35 ∙2.79𝑚

2.47𝑚∙

2𝜋5.04𝑚

𝑔(5.34𝑠)2= 0.173

This is smaller than the threshold value of 0.3, meaning that the waves can be considered

impulsive. Checking the other wave conditions, Table 53, it is clear that these are also impulsive.

This means that the mound affects the wave breaking conditions and thus the overtopping.

Impulsive conditions for composite vertical structures leads to the use of Equation 20, a modified

formula for vertical walls. This formula has another set of boundary conditions which needs to be

met before this formula can be used: The value of 𝑑∗ ∙𝑅𝑐

𝐻𝑚0 should be between 0.05 and 1.0. The

value of this parameter is included in Table 53.

Table 53 Composite impulsiveness parameter and equation 20 boundary conditions for case F wave conditions

𝑐𝑜𝑛𝑑. 𝑑∗[−] 𝑑∗ ∙ 𝑅𝑐/𝐻𝑚0[−]

1 0.113 0.303

2 0.138 0.324

3 0.161 0.366

4 0.173 0.391

5 0.184 0.470

6 0.167 0.476

7 0.099 0.378

8 0.063 0.291

9 0.032 0.178

10 0.003 0.017

Chapter 3 -Case F

82

Only for wave condition 10 the range of application is not met. The water level for this condition

is however very low, with very low wave height as well, making the wave condition irrelevant.

The overtopping is thus calculated with the same formula as the others. Wave condition 4 is used

again as example for the calculation of the overtopping. The results for the other wave conditions

are found in Table 54.

𝑞 = 4.1 ∙ 10−4(0.391)−2.9 ∙ (0.1732 ∙ √𝑔 ∙ (5.04𝑚)3) = 6.53 ∙ 10−3 ∙ 𝑚3/𝑠/𝑚

Table 54 Overtopping results for case F, using EurOtop

condition 𝑅𝑐[𝑚] 𝐻𝑚0[𝑚] 𝐿𝑚−1,0[𝑚] 𝑑∗[−] 𝑞[𝑚3/𝑠/𝑚]

1 6.34 2.37 44.4 0.113 4.69E-03

2 5.94 2.53 44.5 0.138 6.55E-03

3 5.69 2.51 44.5 0.161 6.80E-03

4 5.60 2.47 44.5 0.173 6.58E-03

5 5.67 2.22 44.7 0.184 4.30E-03

6 5.88 2.07 46.5 0.167 3.23E-03

7 6.66 1.75 53.4 0.099 1.69E-03

8 7.21 1.56 55.6 0.063 1.16E-03

9 7.77 1.40 53.6 0.032 0.96E-03

10 8.34 1.26 47.6 0.003 3.97E-03

The overtopping is the largest for wave condition 3. Looking at the overtopping for wave condition

10, it stands out that it is larger than it is for some other wave condition, which have larger waves

and higher still water line, and using the same formula. This is because the boundary conditions

are not met. For condition 10, the application range of 𝑑∗ ∙𝑅𝑐

𝐻𝑚0 is way off. It can be concluded that

the result for wave condition 10 is invalid, and thus ignored in further comparisons.


In the updated papers for composite structures, the chart from Figure 28 is proposed to follow

to reach an equation which should be used. First, it is checked whether the structure can actually

be seen as a composite structure with 𝑑/ℎ < 0.6. Checking this for wave condition 4, with the

largest water height, 𝑑/ℎ = 0.55 which leads to the conclusion that for every wave condition, the

structure is a composite structure (𝑑/ℎ is lower for the other cases). The second criteria is if there

is a foreshore present. For this case, this is true, leading to the following criteria: 𝑑∗ < 0.85 to see

if waves are breaking. For every wave condition, this criteria is met and waves are breaking, and

impulsive conditions are dominant. The final criteria is selecting the most appropriate formula

according to the relative freeboard 𝑅𝑐/𝐻𝑚0. For wave condition 4, with the smallest freeboard,

𝑅𝑐/𝐻𝑚0 = 2.27. This is larger than 1.35 ( the other wave conditions as well), leading to the power

law formula for overtopping, equation 34.b. Results are shown in Table 55.

Chapter 3 -Case F

83

𝑞 = 1.3√2.79𝑚

5.04𝑚0.0014√

2.47𝑚

5.04𝑚 ∙ (2.47𝑚/44.5𝑚) (

5.60𝑚

2.47𝑚)

−3

√𝑔 ∙ (2.47𝑚)3 = 4.20 ∙ 10−3 ∙ 𝑚3/𝑠/𝑚

Table 55 Overtopping results for case F, using updated formulae from van der Meer et al

condition 𝑅𝑐[𝑚] 𝐻𝑚0[𝑚] 𝐿𝑚−1,0[𝑚] 𝑑∗[−] 𝑞𝑉𝑑𝑀[𝑚3/𝑠/𝑚]

1 6.34 2.37 44.4 0.113 2.41E-03

2 5.94 2.53 44.5 0.138 3.94E-03

3 5.69 2.51 44.5 0.161 4.31E-03

4 5.60 2.47 44.5 0.173 4.20E-03

5 5.67 2.22 44.7 0.184 2.51E-03

6 5.88 2.07 46.5 0.167 1.68E-03

7 6.66 1.75 53.4 0.099 5.78E-04

8 7.21 1.56 55.6 0.063 2.66E-04

9 7.77 1.40 53.6 0.032 1.11E-04

10 8.34 1.26 47.6 0.003 1.86E-05

The results are lower than the results from the standard EurOtop. Also, for wave condition 10

a solution is found now that makes sense and seems reliable.


The structure here is not appropriate for PC-O. This section is thus skipped for this case.


In the Neural Network, the toe structure from this case is interpreted as a berm. The lower

slope is the slope angle from the toe. This sloped part has a length of around 1.80m and a width of

2.25m, which makes the anglecot(𝛼𝑑) = 0.8. The berm length is then the length of the toe: 8.5m.

The upper slope is vertical, thus making cot(𝛼𝑢) = 0. The freeboard is limited to 5 ∙ 𝐻𝑚0 which is

a limiting factor for wave condition 9 and 10, setting the freeboard in NN at 7m and 6.3m

respectively. This leads to Table 56, containing the results. The input parameters can be found in

Annex F.

Table 56 Overtopping results for case F, using NN

condition 𝑅𝑐[𝑚] 𝐻𝑚0[𝑚] 𝐿𝑚−1,0[𝑚] 𝑑∗[−] 𝑞𝑁𝑁[𝑚3/𝑠/𝑚]

1 6.34 2.37 44.4 1.13E-01 1.52E-03

2 5.94 2.53 44.5 1.38E-01 2.34E-03

3 5.69 2.51 44.5 1.61E-01 2.86E-03

4 5.60 2.47 44.5 1.73E-01 3.03E-03

5 5.67 2.22 44.7 1.84E-01 2.40E-03

6 5.88 2.07 46.5 1.67E-01 1.81E-03

7 6.66 1.75 53.4 9.94E-02 6.72E-04

8 7.21 1.56 55.6 6.29E-02 2.86E-04

9 7.77 1.40 53.6 3.20E-02 1.36E-04

Chapter 3 -Case F

84

10 8.34 1.26 47.6 2.59E-03 6.47E-05

Every result had remark 6, which is that a correction factor is applied for prototype smooth

vertical structures. Wave condition 7, had an additional remark, stating that the result is less

reliable due to the small value.

3.6.6 CLASH database

To find results in the CLASH database, a similar structure is sought like the one modelled in the

Neural Network, meaning that the toe is seen as the lower slope. The first filter is thus selecting

the cases with the upper slope that is vertical. Setting the height of the water above the berm

greater than 0 and the width of the berm as well is the second step. Now the impulsiveness

parameter 𝑑∗ is calculated with ℎ𝑏 in the file as 𝑑. 𝑑∗ is then set at lower than 0.2 in order to have

similar impulsive conditions. This leaves out 308 results which is still too big. In order to find

similar relative freeboard results, a threshold value is set at 2 at which only larger values are

allowed. 96 results remain. Looking at the lower slope angle possibilities, it is chosen that only the

value cot(𝛼𝑑) = 1. This results in two remaining datasets: 043 and 044 from Pullen, T. (2004).

Dataset 044 is a prototype and is not used in the programming of Neural Network. A last filter is

set on the parameter 𝑑∗ ∙𝑅𝑐

𝐻𝑚0 according to the application range of Equation 20: between 0.05 and

1.0. The end result is 11 results with non-zero overtopping from two different datasets. The reader

is referred to Annex F for these datasets.

3.6.7 Summary

In Figures 68 and 69 are all the results brought together. It’s important to note that the

confidence band could not be plotted as the information given in EurOtop is wrong or insufficient

regarding the confidence interval. The reader is referred to Figure 18 to see the correct interval.

Some conclusions can be drawn by looking at this graph. First, all the results are lower than

those from the traditional EurOtop. For wave condition 10, the highest relative overtopping is

found, (no result for EurOtop calculations due to the application range). Despite the large

freeboard, the relative freeboard is smaller than the other due to the small impulsiveness

parameter 𝑑∗.

Secondly, from Figure 69 it seems that all methods have similar results. All the results seem to

fall within the confidence interval, besides some values from CLASH. The reliability can thus be

considered quite high. This is especially true when comparing this figure physically with figure 18

where the confidence band is drawn. By doing this, it can be seen that most intervals, besides

those for 10 are in the interval or just barely outside. The CLASH datasets are somewhat further

outside the interval band.

Another observation is the weird jumps that the results from van der Meer et al seems to be

making in Figure 69. This is because the formula used, is also a power law such as the one from

Chapter 3 -Case F

85

EurOtop, but it doesn’t have the 𝑑∗ in the root of the power formula. If the graphic was made with

the non-dimensional freeboard 𝑅𝑐

𝐻𝑚0, these jumps would not occur for these results.

Figure 68 Overtopping results for case F

Figure 69 Overtopping for composite vertical structures, impulsive breaking conditions with results from case F

Wave condition 3 has the highest overtopping for the traditional EurOtop formula and for van

der Meer et al. The Neural Network does give a higher overtopping for wave condition 4. The

results for wave condition 3 is found in Table 57 while the other results are found in Annex F.

Table 57 Worst case conditions for overtopping for case F


0,E+00

1,E-03

2,E-03

3,E-03

4,E-03

5,E-03

6,E-03

7,E-03

8,E-03

1 2 3 4 5 6 7 8 9 10

Wave conditions

EurOtop

VdM

PCO

NN

𝑞[𝑚

3/

𝑠/𝑚

]

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

1,E+01

0 0,2 0,4 0,6 0,8 1 1,2

EurOtop(2007)

Van der Meer

Neural Network

CLASH

𝑅𝑐∗ = 𝑑∗

𝑅𝑐

𝐻𝑚0

𝑞∗

=𝑞

𝑑∗

2√

𝑔ℎ

𝑠3

Eq. 20

Chapter 3 -Case M

86

3 6.80 4.31 - 2.86

Annex O can be consulted for the in- and output sheet.

3.7 Case M


The last case provided by EurOtop is an armoured rubble slope protecting the Donegal bay in

Bundoran, Ireland, Figure 70. The bay is exposed to the tides and storm coming from the Atlantic

Ocean. The waves arriving at the bay are assumed to be fully refracted, travelling parallel to the

bed contours and are not effected by offshore wave direction. Wave attack is thus assumed to be

perpendicular.

The slope is composed of two parts, with a walkway in between. Both parts have a steep slope

angle of 1:1.5 covered in two layers of rock armour. The first has rock armour ranging from 450-

1500kg, and the bottom layer 10-60kg. The core is a quarry run, built on a sloping shore. A sloping

foreshore is thus also present in this case. The first part of the slope starts at 1𝑚𝐻𝐷 and goes up

to 5.5𝑚𝐻𝐷 where a 1.4m wide armour berm is, followed by a walkway, 1.5m wide. The second

part then goes up to 7.9𝑚𝐻𝐷 with a 1.3m wide crest. The recreational area is located even higher

at 8.6𝑚𝐻𝐷 leading to a vertical wave wall at the end of the crest. The wave conditions are

presented in Table 58.

Figure 70 Cross section for case M Table 58 Wave conditions for case M

Condtion Water level[mODN] Hm0[m] Tm−1,0[s] Wave direction N[°]

1 2.16 0.85 5.3 270

2 2.7 1.19 5.5 270

3 2.85 1.29 5.4 270

Chapter 3 -Case M

87

4 2 0.77 5.9 270

Depending on the wave condition, the overtopping is calculated at a different spot. Wave

condition 1 has a joint return period of 1:1 and is used to calculate overtopping 𝑞1 at the walkway

at 5.5𝑚𝐻𝐷. Wave condition 2 has a joint return period of 1:50 and wave conditions 3 and 4 a 1:200

joint return period. These conditions are used to calculate overtopping 𝑞2 at the second location,

the recreational area at 8.6𝑚𝐻𝐷. The dike is designed so to limit 𝑞1 at 0.1 𝑙/𝑠/𝑚 while 𝑞2’s limit

is 0.03 𝑙/𝑠/𝑚. In this paper, the overtopping on both locations is calculated for every wave

condition.

The roughness 𝛾𝑓 of this structure is 0.4 as EurOtop mentions this value for 2-layered rock

armour with a permeable core. As mentioned before, the waves are assumed to be fully refracted,

travelling perpendicular to the structure so that 𝛽 = 0° and 𝛾𝛽 = 1.

3.7.2 EurOtop

The calculations for an armoured slope is very similar to normal simple slopes but with a few

alterations. As explained in Chapter 2, there is a limit on 𝛾𝑓 due to the permeability and linear

increase with increasing breaker parameter 𝜉𝑚−1,0. Also the crest width is included in to the

calculations and the difference between the armour crest height and wave wall is hold into

account as well. Due to the different overtopping locations, the calculations are split into two

parts, starting with the overtopping at the walkway.

Overtopping at the walkway

When only considering the first part of the structure, there is no berm present, 𝛾𝑏 = 1, wave

obliquity makes 𝛾𝛽 = 1, 𝑅𝑐 = 𝐴𝑐, and the crest width is 𝐺𝑐 = 1.4𝑚. Taking wave condition 1 as

calculation example, the run-up can be calculated with Equation 3.

𝐿𝑚−1,0 = 43.86𝑚

𝜉𝑚−1,0 = 0.667/√0.85𝑚

43.86𝑚= 4.789

𝛾𝑓,𝑠𝑢𝑟𝑔𝑖𝑛𝑔 = 0.4 + (4.789 − 1.8) ∙1 − 0.4

8.2= 0.619

𝑅𝑢2% = 1.65 ∙ 1 ∙ 0.619 ∙ 1 ∙ 4.789 ∙ 0.85𝑚 = 4.16𝑚

𝑅𝑢2%,𝑚𝑎𝑥1 = 1.00 ∙ 1 ∙ 0.619 ∙ 1 ∙ (4.0 −1.5

√4.789) ∙ 0.85𝑚 = 1.74𝑚

𝑅𝑢2%,𝑚𝑎𝑥2 = 1.97 ∙ 0.85𝑚 = 1.67𝑚

It is clear that due to the high breaker parameter the maximum formula for surging waves should

be applied. However, for slopes with a permeable core, the relative run up 𝑅𝑢2%/𝐻0 is limited to

1.97, which is also the limiting factor for this wave condition. For wave conditions 2 and 3, the

wave steepness is higher and the maximum relative run-up of 1.97 is not reached. The run-up for

all wave conditions are seen in Table 60.

Chapter 3 -Case M

88

The overtopping is calculated with equation 8.b, including the reduction factor 𝐶𝑟 for the crest

width 𝐺𝑐, calculated with Equation 15. The freeboard 𝑅𝑐 is as high as the armour height 𝐴𝑐 , and is

3.34m for wave condition 1. The other crest width reducing factors are in Table 59. The

overtopping for the other conditions is found in Table 60.

𝐶𝑟 = 3.06 ∙ exp (−1.5 ∙1.4𝑚

0.85𝑚) = 0.259

q = 0.259 ∙ 0.2 exp (−2.6 ∙3.34𝑚

0.85𝑚 ∙ 1 ∙ 0.619) ∙ √𝑔 ∙ (0.85𝑚)3 = 8.55 ∙ 10−9𝑚3/𝑠/𝑚

Table 59 Influence factors for overtopping at the first location for case M according to EurOtop

𝑐𝑜𝑛𝑑 𝛾𝑓[−] 𝛾𝑏[−] 𝐶𝑟[−]

1 0.619 1 0.259

2 0.576 1 0.524

3 0.558 1 0.601

4 0.678 1 0.200

Table 60 Overtopping results for case M, using EurOtop, at the first overtopping location


1 3.34 0.85 43.86 4.789 1.675 8.55E-09

2 2.80 1.19 47.23 4.200 2.239 1.03E-05

3 2.65 1.29 45.53 3.961 2.337 3.85E-05

4 3.50 0.77 54.35 5.601 1.517 2.29E-09

Overtopping at recreational area

In this part, not much changes in the calculation, except that the walkway is now considered as

a berm. The structure is now again a composite slope. The slope of the second part does however

have the same slope angle which causes the breaker parameter 𝜉𝑚−1,0 to not change in the

iteration process described in earlier cases. The roughness is also the same for every slope, so it

remains constant through iterations as well. The only thing that is effected by the iterations is the

berm factor 𝛾𝑏 . Excluding the berm factor from the calculations in the first iteration, the run-up

has the same value as in the previous section as every other parameter remains unchanged. To

calculate the effect of the berm now in further iterations, first the definition is checked for 𝛾𝑏 . This

definition states that 𝑟𝑑𝑏 = 1 for a berm lying outside the influence area. As 𝑊𝑎𝑡𝑒𝑟 𝑙𝑒𝑣𝑒𝑙 + 1.0 ∙

𝑅𝑢2% is below the berm for every wave condition, 𝛾𝑏 becomes 1, and the run-up still remains the

same for every wave condition.

For the overtopping, there are some changes. The first one is the crest width, which is 1.3m at

the top, and the wave wall at the top. EurOtop does not give an exact solution method for cases

where the armour freeboard is different from the crown wall height 𝐴𝑐 ≠ 𝑅𝑐. They suggest that

the overtopping formulae should be used by taking the wall freeboard 𝑅𝑐 as the freeboard used in

the formulae, discarding 𝐴𝑐 . Wave condition 1 is taken as example again with a crest width of 1.3m

Chapter 3 -Case M

89

and freeboard of 6.44m. Crest width reducing factors for other wave condtions are found in Table

61 and overtopping in Table 62.

𝐶𝑟 = 3.06 ∙ exp (−1.5 ∙1.3𝑚

0.85𝑚) = 0.309

Table 61 Influence factors for overtopping at the second location for case M according to EurOtop

𝑐𝑜𝑛𝑑 𝛾𝑓,2[−] 𝛾𝑏,2[−] 𝐶𝑟,2[−]

1 0.619 1 0.594378

2 0.576 1 0.594378

3 0.558 1 0.674885

4 0.678 1 0.243156

𝑞 = 0.309 ∙ 0.2 exp (−2.6 ∙6.44𝑚

0.85𝑚 ∙ 1 ∙ 0.619) = 2.25 ∙ 10−15𝑚3/𝑠/𝑚

Table 62 Overtopping results for case M, using EurOtop, at the second overtopping location

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1[−] Ru2%,2[m] 𝑞2[𝑚3/𝑚/𝑠]

1 6.44 0.85 43.86 4.789 1.675 2.25E-15

2 5.90 1.19 47.23 4.200 2.239 9.08E-11

3 5.75 1.29 45.53 3.961 2.337 5.94E-10

4 6.60 0.77 54.35 5.601 1.517 5.49E-16


van der Meer et al do not give any additional advice on rubble slopes so that the formulae for

simple slopes are used, with the influence factors from EurOtop. The slope angle is however steep

so that the updated formulae for steep slopes can be used, equation 26.b. Calculating this for wave

condition 1, at the walkway, this becomes:

𝑎 = 0.09 − 0.01(2 − 1.5)2.1 = 0.0877

𝑏 = 1.5 + 0.42(2 − 1.5)1.5 = 1.648

𝑞 = 0.259 ∙ 0.0877 ∙ exp (− (1.648 ∙3.34𝑚

0.85𝑚 ∙ 1 ∙ 0.619)

1.3

) ∙ √𝑔 ∙ (0.85𝑚)3 = 3.53 ∙ 10−11 ∙ 𝑚3/𝑠/𝑚

The results for the other wave conditions are found in Table 63 and 64 for overtopping location 1

and 2 respectively.

Table 63 Overtopping results for case M, using updated formulae from van der Meer et al, at the first overtopping location

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1[−] Ru2%,𝑉𝑑𝑀1[m] 𝑞𝑉𝑑𝑀1[𝑚3/𝑚/𝑠]

1 3.34 0.85 43.86 4.789 1.675 3.53E-11

2 2.80 1.19 47.23 4.200 2.239 1.21E-06

3 2.65 1.29 45.53 3.961 2.337 7.20E-06

4 3.50 0.77 54.35 5.601 1.517 5.05E-12

Chapter 3 -Case M

90

Table 64 Overtopping results for case M, using updated formulae from van der Meer et al, at the second overtopping location

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1[−] Ru2%,𝑉𝑑𝑀2[m] 𝑞𝑉𝑑𝑀2[𝑚3/𝑚/𝑠]

1 6.44 0.85 43.86 4.789 1.675 1.68E-23

2 5.90 1.19 47.23 4.200 2.239 4.54E-15

3 5.75 1.29 45.53 3.961 2.337 1.10E-13

4 6.60 0.77 54.35 5.601 1.517 1.41E-24


The problem with this case is that the rough armour units and permeable cores cannot be

modelled within PCO. The limit for material roughness in PCO is 0.5, which is not enough for this

case. An overestimation of the overtopping is thus the consequence of this limit. PCO, also

interpolates the roughness factor between 𝛾𝑓 and 1 for large breaker parameters so that the

𝛾𝑓,𝑠𝑢𝑟𝑔𝑖𝑛𝑔, found in previous paragraph, cannot be used as alternative option. PCO also does not

include the crest width which makes the prediction of the overtopping even larger. This last

problem can be fixed by using the same reducing factor 𝐶𝑟 as in EurOtop. The first part of the

structure is modelled in order to calculate the overtopping at the walkway, Figure 71, and the rest

is added later when calculating the overtopping at the second point, Figure 72. For the

overtopping at the recreational area, the slope is extrapolated until the recreational area level as

PCO can’t model this vertical jump. The roughness and berm factors are given in Table 65 and 66

and the overtopping results in Table 67 and 68. The results for overtopping are multiplied by their

respective 𝐶𝑟 from table 65 and 66.

Figure 71 PCO input for case M, wave condition 1 at first overtopping location

Chapter 3 -Case M

91

Figure 72 PCO input for case M, wave condition 1 at second overtopping location Table 65 Influence factors for overtopping at the first location for case M according to PCO

𝑐𝑜𝑛𝑑 𝛾𝑓,𝑃𝐶𝑂1[−] 𝛾𝑏,𝑃𝐶𝑂1[−] 𝐶𝑟,𝑃𝐶𝑂1[−]

1 0.682 1 0.259

2 0.646 1 0.524

3 0.632 1 0.601

4 0.732 1 0.200

Table 66 Influence factors for overtopping at the second location for case M according to PCO

𝑐𝑜𝑛𝑑 𝛾𝑓,𝑃𝐶𝑂2[−] 𝛾𝑏,𝑃𝐶𝑂2[−] 𝐶𝑟,𝑃𝐶𝑂2[−]

1 0.682 1 0.259

2 0.646 1 0.524

3 0.631 0.995 0.601

4 0.732 1 0.200

Table 67 Overtopping results for case M, using PCO, at the first overtopping location

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,PCO1[−] Ru2%,𝑃𝐶𝑂1[m] 𝑞𝑃𝐶𝑂1[𝑚3/𝑚/𝑠]

1 3.34 0.85 43.86 4.788 2.069 2.59E-07

2 2.80 1.19 47.23 4.199 2.707 9.85E-05

3 2.65 1.29 45.53 3.960 2.849 3.11E-04

Chapter 3 -Case M

92

4 3.50 0.77 54.35 5.600 2.042 #N/A

Table 68 Overtopping results for case M, PCO, at the second overtopping location

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,PCO2[−] Ru2%,PCO2[m] 𝑞PCO2[𝑚3/𝑚/𝑠]

1 6.44 0.85 43.86 4.788 2.069 #N/A

2 5.90 1.19 47.23 4.199 2.707 #N/A

3 5.75 1.29 45.53 3.960 2.847 #N/A

4 6.60 0.77 54.35 5.600 2.042 #N/A

The roughness factors follow the same trend as for those in EurOtop, despite the offset due to

the minimum roughness in PCO. For the second location, no overtopping is found due to the low

run-up.


The structure is quite simple to model: a lower slope, berm, and upper slope. The neural

network is also the only method in which a difference in 𝐴𝑐 and 𝑅𝑐 can be modelled. The problem

is though that both these factors are limited to 5 ∙ 𝐻𝑚0. For wave condition 1 and 5 at the second

overtopping location, this limit is reached and 𝐴𝑐 = 𝑅𝑐 for these conditions. For wave condition 2

and 3, the limit for 𝑅𝑐 is not reached and can thus be modelled correctly. Another problem for

overtopping at the second location is that the berm the height of the berm above the still water

line is also limited to 1 ∙ 𝐻𝑚0.For overtopping at the first location, no problems are encountered.

The results are found in table 69 and 70. The input can be found in Annex G.

Table 69 Overtopping results for case M, using NN, at the first overtopping location

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1[−] Ru2%,𝑁𝑁1[m] 𝑞𝑁𝑁1[𝑚3/𝑚/𝑠]

1 3.34 0.85 43.86 4.788 2.069 -

2 2.80 1.19 47.23 4.199 2.707 8.57E-05

3 2.65 1.29 45.53 3.960 2.849 1.74E-04

4 3.50 0.77 54.35 5.600 2.042 -

Table 70 Overtopping results for case M, using NN, at the second overtopping location

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1[−] Ru2%,𝑁𝑁2[m] 𝑞𝑁𝑁2[𝑚3/𝑚/𝑠]

1 6.44 0.85 43.86 4.788 2.069 -

2 5.90 1.19 47.23 4.199 2.707 -

3 5.75 1.29 45.53 3.960 2.847 -

4 6.60 0.77 54.35 5.600 2.042 -

3.7.6 CLASH database

Finding relevant structures is quite easy for this case. A filter is set on upper and lower slope

which have a slope angle of 1:1.5 and with a berm width larger than zero. These filters give 6

Chapter 3 -Case M

93

results with an overtopping result larger than zero, from two data sets: 037 and 326. The

structures from these data sets also have a crest width greater than zero and 𝑅𝑐 > 𝐴𝑐 which

resembles the discussed case very well. Annex G contains these CLASH datasets.

3.7.7 Summary

The results from the different methods can be seen in Figure 74 for the run-up and in Figure

73 and 75 for the overtopping. Important to note is that unlike for the previous cases, the relative

run-up 𝑅∗ in Figure 74 for run-up doesn’t include the influence factor for roughness. This is

because it changes with the breaker parameter and is therefore included in the equation line itself.

Looking at Figure 73, it is more than obvious that the results for overtopping at the second location

are so small that they are left out for the dimensionless figures. For the cases which have a factor

included for the crest width, the relative overtopping 𝑞∗ is multiplied with 1/𝐶𝑟 in order to have

the correct relative value.

Figure 73 Overtopping results for case M

0,00E+00

5,00E-05

1,00E-04

1,50E-04

2,00E-04

2,50E-04

3,00E-04

3,50E-04

1a 2a 3a 4a 1b 2b 3b 4b

Wave conditions

EurOtop

VdM

PCO

NN

𝑞[𝑚

3/

𝑠/𝑚

]

Chapter 3 -Case M

94

Figure 74 Run-up for smooth and simple slopes with results from case M

Figure 75 Overtopping for smooth and sloping dikes, non-breaking waves, with results from case M

The conclusion is pretty clear when studying figures 75 and 74, that all the results comply very

well with the EurOtop equations. Besides a couple of values for the CLASH data, every results falls

within the 90% confidence range. The largest values are found for wave conditions 3 as seen in

Table 71. However, when looking at Figure 73, the overtopping for PCO and NN are much larger

than for the EurOtop results. This is a surprising results as the structures is relatively easy to

model and calculate. Every method does give wave condition 3 as worst case scenario for

overtopping at location 1.

Table 71 Worst case conditions for case M at overtopping location 1

0

0,5

1

1,5

2

2,5

3

0 1 2 3 4 5 6 7 8 9

EurOtop(2007)

PCO

𝜉𝑚 −1,0 = tan(𝛼) /√𝐻𝑚0/𝐿𝑚−1,0[−]

𝑅∗

=𝑅

𝑢2

%

𝐻𝑚

0

[−]

1,E-08

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5

EurOtop(2007)

VdM(2014)

PCO

NN

CLASH

𝑅∗ =𝑅𝑢2%

𝐻𝑚0

1

𝛾𝑓 𝛾𝑏 𝛾𝛽

[−]

𝑞∗

=𝑞

√𝑔

𝐻𝑚

03

∙1 𝐶

𝑟

[−]

Eq. 8.b

5%

95%

Eq. 3

95%

5%

Chapter 3 -Case study Wenduine

95


3 0.039 0.007 0.311 0.174 The limit of 0.1 𝑙/𝑠/𝑚 at overtopping location 1 is fulfilled for wave condition 1. For the other

wave conditions, some methods predict a higher, but still not a lot, than this limit. This indicates

that the first part of the dike is sufficient to protect the mainland. The second overtopping location

is far beneath the limit and thus safe. The in- and output sheet for case M can be found in Annex P.

3.8 Case study Wenduine

The last case study discussed here, is a dike which is found in Wenduine, Belgium. It is one of

the weak links in the Belgian coast line and is currently (2015) being renovated. The renovation

is part of a master plan laid out by the government in order to decrease the risk of flooding and

subsequent damage and loss of lives. Two different wave conditions are considered in order to

evaluate the dike: a 1:1000 year return period storm with water level at 6.84𝑚𝑇𝐴𝑊 and a

superstorm with water level at 7.94𝑚𝑇𝐴𝑊. The first wave condition should limit the overtopping

at 1 𝑙/𝑠/𝑚 in order to reduces the risks for wandering people. The second wave condition should

be limited at 100 𝑙/𝑠/𝑚 so that no structural damage can take place when a superstorm hits. This

extreme storm is estimated to have a return period of over 17,000 years (Van der Biest 2009).

These are very strict measures that are not absolutely not met in the current situation. This case

study is divided into two parts. One part is dedicated to the current existing situation and the other

part is about the future structure, which is actually being built right now.


Current situation

The dike has two different structures according to the location: the roundabout and the

promenade. In this report, the promenade is discussed as it the largest part of Wenduine and thus

the most important. The current situation is described in Figure 76. A beach with slope 1:35 is

leading up to a dike with slope angle 1:2. Then the promenade starts with a slope angle of 1:100.

No crest structures are present making this a simple slope, easy to calculate. The wave conditions

are presented in Table 72. It is already clear that overtopping will be very large as large waves

approach with a very small freeboard.

Figure 76 Current situation Wenduine cross section Table 72 Wave conditions for case Wenduine

Condtion Water level[mTAW] Hm0[m] Tm−1,0[s] Wave direction β[°]


96

1 6.84 4.75 8.6 0

2 7.94 4.97 9 0

Before going further, an issue has to be addressed about where the overtopping is calculated.

UGent and Flanders Hydraulic Research have executed model tests on this structure as well as the

planned structure. In these tests however, they measured the overtopping at the end of the

promenade. By doing this, they considered the promenade as a berm and the crest at the end of

this berm. However, due to the high steepness and the large wave period resulting in a large

breaker parameter, the second part of Equation 8 is used ( see following paragraph). In this

equation, no influence factor for a berm is present and the promenade cannot be hold into account

in the calculations. Therefore, the overtopping is calculated at the start of the promenade, or end

of the dike. This is similar to the previous cases where similar berms at the end of the structure

were not taken into account either. This also simplifies the calculations for some parts but

overestimates the overtopping. EurOtop itself also states that the equations in the manual do not

take into account this crest width and that overtopping values calculated are higher than reality.

The same is true about the beach. It starts at −6𝑚𝑇𝐴𝑊 and has a slope angle of 1:35. It is thus

considered a berm and consequently it does not count towards the composite slope angle of the

structure. This slope of the structure thus remains 1:2, leading again to the high breaker

parameter and again the absence of a berm factor in the overtopping formulae. Comparisons from

the results in this report against those from model test thus should be taken with a grain of salt.

In Table 73 are some overtopping values found in previous tests at the current situation.

Table 73 Current existing overtopping values, calculated at Flanders Hydraulic Lab and UGent(2011) *: extrapolated value

𝐶𝑜𝑛𝑑. 𝑞[𝑙/𝑠/𝑚] 𝑞𝑡𝑜𝑙𝑒𝑟𝑎𝑏𝑙𝑒[𝑙/𝑠/𝑚]

1 50-70 1

2 500* 100

Future situation

Several scenarios were proposed to improve the overtopping at Wenduine. These were tested

at UGent and Flanders Hydraulic Lab. The conclusions of those tests was that the best

improvement was to install a vertical structure 10m in front of the current dike, Figure 77. The

vertical structure goes from the level of the existing beach until the level of the extrapolated

promenade at 8.28𝑚𝑇𝐴𝑊. On top of this structure, a parapet wall is installed with height of 0.8m

and angle of 45°. This parapet wall is limited in height in order to ensure a view for the pedestrians

on the sea. 13m behind this parapet wall, another wave wall is installed, which can be used for the

people as benches. Again, the overtopping measured in the tests from UGent are measured at the

end of the promenade. Here, the overtopping is measured again at the start of the promenade, or

at the vertical wall. The second wave wall is thus not included in the calculations as well as the

length of the promenade. This is again justified as the second wave wall cannot be hold into

account with the standard EurOtop equations, or even the other tools such as Neural Network.


97

Again, the overtopping will be a lot larger than those measured from the model tests. The wave

conditions remain the same as in Table 72. In Table 74, results from the test from UGent are

described.

Figure 77 Cross section of future dike of Wenduine Table 74 Overtopping values found with model tests on the future dike in Flanders Hydraulic Lab and UGent

*: measured with a higher wave height of 5.25m instead of 4.97m

𝐶𝑜𝑛𝑑. 𝑞[𝑙/𝑠/𝑚] 𝑞𝑡𝑜𝑙𝑒𝑟𝑎𝑏𝑙𝑒[𝑙/𝑠/𝑚]

1 0.80 1

2 120* 100

3.8.2 EurOtop

Current situation

It is clear that for both wave conditions, the equation for a simple slope should be used. There

is no artificial roughness so that 𝛾𝑓 = 1 and 𝛾𝛽 = 1 as the waves travel perpendicular to the dike.

Wave condition 1 is taken as example and illustrated as following:

𝐿𝑚−1,0 = 115.47𝑚

𝜉𝑚−1,0 =0.5

√4.75𝑚/115.47𝑚= 2.465

The wave breaker parameter is too high to use the equation for spilling waves so that the

equations for surging waves is used. This is a statement that seems very questionable. Due to the

long beach, it is expected that waves are breaking. The beach is however a berm according to

EurOtop and does not alter the value of 𝜉𝑚−1,0. The length of the beach, 445m, is also too large as

berms are restricted to 0.25 ∙ 𝐿𝑚−1,0. For run-up, there is an influence factor for a berm present in

the formulae, but due to the overtopping being measured at the start of the promenade, and the

boundary conditions on the length of the beach, this factor is ignored. For the beach, there is no

logical way to calculate the factor anyways as the length of the berm is too great, no information

before the beach is available, and so on… For overtopping, no berm factor is present due to the

surging waves, and subsequently, the promenade length or beach does not play a factor in the

theoretical estimation of the overtopping. The same overtopping is thus found with EurOtop

formulas at the start and the end of the promenade. The freeboard is thus 1.54m for wave

condition 1 and 0.44m for the superstorm at 7.94𝑚𝑇𝐴𝑊. Results are shown in Table 75.

𝑅𝑢2% = 1 ∙ 1 ∙ 1 ∙ 1 ∙ (4 −1.54

√2.465) ∙ 4.75𝑚 = 14.46𝑚


98

𝑞 = 0.2 ∙ exp (−2.6 ∙1.54𝑚

4.75𝑚 ∙ 1 ∙ 1) ∙ √𝑔 ∙ (4.75𝑚)3 = 2.791 𝑚3/𝑠/𝑚

Table 75 Overtopping results for Wenduine current situation, using EurOtop

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1[−] Ru2%,current[m] 𝑞𝑐𝑢𝑟𝑟𝑒𝑛𝑡[𝑚3/𝑚/𝑠]

1 1.54 4.75 115.47 2.465 14.46 2.791

2 0.44 4.97 126.47 2.522 15.19 5.514

Future situation

In both wave conditions, the water level reaches the wall, allowing the use of the vertical wall

formulae. First, the impulsiveness parameter ℎ∗ is calculated. Again, the 1000-year storm is used

as calculation example.

ℎ𝑠 = 6.84𝑚𝑇𝐴𝑊 − 6.51𝑚𝑇𝐴𝑊 = 0.33𝑚

ℎ∗ = 1.35 ∙0.33𝑚

4.75𝑚∙

0.33𝑚

115.47𝑚= 0.000268

ℎ∗ ∙𝑅𝑐

𝐻𝑚0= 0.000268 ∙

2.24m

4.75m= 0.000126

The wave are thus considered impulsive and breaking for wave condition 1. For the superstorm,

ℎ∗ = 0.00439 the same is valid. This leads to the use of Equation 18.b for overtopping. Just like for

the old structure, no influence factors for berms are present, which makes the overtopping again

the same at the start or end of the promenade. The freeboard is increased and is now 2.24m for

wave condition 1.

𝑞 = 2.7 ∙ 10−4 ∙ (0.000268 ∙2.24𝑚

4.75𝑚)

−2.7

∙ (0.0002682√𝑔0.333 ) = 0.386 𝑚3/𝑠/𝑚

This overtopping is further reduced with the parapet. Calculating this reduction factor 𝑘 is

done with Figure 19. The height of the parapet is ℎ𝑟 = 0.8𝑚, the width 𝐵𝑟 = 0.64𝑚, and the wall

height 𝑃𝑐 = 1.44𝑚 for a water level at 6.84𝑚𝑇𝐴𝑊.

𝑅0∗ = 0.25 ∙

0.8𝑚

0.64𝑚+ 0.05 ∙

1.44𝑚

2.24𝑚= 0.345

𝑚 = 1.1 ∙ √0.8𝑚

0.64𝑚+ 0.2 ∙

1.44𝑚

2.24𝑚= 1.358

𝑚∗ = 0.8 ∙ 𝑚 = 1.087

𝑅𝑐

𝐻𝑚0= 0.472

𝑘 = 1 −1

1.358∙ (0.472 − 0.345) = 0.907

The reduction is not that big, and for the second wave condition, 𝑅𝑐

𝐻𝑚0 becomes smaller than 𝑅0

∗ so

that 𝑘 = 1, rendering the parapet useless according to EurOtop, see Table 76. The overtopping

results can be seen in Table 77.


99

𝑞 = 0.907 ∙ 0.386 𝑚3/𝑠/𝑚 = 0.350 𝑚3/𝑠/𝑚

Table 76 Reduction factors for the parapet wall in the future dike at Wenduine

condition 𝑘[−]

1 0.907

2 1.000

Table 77 Overtopping results for future dike at Wenduine, using EurOtop

condition Rc[m] Hm0[m] Lm−1,0[m] ℎ∗[−] 𝑞𝑓𝑢𝑡𝑢𝑟𝑒[𝑚3/𝑠/𝑚]

1 2.24 4.75 115.47 0.000268 0.350

2 1.14 4.97 126.47 0.004392 3.442


Current situation

The updated formulae are perfectly applicable on this case as they were designed for cases

with low freeboard. The results from these formulae will thus give a better estimation than those

from the traditional EurOtop equations. The slope of the dike is 1:2 so that the standard values for

the parameters a and b can be picked from equation 26.b, respectively 0.09 and 1.5. As well in the

traditional formulae, no influence factor for the berm is present here.

𝑞 = 0.09 ∙ exp (− (1.5 ∙1.54𝑚

4.75𝑚)

1.3

) ∙ √𝑔(4.75𝑚)3 = 1.972 𝑚3/𝑠/𝑚

Table 78 Overtopping results for current dike at Wenduine, using updated formulae from van der Meer et al

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1[−] Ru2%,cVdM[m] 𝑞𝑐𝑉𝑑𝑀[𝑚3/𝑚/𝑠]

1 1.54 4.75 115.4745 2.465 - 1.9723705

2 0.44 4.97 126.4661 2.522 - 2.717015

Future situation

For a vertical wall, the chart should again be followed from Figure 28. There is no toe structure

besides a foreshore so that wave breaking is examined with the parameter ℎ2/(𝐻𝑚0 ∙ 𝐿𝑚0). This

is the same parameter as ℎ∗ but without the factor 1.35. In previous calculations, it was clear that

both wave conditions are sufficiently below the wave breaking threshold. The relative freeboard

𝑅𝑐/𝐻𝑚0 is also very low leading to the exponential Equation 32.a for both wave conditions:

𝑞 = 0.011 ∙ √4.75𝑚

0.33𝑚 ∙4.75𝑚

115.47𝑚

∙ exp (−2.2 ∙2.24𝑚

4.75𝑚) ∙ √𝑔(4.75𝑚)3 = 2.364 𝑚3/𝑠/𝑚

Applying the same reduction factor 𝑘 from EurOtop, in Table 76, the overtopping then

becomes:

𝑞 = 0.907 ∙ 2.364 𝑚3/𝑠/𝑚 = 2.143 𝑚3/𝑠/𝑚


100

Table 79 Overtopping results for the future dike at Wenduine, using the updated formulae from van der Meer et al

condition Rc[m] Hm0[m] Lm−1,0[m] ℎ∗[−] 𝑞𝑓𝑢𝑡𝑢𝑟𝑒[𝑚3/𝑠/𝑚]

1 2.24 4.75 115.47 0.000268 2.143

2 1.14 4.97 126.47 0.004392 2.167

These results seem very strange as there barely is a difference between both wave conditions

and for the first wave condition, the overtopping is a lot more than for the traditional EurOtop

value. This should not be the case as the standard EurOtop predicted a drastic decrease in

overtopping between the current and future situation. This is also established by the model tests,

Table 73 and 74. The other formulae given by van der Meer for vertical walls in his first paper for

steep slopes, is not valid as a vertical wall in this case should be in relative deep water and requires

unbroken waves. This is clearly not the case with this structure.

Another possibility would be to see the structure as a composite vertical structure. The

problem here is defining ℎ. The beach starts at −6𝑚𝑇𝐴𝑊 but the question is if this is allowed to

take the water height at this point. For the sake of comparison, overtopping is calculated assuming

this hypothesis where the water depth is the depth at the start of the beach. For the first wave

condition:

ℎ = 6.84𝑚𝑇𝐴𝑊 − (−6𝑚𝑇𝐴𝑊) = 12.87𝑚

𝑑 = 6.84𝑚𝑇𝐴𝑊 − 6.51𝑚𝑇𝐴𝑊 = 0.33𝑚

𝑑∗ = 1.35 ∙12.87𝑚

4.75𝑚∙

0.33𝑚

115.47𝑚= 0.0105

this leads to the use of Equation 34.a for wave condition 1. For wave condition 2, 𝑑∗ = 0.0428, also

leading to equation 34.a.

q = 1.3√0.33m

12.87m0.011√

4.75m

2.87𝑚(4.75𝑚

115.47𝑚) exp ( −2.2

2.24m

4.75𝑚) √𝑔 ∙ (4.75m)3 = 0.079𝑚3/𝑠/𝑚

applying the recurve factor from Table 76 leads to the following table:

Table 80 Overtopping results for the future dike at Wenduine, using the updated formulae from van der Meer et al, considering a composite slope

condition Rc[m] Hm0[m] Lm−1,0[m] 𝑑∗[−] 𝑞𝑓𝑢𝑡𝑢𝑟𝑒,𝑐𝑜𝑚𝑝[𝑚3/𝑠/𝑚]

1 2.24 4.75 115.47 0.000268 0.072

2 1.14 4.97 126.47 0.004392 0.289

The results seem far more realistic compared to the model test results. The statement that this

structure is a composite structure however seems very questionable. The length of the beach,

nearly 450m long, does raise awareness if the beach and wall can be seen as a structure all

together. Also, when applying composite parameter to the EurOtop calculations, the boundary

conditions of Equation 20 are not met and the results are a lot higher than assuming just a vertical

structure. A solution might be to limit the water height, but no guidelines are given on this. Despite


101

the better solutions for the updated formulae from van der Meer et al, they are considered

unreliable.


Current situation

There are a few problems which occur when trying to model the current structure. First of all,

the structure cannot begin with a slope milder than 1:8. Second, the length of each segment is

limited to 100m so that the beach cannot be modelled. Last, the structure cannot end in a berm so

that the promenade cannot be modelled as well. This leads to a very simplified model, which is

the same as the simplified model used in the EurOtop calculations by ignoring the beach and

promenade. The overtopping is thus going to be higher than reality again. This leads to the

questions if it has any use on modelling it like this. The results are found in Table 81.

Table 81 Overtopping results for current situation Wenduine, using PCO

condition Rc[m] Hm0[m] Lm−1,0[m] ξm−1,PCO[−] Ru2%,PCO[m] 𝑞𝑃𝐶𝑂[𝑚3/𝑚/𝑠]

1 1.54 4.75 115.47 2.465 15.58 3.076

2 0.44 4.97 126.47 2.410 16.25 5.661

Future situation

Due to the nature of the structure, PCO results for this part are not available.


When trying to model both structures in the Neural Network, a major problem is encountered.

NN has a minimum freeboard requirement of 0.5𝐻𝑚0. This is not met in any wave condition,

current or future situation. This cannot be circumvented by some handy tricks because altering

the freeboard or the wave height has a significant effect on the results. These results would not be

able to be compared with the other results.

When ignoring this, other problems arise as well. When trying to divide the structure in to toe-

slope-crest, several heavy distortions are required for the model to match the Neural Network

application range. Seeing the beach as a slope is not possible as the water height above the toe ℎ𝑡,

at the end of the beach, should be minimum 0.5𝐻𝑚0, which is impossible to do, even for adjusted

wave heights. Seeing the beach as a slope is also not possible as the limit angle is 1:10.

All these problems add up, and the structure, both in future and current situation have to be

altered too much. This is why no results are available for this paragraph as well.

3.8.6 CLASH

Current situation

The first filter is set for the slopes. The beach cannot be modelled as a slope as the mildest slope

angle available in CLASH for the lower slope is 1:7.07. The lower slope is thus set at 1:2 as well as

the upper slope. The next filter is to only have structures where the roughness factor is 1. The last


102

filter is to select the datasets with 𝑅𝑐/𝐻𝑚0 lower than 0.75, excluding those with 0 freeboard. This

leaves 17 results from dataset 030 ( Owen et al 1980) and dataset 042 (Coates et al 1997).

Future situation

For the future situation, the decision is made to check the structures with a lower slope of

cot(𝛼𝑑) = 0 and the upper slope vertical as well. The next filter is set to pick the ones with a

similar low water depth. This is done by checking 𝐻𝑚0/ℎ and setting it higher than 1.5. This lead

to 26 results from 3 datasets: 001, 028 (Herbert 1993) and 802 ( Goda et al 1975).

3.8.7 Summary

The results from previous calculations are put in Figure 79 (current situation) and Figure 80

(future situation). They have different graphs as they are different type of structures which

different formulae. Figure 78 gives all the absolute results.

Figure 78 Overtopping results for case Wenduine

0,E+00

1,E+00

2,E+00

3,E+00

4,E+00

5,E+00

6,E+00

1a 2a 1b 2b

Wave conditions

EurOtop

VdM

PCO

NN


103

Figure 79 Overtopping for smooth and simple slopes, non-breaking waves, with results from case Wenduine, current situation

Figure 80 Overtopping for vertical structures, impulsive conditions, breaking waves, with results from case Wenduine, future situation

Despite the high values, most of the values match the graphs very well, especially the CLASH

data found. The problem is however that compared to the model tests, the values calculated are

too high, for every method. For the current situation, the huge over prediction can partially be

blamed on the lack of a method to incorporate the promenade and the beach berm. For the future

situation, the same explanation can be given as well as the lack of clear definition between

composite structure and pure vertical structure. A better definition for 𝑑∗ or ℎ∗ in these type of

1,E-08

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5

EurOtop(2007)

VdM(2014)

PCO

CLASH

Ugent (2011)

𝑞

√𝑔

𝐻𝑚

03

[−]

𝑅𝑐∗ =

𝑅𝑐

𝐻𝑚0

[−]

1,E+00

1,E+01

1,E+02

1,E+03

1,E+04

1,E+05

1,E+06

1,E+07

1,E+08

1,E+09

0 0,005 0,01 0,015 0,02

EurOtop (2007)

VdM (2014)

Ugent (2011)

CLASH

𝑞∗

=𝑞

ℎ∗

2√

𝑔ℎ

𝑠3[−

]

Eq. 8.b

Eq. 18.b

5%

95%

95%

5%


104

structures, with a beach before the structure, would definitely help in have a more realistic view

on the overtopping. The in- and output sheets for case Wenduine is separated in two sheets, one

for the current, Annex Q, and one for the future situation, Annex R.

Chapter 4 -Comparison of the results

105

Chapter 4 Discussion

4.1 Comparison of the results

In this paragraph, a summary is presented of all the results from every case. In Table 82, the

overtopping results from every case is represented for worst case scenario. Figure 81 contains the

worst case conditions in a bar graphic. Annex I contains the bar graphs for every wave condition

for every case. This Annex can be used to quickly look compare the results between the cases.

Table 82 Worst case wave conditions for each case

𝐶𝑎𝑠𝑒 𝐶𝑜𝑛𝑑. 𝑅𝑐 𝐻𝑚0[𝑚] 𝐿𝑚−1,0[𝑚] 𝑞𝐸𝑢𝑟𝑂𝑡𝑜𝑝[𝑙/𝑠/𝑚] 𝑞𝑉𝑑𝑀[𝑙/𝑠/𝑚] 𝑞𝑃𝐶𝑂[𝑙/𝑠/𝑚] 𝑞𝑁𝑁[𝑙/𝑠/𝑚]

A 3 0.56 0.72 15.81 0.515 0.505(-2%) 0.930(+81%) 0.855(+66%)

B 5b 1.48 0.95 23.75 0.504 15(+2876%) - 0.678(+35%)

C 6 2.9 2 120.91 3.31 3.84(+16%) - 3.97(+20%)

D 1 1.38 1.93 26.13 0.454 0.395(-13%) 0.913(+102%) 15(+3203%)

E 1 5 3.5 290.33 37.4 39(+4%) 90.8(+143%) 39.5(+6%)

F 3 5.69 2.51 44.5 6.8 4.31(-37%) - 2.86(-58%)

M 3 2.65 1.29 45.53 0.039 0.0077(-80%) 0.311(+697%) 0.174(+346%)

W. current 2 0.44 4.97 126.47 5514 2717(-51%) - -

W. future 2 1.14 4.97 126.47 3442 2167(-37%) - -

Figure 81 graphical comparison of overtopping results for the worst wave conditions

Looking at Table 82, the figures from Annex I, the previous results in Chapter 3, and Figure 81,

several conclusions can be drawn. The most obvious conclusion is that there is no clear trend

between the methods. Sometimes some methods predict higher overtopping while for other type

of structures they give a lower results. Only for a few cases, the results from every method are

nearly the same. This is for case A, C and F. However, being nearly the same is quite relative as

some results in these case still differ up to 80%. Considering the other cases, this can however be

seen as similar results. The other cases have very different results with some several magnitudes

0,E+00

1,E+01

2,E+01

3,E+01

4,E+01

5,E+01

6,E+01

7,E+01

8,E+01

9,E+01

1,E+02

A B C D E F M

Wave conditions

EurOtop (2007)

VdM (2013)

PCO

NN

𝑞[𝑚

3/𝑠

/𝑚]


106

larger or smaller: cases B, D and M. For case D and M, this is rather surprising as these are

relatively simple and basic structures.

Case D is a simple slope with the results from EurOtop, van der Meer et al, and PCO having

similar results. The Neural Network however has a much larger value, which is highly

questionable due to the ease of the structure. The calculations in EurOtop have a high reliability

because of this, leading to the conclusion that the value for the Neural Network might be wrong.

It is however so that for wave condition 1 for case D, the period had to be adjusted in order to stay

within the applicability range of the wave steepness in NN. However, a 0.12s increase in wave

period does not justify an increase of an overtopping 30 times larger than those from EurOtop.

Looking at the other results for case D, it also seems that the Neural Network results are a lot

higher and out of place. The PCO results are higher but in the same magnitude of the EurOtop

results. One can conclude that the results from EurOtop, van der Meer et al, and PCO, are the more

correct values and that NN does not correctly process the structure for case D.

For case M, the results from the Neural Network are again a lot higher, as well as the PCO

results. For PCO, this can partly be blamed on the higher roughness factor, but the difference

between 𝛾𝑓 = 0.4 and 𝛾𝑓 = 0.5 should not be the cause of a 700% increase in overtopping. When

looking at Figure 75, it seems that the PCO results are in the same trend line as the NN results. The

results for PCO are more to the left because of the higher roughness influence factor. As both NN

and PCO results are of the same magnitude, questions arise if the EurOtop method is good for

these kinds of structures. This is however unlikely as the overtopping calculated at the walkway

is a very simple structure, consisting of 1 single slope. The crest reducing factor cannot be blamed

as the same factor is applied to the PCO results as to the EurOtop results. Comparing table 59 and

65, a resemblance is seen, with the offset caused by setting 𝛾𝑓 = 0.5 in PCO, meaning the final 𝛾𝑓

is calculated in the same manner. The breaker parameter has almost exactly the same value in

both methods. This leads to the last possible cause which can cause this discrepancy, the equation

on how the overtopping is calculated.

Case B is one of the more complex structures handled in this report. This is definitely made

clear by the huge difference of results for overtopping in the EurOtop result and the van der Meer

results. This is however only for the wave conditions where the water level reaches the vertical

structure. For the lower water level, only NN results are available as no PCO or updated equations

are proposed for emergent structures. The NN results for these lower water levels do show a large

increase compared to the EurOtop values as well. A reason, already mentioned in the case of

Wenduine might be the use of the impulsiveness parameter ℎ∗ in the EurOtop equations for these

kinds of structures: a beach with a vertical structure at the end combined with a very low water

height at the toe. This low water height at the toe makes it unsure that the structure can be seen

as a vertical structure. Comparing wave condition 5b from case B to the wave condition 1 from the

future structure of Wenduine, the same pattern holds true: The new updated formulae for vertical

structures give a lot higher overtopping than the traditional EurOtop formulae. Now, when


107

looking at wave condition 6b for case B and wave condition 2 for future Wenduine, this pattern

does not occur. The results for the overtopping calculated with EurOtop are very similar to those

with the new updated ones from van der Meer et al. In both these wave conditions, the water

height at the toe is higher and thus ℎ∗. This leads to the suggestion that either for the traditional

or for the updated equations, a lower limit should be defined for very low ℎ∗, which is the case for

most beaches running up to a vertical structure. This problem does not occur when using the

equations for a composite vertical structure, as can be seen in cases C and F. This might be because

the equation in EurOtop for composite structures set a lower limit on 𝑑∗ 𝑅𝑐/𝐻𝑚0. In case F, wave

condition 10 is lower than the minimum and it was clear that the value was not correct compared

to the other results.

The other cases, case A, C, E and F, have results for the different methods which are very similar.

For case A, this is not surprising as the structure is very simple. It is however strange that for this

structure NN gives similar results, while case D and M do not. Despite case M being in the non-

breaking waves regime, Case A and D do not have any apparent differences in wave conditions.

They are both in the wave breaking regime, and both have no roughness elements so that no real

cause can be found for the difference in results.

The good results for case C and F can be explained by the clear definition and use of 𝑑∗. In both

cases, it is clear that the structure is a composite structure and the parameter 𝑑∗ can be calculated

without any doubts. This is in contrast with the already mentioned cases of B and future scenario

of Wenduine, where the structure seems composite, but no 𝑑∗ can be calculated.

Despite the complexity of case E, the results are quite satisfying. The different methods seem

to give a same range of results with minor differences. The somewhat larger overtopping for PCO

results can be attributed to the steeper end slope in order to make the structure valid for PCO. As

this does not account for an increase of 130% compared to the EurOtop results, it does have a part

in it.

The cases from Wenduine stand out due to the high value found. This can be partly attributed

to the extreme high return period of 1:17000 year. However, for the 1:1000 year wave condition,

the overtopping is still much larger than for the other cases. The different results for the current

situation are similar, but a lot higher than the model tests results. One important reason for this

is the lack of the ability to model crest complexities in these kind of structures. The length of the

promenade surely plays a role in the actual overtopping amount, but does not show in the EurOtop

equations, and cannot be modelled in PCO or NN. van der Meer does give a significant lower

overtopping for the current situation. When comparing the results with the future structure,

EurOtop predicts a much lower overtopping when considering a vertical wall. However, for the

new updated formulae, wave condition 1 almost gives the same results. This is explained already

above that this might be due to the low water level at the toe. Another problem for the future

structure is that not only the promenade cannot be modelled, but the second wave wall can’t

either. Comparing the results with the model tests results and the problems mentioned here lead

to the conclusion that the results are not correct. Other methods cannot be utilised in order to

Chapter 4 -Safety of the structures

108

have more comparisons as both structures have to be altered too much to be used correctly as a

reference.

4.2 Safety of the structures

In Chapter 2 some limits are mentioned for the overtopping values. In Table 83 the overtopping

results calculated with EurOtop are compared with the limits.

Table 83 Comparing the EurOtop overtopping results with the overtopping limits, taken from EurOtop

case function 𝐿𝑖𝑚𝑖𝑡[𝑙/𝑠/𝑚] 𝑞[𝑙/𝑠/𝑚] Defence behind overtopping location

A rural area-no pedestrians 1-10 0.515 no

B Low speed vehicles+ aware pedestrians+ urban area 0.1 0.504 no

C Industrial area-Trained staff 1-10 3.31 second wave wall, higher freeboard

D Rural area- aware pedestrians 0.1 0.454 no

E Urban area- aware pedestrians 0.1 37.4 crest width

F Aware pedestrians (designed with allowed overtopping) 0.1 6.8 crest width

M Aware pedestrians 0.1 0.039 crest width

W Urban area- aware pedestrians 0.1 3442 crest width+second wave wall

The limits found in Table 83 are based on the table found in EurOtop when pedestrians or vehicles

are present. When the dike is not accessible to people, the limit is based on the table for damage

on the structure.

Structures A,C and M can be considered safe for the functions they serve. For structures B,D,

and F, the overtopping limit is exceeded by a small margin. In all these cases, the overtopping limit

is based on the pedestrians. Overtopping effects are however decreased, the further away the

people are from the point where overtopping happens. EurOtop states that the danger of an

overtopping 𝑞𝑥 at location 𝑥 is inversely proportional with the distance:

𝑞𝑥 = 𝑞/𝑥

In case F, and partially in case B, there is sufficient room for the pedestrians to go further from

this location so that despite the small exceedance, the structures can still be considered safe

according to EurOtop. For case D, the overtopping might be a problem as the walkway on top of

the dike has a small width and can be very dangerous for unaware bikers or people.

For cases E and the ones from Wenduine, serious problems exist. For Wenduine, the values are

skewed as discussed before and the model test state a value of 50𝑙/𝑠/𝑚 which is still a lot. It is for

this reason that the Belgian government decided to build a new structure in order to reduce this

value. For case E, the overtopping is also still high, despite the presence of the roughness elements.

However, when looking at the calculations for this case, the reader can see that this small strip

with roughness blocks, does not give a significant lower roughness influence factor. 𝛾𝑓 is only

about 0.96 while the roughness factor for the strip with blocks is around 0.75. This can be

explained by the high wave heights and large influence area, which is in contrast with the small

length of the roughness strip. The crest does however has a sufficiently large width in order to

decrease the danger for overtopping so that this dike is acceptable but has room for

improvements.

Chapter 4 -Use of the EurOtop manual and van der Meer et al

109

4.3 Use of the EurOtop manual and van der Meer et al

Based on the calculations in chapter 3, some conclusions can be drawn for the use of EurOtop.

The first conclusion is that the most obvious one: the easier the structure, the higher the reliability

of the results. For dikes consisting of one slope, there is much less ambiguity. Not much can go

wrong during the calculation and the results are straightforward. The same is true for the van der

Meer updated formulae. Structures with a composite slope are also easy to calculate and thus

pretty reliable. A berm does not bring any extra complexity to endanger the reliability of the

results. Problems however, do occur when multiple berms are present and when slope angles are

not in application range to be seen as neither a berm nor a slope. This brings a certain complexity

to the case which requires additional calculations and chances for mistakes and/or

misinterpretations.

This misinterpretation is especially a danger for structures where it is not clear it is a clean

vertical structure or a composite structures. One of these structures susceptible to a

misinterpretation is a beach with vertical structure at the end. In EurOtop, they have a solution

for this as an emergent structure, but only in very strict boundary conditions. For higher water

levels, when the SWL reaches the vertical structure, the use of ℎ∗ requires extra attention. EurOtop

also mentions the use of simple slope equations but most of the time, beaches have a mild slope,

which leads to them falling out of the boundary conditions for slope equations. This is made clear

in several cases in Chapter 3 and the author is of opinion that more research should be done in

these kinds of structures.

The complexity of the crests is also a problem which EurOtop does not cover. Second wave

walls, wide crests for smooth slopes with non-breaking waves, other types of crest complexities,

and so on, are not mentioned in EurOtop so that the reliability of the results for structures with

these kinds of elements are pretty low. Even for the crest elements which EurOtop does cover,

such as a recurved wall, or a crest width for armoured slopes, have a low reliability as they often

have very strict application ranges. This is often due to the lack of data so that extra research is

advised on crest elements. Recurve factors for emergent structures can also not be calculated as

this reduction factor 𝑘 is based on the water height above the toe of a vertical structure which

cannot be negative. It is very common for sloped beaches to have a recurved vertical wall at the

end making this a desired research topic as well.

A second conclusion can be drawn about the use of the updated equations for van der Meer.

While these papers have indeed improved the basic equations from EurOtop, they still do not

cover a lot of structures. As pointed out, the equations for vertical walls does have a certain

difficulty in handling structures with very low water height at the toe. The author thinks that some

sort of lower boundary should be set for minimum water height or a better definition for the

impulsiveness parameter where it is not clear it’s a composite or a clean vertical structure. Also,

the updated equations lack the information on how to handle influence factors. This is one of the

biggest drawbacks of the use of these equations. For the vertical structure equations, the


110

equations have a different structure compared to the EurOtop ones and thus the influence factors

cannot be handled in the same manner. In this paper the results from van der Meer et al are

manipulated by a self-defined factor 𝑘𝛽 for example in order to bring wave obliquity into effect in

the van der Meer results. van der Meer also does not give any advice on how to handle emergent

structures.

A third and last conclusion about the use of EurOtop can be drawn: when possible, always

compare the results with other methods. Despite that equations are based on a lot of data, many

structures have different elements making them non-standard structures which cannot be easily

calculated with EurOtop. Relying solely on these results is thus never a good option. It is advised

to consult every possible calculation method as possible and when designing a final structure,

model test should always be advised in order to confirm the results or search for a better or other

structure.


111

Chapter 5 Summary It is clear that overtopping is an important issue that still requires a lot of research. Based on

the results in this report, conclusions could be drawn on the objectives of the thesis.

The first goal was to write out a step-by-step calculation on the several case studies mentioned

in EurOtop. This is done correctly and resulted in discussions per case study. Chapter 3 reported

these calculations as well as a discussion about the results. This eventually resulted in an in- and

output sheet for each case. These sheets are a brief and concise summary on each case with only

the input which is needed to produce the relevant results. They can also be used in order to check

intermediary results used in the calculations such as the breaker parameter.

The second objective of this report was to compare the results from EurOtop with the new

updated van der Meer et al (2013) equations and other methods such as NN or PCO where

applicable. This lead to several conclusions about the use of EurOtop and the results themselves.

It was clear that the EurOtop results were often not in the same range of other results, which lead

to concerns about the reliability of the EurOtop results. This was often the case for the more

complex structures where misinterpretation could happen on how the structures function. Not

only the complex structures, but also some more simple ones, lead to strange results with a huge

difference compared to the Neural Network or PCO. The results from van der Meer et al generally

were in line with the results from EurOtop, with some exceptions such as the vertical structures

with a low water height at the toe and sloping foreshore. All this together lead to the general

conclusion that EurOtop is not suited for designing purposes. It is however good for quick

calculations and compare different kind of structures. This should always be assisted with

calculations from other methods. It is also good to see the effects of small alterations to a structure,

such as adding a recurve, or increasing the slope or freeboard… For actual designing purposes,

the designer should always have model tests done in order to validate structures or compare with

other possible solutions. EurOtop in its own, as well as the other methods, are not good enough to

ensure a safely designed structure.

The third and last objective of the thesis was to give advice on the use of EurOtop and possible

improvements. It is clear from Chapter 3 and 4 that there are still plenty of improvements left for

EurOtop. These improvements are mainly in the focus of structures which a have a mild beach

with a vertical structure on top. Also, crest complexities are not discussed enough in EurOtop.

Things like a second wave wall, or wide crests cannot be calculated with EurOtop. This thus needs

more attention in future editions of EurOtop. Also the topic of influence factors is not detailed

enough, especially for vertical structures. For example, no roughness factor can be implemented

for composite structures with a rough toe and very low water height above the toe, there are only

discrete solutions for oblique wave attack for vertical structures, and so on… The same problem


112

is true for the van der Meer et al equations. In their reports, they make no mention of these

influence factors or how they should be incorporated.

It is clear that despite the wide spread use of EurOtop and its easy to use procedures, a lot can

be improved and that results from this manual can sometimes be misinterpreted. It is good for

easy and simple structures but in real life, structures often have some complexities which are not

covered by the standard EurOtop equations. For these types of structures, extra attention has to

be given.


113

Chapter 6 References

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walls and composite breakwaters, seawalls and low reflections alternatives. Final

report of Monolithic Coastal Structures (MCS) project, Hannover

[2] Allsop, N. W. H., Pullen, T. A., van der Meer, J. W., Bruce, T., Schuttrumpf, H., &

Kortenhaus, A. (2008). Improvements in wave overtopping analysis: the EurOtop

overtopping manual and calculation tool. Proc. COPEDEC VII, Dubai, UAE.

[3] Baelus, L. & De Rouck, J. (2012). Fysische schaalmodelproeven overtopping

reducerende maatregelen promenade Wenduine, Rapportnummer OPW320/005,

Vakgroep Civiele Techniek, UGent & Afdeling kust, Vlaamse Overheid.

[4] Baelus, L., De Rouck, J., & Trouw, K. Physical Modelling of the Sea Defence Structures at

Wenduine for Structural Design.

[5] Battjes, J. A. (1974). Computation of set-up, longshore currents, run-up and

overtopping due to wind-generated waves (pp. 74-2). Department of Civil Engineering,

Delft University of Technology.

[6] Besley, P. (1999). Overtopping of seawalls – Design and assessment manual. R & D

Technical Report W 178, Environment Agency, Bristol, ISBN 1 85705 069 X

[7] Bosman, G., van der Meer, J.W., Hoffmans, G., Schüttrimpf, H., Verhagen, H.J. (2008).

Individual overtopping events at dikes.

[8] Bruce, T., Van der Meer, J. W., Allsop, N. W. H., Franco, L., Kortenhaus, A., Pullen, T., &

Schüttrumpf, H. (2013). EurOtop revisited. Part 2: vertical structures. Proc. ICE, Coasts,

Marine Structures and Breakwaters.

[9] CLASH. Crest Level Assessment of coastal Structures by full scale monitoring, neural

network prediction and Hazard analysis on permissible wave overtopping. Fifth

Framework Programme of the EU, Contract No.EVK3-CT-2001-00058. www.clash-

eu.org

[10] Dehenauw, I. D. (2008). De stormvloed van 1 februari 1953: een historische

terugblik met moderne technieken.

[11] EAK: (2002). Empfehlungen des Arbeitsausschusses Küstenschutzwerke. Die

Küste. H. 65

[12] Franco, L., De Gerloni, M., & Van der Meer, J. W. (1994). Wave overtopping on

vertical and composite breakwaters. Coastal Engineering Proceedings, 1(24).

[13] Hughes, S. (2005). Estimating irregular wave run-up on rough, impermeable

slopes, ERDC/CHL CHETN-III-70, U.S. Army Engineer Research and Develoment Center,

Vicksburg, MS.


114

[14] Nørgaard, J. Q. H., Lykke Andersen, T., & Burcharth, H. F. (2014). Distribution of

individual wave overtopping volumes in shallow water wave conditions.Coastal

Engineering, 83, 15-23.

[15] Pullen, T., Allsop, N.W.H., Bruce, T., Kortenhaus, A., Schüttrumpf, H., van der Meer,

J.W. (2007). EurOtop, Wave Overtopping of Sea Defences and Related Structures:

Assessment Manual, [Online] available at www.overtopping-manual.com

[16] Rimann, M. (2007). Anwendung und kritische beurteilung des neuen EurOtop-

Manuals zum Wellenüberlauf über Küstenschutzbauwerke (Masters’s thesis,

Technische Universität Braunschweig)

[17] Seed, R. B., Bea, R. G., Abdelmalak, R. I., Athanasopoulos, A. G., Boutwell Jr, G. P.,

Bray, J. D., ... & Yim, S. C. (2006). Investigation of the Performance of the New Orleans

Flood Protection System in Hurrican Katrina on August 29, 2005.

[18] Smith, K. (2014). Dawlish destruction: reconnecting the southwest.International

Railway Journal, 54(5).

[19] TAW (1985): Andringa, R. J. (1985). Leidraad voor het ontwerpen van rivierdijken:

Deel 1: Bovenrivierengebied. Rijkswaterstaat, DWW.

[20] TAW (1989): Andringa, R. J. (1989). Leidraad voor het ontwerpen van rivierdijken:

Deel 2: Benedenrivierengebied. Rijkswaterstaat, DWW.

[21] Van Broekhoven, P., & Jumelet, I. (2011). The influence of armour layer and core

permeability on the wave run-up (Doctoral dissertation, Master’s thesis, Delft

University of Technology).

[22] Van der Biest, K., Verwaest, T., Reyns, J., Mostaert, F. (2009) CLIMAR: Deelrapport

2-Kwantificatie van de secundaire gevolgen van de klimmatsverandering in de

Belgische kustvlakte, Flanders Hydraulics Research Report No.814_01.

[23] van der Meer, J. W. (2002). Technical report: wave run-up and wave overtopping

at dikes. Technical Advisory Committee on Flood Defence, Delft, The Netherlands, 43.

[24] van der Meer, J., Bruce, T., Allsop, W., Franco, L., Kortenhaus, A., Pullen, T., &

Schüttrumpf, H. EurOtop revisited. Part 1: sloping structures. Proc. ICE, Coasts, Marine

Structures and Breakwaters.

[25] van der Meer, J. W., Verhaeghe, H., & Steendam, G. J. (2009). The new wave

overtopping database for coastal structures. Coastal Engineering, 56(2), 108-120.

[26] Van Doorslaer, K., & De Rouck, J. (2011). Reduction on Wave Overtopping on a

Smooth Dike by Means of a Parapet. Coastal Engineering Proceedings,1(32),

structures-6

[27] Van Gent, M. R. A. (2002). Low-exceedance wave overtopping events. Delft

Hydraulics project id. DC030202/H3803.

[28] Veale, W., Suzuki, T., Verwaest, T., Trouw, K., Mertens, T. (2012). Integrated design

of coastal protection works for Wenduine, Belgium, Coastal Engineering Proceedings,

1(33), structures, 70.


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[29] Verwaest, T., Hassan, W., Reyns, J., Balens, N., Trouw, K., De Rouck, J., Van Doorslaer,

K., Troch, P. (2011). Hydrodynamic loading of wave return walls on top of seaside

promenades. 6th International conference on coastal structures 2011, Book of

Abstracts (pp. 125–126). Presented at the 6th International conference on coastal

structures 2011.

[30] Victor, L., Van der Meer, J. W., & Troch, P. (2012). Probability distribution of

individual wave overtopping volumes for smooth impermeable steep slopes with low

crest freeboards. Coastal Engineering, 64, 87-101.

[31] Xiaomin, W., Liehong, JU., Treuel, F.M. (2013). The study on wave run-up

roughness and permeability coefficient of stepped slope dike, Proceedings of the 7th

International Conference on Asian and Pacific Coasts, Bali, Indonesia.

Chapter 7 -Annexes

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Chapter 7 Annexes

Chapter 7 -Annexes

117

Annex A: Extra information case A Table A-1 Neural network input for case A with overtopping results

Cond. 𝛽[°] ℎ[𝑚] 𝐻𝑚0[𝑚] 𝑇𝑚−1,0[𝑠] ℎ𝑡[𝑚] 𝐵𝑡[𝑚] 𝛾𝑓[−] 𝑐𝑜𝑡(𝛼𝑑)[−] 𝑐𝑜𝑡(𝛼𝑢)[−] 𝑅𝑐[𝑚] 𝐵[𝑚] ℎ𝑏[𝑚] 𝑡𝑎𝑛(𝛼𝐵)[−] 𝐴𝑐[𝑚] 𝐺𝑐[𝑚] 𝑞[𝑚𝟑/𝑠

/𝑚]

1 36 4.97 0.62 2.455 4.97 0 0.6 6.667 4.5 0.56 0 1.34 0 0.56 0 4.10E-04

2 55 4.97 0.39 1.909 4.97 0 0.6 6.667 4.5 0.56 0 1.34 0 0.56 0 2.00E-05

3 19 4.97 0.72 3.182 4.97 0 0.6 6.667 4.5 0.56 0 1.34 0 0.56 0 8.55E-04

4 50 3.19 1.41 4.182 3.19 0 0.6 6.667 4.5 2.34 0 -0.44 0 2.34 0 1.28E-04

5 56 3.19 1.41 2.818 3.19 0 0.6 6.667 4.5 2.34 0 -0.44 0 2.34 0 0.00E+0

6 22 3.19 1.41 4.091 3.19 0 0.6 6.667 4.5 2.34 0 -0.44 0 2.34 0 1.29E-04

Table A-2 CLASH data for case A

Cond. 𝛽[°] ℎ[𝑚] 𝐻𝑚0[𝑚] 𝑇𝑚−1,0[𝑠] ℎ𝑡[𝑚] 𝐵𝑡[𝑚] 𝛾𝑓[−] 𝑐𝑜𝑡(𝛼𝑑)[−] 𝑐𝑜𝑡(𝛼𝑢)[−] 𝑅𝑐[𝑚] 𝐵[𝑚] ℎ𝑏[𝑚] 𝑡𝑎𝑛(𝛼𝐵)[−] 𝐴𝑐[𝑚] 𝐺𝑐[𝑚] 𝑞[𝑚𝟑/𝑠/𝑚]

109-081 0 0.14 0.0810 1.1345 0.14 0 1 6 6 0.12 0 0 0 0.12 0 3.37E-05

959-006 0 0.20 0.0544 0.9207 0.20 0 1 5.875 5.875 0.05 0 0 0 0.05 0 6.35E-05

101-010 0 0.70 0.1310 1.3345 0.70 0 1 6 6 0.10 0 0 0 0.10 0 4.04E-04

104-013 0 1.00 0.0660 1.0200 1.00 0 1 6 6 0.05 0 0 0 0.05 0 2.40E-04

104-015 0 1.00 0.0670 1.0100 1.00 0 1 6 6 0.05 0 0 0 0.05 0 1.95E-04

217-006 0 4.50 1.3880 4.5455 4.50 0 1 6 6 1.80 0 0 0 1.80 0 9.70E-04

110-022 0 5.01 0.7070 3.3000 5.01 0 1 6 6 0.99 0 0 0 0.99 0 4.00E-04

Table A-3 Summary overtopping results case A for different methods

𝑐𝑜𝑛𝑑. 𝑞[𝑚3/𝑠/𝑚] 𝑞𝑉𝑑𝑀[𝑚3/𝑠/𝑚] 𝑞𝑃𝐶𝑂[𝑚3/𝑠/𝑚] 𝑞𝑁𝑁[𝑚3/𝑠/𝑚]

1 1.60E-02 9.71E-03 3.80E-02 4.10E-01

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118

2 7.21E-06 4.11E-07 - 2.00E-02

3 5.15E-01 5.05E-01 9.30E-01 8.55E-01

4 8.13E-08 1.67E-10 - 1.28E-01

5 1.52E-18 1.75E-29 - - 6 4.93E-08 8.57E-11 - 1.29E-01

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Annex B: Extra information case B Table B-184 Intermediate overtopping results for case B assuming a simple slope, no influence factors but vertical wave wall

condition 𝑅𝑐[𝑚] 𝐻𝑚0[𝑚] 𝐿𝑚−1,0[𝑚] 𝜉𝑚−1,0[−] 𝑅𝑢2%[𝑚] 𝑞[𝑚3/𝑠/𝑚]

1a 3.30 1.32 52.52 0.631 1.374 4.23E-09

2a 2.80 1.60 45.53 0.533 1.408 1.22E-07

3a 2.35 1.55 39.03 0.502 1.283 3.76E-07

4a 1.95 1.25 31.62 0.503 1.037 1.86E-07

5a 1.48 0.95 23.75 0.500 0.784 1.15E-07

6a 1.00 0.50 13.13 0.512 0.423 1.07E-09

1b 3.30 1.32 52.52 0.505 1.099 3.42E-11

2b 2.80 1.60 45.53 0.427 1.127 2.22E-09

3b 2.35 1.55 39.03 0.401 1.027 9.31E-09

4b 1.95 1.25 31.62 0.402 0.830 4.19E-09

5b 1.48 0.95 23.75 0.400 0.627 2.54E-09

6b 1.00 0.50 13.13 0.410 0.338 9.29E-12

Table B-2 Intermediate overtopping results for case B assuming a vertical wall, no influence factors

condition 𝑅𝑐[𝑚] 𝐻𝑚0[𝑚] 𝐿𝑚−1,0[𝑚] ℎ∗[−] ℎ∗ ∙ 𝑅𝑐/𝐻𝑚0[𝑚] 𝑞[𝑚3/𝑠/𝑚]

5b 1.48 0.95 23.75 2.39E-05 3.73E-05 1.24E-03

6b 1.00 0.50 13.13 5.14E-02 1.03E-01 5.07E-04

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Table B-3 Intermediate overtopping results for case B assuming an emergent wall, no influence factors

condition 𝑅𝑐[𝑚] 𝐻𝑚0[𝑚] 𝐿𝑚−1,0[𝑚] cot(𝛼𝑓) ∙ 𝑠00.33[−] 𝑞[𝑚3/𝑠/𝑚]

1a 3.30 1.32 52.52 2.97 4.53E-08

2a 2.80 1.60 45.53 3.31 1.68E-06

3a 2.35 1.55 39.03 3.45 5.14E-06

4a 1.95 1.25 31.62 3.44 2.73E-06

5a 1.48 0.95 23.75 3.46 1.75E-06

6a 1.00 0.50 13.13 3.40 3.21E-08

1b 3.30 1.32 52.52 3.71 7.39E-10

2b 2.80 1.60 45.53 4.14 6.56E-08

3b 2.35 1.55 39.03 4.31 2.73E-07

4b 1.95 1.25 31.62 4.30 1.34E-07

5b 1.48 0.95 23.75 4.32 1.24E-03

6b 1.00 0.50 13.13 4.25 5.07E-04

Table B-4 Neural Network input for case B with overtopping results

𝐶𝑜𝑛𝑑. 𝛽[°] ℎ[𝑚] 𝐻𝑚0[𝑚] 𝑇𝑚−1,0[𝑠] ℎ𝑡[𝑚] 𝐵𝑡[𝑚] 𝛾𝑓[−] 𝑐𝑜𝑡(𝛼𝑑)[−] 𝑐𝑜𝑡(𝛼𝑢)[−] 𝑅𝑐[𝑚] 𝐵[𝑚] ℎ𝑏[𝑚] 𝑡𝑎𝑛(𝛼𝐵)[−] 𝐴𝑐[𝑚] 𝐺𝑐[𝑚] 𝑞[𝑚𝟑/𝑠/𝑚]

1a 30 2.2 1.32 5.8 2.2 0 1 10 -0.3 3.3 0 -1.32 0 3.3 0 1.38E-04

2a 30 2.7 1.6 5.4 2.7 0 1 10 -0.3 2.8 0 -1.6 0 2.8 0 6.34E-04

3a 30 3.15 1.55 5 3.15 0 1 10 -0.3 2.35 0 -1.55 0 2.35 0 9.01E-04

4a 30 3.55 1.25 4.5 3.55 0 1 10 -0.3 1.95 0 -1.25 0 1.95 0 5.58E-04

5a 30 4.02 0.95 3.9 4.02 0 1 10 -0.3 1.48 0 -0.95 0 1.48 0 3.05E-04

6a 30 4.5 0.5 2.9 4.5 0 1 10 -0.3 1 0 -0.5 0 1 0 2.74E-05

1b 30 2.2 1.32 5.8 2.2 0 1 10 -0.1 3.3 0 -1.32 0 3.3 0 1.35E-04

2b 30 2.7 1.6 5.4 2.7 0 1 10 -0.1 2.8 0 -1.3 0 2.8 0 7.26E-04

3b 30 3.15 1.55 5 3.15 0 1 10 -0.1 2.35 0 -0.85 0 2.35 0 1.27E-03

4b 30 3.55 1.25 4.5 3.55 0 1 10 -0.1 1.95 0 -0.45 0 1.95 0 9.06E-04

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121

5b 30 4.02 0.95 3.9 4.02 0 1 10 -0.1 1.48 0 0.02 0 1.48 0 6.78E-04

6b 30 4.5 0.5 2.9 4.5 0 1 10 -0.1 1 0 0.5 0 1 0 1.45E-04

Table B-5 CLASH datasets for case B


036-001 0 0.07 0.0514 1.4556 0.07 0 1 5.455 0 0.09 0.8 0.0068 0.0600 0.09 0 2.72E-06

036-003 0 0.09 0.0538 1.1909 0.09 0 1 5.263 0 0.07 0.48 -0.0052 0.0667 0.07 0 0.00E+00

036-004 0 0.07 0.0499 1.5389 0.07 0 1 7.073 0 0.09 0.4 -0.0162 0.0250 0.09 0 6.08E-06

036-006 0 0.11 0.0597 1.1165 0.11 0 1 5.217 0 0.05 0.32 0.0048 0.0875 0.05 0 8.80E-07

036-007 0 0.07 0.0519 1.6733 0.07 0 1 5.000 0 0.09 0.72 -0.0012 0.0500 0.09 0 3.66E-05

036-008 0 0.09 0.0617 1.5795 0.09 0 1 5.333 0 0.07 0.52 0.0088 0.0923 0.07 0 9.67E-05

036-009 0 0.11 0.0680 1.2927 0.11 0 1 5.238 0 0.05 0.4 0.0188 0.0500 0.05 0 5.86E-05

Table B-6 Summary overtopping results for case B for different methods

𝑐𝑜𝑛𝑑. 𝑞[𝑚3/𝑠/𝑚] 𝑞𝑉𝑑𝑀[𝑚3/𝑠/𝑚] 𝑞𝑃𝐶𝑂[𝑚3/𝑠/𝑚] 𝑞𝑁𝑁[𝑚3/𝑠/𝑚] 1a

4.53E-05 - - 1.38E-01 2a 1.68E-03 - - 6.34E-01 3a 5.14E-03 - - 9.01E-01 4a 2.73E-03 - - 5.58E-01 5a 1.75E-03 - - 3.05E-01 6a 3.21E-05 - - 2.74E-02 1b 7.39E-07 - - 1.35E-01 2b 6.56E-05 - - 7.26E-01 3b 2.73E-04 - - 1.27E+00 4b 1.34E-04 - - 9.06E-01 5b 5.04E-01 1.50E+01 - 6.78E-01

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6b 1.65E-02 3.23E-02 - 1.45E-01

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Annex C: Extra information for case C

table C-1 Overtopping results using EurOtop at second location for case C


1 4.52 0.50 8.26 2.09E+00 8.89E-01 2.55E-24

2 4.32 0.80 48.96 4.03E+00 1.56E+00 3.08E-11

3 4.52 1.20 72.19 3.99E+00 2.34E+00 6.71E-08

4 4.02 0.80 48.96 4.03E+00 1.56E+00 1.57E-10

5 4.52 2.00 120.91 4.00E+00 3.90E+00 9.89E-05

6 4.32 2.00 120.91 4.00E+00 3.90E+00 1.53E-04

7 4.12 1.60 97.44 4.02E+00 3.12E+00 1.81E-05

8 3.72 1.20 72.19 3.99E+00 2.34E+00 1.21E-06

9 3.47 0.80 48.96 4.03E+00 1.56E+00 3.08E-09 condition Rc[m] Hm0[m] Lm−1,0[m] 𝑑∗[−] q[m3/s/m]

10 2.92 0.50 8.26 0.52 1.13E-08

Table C-2 Overtopping results using updated formulae from van der Meer et al at second location for case C


1 4.52 0.50 8.26 2.09 8.89E-01 5.14E-38

2 4.32 0.80 48.96 4.03 1.56E+00 3.19E-14

3 4.52 1.20 72.19 3.99 2.34E+00 3.59E-09

4 4.02 0.80 48.96 4.03 1.56E+00 4.44E-13

5 4.52 2.00 120.91 4.00 3.90E+00 5.98E-05

6 4.32 2.00 120.91 4.00 3.90E+00 1.03E-04

7 4.12 1.60 97.44 4.02 3.12E+00 7.39E-06

8 3.72 1.20 72.19 3.99 2.34E+00 2.23E-07

9 3.47 0.80 48.96 4.03 1.56E+00 4.75E-11

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condition Rc[m] Hm0[m] Lm−1,0[m] 𝑑∗[−] q[m3/s/m]

10 2.92 0.50 8.26 0.52 1.37E-13

Table C-3 Neural Network input and overtopping results for case C at first overtopping location


1 0 6.3 0.5 2.3 3.85 0 0.6 1.94 -0.28 2.5 0.8 -0.5 0 2.5 0 -

2 0 6.5 0.8 5.6 4.05 0 0.6 1.94 -0.28 2.9 0.8 -0.8 0 2.9 0 -

3 0 6.3 1.2 6.8 3.85 0 0.6 1.94 -0.28 3.1 0.8 -1.2 0 3.1 0 9.83E-05

4 0 6.8 0.8 5.6 4.35 0 0.6 1.94 -0.28 2.6 0.8 -0.8 0 2.6 0 -

5 0 6.3 2 8.8 3.85 0 0.6 1.94 -0.28 3.1 0.8 -1.4 0 3.1 0 3.06E-03

6 0 6.5 2 8.8 4.05 0 0.6 1.94 -0.28 2.9 0.8 -1.2 0 2.9 0 3.97E-03

7 0 6.7 1.6 7.9 4.25 0 0.6 1.94 -0.28 2.7 0.8 -1 0 2.7 0 8.81E-04

8 0 7.1 1.2 6.8 4.65 0 0.6 1.94 -0.28 2.3 0.8 -0.6 0 2.3 0 2.44E-04

9 0 7.3 0.8 5.6 4.9 0 0.6 1.94 -0.28 2.05 0.8 -0.35 0 2.05 0 2.27E-05

10 0 7.9 0.5 2.3 5.45 0 0.6 1.94 -0.28 1.5 0.8 0.2 0 1.5 0 -

Table C-4 Neural Network input and overtopping results for case C at second overtopping location


1 0 6.3 0.5 2.3 3.85 0 0.6 1.94 0 2.5 0.8 -0.5 0 2.5 0 -

2 0 6.5 0.8 5.6 4.05 0 0.6 1.94 0 4 0.8 -0.8 0 4 0 -

3 0 6.3 1.2 6.8 3.85 0 0.6 1.94 0 4.52 0.8 -1.2 0 4.52 0 -

4 0 6.8 0.8 5.6 4.35 0 0.6 1.94 0 4 0.8 -0.8 0 4 0 -

5 0 6.3 2 8.8 3.85 0 0.6 1.94 0 4.52 0.8 -1.4 0 4.52 0 0.000532

6 0 6.5 2 8.8 4.05 0 0.6 1.94 0 4.32 0.8 -1.2 0 4.32 0 0.000617

7 0 6.7 1.6 7.9 4.25 0 0.6 1.94 0 4.12 0.8 -1 0 4.12 0 0.000202

8 0 7.1 1.2 6.8 4.65 0 0.6 1.94 0 3.72 0.8 -0.6 0 3.72 0 -

9 0 7.3 0.8 5.6 4.9 0 0.6 1.94 0 3.47 0.8 -0.35 0 3.47 0 -

10 0 7.9 0.5 2.3 5.45 0 0.6 1.94 0 2.5 0.8 0.2 0 2.5 0 -

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Table C-5 CLASH datasets for case C


035-005 0 0.50 0.0910 1.745 0.500 0 1.00 2.000 0 0.200 0.15 -0.055 0 0.200 0.000 1.28E-05

035-024 0 0.50 0.0910 1.745 0.500 0 1.00 2.000 0 0.200 0.00 -0.055 0 0.200 0.000 1.69E-05

035-076 0 0.50 0.0910 1.745 0.500 0 0.40 2.000 0 0.260 0.15 -0.055 0 0.260 0.000 6.30E-08

035-094 0 0.50 0.0910 1.745 0.500 0 0.40 2.000 0 0.140 0.15 -0.055 0 0.140 0.000 3.78E-07

035-113 0 0.50 0.0910 1.745 0.500 0 0.40 2.000 0 0.170 0.15 -0.055 0 0.170 0.000 1.89E-07

035-151 0 0.50 0.0910 1.745 0.500 0 0.40 2.000 0 0.140 0.15 -0.055 0 0.140 0.000 6.30E-08

035-189 0 0.50 0.0910 1.745 0.500 0 0.40 2.000 0 0.140 0.30 -0.055 0 0.140 0.000 1.26E-07

035-225 0 0.50 0.0910 1.745 0.500 0 0.35 2.000 0 0.140 0.15 -0.055 0 0.140 0.000 1.42E-07

509-016 0 0.11 0.0414 1.152 0.111 0 1.00 1.558 0 0.180 0.10 0.030 0 0.080 0.025 5.30E-07

509-018 0 0.09 0.0359 1.129 0.091 0 1.00 1.558 0 0.200 0.10 0.010 0 0.100 0.025 5.76E-08

Table C-6 Overtopping results for case C using NN at second overtopping location


1 4.52 0.50 8.26 2.09 8.89E-01 -

2 4.32 0.80 48.96 4.03 1.56E+00 -

3 4.52 1.20 72.19 3.99 2.34E+00 -

4 4.02 0.80 48.96 4.03 1.56E+00 -

5 4.52 2.00 120.91 4.00 3.90E+00 5.32E-04

6 4.32 2.00 120.91 4.00 3.90E+00 6.17E-04

7 4.12 1.60 97.44 4.02 3.12E+00 2.02E-04

8 3.72 1.20 72.19 3.99 2.34E+00 -

9 3.47 0.80 48.96 4.03 1.56E+00 - condition Rc[m] Hm0[m] Lm−1,0[m] 𝑑∗[−] q[m3/s/m]

10 2.92 0.50 8.26 0.52 -

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Table C-7 Summary overtopping results for case C at first overtopping location


1 4.07E-14 5.30E-21 - -

2 6.75E-05 4.79E-06 - -

3 1.13E-02 4.58E-03 - 9.83E-02

4 3.43E-04 4.89E-05 - -

5 2.14E+00 2.37E+00 - 3.06E+00

6 3.31E+00 3.84E+00 - 3.97E+00

7 8.46E-01 8.60E-01 - 8.81E-01

8 2.04E-01 1.73E-01 - 2.44E-01

9 6.75E-03 2.80E-03 - 2.27E-02

10 9.07E-04 2.77E-05 - -

Table C-8 Summary overtopping results for case C at second overtopping location


1 2.55E-21 5.14E-35 - -

2 3.08E-08 3.19E-11 - -

3 6.71E-05 3.59E-06 - -

4 1.57E-07 4.44E-10 - -

5 9.89E-02 5.98E-02 - 5.32E-01

6 1.53E-01 1.03E-01 - 6.17E-01

7 1.81E-02 7.39E-03 - 2.02E-01

8 1.21E-03 2.23E-04 - -

9 3.08E-06 4.75E-08 - -

10 1.13E-05 1.37E-10 - -

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Annex D: Extra information case D

Table D-1 Neural Network input for case D with overtopping results


1 0 3.0 1.93 4.21 3 0 1 8 8 1.38 0 0 0 1.38 0 1.37E-03

2 0 2.5 1.65 4.09 2.5 0 1 8 8 1.88 0 0 0 1.88 0 2.99E-04

3 0 2.0 1.38 4.09 2 0 1 8 8 2.38 0 0 0 2.38 0 4.89E-05

4 0 1.7 1.21 4.09 1.7 0 1 8 8 2.68 0 0 0 2.68 0 1.60E-05

Table D-2 CLASH datasets for case D


218-047 0 0.50 0.1008 1.432 0.500 0 1.00 7.000 7 0.150 0.00 0.000 0 0.150 0.000 3.64E-06

218-048 0 0.50 0.1013 1.432 0.500 0 1.00 7.000 7 0.133 0.00 0.000 0 0.133 0.000 1.24E-05

218-049 0 0.50 0.0989 1.432 0.500 0 1.00 7.000 7 0.122 0.00 0.000 0 0.122 0.000 2.33E-05

218-050 0 0.50 0.1000 1.432 0.500 0 1.00 7.000 7 0.105 0.00 0.000 0 0.105 0.000 4.76E-05

218-051 0 0.50 0.1037 1.432 0.500 0 1.00 7.000 7 0.080 0.00 0.000 0 0.080 0.000 1.78E-04

218-052 0 0.50 0.1027 1.432 0.495 0 1.00 7.000 7 0.085 0.00 0.000 0 0.085 0.000 1.40E-04

218-053 0 0.50 0.1372 1.432 0.500 0 1.00 7.000 7 0.151 0.00 0.000 0 0.151 0.000 2.27E-05

218-054 0 0.5 0.1403 1.432 0.500 0 1.00 7.000 7 0.133 0.00 0.000 0 0.133 0.000 5.82E-05

218-055 0 0.50 0.1416 1.432 0.500 0 1.00 7.000 7 0.122 0.00 0.000 0 0.122 0.000 9.08E-05

218-056 0 0.50 0.1399 1.432 0.500 0 1.00 7.000 7 0.105 0.00 0.000 0 0.105 0.000 1.97E-04

Table D-3 Summary overtopping results for case D


1 0.454411 0.395212 0.913 14.95

2 0.011782 0.004926 0.033 2.797

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3 0.000151 1.49E-05 0.001 0.382

4 6.25E-06 1.58E-07 - 0.1096

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Annex E: Extra information case E

Table E-1 Neural Network input for case E with overtopping results


1 0 6.27 3.5 13.67 4.445 12.045 1 2.4 7.93 5 7.2 0.23 0.1 5 0 0.0395

2 0 5.41 3.5 13.67 3.585 12.045 1 2.4 7.93 5.86 7.2 -0.63 0.1 5.86 0 0.0187

Table E-2 CLASH datasets for case E


111-001 0 1.82 0.4100 6.498 1.820 0 0.95 4.000 8 1.270 1.45 0.770 0 1.270 0.000 0.00E+00

111-003 0 1.83 0.4030 5.443 1.830 0 0.95 4.000 8 1.260 1.45 0.760 0 1.260 0.000 0.00E+00

111-007 0 1.83 0.6090 5.930 1.830 0 0.95 4.000 8 1.260 1.45 0.760 0 1.260 0.000 5.06E-03

111-010 0 1.83 0.8090 6.384 1.830 0 0.95 4.000 8 1.260 1.45 0.760 0 1.260 0.000 5.42E-02

111-048 0 1.82 0.8030 7.264 1.820 0 0.95 4.000 8 1.270 1.45 0.770 0 1.270 0.000 5.18E-02

111-068 0 1.82 0.7920 8.224 1.820 0 0.95 4.000 8 1.270 1.45 0.770 0 1.270 0.000 6.96E-02

111-076 0 1.82 0.8070 9.858 1.820 0 0.95 4.000 8 1.270 1.45 0.770 0 1.270 0.000 7.78E-02

Table E-3 Summary overtopping results for case E, interpolated values


1 37.4 39 90.8 39.53

2 17.1 17.5 46.1 18.7

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Annex F: Extra information case F

Table F-1 Neural Network input for case F, with overtopping results


1 0 4.30 2.37 5.33 4.3 0 1 0.80 0 6.34 8.5 2.05 0 6.34 0 1.52E-03

2 0 4.70 2.53 5.34 4.7 0 1 0.80 0 5.94 8.5 2.45 0 5.94 0 2.34E-03

3 0 4.95 2.51 5.34 4.95 0 1 0.80 0 5.69 8.5 2.7 0 5.69 0 2.86E-03

4 0 5.04 2.47 5.34 5.04 0 1 0.80 0 5.6 8.5 2.79 0 5.6 0 3.03E-03

5 0 4.97 2.22 5.35 4.97 0 1 0.80 0 5.71 8.5 2.72 0 5.71 0 2.40E-03

6 0 4.76 2.07 5.46 4.76 0 1 0.80 0 5.88 8.5 2.51 0 5.88 0 1.81E-03

7 0 3.98 1.75 5.85 3.98 0 1 0.80 0 6.66 8.5 1.73 0 6.66 0 6.72E-04

8 0 3.43 1.56 5.97 3.43 0 1 0.80 0 7.21 8.5 1.18 0 7.21 0 2.86E-04

9 0 2.87 1.40 5.86 2.87 0 1 0.80 0 7 8.5 0.62 0 7 0 1.36E-04

10 0 2.30 1.26 5.52 2.30 0 1 0.80 0 6.3 8.5 0.05 0 6.3 0 6.47E-05

Table F-2 CLASH datasets for case F


043-008 10 0.17 0.0912 1.307 0.171 0 1.00 1.000 0 0.361 0.40 0.059 0 0.160 0.050 1.40E-06

043-009 10 0.17 0.0628 1.362 0.171 0 1.00 1.000 0 0.361 0.40 0.059 0 0.160 0.050 2.61E-07

043-010 10 0.17 0.0684 1.805 0.171 0 1.00 1.000 0 0.361 0.40 0.059 0 0.160 0.050 1.81E-06

043-011 10 0.17 0.0822 1.236 0.171 0 1.00 1.000 0 0.361 0.40 0.059 0 0.160 0.050 1.31E-06

043-012 10 0.17 0.0851 1.362 0.171 0 1.00 1.000 0 0.361 0.40 0.059 0 0.160 0.050 1.60E-06

043-013 10 0.171 0.0818 1.474 0.171 0 1.00 1.000 0 0.361 0.40 0.059 0 0.160 0.050 2.95E-06

043-014 10 0.17 0.0599 1.467 0.171 0 1.00 1.000 0 0.361 0.40 0.059 0 0.160 0.050 1.12E-06

044-011 4.62 3.43 1.7090 7.278 3.425 0 1.00 1.000 0 7.215 8.00 1.175 0 3.195 1.000 5.98E-04

044-012 2.7 2.865 1.542 7.187273 2.865 0 1.00 1.000 0 7.775 8.00 0.615 0 3.755 1.000 1.91E-04

044-022 14 3.28 1.666 6.171818 3.280 0 1.00 1.000 0 7.360 8.00 1.030 0 3.340 1.000 1.41E-04

044-023 30 2.715 1.537 6.171818 2.715 0 1.00 1.000 0 7.925 8.00 0.465 0 3.905 1.000 1.65E-04

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Table F-3 Summary overtopping results for case F


1 4.689 2.409 - 1.521

2 6.545 3.939 - 2.340

3 6.800 4.310 - 2.858

4 6.576 4.199 - 3.033

5 4.305 2.511 - 2.402

6 3.227 1.682 - 1.807

7 1.688 0.578 - 0.672

8 1.160 0.266 - 0.286

9 0.960 0.111 - 0.136

10 - 0.019 - 0.065

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Annex G: Extra information case M

Table G-1 Neural Network input for case M at overtopping location 1, the walkway


1 0 1.16 0.85 5.3 1.16 0 0.4 1.5 1.5 3.34 0 0 0 3.34 1.4 -

2 0 1.70 1.19 5.5 1.70 0 0.4 1.5 1.5 2.80 0 0 0 2.80 1.4 8.57E-05

3 0 1.85 1.29 5.4 1.85 0 0.4 1.5 1.5 2.65 0 0 0 2.65 1.4 1.74E-04

4 0 1.00 0.77 5.9 1.00 0 0.4 1.5 1.5 3.50 0 0 0 3.50 1.4 -

Table G-2 Neural Network input for case M at overtopping location 2, recreational area


1 0 1.16 0.85 5.3 1.16 0 0.4 1.5 1.5 4.25 2.9 -0.85 0 4.25 1.3 -

2 0 1.70 1.19 5.5 1.70 0 0.4 1.5 1.5 5.90 2.9 -1.19 0 5.20 1.3 -

3 0 1.85 1.29 5.4 1.85 0 0.4 1.5 1.5 5.75 2.9 -1.29 0 5.05 1.3 -

4 0 1.00 0.77 5.9 1.00 0 0.4 1.5 1.5 3.85 2.9 -0.77 0 3.85 1.3 -

Table G-3 CLASH datasets for case M


037-025 0 0.13 0.0515 1.462 0.095 0.030928 0.46 1.500 1.5 0.122 0.12 -0.008 0 0.107 0.078 0.00E+00

037-026 0 0.13 0.0619 1.658 0.095 0.030928 0.46 1.500 1.5 0.122 0.12 -0.008 0 0.107 0.078 2.22E-06

037-027 0 0.13 0.0619 1.841 0.095 0.030928 0.46 1.500 1.5 0.122 0.12 -0.008 0 0.107 0.078 7.99E-06

326-024 0 0.21 0.1138 1.303 0.212 0 0.40 1.500 1.5 0.150 0.17 -0.014 0 0.150 0.278 5.56E-06

326-025 0 0.21 0.1184 1.515 0.212 0 0.40 1.500 1.5 0.150 0.17 -0.014 0 0.150 0.278 6.02E-06

326-026 0 0.21 0.1256 1.364 0.212 0 0.40 1.500 1.5 0.150 0.17 -0.014 0 0.150 0.278 9.26E-06

326-027 0 0.21 0.1354 1.667 0.212 0 0.40 1.500 1.5 0.150 0.17 -0.014 0 0.150 0.278 2.04E-05

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Table G-4Summary overtopping results for case M, at overtopping location 1, the walkway


1 8.55426E-06 3.52526E-08 0.000259 -

2 0.010321289 0.001213808 0.098509 0.08569

3 0.0384707 0.007195197 0.311215 0.1741

4 2.28571E-06 5.05173E-09 - -

Table G-5 Summary overtopping results for case M, at overtopping location 2, recreational area


5 2.25204E-12 1.67984E-20 - -

6 9.08154E-08 4.54155E-12 - -

7 5.93588E-07 1.10497E-10 - -

8 5.49256E-13 1.41163E-21 - -

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Annex H: Extra information case Wenduine Table H-1 CLASH datasets for current situation Wenduine


030-022 0 0.16 0.0699 0.998 0.160 0 1.00 2.000 2 0.040 0.00 0.000 0 0.040 0.000 2.19E-03

030-240 0 0.16 0.0694 1.028 0.160 0 1.00 2.000 2 0.040 0.40 0.000 0 0.040 0.000 7.91E-04

042-110 0 0.40 0.1617 1.643 0.400 0 1.00 2.000 2 0.100 0.00 0.000 0 0.100 0.000 6.07E-03

042-111 0 0.40 0.1595 1.926 0.400 0 1.00 2.000 2 0.100 0.00 0.000 0 0.100 0.000 5.76E-03

042-112 0 0.40 0.1676 2.114 0.400 0 1.00 2.000 2 0.100 0.00 0.000 0 0.100 0.000 6.53E-03

042-113 0 0.40 0.1665 2.241 0.400 0 1.00 2.000 2 0.100 0.00 0.000 0 0.100 0.000 6.62E-03

042-114 0 0.40 0.1755 2.346 0.400 0 1.00 2.000 2 0.100 0.00 0.000 0 0.100 0.000 8.25E-03

042-116 0 0.4 0.1343 1.662 0.400 0 1.00 2.000 2 0.100 0.00 0.000 0 0.100 0.000 3.77E-03

042-117 0 0.40 0.1349 1.822 0.400 0 1.00 2.000 2 0.100 0.00 0.000 0 0.100 0.000 3.32E-03

042-120 0 0.40 0.1355 2.512 0.400 0 1.00 2.000 2 0.100 0.00 0.000 0 0.100 0.000 3.41E-03

042-121 0 0.4 0.1334 2.660 0.400 0 1.00 2.000 2 0.100 0.00 0.000 0 0.100 0.000 3.52E-03

042-122 0 0.4 0.14815 2.753 0.400 0 1.00 2.000 2 0.100 0.00 0.000 0 0.100 0.000 4.95E-03

042-128 0 0.4 0.19635 1.953 0.400 0 1.00 2.000 2 0.100 0.00 0.000 0 0.100 0.000 1.03E-02

042-129 0 0.4 0.16805 2.059 0.400 0 1.00 2.000 2 0.100 0.00 0.000 0 0.100 0.000 6.74E-03

042-130 0 0.4 0.1641 2.067 0.400 0 1.00 2.000 2 0.100 0.00 0.000 0 0.100 0.000 7.56E-03

042-132 0 0.4 0.16345 1.949 0.400 0 1.00 2.000 2 0.100 0.00 0.000 0 0.100 0.000 5.87E-03

042-135 0 0.4 0.14625 1.800 0.400 0 1.00 2.000 2 0.100 0.00 0.000 0 0.100 0.000 5.19E-03

Table H-2 CLASH datasets for future situation Wenduine


001-006 0 0.02 0.0420 1.968 0.023 0 1.00 0.000 0 0.025 0.00 0.000 0 0.009 0.432 2.08E-03

001-012 0 0.02 0.0420 1.968 0.023 0 1.00 0.000 0 0.100 0.00 0.000 0 0.031 0.432 9.69E-05

001-018 0 0.02 0.0420 1.968 0.023 0 1.00 0.000 0 0.025 0.00 0.000 0 0.084 0.432 2.81E-04

001-021 0 0.02 0.0420 1.968 0.023 0 1.00 0.000 0 0.100 0.00 0.000 0 0.031 0.432 1.31E-04

001-024 0 0.02 0.0420 1.968 0.023 0 1.00 0.000 0 0.100 0.00 0.000 0 0.031 1.376 2.03E-05

028-005 0 0.05 0.0820 1.935 0.050 0 1.00 0.000 0 0.050 0.00 0.000 0 0.050 0.000 8.44E-04

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028-014 0 0.05 0.0810 1.930 0.050 0 1.00 0.000 0 0.100 0.00 0.000 0 0.100 0.000 2.13E-04

028-021 0 0.05 0.0810 1.868 0.050 0 1.00 0.000 0 0.150 0.00 0.000 0 0.150 0.000 1.63E-05

028-022 0 0.05 0.0820 1.868 0.050 0 1.00 0.000 0 0.150 0.00 0.000 0 0.150 0.000 5.09E-06

028-027 0 0.05 0.0810 1.930 0.050 0 1.00 0.000 0 0.200 0.00 0.000 0 0.200 0.000 2.21E-05

028-028 0 0.05 0.081 1.930 0.050 0 1.00 0.000 0 0.200 0.00 0.000 0 0.200 0.000 1.61E-05

802-112 0 0.05 0.083 1.644 0.050 0 1.00 0.000 0 0.072 0.00 0.000 0 0.072 0.000 1.33E-03

802-113 0 0.05 0.083 1.644 0.050 0 1.00 0.000 0 0.107 0.00 0.000 0 0.107 0.000 5.77E-04

802-114 0 0.05 0.083 1.644 0.050 0 1.00 0.000 0 0.146 0.00 0.000 0 0.146 0.000 2.11E-04

802-115 0 0.05 0.083 1.644 0.050 0 1.00 0.000 0 0.186 0.00 0.000 0 0.186 0.000 7.58E-05

802-116 0 0.05 0.083 1.644 0.050 0 1.00 0.000 0 0.223 0.00 0.000 0 0.223 0.000 4.61E-05

802-117 0 0.05 0.083 1.644 0.050 0 1.00 0.000 0 0.260 0.00 0.000 0 0.260 0.000 2.23E-05

802-150 0 0.05 0.097 2.223 0.050 0 1.00 0.000 0 0.110 0.00 0.000 0 0.110 0.000 7.99E-04

802-151 0 0.05 0.096 2.223 0.050 0 1.00 0.000 0 0.150 0.00 0.000 0 0.150 0.000 3.57E-04

802-152 0 0.05 0.096 2.223 0.050 0 1.00 0.000 0 0.185 0.00 0.000 0 0.185 0.000 1.45E-04

802-181 0 0.05 0.108 2.701 0.050 0 1.00 0.000 0 0.072 0.00 0.000 0 0.072 0.000 3.30E-03

802-182 0 0.05 0.106 2.701 0.050 0 1.00 0.000 0 0.107 0.00 0.000 0 0.107 0.000 1.97E-03

802-183 0 0.05 0.106 2.701 0.050 0 1.00 0.000 0 0.146 0.00 0.000 0 0.146 0.000 1.07E-03

802-184 0 0.05 0.107 2.701 0.050 0 1.00 0.000 0 0.186 0.00 0.000 0 0.186 0.000 4.84E-04

802-185 0 0.05 0.107 2.701 0.050 0 1.00 0.000 0 0.223 0.00 0.000 0 0.223 0.000 2.41E-04

802-186 0 0.05 0.106 2.701 0.050 0 1.00 0.000 0 0.260 0.00 0.000 0 0.260 0.000 1.28E-04

Table H-3 Summary overtopping results for case Wenduine, current situation


1 2791.371 1972.37 3076 -

2 5513.568 2717.015 5661 -

Table H-4 Summary overtopping results for case Wenduine, future situation


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1 349.9767 2143.28 - -

2 3442.3 2167.324 - -

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Annex I: Graphical summary of overtopping results for every case and method

Figure I-1 Overtopping case A, every method

Figure I-2 Overtopping case B, every method

0,E+00

1,E-04

2,E-04

3,E-04

4,E-04

5,E-04

6,E-04

7,E-04

8,E-04

9,E-04

1,E-03

1 2 3 4 5 6

Wave condition

EurOtop

VdM

PCO

NN

𝑞[𝑚

3/

𝑠/𝑚

]

0,E+00

2,E-03

4,E-03

6,E-03

8,E-03

1,E-02

1,E-02

1,E-02

2,E-02

1a 2a 3a 4a 5a 6a 1b 2b 3b 4b 5b 6b

Wave condition

EurOtop

VdM

PCO

NN

𝑞[𝑚

3/𝑠

/𝑚]

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Figure I-3 Overtopping case C, every method, first location

Figure I-5 Overtopping case D, every method

0,E+00

5,E-04

1,E-03

2,E-03

2,E-03

3,E-03

3,E-03

4,E-03

4,E-03

5,E-03

1 2 3 4 5 6 7 8 9 10

Wave condition

EurOtop

VdM

PCO

NN

𝑞[𝑚

3/

𝑠/𝑚

]

0,E+00

2,E-03

4,E-03

6,E-03

8,E-03

1,E-02

1,E-02

1,E-02

2,E-02

1 2 3 4

Wave conditions

EurOtop

VdM

PCO

NN

𝑞[𝑚

3/

𝑠/𝑚

]

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Figure I-6 Overtopping case E, every method

Figure I-7 Overtopping case F, every method

0,E+00

2,E-02

4,E-02

6,E-02

8,E-02

1,E-01

1,E-01

1,E-01

1_slope 1_berm 2_slope 2_berm

Wave conditions

EurOtop

VdM

PCO

NN

𝑞[𝑚

3/

𝑠/𝑚

]

0,E+00

1,E-03

2,E-03

3,E-03

4,E-03

5,E-03

6,E-03

7,E-03

8,E-03

1 2 3 4 5 6 7 8 9 10

Wave conditions

EurOtop

VdM

PCO

NN

𝑞[𝑚

3/

𝑠/𝑚

]

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Figure I-8 Overtopping case M, every method, first location

Figure I-9 Overtopping case Wenduine, every method, a: current, b: future

0,00E+00

5,00E-05

1,00E-04

1,50E-04

2,00E-04

2,50E-04

3,00E-04

3,50E-04

1a 2a 3a 4a 1b 2b 3b 4b

Wave conditions

EurOtop

VdM

PCO

NN𝑞

[𝑚3

/𝑠/

𝑚]

0,E+00

1,E+00

2,E+00

3,E+00

4,E+00

5,E+00

6,E+00

1a 2a 1b 2b

Wave conditions

EurOtop

VdM

PCO

NN

141

Annex J: Case A-Blyth Sands, Outer Thames, UK: In- and output sheet

Table: Run-up and overtopping results

Wave condition 1 2 3 4 5 6

𝑆𝑊𝐿[𝑚𝑂𝐷𝑁] 5.170 5.170 5.170 3.390 3.390 3.390

𝐻𝑚0[𝑚] 0.620 0.390 0.720 1.410 0.860 1.160 𝑇𝑚−1,0[𝑠] 2.455 1.909 3.182 4.182 2.818 4.091

𝛽[°] 36 55 19 50 56 22

𝛾𝑣[−] 1.000 1.000 1.000 1.000 1.000 1.000

𝛾𝑏[−] 1.000 1.000 1.000 1.000 1.000 1.000 𝛾𝛽,𝑟[−] 0.921 0.879 0.958 0.890 0.877 0.952 𝛾𝛽,𝑞[−] 0.881 0.819 0.937 0.835 0.815 0.927

𝛾𝑓[−] 0.600 0.600 0.600 0.600 0.600 0.600

tan(𝛼) [−] 0.222 0.222 0.222 0.157 0.150 0.158 𝜉𝑚−1,0[−] 0.866 0.849 1.041 0.692 0.568 0.748

ℎ∗[−] - - - - - -

𝑅𝑢2%[𝑚] 4.89E-01 2.88E-01 7.11E-01 8.59E-01 4.24E-01 8.17E-01 𝑅𝑢2%,𝑃𝐶𝑂[𝑚] 5.18E-01 2.98E-01 7.53E-01 9.17E-01 4.52E-01 8.72E-01

𝑞[𝑚3/𝑠/𝑚] 1.60E-05 7.21E-09 5.15E-04 8.13E-11 1.52E-21 4.93E-11

𝑞𝑉𝑑𝑀[𝑚3/𝑠/𝑚] 9.71E-06 4.11E-10 5.05E-04 1.67E-13 1.75E-32 8.57E-14

𝑞𝑃𝐶𝑂[𝑚3/𝑠/𝑚] 3.80E-05 0.00E+00 9.30E-04 0.00E+00 0.00E+00 0.00E+00

𝑞𝑁𝑁[𝑚3/𝑠/𝑚] 4.10E-04 2.00E-05 8.55E-04 1.28E-04 0.00E+00 1.29E-04

Figure: Overtopping for smooth slopes, non-breaking waves

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

0 0,5 1 1,5 2

EurOtop(2007)

VdM & B (2013)

PC-Overtopping

Neural Network

CLASH

01,0

*

3

0

/1

[]

tan

9.8

1

mm

bm

HL

qq

H

𝑅𝑐∗ =

𝑅𝑐

𝐻𝑚0

√𝐻𝑚0/𝐿𝑚−1,0

tan(𝛼)

1


[−]

Eq. 8.a 95%

5%

Remarks:

-Calculated as a composite slope

142

Annex K: Case B-Southend, Outer Thames UK: In- and output sheet


Wave condition 1a 2a 3a 4a 5a 6a 1b 2b 3b 4b 5b 6b

𝑆𝑊𝐿[𝑚𝑂𝐷𝑁] 2.40 2.90 3.35 3.75 4.22 4.70 2.40 2.90 3.35 3.75 4.22 4.70

𝐻𝑚0[𝑚] 1.32 1.60 1.55 1.25 0.95 0.50 1.32 1.60 1.55 1.25 0.95 0.50

𝑇𝑚−1,0[𝑠] 5.80 5.40 5.00 4.50 3.90 2.90 5.80 5.40 5.00 4.50 3.90 2.90

𝛽[°] 30 30 30.00 30 30 30 30 30 30 30 30.00 30

𝛾𝑣[−] - - - - - - - - - - 0.406 0.050

𝛾𝑏[−] - - - - - - - - - - - -

𝛾𝛽,𝑟[−] - - - - - - - - - - - -

𝛾𝛽,𝑞[−] - - - - - - - - - - 1.000 0.651

𝛾𝑓[−] - - - - - - - - - - - -

tan(𝛼) [−] - - - - - - - - - - - -

𝜉𝑚−1,0[−] - - - - - - - - - - - -

ℎ∗[−] - - - - - - - - - - 2.39E-05 5.14E-02

𝑅𝑢2%[𝑚] - - - - - - - - - - - -

𝑅𝑢2%,𝑃𝐶𝑂[𝑚] - - - - - - - - - - - -

𝑞[𝑚3/𝑠/𝑚] 4.53E-08 1.68E-06 5.14E-06 2.73E-06 1.75E-06 3.21E-08 7.39E-10 6.56E-08 2.73E-07 1.34E-07 5.04E-04 1.65E-05

𝑞𝑉𝑑𝑀[𝑚3/𝑠/𝑚] - - - - - - - - - - 1.50E-02 3.23E-05

𝑞𝑃𝐶𝑂[𝑚3/𝑠/𝑚] - - - - - - - - - - - -

𝑞𝑁𝑁[𝑚3/𝑠/𝑚] 1.38E-04 6.34E-04 9.01E-04 5.58E-04 3.05E-04 2.74E-05 1.35E-04 7.26E-04 1.27E-03 9.06E-04 6.78E-04 1.45E-04

Figure: Overtopping for emergent structure

Figure: Overtopping for vertical structure, impulsive conditions

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

0 1 2 3 4 5 6

EurOtop(2007)

VdM & B (2014)

Neural Network

CLASH *

0.33

1,0

3 09.

[81

]

m

m

qq

ms

H

* 0.33

1,0

0

[ ]cc m

m

RR ms

H

1,E-03

1,E-01

1,E+01

1,E+03

1,E+05

1,E+07

1,E+09

0 0,02 0,04 0,06 0,08 0,1 0,12 0,14 0,16 0,18 0,2

EurOtop(2007)

VdM & B (2014)

Neural Network

CLASH

𝑞∗=

𝑞

ℎ ∗2 √𝑔ℎ

𝑠3∙

1

𝑘∙𝑘

𝛽

Remarks:

-Beach has two states: healthy and eroded

conditions (see text)

- Water levels below vertical structure

calculated as an emergent wall

-Water level reaching the vertical structure

calculated as a vertical wall.

-recurve value ( when assuming vertical wall)

can be found in text

143

Annex L: Case C – Dock exit seawall, Dover Harbour, UK- In-and output sheet


Wave cond. 1 2 3 4 5 6 7 8 9 10

𝑆𝑊𝐿[𝑚𝑂𝐷𝑁] 3.4 3.6 3.4 3.9 3.4 3.6 3.8 4.2 4.45 5

𝐻𝑚0[𝑚] 0.5 0.8 1.2 0.8 2 2 1.6 1.2 0.8 0.5

𝑇𝑚−1,0[𝑠] 2.3 5.6 6.8 5.6 8.8 8.8 7.9 6.8 5.6 2.3

𝛽[°] 0 0 0 0 0 0 0 0 0 0

𝛾𝑣[−] 0.648 0.648 0.648 0.648 0.648 0.648 0.648 0.648 0.648 -

𝛾𝑏[−] 1 1 1 1 1 1 1 1 1 -

𝛾𝛽,𝑟[−] 1 1 1 1 1 1 1 1 1 -

𝛾𝛽,𝑞[−] 1 1 1 1 1 1 1 1 1 -

𝛾𝑓[−] 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 -

tan(𝛼) [−] 0.515 0.515 0.515 0.515 0.515 0.515 0.515 0.515 0.515 -

𝜉𝑚−1,0[−] 2.092 4.027 3.992 4.027 4.002 4.002 4.017 3.992 4.027 -

𝑑∗[−] - - - - - - - - - 0.516507

𝑅𝑢2%[𝑚] 0.889 1.561 2.339 1.561 3.900 3.900 3.121 2.339 1.561 -

𝑅𝑢2%,𝑃𝐶𝑂[𝑚] - - - - - - - - - -

𝑞[𝑚3/𝑠/𝑚] 4.07E-17 6.75E-08 1.13E-05 3.43E-07 2.14E-03 3.31E-03 8.46E-04 2.04E-04 6.75E-06 9.07E-07

𝑞𝑉𝑑𝑀[𝑚3/𝑠/𝑚] 5.30E-24 4.79E-09 4.58E-06 4.89E-08 2.37E-03 3.84E-03 8.60E-04 1.73E-04 2.80E-06 2.77E-08

𝑞𝑃𝐶𝑂[𝑚3/𝑠/𝑚] - - - - - - - - - -

𝑞𝑁𝑁[𝑚3/𝑠/𝑚] - - 9.83E-05 - 3.06E-03 3.97E-03 8.81E-04 2.44E-04 2.27E-05 -

Figure: Overtopping for smooth and simple slopes, non-breaking waves

Figure: Overtopping for composite structures, non-impulsive waves

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5

EurOtop(2007)

VdM & B(2014)

Neural Network

EurOtop(2007) location2

Van der Meer (2014) location 2

Neural Network location 2

CLASH

𝑅𝑐∗ =

𝑅𝑐

𝐻𝑚0 ∙ 𝛾𝛽 ∙ 𝛾𝑓

[−]

𝑞∗ =𝑞

√ 𝑔∙𝐻

𝑚03 [−

]

1,E-08

1,E-06

1,E-04

1,E-02

1,E+00

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5

EurOtop(2007)

VdM & B (2014)

𝑞∗ =𝑞

√ 𝑔∙𝐻

𝑚03 [−

]

Remarks:

-Two different overtopping locations

-Here the results are shown for

overtopping at the recurved wall: 𝑞1

- Water levels below 4.80mODN are

calculated as a simple slope with wave

wall

-Water levels above 4.80mODN are

calculated as composite structures with

d=SWL-4.80m and h=SWL-(-2.9m)

-recurve only for first overtopping

location ( value in text)

𝑅𝑐∗ =

𝑅𝑐

𝐻𝑚0 ∙ 𝛾𝛽

[−]

Eq. 17

5%

5%

5%

5%

Eq. 8.b

144

Annex M: Case D – St-Peter Ording, North Sea, Germany – In-and output sheet


Wave cond. 1 2 3 4

𝑆𝑊𝐿[𝑚𝑁𝑁] 6.000 5.500 5.000 4.700

𝐻𝑚0[𝑚] 1.930 1.650 1.380 1.210 𝑇𝑚−1,0[𝑠] 4.091 4.091 4.091 4.091

𝛽[°] 0 0 0 0

𝛾𝑣[−] 1.000 1.000 1.000 1.000

𝛾𝑏[−] 1.000 1.000 1.000 1.000 𝛾𝛽,𝑟[−] 1.000 1.000 1.000 1.000 𝛾𝛽,𝑞[−] 1.000 1.000 1.000 1.000

𝛾𝑓[−] 1.000 1.000 1.000 1.000

tan(𝛼) [−] 0.125 0.125 0.125 0.125 𝜉𝑚−1,0[−] 0.460 0.497 0.544 0.581

ℎ∗[−] - - - -

𝑅𝑢2%[𝑚] 1.46E+00 1.35E+00 1.24E+00 1.16E+00 𝑅𝑢2%,𝑃𝐶𝑂[𝑚] 1.55E+00 1.44E+00 1.31E+00 1.23E+00

𝑞[𝑚3/𝑠/𝑚] 4.54E-04 1.18E-05 1.51E-07 6.25E-09

𝑞𝑉𝑑𝑀[𝑚3/𝑠/𝑚] 3.95E-04 4.93E-06 1.49E-08 1.58E-10

𝑞𝑃𝐶𝑂[𝑚3/𝑠/𝑚] 9.13E-04 3.30E-05 1.00E-06 0.00E+00 𝑞𝑁𝑁[𝑚3/𝑠/𝑚] 1.50E-02 2.80E-03 3.82E-04 1.10E-04

Figure: Overtopping for smooth and simple slopes, breaking waves

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

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

EurOtop(2007)

VdM & B (2014)

PC-Overtopping

Neural Network

CLASH

𝑞∗

=𝑞

√9

.81

∙𝐻𝑚

03 √

𝐻𝑚

0/

𝐿𝑚

−1

,0

tan

(𝛼)

1 𝛾𝑏

[−]

𝑅𝑐∗ =

𝑅𝑐

𝐻𝑚0

√𝐻𝑚0/𝐿𝑚−1,0

tan(𝛼)

1


[−]

Remarks:

-Calculated as a simple slope, with

breaking waves

Eq. 8.a 95%

5%

145

Annex N: Case E – Norderney, North Sea, Germany: In-and output sheet


Wave cond. 1(No berm) 1(Berm) 2(No berm) 2(berm)

𝑆𝑊𝐿[𝑚𝑁𝑁] 5.000 5.000 4.140 4.140

𝐻𝑚0[𝑚] 3.500 3.500 3.500 3.500 𝑇𝑚−1,0[𝑠] 13.636 13.636 13.636 13.636

𝛽[°] 0 0.000 0 0

𝛾𝑣[−] 1.000 1.000 1.000 1.000

𝛾𝑏[−] 1.000 0.849 1.000 0.851 𝛾𝛽,𝑟[−] 1.000 1.000 1.000 1.000 𝛾𝛽,𝑞[−] 1.000 1.000 1.000 1.000

𝛾𝑓[−] 0.965 0.953 0.954 0.940

tan(𝛼) [−] 0.164 0.179 0.167 0.182 𝜉𝑚−1,0[−] 1.493 1.630 1.524 1.655

ℎ∗[−] - - - -

𝑅𝑢2%[𝑚] 8.32E+00 7.62E+00 8.40E+00 7.65E+00 𝑅𝑢2%,𝑃𝐶𝑂[𝑚] 9.36E+00 9.36E+00 9.34E+00 9.34E+00

𝑞[𝑚3/𝑠/𝑚] 4.57E-02 2.63E-02 2.16E-02 1.12E-02

𝑞𝑉𝑑𝑀[𝑚3/𝑠/𝑚] 4.77E-02 2.74E-02 2.23E-02 1.12E-02

𝑞𝑃𝐶𝑂[𝑚3/𝑠/𝑚] 9.08E-02 9.08E-02 4.61E-02 4.61E-02

𝑞𝑁𝑁[𝑚3/𝑠/𝑚] 3.95E-02 3.95E-02 1.87E-02 1.87E-02

Figure: Overtopping on smooth and simple slopes, breaking waves

1,E-08

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

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

EurOtop_Noberm

EurOtop_berm

VdM_noberm

VdM_berm

PCO_noberm

PCO_berm

NN_noberm

CLASH

𝑅𝑐∗ =

𝑅𝑐

𝐻𝑚0

√𝐻𝑚0/𝐿𝑚−1,0

tan(𝛼)

1


[−]

𝑞∗=

𝑞

√9.

81∙𝐻

𝑚0

3 √

𝐻𝑚

0/𝐿

𝑚−

1,0

tan(

𝛼)

1 𝛾 𝑏

[−]

Remarks:

-Both promenades are between a berm

and slope

-First calculation happes assuming both

promenades are slopes

-Second calculation happens assuming 1st

promenade is a berm

-Interpolated values can be found in text

-Wave conditions just below breaking

wave threshold

Eq. 8.a

5%

95%

146

Annex O: Case F – Samphire Doe, Dover, UK: In- and output sheet


Wave cond. 1 2 3 4 5 6 7 8 9 10

𝑆𝑊𝐿[𝑚𝑁𝑁] 1.88 2.28 2.53 2.62 2.55 2.34 1.56 1.01 0.45 -0.12

𝐻𝑚0[𝑚] 2.37 2.53 2.51 2.47 2.22 2.07 1.75 1.56 1.4 1.26

𝑇𝑚−1,0[𝑠] 5.33 5.34 5.34 5.34 5.35 5.46 5.85 5.97 5.86 5.52

𝛽[°] 0 0 0 0 0 0 0 0 0 0

𝛾𝑣[−] - - - - - - - - - -

𝛾𝑏[−] - - - - - - - - - -

𝛾𝛽,𝑟[−] - - - - - - - - - -

𝛾𝛽,𝑞[−] - - - - - - - - - -

𝛾𝑓[−] - - - - - - - - - -

tan(𝛼) [−] - - - - - - - - - -

𝜉𝑚−1,0[−] - - - - - - - - - -

𝑑∗[−] 1.13E-01 1.38E-01 1.61E-01 1.73E-01 1.84E-01 1.67E-01 9.94E-02 6.29E-02 3.20E-02 2.59E-03

𝑅𝑢2%[𝑚] - - - - - - - - - -

𝑅𝑢2%,𝑃𝐶𝑂[𝑚] - - - - - - - - - -

𝑞[𝑚3/𝑠/𝑚] 4.69E-03 6.55E-03 6.80E-03 6.58E-03 4.30E-03 3.23E-03 1.69E-03 1.16E-03 9.60E-04 -

𝑞𝑉𝑑𝑀[𝑚3/𝑠/𝑚] 2.41E-03 3.94E-03 4.31E-03 4.20E-03 2.51E-03 1.68E-03 5.78E-04 2.66E-04 1.11E-04 1.86E-05

𝑞𝑃𝐶𝑂[𝑚3/𝑠/𝑚] - - - - - - - - - -

𝑞𝑁𝑁[𝑚3/𝑠/𝑚] 1.52E-03 2.34E-03 2.86E-03 3.03E-03 2.40E-03 1.81E-03 6.72E-04 2.86E-04 1.36E-04 6.47E-05

Figure: Overtopping for composite vertical wall, impulsive conditions

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

1,E+01

0 0,2 0,4 0,6 0,8 1 1,2

EurOtop(2007)

Van der Meer

Neural Network

CLASH

𝑅𝑐∗ = 𝑑∗

𝑅𝑐

𝐻𝑚0

𝑞∗

=𝑞

𝑑∗

2√

𝑔ℎ

𝑠3

Remarks:

-Calculated as a composite wall, impulsive

conditions

-h=SWL-(-2.42m) and d=SWL-(-0.17m)

-no factors because of composite vertical

wall equation, which don’t have any

influence factors in them

Eq. 20

147

Annex P: Case M – Bundoran, Donegal Bay, Ireland: In- and output sheet


Wave cond. 1 2 3 4 1” 2” 3” 4”

𝑆𝑊𝐿[𝑚𝑁𝑁] 2.16 2.7 2.85 2 2.16 2.7 2.85 2

𝐻𝑚0[𝑚] 0.85 1.19 1.29 0.77 0.85 1.19 1.29 0.77 𝑇𝑚−1,0[𝑠] 5.3 5.5 5.4 5.9 5.3 5.5 5.4 5.9

𝛽[°] 0 0 0 0 0 0 0 0

𝛾𝑣[−] 1 1 1 1 1 1 1 1

𝛾𝑏[−] 1 1 1 1 1 1 1 1 𝛾𝛽,𝑟[−] 1 1 1 1 1 1 1 1 𝛾𝛽,𝑞[−] 1 1 1 1 1 1 1 1

𝛾𝑓[−] 0.619 0.576 0.558 0.678 0.619 0.576 0.558 0.678

tan(𝛼) [−] 0.667 0.667 0.667 0.667 0.667 0.667 0.667 0.667 𝜉𝑚−1,0[−] 4.789 4.200 3.961 5.601 4.789 4.200 3.961 5.601

𝑑∗[−] - - - - - - - -

𝑅𝑢2%[𝑚] 1.675 2.239 2.337 1.517 1.675 2.239 2.337 1.517 𝑅𝑢2%,𝑃𝐶𝑂[𝑚] 2.069 2.707 2.849 2.042 2.069 2.707 2.847 2.042

𝑞[𝑚3/𝑠/𝑚] 8.55E-09 1.03E-05 3.85E-05 2.29E-09 2.25E-15 9.08E-11 5.94E-10 5.49E-16

𝑞𝑉𝑑𝑀[𝑚3/𝑠/𝑚] 3.53E-11 1.21E-06 7.20E-06 5.05E-12 1.68E-23 4.54E-15 1.10E-13 1.41E-24

𝑞𝑃𝐶𝑂[𝑚3/𝑠/𝑚] 2.59E-07 9.85E-05 3.11E-04 - - - - -

𝑞𝑁𝑁[𝑚3/𝑠/𝑚] - 8.57E-05 1.74E-04 - - - - -

Figure: Overtopping at simple slopes, non-breaking waves

1,E-08

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5

EurOtop(2007)

VdM(2014)

PCO

NN

CLASH

𝑅∗ =𝑅𝑢2%

𝐻𝑚0

1

𝛾𝑓 𝛾𝑏 𝛾𝛽

[−]

𝑞∗

=𝑞

√𝑔

𝐻𝑚

03

∙1 𝐶

𝑟

[−]

Remarks:

-𝛾𝑓 = 0.4 with permeable core

-2 overtopping locations: overtopping at

second location, 8.60mHD, noted with “

-Non-breaking waves so no effect of

berms

-Crest width reducing factor can be found

in text

-PCO used 𝛾𝑓 = 0.5

Eq 8.b

95%

5%

148

Annex Q: Case current situation Wenduine, Belgium: In-and output sheet


Wave cond. 1 2

𝑆𝑊𝐿[𝑚𝑁𝑁] 6.84 7.94

𝐻𝑚0[𝑚] 4.75 4.97 𝑇𝑚−1,0[𝑠] 8.6 9

𝛽[°] 0 0

𝛾𝑣[−] 1 1

𝛾𝑏[−] 1 1 𝛾𝛽,𝑟[−] 1 1 𝛾𝛽,𝑞[−] 1 1

𝛾𝑓[−] 1 1

tan(𝛼) [−] 0.5 0.5 𝜉𝑚−1,0[−] 2.46528 2.522196

ℎ∗[−] - -

𝑅𝑢2%[𝑚] 14.46213 15.18584 𝑅𝑢2%,𝑃𝐶𝑂[𝑚] 15.584 16.248

𝑞[𝑚3/𝑠/𝑚] 2.791371 5.513568

𝑞𝑉𝑑𝑀[𝑚3/𝑠/𝑚] 1.97237 2.717015

𝑞𝑃𝐶𝑂[𝑚3/𝑠/𝑚] 3.076 5.661

𝑞𝑁𝑁[𝑚3/𝑠/𝑚] - -

Figure: Overtopping for smooth and simple slopes, non-breaking waves

1,E-08

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5

EurOtop(2007)

VdM(2014)

PCO

CLASH

Ugent (2011)

𝑞

√𝑔

𝐻𝑚

03

[−]

𝑅𝑐∗ =

𝑅𝑐

𝐻𝑚0

[−]

Remarks:

-The beach is too mild and too long to be

considered in the calculations

-The promenade is also not included

-Non-breaking waves so no berm factor

and no way to include crest width

Eq. 8.b

5%

95%

149

Annex R: Case current situation Wenduine, Belgium: In-and output sheet


Wave cond. 1 2

𝑆𝑊𝐿[𝑚𝑁𝑁] 6.84 7.94

𝐻𝑚0[𝑚] 4.75 4.97

𝑇𝑚−1,0[𝑠] 8.6 9

𝛽[°] 0 0

𝛾𝑣[−] - -

𝛾𝑏[−] - - 𝛾𝛽,𝑟[−] - - 𝛾𝛽,𝑞[−] - -

𝛾𝑓[−] - -

tan(𝛼) [−] - -

𝜉𝑚−1,0[−] - -

ℎ∗[−] 0.000268 0.004392

𝑅𝑢2%[𝑚] - -

𝑅𝑢2%,𝑃𝐶𝑂[𝑚] - -

𝑞[𝑚3/𝑠/𝑚] 0.349977 3.4423

𝑞𝑉𝑑𝑀[𝑚3/𝑠/𝑚] 2.14328 2.167324

𝑞𝑃𝐶𝑂[𝑚3/𝑠/𝑚] - -

𝑞𝑁𝑁[𝑚3/𝑠/𝑚] - -

Figure: Overtopping for vertical structures, impulsive conditions, breaking waves

1,E+00

1,E+01

1,E+02

1,E+03

1,E+04

1,E+05

1,E+06

1,E+07

1,E+08

1,E+09

0 0,005 0,01 0,015 0,02

EurOtop (2007)

VdM (2014)

Ugent (2011)

CLASH

𝑞∗

=𝑞

ℎ∗

2√

𝑔ℎ

𝑠3[−

]

Remarks:

-Calculated as a vertical wall, not

composite

-Impulsive breaking wave conditions

-no influence factors in the equation

-recurve factor can be found in the text

Eq. 18.b

150

Wave overtopping of coastal structures in case studies ...lib.ugent.be/fulltxt/RUG01/002/224/511/RUG01... · Wave overtopping of coastal structures in case studies using the EurOtop

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