Hydraulic stability of rubble mound breakwaters in...

Hydraulic stability of rubble mound breakwaters in

breaking wave conditions: a comparative study of existing

prediction formulas

Sander Franco Supervisor: Prof. Josep R. Medina

Co-supervisor: Prof. dr. ir. Peter Troch

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

Science in Civil Engineering Laboratorio de Puertos y Costas

Chair: Prof. Josep R. Medina

Departamento de Ingeniería e Infraestructura de los Transportes

Universidad Politecnica de Valencia

Department of Civil Engineering Chair: Prof. dr. ir. Peter Troch Faculty of Engineering and Architecture

Ghent University

Academic year 2015-2016



prediction formulas









Ghent University


Preface

I would like to thank Prof. Josep R. Medina for giving me the opportunity to do all the research of my

master thesis at the Laboratory of Ports and Coasts of the Polytechnic University of Valencia. Also his

help on the moments I needed it were always helpful

Next, I would like to thank Prof. dr. ir. Peter Troch for bringing me in in contact with Prof. Josep R.

Medina. Also his help before, during and after my stay in Valencia is really appreciated.

I also really appreciated the daily help and guidance by Mapi. When I had doubts, she always helped

and encouraged me. Also a special thanks to Jorge, Ainhoa, Gloria and César for creating a nice

atmosphere in the lab and answer my questions when I had some.

Of course I can’t forget my family. Not only for giving me the opportunity to go abroad and study, but

also for the education and freedom they gave me which made me the person I am now.

Last but not least I would like to thank my Erasmus friends in Valencia for the awesome weekends after

a week of focusing on my thesis. Also my friends in Belgium have always been a great help during my

whole study.

The author gives 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.

Sander Franco

January 18, 2016

Abstract

Hydraulic stability of rubble mound breakwaters in breaking wave conditions: a comparative study

of existing prediction formulas

Supervisor: Prof. Josep R. Medina


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

Master of Science in Civil Engineering

Laboratorio de Puertos y Costas



Polytechnic University of Valencia

Department of Civil Engineering

Chair: Prof. dr. ir. Peter Troch

Faculty of Engineering and Architecture

Ghent University


Summary

Nowadays most breakwaters are built in shallow water. To enlarge the experimental basis of such a

conditions, some tests are carried out in the 30mx1.2mx1.2m wave flume in the ‘Laboratory of Ports

and Coasts’ of the ‘Polytechnic University of Valencia’. Each wave series attacking the rubble mound

breakwater scale model in shallow water consists of 1000 irregular waves with a fixed Iribarren number

in a fixed water depth. After each wave series, the damage is measured. When total destruction occurs,

the scale model is rebuilt and a new test series is initiated with different Iribarren number and/or in a

different water depth. The measured damage is compared with the damage predicted by formulas

based on the Shore Protection Manual (formula by Van der Meer (1988) and Medina (1994)) on the

one hand and predicted by 3 sets of formulas based on Van der Meer (1998) and Van Gent (2004) on

the other. Based on this comparison, some improvements to the original formulas are presented and

verified by a repetition test. Besides this quantitative analysis of the damage, also a qualitative analysis

is performed obtaining the different damage criteria.

Keywords

Rubble mound breakwater, damage, prediction formula, breaking conditions, irregular wave

Hydraulic stability of rubble mound

breakwaters in breaking wave conditions:

a comparative study of existing

prediction formulas Sander Franco

Supervisors: Prof. Josep R. Medina, Prof. dr. ir. Peter Troch

Abstract-- Nowadays most breakwaters are

built in shallow water. To enlarge the experimental

basis of such a conditions, some tests are carried

out in the 30mx1.2mx1.2m wave flume in the

‘Laboratory of Ports and Coasts’ of the

‘Polytechnic University of Valencia’. Each wave

series attacking the rubble mound breakwater

scale model in shallow water consists of 1000

irregular waves with a fixed Iribarren number in

a fixed water depth. After each wave series, the

damage is measured. When total destruction

occurs, the scale model is rebuilt and a new test

series is initiated with different Iribarren number

and/or in a different water depth. The measured

damage is compared with the damage predicted by

formulas based on the Shore Protection Manual

(formula by Van der Meer (1988) and Medina

(1994)) on the one hand and predicted by 3 sets of

formulas based on Van der Meer (1998) and Van

Gent (2004) on the other. Based on this

comparison, some improvements to the original

formulas are presented and verified by a repetition

test. Besides this quantitative analysis of the

damage, also a qualitative analysis is performed

obtaining the different damage criteria.

Keywords-- Rubble mound breakwater,

damage, prediction formula, breaking conditions,

irregular waves

INTRODUCTION

Most breakwaters nowadays are built in shallow

water, but only little research has been done to

improve the knowledge of predicting the damage of

the armour layer caused by wave attack. One of the

reasons of this lack of research is the fact that such a

research is a challenging issue due to the uncertainties

in the different irregular wave parameters in shallow

water.

The scope of this master thesis is to modify and

improve existing damage prediction formulas for

rubble mound breakwaters in breaking wave

conditions based on the comparison between the

measured and predicted damage.

Two different approaches will be used. A first

approach is based on the ‘Shore Protection Manual’

(SPM) [1] with starting point the Hudson formula [2],

another approach is based on the formulas developed

by Van der Meer (VDM) [3] and Van Gent (VG) [4].

EXISTING STABILITY FORMULAS

SPM-approach

Both Van der Meer [3] and Medina [5] developed

a formula to predict the damage of the armour layer of

a rubble mound breakwater under wave attack. The

starting point of these formulas is the Hudson formula

[2]:

𝑊50 =

𝜌𝑟𝑔𝐻3

𝐾𝐷∆3 cot 𝛼 (1)

𝑊50 is the medium weight of an armour stone, 𝜌𝑟 the

apparent rock density, 𝑔 the gravitational

acceleration, 𝐻 the wave height, 𝐾𝐷 the stability

coefficient, ∆= 𝜌𝑠/𝜌𝑤 − 1 the relative buoyant

density and 𝛼 the structure slope angle. Values for the

stability coefficient 𝐾𝐷 can be found in the SPM [1].

Introducing the assumption stated by the SPM [1] that

the highest of 10% of wave heights 𝐻1/10 = 1.27𝐻𝑠

should be used as the design wave height and by

introducing the stability parameter 𝑁𝑠 = 𝐻𝑠/∆𝐷𝑛50

with 𝐷𝑛50 the nominal diameter of the armour stone

and 𝐻𝑠 the significant wave height, this gives:

𝐻𝑠

∆𝐷𝑛50

=(𝐾𝐷𝑐𝑜𝑡𝛼)1/3

1.27 (2)

This formula only corresponds with a damage level of

0-5% (no damage). To overcome this limitation, table

7.9 of the SPM [1] was used by Van der Meer [3] and

Medina [5] and by applying regression analysis,

following formulas were developed:

Van der Meer:

𝐻𝑠

∆𝐷𝑛50

= 0.7(𝐾𝐷𝑐𝑜𝑡𝛼)1/3𝑆𝑆𝑃𝑀0.15 (3)

Medina:

𝐻𝑚0

∆𝐷𝑛50

= 1.15 (cot 𝛼)1/3 𝑆𝑆𝑃𝑀0.2 (4)

𝑆 is the damage parameter in which the ‘SPM’ stands

for the fact that table 7.9 from the SPM [1] was used

to develop the formulas. 𝐻𝑚0 is the significant wave

height obtained from spectral analysis.

VDM-approach

The other approach used to predict the damage of

the armour layer under wave attack is based on Van

der Meer [3]. Van der Meer developed some formulas,

based on tests in a wave flume, taking into account the

number of waves 𝑁, the permeability of the

breakwater 𝑃 and the surf similarity parameter 𝜉𝑚 =

tan 𝛼 /√2𝜋𝐻𝑠/(𝑔𝑇𝑚2 ) with 𝑇𝑚 the mean wave period.

For deep water (ℎ > 3𝐻𝑠,𝑡𝑜𝑒), these formulas are:

Plunging waves (𝜉𝑚 < 𝜉𝑐𝑟𝑖𝑡):

𝐻𝑠

∆𝐷𝑛50

= 𝑐𝑝𝑙𝑃0.18 (

𝑆𝑑

√𝑁)

0.2

𝜉𝑚−0.5 (5)

Surging waves (𝜉𝑚 > 𝜉𝑐𝑟𝑖𝑡):

𝐻𝑠

∆𝐷𝑛50

= 𝑐𝑠𝑃−0.13 (𝑆𝑑

√𝑁)

0.2

√cot 𝛼 𝜉𝑚𝑃 (6)

The values for respectively 𝑐𝑝𝑙 and 𝑐𝑠 are 6.2 and 1.0.

The significant wave height 𝐻𝑠 is the incident wave

height at the toe of the structure based on time domain

analysis. The values of the permeability of the

structure are given in Figure 1.

Figure 1: Values of permeability for different structures

The distinction between plunging and surging waves

is based on the critical breaker parameter 𝜉𝑐𝑟𝑖𝑡:

𝜉𝑐𝑟𝑖𝑡 = [𝑐𝑝𝑙

𝑐𝑠

𝑃0.31√tan 𝛼]

1𝑃+0.5

(7)

It’s recommended to use plunging conditions for

cot 𝛼 ≥ 4 irrespective of whether 𝜉𝑚 is smaller or

larger than 𝜉𝑐𝑟𝑖𝑡.

For shallow water (ℎ < 3𝐻𝑠,𝑡𝑜𝑒), Van der Meer [3]

recommended to use 𝐻2% instead of 𝐻𝑠. With the

know relation 𝐻2%/𝐻𝑠 = 1.4, the formulas become:


𝐻𝑠

∆𝐷𝑛50

= 𝑐𝑝𝑙𝑃0.18 (

𝑆𝑑

√𝑁)

0.2 𝐻𝑠

𝐻2%

𝜉𝑚−0.5 (8)

Surging waves (𝜉𝑚 > 𝜉𝑐𝑟𝑖𝑡):

𝐻𝑠

∆𝐷𝑛50

= 𝑐𝑠𝑃−0.13 (𝑆𝑑

√𝑁)

0.2 𝐻𝑠

𝐻2%√cot 𝛼 𝜉𝑚

𝑃 (9)

Based on his own test, Van Gent [4] modified

formulas (8) and (9) by using the spectral mean energy

wave period 𝑇𝑚−1,0 instead of the mean wave period

𝑇𝑚 for the calculation of the surf similarity parameter

𝜉𝑠−1,0. Also the coefficients 𝑐𝑝𝑙 and 𝑐𝑠 are modified,

these equal respectively 8.4 and 1.3. These formulas

are valid for both deep and shallow water:


𝐻𝑠

∆𝐷𝑛50

= 𝑐𝑝𝑙𝑃0.18 (

𝑆𝑑

√𝑁)

0.2 𝐻𝑠

𝐻2%

𝜉𝑠−1,0−0.5 (10)

Surging waves (𝜉𝑠−1,0 > 𝜉𝑐𝑟𝑖𝑡):

𝐻𝑠

∆𝐷𝑛50

= 𝑐𝑠𝑃−0.13 (𝑆𝑑

√𝑁)

0.2 𝐻𝑠

𝐻2%√cot 𝛼 𝜉𝑠−1,0

𝑃 (11)

Van Gent also developed a new formula. It’s a more

simple formula, not taking into account the wave

period and the difference between plunging and

surging waves:

𝐻𝑠

∆𝐷𝑛50=

1

0.57(1 +

𝐷𝑛50−𝑐𝑜𝑟𝑒

𝐷𝑛50−𝑎𝑟𝑚𝑜𝑢𝑟) (

𝑆𝑑

√𝑁)

0.2

√cot 𝛼 (12)

The permeability of the structure is taken into account

by introducing the ratio of the nominal diameter of the

stones in the core 𝐷𝑛50−𝑐𝑜𝑟𝑒 and the nominal diameter

of the armour layer 𝐷𝑛50−𝑎𝑟𝑚𝑜𝑢𝑟 . This formula can be

used in both deep and shallow water.

Cumulative damage

All formulas according to the VDM-approach are

limited to single storm events. To take into account

subsequent storm events, Van der Meer developed a

method which make directly use of the formulas of the

VDM-approach based on the equivalence hypothesis

(Figure 2) [6].

The procedure to assess the cumulative damage is as

follows:

-Calculate the damage 𝑆𝑑1 for the first wave

condition

-Calculate for the 2nd wave conditions how many

waves would be required to give the same damage as

under the 1st wave conditions. This is denoted by 𝑁1′.

-Add this number of waves, 𝑁1′, to the number of

waves of the second wave conditions: 𝑁𝑡 = 𝑁2 + 𝑁1′

-Calculate the damage under the 2nd wave conditions

𝑆𝑑2𝑡 with this increased number of waves 𝑁𝑡.

-Calculate for the 3th wave condition how many

waves would be required to give the same damage as

caused by the second wave condition etc.

Figure 2: Cumulative damage approach by Van der Meer

EXPERIMENTAL SETUP

Test equipment

The tests are carried out in the 1.2mx1.2mx30m

wave flume in the Laboratory of Ports and Coasts of

the Polytechnic University of Valencia (LPC-UPV).

The wave generating system is controlled by an

Active Wave Absorption Control System to take into

account the reflected waves from the structure on the

other side of the wave flume.

Experimental design

A foreshore slope is created to make the waves

break. This foreshore consists of a first slope of 4%

over a length of 625cm and a second slope of 2% over

1100cm on which the scale model is built on the end.

The slope of the scale model (1:60) of the rubble

mound breakwater is 3/2. The armour layer of the

breakwater is composed out of 2 layers of quarry stone

with nominal diameter equal to 3.18cm covering a

filter layer and the core (Figure 3).

Figure 3: Section of the rubble mound breakwater scale

model, dimensions in cm

Realized experiments

Every test series is initiated with a wave series of

1000 waves with a wave height that doesn’t produce

breaking waves (8𝑐𝑚). By increasing the wave height

with 1 cm every step and by keeping the Iribarren

number constant, more and more waves are breaking

until destruction of the model is observed. Because

the Iribarren number is kept constant in one test series,

a corresponding peak period 𝑇𝑝 can be calculated for

each wave height. The different test series are

performed for different Iribarren numbers (𝐼𝑟 = 3 and

𝐼𝑟 = 5) and water depths (ℎ𝑚𝑜𝑑𝑒𝑙 = 20, 30 and

40𝑐𝑚). One repetition test is carried out (𝐼𝑟 = 3 and

ℎ𝑚𝑜𝑑𝑒𝑙 = 20𝑐𝑚) to use later as a ‘blind test’ to verify

the improved and new developed formulas.

Data analysis

The wave characteristics (wave height, period,…)

are obtained by SwanOne, a model developed by

TUDelft in MatLab to simulate the evolution of a

wave spectrum starting from deep water to shore. It’s

capable to simulate interactions and transformations

of waves.

The porosity has to be calculated to verify if the

initial porosity is about 37% as prescribed [1] and also

to calculate the damage caused by the wave action.

It’s calculated counting the number of stones using the

formula below:

𝑝 =

𝐴𝑣

𝐴𝑡𝑜𝑡

= 1 −𝑁𝐷𝑛50

2

𝐴𝑡𝑜𝑡

(13)

𝐴𝑣 is the area of the voids, 𝐴𝑡𝑜𝑡 is the total area of

the structure slope in which the stones are counted

and 𝑁 is the number of stones.

For the qualitative analysis of the damage, the

damage criteria ([7] & [8]) are distinguished as

follows:

-Initiation of Damage: 5 or more units are displaced

from the original position to a new one at a distance

equal to or larger than a unit length

-Initiation of Iribarren Damage: One stone and his

surrounding stones of the 2nd layer are visible

-Initiation of Destruction: 2 or more stones are forced

out of the lower armour layer.

-Destruction: The filter layer is visible

For the quantitative analysis, the damage is

calculated using the eroded area 𝐴𝑒 which is

calculated using the ‘visual counting method’. For this

method, following formulas are used:

𝑆𝑑 =

𝐴𝑒

𝐷𝑛502 (14)

𝐴𝑒 =

𝑁𝑑𝐷𝑛50³

(1 − 𝑝)𝑅 (15)

𝑆𝑑 is the damage level parameter, 𝑁𝑑 the number of

eroded stones, 𝑝 the porosity of the settled stones and

𝑅 the width of the eroded area.

Comparison measured and predicted damage

The measured damage is compared by the

predicted damage using the damage formulas

mentioned before. For the SPM-approach, 2 formulas

will be used: the Van der Meer formula (3) and the

Medina formula (4) . For the VDM-approach 3 sets of

formulas will be used: the Van der Meer formulas

((5), (6), (8) & (9)), the Van der Meer formulas

modified by Van Gent ((10) & (11)) and the Van Gent

formula (12). For the VDM-approach, cumulative

damage is taken into account.

To evaluate the extent to which measured and

predicted damage are similar, the relative mean

squared error (𝑟𝑀𝑆𝐸) will be calculated:

𝑟𝑀𝑆𝐸 =∑ (𝑆𝑑,𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑,𝑖 − 𝑆𝑑,𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑,𝑖)

2𝑖

∑ (𝑆𝑑,𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ − 𝑆𝑑,𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑,𝑖)

2𝑖

(16)

For the SPM-approach, the 𝐾𝐷 coefficient for the

Van der Meer formula is chosen to be 8 [9]. The

damage will be predicted 5 times for each formula

using 5 different wave heights. First of all, the

significant wave height based on spectral analysis

𝐻𝑚0 obtained from SwanOne will be used, because

this is also the wave height proposed by the authors of

the formulas. But because most of the test were

carried out in breaking conditions, also the breaking

wave height 𝐻𝑏 is calculated 3 times according to the

method explained in the SPM [1] This method is

based on the formula of Goda [10] (Figure 4). 𝐻𝑏 is

calculated using once the mean wave period from time

domain analysis 𝑇𝑚, once the mean energy wave

period 𝑇𝑚−1,0 and once the peak period 𝑇𝑝. Finally,

the damage is predicted using the peak value of the

significant wave height 𝐻1/3 in the surf zone because

an exact breaking wave height can’t be calculated for

irregular waves because the waves break over a wide

zone [11], as Goda stated. This value can be read from

Figure 5

Figure 4: Graph obtaining breaking wave height [10]

Figure 5: Index curves for the maximum value of the

significant wave height

Also, for each used wave height a new formula

will be proposed by calculating the coefficients 𝐶1 and

𝐶2 of the formula below obtaining the lowest 𝑟𝑀𝑆𝐸:

𝐻

∆𝐷𝑛50

= 𝐶1(cot 𝛼)1/3𝑆𝑑𝐶2 (17)

For the VDM-approach, the wave heights as

explained in the formulas will be used. These are

obtained from SwanOne.

RESULTS

Qualitative analysis

By observing the photos taken after each wave

series, the damage criteria of each test series can be

distinguished. By calculating the average value of the

each damage criteria, the following values are

obtained:

Table 1: Damage criteria

Initiation of damage 𝑆𝑑 = 0.7

Initiation of Iribarren damage 𝑆𝑑 = 2.6

Initiation of destruction 𝑆𝑑 = 5.9

Destruction 𝑆𝑑 = 11.0

Quantitative analysis

The values of the 𝑟𝑀𝑆𝐸 of the comparison

between the measured and predicted damage using the

5 different wave heights for the SPM-approach are

shown in Table 2. Also the coefficients of formula

(17) are shown. The new formulas were verified by a

blind test using the repetition test, which resulted in

low 𝑟𝑀𝑆𝐸.

Table 2: 𝑟𝑀𝑆𝐸-values and values of coefficients new

formulas SPM-approach

𝒓𝑴𝑺𝑬

VDM Medina New

Formula 𝐶1 𝐶2

𝑯𝒎𝟎 0.73 0.18 0.07 1.17 0.14

𝑯𝒃 SPM

𝑇𝑚 0.19 0.34 0.17 1.32 0.17

𝑇𝑚−1,0 0.19 0.49 0.16 1.37 0.17 𝑇𝑝 0.33 0.91 0.14 1.46 0.17

Goda 0.25 0.21 0.14 1.26 0.17

For the VDM-approach, it can be observed that all

breaking waves were plunging waves. The

comparison between the measured damage and the

damage predicted by the VDM formula and the VDM

formula modified by VG are similar. As an example,

the comparison using the VDM formula is given in

Figure 6.

Figure 6: Comparison measured damage and predicted

damage by VDM formula

It can be observed that the predicted damage is almost

always lower than the measured damage. It can also

be observed that the lower the Iribarren number and

the higher the water depth, the bigger the difference

between the measured and the predicted damage. So

by introducing the water depth at the toe of the

structure ℎ𝑡𝑜𝑒 and the peak period 𝑇𝑝(which

influences the Iribarren number or surf similarity

parameter), the prediction is improved, as can be seen

on Figure 7. Also the 90% interval is drawn. For the

VDM formula modified by VG the same conclusions

can be made. The 𝑟𝑀𝑆𝐸 decreases from respectively

0.77 to 0.09 for the improved VDM formula and from

0.49 to 0.09 for the improved VDM formula modified

by VG. The improved formulas are given below:

Improved VDM shallow water (𝑐𝑝𝑙∗ = 4.1):

𝐻𝑠

∆𝐷𝑛50= 𝑐𝑝𝑙

∗ (2𝜋ℎ𝑡𝑜𝑒

𝑔𝑇𝑝2 )

−0.2

𝑃0.18 (𝑆𝑑

√𝑁)

0.2 𝐻𝑠

𝐻2% 𝜉𝑚

−0.5 (18)

Improved VDM deep water (𝑐𝑝𝑙∗ = 2.9):

𝐻𝑠



𝑔𝑇𝑝2 )

−0.2

𝑃0.18 (𝑆𝑑

√𝑁)

0.2

𝜉𝑚−0.5 (19)

Improved VDM formula modified by VG(𝑐𝑝𝑙∗ = 4.3):

𝐻𝑠



𝑔𝑇𝑝2 )

−0.2

𝑃0.18 (𝑆𝑑

√𝑁)

0.2 𝐻𝑠

𝐻2% 𝜉𝑠−1,0

−0.5 (20)

Figure 7: Comparison measured and predicted damage by

improved formula of VDM

Finally, the VG formula can be improved by

changing the coefficient (Figure 8 and Figure 9). The

𝑟𝑀𝑆𝐸 decreases from 0.59 to 0.12. The improved

formula is given below:

𝐻𝑠

∆𝐷𝑛50=

1

0.67(1 +

𝐷𝑛50−𝑐𝑜𝑟𝑒

𝐷𝑛50−𝑎𝑟𝑚𝑜𝑢𝑟) (

𝑆𝑑

√𝑁)

0.2

√cot 𝛼 (21)

Figure 8: Comparison measured and predicted damage by formula of VG

0

3

6

9

12

15

0 3 6 9 12 15

Pre

dic

ted

dam

age

Sd

[-]

Measured damage Sd[-]

Comparison VDM formula

hs=20,Ir=5

hs=30,Ir=5

hs=20,Ir=3

hs=30,Ir=3

hs=40,Ir=3

R=0.78rMSE=0.77

0

3

6

9

12

15

0 3 6 9 12 15

Pre

dic

ted

dam

age

Sd

[-]


Comparison improved VDM formula

hs=20,Ir=5

hs=30,Ir=3

hs=40,Ir=3

hs=20,Ir=3

hs=30,Ir=5

R=0.97rMSE=0.09

0

3

6

9

12

15

0 3 6 9 12 15

Pre

dic

ted

dam

age

Sd

[-]


Comparison VG formula

hs=20,Ir=3

hs=30,Ir=3

hs=30,Ir=5

hs=40,Ir=3

hs=20,Ir=5

R=0.94rMSE=0.59

Figure 9: Comparison measured and predicted damage by the improved formula of VG

CONCLUSIONS

First of all, a qualitative analysis was executed

determining the Initiation of Damage, Initiation of

Iribarren damage, Initiation of destruction and

Destruction. The average values based on the tests are

given in Table 1.

Next a quantitative analysis was carried out. For

the SPM-approach, following conclusions can be

made. If the (deep water) wave characteristics are not

known, these can be obtained using SwanOne. The

best formula to use these wave characteristics based

on the executed tests is the new proposed formula (17)

using the significant wave height 𝐻𝑚0 with values for

the coefficients 𝐶1 and 𝐶2 respectively 1.17 and 0.14.

If the deep water characteristics are known, the

breaking wave height can be calculated and used to

predict the damage according to the SPM approach

using the new proposed formula with the mean period

𝑇𝑝 with values for the coefficients 𝐶1 and 𝐶2

respectively 1.32 and 0.17 or by using Goda’s

approach (Figure 5) and with values for the

coefficients 𝐶1 and 𝐶2 respectively 1.26 and 0.17.

Using the VDM formula (3), the breaking wave height

based on 𝑇𝑚 or 𝑇𝑚−1 gives the best prediction of the

damage while using the Medina formula (4) the

significant wave height 𝐻𝑚0 at the toe of the structure

obtained by SwanOne gives the best approximation.

For the VDM approach, the improved VDM

formulas (18) & (19), the improved VDM formula

modified by VG (20) and improved VG formula (21)

are a very good improvement.

If only a few wave characteristics are known (no wave

period, 𝐻2%,..), the improved VG formula can be

used. In other cases, both the improved VDM formula

and improved VDM formula by VG will give the best

approximation.

A really important remark should be made. The

qualitative analysis, all improvements and new

proposed formulas are based only a few test series. All

formulas should be verified and tested with other tests

and more repetition tests (also in other laboratories) as

well as the qualitative analysis. These repetition tests

are necessary because damage is a very sensitive

parameter. The difference in measured damage for 2

equal tests can be 30% [11].

FURTHER RESEARCH

Besides the repetition tests discussed above, other

future research can be done by changing the

parameters. For instance the nominal armour

diameter, structure slope, foreshore slope,…

The conditions could also be changed in such a

way that surging waves occur (instead of plunging

waves as in these conditions).

Finally also other armour units could be

investigated.

REFERENCES

[1] CERC, Shore protection manual: Washington

(D.C.) : Government printing office, 1984.

[2] R. Y. Hudson, "Laboratory investigation of

rubble-mound breakwaters," Journal of the

Waterways and Harbors Division, vol. 85, pp.

93-122, September 1959 1959.

[3] J. W. Van der Meer, "Rock slopes and gravel

beaches under wave attack," PhD thesis, Delf

University of Technology, 1988.

[4] M. A. Van Gent, A. Smale, and C. Kuiper,

"Stability of rock slopes with shallow

foreshores," presented at the Coastal Structures

2003, Portland, 2004.

[5] J. R. Medina, R. T. Hudspeth, and C. Fassardi,

"Breakwater Armor Damage due to Wave

Groups," Journal of Waterway, Port, Coastal,

and Ocean Engineering, vol. 120, pp. 179-198,

1994.

[6] J. W. Van der Meer, "Design of concrete

armour layers," in Proc 3rd dint con. coastal

structures, Santander, Spain, 2000, pp. 213-

221.

[7] M. Losada, J. M. Desire, and L. M. Alejo,

"Stability of blocks as breakwaters armour

units," Journal of Structural Engineering, pp.

2392-2401, 1986.

[8] C. Vidal, M. Losada, and J. R. Medina,

"Stability of mound breakwater's head and

trunk," Journal of Waterway, Port, Coastal and

Ocean Engineering, pp. 570-587, 1991.

[9] CIRIA, The rock manual : the use of rock in

hydraulic engineering: London : CIRIA, 2007.

[10] Y. Goda, "A synthesis of breaker indices,"

Transactions of the Japan Society of Japan

Engineers, vol. 2, pp. 227-230, 1970.

[11] Y. Goda, Random seas and design of maritime

structures: Singapore : World scientific, 2000.

0

3

6

9

12

15

0 3 6 9 12 15

Pre

dic

ted

dam

age

Sd

[-]


Comparison improved VG formula

hs=20,Ir=5

hs=40,Ir=3

hs=30,Ir=5

hs=30,Ir=3

hs=20,Ir=3

R=0.94rMSE=0.12

Table of Contents

CHAPTER 1: INTRODUCTION ................................................................................................................... 1

CHAPTER 2: LITERATURE STUDY .............................................................................................................. 2

1. Wave reflection ........................................................................................................................... 2

a. Reflection coefficient .............................................................................................................. 2

b. Separation methods ................................................................................................................ 3

c. LASA method ........................................................................................................................... 3

2. Breaking waves ............................................................................................................................ 5

a. General .................................................................................................................................... 5

b. Wave breaking criteria ............................................................................................................ 6

Regular waves.............................................................................................................................. 7

Irregular waves ............................................................................................................................ 9

3. Damage ...................................................................................................................................... 11

a. Failure modes breakwater .................................................................................................... 11

b. Armour damage ..................................................................................................................... 12

Qualitative analysis.................................................................................................................... 12

Quantitative analysis ................................................................................................................. 13

4. Stability formulae ...................................................................................................................... 16

a. Hudson .................................................................................................................................. 16

b. Van der Meer ......................................................................................................................... 19

Deep-shallow water .................................................................................................................. 19

Deep water conditions .............................................................................................................. 19

Shallow water conditions .......................................................................................................... 22

c. Van der Meer modified by Van Gent .................................................................................... 22

d. Van Gent ................................................................................................................................ 23

e. Cumulative damage ............................................................................................................... 24

Melby method ........................................................................................................................... 24

Van der Meer method ............................................................................................................... 24

CHAPTER 3: EXPERIMENTAL SETUP ....................................................................................................... 26

1. Test equipment ......................................................................................................................... 26

a. Wave generator ..................................................................................................................... 27

b. Energy dissipation system ..................................................................................................... 28

c. Wave measurement .............................................................................................................. 28

d. Cameras ................................................................................................................................. 29

2. Experimental design .................................................................................................................. 30

a. Foreshore slope ..................................................................................................................... 30

b. Model .................................................................................................................................... 30

Core ........................................................................................................................................... 30

Filter........................................................................................................................................... 31

Armour layer.............................................................................................................................. 31

3. Realized experiments ................................................................................................................ 32

4. Data analysis .............................................................................................................................. 35

a. Wave analysis ........................................................................................................................ 35

SwanOne ................................................................................................................................... 35

Measurements by wave gauges in canal without a model ....................................................... 36

Measurements by wave gauges in canal with model. .............................................................. 37

Comparison between SwanOne and measurement in canal without model ........................... 37

b. Porosity measurement .......................................................................................................... 38

c. Damage calculation ............................................................................................................... 38

d. Comparison measured damage with predicted damage ...................................................... 39

SPM-approach ........................................................................................................................... 39

VDM-approach .......................................................................................................................... 40

CHAPTER 4: RESULTS ............................................................................................................................. 42

1. Wave data ................................................................................................................................. 42

2. Measured damage and porosity ............................................................................................... 44

3. Comparison measured damage with predicted damage .......................................................... 46

a. SPM-approach ....................................................................................................................... 46

b. VDM-approach ...................................................................................................................... 49

CHAPTER 5: CONCLUSIONS ................................................................................................................... 52

REFERENCES .......................................................................................................................................... 54

APPENDIX A: SUMMARY DATA SWANONE ........................................................................................... 59

APPENDIX B: PHOTOS MODEL AFTER WAVE ACTION ........................................................................... 60

List of Figures Figure 1: Breaker types............................................................................................................................ 5

Figure 2: Limiting steepness in deep water (CERC, 1984) ....................................................................... 6

Figure 3: Graph showing formula by Goda (1970) .................................................................................. 8

Figure 4: Graph showing formula by Weggel (1972) .............................................................................. 8

Figure 5: Breaker index as a function of the Iribarren number............................................................... 9

Figure 6: Index curves for the maximum value of the significant wave height .................................... 10

Figure 7: Different failure modes according to Bruun (1978) ............................................................... 11

Figure 8: Different breakwater failure modes according to Burcharth (1992) ..................................... 12

Figure 9: Eroded area ............................................................................................................................ 14

Figure 10: Values of permeability for different structures ................................................................... 20

Figure 11: Sensitivity analysis for damage level parameter and permeability ..................................... 21

Figure 12: Cumulative damage approach by Van der Meer ................................................................. 25

Figure 13: Wave flume at LPC-UPV ....................................................................................................... 26

Figure 14: Wave generator (Herrera, 2013) .......................................................................................... 27

Figure 15: Energy dissipation system and schematically overview of the frameworks ........................ 28

Figure 16: Position of the sensors, dimensions in meter ...................................................................... 29

Figure 17: Section of the rubble mound breakwater, dimensions in cm .............................................. 30

Figure 18: Sieve curve of the core ......................................................................................................... 30

Figure 19: Distribution of the filter layer stones ................................................................................... 31

Figure 20: Distribution of the armour layer stones ............................................................................... 31

Figure 21: Rubble mound model constructed with an initial porosity of approximately 37% ............. 31

Figure 22: Screenshot of LPCLab 2.0 ..................................................................................................... 36

Figure 23: Comparison between significant wave height obtained from SwanOne and measurements

in the canal without the model ............................................................................................................. 37

Figure 24: Marked stones by AutoCAD ................................................................................................. 38

Figure 25: Comparison formulas based on .................................................................................. 47

Figure 26: Comparison formulas based on the (Goda, 2000) ......................................................... 47

Figure 27: Comparison formulas based on ............................................................................ 47

Figure 28: Comparison formulas based on .................................................................. 47

Figure 29: Comparison formulas based on ............................................................................. 47

Figure 30: Verification new formulas (blind test) ................................................................................. 47

Figure 31: Comparison measured and predicted damage by formula of VDM .................................... 49

Figure 32: Comparison measured and predicted damage by improved formula of VDM .................... 49

Figure 33: Comparison measured and predicted damage by formula of VDM modified by VG .......... 50

Figure 34: Comparison measured and predicted damage by improved formula of VDM modified by

VG .......................................................................................................................................................... 50

Figure 35: Comparison measured and predicted damage by formula of VG........................................ 51

Figure 36: Comparison measured and predicted damage by the improved formula of VG ................. 51

Figure 37: Verification new formulas .................................................................................................... 51

List of Tables Table 1: Values surf similarity parameter for different breaker types.................................................... 6

Table 2: Design values of for a two-diameter thick armour layer ................................................... 15

Table 3: as a function of armour layer damage and armour type ..................................... 18

Table 4: Range of parameters VDM formula ........................................................................................ 21

Table 5: Range of parameters VDM formula modified by VG and VG formula .................................... 23

Table 6: Range of parameters Melby method ...................................................................................... 24

Table 7: Test data for Ir=3 and h=20cm ................................................................................................ 32



Table 10: Test data for Ir=5 and h=30cm .............................................................................................. 34

Table 11: Test data for Ir=3 and h=40cm .............................................................................................. 34

Table 12: Summary formulas VDM-approach ....................................................................................... 41

Table 13: Summary of the SwanOne data at the toe of the structure ( ....................... 42

Table 14: Summary of the SwanOne data at the toe of the structure ( ) ....................... 43

Table 15: Summary of the SwanOne data at the toe of the structure ( ) ....................... 43

Table 16: Measured damage, porosity and qualitative damage analysis for ................ 44

Table 17: Measured damage, porosity and qualitative damage analysis at ................. 45

Table 18: Measured damage, porosity and qualitative damage analysis for and Ir=3 . 45

Table 19: Measured damage, porosity and qualitative damage analysis for the repetition test (

, Ir=3) ............................................................................................................................................ 46

Table 20: Proposed new formulas based on SPM-approach ................................................................ 48

Table 21: of all formulas ............................................................................................................. 48

Table 22: Qualitative damage analysis .................................................................................................. 52

List of Symbols Symbol Description/Explanation Unit

Eroded area

Height of one strip

Width of one strip

Damage

Particle size for which 15% of the particles are smaller

Particle size for which 95% of the particles are smaller

Damage according to the virtual net method

Damage of one strip

Nominal block diameter or equivalent cube size

Median nominal diameter or equivalent cube size

Median nominal diameter of the armour stones

Median nominal diameter of the core material

Incident wave energy

Reflected wave energy

Gravitational acceleration

Wave height

Wave height in deep water conditions

Significant wave height based on time domain analysis equal to the

average of highest 1/3 of all waves heights

Average of highest 1/10 of all wave heights

Wave height exceeded by 2% of the waves

Breaking wave height

Incident wave height

Critical wave height over which all waves are breaking (irregular waves)

Mean wave height

Significant wave height based on spectral analysis

Root-mean-square wave height

Significant wave height

Significant wave height at the toe of the structure

Reflected wave height

Wave height at toe of the structure

Total wave height

Water depth

Breaking water depth

Water depth at the toe of the model

Water depth at the wave paddle of the wave generating system

Water depth at toe of the structure

Stability coefficient

Shoaling coefficient

Wave number

Wave length

Wave length in deep water condition

Slope of the beach (gradient)

Number of stones

Number of waves

Number of armour units in 1 strip

Damage

Number of eroded stones

Stability parameter

Damage

Time counter

Permeability of the structure

Porosity of settled stones

Initial porosity

Porosity strip i after wave attack

Width of the eroded area

Crest freeboard, level of crest relative to SWL

Reflection coefficient

Relative mean squared error

Damage level parameter

Damage level parameter based on table 7.9 in the SPM (CERC, 1984)

Wave steepness

Wave period

Significant wave period based on time domain analysis

Mean wave period based on time domain analysis

Mean energy wave period

Spectral peak period

Duration time of storm to reach damage level

Duration time of additional storm

Medium weight of an armour stone

Structure slope angle

Slope of the beach

JONSWAP spectral shape parameter

Breaker depth index

Relative buoyant density

Spectral shape parameter

Surf similarity parameter/Iribarren number

Critical breaker parameter

Surf similarity parameter using the mean wave period

Surf similarity parameter using the mean energy wave period

Mass density of stone

Bulk density of material as laid on slope

Apparent rock density

Density of water

Effective friction angle between rock

Breaker height index

Angular frequency of waves

List of Abbreviations Abbreviation Explanation

AWACS Active Wave Absorption Control System

CEM Coastal Engineering Manual

De Destruction

DFT Discrete Fourier transform

ESCOLIF Estabilidad hidráulica de los mantos de escollera, cubos y Cubípodos

frente a oleaje limitado por el fondo/Hydraulic stability of cube,

Cubipod and rubble mound breakwaters in depth limited conditions

FFT Fast Fourier Transform

HeP Heterogeneous packing

IDa Initiation of damage

Ide Initiation of destruction

IIDa Initiation of Iribarren damage

JONSWAP Joint Northsea Wave project

LASA Local approximation using simulated annealing

LPC Laboratory of Ports and Coasts/Laboratorio de Puertos y Costas

SPM Shore Protection Manual

SWL Still water level

TUDelft Delft University of Technology

UPV Polytechnic University of Valencia/Universidad Politécnica de Valencia

VDM Van der Meer

VG Van Gent

1

Chapter 1: Introduction Rubble mound breakwaters are structures protecting a coastal area from excessive wave action

as there are ports, port facilities and coastal installations. They consist of mainly quarried rock.

Generally armour stone or artificial concrete armour units are used for the outer armour layer

which ensures the protection against wave attack

Most breakwaters nowadays are built in shallow water, but only little research has been carried

out to improve the knowledge of predicting the damage of the armour layer caused by wave

attack. One of the reasons of this lack of research is the fact that such a research is a challenging

issue due to the uncertainties in the different irregular wave parameters in shallow water.

The scope of this master thesis is to modify and improve existing damage prediction formulas

for rubble mound breakwaters in breaking wave conditions based on the comparison between

the measured and predicted damage. Some model tests in the wave flume in the Laboratory of

Ports and Coasts of the Polytechnic University of Valencia were carried out for this purpose.

Two different approaches will be used to predict the damage. A first approach is based on the

Shore Protection Manual. The starting point of this approach is the Hudson formula, another

approach is based on the formulas developed by Van der Meer.

In chapter two, the theoretical background of these approaches will be given together with a

resume of the existing literature dealing with the hydraulic stability of rubble mound

breakwaters including wave reflection, wave breaking and (armour layer) damage.

Chapter three describes the test equipment together with an explanation of the experimental

design. Also data analysis will be extensively discussed.

In chapter four, the results are presented. The best existing damage prediction formulas are

chosen and some improvements of these formulas are presented.

Chapter 5 finally contains the conclusions of the realized work.

2

Chapter 2: Literature study In this chapter, an overview of the existing literature about the stability of rubble mound

breakwaters in breaking conditions will be given.

First of all, an important process that influences all experiments will be discussed, i.e. wave

reflection. Also some methods to overcome this problem will be given.

Next the phenomenon of wave breaking will be discussed. The different breaker types and the

methods to calculate the breaking wave height in both regular and irregular conditions will be

defined.

Further also the damage that waves cause on the armour layer of a breakwater is treated by

explaining the different failure modes that can damage a breakwater. Also the qualitative and

quantitative analyses of armour damage will be reviewed.

Finally the heart of the matter will be discussed, i.e. the different stability formulas. A historical

overview is given and the most important formulas will be discussed.

1. Wave reflection

a. Reflection coefficient

When waves attack a structure, 3 different processes will take place:

· Dissipation of energy on the permeable medium and transmission of a part of the

energy through the structure to the other side of the structure

· Breaking of the waves over the slope of the structure

· Reflection of the waves on the structure

The breaking of the waves will be introduced in calculations by adjusting coefficients or taking

into account the maximum breaking wave height. Also the permeability will be taken into

account by different coefficients in empirical formulas.

The amount of the wave energy that will be reflected on a sloping structure depends on the

slope, permeability and roughness of the structure. Also the wave steepness and angle of

wave attack influence the reflected wave energy.

A measure to take these influences into account is the reflection coefficient which is equal to

the ratio of the reflected wave height and the incident wave height. Because of the proportional

relationship between the wave energy and the square of the wave height:

(1)

The reflected, incident and total wave height and energy are related as follows:

(2)

This gives following formulas for the incident and reflected wave height:

3

(3)

(4)

b. Separation methods

The problem is that this reflection coefficient is unknown. To obtain the incident or reflected

wave height, some methods are developed. Most of these methods are based on the linear wave

theory: it’s assumed that an irregular wave train can be represented as a superposition of a finite

number of regular waves of different amplitude, emphasis and frequency (Mansard and Funke,

1980). These methods will be discussed briefly and only the characteristics of these methods will

be given. An extensive discussion would lead us too far. If more information is needed, the

references can be consulted.

The classic method to separate the incident waves and reflected used in most laboratories is the

‘Two point method’ by Goda and Suzuki (1976) generalized using 3 sensors as the ‘Three least

squares method’ by Mansard and Funke (1980) and Gaillard et al. (1980). Each lab can have his

own variations and adaptations on these methods.

The method of Goda and Suzuki (1976) is based on earlier work of Kajima (1969) and Thornton

and Calhoun (1972). Goda & Suzuki introduce the ‘Fast Fourier transform (FFT)’ in the method

of Thornton & Calhoun to identify the linear wave component in each signal. This two point

method has some assumptions that limit its functionality:

· Linear dispersion:

· Only stationary waves are considered

· Linear superposition is assumed to calculate the irregular wave components

· No noise: the components of high and low frequency are eliminated previously

· Global estimation: It’s necessary to use the complete registration of the waves to

estimate the incident and reflected wave.

Later, the tow point method of Goda and Suzuki was extended to the Three point method.

As mentioned before, each laboratory has its own variation and adaptation on these methods.

The method that will be discussed more in detail is the one used in the Laboratory of Ports and

Coasts (LPC), i.e. the LASA method

c. LASA method

The LASA method (Local Approximation using Simulated Annealing) is a method developed by

Medina (2001) and improved by Figueres and Medina (2004) to separate reflected and incident

waves. The method is intended to overcome some of the limitations of previous methods, i.e.

the limitations of linearity and stationarity. To do so, the LASA method (Medina, 2001) is based

on a local approximation model considering linear and Stokes II non-linear components and uses

simulated annealing to calculate and optimize the model parameters of the local approximation

model. The main characteristics of this method are the possibility to take into account the data

of n sensors (n≥2), the analysis of both non-stationary regular and irregular waves and the

4

discretization of the wave analysis. The LASA method has been verified with the two point

method of Goda and Suzuki (1976), Kimura (1985) and others. It became clear that the LASA

method is very robust and consistent method in numerical and physical test both for regular as

irregular waves.

The general process of the LASA method to realize the separation of incident waves and

reflected waves can be divided in three steps:

· Elimination of the noise

· Establishment of the frames for the estimation of the central points

· Definition of a local approximation model

Figueres and Medina (2004) optimized the original LASA method, based on Stokes II

components, and developed ‘LASA-V’, a method that is based on a local approximation model

considering non-linear Stokes-V components. The LASA-V method can be used for waves with a

high steepness. This model allows the analysis of tests with non-linear and non-stationary

waves.

5

2. Breaking waves

a. General

When waves arrive in the surf zone, they start to break. Simply said, this is because when a wave

approaches a beach, its length starts to decrease and the wave height increases.

Galvin (1968) classified wave breaking in four categories: Spilling, plunging, collapsing and

surging breakers (Figure 1).

Spilling breakers produce a foamy water surface because of an unstable wave crest. Spilling

breakers are also characterized by their symmetrical wave contours. This type of breaker is

typical for very gentle beach slopes.

Plunging waves produce a high splash, coming from the crest that curls over the shoreward face

of the wave. The wave front is first very vertical, starts to curl and finally falls. A lot of energy is

dissipated during this process. This kind of breaking is observed on gentle to intermediate beach

slopes.

In very steep beaches, surging waves occur: the wave will not break. The front of the wave

arrives on the beach with minor breaking. The wave goes up and down on the slope only forming

a little bit of foamy water.

Collapsing waves are classified somewhere between surging and plunging waves. The crest is

not breaking, but the lower part of the shoreward face steepens up and falls. An irregular

turbulent water face is created.

Figure 1: Breaker types

The classification of the different breaking waves is based on the surf similarity parameter or

Iribarren number (Iribarren and Nogales, 1949). This number is proportional to the tangent of

the slope of the beach and inversely proportional to the root of the wave steepness:

(5)

With Slope of the beach [ ]

Wave height in deep water conditions [ ]

Wavelength in deep water conditions [ ]

Wave period

Gravitational acceleration

6

The values for the surf similarity parameter for the different breaker types (Battjes, 1974) are

given in Table 1.

Table 1: Values surf similarity parameter for different breaker types

Surging/collapsing

Plunging

Spilling

b. Wave breaking criteria

The used criteria for wave breaking in coastal engineering are mostly semi-empirical formulas

obtained by laboratory tests.

A wave remains stable and only doesn’t break if the velocity of the water particles is lower than

the wave celerity (Stokes, 1880). Michell (1893) observed that the limiting wave steepness,

which is the ratio of the wave height and the wavelength, is 0.142 in deep water. This occurs

when the crest angle is equal to 120° (Figure 2).

Figure 2: Limiting steepness in deep water (CERC, 1984)

When a wave is moving into shoaling water, the limiting steepness which it can attain, will

decrease (CERC, 1984). Miche (1944) observed that the limiting wave steepness for waves in

depths less than equal is to , with the water depth.

A wave that moves from deep water to shoaling water will move towards the shore until it

breaks. The wave height at breaking is commonly defined by the breaker (depth) index

defined as the maximal wave height to depth ratio , with the subscript standing for the

breaking point.

(6)

A second parameter, the breaker height index, is also widely used:

(7)

However, the latter induces a greater uncertainty in the prediction of according to present

authors (Camenen and Larson, 2007).

7

Regular waves

The first value of the breaker depth index for regular waves was estimated by McCowan (1891)

and equal to 0.78 for a solitary wave traveling over a horizontal bottom.

Munk (1949) defined the breaker height index of a solitary wave as:

(8)

and the breaker depth index:

(9)

Subsequent observations and investigations by Iversen (1952), Galvin (1968), Goda (1970),

Weggel (1972), and others have shown that and depend on incident wave steepness and

beach slope. Thus for regular waves and uniform beach slope, next formulas are proposed:

Goda (1970)

(10)

Weggel (1972)

(11)

Battjes (1974)

(12)

Ostendorf and Madsen (1979)

(13)

Singamsetti and Wind (1980)

(14)

Smith and Kraus (1990)

(15)

8

Rattanapitikon and Shibayama (2000)

(16)

Those formula were compared with a compiled data set (Camenen and Larson, 2007), but there

was not one best formula. The best results for the existing formulas and use of all data were

given by Battjes (1974), Goda (1970) and Ostendorf and Madsen (1979). For steep beaches, the

Weggel (1972) formula gave the best prediction. Also some general remarks were made. Weggel

(1972) and Singamsetti and Wind (1980) overestimate the breaker depth index and produce

considerable dispersion of the results while Smith and Kraus (1990) underestimate the breaker

depth index. A disadvantage of the Ostendorf and Madsen (1979) and Battjes (1974) formulas

are that no value can be calculated for a beach with a slope equal to zero.

A semi-empirical relationship for the breaker height index is derived from linear wave theory by

Gaughan and Komar (1974) and equals:

(17)

Finally another criterion is given by Rattanapitikon and Shibayama (2000) and Rattanapitikon et

al. (2003), with the wavelength when the wave breaks calculated using the linear wave

theory:

(18)

Both Goda (1970) and Weggel (1972) included their formulas in a graph. These are given in

Figure 3 and Figure 4.

Figure 3: Graph showing formula by Goda (1970)

Figure 4: Graph showing formula by Weggel (1972)

9

To calculate the breaking wave height from the breaker height index , The deep water

wave height has to be calculated if this is not known. This can be done using the shoaling

formula (Goda, 2000):

(19)

With Shoaling coefficient by small amplitude wave theory [ ]

wavenumber

Irregular waves

For irregular waves, the breaking point as well as the breaking wave height can start over a wide

zone, in contrast to the case of regular waves (Goda, 2000). In the zone where more or less all

waves are breaking (saturated breaking zone), the root-mean-square breaking wave height and

the zero-moment wave height depend on the local depth (Thornton and Guza, 1983):

(20)

(21)

CIRIA (2007) states that the typical values for the breaker depth index are 0.5 to 0.6. These values

mainly depend on the Iribarren number , and they can reach 1.5 for individual waves. Data

from different authors are shown in Figure 5.

Figure 5: Breaker index as a function of the Iribarren number

The critical wave height over which all the waves are broken can be found by applying an

energy flux balance and is given by (Battjes and Stive, 1985):

(22)

10

This formula was later modified by Nairn (1990):

(23)

A linear relationship between the product of the local wave number and the water depth was

found by Ruessink et al. (2003):

(24)

Goda (2010) changed his formula (10) for irregular waves:

(25)

The deep water wavelength is calculated using the period obtained from time-domain

analysis. This formula was verified starting with regular breaking waves for 6 different slopes

(including 0%).

As final remark, it can be stated that, as said before, the breaking point can’t be defined clearly.

Therefore, Goda (2000) proposes to use the peak value of the significant wave height within

the surf zone as an alternative to the breaker height. These values are depicted on Figure 6.

Figure 6: Index curves for the maximum value of the significant wave height

11

3. Damage

a. Failure modes breakwater

To be able to talk about the stability of a mound breakwater, it’s necessary to consider both the

individual stability of an armour unit as the stability of all units together. The loss of stability can

have different reasons which should be understood. Bruun (1978) is one of the authors who has

worked with it. He stated 11 different principal failure modes. They are summarized as follows

(Vanhoutte, 2009):

I. Loss of armour units from the principal armour layer (increasing porosity)

II. Rocking of the armour units; breaking occurs due to fatigue

III. Damage of the inner slope by wave overtopping

IV. Sliding of the armour layer due to a lack of friction with the layer below

V. Lack of compactness in the underlying layers, causing transmission of energy to

the interior of the breakwater; this might lift the cap and the interior layers.

VI. Undermining of the crone wall

VII. Breaking of the armour units caused by impact, simply by exceeding its

structural resistance or by slamming into other units

VIII. Settlement or collapsing of the subsoil

IX. Erosion of the breakwater toe or the breakwater interior

X. Loss of the mechanical characteristics of the materials

XI. Construction errors

These failure modes are depicted in Figure 7.

Figure 7: Different failure modes according to Bruun (1978)

All these 11 different failure modes can be classified in 5 groups (Gomez-Martin, 2002):

· Unit stability: refers to the capacity of each armour unit to resist movement subjected

to the wave action. (I,II,III)

· Global stability: it’s the stability of the complete breakwater or more specific the

complete armour layer. (IV,V,VI)

· Structural stability: refers to the capacity of each unit to resist (without breaking) the

tensions caused by transport, construction, wave action, used granular and movements

caused by currents. (II, VII)

· Geotechnical stability: stability of the underground. It includes the carrying capacity and

the sensitivity to erosion of the breakwater toe.(VIII, IX)

· Construction errors (X,XI)

12

The global and unit stability are the stability that will be considered here. Together they can be

called the hydraulic stability. Burcharth (1992) also enumerate the different failure modes of a

breakwater (Figure 8).

Figure 8: Different breakwater failure modes according to Burcharth (1992)

In this thesis, most attention will go to the hydraulic stability of the armour units. This is also the

main mode of failure. It’s classified in the failure group of ‘Unit Stability’ according to Gomez-

Martin (2002). This failure mode can be caused by 2 different reasons: the simple extraction of

the units under wave action and the settlement caused by the heterogeneous packing. The

heterogeneous packing (HeP) failure mode is a failure mode that is significant in the case of

regular armour units (Gomez-Martin, 2006). The HeP failure mechanism reduces the packing

density of the armour layer near the SWL without extracting elements, generating zones with

low porosity and corresponding zones with high porosity. The impact of the HeP failure mode

depends on:

I. Type of armour unit

II. Difference between the initial porosity and the minimum porosity

III. Slope of the armour layer

IV. Friction coefficient between the armour layer and the secondary layer

b. Armour damage

There are two ways to quantify the damage: the qualitative and the quantitative way.

Qualitative analysis

To do a qualitative analysis of the damage, several stages of damage should been distinguished.

Losada et al. (1986) created three hydrodynamic criteria: incipient damage, Iribarren’s damage

and destruction. Vidal et al. (1991) further developed these criteria by adding a fourth criteria:

Initiation of destruction. Hence the four criteria are:

· Initiation of damage (IDa): certain number of units are displaced from their original

position to a new one at a distance equal to or larger than a unit length.

13

· Initiation of Iribarren damage (IIDa): wave action may extract armour units placed on

the lower armour layer. This can be defined as the moment when one stone and his

surrounding stones of the 2nd layer are visible.

· Initiation of destruction (IDe): small number of units (two or three) in the lower armour

layer are forced out.

· Destruction (De): more pieces of the secondary layer are removed and the filter layer is

visible.

A disadvantage of these criteria, is the fact that HeP is not taken into account, because only the

units extracted are considered.

Quantitative analysis

The damage in the quantitative analysis is measured by counting the number of displaced units

or by measuring the eroded surface profile of the armour slope (USACE, 2002). If the damage is

measured by counting the displaced units, which is mostly done in case of (complex structures

of) concrete armour units, the damage can be given as a percentage displaced units within a

reference area:

(26)

The damage can also be given as a dimensionless parameter:

(27)

Another way to calculate the damage is making use of the eroded area. This is mostly done in

case of rock armour. One of the first who used this way of calculating were Iribarren (1938) and

Hudson (1959). Hudson defined the damage as the percent erosion of original volume:

(28)

Thompson and Shuttler (1975) defined another damage parameter :

(29)

With Erosion area in a cross-section

Bulk density of material as laid on the slope

Mass density of stone

Diameter of stone that exceeds the 50% value of sieve curve

The advantage of these formulas is the fact that the damage is independent of the size of the

armour layer, compared to a percentage of damage. The disadvantages of the last formula are

the measurement of the bulk density and the use of the sieve diameter instead of the actual

14

mass of the stone. To improve this formula, Broderick (1983) deleted the bulk density and

defined the damage level parameter as follows :

(30)

With Eroded area around SWL [ ]

Medium weight of an armour stone [ ]

By introducing the median nominal diameter the damage can be written

as:

(31)

The eroded area is depicted in Figure 9. It takes into account both settlement and displacement.

The eroded area can be seen as the number of squares with a side that fits into the erosion

area. Another physical description of the damage of is the number of cubic stones with a side

of eroded within a -wide strip of the structure. The actual number of stones eroded

within this strip can be more or less than , depending on the porosity, the grading of the

armour rocks and the shape of the rocks. Generally the actual number of rocks eroded in a -

wide strip is equal to 0.7 to 1 times the damage .(Van der Meer, 1998)

Figure 9: Eroded area

The slope angle of the structure has a big influence on the limits of . These limit values are

characterized as follows:

· Start of damage/initial damage: corresponding to no damage in the Hudson formula (see

later)

· Intermediate damage

· Failure, when the filter layer is visible

15

For design purposes of a double layer armour stone breakwater, these values are given in Table

2. For S-values higher than 15-20, the deformation of the structure results in an S-shaped profile

(Van der Meer, 1998).

Table 2: Design values of for a two-diameter thick armour layer

Slope Start of

damage

Intermediate

damage

Failure (under

layer visible)

1:1.5 2 3-5 8

1:2 2 4-6 8

1:3 2 6-9 12

1:4 3 8-12 17

1:6 3 8-12 17

There are different ways to measure the eroded area. It can be measured by a surface profiler

(mechanic/laser profiler), by computing the planar eroded area on the outer layer of the armour,

using a digital image processing technique or by counting the removed armour stones settled

over the original armour layers (Vidal et al., 2006) . If the latter is used, the eroded area is given

by:

(32)

With the number of eroded stones, the porosity of the settled stones and the width of

the eroded area. This method is called ‘the visual counting method’.

The disadvantage of all formulas above is that they do not take into account the heterogeneous

packing. Therefore, a new method was necessary that takes into account the changes in

porosity. This method is called the new Virtual Net Method (Gómez-Martín and Medina, 2006).

In this method, the armour layers are divided into strips with each a width of times the

equivalent cube size and a length . The number of armour units in every strip is

counted and with this number, the porosity of every strip after wave attack is calculated using

the formula below.

(33)

Next, the dimensionless damage in each strip can be calculated taking into account the initial

porosity :

(34)

By summing up the different damages over the different strips, the equivalent dimensionless

armour damage could be obtained:

(35)

16

4. Stability formulae

Until 1933 no methods to calculate a rubble mound breakwater existed. The breakwaters were

build using the experience obtained from the construction of older ones. Obviously, this

qualitative knowledge wasn’t sufficient for the construction of breakwaters. Also, the

complexity of the phenomena that were involved (wave characteristics, wave behavior,…)

impeded the development of this qualitative knowledge.

The first formula for the calculation of rubble mound breakwaters was formulated by Castro

(1933). Castro showed that the wave forces are the reason of the destruction of the breakwaters

and that the waves push the rock over the breakwater. However, this almost never happens

Iribarren (1938) published a new formula for the weight of the rocks resisting a certain wave

height for the principal layer of the breakwater. This formula would be the starting point for the

later developed formula by Hudson.

From 1949, a big development in the knowledge and of the formulas started. Also the

phenomena related with the water flow over the slope of the breakwater were studied. The

discussion of all the different formulas and modified formulas would lead us so far, so only the

most important and the formulas that have most to do with this master thesis will be discussed.

a. Hudson

One of the most known and used stability formula is Hudson`s formula (Hudson, 1959). It’s based

on model tests with regular waves on non-overtopped rock structures with a permeable core

and based on the pioneering work of Iribarren (1938) and Hudson and Moore (1951):

Iribarren (1938)

(36)

With Medium weight of armour stone [ ]

Apparent rock density [ ]

=0.015 & 0.019 (rock-fill & concrete blocks randomly dumped)

Wave height [ ]

[-]

Density of water [ ]

Structure slope angle [rad]

Hudson and Moore (1951)

(37)

With Wave height at the toe of the breakwater [m]

constant between 0.0035 and 0.0300, depends on the slope

Effective friction between rock [-]

17

Hudson (1959)

(38)

The parameters are equal to the ones stated before.

The Hudson formula can be applied for structures with a slope from 1:1.5 to 1:5.

The values of the stability coefficient depend on many characteristics:

· Shape of the element of the armour layer

· Nature of the element of the armour layer (concrete-rock)

· Number of armour layers

· Roughness of the element

· Degree of interlocking obtained in placement

· Water depth near the structure (breaking or non-breaking)

· Part of the mound breakwater (head or body)

· Angle of incident wave

· Porosity of the core

· Size of the core

· Width of the crest

· …

In the ‘Shore Protection Manual (SPM)’ (CERC, 1977), the values of the stability coefficient for

rough angular quarry stone armour units are and for respectively breaking

waves on the foreshore and non-breaking waves on the foreshore, corresponding with a damage

level of 0-5% (no damage condition). These values are valid for rough, angular, randomly placed

armour stone in two layers on a breakwater trunk. Also it is suggested to use the significant wave

height in equation (37). In addition other values were suggested for a wide range of

armour units and other conditions.

In SPM (CERC, 1984) some modifications to the use of Hudson’s formula are made. One of these

modifications is the use of the average of the highest 10% of wave heights as the design

wave height which is equal to 1.27 instead of . Also the value for breaking waves is

changed from 3.5 to 2.0; the value for non-breaking waves remains the same.

The original Hudson formula (37) can be rewritten, in terms of the stability parameter :

(39)

With Stability parameter [-]

Significant wave height [ ]

Mean nominal diameter of the armour stone [ ]

18

This gives:

(40)

The formula of Hudson has its limitations. First, it can only be used for regular waves. Also the

wave period and storm duration are not taken into account. Besides these limitations, Hudson’s

formula can only be used for non-overtopped and permeable structures only. Finally there is no

description of the damage level, because the formula only corresponds with a damage level of

0-5% (no damage).

To overcome this last limitation, higher damage percentages have been determined as a

function of the wave height for several types of armour units. The values for armour stone are

given in Table 3 (CERC, 1984).

Table 3: as a function of armour layer damage and armour type

The notation instead of is used in this case, because it’s not clarified in the SPM how

the damage is measured and the conversion from the damage to the damage level parameter

is only an assumption. Also a slightly different conversion could be made.

Van der Meer (1988) modified equation (40) by using Table 3 (angular armour stone) and

applying regression analysis. This modified formula for the stability number equals:

(41)

According to the Rock Manual (CIRIA, 2007), the values for an impermeable and permeable

core for both breaking and non-breaking conditions are respectively 1 and 4, accepting that 5%

of the data will lead to a higher damage level than predicted. values of respectively 4 and 8

can be used to describe the main trend. This was concluded after comparing data used by Van

der Meer (1988) and Van Gent et al. (2004) with the equation above using different values.

Medina et al. (1994) proposes another empirical formula to include the data in Table 3 in

equation (40):

Armour Type

Relative

wave

height

Damage with corresponding damage level

[-] 0-5 5-10 10-15 15-20 20-30 30-40 40-50

[-] 2 6 10 14 20 28 36

Smooth

armour

stone

1.0 1.08 1.14 1.20 1.29 1.41 1.54

Angular

armour

stone

1.0 1.08 1.19 1.27 1.37 1.47 1.56

19

(42)

One of the reasons of the difference between this formula and formula (41) is because Medina

used a different conversion between the damage and the damage level parameter . But

because the formula is still based on the data from the SPM (CERC, 1984), the notation is

still used.

b. Van der Meer

Deep-shallow water

While the classification according to wave propagation of water waves (deep water waves,

transitional water waves and shallow water waves) is based on the relative depth criterion

(respectively , and ) (USACE, 2002), the definition of

shallow water is defined by for the limit of the field of application of the Van der

Meer formulae (CIRIA, 2007). More recently, Van Gent developed even another classification,

based on the ratio of the significant wave height at the structure to the one observed offshore.

If the ratio is larger than 0.9 or smaller than 0.7, the structure is in respectively deep and very

shallow water (with a considerable amount of wave breaking). If the ratio is between 0.7 and

0.9, the structure is said to be in shallow water conditions and some (limited) wave breaking will

already occur (Van Gent et al., 2004).

Deep water conditions

Besides the modified Hudson formula, Van der Meer (1988) also developed also some formulas,

He based his study on the earlier work of Thompson and and Shuttler (1975). An extensive series

of model tests was conducted at Delft Hydraulics. This series include structures with a wide

range of core/underlayer permeabilities and a wide range of wave conditions.

The Van der Meer (VDM) formula takes, in contrast to the Hudson formula, into account the

effects of storm duration, wave period, the structure’s permeability and a clearly defined

damage level. VDM considers 2 types of breaking waves: plunging and surging waves. The

transition from plunging conditions to surging conditions is given by the critical breaker

parameter and depends on the structure slope :

(43)

For plunging waves ( ):

(44)

20

For surging waves ( ):

(45)

With Significant wave height [m], of the incident waves at the

toe of the structure

6.2

1.0

P Permeability of the structure [-]

N Number of incident waves at the toe of the structure [-]

surf similarity parameter based on the mean wave period

from time domain analysis and significant wave height

It is recommended to assume plunging conditions for irrespective of whether the surf-

similarity parameter is smaller or larger than .

The values for the permeability of different structures are given in Figure 10. The lower boundary

is equal to 0.1. This is the case when the armour layer has a thickness of 2 diameters of the

armour unit and the layer under it is impermeable. This is often the case for seawalls and

revetment. The upper boundary is given by a homogeneous structure without filter and core. It

only consists of rock. The permeability is then equal to 0.6.

The considered range of parameters by Van der Meer (1988b) is given in Table 4.

Figure 10: Values of permeability for different structures

21

Table 4: Range of parameters VDM formula

Parameter Symbol Range

Stability number 1-4

Wave steepness 0.01-0.06

Surf similarity parameter 0.7-7

Damage as a function of the

number of waves <0.9

Ratio deep water wave

height and water depth at

toe

0.18-2.7

Armourstone gradation 1-2.5

Permeability 0.1-0.6

Slope angle 1:1.5-1:6

Spectral shape parameter 0.3-0.9

Crest height -1-2

The VDM formulas are related to a single storm event. The maximum number of waves is equal

to 7500 because after this number of waves, the damage reaches more or less an equilibrium

This means that the damage for more than 7500 waves is found by using . For a

number of waves smaller than 1000, the formulas give a slight overestimation.

The limits of the damage level parameter mainly depend on the slope of the structure

If the Van der Meer formula is used to for design proposes, it’s required to do a sensitivity

analysis for all parameters. Also a sensitivity analysis of the constants should be done. Two

graphs are given as an example of such an analysis (Figure 11). The influence on the changes of

the damage level parameter and permeability on the design wave height for different breaker

parameters is analysed.

Figure 11: Sensitivity analysis for damage level parameter and permeability

22

Shallow water conditions

Some of the tests carried out by Van der Meer were in shallow water. Van der Meer (1988)

recommends to use the wave height exceeded by 2% of the waves instead of the significant

wave height . This because the distribution of the wave heights deviate from the Rayleigh

distribution (due to wave breaking) so the stability of the armour layer in depth limited waters

is better described by the higher characteristic value of the wave height distribution than

by . The known ratio is equal to 1,4 for a Rayleigh distribution, so the deep water

values of the coefficients and should be changed to respectively 8.7 and 1.4 for shallow

water. The transition between plunging and surging waves is the same as equation (43) using

the coefficients and valid for shallow water.


(46)


(47)

In conclusion, a remark should be made. A safer approach for design purposes is to use formulas

(44) and (45) with . In that case the truncation of the wave height exceedance curve due to

wave breaking is not taken into account (Van der Meer, 1988).

The equations (44) and (45) will give the same results as (46) and (47) if the waves are Rayleigh

distributed due to the know Rayleigh-distribution-based ratio that was introduced

before.

To do the correct calculation, the actual ratio should be obtained, which is not always

possible because often only the wave heights based on energy ( is based on the zero-est

moment of the spectrum ) are obtained. This means that (actually ) is different from

and so it’s difficult to find a good estimation for . To overcome this issue, a good

approximation for based on is given by Battjes and Groenendijk (2000) who proposed

a composed Weibull-Rayleigh distribution for the waves. Van Gent et al. (2004) concluded that

both wave heights and can be used, leading to almost the same accuracy on average.

c. Van der Meer modified by Van Gent

Van Gent et al. (2004) modified, based on his own tests, the Van der Meer formula for shallow

foreshore. First, he uses the spectral mean energy wave period instead of the mean wave

period obtained from time-domain analysis . Also the coefficients and are recalibrated

and have a value of respectively 8.4 and 1.3. Thus the final formulas are:


(48)

23


(49)

With

The formulas were developed using tests in the ranges given in Table 5 and are valid in both

deep and shallow water

d. Van Gent

Another (more simple) formula was developed by Van Gent et al. (2004), based on a earlier

series of tests with a 1:100 foreshore (Smith et al., 2002) and additional tests with a considerably

steeper foreshore (1:30). The structure slopes were 1:2 and 1:4. Both test series were combined

and analysed to develop the new formula.

In the new formula the permeability is directly related to a structure parameter (mean nominal

diameter of the core material ). The influence of the period isn’t taken into account.

On the one hand the wave period influences the damage, but on the other this influence is small

compared to the amount of scatter in the data. For the same reason, there is also no difference

between plunging and surging waves. Besides this, the ratio also influences the damage,

but again, these influence is considered small. The Van Gent formula is given by:

(50)

The range of tests used to develop the formula is the same as mentioned above in Table 5. The

formula is valid in both shallow and deep water.

Table 5: Range of parameters VDM formula modified by VG and VG formula


Surf similarity parameter 1-5

Ratio deep water wave height and

water depth at toe

0.18-2.72

Ratio 2% and significant wave height 1.21-1.42

Ratio significant wave height and

water depth at toe

0.15-0.78

Ratio significant and deep water

wave height

0.17-1.1

Armour diameter 0.022m-0.035m

Slope angle 0.01-0.033

Slope structure 1:2 & 1:4

Crest height 0.6-4.3

24

e. Cumulative damage

All previous formulas are limited to single storm events. Some studies were performed to take

into account the phenomenon of progressive damage due to subsequent storm event. 2

different methods taking into account subsequent storm events will be discussed: the method

developed by Melby (2001) and the method developed by Van der Meer (2000).

Melby method

The cumulative damage is evaluated by

(51)

With Damage level at time [ ]

Damage level at time [ ]

Duration time of additional storm [ ]

Duration time of storm to reach a damage level [ ]

The stability number, based on the significant wave height

from time domain analysis [ ]

Mean wave period [ ]

Time counter [ ]

Coefficient determined in experiments,

The range of validity of the parameters of the laboratory tests on which the formula is based is

limited and is given in Table 6. Also the depth-limited wave conditions and the wave conditions

of the subsequent events are relatively constant.

Table 6: Range of parameters Melby method


Surf similarity parameter 2-4

Structure slope 0.5

Ratio of armour and

filter stone sizes 2.9

Permeability <0.4

Van der Meer method

The approach of Van der Meer to take into account cumulative damage makes directly use of

the Van der Meer deep water stability formula (44) and (45). Nevertheless this approach can

also be applied to the Van der Meer formula in shallow water ((46) and (47)), the Van der Meer

formula modified by Van Gent (48) and (49) and the Van Gent formula (50). It’s based on the

equivalence hypothesis.

The procedure to assess the cumulative damage is as follows:

25

· Calculate the damage for the first wave condition

· Calculate for the second wave conditions how many waves would be required to give

the same damage as under the first wave conditions. This is denoted by .

· Add this number of waves, to the number of waves of the second wave conditions:

· Calculate the damage under the second wave conditions with this increased

number of waves

· Calculate for the third wave condition how many waves would be required to give the

same damage as caused by the second wave condition etc.

Figure 12: Cumulative damage approach by Van der Meer

26

Chapter 3: Experimental setup In this chapter, the test equipment and experimental model will be discussed. Also the realized

experiments and calculations will be explained.

First of all, the wave flume will be described including the wave generating system, the system

that dissipates the energy, the sensors that measure the wave elevations and the visual

equipment.

Next the design of the foreshore and model will be set out together with the properties of the

used quarry stone.

Furthermore, the characteristics of the realized experiments will be given.

Finally this chapter will be concluded by the different steps taken to analyse the data. The

different methods to obtain the wave characteristics will be explained, the porosity

measurement method will be discussed together with the damage calculation method. Also it

will be explained how the original formulas will be improved.

1. Test equipment

All tests are performed in the 2D wave flume in the Laboratory of Ports and Coast of the

Polytechnic University of Valencia(LPC-UPV)(Figure 13). The length of the wave flume is 30

meters, the height is 1.2 meters and the width is 1.2 meters. There is a false bottom of 25 cm

for the circulation of the water. The wave generator is installed at one side of the wave flume.

The system to dissipate the energy of the waves is installed at the other side. The breakwater

model is placed in front of the energy dissipater. A part of the walls of the wave flume is

transparent to give the researchers the possibility to see what happens inside the wave flume.

Figure 13: Wave flume at LPC-UPV

27

a. Wave generator

As mentioned before, the wave generator is installed at one end of the wave flume. It consists

of and metal plate that is installed vertical and connected to a piston which generates the

translation of the plate. This piston is driven by an electrical servomotor (Figure 14).

Figure 14: Wave generator (Herrera, 2013)

The wave generating system is controlled by an Active Wave Absorption Control System

(AWACS). This is a digital control system that absorbs the reflection of the waves and generates

the desired waves. This is needed to take into account the reflected waves from the structure

on the other side of the wave flume. Since incident waves on the model are reflected, they could

re-reflect on the wave maker, which results in an uncontrollable, and undesirable nonlinear

distortion of the desired waves impinging on the test structure, because the wave generator

keeps on having the same movement. So the reflected waves from the breakwater are taken

into account by the AWACS. The principle of the AWACS is as follows. It measures the surface

elevation by 2 wave gauges integrated in the paddle front. This measured surface elevation is in

fact the superposition of the desired wave and the reflected wave. The reflected wave is

identified by the digital recursive filter of the AWACS and absorbed by the wave maker. The

specific type of AWACS installed in LCP-UPV is DHI AWACS2, from Denmark.

The wave making system has 3 options. First of all, it can reproduce previous generated waves.

Secondly, it can produce regular waves (Stokes 1st order). Last it can also generate irregular

waves. The parameters that has to be given in into the system are (depending if regular or

irregular waves will be generated):

· Scale: it’s used to make the conversion from the parameters of the prototype to the

model

· Water depth: water depth in front of the wave paddle.

· Wave height and period

· Spectrum(only for irregular waves)

· and (only for irregular waves)

· Skewness (only for irregular waves)

· Duration

28

b. Energy dissipation system

The energy dissipation system is placed at the end of the wave flume, on the opposite side of

the wave generating system. It consists of five groups of three grooved metal frameworks and a

plastic plate which is perforated. The metal frameworks have 3 different porosities: 70%, 50%

and 30%. The frameworks are placed in such a way so the framework with the highest porosity

is the closest to the approaching wave. The first group of 3 metal frameworks with a porosity of

70% is followed by 2 groups with a porosity of 50%. The porosity of the last 2 groups is 30%. The

voids between the 3 frameworks of this last group are filled with quarry stone (Figure 15).

Figure 15: Energy dissipation system and schematically overview of the frameworks

c. Wave measurement

The wave flume is equipped with a series of wave gauges and run-up sensors. Because there is

no run-up during the test carried out for this master thesis, only the wave gauges will be

discussed.

The wave gauges measure the surface elevation. They consist of two vertical electrodes. When

they are submerged in the water, the sensors measure the conductivity of the volume of water

between the electrodes. The conductivity changes as a function of the surface elevation

between the electrodes.

The gauges has to be calibrated every time before the tests to overcome any errors caused by

for instance changes in water level, caused by leaks and evaporation.

The wave gauges are placed in different groups: One group close to the wave paddle to measure

the wave height generated by the wave generating system in deep water (S1, S2, S3 and S4),

one group close to the model to measure the wave height at the toe of the structure (S11 and

S12). Between these 2 groups, there are placed some other sensors along the slope of the beach

(Figure 16).

29

Figure 16: Position of the sensors, dimensions in meter

Mansard and Funke (1980) defined some criteria for the distance between 3 wave gauges, with

and the distances between the 3 wave gauges:

(52)

Using these criteria, a distance between the waves gauges is chosen for all the wave periods

that will be used, so it wouldn’t be necessary to change them every time after one test.

d. Cameras

The wave flume is equipped with different hardware for visual registration. Both digital pictures

and digital videos are made. They are not only used to have a good view of them in the office

during the test, but also to take the pictures used to calculate the damage.

30

2. Experimental design

a. Foreshore slope

A foreshore slope is created to make the waves break (Figure 16). The slope of the first one

located 545 cm in front of the wave paddle and with a length of 625cm is 4%. Behind this slope,

another foreshore slope of 2% is constructed. Its length is 1100 cm and the model is built on the

end of this slope. The water level varies according to the test series between 61.7 and 81.7 in

front of the wave paddle with steps of 10 cm. This corresponds with a water level between 20

and 40 cm at the toe of the breakwater model.

b. Model

The used model (scale 1:60) is a rubble mound breakwater model with a slope of 3/2 at the

side exposed to the waves (Figure 17) . The armour layer of the breakwater is composed out of

2 layers of quarry stone with a porosity of approximately 37% without a toe berm. The model

has 3 different parts: The core, a filter layer and an armour layer. The characteristics of the 3

will be briefly discussed.

Figure 17: Section of the rubble mound breakwater, dimensions in cm

Core

The characteristics of the core are not really relevant. But for the completeness, the sieve

analysis is given in Figure 18.

Figure 18: Sieve curve of the core

31

Filter

The size of the stones of the filter used in the model have a nominal diameter of .

The distribution for a sample of 20 filter layer stones is given in Figure 19.

Figure 19: Distribution of the filter layer stones

Armour layer

The mean nominal diameter of the stones used is equal to . The distribution of

a sample of 25 stones is given by Figure 20.

Figure 20: Distribution of the armour layer stones

The initial porosity of the armour stones has to be approximately 37% (CERC, 1984) before the

test series start. The model with this initial porosity is depicted on Figure 21.

Figure 21: Rubble mound model constructed with an initial porosity of approximately 37%

32

3. Realized experiments

The realized experiments on the rubble mound breakwater model are part of a 2 years long

ongoing project called ‘ESCOLIF’ (Estabilidad hidráulica de los mantos de escollera, cubos y

Cubipodos frente a oleaje limitado por el fundo/ Hydraulic stability of cube, Cubipod and rubble

mound breakwaters in depth limited conditions) which is carried out at the LPC-UPV. The goal of

the project ESCOLIF is to improve the knowledge of the hydraulic stability of single- and double-

layer Cubipod armours (an armour unit developed by LPC-UPV) in depth limiting conditions.

Another goal of ESCOLIF is to increase the experimental basis of double-layer cubes and quarry

stone in depth limiting conditions:

· · ·

The initial testing conditions (foreshore slope of 0, 2 and 4%) were modified by adding also a

foreshore slope of 10% to the tests because a high influence of the slope of the foreshore was

observed.

The experiments discussed in this master thesis are carried out with 1000 irregular waves with

a JONSWAP spectrum ( ). The foreshore for all test is equal to 2%. The typical breaker

type under such a foreshore slope is spilling breakers.

Every test series is initiated with a wave series of 1000 waves with a wave height that doesn’t

produce breaking waves ( ). By increasing the wave height with 1 cm every step and by

keeping the Iribarren number constant, more and more waves are breaking until destruction of

the model is observed. Because the Iribarren number is kept constant in one test series, a

corresponding peak period can be calculated for each wave height. The different test series

are performed for different Iribarren numbers ( and ) and in different water depths

( and ). One repetition test is carried out ( and ) to

use later as a ‘blind test’ to verify the improved and new developed formulas.

An overview of the input data for the different test series is given in Table 7 to Table 11

Table 7: Test data for Ir=3 and h=20cm

Model scale Prototype scale

8 1.02 61.70 20 4.8 7.89 37.02 12

9 1.08 61.70 20 5.4 8.37 37.02 12

10 1.14 61.70 20 6 8.82 37.02 12

11 1.19 61.70 20 6.6 9.25 37.02 12

12 1.25 61.70 20 7.2 9.66 37.02 12

13 1.30 61.70 20 7.8 10.06 37.02 12

14 1.35 61.70 20 8.4 10.44 37.02 12

15 1.39 61.70 20 9 10.80 37.02 12

33

16 1.44 61.70 20 9.6 11.16 37.02 12

17 1.48 61.70 20 10.2 11.50 37.02 12

18 1.53 61.70 20 10.8 11.84 37.02 12

19 1.57 61.70 20 11.4 12.16 37.02 12



8 1.70 61.70 20 4.8 13.15 37.02 12

9 1.80 61.70 20 5.4 13.95 37.02 12

10 1.90 61.70 20 6 14.70 37.02 12

11 1.99 61.70 20 6.6 15.42 37.02 12

12 2.08 61.70 20 7.2 16.11 37.02 12

13 2.16 61.70 20 7.8 16.76 37.02 12

14 2.25 61.70 20 8.4 17.40 37.02 12

15 2.32 61.70 20 9 18.01 37.02 12



8 1.02 71.70 30 4.8 7.89 43.02 18

9 1.08 71.70 30 5.4 8.37 43.02 18

10 1.14 71.70 30 6 8.82 43.02 18

11 1.19 71.70 30 6.6 9.25 43.02 18

12 1.25 71.70 30 7.2 9.66 43.02 18

13 1.30 71.70 30 7.8 10.06 43.02 18

14 1.35 71.70 30 8.4 10.44 43.02 18

15 1.39 71.70 30 9 10.80 43.02 18

16 1.44 71.70 30 9.6 11.16 43.02 18

17 1.48 71.70 30 10.2 11.50 43.02 18

34



8 1.70 71.70 30 4.8 13.15 43.02 18

9 1.80 71.70 30 5.4 13.95 43.02 18

10 1.90 71.70 30 6 14.70 43.02 18

11 1.99 71.70 30 6.6 15.42 43.02 18

12 2.08 71.70 30 7.2 16.11 43.02 18

13 2.16 71.70 30 7.8 16.76 43.02 18

14 2.25 71.70 30 8.4 17.40 43.02 18



8 1.02 81.70 40 4.8 7.89 49.02 24

9 1.08 81.70 40 5.4 8.37 49.02 24

10 1.14 81.70 40 6 8.82 49.02 24

11 1.19 81.70 40 6.6 9.25 49.02 24

12 1.25 81.70 40 7.2 9.66 49.02 24

13 1.30 81.70 40 7.8 10.06 49.02 24

14 1.35 81.70 40 8.4 10.44 49.02 24

15 1.39 81.70 40 9 10.80 49.02 24

16 1.44 81.70 40 9.6 11.16 49.02 24

35

4. Data analysis

a. Wave analysis

The wave characteristics (wave height, period…) have to be obtained to estimate the damage

using the different prediction formulas. These values can be obtained using different methods.

Their registered values can’t be used directly because these values consist of 2 different

influences, namely the incident wave and the reflected wave by the structure. The

characteristics of the incident wave are the characteristics that will be used to do all calculations.

There are 3 different methods to obtain the characteristics of the incident waves with each its

advantages and disadvantages: SwanOne, measurements by wave gauges in canal without

model and measurements by wave gauges in canal with model. Each one will be discussed.

SwanOne

SwanOne is a model developed by TUDelft in MatLab to simulate the evolution of the wave

spectrum starting from deep water to shores. The model is capable to simulate interactions and

transformations of waves (TUDelft, 2015).

The different input parameters are:

· Bottom profile

· Current

· Wave direction

· Water level

· Wind velocity and direction

· Boundary conditions: arbitrary spectrum file or definition input diameters (significant

wave height based on spectral analysis [m] and the peak period [s]).

Also the output locations should be provided, i.e. the locations along the profile where you want

to obtain the output parameters.

The output parameters are:

· Significant wave height [m]

· Root-mean-square wave height [m]

· Peak period [s]

· Mean absolute period [s]

· Mean period [s]

· Mean energy wave period [s]

· Significant wave height calculated using the Battjes and Groenendijk method [m]

· Wave height exceeded by 2% of the waves [m]

· Mean wave height of highest 1/10 fraction of waves [m]

No breakwater model is used in SwanOne thus also no reflection by the model is possible.

Therefore the obtained parameters are the incident characteristics.

36

Measurements by wave gauges in canal without a model

A second method to measure and calculate the wave characteristics is using directly the surface

elevations measured by the wave gauges in the wave flume. To avoid reflection by the model,

no model is placed in the canal. Theoretically the waves will not be reflected by the end side of

the canal since the energy of the waves is dissipated there by the energy dissipating system.

The wave characteristics are obtained by the software developed in the LPC and is called LPCLab

2.0. LPCLab 2.0 calculates different wave heights and period spectra and moments for later

calculations using the surface elevations measured by the wave gauges.

The wave characteristics are analysed by LPCLab 2.0 in both the time-domain and frequency

domain and generates information about all relevant parameters and gives also some graphs.

i. Time-domain analysis

In the time-domain, each individual wave is defined by the downward crossing of the zero-line

by the surface elevation (zero down-crossing). The mean wave height is calculated from

time-series of individual waves.

ii. Frequency domain analysis

In addition to the time domain analysis , the wave spectrum of the realized test is calculated

using the discrete Fourier transform (DFT) of the measured surface elevation.

The total registered waves are divided in different time-windows. The fast Fourier transform

(FFT) is done over the different windows and the results are summed at the end. The width of

the time-window, the number of points to represent the spectrum and the percentage of

overlap between the different windows can be chosen by the user. Another important

parameter to put in LPCLab 2.0 is a range of frequencies so useless frequencies can be

eliminated. A screenshot of LPCLab 2.0 is given in Figure 22.

Figure 22: Screenshot of LPCLab 2.0

37

Measurements by wave gauges in canal with model.

LPCLab 2.0 can also be used to process the surface elevations that are measured by the wave

gauges in the wave flume when the model is present. The only problem is that the model will

reflect the waves so LPCLab 2.0 won’t generate the incident wave characteristics, but only the

registered wave characteristics. To overcome this problem, the LASA method can be used.

However, the LASA or LASA-V method can’t be used in breaking wave conditions, so the LASA

method can only be applied on the wave gauges in front of the wave paddle, where the waves

aren’t breaking. Hence the incident and reflected waves are known for the sensors in front of

the wave paddles and only the total registered waves are known for the rest of the sensors.

Starting from the calculated reflection coefficient in front of the wave paddle (mean value of the

3 wave gauges in front of wave paddle) obtained from the LASA method, the height of the

incident and reflected waves can be calculated using equations (3) and (4) starting from the total

measured wave height by the other wave gauges where the waves are breaking. The problem

with this method is that the assumption is very rough (it’s assumed that the reflection coefficient

in front of the wave paddle is equal for the whole canal). Of course this is not the case.

Comparison between SwanOne and measurement in canal without model

A comparison between the significant wave height obtained from SwanOne and the significant

wave height obtained from the measurements in the canal without the model is depicted in

Figure 23.

It can be stated that the values of the significant wave height obtained from SwanOne are

slightly higher than the ones obtained from the measured water elevations in the canal without

model.

For this master thesis, the wave characteristics are obtained from SwanOne.

Figure 23: Comparison between significant wave height obtained from SwanOne and measurements in the canal

without the model

0,00

5,00

10,00

15,00

20,00

0,00 5,00 10,00 15,00 20,00

Hm

0,

t (S

WA

N)[

cm]

Hm0,t (without model) [cm]

38

b. Porosity measurement

The porosity has to be measured for 2 purposes. First, the initial porosity has to be measured to

verify if this one is about 37% as prescribed (CERC, 1984). Also, the porosity after each test

should be measured, because this porosity is used to calculate the damage.

The porosity , equal to the ratio of the area of the voids and the total area , is calculated

counting the number of stones. The counting of the stones is done using the software AutoCAD.

First a virtual net is drawn over the picture of the breakwater. Different strips are drawn. This is

done because the Virtual Net Method requires this (which is used for other test in the ESCOLIF

project, but doesn’t has a real meaning for this master thesis). Then each stone is marked (Figure

24). Finally, by the command PRICAPAXYZ, developed by the LPC-UPV, the marks are counted

( ) and saved. Finally the porosity can be calculated:

(53)

Figure 24: Marked stones by AutoCAD

c. Damage calculation

The damage of the armour layer is calculated according to the visual counting method. This

method is discussed in the literature study and makes use of equations (31) and (32). The

number of eroded stones is calculated by counting the difference of stones between the initial

state and the state after wave attack. To do so, again AutoCAD is used as described above.

To do the qualitative analysis of the damage, the damage criteria are distinguished as follows:

· Initiation of damage (IDa): 5 or more units are displaced from their original position to

a new one at a distance equal to or larger than a unit length

· Initiation of Iribarren damage (IIDa): One stone and his surrounding stones of the 2nd

layer are visible

· Initiation of destruction (IDe): 2 or more stones in the lower armour layer are forced out

· Destruction (De): The filter layer is visible

39

d. Comparison measured damage with predicted damage

Once the damage is calculated, the measured damage will be compared with the predicted

damage. This predicted damage will be calculated using 2 approaches: The SPM-approach and

the VDM approach.

SPM-approach

The first approach is the one based on the SPM (CERC, 1984) by Van der Meer (1988) and Medina

et al. (1994):

Van der Meer (1988)

(54)

Medina et al. (1994)

(55)

All parameters are explained in the literature study. For the Van der Meer (1988) formula, the

value for the stability coefficient is taken to be equal to 8 because the goal is to predict the

damage so the value for an permeable core describing the main trend is chosen.

The damage will be predicted using both prediction formulas for 5 different wave heights. First

the significant wave height obtained from spectral analysis will be used because this is also the

wave height proposed by the authors of the formulas. But because the tests are performed in

breaking conditions, the damage will also be calculated using the breaking wave height .

There are different methods to calculate the breaking wave height for regular waves, but for

irregular waves the waves can break in a wide range as discussed before so there is not one right

value. Because of this, the breaking wave height is calculated in 4 different ways. The first 3 are

based on the method described in the Shore Protection Manual(CERC, 1984). For this method,

Figure 3 based on Goda (1970) can be used. It’s not clearly defined which period should be used

to calculate the deep water wave steepness, so the breaking wave height is calculated 3

times based on the mean absolute period , the mean energy wave period

and the peak period with all periods obtained from SwanOne. As 5th

used wave height, the wave height proposed by Goda (2000) for irregular conditions is used, i.e.

the peak value of the significant wave height in the surf zone which can be read from Figure

6.

Next the predicted and measured damage will be compared for each formula and each used

wave height. Furthermore 5 new formulas will be presented using the 5 different wave heights

based on all the tests except for the repetition test ( and ).

40

The new formula will be developed by calculating the values of the constant coefficients and

of:

(56)

The coefficients will be calculated using the Solver function in excel obtaining the lowest relative

mean squared error . The is used to evaluate the extent to which measured and

predicted damage are similar. It’s equal to:

(57)

Also the correlation coefficient will be calculated, but this is only a measure of the linear

correlation between the measured and predicted damage, so won’t be really used in the

discussion of the comparison between the predicted and measured damage..

The repetition test will be used to verify the new formulas as a blind test. In conclusion the best

of the 5 different formulas will be chosen to predict the damage.

No cumulative damage is taken into account because the formulas from the SPM-approach do

not include the number of waves. So it can be assumed that the damage of a lower wave height

which is already present before the initiation of the wave series with a higher wave height also

will be caused by a certain (lower) number of waves with the higher wave height than the

previous waves.

VDM-approach

The second approach is based on Van der Meer (1988) and Van Gent et al. (2004). The damage

for each test will be predicted (taking into account cumulative damage as discussed before) and

compared to the measured damage (the data points with a measured damage higher than 15

were deleted, because it doesn’t make sense to take them into account; destruction will already

have occurred at such a high value). This is also shown in Table 2. The prediction of the damage

will be done 3 times. Once according to the Van der Meer formula (in deep or shallow water

according to the conditions), once according to the Van der Meer formula modified by Van Gent

and finally once according to the formula by Van Gent. A summary of all formulas is given in

Table 12.

41

Table 12: Summary formulas VDM-approach

VDM Deep

water Plunging

(58)

VDM Deep

water Surging

(59)

VDM Shallow

water Plunging

(60)

VDM Shallow

water Surging

(61)

VDM

(mod by

VG)

Shallow

and deep

water

Plunging

(62)

VDM

(mod by

VG)

Shallow

water Surging

(63)

VG

Shallow

and deep

water

(64)

The 3 formulas will be compared using again the and in a smaller extent the correlation

coefficient and some improvements to the original formulas will be presented. Finally the

modified original formulas will be verified with a blind test.

42

Chapter 4: Results In this chapter the results of the comparison between the measured and predicted damage are

presented. The comparison is based on all test but the repetition test series. This repetition test

is used later as a blind test to verify the new and improved formulas.

The comparison is done for the 2 approaches as discussed before, i.e. The SPM-approach and

the VDM-approach.

To quantify the error between the measured and predicted damage, the relative mean squared

error ( is calculated for each case.

Finally a new formula or improvement to the original formula is proposed based on the

comparison between measured and predicted damage. The repetition test is used to verify these

new and improved formulas (blind test).

1. Wave data As mentioned before, the used wave data is obtained from SwanOne at the toe of the model

(unless otherwise mentioned). The most important parameters are listed in Table 13, Table 14

and

Table 15. All data from SwanOne at the toe of the structure is listed in Appendix A.

Table 13: Summary of the SwanOne data at the toe of the structure (

Theoretical values SwanOne (at toe of

structure) Theoretical values

SwanOne (at toe of

structure)

3

8 1.02 0.05 1 0.08

5

8 1.7 0.07 1.74 0.11

9 1.08 0.06 1 0.09 9 1.8 0.08 1.74 0.12

10 1.14 0.07 1.12 0.11 10 1.9 0.09 1.94 0.14

11 1.19 0.08 1.25 0.12 11 1.99 0.1 1.94 0.15

12 1.25 0.1 1.25 0.14 12 2.08 0.11 1.94 0.17

13 1.3 0.1 1.25 0.16 13 2.16 0.11 2.16 0.18

14 1.35 0.11 1.39 0.16 14 2.25 0.12 2.16 0.19

15 1.39 0.11 1.39 0.17 15 2.32 0.13 2.41 0.2

16 1.44 0.11 1.39 0.17

17 1.48 0.11 1.56 0.18

18 1.53 0.12 1.56 0.18

19 1.57 0.12 1.56 0.19

43

Table 14: Summary of the SwanOne data at the toe of the structure ( )


structure) Theoretical values

SwanOne (at toe of

structure)

3

8 1.02 0.06 1.12 0.08

5

8 1.7 0.07 1.74 0.1

9 1.08 0.06 1 0.09 9 1.8 0.08 1.74 0.12

10 1.14 0.07 1.12 0.11 10 1.9 0.1 1.94 0.14

11 1.19 0.09 1.25 0.14 11 1.99 0.11 1.94 0.16

12 1.25 0.1 1.25 0.14 12 2.08 0.12 1.94 0.17

13 1.3 0.11 1.39 0.17 13 2.16 0.12 2.16 0.18

14 1.35 0.12 1.39 0.17 14 2.25 0.14 2.16 0.21

15 1.39 0.12 1.39 0.18

16 1.44 0.13 1.39 0.2

17 1.48 0.13 1.56 0.2

Table 15: Summary of the SwanOne data at the toe of the structure ( )


structure)

3

8 1.02 0.06 1 0.08

9 1.08 0.07 1 0.09

10 1.14 0.07 1.12 0.1

11 1.19 0.09 1.25 0.13

12 1.25 0.1 1.25 0.14

13 1.3 0.11 1.25 0.16

14 1.35 0.12 1.39 0.17

15 1.39 0.13 1.39 0.19

16 1.44 0.14 1.39 0.2

44

2. Measured damage and porosity

The measured damages and porosities for the initial state and after wave series are listed in

Table 16 to Table 19

The initial porosity of each test series was 36% or 37%. So this is in line with the requirements

(CERC, 1984).

Also the different damage criteria are identified. This is based on the photos taken after each

wave series. The photos are listed in Appendix B. The average damage parameter for each

damage criteria is calculated and the following values are obtained:

· Initiation of damage:

· Initiation of Iribarren damage:

· Initiation of destruction:

· Destruction:

Table 16: Measured damage, porosity and qualitative damage analysis for

Q.A.

Q.A.

3

initial 0 37%

5

initial 0.0 37%

8 0.1 37% 8 0.4 37%

9 0.1 38% 9 0.9 38% IDa

10 0.2 38% 10 1.5 40% IIDa

11 0.5 38% IDa 11 2.6 42%

12 0.5 38% 12 3.4 44% IDe

13 1.1 40% IIDa 13 7.1 52%

14 2.0 41% 14 9.0 55%

15 4.1 46% 15 11.4 61% De

16 5.5 49% IDe

17 8.0 54% De

18 10.9 60%

45

Table 17: Measured damage, porosity and qualitative damage analysis at

Q.A.

Q.A.

3

initial 0.0 36%

5

initial 0 36%

8 0.0 36% 8 0.1 36%

9 0.0 36% 9 0.6 37% IDa

10 0.3 37% 10 0.9 38%

11 0.7 37% IDa 11 3.8 44% IIDa

12 1.7 40% 12 7.8 53% IDe

13 3.8 44% IIDa 13 13.4 64% De

14 5.0 47% IDe 14 21.6 82%

15 7.0 51%

16 11.0 59%

17 17.4 73% De

initial 0.0 36%

Table 18: Measured damage, porosity and qualitative damage analysis for and Ir=3

Q.A.

initial 0.0 36%

8 0.0 36%

9 0.0 36%

10 0.3 36%

11 0.8 37% IDa

12 1.3 38%

13 3.0 42% IIDa

14 6.7 50% IDe

15 10.6 58% De

16 21.0 80%

46

Table 19: Measured damage, porosity and qualitative damage analysis for the repetition test ( , Ir=3)

Q.A.

initial 0.0 36%

8 0.0 36%

9 0.1 36%

10 0.2 36%

11 0.6 37% IDa

12 0.9 38%

13 2.0 40%

14 2.6 41% IIDa

15 3.0 42%

16 4.7 46%

17 5.2 47% IDe

18 7.3 51%

19 10.5 58% De

3. Comparison measured damage with predicted damage

a. SPM-approach

The comparisons between the measured damage and the damage predicted by the Medina and

VDM formula using the 5 different wave heights are shown in Figure 25 to Figure 29. The

proposed new formulas are also depicted on each graph, and are presented in Table 20.

The verification of these new formulas is done using the repetition test (blind test) which is

depicted in Figure 30. The blue data points represent the comparison between the measured

damage and the predicted damage calculated using the Medina formula, the grey data points

the comparison between the measured damage and predicted damage by the VDM formula

while the orange data points represent the comparison between the measured damage and the

predicted damage according to the new formula. Of course this formula will give the best

comparison because the coefficients of this formula were chosen in such a way that the

measured and predicted damage agree in the best way.

47

Figure 25: Comparison formulas based on

Figure 26: Comparison formulas based on the (Goda,

2000)




Figure 30: Verification new formulas (blind test)

0,0

0,4

0,8

1,2

1,6

2,0

0,0 0,4 0,8 1,2 1,6 2,0

Pre

dic

ted

da

ma

ge

SS

PM

^X

[-]

Measured damage Sd^X[-]

Comparison Medina, VDM and New formula

based on Hm0 (SwanOne)

Medina(X=0.2)

VDM (X=0.15)

New Formula (X=0.14)

R=0.96

rMSE=0.18

R=0.97

rMSE=0.73

R=0.97

rMSE=0.070,0

0,4

0,8

1,2

1,6

2,0

0,0 0,4 0,8 1,2 1,6 2,0

Pre

dic

ted

da

ma

ge

SS

PM

^ [

-]



based on Hbr (Goda, 2000)

Medina (X=0.2)

VDM (X=0.15)

New formula (X=0.17)

R=0.92

rMSE=0.21

R=0.93

rMSE=0.25

R=0.99

rMSE=0.14

0,0

0,4

0,8

1,2

1,6

2,0

0,0 0,4 0,8 1,2 1,6 2,0

Pre

dic

ted

da

ma

ge

SS

PM

^X

[-]



based on Hbr (Tm) (CERC, 1984)

Medina (X=0.2)

VDM (X=0.15)


R=0.91

rMSE=0.34

R=0.92

rMSE=0.19

R=0.99

rMSE=0.170,0

0,4

0,8

1,2

1,6

2,0

0,0 0,4 0,8 1,2 1,6 2,0

Pre

dic

ted

da

ma

ge

SS

PM

^X

[-]



based on Hbr (Tm-1) (CERC, 1984)

Medina (X=0.2)

VDM (X=0.15)


R=0.92

rMSE=0.49

R=0.92

rMSE=0,19

R=0.99

rMSE=0,16

0,0

0,4

0,8

1,2

1,6

2,0

0,0 0,4 0,8 1,2 1,6 2,0

Pre

dic

ted

da

ma

ge

SS

PM

^X

[-]



based on Hbr (Tp) (CERC, 1984)

Medina (X=0.2)

VDM (X=0.15)

New Formula(X=0.17)

R=0.92

rMSE=0.91

R=0.93

rMSE=0.33

R=1.00

rMSE=0.14

0,5

1,0

1,5

2,0

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

Pre

dic

ted

da

ma

ge

SS

PM

^X

[-]


Verification new formulas by repeated test

Hm0 SWAN

Hb (Tm), SPM

Hb (Tm-1), SPM

Hb (Tp), SPM

Hb, Goda

rMSE=0.05

rMSE=0.10

rMSE=0.12

rMSE=0.15

rMSE=0.16

48

Table 20: Proposed new formulas based on SPM-approach

Used wave height New formula

(65)

(66)

based on (67)

based on (68)

based on (69)

The of all formulas using all the different wave height are given in Table 21. It can be

concluded that for the existing formulas the Medina formula using the significant wave height

is the best formula to predict the damage ( . Also the VDM formula using

breaking wave height gives a low , which is equal to 0.19 using both or .

The best fitting proposed new formula is equation (65) using . The equals 0.07. All

proposed new formulas were verified by a blind test (Figure 30), and it can be concluded that

also here, the best fitting formula is equation (65) with a of 0.05.

Table 21: of all formulas

VDM Medina New Formula Blind test

SwanOne 0.73 0.18 0.07 0.05

CERC (1984)

0.19 0.34 0.17 0.10

0.19 0.49 0.16 0.12

0.33 0.91 0.14 0.15

Goda (2000) 0.25 0.21 0.14 0.16

49

b. VDM-approach

The comparison between the predicted and the measured damage calculated by the original

Van der Meer formula is depicted in Figure 31. As can be seen on the figure, it’s clear that all

predicted damage is lower than the measured damage. It can also be observed that the lower

the Iribarren number and the higher the water depth, the bigger the difference between the

measured and the predicted damage. That is why the original formula will be improved by

introducing the water depth at the toe of the structure and the peak period which

influences the Iribarren number. All waves are plunging waves so only the formulas for plunging

waves can be improved. The power to which the new dimensionless parameter is raised and the

value of the constant are calculated using the solver function in excel obtaining the smallest

:

For shallow water:

(70)

For deep water:

(71)

By introducing the dimensionless parameter that takes into account the influence of the water

depth and Iribarren number, the decreases from 0.77 to 0.09 which is a very big

improvement (Figure 32). Also the correlation coefficient improves. The interval in which 90%

of the values are located is also indicated on Figure 32.

Figure 31: Comparison measured and predicted damage

by formula of VDM


by improved formula of VDM

The same improvement was executed for the VDM formula modified by VG. The same

observations and conclusions can be made: Almost everywhere the measured damage is higher

than the predicted damage, and the lower the Iribarren number & higher the water depth, the

higher the difference between measured and predicted damage (Figure 33). Again the same

improvement is made by introducing a dimensionless parameter.

0

3

6

9

12

15

0 3 6 9 12 15

Pre

dic

ted

da

ma

ge

Sd

[-]


Comparison VDM formula

hs=20,Ir=5

hs=30,Ir=5

hs=20,Ir=3

hs=30,Ir=3

hs=40,Ir=3

R=0.78

rMSE=0.77

0

3

6

9

12

15

0 3 6 9 12 15

Pre

dic

ted

da

ma

ge

Sd

[-]


Comparison improved VDM formula

hs=20,Ir=5

hs=30,Ir=3

hs=40,Ir=3

hs=20,Ir=3

hs=30,Ir=5

R=0.97

rMSE=0.09

50

(72)

Again the comparison is improved by introducing the dimensionless parameter. The

decreases from 0.49 to 0.09. The interval in which 90% of the values are located is indicated on

the figure.


by formula of VDM modified by VG


by improved formula of VDM modified by VG

If the comparison between the measured and predicted damage by the VG formula is observed,

there can’t be really made a distinction between the different test series. The only conclusion

that can be made is that the predicted damage is almost always lower than the measured

damage (Figure 35). That’s the reason why only the constant coefficient is changed:

(73)

By changing the coefficient from 0.57 to 0.67, the results are improved from a of 0.59 to

0.12 (Figure 36). The interval in which 90% of the values are located is indicated on the figure.

0

4

8

12

16

0 4 8 12 16

Pre

dic

ted

da

ma

ge

Sd

[-]


Comparison VDM formula (modified by VG)

hs=20,Ir=5

hs=30,Ir=5

hs=20,Ir=3

hs=30,Ir=3

hs=40,Ir=3

R=0.77

rMSE=0.49

0

3

6

9

12

15

0 3 6 9 12 15

Pre

dic

ted

da

ma

ge

Sd

[-]


Comparison improved VDM formula (modified

by VG)

hs=20,Ir=5

hs=40,Ir=3

hs=30,Ir=5

hs=20,Ir=3

hs=30,Ir=3

R=0.97

rMSE=0.09

51


by formula of VG


by the improved formula of VG

Finally, the improved formulas of VDM, VDM modified by VG and VG can be verified by a ‘blind

test’ using the repetition test. The comparison between the measured damage and the

predicted damage by the 3 new formulas is given in Figure 37. To make the improvement clearer

also the predicted values by the original formulas are depicted in the graph.

Figure 37: Verification new formulas

It can be concluded that the new formulas give a good improvement of the original formulas.

0

3

6

9

12

15

0 3 6 9 12 15

Pre

dic

ted

da

ma

ge

Sd

[-]


Comparison VG formula

hs=20,Ir=3

hs=30,Ir=3

hs=30,Ir=5

hs=40,Ir=3

hs=20,Ir=5

R=0.94

rMSE=0.59

0

3

6

9

12

15

0 3 6 9 12 15

Pre

dic

ted

da

ma

ge

Sd

[-]


Comparison improved VG formula

hs=20,Ir=5

hs=40,Ir=3

hs=30,Ir=5

hs=30,Ir=3

hs=20,Ir=3

R=0.94

rMSE=0.12

0

2

4

6

8

10

0 2 4 6 8 10

Pre

dic

ted

da

ma

ge

Sd

[-]

Measured damage S[-]

Comparison of old and new formulas with

repeated test

VDM

VDM (mod by VG)

VG

Improved VDM

Improved VDM

(mod by VG)

Improved VG

52

Chapter 5: Conclusions The aim of this thesis was to compare and improve different prediction formulas. 2 different

approaches were used: the approach based on the SPM (CERC, 1984) and the approach based

on Van der Meer (1988).

First of all, a qualitative analysis was executed determining the Initiation of damage, Initiation

of Iribarren damage, Initiation of destruction and destruction. The average values based on the

tests are given in Table 22.

Table 22: Qualitative damage analysis

Initiation of damage

Initiation of Iribarren damage

Initiation of destruction

Destruction

If the (deep water) wave characteristics are not known, these can be obtained using SwanOne.

The best formula to use these wave characteristics based on the executed tests is the new

proposed formula using the significant wave height :

If the deep water characteristics are known, the breaking wave height can be calculated and

used to predict the damage according to the SPM approach using the new proposed formula

with the mean period :

Also the new proposed formula using the peak value of the significant wave height in the surf

zone can be used if the deep water characteristics are known:

Using the VDM formula, the breaking wave height based on or gives the best prediction

of the damage while using the Medina formula the significant wave height at the toe of the

structure obtained by SwanOne gives the best approximation.

For the approach based on Van der Meer (1988), the improved VDM formula (deep & shallow

water) and improved VDM formula modified by VG for plunging waves and improved VG formula

are given respectively by:

53

The red parameters are the introduced ones to improve the prediction. If only a few wave

characteristics are known (no wave period, ,..), the improved VG formula can be used. In

other cases, both the improved VDM formula and improved VDM formula by VG will give the

best approximation.

A really important remark should be made. The qualitative analysis, all improvements and new

proposed formulas are based only a few test series. All formulas and qualitative analysis should

be verified and tested with other test series and more repetition tests (also in other

laboratories). These repetition tests are necessary because damage is a very sensitive

parameter. The difference in measured damage of 2 equal tests can be 30% (Van der Meer,

2000).

Other future research can be done by changing the parameters, for instance the nominal armour

diameter, structure slope, foreshore slope,…

The conditions could also be changed in such a way so surging waves occur (instead of plunging

waves as in these conditions).

Finally also other armour units should be investigated.

54

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59

Appendix A: Summary data SwanOne

60

Appendix B: Photos model after wave

action

hs=20cm and Ir=3

hs=20cm and Ir=5

61

hs=30cm and Ir=3

62

hs=30cm and Ir=5

hs=40cm and Ir=3

63

hs=20cm and Ir=3 (repetition test)

64



prediction formulas









Ghent University


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