numerical modelling of wave overtopping of cubipod armored

UntitledBrecht De Vos
Prof. dr. ir. Andreas Kortenhaus
External 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
Chair: Prof. Josep R. Medina
Departamento de Ingeniería e Infraestructura de los Transportes
Universidad Politecnica de Valencia
Department of Civil Engineering
Faculty of Engineering and Architecture
Ghent University
Preface
I would like to thank Prof. Troch for placing me in contact with Prof. Medina, for
making the exchange possible and assisting me whenever I had questions.
I would like to thank Prof. Medina for allowing me to participate in the ESBECO project
currently being conducted in the Laboratory of Ports and Coasts at the Polytechnic
University of Valencia. Also his guidance during the writing of this thesis was greatly
appreciated.
Special thanks to Mapi, Gloria, Jorge and Pepe for their help with the experiments, for
sharing their knowledge while making this thesis and creating a very pleasant
atmosphere in the laboratory.
Finally, I want to express my gratitude to my parents for allowing me to seize this
opportunity to have an international experience. Of course, I can’t forget to thank my
entire family for their endless support.
Departamento de Ingeniería e Infraestructura de los Transportes.
ETSI Caminos, Canales y Puertos (ETSICCP)
Content
2.1 Low-crested breakwaters ....................................................................................... 2
2.2 Breaking conditions ................................................................................................ 3
2.4.1 Rock armour units .......................................................................................... 10
2.4.2 Artificial units ................................................................................................. 21
2.6 Overtopping .......................................................................................................... 32
Chapter 3 Experimental setup ........................................................................................ 47
3.1 Equipment for experiments .................................................................................. 47
3.1.1 Wave flume .................................................................................................... 47
3.1.4 Overtopping measurements .......................................................................... 49
3.2 Experiments .......................................................................................................... 52
3.2.1 Design ............................................................................................................. 52
3.2.2 Construction ................................................................................................... 53
3.2.4 Tests without breakwater model (regular and irregular) .............................. 55
3.2.5 Tests with breakwater model (regular and irregular) .................................... 55
3.3 Data processing ..................................................................................................... 60
3.3.2 Analysis of the waves ..................................................................................... 61
3.3.3 Porosity & damage ......................................................................................... 61
4.2.3 Regular wave tests ......................................................................................... 78
Chapter 5 Conclusion ...................................................................................................... 82
Chapter 6 References ..................................................................................................... 84
Annex C: Mean overtopping discharges ..................................................................... 93
List of figures
Figure 3: Failure modes .................................................................................................... 5
Figure 4: Heterogeneous packing ..................................................................................... 6
Figure 5: Cubipod armour unit ......................................................................................... 7
Figure 6: Transition heights (van der Meer, 1990) ......................................................... 13
Figure 7: Reduction factor rD (van der Meer, 1990) ....................................................... 14
Figure 8: Breakwater geometry (Vidal et al., 1995) ....................................................... 15
Figure 9: Stability graphs for different sections at start of damage (Burger, 1995) ...... 17
Figure 10: Cross-section according to Burger (1995) ..................................................... 18
Figure 11: Detail roundhead layout (Kramer and Burcharth, 2004) .............................. 19
Figure 12: Assembled data (Kramer and Burcharth, 2004) ............................................ 20
Figure 13: Cross-section breakwater (Muttray et al., 2012) .......................................... 21
Figure 14: Overview of displaced armour units (all tests) (Muttray et al., 2012) .......... 22
Figure 15: Stability number Ns at start of damage: overall damage (top left), seaward
slope (top right), crest (bottom left) and rear slope (bottom right) (Muttray et al.,
2012) ............................................................................................................................... 24
Figure 16: Stability number Ns at start of damage (Muttray et al., 2012) ..................... 25
Figure 17: Influence of toe berm on double layered breakwaters (Vanhoutte, 2009) .. 27
Figure 18: Influence of toe berm on single layered breakwaters (Vanhoutte, 2009) .... 27
Figure 19: Linearized dimensionless damage using qualitative KD (Vanhoutte, 2009) .. 28
Figure 20: Linearized dimensionless damage using quantitative KD (Vanhoutte, 2009) 28
Figure 21: Linearized equivalent dimensionless armour damage in function of stability
number (Gómez-Martín, 2015) ...................................................................................... 29
Figure 22: Measured stability numbers of double-layer cube and Cubipod armour
layers (Gómez-Martín, 2015).......................................................................................... 30
Figure 23: Measured stability numbers of single- and double-layer Cubipod armour
layers (Gómez-Martín, 2015).......................................................................................... 31
Figure 24: Conventional cross-section with parameters ................................................ 32
Figure 25: Design graph (Goda, 1985) ............................................................................ 33
Figure 26: Comparison of EurOtop (2007) formula with EurOtop II (2016) formula ..... 36
Figure 27: Cross section (Gómez-Martín and Medina, 2008) ........................................ 36
Figure 28: Measured versus estimated overtopping rates (Gómez-Martín, 2008) ....... 37
Figure 29: IH-2VOF Preprocessing interface .................................................................. 42
Figure 30: New geometry and mesh with Coral ............................................................. 43
Figure 31: Calculation progress ...................................................................................... 43
Figure 32: Wave gauge registrations .............................................................................. 44
Figure 33: Overtopping analysis ..................................................................................... 45
Figure 34: Visualisation of the model ............................................................................. 45
Figure 35: 2D Wave flume .............................................................................................. 47
Figure 36: Wave generator ............................................................................................. 48
Figure 37: Wave Paddle .................................................................................................. 48
Figure 38: Piston ............................................................................................................. 49
Figure 39: Servomotor .................................................................................................... 49
Figure 42: Wave gauge locations near wave paddle ...................................................... 50
Figure 43: Wave gauge locations near breakwater model ............................................ 51
Figure 44: Video camera locations ................................................................................. 51
Figure 45: Breakwater cross-section .............................................................................. 52
Figure 46: Drawn cross-section ...................................................................................... 54
Figure 47: Placement of the core ................................................................................... 54
Figure 48: Placement of the filter layer .......................................................................... 54
Figure 49: Placed armour units ...................................................................................... 54
Figure 50: Top view armour layer .................................................................................. 54
Figure 51: LASA-V software ............................................................................................ 60
Figure 52: LPCLab 3.7.1................................................................................................... 61
Figure 55: Wave height comparison at the toe .............................................................. 66
Figure 56: H*m,inc at the toe in function of Hm,inc at deep water (for hs=20 cm) ............ 67
Figure 57: H*m,inc at the toe in function of Hm,inc at deep water (for hs=25 cm) ............ 67
Figure 58: Mean wave overtopping discharge q in function of relative freeboard /, ∗ ................................................................................................................. 68
Figure 59: Mean wave overtopping discharge q in function of incident mean wave
height at the toe , ∗ ............................................................................................ 69
Figure 60: Mean wave overtopping discharge q in function of mean wave period Tm . 69
Figure 61: Hm and Tm comparison - Sensor at deep water ............................................. 71
Figure 62: Hm and Tm comparison - Sensor 5 ................................................................. 71
Figure 63: Hm and Tm comparison - Sensor 10 ............................................................... 72
Figure 66: Influence of β ................................................................................................. 74
Figure 67: Influence of α ................................................................................................. 75
Figure 68: Influence of porosity ..................................................................................... 76
Figure 69: Overtopping comparison of case E with laboratory...................................... 77
Figure 70: Mean overtopping discharge comparison for hs=20cm ............................... 80
Figure 71: Mean overtopping discharge comparison for hs=25cm ............................... 80
Figure 72: Comparison of overtopping for all tests ........................................................ 81
List of tables
Table 1: Breaker classification according to Galvin (1968) .............................................. 3
Table 2: Empirical coefficients a and b by Allsop (1983) ................................................ 10
Table 3: Summary of test conditions (van der Meer, 1998b) ........................................ 12
Table 4: Damage levels with damage level parameter Sd according to Vidal et al., 1995.
........................................................................................................................................ 16
Table 5: Coefficients by Vidal et al. (1995) ..................................................................... 16
Table 6: Summary of test conditions (Vidal et al., 1995) ............................................... 16
Table 7: Different types of tests (Burger, 1995) ............................................................. 18
Table 8: Summary of conditions by Kramer and Burcharth (2004) ................................ 19
Table 9: Summary of conditions by Muttray et al. (2012) ............................................. 22
Table 10: Incident wave heights resulting in damage levels (Vanhoutte, 2009) ........... 26
Table 11: Coefficients a and b for straight slopes (Owen, 1980) ................................... 33
Table 12: Tolerable overtopping according to Eurotop II (Van der Meer et al., 2016) .. 39
Table 13: Regular wave characteristics hs=20cm (20 waves per test) ........................... 56
Table 14: Regular wave characteristics hs=25cm (20 waves per test) ........................... 57
Table 15: Irregular wave characteristics (1000 waves per test) .................................... 58
Table 16: Mesh characteristics ....................................................................................... 62
Table 17: Initial material characteristics ........................................................................ 64
Table 18: Porosity measurements .................................................................................. 64
Table 19: Wave characteristics of cases without model ................................................ 70
Table 20: Initial characteristics ....................................................................................... 74
Table 21: Average overtopping discharges .................................................................... 75
Table 22: Average overtopping discharges .................................................................... 75
Table 23: Porosity combinations for calibration ............................................................ 76
Table 24: Resulting overtopping discharges ................................................................... 77
Table 25: Wave parameters introduced into the IH-2VOF model for hs=20cm ............ 78
Table 26: Wave parameters introduced into the IH-2VOF model for hs=25cm ............ 79
1
Chapter 1 Introduction
This thesis was made during the start of a project called ESBECO which is an acronym
for “EStabilidad hidráulica del manto, BErma y COronacion de dique en talud con
rebase y rotura por fondo”. The translation of this title is “Hydraulic stability of armour
layer, toe berm and crest of breakwaters with overtopping and breaking wave
conditions”. Breakwaters with a significant amount of overtopping in breaking wave
conditions are not abundantly studied in literature. The majority of design rules and
formulae are based on small-scale experiments, conducted in non-breaking wave
conditions and either negligible overtopping or enormous overtopping (low-crested
structures). Nowadays, breakwaters in breaking or partially breaking conditions (Hs >
0.4 h) and with relevant overtopping (0.5 < Rc/Hs < 1.0) are becoming important due to
the desire of reduced visual impact and the imminent consequences of climate change.
Both overtopping and hydraulic stability will be investigated during the project.
The objective of this thesis was to provide a numerical model that represents the wave
flume in the laboratory and that is able to achieve comparable results obtained from
2D small scale physical tests.
In the second chapter a literature study is presented of the different aspects related to
the subject. Breaking wave conditions, damage quantification, hydraulic stability of
low-crested structures and Cubipod armour units, overtopping formulae and the
numerical software IH-2VOF will be discussed.
The experimental setup, characteristics of the experiments and the methodology to
analyse the data will be elaborated in chapter three.
In chapter four, the results from the tests conducted in the laboratory and executed
with the software will be compared and discussed.
Finally, in chapter five a conclusion will be formulated about the obtained findings and
recommendations for future research will be given.
2
Low-crested structures differ from conventional breakwaters in that way that they
allow some or even severe overtopping. Hydraulic stability of armour units on the
seaward slope can be higher than for non-overtopped structures since a lot of wave
energy can pass over the structure. Armour units on the crest and the rear slope,
however, are subjected to wave attack resulting in a lower hydraulic stability.
Three types of low-crested breakwater can be distinguished according to Van der Meer
and Pilarczyk (1990). Dynamically stable reef breakwaters are mounds of stones
without core or filter layer which are allowed to reshape when subjected to wave
attack. Statically stable low-crested breakwaters have a smaller freeboard compared
to conventional breakwaters and are not allowed to undergo changes of the cross-
section. Finally, statically stable submerged structures have a crest below the still
water level. In this thesis, only the second category is dealt with so the other types are
not discussed. (Figure 1)
3
2.2 Breaking conditions
When waves propagate from deep to shallow waters, various transformations that can
occur are refraction, shoaling, diffraction, dissipation due to friction & percolation,
breaking, additional growth due to the wind, wave-current interaction and wave-wave
interaction.
The surf zone can be defined as the region between the seaward boundary of wave
breaking and the limit of wave uprush. In this zone, wave breaking is the dominant
hydrodynamic process.
Waves approaching the coast experience a decrease in wave length L and possibly an
increase in height H, causing the steepness H/L to increase with decreasing water
depth. Waves break as they reach a limiting steepness, which is a function of the
relative depth d/L and beach slope tanβ.
Galvin (1968) proposed four types of breaking waves: spilling, plunging, collapsing and
surging waves. The wave may be classified in one of these groups in function of the
breaker or surf similarity parameter, ξ0, defined as:
=
Eq. 1
In this equation, α represents the angle of the beach or bottom, H0 denotes the wave
height in deep water and L0 is the wave length in deep water conditions. The surf
similarity parameter can be seen as the ratio between the steepness of the bottom
slope and the wave steepness.
Table 1: Breaker classification according to Galvin (1968)
Breaker type Surf similarity parameter range
Spilling ξ < 0.5
Collapsing/Surging 3.3 > ξ
In spilling breakers, the wave crest becomes unstable and runs down the shoreward
face of the wave, creating a foamy water surface. This tends to occur for waves with a
4
high steepness on gently sloping beaches. Plunging breakers occur on steeper beaches
with intermediately steep waves. The crest curls over the shoreward face of the wave
and plunges into the base of the wave which results in a large splash. Finally, surging
and collapsing breakers occur for waves with a low steepness on steep beaches. A
visual distinction can be made between both types which enables a more precise
classification. In collapsing breakers, the crest remains unbroken while the lower part
of the face steepens and collapses. In surging breakers, the crest remains unbroken
and the front face advances up the beach with minimal foaming or breaking. (USACE,
2006)
5
2.3 Damage
Breakwaters are the most commonly used structures to protect coastal areas or port
entrances. They dissipate wave energy by friction, wave breaking and partly wave
transmission.
2.3.1 Failure modes
During the design of a breakwater, all failure modes need to be taken into account.
Eleven different failure mechanisms were distinguished by Bruun (1979). They are
drawn in Figure 3 and are the following:
• Rocking of the armour units
• Loss of armour units which results in increased porosity
• Sliding of the armour layer due to lack of friction with underlying layers
• Breaking of the armour units caused by the wave impact, either the structural
resistance was exceeded or the unit was slammed into another unit
• Undermining the crown wall
• Damage of the inner slope by wave overtopping
• Excessive transmission of energy to the interior of the breakwater due to the
lack of compacted underlying layers
• Erosion of the breakwater toe
• Settlement or collapsing of the subsoil
• Loss of characteristics of the used materials
• Mistakes in the construction
Figure 3: Failure modes
Bruun (1979) proposed four different groups of failure mechanisms for breakwaters:
• Hydrodynamic stability of armour units
• Structural integrity of armour units
• Geotechnical stability of different components of the structure
• Constructions mistakes
Erosion of the armour layer can be caused by unit extractions, sliding of the entire
armour layer or heterogeneous packing.
Unit extraction exposes the under-layer. Armour damage is often quantified by the
amount of units that are extracted from the armour layer. Armour sliding as a whole is
usually related to steep slopes and low friction with the under-layer.
Cubes are difficult to place randomly in a breakwater with two armour layers. Under
wave attack, they tend to position themselves face-to-face without any unit extraction.
Due to gravity, the porosity in the lower part of the armour layer is reduced which
then causes a higher porosity in the upper part. This results in an exposure of the layer
underneath which is then susceptible to extraction. This process was called the
Heterogeneous Packing failure mode (Figure 4) (Gómez-Martín, 2007).
Figure 4: Heterogeneous packing
Gomez-Martin and Medina (2007) designed a new armour unit, the Cubipod (Figure 5).
It is a massive concrete unit with protuberances on each face. Their function is to
prevent face-to-face arrangements, separate adjacent units and increase the friction
with the under-layer. Tests show a significant increase in hydraulic stability compared
to those with cubic units. Wave run-up and overtopping are also reduced, the friction
7
with the under-layer is increased and the units tend to position themselves randomly
over time.
2.3.2 Measuring damage
Damage can be calculated by profiling the armour layer or by counting the number of
displaced units. Various definitions exist for the displacement of an armour unit. The
Shore Protection Manual (USACE, 1984) and Coastal Engineering Manual (USACE,
2006) define the percentage of damage, D%, as the ratio of the number of displaced
units and the total number of units in a predetermined area. Van der Meer (1988a)
proposed the relative damage number, Nod, which represents the number of units
displaced out of the armour layer within a vertical strip with width Dn. Broderick (1983)
suggested the dimensionless armour damage parameter, S, which is equal to the ratio
of the average eroded cross sectional area, Ae, and the equivalent cube area, Dn 2 (with
Dn=(M/ρr) 1/3
Aside from these quantitative armour damage assessments also qualitative
assessments exist. Vidal (1995) used four criteria to visually assess armour damage
after each test:
• Initiation of damage: When a certain number of armour units are displaced a
distance equal to or larger than the unit nominal diameter from their original
position. The outer armour layer displays large holes.
• Iribarren’s damage: When the extent of the failure area of the outer armour
layer is large enough for degrees of waves to act directly on the lower armour
layer, with its unit susceptible to being displaced.
• Start of destruction: Defined as the initiation of damage of the lower armour
layer, whereby a number of units of the inner armour layer are displaced,
causing large holes.
8
• Destruction: Material from the secondary (or filter) layer is removed. If the
wave conditions do not decrease, the breakwater will not be able to provide
the necessary protection.
Quantitative assessments may lead to objective values for damage, but they lack the
information of the severity of damage as this depends on the spatial distribution of
damage on the armour layer. This is exactly the advantage of qualitative assessments
so using a combination of both would be beneficial.
Gómez-Martín and Medina (2006) proposed to the use of the Virtual Net Method. A
grid is drawn over a photograph of the damaged armour layer. It is divided in a number
of strips with length a=m*Dn and constant width b=k*Dn. The amount of armour units
whose center is inside each strip i (Ni) is counted and the porosity nvi of strip i is
calculated according to equation x:
= 1 − = 1 − ! Eq. 2
With the porosity nvi, the dimensionless armour damage Si can be calculated:
∀ = 1 − 1 − 1 − = − 1 − Eq. 3
in which nv0i is the initial porosity in strip i.
When these values for dimensionless armour damage are integrated for all the strips i,
the equivalent dimensionless armour damage Se is obtained:
∀# = ∃∀ %
2.4 Hydraulic stability of low-crested structures
Various experiments have been conducted in order to develop formulae concerning
hydraulic stability of low-crested structures. Originally, rock units were used to armour
the slopes of low-crested breakwaters. In the recent past, also concrete units found
their application in low-crested structures. The evolution of formulae for both will be
described in this chapter.
According to the Coastal Engineering Manual (CEM) (USACE, 2006), a variety of
parameters is necessary to quantify hydraulic stability of armour layers in breakwaters.
Two categories can be distinguished.
The first category is related to the state of the sea. Wave conditions are usually
defined by incident wave height at the toe of the structure, H, for which often the
significant wave height, H1/3 (average wave height of the 1/3 highest waves in time
domain) or Hm0 (extracted from the frequency domain), is used; the mean or peak
wave periods, Tm or Tp; water depth, h; angle of incident wave, β; number of waves, N
and the density of water, ρw.
The second category consists of parameters related to the geometry of the structure:
the mass density of armour units, ρs; grading of rock armour Dn50, Dn15 and Dn85 or
nominal diameter of concrete armour units Dn; armour layer slope angles, α; crest
freeboard, Rc; permeability of the armour layer, P; unit placement pattern and packing
density.
An important parameter with respect to the hydraulic stability is the stability number
Ns because it relates the structure to the wave conditions. It is defined as:
N− = .Δ Eq. 5
2.4.1 Rock armour units
One of the first formulae to design a rubble mound breakwater was developed by
Hudson (1959): 0123456 = 7389:; < =⁄ Eq. 6
In which Hs represents the significant wave height, Dn50 the nominal diameter and Δ
the relative buoyant density and KD a stability coefficient. The suggested values for KD
correspond to a condition without damage that allows 5% of the armour units to be
displaced. The Shore Protection Manual (USACE, 1973) proposed KD = 3.5 for breaking
waves and KD = 4.0 for non-breaking waves for angular stones and significant wave
height Hs. In the version 1984 some changes were made. For breaking waves, KD = 2.0
was proposed and H1/10, the average of the highest ten percent of the wave heights, as
design wave height.
Powell and Allsop (1985) analysed data from Allsop (1983) and a formula was
proposed to relate the stability number to the damage level:
013456 = 19≅</=Α Β4<ΧD9ΕDΧ Eq. 7
Empirical coefficients a and b are given in Table 2 as functions of freeboard Rc and
water depth h, deduced by exponential regression analysis to fit curves of the form
A·exp (B.Ns)
. Furthermore, Nod is the number of armour units displaced out of the armour
layer per width Dn50 across the armour face, Na is the total number of armour units in
that same area and sop is wave steepness defined as Hs/Lop.
Table 2: Empirical coefficients a and b by Allsop (1983)
Rc/h a b sop
11
Van der Meer (1988b) developed stability formulae for non-overtopped structures
based on many wave flume tests.
For plunging waves (ξm < ξmc):
.Φ = 6.2 ∀√. Κ.Λ Μ.Φ Eq. 8
For surging waves (ξm > ξmc):
.Φ = 1.0 ∀√. Κ.Ν√ Ο ΜΠ Eq. 9
With:
Where:
HS = significant wave height at the toe of the structure
Validity of the equation:
1) Both equations above are valid for non-depth limited waves. Hs should be
replaced by H2%/1.4 for the case of depth-limited waves.
2) The number of waves N should be smaller than or equal to 7500. Equilibrium
occurs when this amount of waves is surpassed.
3) 0.005 ≤ som ≤ 0.06; 2000 kg/m 3 ≤ r ≤ 3100 kg/m
3 ; 0.1 ≤ P≤ 0.6
4) For cot α ≥ 4, the equation for plunging waves should be used.
12
There is uncertainty due to randomness of the rock slopes and inaccuracy of damage
measurements. Therefore, the coefficients in the equations are given a normal
distribution with a standard deviation. The coefficients of variation on factors 6.2 and
1.0 are respectively 6.5% and 8% (or, respectively, a standard deviation of 0.4 and
0.08).
Van der Meer (1988b) also investigated low-crested structures to compare directly
with the non-overtopped breakwaters. Tests were conducted in a wave flume of 1.0 m
wide, 1.2 m deep and 50.0 m long. A foreshore of 1:30 was constructed in front of the
breakwater. Reflected waves were measured and compensated at the wave board by a
specially developed system. The incident significant wave height was registered using
two wave gauges. The surface profile was measured with nine gauges placed on a
computer-controlled carriage in order to determine the erosion damage, S. The front
slope of the breakwater was 1:2 and the crest width was equal to 8 Dn50.
A test series consisted of five tests with the same wave period but with different
significant wave heights. Wave heights ranged from 0.05 m to 0.25 m and wave
periods from 1.3 s to 3.2 s. Each test started with a pre-test sounding, a test of 1000
waves, an intermediate sounding, 2000 more and a final sounding.
Table 3: Summary of test conditions (van der Meer, 1998b)
Parameter Symbol Range
Mean nominal diameter of armour [cm] Dn50 3.44
Mean nominal diameter of core [cm] Dn50,core 1.1
Relative buoyant density Δ 1.61
Number of waves N 1000 - 3000
Fictitious wave steepness sop 0.010 - 0.036
Non-dimensional freeboard Rc/Dn50 -2.9 - 3.6
Non-dimensional crest width B/Dn50 8.0
Non-dimensional structure height d/Dn50 9 - 15
Stability number Hs/(ΔDn50) 1.4 - 4.0
13
The stability of the front slope of a low-crested emergent structure can be related to
the stability of a non-overtopped structure by applying a reduction factor rD on the
nominal diameter Dn50. Van der Meer (1990) used data sets from Ahrens (1987),
Powell and Allsop (1985) and van der Meer (1988b) to do so.
Figure 6: Transition heights (van der Meer, 1990)
A decrease in structure height results in an increase in stability. A long period gives
higher run-up on a slope than a short period. Therefore more energy is dissipated by
overtopping for a long period at the same crest height as for a short period. The
transition height where the increase in stability starts should be a function of the wave
period (or wave steepness). These transition heights were extracted from the
aforementioned datasets (Figure 6).
W801 = 6. <= 19≅6.5 Eq. 10
Powell and Allsop (1985) introduced the dimensionless crest height Rp * as:
W≅∗ = W801 Ξ19≅ΨΖ Eq. 11
Equations 10 and 11 combined give a simple transition crest height Rp * = 0.052. If the
average increase in stability for a structure with the crest at the water level, in
comparison with a non-overtopped structure, is set at 25%, independent of the wave
14
* =
0, the increase in stability can be described as a function of Rp * only. And when the
reduction in required nominal diameter Dn50 is taken as a measure instead of the
increase in stability, the final equation becomes:
[3 = <<. Ψ5 − ∴. ]W≅∗
For 6 < ≅∗ < 0.052
Eq. 12
Equation 12 is shown in Figure 7. The suggested reduction factor to apply on the
required Dn50 of the equivalent non-overtopped structure can be read from this graph.
An average reduction factor of 0.8 in diameter is obtained for a structure with the
crest at still water level. This results in a required mass of 51% of that required for a
non-overtopped structure.
15
Vidal et al. (1995) proposed a stability formula for both emergent and submerged low-
crested structures. The stability of the roundheads is not discussed.
Figure 8: Breakwater geometry (Vidal et al., 1995)
The wave basin was 37 m long and 14 m wide. A 1:15 sloped gravel beach was built for
wave energy dissipation, the rest of basin had a horizontal bottom. Water surface
elevations were measured at 11 different locations. The cross section was composed
of a permeable core armoured with two layers of rocks. The slopes were 1:1.5 and the
crest width was equal to six times Dn50. Tests were run with irregular waves using
Jonswap spectra with a peak enhancement factor of γ = 3.3. Peak periods Tp of 1.4 s
and 1.8 s were used for varying significant wave heights Hm0.
The trunk armour layer was divided into four sections (front slope, back slope, crest
and total section) as can be seen in Figure 8 by installing a steel frame. A steel wire
mesh was used to cover the parts of the armour layer where damage would not be
measured. As a result, damage interaction between different sections was prevented
and not the entire structure needed to be rebuilt.
Four different damage levels, which were defined in the previous chapter, were used
and approximated by damage level parameter Sd which is displayed in Table 4:
16
Table 4: Damage levels with damage level parameter Sd according to Vidal et al.,
1995.
Initiation of
Iribarren’s
Start of
Destruction 9.0 10.0 - 12.0
Vidal et al. (1995) proposed formulae of the following shape:
013456 = + β W83456 + χ W83456 Ψ
Eq. 13
For the different sections, following coefficients for initiation of damage were found to
give the best results:
Segment A B C
Crest 1.652 0.0182 0.1590
Table 6: Summary of test conditions (Vidal et al., 1995)
Number of waves N 2600 - 3000
17
Non-dimensional structure height d/Dn50 16 - 24
Burger (1995) reanalysed the data of both Vidal et al. (1995) and Van der Meer
(1988b). The different sections of the breakwater were front slope, crest, rear-side
slope and the total section. For each section, a graph was fitted through both data sets
at start of damage. The data was extrapolated outside of the range of the test data so
these graphs should be used with caution (Figure 9).
Figure 9: Stability graphs for different sections at start of damage (Burger, 1995)
The rear-side slope is never the least stable section. For positive freeboards the front
slope is the critical section while for submerged structures the crest is critical. It must
be taken into account that these tests were conducted with slopes of 1:1.5 and 1:2 and
with a crest width of 6 to 8 Dn50.
Burger (1995) also conducted some tests in order to derive the influence of the type of
rock used in an armour layer. In Figure 10 the cross-section of the breakwater is
shown. Two sections of 1 m width were tested simultaneously to reduce the number
of tests by half.
Figure 10: Cross-section according to Burger (1995)
Six types of rock with varying parameters (angular or rounded, grading, shape) were
used in the investigation. Waves were generated according to a Jonswap spectrum (γ =
3.3). Each type of rock was tested with six different wave heights and two different
wave steepnesses. Table 7 gives an overview of the tests that were executed.
Table 7: Different types of tests (Burger, 1995)
Rock type s [-] Hs [cm]
1&2 0.02 7,1 10,0 12,4 14,4 17,0 17,8
1&2 0.04 7,4 9,5 12,2 13,5 16,2 18,3
3&4 0.02 7,8 10,1 11,9 14,0 16,0 18,1
3&4 0.04 8,4 10,2 12,0 14,3 16,0 18,1
5&6 0.02 5,9 8,0 10,2 12,0 14,2 16,0 18,8
5&6 0.04 5,9 8,1 10,1 12,2 14,3 16,0 18,2
The various parameters appear to have a low influence on the stability of the armour
layer of a low-crested breakwater. The influence of the grading was largest was the
most noticeable. A wide grading of D85/D15 = 2.5 results in a lower stability than a
grading of D85/D15 = 1.25. Rounded stones had less stability for higher damage levels
than angular stones. The shape of the stones (measured with the percentage of a
predetermined L/D ratio present in the rock type) did not show any differences in
stability.
Kramer and Burcharth (2004) conducted 3D laboratory experiments on low-crested
breakwaters in a wave basin, with length 18 m and width 12 m, at Aalborg University,
Denmark for the DELOS project. Waves were recorded by an array of five wave gauges
located 1.5 m from the roundhead (Figure 11). The influence of the roundhead on the
19
incoming waves was believed to be negligible. However, the trunk reflects some wave
energy which is re-reflected by the paddles. Therefore measurements from the 3-
gauge system and visual observations were performed to quantify this effect.
Figure 11: Detail roundhead layout (Kramer and Burcharth, 2004)
A foreshore with slope 1:20 was constructed in front of the breakwater. The structure
had a front slope of 1:2 and crest widths of both 10 cm and 25 cm were investigated.
Jonswap spectra were used with a peak enhancement factor of γ = 3.3. A test block
was defined by a fixed water level, wave direction, wave steepness and spreading. In
each test block the significant wave height HS was increased in steps until severe
damage was observed.
Table 8: Summary of conditions by Kramer and Burcharth (2004)
Number of waves N 1000
Non-dimensional crest width B/Dn50 3.1 and 7.7
Non-dimensional structure height d/Dn50 9.1
20
In Figure 12, Kramer and Burcharth (2004) plotted data of Vidal et al. (1992), Burger
(1995) and Kramer and Burcharth (2003) in one graph. The line represents the lower
limit of the test results. It has a similar shape as equation 13 and is given in equation
14.
013456 = <. =δ − 6. Ψ= W83456 + 6. 6δ W83456 Ψ
For −= ≤ W83456 < 2
2.4.2 Artificial units
Van der Linde (2010) performed 200 wave flume tests in Utrecht, the Netherlands.
These data were analysed by Muttray et al. (2012). Both submerged and low-crested
structures with single layer interlocking concrete Xbloc armour units were tested. A
foreshore with 1:30 slope was installed. The water depth at the toe remained 33.9 cm
but the toe height varied to achieve the different freeboards of 0, ±4.4 and ±8.9 cm
(Figure 13). By doing so, the total number of armour units was fixed namely 326 or 389
(which depended on whether 3 or 9 armour units were installed on the crest). A
number of tests were also repeated with a 3% lower packing density by reducing the
crest width by 3 cm.
Figure 13: Cross-section breakwater (Muttray et al., 2012)
Stability tests were performed with irregular waves according to a Jonswap spectrum
with peak enhancement factor of γ = 3.3. Test series comprised several tests, starting
with a nominal wave height 0.6 Hs,D and stepwise increasing up to 1.8 Hs,D.
Displacement of a single armour unit by more than 0.5 D was considered as start of
damage and after the displacement of 10 armour units testing was stopped (0.3% and
3% damage respectively).
Table 9: Summary of conditions by Muttray et al. (2012)
Slopes tanα 3:4
Wave steepness s 0.02 - 0.04
Number of Waves N 1000
Non-dimensional crest width B/Dn50 3 and 9
Non-dimensional structure height d/Dn50 varied
Stability number N 1.15 – 3.5
For the design Xbloc breakwaters, a stability number Ns of 2.77 is often used. When Ns
is higher than 3.5, damage can be expected for non-overtopped structures. The
accumulated damage results of the test series are depicted in Figure 14. For the low-
crested structures, damage starts if Ns > 3.0. Therefore, if the same design rules would
be applied a smaller safety margin would be obtained for low-crested structures
compared to non-overtopped breakwaters.
Figure 14: Overview of displaced armour units (all tests) (Muttray et al., 2012)
23
In Figure 15 the stability numbers at start of damage are displayed in function of the
relative freeboard for the different breakwater sections:
• The stability for the total structure decreases and reaches a minimum for
breakwaters with zero freeboard (Ns = 3.0) after which the stability increases
for submerged structures (Ns = 3.6).
• The seaward slope is most critical for emergent structures (Rc/Hs = +0.4) with a
stability number Ns = 3.1. For submerged structures the stability increases to Ns
= 3.6.
• A minimum stability of the crest is obtained when structures have zero
freeboard (Ns = 3.0). The stability of the crest increases for both submerged and
emergent breakwaters.
• Only for structures with a relative freeboard of +0.0 and +0.4, damage at the
rear slope was observed.
For non-overtopped Xbloc breakwaters, a nominal stability number Ns of 3.5 is used for
start of damage. It is clear that start of damage occurs at lower stability numbers for
low-crested Xbloc breakwaters.
24
Figure 15: Stability number Ns at start of damage: overall damage (top left),
seaward slope (top right), crest (bottom left) and rear slope (bottom right)
(Muttray et al., 2012)
The trend lines of the stability numbers at start of damage are combined in figure
Figure 16. Guidelines recommend an increase in unit weight of 50% for crest levels
Rc/Hs < +1.0 and 100% for crest levels Rc/Hs < +0.5 (DMC, 2011). The resulting design
stability numbers are Ns,D = 2.77 for Rc/Hs > +1.0, Ns,D = 2.42 for +0.5 < Rc/Hs < +1.0 and
Ns,D = 2.20 for Rc/Hs < +0.5. For non-overtopped breakwaters, the safety margin
between the design value of 2.77 and the lower bound of the expected start of
25
damage of 3.5 amounts to 26%. This safety margin corresponds to twice the unit
weight. The same margin has been used for the stability numbers at start of damage of
low-crested breakwaters. Values of Ns = 3.05 for +0.5 < Rc/Hs < +1.0 and Ns = 2.77 for
Rc/Hs < +0.5 were found.
For comparison the stability of structures with a rock armour layer (according to the
Rock Manual) were drawn in the Figure 16.
Figure 16: Stability number Ns at start of damage (Muttray et al., 2012)
26
2.5 Hydraulic stability of Cubipods
Vanhoutte (2009) wrote a thesis about the Cubipod armour units to find a first
estimate of the hydraulic stability of the newly developed armour units. Non-
overtopped breakwaters were subjected to both regular and irregular waves but only
regular waves were studied since they cause more damage. Armour layers with a
single Cubipod unit, with two Cubipod units and with a combination of cube and
Cubipod armour unit were investigated. Both a quantitative and qualitative damage
assessment were performed.
In the qualitative analysis photos are visually evaluated. The value of the incident wave
height is sought for various damage levels (Initiation of Damage, Initiation of Iribarren
Damage and Initiation of Destruction).
Table 10: Incident wave heights resulting in damage levels (Vanhoutte, 2009)
ID IID
Double layer Cubipods without toe berm 1.20 16.5 2.96 23.4
Single layer Cubipods without toe berm 1.26 19.8
Double layer Cubipods with toe berm 1.32 19.7 2.97 23.6
Single layer Cubipods with toe berm 1.10 18.4
Combined layer of cube and Cubipod 1.37 16.9 3.15 23.6
The averages of the values for dimensionless damage De were calculated for both
damage levels. They were found to be 1.2 for Initiation of damage and 3.0 for Initiation
of Iribarren damage.
The values for the incident wave height were used in the Hudson formula to calculate
the hydraulic stability coefficient KD. For a double layer of Cubipods KD = 43 was found,
for a single layer of Cubipods KD = 35 was found and KD = 23 for a combined armour
layer with cubes and Cubipods.
Also a quantitative analysis was performed using the Virtual Net Method. This method
compares the porosity of the different zones of the net to the previous porosity which
results in a dimensionless value for the equivalent damage D. When the fifth root of D
is taken, the linearized dime
18 the results are shown.
Figure 17: Influence of t
double layered break
layers, the opposite phenom
armour layer. As a result the
clear conclusion could be dr
In Figure 19 and Figure 20
shown. The double Cubipod
armour layer with combined
of toe berm on
double layer of Cubipods and with a toe berm i
the case when no toe berm is present. For sing
nomenon appears but this can be due to the fac
executed by removing the upper layer after test
t the armour units had more friction with the fil
drawn.
ined units gives the lowest stability.
27
erent sections are
(Vanhoutte, 2009)
In Figure 19, equations are drawn according to the qualitative analysis namely with KD
= 43, KD = 35 and KD = 23 for a double Cubipod layer, single Cubipod layer and the
combined layer respectively. The damage appears to be higher than predicted by the
equations. It would be safer to adjust the equation to KD = 28, KD = 23 and KD = 18 as
shown in Figure 20.
29
Gómez-Martín (2015) wrote a doctoral thesis about the damage to armour layers of
breakwaters with rock and cube and Cubipod breakwaters. Hydraulic stability was
investigated under non-breaking and non-overtopping conditions.
The linearized equivalent dimensionless armour damage Se* = Se 1,5
is shown in Figure
21 in function of the measured stability number Ns=Hm0/DDn for experiments with
double cube and Cubipod armour layers. The failure functions (Eq. 15) corresponding
to cubes and Cubipods are drawn, using KD=6 and KD=28 according to Medina and
Gómez-Martín (2012). The qualitative damage levels for double armour layers are
represented by the horizontal lines: Initiation of damage (Se=1.0 for cubes and
Cubipods), Initiation of Iribarren damage (Se=3.4 for cubes and Cubipods) and Initiation
of Destruction (Se=8.3 for cubes and Se=24.1 for Cubipods). This high difference in
dimensionless damage for Initiation of destruction is due to the fact that many
Cubipods of the upper layer need to be removed before a unit from the bottom
armour layer can be extracted (which is defined as the Initiation of destruction damage
level).
Figure 21: Linearized equivalent dimensionless armour damage in function of
stability number (Gómez-Martín, 2015)
30
∀#∗ = ∀#/Φ = 1.6 /Φ 4γη /Ν .4 Ο /Ν ≈ 0.96 .γη/Ν Eq. 15
In Figure 22 the measured stability numbers are represented in function of the wave
steepness. The measured stability numbers corresponding to Initiation of damage,
Initiation of Iribarren damage and Initiation of Destruction for double-layer cube and
Cubipod armour layers. The hydraulic stability of double-layer Cubipod armours with
porosity of 41% is much higher than double layer cube armours with 37% porosity.
Figure 22: Measured stability numbers of double-layer cube and Cubipod armour
layers (Gómez-Martín, 2015)
Initiation of damage and Initiation of destruction for double-layer Cubipod armour
layers showed Ns(IDa)>3.0 and Ns(IDe)>4.0. Initiation of damage and Initiation of
destruction for double-layer cube armour layers showed Ns(IDa)<2.0 and Ns(IDe)<3.1.
Figure 23 shows the measured stability numbers of single- and double-layer Cubipod
armour for Initiation of damage and Initiation of destruction. The Initiation of damage
limit for double-layer Cubipods seems to be independent of the wave steepness,
Ns(IDa)=3.4 while the Initiation of destruction limit for double-layer Cubipods appears
to be dependent on the wave steepness, a minimum value of Ns(IDe)=4.0 with the
highest s0p is observed. For the stability numbers, minimum values of Ns(IDa)=2.8 and
31
Ns(IDe)=3.4 were found for single-layer Cubipods. The values for single-layer Cubipods
are lower than for double-layer Cubipods but higher than the double-layer cube
armour layers.
Figure 23: Measured stability numbers of single- and double-layer Cubipod
armour layers (Gómez-Martín, 2015)
2.6 Overtopping
Overtopping occurs when waves run up the seaward face of coastal defences, reach
the crest of the structure and pass over it. The overtopping discharge plays a primary
role in the design of coastal structures. Its importance is linked to the fact that it is the
main parameter to determine the crest freeboard of the breakwater.
In case of quasi-stationary wave conditions and water level, it is possible to calculate
an average wave overtopping discharge, defined as the average discharge per unit
width of the structure, q, expressed in m³/s/m or l/s/m. This value can depend both on
the wave characteristics such as the significant incident wave height Hm,0 or significant
wave period Tm1-0 and on geometrical and structural features of the breakwater such
as slope angle α, crest freeboard Rc, surface roughness gf or the presence of a berm.
In Figure 24, main parameters are displayed on a conventional breakwater cross-
section.
2.6.1 Empirical formulae
The overtopping of breakwaters due to random waves can be estimated using several
methods such as design diagrams (Goda, 1985) or empirical formulae (Owen, 1980;
van der Meer and Janssen, 1994; Pullen et al., 2007; van der Meer and Bruce, 2014;
Molines, 2015).
Goda (1985) designed a set of design diagrams (Figure 25) in order to estimate the
mean wave overtopping discharge for vertical and sloped sea walls. The mean
overtopping discharge was defined as the volume of overtopped water divided by the
33
time duration of some hundreds of waves. In the diagrams, the mean overtopping
discharge is plotted in function of the relative water depth h/Hm,0 for different values
of relative crest freeboard Rc/Hm,0.
Figure 25: Design graph (Goda, 1985)
Owen (1980) determined an empirical formula based on small-scale tests. The
overtopping discharge was evaluated for irregular waves on straight and bermed
seawalls.
κΘλ.Ν ΞΡΜ2m = νοπθ− Σ. ΞΡΜ2m 1ρστ Eq. 16
Coefficients a and b change in function of the structure geometry and the slope angle.
In Table 11 the values are shown for straight slopes. The roughness factor, ρf, was
defined as the ratio between the run-up of a given wave on a rough slope and a
smooth slope. For rock slopes, ρf = 0.50-0.60 was proposed while for smooth slopes a
value ρf = 1.00 was used.
Table 11: Coefficients a and b for straight slopes (Owen, 1980)
Slope Χ Α
1:1 0.0079 20.12
1:1.5 0.0102 20.12
1:2 0.0125 22.06
1:2.5 0.0145 26.10
1:3 0.0163 31.90
1:3.5 0.0178 38.90
1:4 0.0192 46.96
1:4.5 0.0215 55.70
1:5 0.0250 65.20
Van der Meer and Janssen (1995) presented other formulae for impermeable slopes
including reduction factors due to the influence of a berm gb, the roughness of the
surface gr, shallow water conditions gh and oblique waves gβ. For ξop < 2 (breaking
waves):
Eq. 17
κΘλ.Ν = 0.2 νοπ θ−2.6ψ. 1ρσρζρ{ρ|τ
Eq. 18
The reliability of eq. 17 is expressed by taking coefficient 5.2 as a normally distributed
stochastic variable with an average of 5.2 and a standard deviation s = 0.55 while for
eq. 18 the standard deviation s for coefficient 2.6 is equal to 0.35.
The EurOtop (Pullen et al., 2007) manual presented overtopping formulae for dikes
and rubble mound breakwaters. Since the latter often have steep slopes, formulae for
the maximum overtopping of dikes are used for rubble mound structures while the
roughness factor takes the rough surface into account. For probabilistic design and
prediction or comparison of measurements, equation 19 is given:
κΘλ 03 = 0.2 νοπθ−2.6 0 ρ∼ ρξτ Eq. 19
The reliability of the formula is described by a standard deviation s of 0.35 for
coefficient 2.6. For deterministic design or safety assessment it is recommended to
increase the average discharge by about one standard deviation, resulting in equation
20:
35
κΘλ 03 = 0.2 νοπθ−2.3 0 ρ∼ ρξτ Eq. 20
Besley (1999) conducted tests on rock- and Accropode-armoured mound breakwaters
with Rc = Ac to determine the influence of the crest berm width on the overtopping
discharges. A correction factor for the overtopping discharge was proposed:
ϖ = νοπ Σ. Eq. 21
For rock armour layers, a = 3.06 and b = -1.5 was proposed while for Accropode
armour layers a = 4.35 and b = -2.1 was suggested. It was advised to use the values for
rock armour layers if a conservative approach was desired. No reduction was found for
rock slopes if Gc/Hs < 0.75.
The EurOtop (2007) manual formulae fit the overtopping data well for values of RC/Hm0
> 0.5. In the case of 0 < RC/Hm0 < 0.5, a significant overprediction was observed. Van
der Meer and Bruce (2014) proposed a Weibull-shaped formula (Eq. 22) to fit the
overtopping expressions suggested in EurOtop (2007) for low-crested structures.
κΘλ 03 = νοπ − 0
Eq. 22
The main difference of equation 22 is the shape factor c. The best fitting value was
found to be 1.3, after which also a and b could be estimated. In the recently presented
EurOtop II (2016) manual, equations 23 and 24 were proposed. For predictions and
comparisons with measurements:
Eq. 23
The reliability of formula 23 is described by standard deviations of 0.0135 and 0.15 for
coefficients 0.09 and 1.5 respectively. For a design and assessment approach, the
average discharge is increased by about one standard deviation which results in
equation 24:
Eq. 24
The main differences with the equations from EurOtop (2007) are the coefficients and
the exponent 1.3. In Figure 26, both versions of the EurOtop formulae are plotted. The
difference is largest for relative freeboards Rc/Hm0 < 0.5.
Figure 26: Comparison of EurOtop (2007) formula with EurOtop II (2016) formula
Gómez-Martín and Medina (2008) performed overtopping tests on single- and double-
layer Cubipod armour layers with two different crown wall elevations and two
different water depths. Figure 27 shows the cross section of the model. Irregular wave
series of 1000 waves were launched following a JONSWAP spectrum with γ = 3.0.
Figure 27: Cross section (Gómez-Martín and Medina, 2008)
37
Smolka et al. (2009) proposed an overtopping formula for Cubipod-armoured
breakwaters:
κ Ξλ ΜΝ = 0.2 νοπ θ0.53 − 3.27 ΣΣ − 2.16 ΣΜ
1ρστ Eq. 25
With γf = 0.46 for single layer Cubipods and γf = 0.44 for double layer Cubipods. This
formula is valid in following conditions: 2.7 < ξ0p < 7.0, cotα = 1.5, 1.30 < Rc/Hm0 < 2.80,
0.40 < Ac/Rc < 0.65 for single layer Cubipods and 0.58 < Ac/Rc < 0.80 for double layer
Cubipods. In Figure 28 the experiment results are shown together with the equation
25.
Figure 28: Measured versus estimated overtopping rates (Gómez-Martín, 2008)
Molines and Medina (2015) estimated the roughness factor, gf, for different armour
units to be used in various prediction formulas. Authors might decide to use different
values for the roughness factor even if the same armour unit is used. Therefore, a
methodology was used to estimate the roughness factor which resulted in the smallest
relative mean squared error (rMSE = MSE/Var) using data from the CLASH Neural
Network (CLASH NN).
38
Using the new roughness factors, it was concluded that the CLASH NN gave the best
estimations. A new explicit formula, given in equation 26, was developed which follows
the behaviour of the CLASH NN. The formula estimated overtopping almost as well as
the CLASH NN, but on the contrary not being a “black-box”.
κ ΞλΜ,Ν = exp ΝΦ θ−1.6 − 2.6 ΣΜ,
1σ|τ Eq. 26
In which:
= 1.20 − 0.05Μ, ΣΜ,
Ν = 1.0 + 2.0exp −35Σ = ο 0.95; 0.85 + 0.13 ΣΜ,
Φ = 0.85 + 0.15ΣΣ
= ο 1; 1.2 − 0.5Σ ∼ > 0 1 ∼ = 0
2.6.2 Tolerable overtopping
According to EurOtop (2016), risks related to overtopping can be attributed to several
parameters. An important insight since EurOtop (2007) is that the tolerable
overtopping depends very strongly on the peak volume, and hence the wave height
causing the overtopping. Therefore, not only the mean overtopping discharge, q, is of
importance but as well the individual maximum overtopping volume, Vmax.
In Table 12 the tolerable values are gives for different kinds of structures. Acceptable
levels of overtopping, however, also depend on the use of the defence structure, the
access to the public or staff and the use of the land or water behind the structure.
39
Table 12: Tolerable overtopping according to Eurotop II (Van der Meer et al.,
2016)
Rubble mound breakwaters; Hm0 > 5 m; no damage 1 2000-3000
Rubble mound breakwaters; Hm0 > 5 m; rear side designed for
wave overtopping 5-10 10000-20000
grass cover; Hm0 = 1-3 m 5 2000-3000
Grass covered crest and landward slope; not maintained grass
cover, open spots, moss, bare patches; Hm0 = 0.5-3 m 0.1 500
Grass covered crest and landward slope; Hm0 < 1 m 5-10 500
Grass covered crest and landward slope; Hm0 < 0.3 m No limit No limit
40
2.7 IH-2VOF
2.7.1 Introduction
IH-2VOF is a numerical software developed in El Instituto de Hidráulica Ambiental de la
Universidad de Cantabria. The software solves the two-dimensional Reynolds Averaged
Navier-Stokes (RANS) equations at clear fluid regions. In a turbulent flow, the
instantaneous velocity field ui and pressure field p can be divided into two parts, the
mean velocity and pressure components, and π, on the one hand and the turbulent
velocity and pressure fluctuations, and πon the other:
= + ; π = π + π Eq. 27
Where i = 1, 2 for a bidimensional flow. Applying this decomposition to the Navier-
Stokes equations and assuming incompressible fluid, the RANS equations are derived:
ο = 0
Eq. 28a
Eq. 28b
Where ρ is the fluid density, gi is the i th
component of the gravitational acceleration
and is the mean viscous stress tensor. To calculate the Reynolds stress tensor, the
algebraic non-linear k – ε model is used with k the turbulent kinetic energy and ε the
dissipation rate of the turbulent kinetic energy.
The software calculates the Volume-Averaged Reynolds Averaged Navier-Stokes
(VARANS) equations inside the porous media regions. These equations are derived by
integration of the RANS equations over a control volume. Inside the porous media, the
Forchheimer’s relationship is used to calculate the last term in Eq. 28b. Following
VARANS equations are obtained:
⟨⟩ο = 0
⟨⟩ + ⟨⟩1 + ¤ ⟨⟩ο = 11 + ¤ −⟨Κ⟩ο − ⟨⟩ο + ⟨⟩ο + λ − 11 + ¤ ∞1 − Φ ⟨⟩ + ξ1 − Φ Θ⟨⟩ + ⟨⟩ ⟨⟩
Eq. 29a
Eq. 29b
Where “⟨… ⟩” denotes the intrinsic volume averaging, ¤the added mass coefficient, α
and β two empirical coefficients associated with the linear and non-linear drag force
respectively and n the porosity. In the free fluid region, i.e. with n = 1 and ¤= 0, the
VARANS equations obviously convert to the original RANS equations.
The free surface movement of the water is tracked by the volume of fluid (VOF)
method. This method does not consist in pursuing the exact location of the free
surface but instead in identifying the free surface location by tracking the density
change in each cell. Different cell types are identified: empty, surface or interior cells
depending on the value of the VOF function, which is defined as:
♣ = σ Eq. 30
= σ♦σ♦σ + ♦♥ Eq. 31
With ρf being the fluid density Vf the volume of fluid in the cell and Va the volume of air
in the cell: interior, empty and surface cells are defined as the F=1, F=0 and F>0 cells
respectively. The introduction of the VOF function in the equation of mass
conservation yields the transport equation for F(x,y,t):
ο, ♠, = ♣ο, ♠, σ Eq. 32 ♣ + ο ↔♣ + ♠ ∞♣ = 0 Eq. 33
42
When using the software, three different stages can be distinguished: Preprocessing
(Figure 29), actual Calculation and Postprocessing.
Preprocessing is the complete preparation of the software test and consists of four
steps. The first step is to draw the geometry, to insert the properties of the materials
and to generate the mesh (Figure 30). In the second step, the wave serie is defined by
prescribing its parameters. Next, the (static or dynamic) wave paddle is generated. And
finally, the input file is created by specifying the test conditions such as absorption
boundaries, frequency of the measurements, wave gauge locations, parameters to
save, etc.
43
Figure 30: New geometry and mesh with Coral
When the test is completely created, it has to be calculated (Figure 31). The progress
of the calculation can be observed at all times. The duration of the calculation depends
on the length of the simulated test and the accuracy of the mesh.
Figure 31: Calculation progress
44
When calculation is successful, in the Postprocessing menu all data can be processed
and results can be observed. The free surface profiles of all wave gauges can be
visualized (Figure 32) and found in a .txt-file, as well as horizontal and vertical
velocities, overtopping discharges (Figure 33), acting pressures on specified surfaces
and finally also visual representation of the propagating wave in the flume (Figure 34).
Figure 32: Wave gauge registrations
45
46
2.7.3 Calibrating the breakwater model
The representation of the flow resistance forces inside the porous medium is based on
the extended Darcy-Forchheimer equation:
♣↑→ = + ||² + ¤ ± ± Eq. 34
where u is the seepage velocity through the voids. The linear term represents a Darcy’s
type of flow for a laminar flow behavior, the non-linear term considers the turbulent
flow characteristics and the inertial term accounts for the added mass effect due to
transient effects.
Burcharth and Andersen (1995) proposed analytical expressions to calculate a and b:
= ″1 − Ν Eq. 35
= ξ 1 − Ν Eq. 36
Where α and β are empirical parameters and ν is the kinematic viscosity. Despite the
numerous amount of research, the problem of flow in a porous medium has not been
completely solved (Vílchez et al., 2016). The Forchheimer coefficients depend on
several parameters such as the Reynolds number, the shape of the stones, the grade of
the porous material, the permeability and of course the flow characteristics. These
empirical parameters could be used as tuning parameters to obtain better adjustment
between numerical calculations and experimental data (Losada et al., 2008).
Rubens (2014) performed simple tests to quantify the porosity of the materials used in
laboratory tests. A bucket of known volume was filled with stones and the weight was
determined. Water was added and again the bucket was weighed. Also the weight of
an empty bucket and a bucket filled only with water were determined. The porosity
between the stones was then calculated by:
= ♦≥♥#×♦∝♥∂ Eq. 37
47
Chapter 3 Experimental setup
In this chapter, the setup of the experiments will be discussed. Firstly the
infrastructure and used equipment will be described. The geometry and characteristics
will be listed, the experiments will be described and the software for analysis will be
elaborated shortly.
3.1 Equipment for experiments
The experiments are conducted in Laboratorio de Puertos y Costas de la Universidad
Politécnica de Valencia (LPC-UPV). In this laboratory a (wind and) wave flume is able to
test the stability of coastal structures. At one side, a wave generator is present. A wave
dissipation system is installed at the other side. The model of the breakwater is
constructed on the opposite side of the wave generator so that waves have time and
space to propagate.
3.1.1 Wave flume
The wave flume has a cross section of 1.2 m x 1.2 m and has a total length of 30 m. The
foreshore has an upward slope of 4% over a length of 6.15 m followed by a gentler
slope of 2% until the end of the wave flume. The model is built on a scale of 1/60 and
located 2 m in front of the dissipation system. A drawing of the layout of the wave
flume is given in Figure 35.
Figure 35: 2D Wave flume
In order to maintain an equal average water level in front of and behind the
breakwater during tests, recirculation of the water is necessary. This is realized by the
presence of a false bottom of 25 cm in the wave flume. By doing so, a return flow of
the water is able to establish and sustain the desired water depth at the toe of the
48
structure. A water depth of 64.8 cm or 69.8 cm is maintained in front of the wave
generator which results in a wave depth of respectively 20.0 cm or 25.0 cm at the toe
berm.
3.1.2 Wave generator
An important component of the wave flume is the wave generation system (Figure 36).
One part of the generator is the wave paddle (Figure 37) which transmits its horizontal
movement to the water. The vertical paddle is brought into motion by a piston (Figure
38) that is powered by an electric servomotor (Figure 39).
The metal plate moves over bronze bearings that slide over steel bars. These are
supported by a rigid metal structure.
Parts of the generated waves can be reflected by the breakwater model. If these
reflected waves reach the wave paddle, they could be re-reflected in the wave flume in
the direction of the model. Active Wave Absorption Control Systems (AWACS) are
developed to take into account these re-reflected waves. The wave paddle is provided
with two wave gauges as can be seen in Figure 37. In this way, the generated waves
are immediately known and the digital system adapts the movements of the wave
paddle in order to take into account the re-reflected waves and realize the actual
desired wave heights.
49
3.1.3 Wave dissipation system
The wave dissipation system is located opposite of the wave generator and behind the
breakwater model in order to dissipate energy gradually. Originally, five groups of
three metal frames with decreasing porosity were installed (Figure 40). Due to the
measuring of overtopping however, a basin, to intercept the discharge of the
overtopping channel, had to be installed. Therefore, some frames had to be removed.
The final layout is shown in Figure 41.
Figure 40: Original wave dissipation
system
layout
3.1.4 Overtopping measurements
In order to measure the overtopping, a channel made of methacrylate, is installed
behind the crest of the breakwater. It is 5 cm wide and has a small downward slope.
The end of the channel rests above a basin, which is placed on a scale that sends the
weight in real-time to the central computer. By analyzing the observed weight of water
in the basin during a test, the overtopping discharge, expressed in l/s/m, can be
calculated.
50
3.1.5 Wave height registration
Inside the wave flume, various wave gauges are installed in the center of the cross
section. A group is installed after the wave paddle and another in front of the model.
The wave height is registered with a frequency of 20 Hz. This data is used to separate
the registered waves into incident and reflected waves.
A wave gauge consists of two vertical metal rods. When a current is sent through these
rods, the encountered resistance caused by the water mass is a measure for the
immersion depth of the wave gauge.
The first group consists of wave gauges (or sensors) S1, S2, S3, S14 and S4. These
register the generated waves close to the wave paddle (Figure 42). The second group
of sensors, S5 – S11, is located on the 2% slope to observe the alterations in wave
height during propagation and in front of the breakwater model to measure the
incident wave height at the toe of the structure. A sensor (S12) is placed on the crest
of the breakwater to measure the overtopping thickness during tests. Finally, one
sensor (S13) is placed behind the breakwater to observe the water level to verify if
recirculation through the false bottom occurs sufficiently (Figure 43).
Figure 42: Wave gauge locations near wave paddle
51
3.1.6 Photo cameras and video cameras
Three cameras are installed perpendicularly to the front slope, crest and back slope
(see Figure 44: purple). Before testing and after each test a photograph is taken. The
Virtual Net Method will be carried out afterwards to assess the damage. Additionally,
three video cameras are placed to record anything that might occur (Figure 44: red).
Figure 44: Video camera locations
52
3.2 Experiments
3.2.1 Design
In this project, low-crested breakwaters are subjected to wave attack. For this thesis
only single layer Cubipods are tested. The nominal diameter of the armour units is 3.89
cm. In a later stage of the project also other armour layers will be investigated such as
single and double cube layer, double Cubipod layer and combinations of cube and
Cubipods.
The model is constructed on a foreslope of 2%. In the future, also foreslopes of 4% or
10% might be studied.
The front and back slope both have an inclination of cot α = 3/2. Two water depths at
the toe hs were placed namely 20 cm and 25 cm. Since the height of the constructed
breakwater was kept constant at 32 cm, the resulting crest freeboards were 12 cm and
7 cm. The breakwater is drawn in Figure 45.
Different breakwater crest widths will be investigated in this project but this thesis will
focus on a crest width of six times the nominal armour unit diameter Dn which equals
24 cm. Later, this dimension will be enlarged to 30 cm, 36 cm, 40 cm and 48 cm to
quantify its influence on hydraulic stability and overtopping discharge.
At the front and back slope, a toe berm is installed. Its dimensions are 2Dn50 of height
and 4Dn50 of width. Since it is a low-crested structure with important overtopping, it is
recommended to install a toe berm at the back slope as well since ending the back
slope armour layer early around the water level could cause early failure of this back
slope and consequently destruction of the entire structure.
Figure 45: Breakwater cross-section
3.2.2 Construction
The breakwater model needs to be constructed on the end of the 2% slope. To
facilitate this process, the section was printed in order to draw the points of
importance on the inside of the walls of the wave flume (Figure 46). After connecting
these points, a good reference is obtained to construct the breakwater section.
First, the core material is placed on the wave flume slope (Figure 47). The mound is
compressed manually and material is added until the core section is filled. This core is
important as it will support the layers above during the tests. The nominal diameter of
the core, Dn50, used in this project is 0.7 cm. Afterwards, the filter layer can be
deposited on top of the core (Figure 48). The nominal diameter, Dn50, is 2.0 cm and a
layer with thickness of 3.3 cm is desired.
Finally, the armour units are placed meticulously on the filter layer. A thickness equal
to the nominal diameter, Dn50 = 3.89 cm, is foreseen.
The Cubipod units are not placed randomly. The number of units in each row depends
on the theoretical porosity which is pursued namely 41%. The amount of Cubipods in
each row can be calculated since the width of the wave flume is constant. The number
of units in each row constantly alternates between 18 and 19 units. In this way, a
porosity of 41% is obtained.
Armour units were painted in different colors to distinguish easily during damage
measurement and, if desired, even determine its original location. Additionally, red
“targets” are glued inside the wave flume walls. These help when collocating the
armour units during construction and provide the grid for damage calculation.
54
Figure 46: Drawn cross-section Figure 47: Placement of the core
Figure 48: Placement of the filter layer Figure 49: Placed armour units
Figure 50: Top view armour layer
3.2.3 Calibration of the wave gauges
Before starting the tests, the water level inside the flume is verified and adjusted if
necessary. Afterwards, the wave gauges need to be calibrated. The sensors are lifted
and lowered over a distance of 15 cm while in the central computer the readings are
corrected until each sensor displays approximately 0.00 cm.
55
3.2.4 Tests without breakwater model (regular and irregular)
In the wave flume without breakwater, the tests are executed in order to verify the
generated wave properties. Since no reflection by the breakwater can occur, reflection
coefficients should remain low. Two types of tests can be distinguished. A number of
tests with regular waves (N = 50) are launched to compare with the flume modeled in
IH-2VOF. All tests with irregular waves are launched in the empty wave flume as well.
For these tests, a storm duration of 1000 waves is launched.
3.2.5 Tests with breakwater model (regular and irregular)
First, regular waves were launched to measure overtopping that will be used to
calibrate the model in IH-2VOF. For each test, 20 waves are launched since these
waves have a fixed wave height and computation time can be kept low. Afterwards,
irregular waves will be launched to analyze the hydraulic stability and the overtopping.
Approximately 1000 waves are launched in each test with irregular waves. In Table 13
and Table 14, the wave characteristics are shown for the regular wave tests and in
Table 15 for irregular wave tests.
56
Table 13: Regular wave characteristics hs=20cm (20 waves per test)
Model Water
depth hs
(cm) Scale
hs=20cm
Ir5
57
Table 14: Regular wave characteristics hs=25cm (20 waves per test)
Model Water
depth hs
(cm) Scale
hs=25cm
Ir5
58
Model Water depth
hs (cm) Scale
hs=20cm
Ir5
59
hs=25cm
Ir5
60
3.3.1 Incident wave height
Design of coastal structures is highly dependent on the incident waves. In laboratory
model tests however, only the registered waves, which is the sum of the incident and
reflected waves, can be measured. Thus arises the necessity to separate the reflected
waves from the incident waves in order to be able to study the response of the model
structure.
Various software is available to execute the separation of the waves. In this project,
the LASA-V method, which is developed by Figueres and Medina (2004), is used. The
program (Figure 51) uses an approximate Stokes-V wave model which is able to
analyze highly non-linear waves. These types of waves occur especially with high crests
compared to the wave length or in shallow water conditions, both with regular and
irregular waves. LASA is an acronym that stands for Local Approximation using
Simulated Annealing. LASA-V can be used in the same conditions as the original LASA
software but additionally, it is suitable for steeper waves. Only for breaking waves in
combination with reflected waves due to the breakwater model, problems occur
during the separation of the waves. This is the reason for registering the wave
characteristics during tests without breakwater model as well.
Figure 51: LASA-V software
3.3.2 Analysis of the waves
After the incident and reflected waves have been separated by LASA-V, the waves are
analysed by LPCLab 3.7.1 (Figure 52). This software also has been developed by the
Laboratory of Ports and Coasts in Valencia. The program analyses the wave data in the
time and frequency domain.
Figure 52: LPCLab 3.7.1
3.3.3 Porosity & damage
The porosity is measured for two reasons. Firstly, the initial porosity has to be equal to
the fixed value of approximately 41% for Cubipod units. This is nearly always fulfilled
since the amount of armour units placed is predetermined for every row. After each
test, the porosity is calculated again by taking a photograph and processing it using the
developed PRICAPAXYZ command in Autocad, after every Cubipod in the armour layer
has been located. The damage is calculated with the Virtual Net Method using the
porosity values as explained in 2.3.2.
62
3.4 IH-2VOF
3.4.1 Mesh
When using a variable mesh, changes in the dimensions of each cell have to be less
than 5%. This means that 2 x < 0.05 and
2 y < 0.05 should be fulfilled. The wave height
should be at least ten times y and x should be smaller than 2.5 times y. In breaking
wave conditions, it is advised to use at least 100 cells per wave length. It is also
recommended to have at least 1.5 times the wave length in front of the breakwater.
Taking these guidelines into account, the following parameters were used in the
software.
The domain has a width of 22.75 m and a height of 1.00 m. A variable grid is chosen to
mesh the domain. The part of greatest interest (i.e. around the breakwater crest) has
the maximum resolution, namely a width x of 10 mm and a height y of 5 mm.
Towards the boundaries of the mesh, the size of the cells was allowed to increase. This
was realized by introducing a fixed number of cells in the subzones at the edges. The
final values introduced into the software to create the mesh are shown in Table 16.
This was the finest mesh possible since the number of cells quickly rises with smaller
cell dimensions, resulting in excessive computation times.
Table 16: Mesh characteristics
Num. cells left 800 36 1 20 1 1
Num. cells right 1 36 40 1 51 78
Max. sep.
63
Figure 53: Displayed mesh
The mesh quality can be verified for both x- and y-direction (Figure 54). On the left axis
the cell size can be read (black graph) and on the right axis the value of second
derivative is displayed (green graph). The latter should be smaller than 0.05, as
explained earlier. If this condition was not fulfilled, the graph would partly have a red
color in the area of the issue.
Figure 54: Mesh quality
Δx=0.010 m Δx=variable
Δx=variable
Zone 1
3.4.2 Geometry
Different objects are inserted in the domain. For the tests with an empty wave flume,
only the bottom slope of the flume is introduced as an impermeable obstacle and the
water depth is given. The breakwater is introduced as a combination of porous media
with different characteristics for the calibration tests and with exactly the same
dimensions as in the laboratory.
3.4.3 Characteristics
The initial characteristics are shown in Table 17 and are based on the findings of
Losada et al. (2008).
Porosity D50 α β
Core 0.2 0.007 200 0.8
Filter 0.3 0.02 200 1
Armour layer 0.41 0.0389 200 1.1
As explained in 2.7.3, the characteristics need to be calibrated by changing various
parameters. The influence of various parameters was investigated by introducing
different values. Parameter α for the armour layer was changed from 200-600 with
steps of 100 and β from 1.1-1.5 with steps of 0.1.
As explained at the end of 2.7.3, the porosity of a stone layer can be quantified using
the described method. The results are shown in Table 18.
Table 18: Porosity measurements
Wstones [g] Wstones+water [g] Wwater [g] Wbucket [g] Vwater [l] Vtotal [l] n
Core 3946.2 4992.6 2615.7 371.3 1.046 2.244 0.4662
Filter 4054.1 5102.9 2615.7 367.2 1.049 2.249 0.4664
Finally, also the influence of the porosity was investigated. With the optimal
characteristics all the tests were run with the IH-2VOF software. The results are shown
in the next chapter.
Chapter 4 Results
Hereafter, following statistical parameters will be used to quantify the quality of the
obtained results:
The relative mean square error (rMSE) is defined as the mean square error between
observed and estimated data divided by the variance of the observed data and it is
situated between 0 and 1.
Coefficient of correlation (r) is defined as the covariance of two variables divided by
the product of their standard deviations. It is a measure of the linear dependence
between two variables and -1 ≤ r ≤ 1.
4.1 Laboratory tests
As stated earlier, the software to separate incident and reflected waves does not
function properly in breaking conditions near the model. The incident wave height at
the toe of the breakwater, however, is an important parameter.
If the reflection coefficient, determined at deep water conditions, is assumed to be
constant over the entire wave flume, an estimated incident wave height, denoted as
H*, at the toe can be calculated by using the following formula:
Σ∅#∗ = Ξ×#∩.#×#∅ − ×#σ∂#Σ#∅
For tests without breakwater model present, the incident wave height is equal to the
registered wave height since no reflection occurs. Therefore, the registered wave
height without breakwater model can be used as verification if the above mentioned
formula is justified. As irregular wave tests were executed both with and without
breakwater model, the data is available to make this comparison. It is presented in
Figure 55. The difference between both wave heights appears to be small with a rMSE
< 10% and a coefficient of corre

numerical modelling of wave overtopping of cubipod armored

Documents