raosa andrea natalia tesi

raosa andrea natalia tesi
Alma Mater Studiorum – Università di Bologna
DOTTORATO DI RICERCA IN
INGEGNERIA CIVILE E AMBIENTALE
Ciclo XXVI
Settore Concorsuale di afferenza: 08/A1
Settore Scientifico disciplinare: ICAR/01
ANALYSIS AND MATHEMATICAL MODELING
OF WAVE-STRUCTURE INTERACTION
Presentata da:
Andrea Natalia Raosa
Coordinatore Dottorato
Relatore
Chiar.mo Prof. Alberto Lamberti
Prof. Barbara Zanuttigh
Relatore
Prof. Javier L. Lara
Esame finale anno 2013-2014
1
Index
List of figures ..................................................................................................................... 5
List of tables ..................................................................................................................... 12
Chapter 1 INTRODUCTION........................................................................................ 14
1.1 Motivations ............................................................................................................ 14
1.2 Background ............................................................................................................ 15
1.3 Definition of the objectives .................................................................................... 17
1.4 Outline .................................................................................................................... 18
Chapter 2 SEDIMENT TRANSPORT MODELLING IN THE SWASH ZONE. STATE OF
THE ART. ........................................................................................................................ 19
2.1 Parametric and empirical modeling of cross-shore swash zone sediment transport20
2.2 Longshore sediment transport rate (LSTR) ............................................................ 23
2.3 Process-based numerical modeling of swash zone sediment transport .................. 26
2.3.1 Non-linear shallow water equations (NLSWE) ............................................... 26
2.3.2 Boussinesq equations ...................................................................................... 29
2.3.3 Navier-Stokes equations (NSE) ...................................................................... 32
Chapter 3 THE IH-2VOF NUMERICAL MODEL. ................................................... 40
3.1 Governing equations in the fluid domain: the Reynolds Averaged Navier-Stokes (RANS)
equations ....................................................................................................................... 41
3.2 Initial and boundary conditions .............................................................................. 45
3.2.1 Initial conditions.............................................................................................. 45
3.2.2 Boundary conditions ....................................................................................... 46
3.3 Free surface tracking trough the volume of fluid (VOF) method .......................... 49
3.4 Partial cell treatment............................................................................................... 50
3.5 Governing equations for the flow in porous media (VARANS equations) ............ 50
3.6 Schematics of computational domain .................................................................... 57
2
3.7 Numerical resolution .............................................................................................. 59
3.8 Resolution procedure.............................................................................................. 61
Chapter 4 REPRESENTATION OF THE WAVE OVERTOPPING PROCESS. .. 62
4.1 Design criteria for coastal defences structures ....................................................... 63
4.2 Overtopping over a seadikes .................................................................................. 66
4.2.1 Wave overtopping discharge ........................................................................... 66
4.2.2 Overtopping flow velocities and overtopping flow depth ............................... 70
4.3 The numerical database .......................................................................................... 73
4.4 Validation of the model .......................................................................................... 74
4.4.1 Wave reflection coefficient ............................................................................. 75
4.4.2 Overtopping discharge .................................................................................... 77
4.5 Flow height evolution over the dike crest .............................................................. 80
4.5.1 Influence of the dike submergence and geometry ........................................... 80
4.5.2 Comparison with the theory ............................................................................ 83
4.5.3 Formula for the determination of the decay coefficient .................................. 89
4.6 Flow velocity evolution over the dike crest ........................................................... 91
4.6.1 Influence of the dike submergence and geometry ........................................... 91
4.6.2 Approximation of the velocity trend with a fitting function ........................... 94
4.7 Statistical characterization of extreme overtopping wave volumes ..................... 100
Chapter 5 TWO-PHASE APPROACH FOR SEDIMENT TRANSPORT MODELLING. 103
5.1 Governing equations for fluid and particle phase: RANS equations ................... 103
5.2 Closure of fluid stresses ....................................................................................... 106
5.3 Closure of sediment stresses ................................................................................ 110
5.4 Model implementation ......................................................................................... 117
5.5 Spatial discretization in finite different form ....................................................... 120
5.5.1 Advection terms ............................................................................................ 122
3
5.5.2 New advection terms ..................................................................................... 125
5.5.3 Tangential terms ............................................................................................ 127
5.5.4 Drag force terms ............................................................................................ 132
5.5.5 Pressure terms ............................................................................................... 133
Chapter 6 WAVE-INDUCED EROSION AND DEPOSITION PATTERNS: VERIFICATION
OF MODEL RESULTS. ............................................................................................... 136
6.1 Computational set-up ........................................................................................... 136
6.2 Results for test P1................................................................................................. 137
6.3 Results for test P2................................................................................................. 142
Chapter 7 CONCLUSIONS AND FUTURE WORK. .............................................. 146
REFERENCES .............................................................................................................. 150
4
LIST OF FIGURES
Figure 2.1. Schematic illustration of swash zone (Elfrink and Baldock, 2002).
……………………………………………………………………………………………………….19
Figure 3.1. Schematics of computational domain with the different cell types on the
information of the VOF function and definition of the computed magnitudes.
……………………………………………………………………………………………………….58
Figure 3.2. Schematics of solid boundaries definition through the partial cell treatment.
……………………………………………………………………………………………………….58
Figure 4.1. Severe wave overtopping at the Samphire Hoe seawall, UK (CLASH project,
www.clash-eu.org, 2001).
……………………………………………………………………………………………………….62
Figure 4.2. Wave overtopping and overflow for positive, zero and negative freeboard (by
Eurotop 2007).
……………………………………………………………………………………………………….68
Figure 4.3. Definition sketch for overtopping flow parameters on the dike crest (by
Eurotop 2007).
……………………………………………………………………………………………………….71
Figure 4.4. Overtopping flow velocity data vs overtopping flow velocity formulae (by
Eurotop 2007).
……………………………………………………………………………………………………….72
Figure 4.5. Left side: influence of overtopping flow depth on overtopping flow velocity;
right side: influence of bottom friction on overtopping flow velocity (by Eurotop 2007).
……………………………………………………………………………………………………….72
5
Figure 4.6. Tested levee cross section (model-scale units).
……………………………………………………………………………………………………….74
Figure 4.7.
obtained by the experimental database for smooth straight (Zanuttigh and
Van der Meer, 2006) and the numerical simulation characterized by
and
.
……………………………………………………………………………………………………….75
Figure 4.8.
obtained by the experimental database for smooth straight (Zanuttigh and
Van der Meer, 2006) and the numerical simulation characterized by
and
.
……………………………………………………………………………………………………….76
Figure 4.9.
obtained by the experimental database for smooth straight (Zanuttigh and
Van der Meer, 2006) and the numerical simulation characterized by
and
.
……………………………………………………………………………………………………….76
Figure 4.10. Total numerical discharge
versus total theoretical discharge
for
emerged cases.
……………………………………………………………………………………………………….78
Figure 4.11. Total numerical discharge
versus total theoretical discharge
for
cases with zero freeboard.
……………………………………………………………………………………………………….78
Figure 4.12. Wave height trend on the dike crest for test T17C (
(
, blue), T1A (
, green), and T25D (
, red), T9B
, yellow).
……………………………………………………………………………………………………….82
6
Figure 4.13. Wave height trend on the dike crest for tests characterized by landward
slope 1:3 (circles) and 1:2 (crosses) and different freeboard.
……………………………………………………………………………………………………….82
Figure 4.14. Wave height trend on the dike crest for tests characterized by seaward slope
1:4 (circles) and 1:6 (crosses) and different freeboard.
……………………………………………………………………………………………………….83
Figure 4.15. Wave height decay on the dike crest with
Squares: cases with
and
and
; triangles: cases with
; diamonds: cases with
and
.
and
.
……………………………………………………………………………………………………….86
Figure 4.16. Wave height decay on the dike crest with
Squares: cases with
and
and
; triangles: cases with
; diamonds: cases with
and
.
and
.
……………………………………………………………………………………………………….86
Figure 4.17. Wave height decay on the dike crest with
Squares: cases with
and
and
; triangles: cases with
; diamonds: cases with
and
.
and
.
……………………………………………………………………………………………………….87
Figure 4.18. Wave height decay on the dike crest with
Squares: cases with
and
and
; triangles: cases with
; diamonds: cases with
and
.
and
.
……………………………………………………………………………………………………….87
7
Figure 4.19. Wave height decay on the dike crest with
Squares: cases with
and
and
; triangles: cases with
; diamonds: cases with
and
.
and
.
……………………………………………………………………………………………………….88
Figure 4.20. Wave height decay on the dike crest with
Squares: cases with
and
and
; triangles: cases with
; diamonds: cases with
and
.
and
.
……………………………………………………………………………………………………….88
Figure 4.21. Wave decay coefficients against new Equation 4.14. Tests with
.
……………………………………………………………………………………………………….90
Figure 4.22. Wave decay coefficients against new Equation 4.14. Tests with
.
……………………………………………………………………………………………………….90
Figure 4.23. Wave decay coefficients against new Equation 4.14. Tests with
.
……………………………………………………………………………………………………….91
Figure 4.24. Flow velocity trend on the dike crest for test T17C (Rc/Hs=0.5, red), T9B
(
, blue), T1A (
, green), and T25D (
, yellow).
……………………………………………………………………………………………………….92
Figure 4.25. Flow velocity trend on the dike crest for tests characterized by landward
slope 1:3 (circles) and 1:2 (crosses) and different freeboard.
……………………………………………………………………………………………………….93
Figure 4.26. Flow velocity trend on the dike crest for tests characterized by seaward
slope 1:3 (circles) and 1:6 (crosses) and different freeboard.
……………………………………………………………………………………………………….93
8
Figure 4.27. Wave velocity evolution on the crest of the structure for tests with
and
. Squares: cases with
cases with
and
and
; triangles:
; diamonds: cases with
and
.
……………………………………………………………………………………………………….95
Figure 4.28. Wave velocity evolution on the crest of the structure for tests with
and
. Squares: cases with
cases with
and
and
; triangles:
; diamonds: cases with
and
.
……………………………………………………………………………………………………….95
Figure 4.29. Wave velocity evolution on the crest of the structure for tests with
and
with
. Squares: cases with
and
and
; triangles: cases
; diamonds: cases with
and
.
……………………………………………………………………………………………………….96
Figure 4.30. Wave velocity evolution on the crest of the structure for tests with
and
with
. Squares: cases with
and
and
; triangles: cases
; diamonds: cases with
and
.
……………………………………………………………………………………………………….96
Figure 4.31. Wave velocity evolution on the crest of the structure for tests with
and
cases with
. Squares: cases with
and
and
; diamonds: cases with
; triangles:
and
.
……………………………………………………………………………………………………….97
9
Figure 4.32. Wave velocity evolution on the crest of the structure for tests with
and
. Squares: cases with
cases with
and
and
; triangles:
; diamonds: cases with
and
.
……………………………………………………………………………………………………….97
Figure 4.33. Comparison numerical results (
) with smooth structures
against formula 4.16.
……………………………………………………………………………………………………101
Figure 4.34. Comparison numerical results (
, pink;
, orange;
, green;
, blue) with smooth structures against formula 4.16.
…………………………………………………………………………………………………….102
Figure 6.1. Sketch of the cases tested.
…………………………………………………………………………………………………….137
Figure 6.2. Water depth trend measured at
from the beginning of the channel.
…………………………………………………………………………………………………….137
Figure 6.3. Water depth trend along the channel at
(red),
(green) and
(blue).
…………………………………………………………………………………………………….138
Figure 6.4. Bottom level along the channel at
(red),
(green) and
(blue).
…………………………………………………………………………………………………….138
Figure 6.5. Horizontal velocity at the gauge sets at
from the begin of the
channel.
…………………………………………………………………………………………………….139
10
Figure 6.6. Horizontal velocity at the gauge sets at
from the begin of the
channel.
…………………………………………………………………………………………………….139
Figure 6.7. Sediment concentration at the gauge sets at x=1 m from the begin of the
channel.
…………………………………………………………………………………………………….140
Figure 6.8. Sediment concentration at the gauge sets at x=8 m from the begin of the
channel.
…………………………………………………………………………………………………….141
Figure 6.9. Water depth trend along the channel at t = 20 s (red), t = 50 s (green) and
t = 80 s (blue).
…………………………………………………………………………………………………….142
Figure 6.10. Bottom level along the channel at t = 20 s (red), t = 50 s (green) and
t = 80 s (blue).
…………………………………………………………………………………………………….142
Figure 6.11. Sediment concentration at the gauge sets at
from the begin of the
channel.
…………………………………………………………………………………………………….143
Figure 6.12. Sediment concentration at the gauge sets at
from the begin of the
channel.
…………………………………………………………………………………………………….144
Figure 6.13. Sediment concentration at the gauge sets at
from the begin of the
channel.
…………………………………………………………………………………………………….144
11
LIST OF TABLES
Table 4.1. Characteristics of simulated tests.
……………………………………………………………………………………………………….73
Table 4.2. Model settings adopted for the numerical simulations.
……………………………………………………………………………………………………….74
Table 4.3. Model settings adopted for the numerical simulations.
……………………………………………………………………………………………………….77
Table 4.4. Model settings adopted for the numerical simulations.
……………………………………………………………………………………………………….77
Table 4.5. Model settings adopted for the numerical simulations.
……………………………………………………………………………………………………….79
Table 4.6. Flow characteristics at the dike off-shore edge (
).
……………………………………………………………………………………………………….81
Table 4.7. Wave decay coefficients of the best fitting and relative standard deviation.
……………………………………………………………………………………………………….84
Table 4.8. Average wave decay coefficients and relative standard deviation for each
tests.
……………………………………………………………………………………………………….85
Table 4.9. Coefficients obtained with the best fitting curve for the wave velocity on the
dike crest and relative standard deviation.
……………………………………………………………………………………………………….98
12
Table 4.10. Average coefficients for wave velocity evolution on the dike crest and relative
standard deviation for each tests.
……………………………………………………………………………………………………….99
Table 6.1. Characteristics of cases tested.
…………………………………………………………………………………………………...…136
13
CHAPTER 1
INTRODUCTION
1.1 Motivations
Shore protection against erosion has turned to be a major issue in a great part of the
worldwide coastal areas. About 80% of the world shorelines are under an erosion process
while 70% of the global human population, representing 4 billions of persons, live in a 60
km-wide strip contiguous to the sea. The natural tendency of coastal erosion has been
dramatically accelerated by the impact of human activities, and the average rate of
shoreline recession in some coastal sites of the world reaches values of tens of meters a
year (Komar, 1998; Pilkey and Hume, 2001). In particular, in the countries of the
European Union, about 20.000 km of coasts, i.e. 20% of the European coasts, are to some
extent affected by erosion, being most of them actively retreating. The area lost or
2
seriously impacted by erosion is estimated to be 15 km per year. Extensive coastal areas
in the Netherlands, England, Germany, Poland and Italy are already at an altitude lower
than the levels of the high tide and therefore inherently more vulnerable to flooding. At
the same time, over the past 50 years, the population living in European coastal
municipalities has more than doubled to reach 70 millions inhabitants in 2001
(EUROSION UE project).
Hence, the population and development pressures on the coastal zones require life and
property defense policies from coastal hazards and the definition of technical alternatives
for shore protection. The response options to eroding coasts are multifold, and have been
basically split into “soft” solutions, mainly beach nourishment designs, and “hard”
solutions of shoreline stabilization using seawalls, grayness and offshore breakwaters.
Climate change will have significant impacts on coastal areas, due to the sea level rise
(based on the SRES scenarios in the range 0.2-0.8 m/century) and the increased frequency
Chapter 1
and intensity of extreme events. So, the large stretches of the coast that are protected by
hard structures, are very sensitive to rising average sea level because these kind of
structures will be more frequently overflowed and the shoreline will consequently subject
to greater wave energy remaining. These elements make essential an accurate knowledge
of the phenomena of wave-structure interaction and the rising wave on the beach and / or
dunes which over time can be eroded until the formation of a breach.
The University of Bologna has leaded the investigation tasks of the THESEUS project
and the present PhD Thesis has been realised in the frame of this project. One of the
main objectives of the project was to analyse innovative technologies aimed at the
mitigation of the flood and erosion risk (as resilient dikes or over-washed structures).
The increase of frequency and intensity of storms, combined with the uncertainties related
to extreme events and climate change, carries an increased the risk of flooding of low
lying areas, an accelerate erosion of exposed soft beaches and a challenges in the long
term design of coastal protection structures. The aim of this thesis work, included within
the THESEUS project, is the development of a mathematical model 2DV two-phase,
based on an existing code, able to represent the real conditions of inundation i.e able to
represent together the overtopping phenomenon on emerged/submerged structures and the
sediment transport.
1.2 Background
Traditionally wave-structure interaction has been studied through physical tests (two- and
three-dimensional, small- and large- scale model tests). Empirical formulations arisen
from physical modeling present several restrictions and a narrow range of applicability.
Many other issues related with scale factors or processes such as porous media flow,
wave impacts or viscous effects are not correctly represented in the experiments. A great
effort has been made over the last decade in the numerical modeling of wave interaction
with coastal structures to overcome these limitations.
15
Chapter 1
A similar consideration can be made also for the study of the sediment transport. In fact,
field experimentation is challenging due to the difficulty and expense in deploying
equipment, obtaining robust data and the variability of meteorological conditions.
Although, laboratory measurements have some limitations, due to the constrained
circumstances compared with the field, the data obtained are the most reliable for
investigating processes and for validating models.
Meanwhile, developments in
computational hardware and numerical solution methods have driven the popularity of
numerical modeling of coastal hydrodynamics.
Several approaches have been followed to study the wave-structure interaction, the
induced hydrodynamic and the consequently mixing and sediment suspension. Among
other existing approaches, Nonlinear Shallow Water (NSW), Boussinesq-type or NavierStokes equations models have traditionally been used.
Good results in terms of averaged magnitudes have been obtained using NSW equation
(Kobayashi et al., 2007), though vertical velocity structure cannot be resolved using this
approach and the energy transfer to higher frequencies occurring before wave breaking
cannot be reproduced accurately due to the lack of dispersion.
Boussinesq-type models are able to include frequency dispersion, a depth-dependent
velocity profile, and they can be applied to both breaking and non-breaking wave
conditions. A great effort has been made in order to relax the original equations by
deriving the extended Boussinesq equations (Kirby, 2003). However, this type of models
requires setting both the triggering wave breaking mechanism and the subsequent wave
energy dissipation due to wave breaking. Moreover, these models fail to reproduce the
strong nonlinear shoaling prior to wave breaking and the free-surface and velocity higher
order statistics which are thought to be relevant for structure stability.
Navier-Stokes equations models assume a number of simplifications in the equations
lower than in other approaches. These models are able to calculate flows in complex
geometries and provide very refined information on the velocity, pressure and turbulence
field.
Models based on a two dimensional eulerian Navier-Stokes set of equations
(Losada et al., 2008; Lara et al., 2008; Guanche et al., 2009; Lara et al., 2011) have
proven to be powerful to address wave-induced processes.
16
Wave reflection and
Chapter 1
overtopping have been reproduced numerically with a high degree of accuracy,
introducing new models to be used as a complementary tool in the design process. These
types of models are so accurate and promising that 3D applications were developed (P.
Higuera et al., 2013).
Wave-structure interaction and wave run-up on beaches and/or dunes require models
capable to deal with steep and emerged slopes. Traditional 2DH numerical models, such
as Mike21 Shallow Water equations model (SW+HD modules), can be applied only when
the structure/beach is submerged. Other models, such as the Boussinesq models (as Mike
21 BW module) can be powerful tools for run-up and overtopping in case of steep slopes
up to 1:3 but so far are time consuming tools (high spatial resolution, low Courant
number) and need the introduction of a lot of artificial dissipation usually to avoid
instabilities: application thus depends on the extension of the area to be modelled and on
the phenomena to be included (wave breaking, wave run-up).
Moreover, existing
Boussinesq models do not include the representation of sediment transport do that
beach/dune reshaping during storms and possible breaching cannot be reproduced. So far
only RANS-VOF models can deal with wave run-up and overtopping on steep slopes
(also structures, slopes 1:2) without the inclusion of many artifices.
1.3 Definition of the objectives
The overall aim of this thesis is to develop a tool that can represent wave run-up and
overtopping together with beach reshaping during storms. Actually what can be a more
promising research field, due to the lack of good representation for many of the related
processes, is the modellisation of the swash zone that is an area of greatest importance
both for flooding issues and ecosystem conservation.
The specific objectives of the present study are:

to characterize the flow (velocities and layer thicknesses) on the crest of the
structure in order to extend the theoretical models and provide criteria for the
design of structures close to mean sea level or overwashed;
17
Chapter 1

to introduce in a two-dimensional numerical model based on the Reynolds
Averaged Navier–Stokes equations (RANS), called IH-2VOF (Losada et al.,
2008), new equations for the representation of the sediment transport;

to verify the model as a reliable tool for the simulation of wave-structures
interactions and sediment transport dynamics.
1.4 Outline
The present thesis is organized following the objectives listed above.
In chapter 1, an introductive description of the work and the objectives of the study are
presented.
In chapter 2, a state-of-the-art review of both experimental and mathematical the
modelling of sediment transport is included.
In chapter 3, the characteristics of the numerical model used to carry out the present work
are described. The governing equations and main mathematical assumptions, free surface
tracking method and resolution procedure are presented.
In chapter 4, previously the existing theories for the overtopping process are described,
than the numerical tests and its set-up for the study of wave overtopping process above a
particular kind of coastal defense structure are introduced. The key results obtained by
the numerical simulation (for example the influence of the seaward-landward slope and of
the dike submergence), the analysis of the wave flow characteristics above the structure
and the comparison with the theoretical approach are reported.
In chapter 5, the modifications of the initial code are reported. New equations
implemented for the representation of the sediment transport in a two-phase model are
shown and described.
In chapter 6, the stages and results of the two-phase model verification process are
presented.
In chapter 7, conclusions and discussion are finally drawn.
18
CHAPTER 2
SEDIMENT TRANSPORT MODELLING IN
THE SWASH ZONE. STATE OF THE ART.
The surf and swash zones are hydrodynamically active regions. Nearshore breaking
waves play a paramount role in coastal morphology and influence most coastal processes.
These waves produce highly turbulent regions causing significant mixing and sediment
suspension. The suspended sediments are transported by the nearshore currents induced
by breaking waves. Moreover, breaking waves impact offshore structures and should be
considered in their design. Fluid and sediment interactions occurring in the swash zone
determine the erosion or accretion of a beach and act as boundary conditions for
nearshore hydrodynamic and morphodynamic models. A schematic illustration of the
surf and swash zones is shown in Figure 2.1.
Figure 2.1. Schematic illustration of swash zone (Elfrink and Baldock, 2002).
In this chapter a critical review of conceptual and mathematical models developed in
recent decades on sediment transport in the swash zone is presented.
Chapter 2
Evidently, the hydrodynamics of the swash zone are complex and not fully understood.
Key hydrodynamic processes include both high-frequency bores and low-frequency
infragravity motions, and are affected by wave breaking and turbulence, shear stresses
and bottom friction. The prediction of sediment transport that results from these complex
and interacting processes is a challenging task.
Besides, sediment transport in this
oscillatory environment is affected by high-order processes such as the beach ground
water flow. Most relationships between sediment transport and flow characteristics are
empirical, based on laboratory experiments and/or field measurements.
Analytical
solutions incorporating key factors such as sediment characteristics and concentration,
waves and coastal aquifer interactions are unavailable. Therefore, numerical models for
wave and sediment transport are widely used by coastal engineers.
2.1 Parametric and empirical modeling of cross-shore swash zone
sediment transport
The swash zone sediment transport and foreshore evolution have been analyzed in several
studies (e.g., Masselink et al., 2005; Miles et al., 2006). Empirical formulas based on
numerous experiments on steady flow have been implemented to describe the amount of
sediment transport (Nielsen, 1990).
Most of these formulas were based on the
relationship between the Shields parameter (Shields, 1936) and the dimensionless
sediment transport rate.
Madsen (1991) derived a sediment transport rate formula for the instantaneous bed-load
that was further generalized by Madsen (1993). Masselink and Hughes (1998)
found that Bagnold’s energetics-based bed-load sediment transport equations fitted their
field data, and concluded that formulae based on a modified Shields parameter could also
be used. Since physically the up-rush and back wash flows are different, it seems logical
that the associated sediment transport processes can also be different. Masselink and
Hughes (1998) showed that swash zone sediment transport rate formula required different
empirical constants ( ) in order to fit measured velocities and sediment transport rates in
both up-rush and back wash phases of the swash.
20
Chapter 2
On one hand, the modeling approach using two different values of model coefficient is
consistent with differences between the up wash and back wash (flow characteristics and
sediment transport modes) as speculated by previous researchers (Nielsen, 2002). On the
other hand, our limited understanding of the up wash and back wash hydrodynamics
prevents us from quantifying further these coefficients for different conditions (different
values have been obtained for different data sets; i.e., they remain empirical). Nielsen
(2002) presented two mechanisms for the sediment transport during up-rush:‘‘(i) the
existence of higher shear stresses during up-rush; and (ii) the existence of pre-floating
sediment from the bore collapse (Masselink and Hughes, 1998)’’. The shear stress of the
bed was modeled by the time series of the free stream velocity in terms of the wave
boundary layer model plus a phase lead
of the bed shear stress, compared with the
free stream velocity at the peak frequency. He postulated that the total amount of
sediment transport during up-rush and back wash is well estimated by the model without
the need for incorporating different multipliers for up-rush and back wash and suggested
that the range in
values is 9.7 0.2 (Nielsen,2002). It should be noted that although the
total amount of sediment transport is the same, the timing of sediment transport rate has
not yet been accurately modeled due to the very unsteady nature of the swash zone and
the existence of pre-floated sediments.
Besides, some mechanisms have not been
considered in this formulation. For example, the observed lag between the instantaneous
bed shear stress and the rate of sediment transport has not been considered
(Nielsen,2002).
Larson et al. (2004) developed the sediment transport formulae to predict the net transport
rate over many swash cycles and compared the predictions with field data. Net sediment
transport rate sand the formulae developed to calculate the net sediment transport in the
swash zone showed good agreement with transport rate measurements at the seaward end
of the swash.
Drake and Calantoni (2001) added an extra term to the Bailard formula to account for
acceleration effects and showed that the inclusion of acceleration effects improved the
performance of the transport model. Puleo et al. (2003) also modified an energetics
model for sediment transport to include the effect of fluid acceleration and were able to
21
Chapter 2
strongly reduce the prediction error. Pedrozo-Acuña et al. (2007) modified the bed-load
formulation to include an acceleration term similar to that proposed by Drake and
Calantoni (2001):
(2.1)
{
where
|
is the Shields parameter;
moving grain;
threshold and
|
is the local beach angle;
is the friction angle for a
is the horizontal velocity at the sea floor;
is the acceleration
is the efficiency. The acceleration is calculated by differentiating the
velocity time series from the hydrodynamic model. The addition of an acceleration term
does not by itself improve the prediction. However, it enables morphological models to
predict on shore migration of bars, in accordance with results shown by previous
researchers (Pedrozo-Acuña etal., 2007).
Karambas (2003,2006) derived the non-dimensional sediment transport rate based on
modified Meyer-Peterand Muller (1948) formula using different values the multiplier
for up-rush and back wash that includes infiltration/exfiltration effects.
Note that a Shields-type transport formula does not account for inertial forces, which may
become significant for coarse grains due to the high fluid accelerations during up wash
(Baldock and Holmes, 1997). In summary, none of the above models can resolve all
potentially important details of the flow and sediment transport in the swash zone, such as
the wave boundary layer, percolation, flow separation at the beach step and the 2D or 3D
distribution of suspended sediments. Despite these efforts, the energetics-based models
are unable to account for the phase difference between the sediment transport rate and
hydrodynamic forcing parameters.
Hsu and Raubenheimer (2006) indicated that
sediment transport in the swash zone might not correlate to the instantaneous forcing
computed in a specific location, so such equations might not be valid in the swash. A
means to make progress on this issue are process-based and a two-phase modeling
approach.
22
Chapter 2
2.2 Longshore sediment transport rate (LSTR)
The longshore current generated by obliquely incident breaking waves plays an important
role in transporting sediment in the swash zone and is a key component of most coastal
engineering studies (Kumar et al.,2003). Under obliquely incident waves, near shore
sediment moves in a zig-zag way that results in LST in the swash zone (Asano, 1994).
The cross-shore distributions of LST indicate three distinct zones of transport: the
incipient breaker zone, the inner surf zone and the swash zone (Smith et al., 2004). A
peak in transport occurs for plunging waves in the incipient breaker zone, indicating that
this breaker type suspends more sediment for transport. The breaker is a function of
wave height, period and beach slope.
In the inner surf zone, wave height is the
dominating factor in controlling sediment transport, which depends less on wave period.
Swash zone sediment transport, which accounts for a significant percentage of the total
transport, shows a dependence on wave height, period and beach slope. The occurrence
of the increased longshore flow velocities in the swash zone is related to differences in
fluid motion between the inner surf zone and swash zone (Smith et al., 2004).
In the surf zone, the oscillatory part of the flow is directed more or less perpendicular to
the wave crests. In the swash zone, however, the flow direction during up-rush is
perpendicular to the wave crest, but perpendicular to the beach orientation during
backwash, in the absence of longshore current. This effect increases with increasing bed
slope or, rather, the surf similarity parameter (Elfrink and Baldock, 2002).
Bijker’s (1971) LST formula is one of the earliest formulae developed for waves and
currents in combination. It is based on a transport formula for rivers proposed by
Kalinske–Frijlink (Frijlink, 1952). Bailard (1981) developed an energy-based surf zone
sediment transport model based upon Bagnold (1963, 1966) steady flow models. His
model has both bed-load and suspended load components.
Both components were
expressed in terms of various instantaneous velocity components, which limited the
model’s usefulness. The gross LST is mainly computed with the CERC formula (Shore
Protection Manual, 1984) in engineering applications. This model, which is based on the
assumption that the total LSTR is proportional to longshore energy flux, was developed
23
Chapter 2
from the pioneering work of Bagnold in the early 1960s and further developed by Komar
and Inman (1970). The sediment transport rate was also calculated using the breaker
height, surf zone width and average longshore current velocity in the surf zone (Walton
and Bruno, 1989). Kamphuis (1991) performed laboratory experiments on sediment
transport due to oblique wave attack and found two peaks in the LSTR: one in the surf
zone and one in the swash zone. Kamphuis (1991) expanded his earlier work and
developed a relationship for estimating LSTR based upon dimensional analysis and
calibrated it using experiments within a physical model. Kamphuis (2002) found the
equation to be applicable to both field and laboratory data. Watanabe (1992) proposed a
formula for the total load. The Watanabe formula and its coefficient values have been
calibrated and verified for a variety of laboratory and field data sets. Nevertheless, it has
not yet been recognized whether the value of the non-dimensional coefficient in the
formula is a constant or it depends on the wave and sediment conditions.
Bayram et al. (2001) studied the cross-shore distribution of LST and evaluate the
predictive capability of well-known sediment transport formulae, based upon field data
sets. They pointed out that no existing sediment transport formula has taken into account
all the different factors that control LST in the surf and swash zones. Kumar et al. (2003)
compared measurement and estimation of LSTR for data from the central west coast of
India.
Tajima (2004) developed a computer routine to model surf zone sediment
transport.
The code is in the form of two programs that run sequentially: a
hydrodynamics model and a sediment transport model.
The hydrodynamic model
calculates the forcing functions needed to drive the sediment transport model at each
point in the profile, and includes modules for nonlinear wave propagation, wave
breaking, surface rollers and nearshore cur- rents.
The sediment transport model
calculates the transport at each profile point and includes bed-load and suspended load
modules. The models selected are not intended to be inclusive, but merely representative
of classes of models. The CERC equation, containing one term for the calculation of
total load (combined bed-load and suspended load), is the simplest formula in general
use. The Kamphuis formula is also a one-term, total load model, but explicitly includes
the effects of wave period, beach slope and grain size.
The Bailard formula is
representative of models that divide the transport into bed-load and suspended load
24
Chapter 2
transport. The Tajima model is representative of the complex computer routines that
provide a stepwise model of hydrodynamics and sediment dynamics across the surf zone,
and thus predict not only the total bed-load and suspended load LST, but also its crossshore distribution.
Although some of the above models included the swash zone component, in most LST
models, the swash transport contribution is either completely ignored or merely
accounted for as part of the total sediment transport budget. Van Wellen et al. (2000)
developed an engineering model, STRAND, to provide a simple engineering model of
swash sediment transport on steep, coarse- grained beaches. Although a good correlation
between their predictions and Kamphuis’ laboratory data was obtained, new laboratory
and field data are required to validate the model further. Kobayashi et al. (2007)
developed a numerical model based on the time-averaged continuity, cross-shore
momentum, longshore momentum, and energy equations to predict the longshore current
and sediment transport on a sand beach of alongshore uniformity under unidirectional
irregular breaking waves. For obliquely incident waves, the water particles in the run-up
flow move on saw-tooth trajectories with net longshore displacement. There are three
significant works on this procedure: Leont’yev (1999), Antuono et al. (2007), and Baba
and Camenen (2008). Leont’yev (1999) studied the contribution of the swash zone to the
total sediment transport and showed that the mean longshore transport velocity at the
shoreline is proportional to the net longshore displacement per wave period. Antuono et
al. (2007) investigated the integral properties of the swash zone and defined longshore
shoreline boundary conditions for wave- averaged nearshore circulation models and
found two main terms to contribute to the longshore drift velocity: (i) a drift-type term
representing the momentum transfer due to wave breaking; and (ii) a term proportional to
the shallow water velocity, accounting for short wave interactions, frictional swash forces
and continuous forcing due to non-breaking wave nonlinearities. Baba and Camenen
(2008) implemented a LST model for the swash zone in a beach evolution model based
on the N-line approach. The erosion and the accumulation around the shoreline are
clearly represented by the introduction of sediment transport in the swash zone. It was
found that sediment transport in the swash zone has an important effect on beach
evolution and could be one of the main contributors for the erosion/accumulation
25
Chapter 2
processes close to the shoreline. Bakhtyar et al. (2008) calculated the LSTR in the
nearshore using an Adaptive-Network-Based Fuzzy System (ANFIS). Their results reveal
that the ANFIS model provides higher accuracy and reliability for LSTR estimation than
empirical formulae.
The sediment transport models described include some aspects of a detailed deterministic
approach. The main short- coming of these models is that they give a wide range of
different predictions and, consequently, their reliability under changing wave conditions
is uncertain.
2.3 Process-based numerical modeling of swash zone sediment
transport
Numerical models become powerful tools for the understanding of sediment transport,
hydrodynamics and morphology in the coastal areas, yet most of the sediment transport
relationships between the sediment transport rate and flow parameters relations are based
on empirical and experimental studies. Process-based numerical models simulate the
major processes in the swash zone (interacting wave motion on the beach, coastal ground
water flow, sediment transport) using a hydrodynamic model coupled with as wash zone
sediment transport, beach profile change sand porous flow models. Different numerical
techniques have been devised and practiced. In the following sections, the numerical
methods frequently implemented in the swash zone analysis are reviewed.
2.3.1 Non-linear shallow water equations (NLSWE)
The solution to the shallow water wave equations is one of the classic problems for
coastal engineers. This model describes the evolution of water surface elevation and
depth-averaged velocity induced by small amplitude waves with large wave lengths
compared to the water depth.
The model assumes that the pressure distribution is
hydrostatic everywhere, i.e., there is no variation of flow variables with depth other than
the pressure. Swash hydrodynamics and run-up are traditionally modeled using the
NLSWE, a simplification to the full Navier-Stokes equations. One general form of the
NLSWE is
26
Chapter 2
(2.2)
(2.3)
(2.4)
where
is the total water depth;
and
are the cross-shore and longshore velocity
components.
Breaking waves and bore motions on a sloping beach were investigated by Carrier and
Greenspan (1958) and Shen and Meyer (1963). The focus was on the collapse of the bore
at the beach and the subsequent motion of the thin up-rush tongue and backwash flows.
These studies led to analytical descriptions of the location of the leading swash edge as a
function of space and time through ballistic motion equations and the shape of the swash
lens during its cycle. Using the analytical solution of Carrier and Greenspan (1958),
Baldock and Huntley (2002) and Jensen et al. (2003) described the run-up of standing
long waves and the run-up of non-breaking solitary waves, respectively. While these
investigations have shown that the analytical solution provides a good overall model for
motion at the shoreline, the internal hydro- dynamics are less well described. For
example, for real swash, flow reversal tends to occur later than predicted by the analytical
solution. Also, the prediction of flow depth is unrealistically small in comparison with
laboratory and field data (Baldock et al., 2005). Moreover, the swash prediction given by
the analytical solution is hydrodynamically similar for all swash events, i.e., the internal
flows are independent of the incident wave conditions at the seaward swash boundary
after the initial bore collapse. Guard and Baldock (2007) presented numerical solutions
for swash hydrodynamics for the case of breaking wave bores on a plane beach and found
significant difference from the standard analytical solution of Shen and Meyer (1963).
The results are important in terms of determining overwash flows, flow forces and
sediment dynamics in the swash zone and show that the analytical solution gives a very
shallow swash lens in comparison to the field measurements.
27
Chapter 2
Brocchini and Peregrine (1996) proposed a flow model in which swash zone motions are
described in terms of integral properties, i.e., spatially averaged over the swash width.
Their solution is a 3D extension of that given by Carrier and Greenspan (1958) for the
shallow water equations for a wave reflecting on an inclined plane beach. The integral
model seems very valuable for numerical integration, as long as details of swash zone
behavior are not required. When the full swash zone is included in a computation, it not
only involves a larger domain of integration with a special boundary condition at the
shoreline, but also frequently determines the maximum permitted time step.
The
changing position of the swash zone boundary and the longshore flow in the swash zone
may be determined. Archetti and Brocchini (2002) used numerical analyses to assess the
validity and potentialities of the integral swash zone model of Brocchini and Peregrine
(1996), which was extended to include seabed friction effects. They concluded that the
model was useful for two main purposes: (i) it can provide swash zone boundary
conditions for both wave-resolving and wave-averaging models of nearshore flows; and
(ii) an integral version of available sediment transport models, using as input conditions
the integral hydrodynamic properties computed by means of the proposed model, might
represent an improvement over currently used models as it would not require local values
of seabed friction inside the swash zone. Alsina et al. (2005) presented a numerical
model for sediment transport in the swash zone based on the classical ballistic motion for
the shoreline described by Shen and Meyer (1963), and the hydrodynamic-kinematic
model of Hughes and Baldock (2004). In the sediment transport module, the suspended
load is calculated by a Lagrangian scheme, whilst the variation of suspended sediment
concentration is computed with the advection–diffusion equation along particle
trajectories.
Kobayashi et al. (1989) and Kobayashi and Poff (1994) developed a 1D depth-averaged
nonlinear shallow water model, known as RBREAK, to predict the wave transformation
in the surf and swash zones on gentle slopes. The numerical simulations covered a range
of incident wave conditions between spilling and plunging waves. It has compared well
with laboratory data in terms of time-averaged hydrodynamic parameters. Dodd (1998)
developed an upwind finite volume scheme to solve the NLSWE for wave run-up and
overtopping. The model tends to over-predict the water depth on the revetment. Asano
28
Chapter 2
(1994) developed a numerical model to predict the flow characteristics in the swash zone
for obliquely incident wave trains. In his study, the 2D shallow water equations were
decoupled into independent equations each for on–off shore and for longshore motion.
Hu et al. (2000) presented a high-resolution NLSWE model for wave propagating in the
surf zone and wave overtopping of coastal structures. Although they indicated that the
use of NLSWE to model wave overtopping is computationally efficient, model has not
been tested for the up-rush of breaking wave and the detailed structure of wave breaking
is ignored. Shiach et al. (2004) implemented a numerical model based on NLSWE to
model a series of experiments examining violent wave overtopping of a near-vertical
sloping structure. They pointed out that this model needs to extend to include dispersive
terms for improving the model capability.
However, these models are unable to simulate details of the flow and turbulence fields
necessary for predictions of sediment transport in the swash zone. Raubenheimer (2002)
compared the observations and predictions of fluid velocities using nonlinear shallow
water equations in the surf and swash zones and proposed that velocity skewness, up-rush
and backwash velocities were over-predicted in the swash zone. Therefore, the
applicability of these equations to sediment transport modeling in the swash zone had not
been adequately investigated.
2.3.2 Boussinesq equations
Applications of the Boussinesq equations cover a variety of ocean and coastal problems
of interest: from wind wave propagation in intermediate and shallow water depths to the
study of tsunami wave propagation across large ocean basins (Sitanggang and Lynett,
2005). The governing equations consist of the 2D depth-integrated continuity equation
and the horizontal momentum equation. In the dimensional form, the nonlinear
Boussinesq equations are (Lynett et al., 2002)
)
{ [(
]
(2.5)
(
)
29
}
Chapter 2
(
[
)]
(2.6)
[
]
⁄ ;
where
elevation;
,
is local water depth;
is horizontal velocity vector and
is free surface
is the reference depth.
Many researchers have modified the Boussinesq equations. Madsen et al. (1997a, b)
discussed results from a Boussinesq-type wave model of swash oscillations induced by
bichromatic wave groups and irregular waves on gentle beach slopes. They speculated
that the shoreline motion consists of a significant low-frequency component at the group
frequency and individual swash of the primary waves.
Sørensen et al. (2004) presented a numerical model for solving a set of extended time
domain Boussinesq-type equations including the breaking zone and the swash zone. The
model is based on the unstructured finite element technique. The model has been applied
to a number of test cases, and found to compare well with laboratory measurements
showed good agreement.
The use of unstructured meshes offers the possibility of
adapting the mesh resolution to the local physical scale and reduces the number of nodes
in the spatial discretisation.
Kennedy et al. (2000) used a numerical model based on weakly nonlinear Boussinesq
equations with a slot-type shoreline boundary. The model was further enhanced to
improve numerical stability on steep beach slopes. Both infragravity and wind wave
frequency swash are significant on steep beach slopes, while their relative dominance
depends on the frequency of the incident waves. Karunarathna et al. (2005) studied
swash motions on steep and gentle beaches based on numerical simulations and found
swash excursions on any given slope were highest when individual bores from a partially
saturated surf zone rode on top of low-frequency waves. A poor correlation was found
30
Chapter 2
between swash excursion and the surf similarity parameter due to the involvement of
infragravity wave energy in the swash.
The Boussinesq hydrodynamic model of Rakha et al. (1997) was coupled with a bed-load
formulation to calculate changes across the beach profile.
It showed reasonable
agreement with observed elevation changes but under-predicted the observations.
Karambas (2006) investigated numerically the sediment transport rate in the swash using
a nonlinear wave model equation that incorporated infiltration/exfiltration effects. The
model is based on the Boussinesq equations and is able to describe breaking and nonbreaking wave propagation and run-up (Karambas and Koutitas, 2002). It was coupled
with a porous flow model to account for infiltration/exfiltration effects on the sediment
transport rate (Karambas, 2003). The authors suggest that their nonlinear model better
describes sediment motion than other simplified approaches.
Pedrozo-Acuña et al.(2006) presented a numerical–empirical investigation of the
processes that control sediment transport in the swash zone on steep gravel beaches. This
was based on a sensitivity analysis of a sediment transport/profile model driven by a
highly non-linear Boussinesq model that was compared to nearly full-scale measurements
performed in a large wave flume. Pedrozo-Acuña et al. (2007) extended their analysis to
compare these earlier results with those relating to a mixed sediment (gravel and sand)
beach. The parametric sensitivity analysis incorporated a discussion of the effects of
acceleration about which there is much debate. The sensitivity analysis suggests that
fluid acceleration can contribute to the onshore movement of sediment that causes
steepening of initially flat beach faces composed of coarse sediment. A complex balance
of processes is responsible for the profile evolution of coarse-grained beaches with no
single dominant process.
The accuracy of nearshore wave modeling using high-order Boussinesq-type models
compared with typical order models was examined by Lynett (2006), who used the highorder two-layer model of Lynett and Liu (2004). For regular wave evolution over a bar,
high-order models are in good agreement with experiments, correctly modeling the free
31
Chapter 2
short waves behind the step. Under irregular wave conditions, it was shown that highorder non- linearity is important near the breaker line and the outer surf zone.
Fuhrman and Madsen (2008) simulated nonlinear wave run-up with a highly accurate
Boussinesq-type model. A new variant of moving wet-dry boundary algorithms based on
so-called extra- polating boundary techniques were utilized in 2D. Computed results
involving the nonlinear run-up of periodic as well as transient waves on a sloping beach
were considered in a single horizontal dimension, demonstrating excellent agreement
with analytical solutions for both the free surface and horizontal velocity, with some
discrepancies near the breaking point.
2.3.3 Navier-Stokes equations (NSE)
Another framework for numerical simulation of wave breaking and wave run-up/rundown is the implementation of models based on the Navier-Stokes equations (NSE).
These equations have become more common with the improvement in computational
techniques and facilities.
The mass and momentum conservation equations are as
follows:
(2.7)
[
]
[
where
concentration;
]
is the kinematic pressure;
is the diffusivity;
product of , and
is the body force;
the velocity vector;
(2.8)
(2.9)
is a scalar quantity like
stands for the tonsorial
is a source term. Unlike the depth-averaged models, NSE are able to
simulate details of the flow and turbulence fields, and vertical velocities can be
determined directly.
32
Chapter 2
For free-surface flow simulations, it is important to numerically describe the moving
boundary. Several methods have been successfully incorporated in the NSE, e.g., the
marker and cell (MAC) method (Park et al., 1999), the volume of fluid (VOF) method
(Hirt and Nichols, 1981; Shen et al., 2004; Nielsen and Mayer, 2004), and the Arbitrary
Lagrangian–Eulerian (ALE) method (Zhou and Stansby, 1999). These methods can deal
with complicated free surfaces (e.g., breaking waves), yet their major drawback is that
they require strict stability requirements and are computationally expensive. The free
surface elevation can be calculated using either the free surface equation or kinematic free
surface boundary condition.
To better simulate the flow and turbulence fields at the time of wave breaking, all
hydrodynamic governing equations should be investigated.
In principle, Direct
Numerical Simulation (DNS) can be implemented for the simulation of wave breaking.
However, computational demands are high for DNS methods. Considering turbulent
flows with a high Reynolds number, such as wave breaking and wave run-up, since the
turbulence oscillations should be computed in very fine time steps, the computational
process would be time consuming. Also, it remains the case that even as computers
become more and more powerful, DNS is still possible only with low Reynolds numbers
in the foreseeable future. Another framework for numerical simulation of wave breaking
is the implementation of models based on the Reynolds-Averaged Navier-Stokes (RANS)
equations. In the RANS equations, the average motion of flow is described, and the
effects of turbulence on the average flow are considered by the Reynolds stresses. In
order to compute the Reynolds stresses and the turbulence characteristics, turbulence
closure models are used. One of the solutions to the analysis of the NSE and the closure
problem is the use of Boussinesq’s eddy viscosity. The eddy viscosity is a characteristic
defined by the local conditions of turbulence and hence is variable with time and location.
The linear eddy viscosity model considers the relation between the Reynolds stresses and
the rate of flow shape change. In order to acquire an approximation of local turbulence
conditions and the related parameters, one can obtain and solve the equations governing
the transformation of turbulence parameters
and
(
closure models). Liu and Lin
(1997) and Lin and Liu (1998a,b) developed a VOF-RANS model including a
turbulence closure scheme based on the non-linear Reynolds stress model to model the
33
Chapter 2
turbulence levels in the surf zone. They implemented the model to study the propagation,
shoaling and breaking in the nearshore, up-rush and backwash of wave train under
breaking waves and discussed the turbulence mechanism in the surf zone. Their results
yielded strong correspondence with free surface displacement and turbulence intensity
from a laboratory experiments. Lin and Liu (1999) proposed a new general method for
generating essentially any waves in a numerical wave tank based on the NSE by using
designed mass source functions for the equation of mass conservation. The precision of
this method in comparison with theories is very good. Although these models are able to
forecast free surface displacement, velocity and turbulent fields, Elfrink and Baldock
(2002) revealed that the resolution of these studies was too coarse to simulate the physical
processes like wave boundary layer in the swash zone.
Drago and Iovenitti (1995) used the eddy viscosity approach, evaluating the eddy
viscosity by a
equation model (where is
the turbulent kinetic energy (TKE) and
represents the turbulence eddy scale length). Though the parameters incorporated in
these models needed calibration, all researchers found their results corresponding to
laboratory data of wave height growth in the surf zone. They acquired good results for
spilling breakers but not for plunging. A key point in better comprehending the swash
zone hydrodynamics is finding the accurate velocity distribution in the inner surf and
swash zones.
Kothe et al.(1991) presented the RIPPLE computer program for modeling a transient, 2D,
incompressible fluid flow. The free surface was computed using the VOF method. Puleo
et al. (2002) studied breaking waves and run-up using RBREAK2 and RIPPLE models
and showed the RIPPLE model more accurately displays wave breaking and wave runup. However, the velocity estimates from the RIPPLE models how lag relationships as
compared to the laboratory measurements.
Bradford (2000) compared the performance of the
Renormalized Group extension of the
model, linear
model and a
model (RNG model) in the surf zone. It was
found that all these models predict wave breaking far earlier than that observed in
experiments, while also underestimating the undertows. Pope (2000) used a large eddy
34
Chapter 2
simulation (LES) approach, which results from the calculation of stresses at the
resolvable scales and modeling them at the sub-grid scales (SGS), since complex flows
and adverse pressure gradients cause difficulties for the
turbulence closure
schemes. Their model considered the swash zone but, like other surf zones studies,
emphasized wave breaking processes. Christensen and Deigaard (2001) developed the
numerical model to simulate the large-scale wave motions and turbulence induced by the
breaking process. Their hydrodynamic model has been combined with a free surface
model based on the surface markers method to simulate the flow field in breaking waves,
where the large turbulent eddies have been simulated by the LES method and the smallscale turbulence is represented by a simple Smagorinsky sub-scale model. Wood et al.
(2003) incorporated a VOF technique in a FLUENT code to model run- up of steep nonbreaking waves. While this model qualitatively explained the development of the wave
and the fluid velocity and acceleration during the up-rush, maximum run-up heights could
not be obtained owing to limited accuracy of the VOF algorithm. Puleo et al. (2003) used
LES to describe the turbulent eddy viscosity. In the model improvement, the effects of the
LES were neglected due to the small grid scales used, but there was an excellent
agreement for both sea surface and velocities in the inner surf and swash zones. Zhao et
al. (2004) used the multi- scale turbulence model to simulate breaking waves and found
good agreement with the wave set-up; however, the shape of the undertow profile does
not seem to follow the measured profiles in all cases. The turbulence level near the
breaking point was too high in all these studies. Christensen (2006) studied the LES of
spilling and plunging breakers based on a model solving the NSE and found that the
turbulence levels in general were too high compared with measurements, especially in
plunging breakers. Also, the model requires a very long computational time and a fine
grid to predict the details of hydrodynamics.
Zhang and Liu (2008) investigated
numerically the swash flows generated by bores using RANS model equations. Their
results showed that the weak bore does not break, while the strong bore breaks as a
plunger before it reaches the still-water shoreline.
Chopakatla1 et al. (2008) used
FLOW3D code to simulate 2D wave transformation and wave breaking and found good
agreement between modeled and observed wave height, mean cross-shore flow and wave
breaking variability. However, their model has been applied in the surf zone and not in
the swash zone. Bakhtyar et al. (2007, 2009) presented a 2D numerical model for the
35
Chapter 2
simulation of wave breaking, run-up and turbulence in the surf and swash zones. The
numerical simulations covered a range of incident wave conditions between spilling and
plunging waves. Their model provides a precise and efficient tool for the simulation of
the flow field and wave transformations in the nearshore area, especially the swash zone.
Drake and Calantoni (2001) presented a discrete particle model (DPM) for sheet flow
sediment transport in the nearshore zone. Due to memory requirements, they used only
1600 particles. Calantoni et al. (2006) used a VOF NSE solver (RIPPLE) to simulate
inner surf zone and swash zone flow with a 3-s wave period and wave height of 0.14m on
a planar, 1:10 sloping beach. In their work, RIPPLE was used to provide high-resolution
predictions of the pressure gradient and fluid velocity in the horizontal and vertical
directions, which were linked to a DPM. Coupling between RIPPLE and the DPM was
one-way such that particle– particle and fluid–particle interactions in the DPM did not
provide feedback to alter the flow predicted by RIPPLE. RIPPLE was derived from the
mean 2D NSE, and the governing equation used for translational particle motion was
(Madsen, 1991)
|
where
and
and
|
(2.10)
are, respectively, the particle and fluid densities;
are the particle and fluid velocities, respectively;
the drag coefficient;
is the particle volume;
is the fluid pressure;
is the projected area of the spherical particle and
is
represents the
forces from inter-particle collisions. The numerical simulation showed a significant
amount of sediment suspended locally under vortices that reached the bed. They
demonstrated the model’s ability to simulate sediment suspension events, while
producing high-resolution predictions of motions of each sediment particle in the
simulation.
In this chapter, we have discussed mainly process-based flow models.
Models of
sediment transport can also be more sophisticated than those based on a parametric
relationship between sediment transport rate and flow parameters. The above review of
the current research status demonstrates that cross- shore beach processes are intrinsically
36
Chapter 2
nonlinear, unsteady and coupled.
Therefore, in developing an improved modelling
approach, the key challenge is to resolve the temporal and spatial phase variations of the
fluid and sediment parameters.
Among the main processes, the least well-understood and most difficult to predict are the
dynamics of near-bed sediment motions on beaches. This is partly due to the lack of
detailed measurements of the flow and sediment transport in this region, and also
constrained by the weakness of the conventional local transport modelling approach in
calculating beach evolution. As shown by recent experimental and numerical studies
(Ribberink and Al-salem, 1995; Davies et al., 1997; Dong and Zhang, 1999), the local
models, whether they are based on the turbulent diffusion concept making use of an
empirically derived bottom reference concentration as the boundary condition or the
energetics concept, are too simplistic to truly represent the unsteady, nonlinear and twophase nature of the sediment motions.
Two-phase flow modelling is capable of simulating fluid and sediment phases separately
although the interphase coupling needs to be considered with some care. For the twophase flow model, the governing equations of fluid phase are generally described in
Eulerian form; whereas, the governing equations of the sediment phase can be written in
either Eulerian or Lagrangian form. Furthermore, by coupling the governing equations of
both phases, a system of the Eulerian equations or Euler–Lagrange coupled ones, is
obtained to analyze the sediment-laden flow.
In the Euler–Euler coupling model, the sediment phase is treated as a continuum, which
follows different constitutive laws to those for the clear water. In these models mainly
the fluid–particle interaction of bed-load is taken in to account; whereas, the fundamental
characteristics of the sediment motion cannot be expressed well. For this model, four
essential equations for the modelling of the mass and momentum fields and the sediment
are required.
These equations are valid for both phases; therefore two additional
equations are required for the mass and momentum exchanges between the phases
(Crowe,2006). The general form of the equations for phase is as follows (Crowe, 2006):
37
Chapter 2
(2.11)
̂]
∑[
(2.12)
(2.13)
∑
In these equations,
is the unit tensor,
[
̂]
.
is the stress tensor,
is the gravity acceleration,
the phases, ̂ is the unit normal to phase ,
(2.14)
is the thermodynamic pressure,
is the source of momentum between
is the velocity of the common interface and
is the velocity vector of each phase.
Sheet flows widely occur in the swash zones (Hughes et al., 1997). Since the sheet flow
in the swash zone is a highly concentrated combined flow of fluid and sediments under
high shear stress, the dominating mechanism is very complicated. The location of the
particles in the sheet flow is defined by the collision and the contact of the grains which
differs from the usual turbulence-generated suspension (Asano, 1990). Sheet flow is an
unsteady flow regime since it yields a vertical distribution and sporadic variations in the
velocity and concentration fields.
Over the last two decades, the two-phase flow
technique has been used by several researchers to model sediment transport in sheet flow
conditions. Asano (1990) presented a two-phase flow model based on the principles of
the Kobayashi and Seo (1985) model in which the vertical velocity of particles was
approximated by empirical relations. Ono et al. (1996) devised a model where the
horizontal velocities of the fluid and the particles where considered to be identical. Dong
and Zhang (1999) presented a two-phase flow model capable of simulating the fluid and
particle motions in the sheet flows and oscillatory conditions. Their model is based on
the principles of eddy viscosity model which is very restricted for modelling this complex
flow. Hsu et al. (2003, 2004) applied a two-phase flow model to steady open channel
38
Chapter 2
flow and unsteady oscillatory flow. Liu and Sato (2006) applied a two-phase flow model
to simulate the net transport rate under combined wave/ current flow and various
asymmetric sheet flow conditions. Their turbulent enclosure model was based on the
parabolic eddy viscosity distributions.
To improve the Eulerian models deficiency, a granular material model can be employed
to simulate the inter-particle collision mechanism of the bed-load transport, in Euler–
Lagrange coupling model. A major development in modelling two-phase flow was use of
Discrete Element Method (DEM) (Cundall and Strack, 1979) to simulate sediment
transport during sheet flow in the swash zone as the motion of granular materials. In this
approach, inter- particle collisions and forces can be quantified in great detail. Gotoh and
Sakai (1997) performed pioneering work on simulation of the bed-load from the
viewpoint of granular material dynamics.
In this model, the Lagrangian sediment
behaviour is modelled based on the DEM. Yeganeh-Bakhtiary et al. (2000) presented an
Euler–Lagrange coupling two-phase flow model to bed-load transport under high bottom
shear. Although the predominant particle–particle interaction is described in their model,
the sediment particle has been traced as moving disk in the 2D coordinates, which has
different character than real sand grains.
To date, the existing two-phase flow approaches are focusing on describing timedependent and time-averaged concentration distributions. For practical purposes, the
magnitude and direction of net sand transport are more attractive and important. This
review shows that none of the existing numerical models can describe the wave breaking
satisfactorily and none of them studied the surf and swash zones mutually and
comprehensively. In particular, none has been verified carefully for both the turbulent
and mean velocity field.
39
CHAPTER 3
THE IH-2VOF NUMERICAL MODEL.
Numerical models of fluid/wave-structure interactions are increasingly becoming a viable
tool in furthering our understanding of the complicated phenomena that govern the
hydraulic response of breakwaters, including effects of permeability (Losada, 2003).
These include Lagrangian models with particle-based approaches such as the Moving
Particle Semi-Implicit method (Koshizuka et al., 2004) and Smooth Particle
Hydrodynamics (Dalrymple et al., 2009).
For reasons ranging from computational
efficiency to an accurate representation of the physical processes, Reynold Averaged
Navier Stokes-Volume Of Fluid (Rans-Vof) models have become an attractive choice to
model wave interactions with both solid as well as porous structures. This kind of models
solves the 2DV Reynolds Average Navier–Stokes (RANS) equations, based on the
decomposition of the instantaneous velocity and pressure fields, into mean, and turbulent
components and the free surface movement is tracked by the Volume of Fluid (VOF)
method.
Lin and Liu (1998), based on a previously existing model called RIPPLE (Kothe et al.,
1991; originally designed to provide a solution of two-dimensional versions of the
Navier-Stokes equations in a vertical plane with a free surface), presented COBRAS
(Cornell Breaking Waves and Structures) for simulating breaking waves and wave
interaction with coastal structures. The model has been under a continuous development
process based on an extensive validation procedure, carried out for low-crested structures
(Garcia et al., 2004, Losada et al., 2005; Lara et al., 2006a), wave breaking on permeable
slopes (Lara et al., 2006b), surf zone hydrodynamics on natural beaches (TorresFreyermuth et al., 2010) and overtopping on rubble mound breakwaters and low-mound
breakwaters (Losada et al., 2008; Lara et al., 2008). In this work, a modified and
improved version of COBRAS, named IH-2VOF (Lara et al., 2011), is used.
Chapter 3
In this chapter a synthetic description of the main features of the IH-2VOF (mathematical
formulation, boundary and initial conditions, computational domain, wave generation
method, free surface tracking method and numerical resolution) are presented. The
description is mostly based in Liu and Lin (1997), Lin and Liu (1998), Hsu et al. (2002)
and Lara et al. (2011).
3.1 Governing equations in the fluid domain: the Reynolds
Averaged Navier-Stokes (RANS) equations
The governing Reynolds Averaged Navier-Stokes (RANS) equations of the IH-2VOF
model represent the general principle of mass and momentum conservation. Due to the
less assumption involved in the governing equations, RANS models are able to simulate
the frequency dispersive nature of gravity waves in deep waters as well as the nonlinear
wave transformations in shallow waters over a sloping bottom. With the inclusion of a
proper turbulent model, they are able to describe difficult wave problems such as
breaking waves, energy transfer between wave components, wave-current interactions
and wave-structure interactions. The refined information on the velocity, pressure and
turbulence field makes them suitable to study surf zone hydrodynamics. Wave breaking
and its evolution along the surf zone are directly solved without any imposed forcing.
The IH-2VOF model solves the two-dimensional Reynolds Averaged Navier-Stokes
(RANS) equations base on the assumption that in a turbulent flow the instantaneous
velocity field
and the pressure field
can be split in two parts, the ensemble-averaged
(mean) velocity and pressure components, ̅ and
pressure fluctuations,
and
:
̅
where
̅ , and the turbulent velocity and
̅
(3.1)
for a bidimensional flow.
Applying the former decomposition to the Navier-Stokes equations and assuming
incompressible fluid, the RANS equations are derived:
41
Chapter 3
̅
̅
where
(3.2)
̅
̅
̅
̅
(3.3)
is the i-th component of the gravitational acceleration,
is the density of the
fluid, ̅ is the sum of the viscous stress tensor of the mean flow and the Reynolds stress
tensor. For a Newtonian fluid
̅
where
̅
(3.4)
is the Reynolds stress tensor and ̅ is the rate of
is the molecular viscosity,
strain tensor of mean flow given by:
̅
The Reynolds stress term
(
̅
̅
)
(3.5)
in the momentum equation represents the influence of
turbulent fluctuations on the mean flow
(̅ ̅ )
(3.6)
In the IH-2VOF model, the Reynolds stress tensor is assumed to be related to the strain
rate of the mean flow through the algebraic non-linear
Lin and Liu, 1998):
42
model (Shih et al., 1996;
Chapter 3
̅ ̅
(
̅
(
(
[
in which
,
,
and
̅
̅
̅
̅
̅
̅
(
̅
̅
̅
̅
̅
)
̅
(3.7)
̅
̅
)
)
̅
are empirical coefficients,
)
]
is the Kronecker delta,
is the
turbulent kinetic energy and is the dissipation rate of turbulent kinetic energy.
(̅ ̅ )
̅
(
where
)
(3.8)
(3.9)
is the molecular kinematic viscosity.
The condition:
(3.10)
in equation (3.7) leads to the conventional linear (isotropic) eddy viscosity model for the
Reynolds stresses closure:
̅ ̅
with
̅
(3.11)
is the eddy viscosity expressed as:
(
)
43
(3.12)
Chapter 3
Contrary to the conventional eddy viscosity models as expressed by equation (3.12), valid
for the description of isotropic-eddy-viscosity turbulent flows, the non-linear Reynolds
stress model implemented in the IH-2VOF model and expressed by equation (3.7) can be
applied to general anisotropic turbulent flows.
The values for the coefficients
and
are obtained from experimental results on
turbulent shear flow by Champagne et al. (1970). A value for
(1980). Finally, the
is proposed by Rodi
coefficient is deduced from the assumption by Shih et al. (1996):
. The whole set of coefficients are summarized as follows:
(3.13)
However, considering constant values for these coefficients may lead under some
extreme circumstances to inconsistent physical situations in equation (3.7), such as
negative turbulence energy or infinite non-linear contributions. Hence, modified
expressions for the empirical coefficients have been implemented in the IH-2VOF model:
(
)
(3.14)
where
[|
̅
|]
[|
̅
all coefficients take their originally proposed values when
The governing equation for
̅
|]
(3.15)
and
are zero.
and are (Rodi, 1980; Lin and Liu, 1998):
[(
)
]
44
(̅ ̅ )
̅
(3.16)
Chapter 3
̅
[(
)
]
(
̅
̅
)
̅
(3.17)
In equation (3.16), the first and second terms of the left-hand side represent convection
and diffusion respectively. The second and third terms of the right-hand side represent
production and dissipation of kinetic turbulent energy respectively. The Reynolds stress
only appears as the turbulence production term. The empirical coefficients in equation
(3.17) have been determined by performing many simple experiments. Recommended
values for these coefficients are (Rodi, 1980; Lin and Liu, 1998):
(3.18)
As outlined by Lin and Liu (1998), the RANS equations along with the k −ε transport
equations using the former values for the empirical coefficients were found to adequately
simulate many complex turbulent flows.
3.2 Initial and boundary conditions
For the initial time and at boundary of the spatial domain, additional constrains or
equations are required by the physics of the problem.
3.2.1 Initial conditions
The model considers as initial conditions for the mean flow in the whole domain still
water with no wave or current motion, i.e. zero velocities and hydrostatic pressure. For
the turbulence field, due to the fact that the production term in the
proportional to
equation is
itself, no turbulence will be produced if the initial value for k is zero.
Therefore, a finite but very small value of
is imposed. This initial value (“seed”) for the
turbulence energy produces a numerical perturbation:
(3.19)
45
Chapter 3
with
, where
is the wave celerity in the generation zone and
is a constant
equal to 0.0025 (Lin, 1998).
For the turbulent dissipation rate, the model considers the following expression:
(3.20)
where
and is a constant equal to 0.1 (Lin, 1998).
Variation of the
and
values were found to have a negligible effect on the final results
of the computation (Lin, 1998). Numerical simulations performed by Lin and Liu (1998)
showed that the influence of
on the flow conditions reduces to a slight delay in the
initiation of breaking for smaller values of .
3.2.2 Boundary conditions
As regards boundary conditions, it is possible identify three different cases.
1. Solid boundaries
At the solid boundaries two types of conditions for the mean flow can be considered:
̅̅̅̅
̅̅̅̅
with
̅̅̅
̅̅̅
(3.21)
(3.22)
and are the directions normal and parallel to the boundary respectively.
In the case of turbulent flows, the model considers a log-law distribution for the mean
tangential velocity in the turbulent boundary layer:
46
Chapter 3
̅
where
and
(3.23)
is the von Karman constant
,
is the distance from the solid domain
is a friction velocity. Integrating equation (3.23) and assuming that production and
dissipation are equivalent in the boundary layer leads to the following equations for
and
at the solid boundary:
(3.24)
√
(3.25)
The value of
is obtained from the values of mean flow:
̅
|
(3.26)
1. Free surface
The application of an appropriate boundary condition at the mean free surface in turbulent
flows is quite complex as the mean free surface is not clearly defined (Brocchini and
Peregrine, 1995; Lin and Liu, 1997; Lin and Liu, 1998). In the IH-2VOF model, the mean
density fluctuations near the free surface due to mixing and air intrusion are neglected,
and similarly to situations of laminar flow, the zero stress and zero pressure conditions
are imposed at the free surface:
̅
̅̅̅
(3.27)
For the turbulent field, the zero-gradient boundary condition is applied for both
on the free surface:
47
and
Chapter 3
(3.28)
with
the unit normal on the free surface, based on the assumption of no turbulence
exchange between water and air.
With respect to the lateral conditions, the model can consider a closed boundary, regarded
as a solid boundary, in which the conditions described before are applied, or allow the
flow to go out of the domain, as an open boundary or radiation condition.
1. Open boundaries
The open boundary condition in the IH-2VOF model is expressed as:
(3.29)
where
represents the variable to be evaluated ̅ ̅
and
the wave celerity
at the considered position expressed as
√
(3.30)
for long waves, and
√
for short waves, where
(
)
is the wave amplitude,
is the water depth and
(3.31)
is the wave
length for this depth. This radiation condition has been checked to adequately reproduce
theoretical results for nonlinear waves at indefinite and intermediate depths
⁄
.
More details on the boundary conditions can be found in Rodi (1980) and Liu and Lin
(1997).
48
Chapter 3
3.3 Free surface tracking trough the volume of fluid (VOF)
method
In the COBRAS model, the free surface is tracked using the Volume Of Fluid (VOF)
method. The method does not consist in pursuiting the exact location of the free surface,
but in identifying the free surface location tracking the density change in each cell. The
model identifies the different cell types: empty (E), surface (S) or interior (I) cells
depending on the value of the VOF function defined as follows:
(3.32)
where
(3.33)
being
the fluid density,
the volume of fluid in the cell and
the cell. Interior, empty and surface cells are defined as the
the volume of air in
,
and
cell
respectively. Cell types are shown in the following figure.
The introduction of the VOF function in the equation of mass conservation yields the
transport equation for
:
(3.34)
̅
̅
(3.35)
Hirt and Nichols (1981) developed an algorithm to avoid errors in the convection of
Their method consists in evaluating
.
gradients in both directions in order to identify the
free surface location. Lin (1998) improved the algorithm with a new method that solves
the problem of the
cells. A detailed analysis of the VOF method can be found in
Hirt and Nichols (1981) and Lin (1998).
49
Chapter 3
3.4 Partial cell treatment
The IH-2VOF model allows the introduction in the computational domain of solid
boundaries of arbitrary shape, using a partial cell treatment. This method avoids the
potential spurious reflection at solid boundaries defined as sawtooth-shape surfaces fitting
the cell boundaries (Lemos, 1992). It consists in modelling the solid object as a special
case of fluid with an infinite density, introducing openness functions at the cell centre and
at the cell faces. At the cell centre,
is defined as the ratio of space not occupied by the
solid object (thus open to the fluid) to the whole cell area. On the cell faces,
( ) is
defined as the length open to the fluid to the whole length of the right (top) boundary.
Therefore, similar to the VOF free surface tracking method, the model can identify
whether the cell corresponds to the solid object or obstacle (O), the fluid (air)-solid
boundary (FA-O) or the fluid (air) domain (FA). In order to solve the magnitudes defined
at the right face of the cells, the parameter identifies whether the cell face belongs to a
solid boundary or not. The only difference between the VOF function and the openness
functions is that the former is time-varying and the latter are not.
3.5 Governing equations for the flow in porous media (VARANS
equations)
To make the fluid/porous structure interaction modelling easier, a volume-averaging
process has been applied to the RANS and the
and
equations. The flow in porous
media is obtained in the IH-2VOF model through the resolution of the Volume-Averaged
Navier-Stokes (VARANS) equations. These equations are derived by integration of the
RANS equations over a control volume. The size of the averaging volume is chosen
much larger than the characteristic pore size but much smaller than the characteristic
length scale of the flow, i.e. the scale of the spatial variation of the physical variables in
the fluid domain.
The mathematical process of volume averaging of a certain quantity “a” is defined by the
following expression:
50
Chapter 3
〈 〉
∫
(3.36)
where “〈 〉” denotes the intrinsic volume averaging,
is the volume in
is the total averaging volume,
which is occupied by the fluid phase and 〈 〉 is the averaged
magnitude. The intrinsic averaging operator defined by equation (3.36) can be related to
the Darcy’s volume averaging operator defined as follows:
〈 〉
∫
(3.37)
through the simple relationship:
〈 〉
〈 〉
(3.38)
where
(3.39)
is the porosity and is assumed for simplicity to be a constant in the present model. In
terms of velocity, 〈 〉
would be the seepage velocity and 〈 〉 the filtration velocity.
Hereafter, unless specified, volume averaging will be understood as intrinsic volume
averaging, as defined by expression (3.36).
To quantify the flow within the porous medium, the pore Reynolds number is defined as:
| |
where
is the equivalent mean diameter of the porous material,
viscosity and | | is a typical velocity scale around the pore.
51
(3.40)
is the fluid kinematic
Chapter 3
The VARANS equations are obtained by applying the intrinsic volume average to the
RANS equations. The ensemble averaged velocity of the RANS equations is assumed to
be:
〈̅ 〉
̅
̅
(3.41)
where 〈 ̅ 〉 is the ensemble-volume averaged velocity field and ̅ is the fluctuation with
respect to volume averaging, in other words the residual velocity field between ensemblevolume averaging and ensemble averaging.
Applying this decomposition to the equations of continuity (3.2) and momentum
conservation (3.3), we obtain:
〈̅ 〉
〈̅ 〉
〈 ̅ 〉〈 ̅ 〉
〈 ̅〉
(3.42)
〈̅ ̅ 〉
∫[ ̅ ̅
〈 ̅ 〉
̅
is the total fluid-solid interface and
̅
〈̅ ̅ 〉
(3.43)
]
the jth component of the unit vector pointing
normally outward from the fluid to solid phase. The last term in equation (3.43) accounts
for the jump at the interface and represents the interfacial momentum transfer between the
fluid phase and the solid skeleton. This term is crucial in the modelling of the flow in
porous media. The previous term in the equation (3.44) is the residual stress due to
volume averaging: it results from the volume averaging of the convective term ̅ ̅ as:
〈̅ ̅ 〉
〈 ̅ 〉〈 ̅ 〉
〈̅ ̅ 〉
analogously to the stress term in the Reynolds decomposition of the product
(3.44)
. These
last two terms are unclosed and need to be modelled. Inside the porous media, these
52
Chapter 3
terms are modelled collectively using the Forchheimer’s relationship with the inclusion of
unsteady effects (Liu et al., 1999a):
〈̅ ̅ 〉
̅
∫[ ̅ ̅
̅
]
(3.45)
〈̅ 〉
[
where
√〈 ̅ 〉
is the added mass coefficient and
〈̅ 〉
]
〈̅ 〉 〈̅ 〉
and
two empirical coefficients associated
with the linear and nonlinear drag force respectively. The third term of the right-hand
side of equation (3.45) accounts for the inertial effects.
The precise descriptions of the
,
and
coefficients are still not fully understood.
They depend a priori on the pore Reynolds number and flow directions. In their recent
study of wave motions and turbulent flows in front of a composite breakwater using the
COBRAS model, Hsu et al. (2002) propose the following values for these coefficients,
based on previous works by van Gent (1994), Liu et al. (1999a) or Nield and Bejan
(1999):
(3.46)
Finally, given the former closure expression for the residual stress term due to volume
averaging and the momentum transfer at the interface of equation (3.38), the complete
VARANS equations (3.38) and (3.39) can be rewritten as:
〈̅〉
〈̅〉
〈̅〉
〈̅〉
[
(3.47)
〈 ̅〉
〈̅̅̅̅̅〉
〈̅̅̅〉
]
(3.48)
[
〈̅〉
√〈 ̅ 〉
53
〈̅̅̅〉 〈 ̅ 〉]
Chapter 3
In the free fluid region, i.e. with
and
, the VARANS equations obviously
return to the original RANS equations.
The volume-averaged Reynolds stress is closed using an assumption similar to Shih et al.
(1996):
〈̅ ̅ 〉
(
〈 〉
̅
〈 〉
〈 〉
̅
(
(
[
̅
̅
̅
̅
̅
〈 〉(
̅
̅
̅
̅
̅
)
̅
(3.49)
̅
̅
)
)
̅
)
]
where 〈 〉 is the volume-averaged eddy viscosity, 〈 〉 is the volume-averaged turbulence
kinetic energy and 〈 〉 is the volume-averaged turbulent dissipation rate.
Similarly to the
model, the volume-averaged eddy viscosity is expressed as:
〈 〉
with
〈 〉
〈 〉
(3.50)
a coefficient depending on the local strain rate.
Equations (3.49) and (3.50) can be regarded as the result of a first-order approximation of
the volume averaging of the original nonlinear eddy viscosity model expressed by
equations (3.7) and (3.12). Any higher correlations related to the volume averaging
process have been ignored.
The governing equations for the turbulence in the porous media can be similarly obtained
by taking the volume-averaged turbulent kinetic energy 〈 〉 and its dissipation rate 〈 〉
can then be written as:
54
Chapter 3
〈 〉〈 ̅ 〉
〈 〉
〈̅̅̅̅̅
〈 〉
〈 〉
〈 〉
̅
〈̅〉
〈̅̅̅̅̅〉
[(
)
〈 〉
]
〈 〉
(3.51)
〉
∫
〈 〉〈 ̅ 〉
(
)
〈 〉 ̅̅̅̅̅ 〈 ̅ 〉
〈
〉
〈 〉
〈 〉 ̅̅̅̅̅
〈
〈 〉
̅
[(
〈̅
〉
)
〈 〉
]
(3.52)
〉
∫
(
)
〈̅
〉
in which the following decompositions related to volume averaging have been assumed:
̅̅̅̅̅
〈 〉
(3.53)
〈 〉
(3.54)
̅̅̅̅̅̅̅̅
〈
〉
̅̅̅̅̅
(3.55)
In both (3.51) and (3.52) equations, the fourth term on the right-hand side represents an
additional source term due to volume averaging. These terms describe the effects of
turbulence at a scale smaller than the volume-averaging scale, for instance the turbulence
generation in the wake region around the solid skeleton in high pore Reynolds number
conditions.
The fifth term represents the interfacial exchange of turbulence and is viewed as an
additional source or sink of turbulence due to the presence of solid materials. These two
terms are modelled collectively according to Nakayama and Kuwahara (1999),
substituted by
in the 〈 〉 transport equations and by the term:
(3.56)
in the 〈 〉 transport equation.
55
Chapter 3
The last term on both 〈 〉 and 〈 〉 equations is the additional diffusion term due to the
volume averaging and can be combined with the existing diffusion terms in the 〈 〉 and
〈 〉 equations. The overall effect can be modelled by adjusting the values of
However, due to a lack of experimental information, the values of
and
and
.
are kept
unchanged in the IH-2VOF model. Therefore, the equations (3.51) and (3.52) can be
rewritten as:
〈 〉
〈 〉
〈 〉〈 ̅ 〉
〈 〉〈 ̅ 〉
〈̅ 〉
〈̅̅̅̅̅〉
[(
)
〈 〉 ̅̅̅̅̅ 〈 ̅ 〉
〈
〉
〈 〉
〈 〉
]
[(
〈 〉
)
(3.57)
〈 〉
]
(3.58)
〈 〉
〈 〉
The values of the closure coefficients, due to a lack of information here again, are kept
the same as those proposed by the standard
model equations and the nonlinear eddy
viscosity model.
Expressions for the small-scale turbulence terms
and
have been proposed by
Nakayama and Kuwahara (1999), resulting from numerical simulations of flow passing
an array of square rods for pore Reynolds numbers between 105 and 107. The work by
Nakayama and Kuwahara (1999) for this range of
values led to the following closure
forms:
〈̅̅̅〉
〈̅̅̅〉
〈̅̅̅〉
〈̅̅̅〉
To date, no work on the effect of the small-scale turbulence and expressions of
(3.59)
(3.60)
and
for small values of the pore Reynolds number is available in the literature. However, it
can be verified that the small-scale turbulence represented by equations (3.59) and (3.60)
56
Chapter 3
has a negligible effect when the pore Reynolds number becomes small. Equations (3.59)
and (3.60) are thus expected to be still applicable in low pore Reynolds number
conditions.
3.6 Schematics of computational domain
The computational domain in the IH-2VOF model is discretised in rectangular cells as
sketched in Figure 3.1. The computing mesh can be divided into sub-mesh regions,
which allows a variable cells spacing: a finer grid can be defined for the representation of
specific study zones.
The different quantities in each of the cells are defined as follows: all scalar quantities,
i.e. pressure ( ), turbulent kinetic energy ( ), dissipation rate ( ), VOF function ( ) and
the hereafter specified openness function ( ) are defined in the centre of the cells. The
vector and vector-related quantities, i.e. the components of the mean velocity ̅ and the
additional openness functions (
and
), are defined on the cell faces as shown in
Figure 3.2. The x-component of the mean velocity is defined at the left face, the ycomponent of the mean velocity is defined at the top face.
The IH-2VOF model allows the introduction in the computational domain of solid
boundaries of arbitrary shape, using a partial cell treatment. This method avoids the
potential spurious reflection at solid boundaries defined as sawtooth-shape surface4s
fitting the cell boundaries (Lemos, 1992). It consists in modelling the solid object as a
special case of fluid with an infinite density, introducing openness functions at the cell
centre and at the cell faces. At the cell centre,
Figure 3.1 and Figure 3.2 illustrate the definition of the different cell types based on the
information of the VOF function and openness functions.
57
Chapter 3
Figure 3.1. Schematics of computational domain with the different cell types on the information of the VOF
function and definition of the computed magnitudes (Lin, 1998).
Figure 3.2. Schematics of solid boundaries definition through the partial cell treatment (Lin, 1998).
58
Chapter 3
3.7 Numerical resolution
The finite difference method is used to solve the governing equations. A grid of points at
fixed locations is introduced in the spatial domain and the dependent variables are
initially defined and subsequently computed at these points. Approximate expressions for
derivatives appearing in the governing equations are found. These approximations are
formed using differences of dependent variables over finite space and time intervals. The
time step used in advancement of the evolution equations is the time scale of the
averaging of original Navier-Stokes equations. Then a system of algebraic equations that
approximates the governing partial differential equations is constructed.
The RANS equations are solved using the finite differences two-step projection method.
1. The first step is to introduce an intermediate velocity u~i that does not satisfy the
continuity equation
̃
(3.61)
where the superscripts denote the time level. This is the forward time difference
equation of the momentum equation in the RANS equation without the pressure
gradient term.
2. The second step is to project the intermediate velocity into a divergence free
plane to obtain the final velocity
̃
(3.62)
(3.63)
59
Chapter 3
Taking the sum of the first and second step equations, the RANS momentum
equations are satisfied with the pressure gradient being evaluated at the (n+1)-th
time level
̃
(3.64)
Taking the divergence of second step equation and applying continuity equation
yields we have
(
̃
)
(3.65)
which is called the Poisson Pressure Equation (PPE).
In the two-step projection method, the spatial derivations of the velocity components and
the pressure field need to be expressed in finite-difference forms. The convection terms
are discretised by the combination of the central difference method and upwind method.
The combination of both is aimed at preventing their respective drawbacks of significant
numerical damping and numerical instability. A weighting factor is introduced in the
spatial derivative discretisation expressions in order to adjust the influence of each one of
the two schemes in the computation and to obtain stable and accurate solutions. Only the
central difference method is employed to discretise the pressure gradient terms and stress
terms.
Similarly to the Reynolds equations, the
equations are solved by discretising the
convective terms with the combined central difference and upwind methods.
The detailed implementation of the numerical model can be found in Liu and Lin (1997)
and Lin (1998).
60
Chapter 3
3.8 Resolution procedure
The basic resolution procedure of IH-2VOF to update the field variables at a given time
step is summarised as follows:

compute intermediate velocities ̃ using equation (3.51);

apply the boundary conditions at the free surface and definition of the source
function;

compute the pressure value from equation (3.55);

obtain the final values of velocities from equation (3.54);

apply newly the boundary conditions at the free surface;

update VOF function values;

apply the boundary conditions at the newly fluid cells (empty at the previous time
step) consecutively to VOF function updating.
61
CHAPTER 4
REPRESENTATION OF THE WAVE
OVERTOPPING PROCESS.
The goal of this chapter is to analyze the overtopping process, in particular the layer
thicknesses and the velocities of the flow over the crest of a emerged/submerged
structures, in order to extend the existing theoretical approaches and provide criteria for
the design of structures close to mean sea level or overwashed. This analysis was carried
out on a numerical database derived by running the Rans-Vof code (IH-2VOF) in
presence of impermeable structures characterized by different slopes and freeboard.
In particular, we chose to investigate seadikes in order to compere our work with the
extensive work that it is possible find in literature (Schüttrumpf, 2001; Van Gent,2002;
Schüttrumpf and Van Gent, 2003; Schuttrumpf and Oumeraci, 2005; Eurotop, 2007).
However, the analysis can be extended to other types of coastal defences, for example,
breakwaters that are more frequently present in Italian coasts.
Figure 4.1. Severe wave overtopping at the Samphire Hoe seawall, UK (CLASH project, www.clash-eu.org,
2001).
Chapter 4
4.1 Design criteria for coastal defences structures
Different types of coastal structures are built worldwide to protect low lying areas from
coastal flooding in coastal areas. Steep sea walls (slopes between 1:1.5 and 1:3) are used
in some coastal areas which are in general constructed of concrete, blocks or placed
stones. In Belgium, the Netherlands, Denmark, Germany and Poland, smooth seadikes
(slopes between 1:3 and 1:7) which are built of clay and sand are preferred. Thus, the
failure mechanisms of seadikes and seawalls differ significantly and different design
methods are used. At present, wave overtopping is the most important design criteria for
seawalls, while wave run-up still represents the most important design criteria for
seadikes.
The history of seadike design has recorded many heavy storm surges and dike breaches
up to the middle of the last century. The heavy storm surges in the Netherlands in 1953,
and in Germany in 1962, with many fatalities in both countries, have significantly
changed the design philosophy of seadikes. The design of seadikes was essentially based
on the highest water level ever observed plus a safety margin during previous centuries.
This procedure was replaced by a deterministic design philosophy as a consequence of
the storm surges in 1953 and 1962 and the crest level of seadikes are now determined
using a design water level with specific exceedance frequencies and the corresponding
wave run-up height.
Nevertheless, a design water level with specific exceedance
frequencies and the corresponding wave run-up height are subject to large uncertainties,
so that wave overtopping cannot always be avoided. Therefore, wave overtopping has to
be considered for the design of seadikes and a deterministic design philosophy should be
replaced by a probabilistic design philosophy for seadikes in the future.
Probabilistic design requires an improved under-standing of all physical processes
responsible for dike failures. One important aspect is the determination of those wave
overtopping parameters which are relevant for infiltration, erosion and consequently dike
failures. The present design of seadikes does not consider the interaction of wave
overtopping flow and soil mechanic properties. The design water level is calculated from
measured water levels and extrapolated by fitted statistical distributions including an
exceedance frequency which is specified as a function of the local conditions and the
63
Chapter 4
vulnerability of the flood prone area. The corresponding wave run-up height is calculated
from the incoming wave parameters at the design water level. Important investigations
about the determination of the wave run-up height were carried out by Wassing (1957),
Hunt (1959), Battjes (1974), Owen (1980), Tautenhain (1981) and Van der Meer and
Janssen (1995). The wave run-up and wave overtopping formulas by Van der Meer and
Janssen (1995) have now been adopted in many national recommendations for the design
of seadikes.
As mentioned above, a wave run-up height and an average overtopping rate are
inappropriate parameters for the determination of infiltration and erosion. Therefore, all
processes relevant to the wave overtopping flow parameters have to be determined.
These processes must be understood to describe the overtopping flow which will provide
the hydraulic boundary conditions for erosion, infiltration and slip failure analysis.
Empirical equations describing wave overtopping processes in terms of incident wave
conditions, structure geometry, and crest freeboard have been developed based on smalland large-scale physical model tests of common structure geometries. In particular, in
literature it is possible to find some theories that describe flow depths and velocities over
the dike crest for emerged conditions, whereas for zero freeboard and submerged
conditions a theoretical approach that allows to know the flow characteristics over the
dike crest is not available. Economic constraints and environmental and aesthetic impact
often impose more practical levee designs having lower crown elevations with the
associated risk that some wave/surge overtopping will occur during extreme events. In
addition, the increase of frequency and intensity of storms, combined with the
uncertainties related to extreme events and climate change, pose serious challenges in the
long term design of defences from coastal flooding. Therefore it is more likely that in the
next future many dikes will operate for longer times at lower crest freeboards, i.e. close to
mean sea level or even overwashed. For design purposes, accurate estimates of the
statistics of overtopping waves in terms of flow depths, duration and especially velocities
for a set of climate conditions are needed and have to be combined with consolidated
criteria for identifying tolerable overtopping threshold. At the moment, the information
64
Chapter 4
available for overwashed dikes can be derived from the (limited) set of tests performed by
Hughes et al. (2012).
As regards emerged structures, recently formulae have been derived for maximum flow
depth and velocities on the crest and inner slope. These formulae are based on the
difference fictive wave run-up and the crest freeboard. This is a good measure to
determine the flow depths and velocities on the dike.
These formulae have been
calibrated by two independent physical model test programs in different wave flumes.
One of the studies has been carried out by Schüttrumpf in Germany (Schüttrumpf, 2001).
He performed large and small scale model tests and determined the empirical coefficients.
The other study has been carried out on small scale by Van Gent (Van Gent, 2002). He
calibrated the formulae and determined the empirical coefficients as well. When these
two studies are compared appears a large difference. In fact, the empirical coefficient of
the flow depth equation was determined to be a factor 2.2 higher by Schuttrumpf than by
Van Gent, whereas they agreed about the empirical coefficient of the velocity equation.
They collectively wrote a paper (Schuttrumpf and Van Gent, 2003) and found the test setup as primary cause for the discrepancy in the flow depth coefficient (see also
Schuttrumpf and Oumeraci, 2005).
Bosman et al. (2008) used the raw data of
Schüttrumpf (2001) and van Gent (2002) and analyzed them using a new approach. He
shown that the outer slope is of great importance in the flow depths and velocities on the
crest. Whereas Schüttrumpf performed his tests on a dike model with an outer slope of
1:6, Van Gent used a dike model with outer slope of 1:4. Bosman verified that the
empirical coefficients are depending on the outer slope steepness. Therefore formulae for
maximum flow depth and velocity were adapted: the coefficients are written as a function
of the outer slope angle.
Moreover full-scale testing of dike landward slopes under a given sequence of
overtopping wave volumes through the Wave Overtopping Simulator (Van der Meer et
al., 2006, 2008, 2009) significantly extended our knowledge of landward-side dike
resiliency (Hoffmans et al., 2008; Van der Meer et al., 2010). However, the WOS
requires an accurate estimate of the wave overtopping volumes and related statistics to
65
Chapter 4
allow in cascade an accurate reproduction of flow depths and velocities on landward
slopes.
All formulae reported in this chapter are proposed by Eurotop manual (2007) which
summarizes all the existing theories.
4.2 Overtopping over a seadikes
Wave overtopping occurs if the crest level of the dike is lower than the highest wave runup level. In that case, the freeboard
water level
, defined as the vertical difference between the still
and the crest height, becomes important. Hence wave overtopping
depends on the freeboard
and increases for decreasing freeboard height
.
An exact mathematical description of the wave run-up and wave overtopping process for
coastal dikes is not possible due to the stochastic nature of wave breaking and wave runup and the various factors influencing the wave run-up and wave overtopping process.
Therefore, wave run-up and wave overtopping for coastal dikes are mainly determined by
empirical formulas derived from experimental investigation. The influence of roughness
elements, wave wall, berms, etc. is taken into account by introducing influence factors.
4.2.1 Wave overtopping discharge
In the case of emerged structures (positive freeboard) the Eurotop 2007 suggests the
following distinction (Figure 4.2).

Probabilistic design and prediction or comparison of measurements
(
).
The following overtopping formulae for breaking and non-
breaking waves describes the average overtopping discharge:
√
√
(
with a maximum of:
66
)
(4.1)
Chapter 4
(
)
(4.2)
√
The reliability of Equations 4.1 and 4.2 are described by taking the coefficients
4.75 and 2.6 as normally distributed stochastic parameters with means of 4.75 and
2.6 and standard deviations
and 0.35 respectively. For probabilistic
calculations Equations 4.1 and 4.2 should be taken together with these stochastic
coefficients. For predictions of measurements or comparison with measurements
also Equations 4.1 and 4.2 should be taken with, for instance, 5% upper and
lower exceedance curves.

Deterministic design or safety assessment (
). For deterministic
calculation in design or safety assessment it is strongly recommended to increase
the average discharge by about one standard deviation. Thus, Equation 4.3
should be used for deterministic calculations in design and safety assessment:
(
√
√
)
(4.3)
with a maximum of:
(
)
√
(4.4)
In the case of very heavy breaking on a shallow foreshore the wave spectrum is
often transformed in a flat spectrum with no significant peak. In that case, long
waves are present and influencing the breaker parameter
. Other wave
overtopping formulae are recommended for shallow and very shallow foreshores
to avoid a large underestimation of wave overtopping by using formulae 4.1, 4.2,
67
Chapter 4
4.3 and 4.4. Since those formulae are valid for breaker parameters
linear interpolation is recommended for breaker parameters

Deterministic design or safety assessment (
a
.
).
The following
formula is recommended including a safety margin for deterministic design and
safety assessment.
[
(
√

]
)
(4.5)
Probabilistic design and prediction or comparison of measurements
(
). The following formula was derived from measurements with a
mean of -0.92 and a standard deviation of 0.24:
[
(
√
]
)
(4.6)
Figure 4.2. Wave overtopping and overflow for positive, zero and negative freeboard (by Eurotop 2007).
68
Chapter 4
Wave overtopping for zero freeboard (Figure 4.2) becomes important if a dike is
overtopping resistant (for example a low dike of asphalt) and the water level comes close
to the crest. Schüttrumpf (2001) performed model tests for different straight and smooth
slopes in between 1:3 and 2:6 to investigate wave overtopping for zero freeboard and
derived the following formula
, which should be used for probabilistic design
and prediction and comparison of measurements:
√
(4.7)
√
(4.8)
If the water level is higher than the dike crest, large overtopping quantities
overflow/overtop the structure. In this situation, the amount of water flowing to the
landward side of the structure is composed by a part which can be attributed to overflow
and a part which can be attributed to overtopping
. The part of overflowing
water can be calculated by the well known formula for a broad crested structure:
√
where
|
|
(4.9)
is the (negative) relative crest height and
The effect of wave overtopping
zero freeboard
is the overflow depth.
is accounted for by the overtopping discharge at
in Equations 4.7 and 4.8 as a first guess. The effect of combined
wave run-up and wave overtopping is given by the superposition of overflow and wave
overtopping as a rough approximation for
√
|
:
|
√
69
(4.10)
Chapter 4
4.2.2 Overtopping flow velocities and overtopping flow depth
Average overtopping rates are not appropriate to describe the interaction between the
overtopping flow and the failure mechanisms (infiltration and erosion) of a clay dike.
Therefore, research was carried out recently in small and large scale model tests to
investigate the overtopping flow velocities and the related flow depth on the seaward
slope, the dike crest and the landward slope. Results are summarized in Schüttrumpf and
Van Gent (2003). Empirical and theoretical functions were derived and verified by
experimental data in small and large scale. These parameters are required as boundary
conditions for geotechnical investigations, such as required for the analysis of erosion,
infiltration and sliding.
The parameters of overtopping flow velocities and overtopping flow depth will be
described separately for the seaward slope, the dike crest and the landward slope. Here
only the considerations about the flow over the dike crest is reported.
The overtopping tongue arrives as a very turbulent flow at the dike crest. Maximum flow
depth and overtopping velocities were measured in this overtopping phase over the crest.
The overtopping flow separation occurs at the middle and at the dike surface at the front
edge of the crest. No flow separation occurs at the middle and at the rear edge of the
crest. In the second overtopping phase, the overtopping flow has crossed the crest. Less
air in the overtopping flow but the flow itself is still very turbulent with waves in flow
direction and normal to flow direction. In the third overtopping phase, a second peak
arrives at the crest resulting in nearly the same flow depth as the first peak. In the fourth
overtopping phase, the air has disappeared from the overtopping flow and both
overtopping velocity and flow depth are decreasing. Finally, the overtopping flow nearly
stops on the dike crest for small overtopping flow depths. Few air is in the overtopping
water. At the end of this phase, the overtopping water on the dike crest starts flowing
seaward.
The overtopping flow depth on the dike crest depends on the width of the crest
co-ordinate on the crest
and the
(Figure 4.4) and the flow depth decreases due to the fact that
70
Chapter 4
the overtopping water is deformed. Thus, the decrease of the overtopping flow depth
over the dike crest can be described by an exponential function:
(
with
)
the overtopping flow depth on the dike crest,
dike crest with
(4.11)
the horizontal coordinate on the
at the beginning of the dike crest,
for TMA spectra
the width of the dike crest (for
the dimensional coefficient
and 1.11 for natural wave spectra
, and
to 3 m in prototype scale).
Figure 4.3. Definition sketch for overtopping flow parameters on the dike crest (by Eurotop 2007).
A theoretical function for overtopping flow velocities on the dike crest has been
developed by using the simplifies Navier-Stokes equations and the following
assumptions: the dike crest is horizontal; velocities vertical to the dike slope can be
neglected; the pressure term is almost constant over the dike crest; viscous effects in flow
direction are small; bottom friction is constant over the dike crest.
The following formula was derived from the Navier-Stokes equations and verified by
small and large scale model tests (Figure 4.4):
(
71
)
(4.12)
Chapter 4
with
the overtopping flow velocity on the dike crest;
velocity at the beginning of the dike crest
crest;
the friction coefficient; and
;
the flow depth at
the overtopping flow
the coordinate along the dike
. From the Equation (4.12) it
is obviously that the overtopping flow velocity decreases from the beginning to the end of
the dike crest and this decrease is more marked for increasing surface roughness (Figure
4.6). But for flow depth larger than about 0.1 m and dike crest widths around 2-3 m, the
flow depth and velocity hardly change over the crest.
Figure 4.4. Overtopping flow velocity data vs overtopping flow velocity formulae (by Eurotop 2007).
Figure 4.5. Left side: influence of overtopping flow depth on overtopping flow velocity; right side: influence
of bottom friction on overtopping flow velocity (by Eurotop 2007)
72
Chapter 4
4.3 The numerical database
The simulations were performed in 1:10 scale in a numerical flume 52.3 m long and 1.5
m deep under irregular waves with different significant wave heights
and peak period
characterized by Jonswap spectrum (Table 4.1).
Table 4.1. Characteristics of simulated tests.
αoff = 1:4 ; αin = 1:3
Rc/Hs
Tests
T1A
T2A
-1
T3A
T4A
T5A
T5A
T7A
T8A
T9B
T10B
0
T11B
T12B
T13B
T14B
T15B
T16B
T17C
T18C
0.5
T19C
T20C
T21C
T22C
T23C
T24C
T25D
T26D
-1.5
T27D
T28D
T29D
T30D
T31D
T32D
Hs
[m]
0.10
0.20
0.10
0.20
0.10
0.20
0.10
0.20
Tp
[s]
1.74
2.18
1.58
2.25
3.29
3.10
4.36
4.43
1.74
2.18
1.58
2.25
3.29
3.10
4.36
4.43
1.74
2.18
1.58
2.25
3.29
3.10
4.36
4.43
1.74
2.18
1.58
2.25
3.29
3.10
4.36
4.43
αoff = 1:4 ; αin = 1:2
wd
[m]
0.80
0.90
0.70
0.70
0.65
0.60
0.85
1.00
Tests
T1E
T2E
T3E
T4E
T5E
T5E
T7E
T8E
T9F
T10F
T11F
T12F
T13F
T14F
T15F
T16F
T17G
T18G
T19G
T20G
T21G
T22G
T23G
T24G
T25H
T26H
T27H
T28H
T29H
T30H
T31H
T32H
Hs
[m]
0.10
0.20
0.10
0.20
0.10
0.20
0.10
0.20
73
Tp
[s]
1.74
2.18
1.58
2.25
3.29
3.10
4.36
4.43
1.74
2.18
1.58
2.25
3.29
3.10
4.36
4.43
1.74
2.18
1.58
2.25
3.29
3.10
4.36
4.43
1.74
2.18
1.58
2.25
3.29
3.10
4.36
4.43
αoff = 1:6 ; αin = 1:3
wd
[m]
0.80
0.90
0.70
0.70
0.65
0.60
0.85
1.00
Tests
T1I
T2I
T3I
T4I
T5I
T5I
T7I
T8I
T9J
T10J
T11J
T12J
T13J
T14J
T15J
T16J
T17K
T18K
T19K
T20K
T21K
T22K
T23K
T24K
T25L
T26L
T27L
T28L
T29L
T30L
T31L
T32L
Hs
[m]
0.10
0.20
0.10
0.20
0.10
0.20
0.10
0.20
Tp
[s]
1.74
2.18
1.58
2.25
3.29
3.10
4.36
4.43
1.74
2.18
1.58
2.25
3.29
3.10
4.36
4.43
1.74
2.18
1.58
2.25
3.29
3.10
4.36
4.43
1.74
2.18
1.58
2.25
3.29
3.10
4.36
4.43
wd
[m]
0.80
0.90
0.70
0.70
0.65
0.60
0.85
1.00
Chapter 4
For each wave attack the wave steepness remains constant and close to 2%.
All
simulations were carried out without the implementation of the turbulent model. The
model settings adopted are shown in Table 4.2.
Table 4.2. Model settings adopted for the numerical simulations.
Boundary Condition
Seaward
Hs - Tp
(Jonswap spectrum)
Mesh resolution
Landward
cell width [m]
absorption
0.02
cell height [m]
Off-shore
Dike crest
In-shore
0.02÷0.05
0.02
0.02÷0.04
The dimensions of the structure (crest width and height) are constant while crest
freeboard
, seaward
and landward
slopes are variable (Figure 4.6).
By combining different geometries and wave attacks, a total of 96 tests were carried out.
Figure 4.6. Tested levee cross section (model-scale units).
4.4 Validation of the model
In order to verify the numerical simulations, selected numerical results were compared
against experimental data and consolidated theoretical formulae. In particular, it was
verified that the wave reflection coefficients calculated numerically for each test were
compatible with the experimental results obtained by Zanuttigh et al. (2006). Moreover,
also the overtopping discharges were compared with the theoretical formula by Van der
Meer (Eurotop, 2007).
74
Chapter 4
4.4.1 Wave reflection coefficient
The reflection coefficients
obtained by the simulations were compared in Figure 4.7,
4.8 and 4.9 with the data for smooth straight slopes that are included in the reflection
database by Zanuttigh and Van der Meer (2006). It is worthy to remark that this database
includes structures in design conditions (
,
, sop>1%).
In these graphs all the numerical tests for the structures with different seaward and
landward slopes are shown. The values of
for emerged and zero freeboard cases fall
perfectly in the range of the experimental values. As expected, submerged conditions
give lower values of
, falling under the range of the experimental data or at least in its
bottom part. Notwithstanding the numerical values of
appear to be slightly greater
than the experimental values. However, the numerical trends show two key issues in
agreement with the physical process: the greater the submergence and/or the lower the
wave height, the lower the reflection.
Figure 4.7.
obtained by the experimental database for smooth straight (Zanuttigh and Van der Meer,
2006) and the numerical simulation characterized by
and
75
.
Chapter 4
Figure 4.8.
obtained by the experimental database for smooth straight (Zanuttigh and Van der Meer,
2006) and the numerical simulation characterized by
Figure 4.9.
and
.
obtained by the experimental database for smooth straight (Zanuttigh and Van der Meer,
2006) and the numerical simulation characterized by
and
76
.
Chapter 4
4.4.2 Overtopping discharge
Tables 4.3, 4.4 and 4.5 report the discharge calculated numerically and theoretically for
emerged structures, structures with freeboard zero and submerged structures respectively.
For emerged cases the theoretical discharge is calculated by Equation (4.3), for cases with
freeboard zero the formulae used are Equations (4.7) and (4.8) and, finally, Equation (4.3)
is used for submerged cases.
Both for emerged cases and cases with freeboard zero, the theoretical discharge is well
represented and the relative error remains always under about 28%. The following
figures (4.10 and 4.11) report this good agreement showing the theoretical discharge vs
the numerical one.
Table 4.3. Emerged cases: theoretical total discharge qth, numerical total discharge qnum, relative error Err
(%).
Tests
T17C
T18C
T19C
T20C
T21C
T22C
T23C
T24C
qth
0.005
0.007
0.004
0.007
0.014
0.013
0.020
0.020
qnum
0.005
0.009
0.005
0.008
0.015
0.015
0.022
0.024
Err(%)
8.81
24.21
25.89
5.43
6.01
17.89
11.22
17.66
Tests
T17G
T18G
T19G
T20G
T21G
T22G
T23G
T24G
qth
0.005
0.007
0.004
0.007
0.014
0.013
0.020
0.020
qnum
0.004
0.008
0.005
0.009
0.017
0.016
0.022
0.022
Err(%)
21.99
15.44
28.31
27.92
16.61
20.19
8.74
4.97
Tests
T17K
T18K
T19K
T20K
T21K
T22K
T23K
T24K
qth
0.002
0.003
0.002
0.003
0.006
0.006
0.010
0.010
qnum
0.003
0.003
0.002
0.004
0.008
0.007
0.010
0.010
Err(%)
26.56
10.62
25.89
23.81
20.60
17.60
3.75
4.59
Table 4.4. Cases with freeboard zero: theoretical total discharge qth, numerical total discharge qnum,
relative error Err (%).
Tests
T9B
T10B
T11B
T12B
T13B
T14B
T15B
T16B
qth
0.009
0.010
0.008
0.010
0.025
0.025
0.029
0.030
qnum
0.008
0.010
0.010
0.011
0.021
0.027
0.034
0.030
Err(%)
12.19
1.88
17.33
3.08
16.69
8.50
16.76
1.06
Tests
T9F
T10F
T11F
T12F
T13F
T14F
T15F
T16F
qth
0.009
0.010
0.008
0.010
0.025
0.025
0.029
0.030
qnum
0.010
0.011
0.010
0.012
0.030
0.022
0.031
0.033
77
Err(%)
15.53
4.88
17.33
18.78
16.47
9.45
5.90
10.83
Tests
T17J
T18J
T19J
T20J
T21J
T22J
T23J
T24J
qth
0.006
0.007
0.005
0.007
0.017
0.016
0.020
0.020
qnum
0.005
0.007
0.006
0.008
0.016
0.017
0.023
0.022
Err(%)
8.15
7.87
13.01
14.86
6.43
0.96
18.12
11.67
Chapter 4
Figure 4.10. Total numerical discharge
Figure 4.11. Total numerical discharge
versus total theoretical discharge
versus total theoretical discharge
for emerged cases.
for cases with zero
freeboard.
As regards submerged cases, by calculating the relative error, we can observe that there is
a good agreement between the numerical and theoretical overflow discharge, whereas the
differences for the overtopping discharge are more marked. However, this contribute is
very small and so it not affects too much the total results.
78
Chapter 4
Table 4.5. Submerged cases: theoretical/numerical overtopping discharge qwth/qwnum, theoretical/numerical
overflow discharge qsth/qsnum, relative error E(%).
Tests
T1A
T2A
T3A
T4A
T5A
T6A
T7A
T8A
T25D
T26D
T27D
T28D
T29D
T30D
T31D
T32D
T1E
T2E
T3E
T4E
T5E
T6E
T7E
T8E
T25H
T26H
T27H
T28H
T29H
T30H
T31H
T32H
T1I
T2I
T3I
T4I
T5I
T6I
T7I
T8I
T25K
T26K
T27K
T28K
T29K
T30K
T31K
T32K
qwth
0.009
0.010
0.008
0.011
0.027
0.026
0.003
0.003
0.009
0.010
0.008
0.011
0.027
0.027
0.003
0.003
0.009
0.010
0.008
0.01
0.027
0.026
0.003
0.003
0.009
0.010
0.008
0.011
0.027
0.027
0.003
0.003
0.006
0.007
0.005
0.007
0.018
0.017
0.021
0.021
0.006
0.007
0.006
0.007
0.018
0.018
0.021
0.021
qwnum
0.011
0.011
0.009
0.011
0.019
0.022
0.004
0.004
0.010
0.012
0.009
0.011
0.028
0.030
0.003
0.003
0.010
0.011
0.009
0.012
0.024
0.022
0.004
0.003
0.008
0.012
0.009
0.010
0.023
0.024
0.003
0.003
0.0058
0.0085
0.0049
0.0082
0.0184
0.0205
0.0239
0.0231
0.0060
0.0075
0.0058
0.0057
0.0205
0.0206
0.0211
0.0276
E(%)
21.12
6.13
12.14
5.21
27.60
13.94
23.78
16.28
11.10
11.69
4.83
4.02
2.85
13.57
11.01
16.65
8.67
8.08
4.37
12.87
9.68
15.10
20.08
20.65
9.10
11.69
12.06
7.33
15.02
8.31
9.19
16.27
1.52
24.14
10.80
17.65
3.01
18.67
14.69
10.01
1.00
8.32
4.83
19.15
12.15
16.59
1.20
29.24
79
qsth
1.879
1.879
1.879
1.879
1.879
1.879
1.879
1.879
3.452
3.452
3.452
3.452
3.452
3.452
3.452
3.452
1.879
1.879
1.879
1.879
1.879
1.879
1.879
1.879
3.452
3.452
3.452
3.452
3.452
3.452
3.452
3.452
1.8793
1.8793
1.8793
1.8793
1.8793
1.8793
1.8793
1.8793
3.4524
3.4524
3.4524
3.4524
3.4524
3.4524
3.4524
3.4524
qsnum
1.787
1.724
1.812
1.764
1.877
1.865
1.873
1.907
3.121
3.307
3.316
3.398
3.298
3.320
3.182
3.247
1.771
1.739
1.912
1.788
1.873
1.748
1.858
1.844
3.290
3.187
3.278
3.303
3.081
3.208
3.126
3.247
1.8965
1.8995
1.9052
1.9041
1.8053
1.9058
1.8784
1.9090
3.4225
3.4017
3.3927
3.2586
3.2333
3.6168
3.5116
3.1296
E(%)
4.89
8.25
3.59
6.15
0.10
0.76
0.32
1.48
9.61
4.21
3.95
1.57
4.47
3.83
7.84
5.95
5.74
7.49
1.73
4.85
0.33
6.99
1.11
1.89
4.70
7.69
5.06
4.33
10.77
7.09
9.45
5.96
0.9176
1.0773
1.3806
1.3221
3.9353
1.4125
0.0455
1.5828
0.8664
1.4689
1.7296
5.6138
6.3466
4.7615
1.7144
9.3503
Chapter 4
4.5 Flow height evolution over the dike crest
In this paragraph the effects of the structure design parameters on the trend of depths of
the flow over the crest are investigated. The values of
of the waves) and
%
%
(flow depth exceeded by 2%
(wave period exceeded by 2% of the waves) at the dike off-shore
edge are summarized in Table 4.6 for all the tests. All the numerical results are compared
with the theory reported in the paragraph 4.2.
4.5.1 Influence of the dike submergence and geometry
Figure 4.12 shows the wave height trend over the crest structure for a test characterized
by
s,
,
and
𝑆
and
. It appears that in case of
, the evolution of the wave height is similar to the
literature results in case of emerged structures (Schuttrumpf, 2001; Van Gent, 2002;
Schuttrumpf and Van Gent, 2003; Schuttrumpf and Oumeraci, 2005; Bosman, 2008), i.e
the wave height tends to decrease along the crest. By increasing the submergence, the
decay of the wave height is less marked and it completely disappears when
.
In Figure 4.13, the evolution of the overtopping flow depth on the dike crest is reported
for the structures with different
. In the same graph the results obtained for the same
wave attack and different submergences are shown. As expected, irrespectively of the
submergence, the influence of the
appears to be negligible. This means that the flow
remains always subcritical over the crest under a sufficient hydraulic head.
The same comparison is reported in Figure 4.14 between structures characterized by
different
. As in Figure 4.14, the results obtained for the same wave attack and
different submergences are reported. It is possible to observe that, only in case of
and
the variation of
affect the evolution on the dike crest
of the overtopping flow depth and a slight discrepancy among the results is present.
Hence for emerged structures the wave decay over the crest is less marked decreasing the
seaward steepness.
80
Chapter 4
Table 4.6. Flow characteristics at the dike off-shore edge (xc=0).
h2%
T2%
u2%
h2%
T2%
u2%
h2%
T2%
u2%
(xc=0)
(xc=0)
(xc=0)
(xc=0)
(xc=0)
(xc=0)
(xc=0)
(xc=0)
(xc=0)
T1A
0.114
1.200
0.319
T1E
0.116
1.187
0.315
T1I
0.114
1.158
0.332
T2A
0.142
1.529
0.391
T2E
0.146
1.468
0.373
T2I
0.114
1.178
0.334
T3A
0.099
1.103
0.284
T3E
0.100
1.108
0.286
T3I
0.100
1.064
0.298
T4A
0.143
1.589
0.385
T4E
0.147
1.535
0.365
T4I
0.139
1.552
0.408
T5A
0.279
2.156
0.607
T5E
0.277
2.147
0.614
T5I
0.284
2.122
0.611
T6A
0.265
2.086
0.574
T6E
0.262
2.024
0.583
T6I
0.271
1.991
0.574
T7A
0.261
2.533
0.639
T7E
0.260
2.520
0.656
T7I
0.275
2.410
0.656
T8A
0.261
2.507
0.624
T8E
0.258
2.426
0.643
T8I
0.272
2.474
0.634
T9B
0.043
0.958
0.672
T9F
0.043
0.959
0.665
T9J
0.039
1.032
0.667
T10B
0.054
1.197
0.765
T10F
0.043
0.959
0.678
T10J
0.048
1.254
0.771
T11B
0.039
0.929
0.616
T11F
0.039
0.899
0.624
T11J
0.036
0.958
0.619
T12B
0.114
2.784
0.992
T12F
0.114
2.785
0.904
T12J
0.096
3.144
0.963
T13B
0.116
1.855
1.162
T13F
0.117
1.859
1.131
T13J
0.093
1.960
1.257
T14B
0.111
1.737
1.137
T14F
0.112
1.725
1.105
T14J
0.090
1.837
1.223
T15B
0.149
2.097
1.148
T15F
0.150
2.077
1.142
T15J
0.137
2.132
1.256
T16B
0.154
2.012
1.131
T16F
0.153
1.996
1.130
T16J
0.142
2.102
1.229
T17C
0.029
1.284
0.679
T17G
0.023
2.055
0.679
T17K
0.026
1.644
0.553
T18C
0.037
1.300
0.679
T18G
0.034
1.364
0.680
T18K
0.032
1.385
0.679
T19C
0.035
1.294
0.676
T19G
0.033
1.250
0.679
T19K
0.016
1.300
0.679
T20C
0.050
1.342
0.680
T20G
0.040
1.342
0.681
T20K
0.040
1.342
0.681
T21C
0.073
1.594
1.001
T21G
0.073
1.594
1.001
T21K
0.073
1.594
1.002
T22C
0.069
1.579
1.174
T22G
0.069
1.582
1.130
T22K
0.069
1.582
1.140
T23C
0.093
1.735
1.104
T23G
0.095
1.730
1.110
T23K
0.093
1.735
1.110
T24C
0.095
1.725
1.121
T24G
0.091
1.725
1.132
T24K
0.093
1.712
1.532
T25D
0.112
1.177
0.265
T25H
0.115
1.163
0.256
T25L
0.112
1.141
0.279
T26D
0.142
1.512
0.307
T26H
0.147
1.480
0.288
T26L
0.144
1.479
0.319
T27D
0.097
1.047
0.242
T27H
0.099
1.048
0.239
T27L
0.098
1.045
0.255
T28D
0.150
1.500
0.327
T28H
0.155
1.441
0.308
T28L
0.153
1.458
0.338
T29D
0.270
2.098
0.465
T29H
0.268
2.048
0.473
T29L
0.286
2.044
0.472
T30D
0.275
2.036
0.449
T30H
0.274
1.981
0.455
T30L
0.288
1.948
0.453
T31D
0.263
2.314
0.552
T31H
0.258
2.315
0.572
T31L
0.276
2.284
0.558
T32D
0.254
2.567
0.541
T32H
0.250
2.571
0.559
T32L
0.264
2.525
0.544
Tests
Tests
81
Tests
Chapter 4
Figure 4.12. Wave height trend on the dike crest for test T17C (
T1A (
, green), and T25D (
, red), T9B (
, blue),
, yellow).
Figure 4.13. Wave height trend on the dike crest for tests characterized by landward slope 1:3 (circles) and
1:2 (crosses) and different freeboard.
82
Chapter 4
Figure 4.14. Wave height trend on the dike crest for tests characterized by seaward slope 1:4 (circles) and
1:6 (crosses) and different freeboard.
4.5.2 Comparison with the theory
The results obtained by the numerical analysis, i.e. wave height development on the dike
crest, are presented in the same way in which are presented in the theoretical approach
reported in Eurotop 2007 (Paragraph 4.2). The purpose of this analysis is to fit the
variation of the wave height over the dike crest by means of an appropriate curve and
identify the key parameters of such fitting.
The numerical results show that, as well as in semi-empirical formulation, the
overtopping wave height on the crest of the structure tends to exponentially decrease.
The wave height decay for each test was therefore fitted with an exponential curve and
the best fitting coefficient is reported in Table 4.7 for tests with
and
𝑆
𝑆
,
𝑆
. Even if each test is characterized by a specific decay coefficient, it
is possible to observe common trends of decay by grouping the tests with similar
submergence and wave attacks.
83
Chapter 4
The coefficients and the trend decay depend on the wave height whereas they do not
depend on the seaward/landward slope, hence six different average coefficients were
calculated for the three values of
and the two of
(see Table 4.8).
Table 4.7. Wave decay coefficients of the best fitting and relative standard deviation.
Tests
ch
σ'
Tests
ch
σ'
Tests
ch
σ'
T17C
0.65
2.37
T17G
0.65
2.09
T17K
0.49
1.97
Rc/Hs=0.5
T18C
0.85
3.14
T18G
0.85
2.25
T18K
0.79
0.75
Hs=0.1m
T19C
0.64
1.79
T19G
0.65
3.34
T19K
0.49
1.79
T20C
0.88
1.46
T20G
0.89
2.05
T20K
0.78
2.17
T21C
0.36
2.97
T21G
0.37
2.87
T21K
0.32
3.51
Rc/Hs=0.5
T22C
0.36
0.60
T22G
0.39
3.37
T22K
0.32
4.21
Hs=0.2m
T23C
0.59
2.31
T23G
0.58
2.48
T23K
0.45
2.56
T24C
0.61
2.37
T24G
0.61
2.62
T24K
0.48
2.21
T9B
0.38
1.03
T9F
0.39
0.90
T9J
0.44
4.83
Rc/Hs=0
T10B
0.41
0.95
T10F
0.42
1.03
T10J
0.47
1.47
Hs=0.1m
T11B
0.36
0.80
T11F
0.37
0.60
T11J
0.43
0.95
T12B
0.42
0.63
T12F
0.42
0.41
T12J
0.46
1.22
T13B
0.25
1.76
T13F
0.24
1.82
T13J
0.30
0.67
Rc/Hs=0
T14B
0.24
1.57
T14F
0.24
1.83
T14J
0.29
2.49
Hs=0.2m
T15B
0.38
0.99
T15F
0.29
0.88
T15J
0.39
0.42
T16B
0.28
1.49
T16F
0.29
1.62
T16J
0.33
3.98
T1A
0.12
0.80
T1E
0.11
0.67
T1I
0.10
0.61
Rc/Hs=-1
T2A
0.17
0.32
T2E
0.17
0.36
T2I
0.13
0.42
Hs=0.1m
T3A
0.11
0.20
T3E
0.12
0.21
T3I
0.09
0.39
T4A
0.17
0.21
T4E
0.27
0.28
T4I
0.15
0.23
T5A
0.07
0.28
T5E
0.08
0.35
T5I
0.06
0.28
Rc/Hs=-1
T6A
0.06
0.23
T6E
0.06
0.27
T6I
0.05
1.07
Hs=0.2m
T7A
0.12
0.47
T7E
0.12
0.44
T7I
0.10
0.19
T8A
0.12
0.37
T8E
0.12
0.39
T8I
0.10
5.21
In Figures 4.15 and 4.16 the tests with
and
m and
are shown respectively. Figure 4.17 and 4.18 show the numerical results for tests with
and with
and
respectively. Finally, Figure 4.19
and 4.20 show the results for tests with
and with
and
. Black curves are the fitting curves (obtained by the average coefficients of
84
Chapter 4
Table 4.8) that provides an overall fair approximation, with a decreasing of the quality
(for both wave height) in the second half of the crest.
By observing these graph, it immediately appears that the decay is much more marked as
the structure is emerged. For this reason this analysis has not been done for the cases
with
because for those tests the decay is negligible.
Table 4.8. Average wave decay coefficients and relative standard deviation for each tests.
Average
coefficient
Rc/Hs=0.5
Hs=0.1m
Rc/Hs=0.5
Hs=0.2m
Rc/Hs=0
Hs=0.1m
Rc/Hs=0
Hs=0.2m
Rc/Hs=-1
Hs=0.1m
Rc/Hs=-1
Hs=0.2m
0.72
0.45
0.38
0.28
0.13
0.09
Tests
σ'
Tests
σ'
Tests
σ'
T17C
3.79
T17G
3.42
T17K
8.62
T18C
5.24
T18G
4.77
T18K
2.44
T19C
3.50
T19G
4.21
T19K
9.13
T20C
5.74
T20G
5.87
T20K
3.17
T21C
4.21
T21G
3.93
T21K
5.45
T22C
4.01
T22G
3.92
T22K
6.24
T23C
6.05
T23G
6.37
T23K
2.55
T24C
7.19
T24G
7.62
T24K
2.48
T9B
1.04
T9F
0.95
T9J
2.72
T10B
1.82
T10F
2.13
T10J
3.99
T11B
1.17
T11F
0.82
T11J
2.53
T12B
1.72
T12F
1.53
T12J
3.93
T13B
2.47
T13F
2.78
T13J
1.05
T14B
2.52
T14F
2.33
T14J
2.17
T15B
0.94
T15F
0.92
T15J
2.81
T16B
1.47
T16F
1.58
T16J
1.36
T1A
0.66
T1E
1.34
T1I
1.55
T2A
1.89
T2E
2.03
T2I
0.43
T3A
1.17
T3E
1.09
T3I
2.57
T4A
1.88
T4E
2.02
T4I
0.90
T5A
0.95
T5E
1.00
T5I
1.69
T6A
1.67
T6E
1.54
T6I
1.31
T7A
1.72
T7E
1.80
T7I
0.73
T8A
1.65
T8E
1.76
T8I
0.72
85
Chapter 4
Figure 4.15. Wave height decay on the dike crest with Hs=0.1m and
and
and
; triangles: cases with
and
; diamonds: cases with
.
Figure 4.16. Wave height decay on the dike crest with
and
and
. Squares: cases with
; triangles: cases with
and
and
.
86
. Squares: cases with
; diamonds: cases with
Chapter 4
Figure 4.17. Wave height decay on the dike crest with
and
and
; triangles: cases with
and
and
; diamonds: cases with
.
Figure 4.18. Wave height decay on the dike crest with
and
and
. Squares: cases with
; triangles: cases with
and
and
.
87
. Squares: cases with
; diamonds: cases with
Chapter 4
Figure 4.19. Wave height decay on the dike crest with
and
and
; triangles: cases with
and
and
; diamonds: cases with
.
Figure 4.20. Wave height decay on the dike crest with
and
and
. Squares: cases with
; triangles: cases with
and
and
.
88
. Squares: cases with
; diamonds: cases with
Chapter 4
To understand the degree of approximation at the fitted formulation to the numerical data,
the standard deviation for each test was calculated (Table 4.5). The values of the standard
deviation remains always under about 9% suggesting that the approximation is very good
and in particular is more accurate for tests with
= 0.1 m.
It is possible to observe that by keeping in constant the submergence, the wave height
decay tends to decrease, almost halved, when the significant wave height increases.
Similarly, the numerical results show that the wave height decay increases with
increasing the submergence when the significant wave height is constant.
4.5.3 Formula for the determination of the decay coefficient
The aim of this sub-paragraph is to understand if it is possible to predict the decay
coefficient that controls the decrease of the wave height over the dike crest. For example
it could exist a dependence on the wave attack, the characteristics structure and/or the
submergence.
An equation describing the trend of the dimensionless coefficient
formula (Equation 4.14) shows the dependency of
-
overtopping discharge;
-
wave height;
-
wave peak period;
-
break parameter.
𝑆
where
from:
(4.14)
√
the total discharge (in case of submerged structures the sum for the contribute of
overflow and the overtop),
and
was derived. The
is the wave height,
the peak period,
the water depth
the Irribarren coefficient.
The three following Figures plot all data against Equation 4.14 and show that the trend is
different for different submergences. In fact, in Figure 4.21 the cases characterized by
are reported, whereas in Figure 4.22 and 4.23 the cases with
and
are respectively represented. It is possible to approximate all points
89
Chapter 4
with a negative exponentially curve and, overall, there is a modest scatter with a
determination coefficient close to 0.95.
Figure 4.21. Wave decay coefficients against new Equation 4.14. Tests with
Figure 4.22. Wave decay coefficients against new Equation 4.14. Tests with
90
.
.
Chapter 4
Figure 4.23. Wave decay coefficients against new Equation 4.14. Tests with
.
4.6 Flow velocity evolution over the dike crest
In this paragraph the evolution of the flow velocity over the dike crest is described; in
particular the effects of the structure design parameters on the trend of velocities are
investigated. The values of
%
(flow velocity exceeded by 2% of the waves) at the dike
off-shore edge are summarized in Table 4.6 for all the tests. As for the wave height trend,
also the numerical results relative to the velocity are compared with the theory reported in
the paragraph 4.2.
4.6.1 Influence of the dike submergence and geometry
Figure 4.24 shows the evolution of the overtopping flow velocity by varying the
submergence. It can be observed that the velocity increases while the wave travels over
the crest, and specifically the growth rate decreases with increasing dike submergence.
Moreover, in the submerged cases, the decrease of flow depth (see Figure 4.12) and the
increase of velocity start from about the middle of the crest of the structure and however
are very modest.
91
Chapter 4
In Figure 4.25, the trends of the overtopping flow velocities on the dike crest are reported
for the structures with both landward slope 1:3 (represented as void circles) and 1:2
(represented as crosses). In the same graph the results obtained for different submergence
are shown. As regards the overtopping wave height, the influence of the landward slope
appears to be negligible. Only in case of
(red color) a slight discrepancy among
the velocity results obtained for different seaward slopes is present.
Figure 4.26 compares the evolution of the overtopping flow velocities on the dike crest
for the 1:4 (represented as void circles) and 1:6 seaward slopes (represented as crosses).
It is possible to observe that, irrespectively of the submergence, the seaward slope does
not significantly affect the evolution on the dike crest of the overtopping flow velocity.
Figure 4.24. Flow velocity trend on the dike crest for test T17C (
T1A (
, green) and T25D (
, yellow).
92
, red), T9B (
, blue),
Chapter 4
Figure 4.25. Flow velocity trend on the dike crest for tests characterized by landward slope 1:3 (circles) and
1:2 (crosses) and different freeboard.
Figure 4.26. Flow velocity trend on the dike crest for tests characterized by seaward slope 1:3 (circles) and
1:6 (crosses) and different freeboard.
93
Chapter 4
4.6.2 Approximation of the velocity trend with a fitting function
The numerical tests show a slight increase of the velocity over the dike crest that can be
approximated with a second-order polynomial function. The best fitting coefficients for
each tests are reported in Tables 4.9. As for the wave heights, also the coefficients of the
velocity are found to be dependent on the wave height and not on the seaward/landward
slope. Hence, the average coefficients are reported in Table 4.10.
In Figure 4.27 and 4.28 tests with
and the polynomial function obtained by
the average coefficients of Table 4.10 are respectively shown. The trend of the velocity
on the dike crest is very different for the tests with
cases with
and
. In the
the velocity tends to increase from the beginning of the crest, in
the cases with
there is a first phase of decrease and only after the middle of
the crest width the velocity starts to increase. The values of the standard deviation
reported (Table 4.10) remains under about 5% showing that the approximation is always
very good (irrespectively of the wave height).
Figure 4.29 and 4.30 compares the numerical results for tests with
and the
polynomial function obtained by the average coefficients of Table 4.10. Tests with
and
are respectively reported. Also in these cases, the trend of
the velocity on the dike crest is very different for the tests with different wave height. In
particular a continuous velocity growth is found for the tests with
the tests with
while for
the velocity tends first to decrease and then to increase starting
from the middle of the dike crest. By calculating the standard deviation for each test
(Table 4.10), it appears that the discrepancy from the curve remains always very low. In
particular, the approximation is better for tests with
and the standard
deviation remains always under about 2.4%.
In the submerged case (Figures 4.31-4.32), for both wave heights, the trend of the
velocity on the dike crest tends to decrease and to increase starting from the middle of the
crest. The standard deviation for each tests (Table 4.10) remains under about 5%.
94
Chapter 4
Figure 4.27. Wave velocity evolution on the crest of the structure for tests with
. Squares: cases with
diamonds: cases with
and
and
; triangles: cases with
diamonds: cases with
and
and
and
;
.
Figure 4.28. Wave velocity evolution on the crest of the structure for tests with
. Squares: cases with
and
; triangles: cases with
.
95
and
and
;
Chapter 4
Figure 4.29. Wave velocity evolution on the crest of the structure for tests with
Squares: cases with
cases with
and
and
; triangles: cases with
and
and
cases with
and
and
; diamonds:
.
Figure 4.30. Wave velocity evolution on the crest of the structure for tests with
Squares: cases with
.
; triangles: cases with
.
96
and
and
.
; diamonds:
Chapter 4
Figure 4.31. Wave velocity evolution on the crest of the structure for tests with
. Squares: cases with
diamonds: cases with
and
and
; triangles: cases with
diamonds: cases with
and
and
and
;
.
Figure 4.32. Wave velocity evolution on the crest of the structure for tests with
. Squares: cases with
and
; triangles: cases with
.
97
and
and
;
Chapter 4
Table 4.9. Coefficients obtained with the best fitting curve for the wave velocity on the dike crest and
relative standard deviation.
Rc/Hs=0.5 Hs=0.1m
Rc/Hs=0.5
Hs=0.2m
Tests
au
bu
σ'
Tests
au
bu
σ'
Tests
au
bu
σ'
T17C
0.15
0.26
0.68
T17G
0.13
0.30
1.32
T17K
0.09
0.34
3.92
T18C
0.32
0.03
0.94
T18G
0.33
0.02
0.88
T18K
0.29
0.08
1.04
T19C
0.08
0.36
1.28
T19G
0.11
0.32
1.18
T19K
0.32
0.14
1.23
T20C
0.07
0.29
2.94
T20G
0.17
0.23
1.79
T20K
0.09
0.33
1.92
T21C
0.43
-0.22
1.51
T21G
0.45
-0.24
1.73
T21K
0.44
-0.23
2.44
T22C
0.54
-0.41
1.98
T22G
0.54
-0.41
2.17
T22K
0.49
-0.23
1.81
T23C
0.55
-0.36
2.12
T23G
0.56
-0.37
2.34
T23K
0.58
-0.40
2.42
T24C
0.49
-0.36
1.93
T24G
0.49
-0.38
1.33
T24K
0.53
-0.43
1.51
T9B
0.04
0.11
0.41
T9F
0.04
0.12
0.34
T9J
0.04
0.11
0.36
Rc/Hs=0
T10B
0.08
0.08
0.85
T10F
0.08
0.09
0.70
T10J
0.08
0.09
0.97
Hs=0.1m
T11B
0.04
0.16
0.21
T11F
0.04
0.13
0.60
T11J
0.05
0.13
0.36
T12B
0.09
0.09
0.92
T12F
0.09
0.09
0.67
T12J
0.09
0.10
0.81
T13B
0.26
-0.19
0.49
T13F
0.26
-0.19
0.57
T13J
0.28
-0.20
0.52
Rc/Hs=0
T14B
0.26
-0.19
0.39
T14F
0.25
-0.18
0.55
T14J
0.26
-0.18
0.37
Hs=0.2m
T15B
0.25
-0.12
0.62
T15F
0.25
-0.12
0.54
T15J
0.25
-0.12
0.58
T16B
0.25
-0.12
0.45
T16F
0.24
-0.11
0.63
T16J
0.23
-0.10
0.64
T1A
0.13
-0.04
0.68
T1E
0.13
-0.03
0.73
T1I
0.13
-0.04
0.73
Rc/Hs=-1
T2A
0.13
-0.07
0.76
T2E
0.14
-0.06
0.80
T2I
0.13
-0.06
0.60
Hs=0.1m
T3A
0.13
-0.04
0.77
T3E
0.13
-0.03
0.73
T3I
0.13
-0.04
0.69
T4A
0.13
-0.05
0.56
T4E
0.12
-0.04
0.64
T4I
0.12
-0.04
0.48
T5A
0.08
-0.02
0.38
T5E
0.08
-0.02
0.47
T5I
0.08
-0.02
0.47
Rc/Hs=-1
T6A
0.10
-0.02
0.23
T6E
0.11
-0.02
0.21
T6I
0.11
-0.02
0.23
Hs=0.2m
T7A
0.10
-0.01
0.20
T7E
0.10
-0.02
0.20
T7I
0.10
-0.01
0.13
T8A
0.10
-0.01
0.14
T8E
0.10
-0.02
0.22
T8I
0.11
-0.02
5.13
98
Chapter 4
Table 4.10.
Average coefficients for wave velocity evolution on the dike crest and relative standard
deviation for each tests.
Average coefficient
au
Rc/Hs=0.5 Hs=0.1m
Rc/Hs=0.5
Hs=0.2m
Rc/Hs=0
Hs=0.1m
Rc/Hs=0
Hs=0.2m
Rc/Hs=-1
Hs=0.1m
Rc/Hs=-1
Hs=0.2m
0.18
0.51
0.06
0.25
0.13
0.09
bu
0.22
-0.35
0.11
-0.15
-0.05
-0.02
Tests
σ'
Tests
σ'
Tests
σ'
T17C
0.82
T17G
1.80
T17K
1.27
T18C
5.31
T18G
5.18
T18K
4.19
T19C
2.71
T19G
2.20
T19K
2.48
T20C
3.62
T20G
1.93
T20K
1.44
T21C
3.27
T21G
2.70
T21K
2.53
T22C
3.94
T22G
4.20
T22K
2.34
T23C
1.84
T23G
2.19
T23K
2.73
T24C
3.30
T24G
3.83
T24K
5.02
T9B
1.00
T9F
0.83
T9J
0.93
T10B
1.38
T10F
1.18
T10J
1.40
T11B
1.89
T11F
0.86
T11J
0.59
T12B
0.95
T12F
0.74
T12J
0.78
T13B
2.01
T13F
1.86
T13J
1.94
T14B
1.61
T14F
1.32
T14J
1.33
T15B
1.93
T15F
1.92
T15J
1.97
T16B
2.08
T16F
2.19
T16J
2.41
T1A
1.23
T1E
1.39
T1I
1.33
T2A
0.89
T2E
0.78
T2I
0.79
T3A
1.33
T3E
1.23
T3I
1.25
T4A
0.57
T4E
0.82
T4I
0.71
T5A
4.96
T5E
4.92
T5I
4.93
T6A
4.12
T6E
4.06
T6I
4.02
T7A
4.10
T7E
4.07
T7I
4.09
T8A
4.11
T8E
4.00
T8I
3.99
99
Chapter 4
4.7 Statistical characterization of extreme overtopping wave
volumes
In the design of coastal defences and in the estimate of their vulnerability a key aspect is
the realistic prediction of the characteristics of the overtopping waves.
In fact
hydrodynamic forces on landward-side slopes largely depend on the distribution of
instantaneous overtopping wave volumes, flow thicknesses and flow velocities (Van der
Meer et al., 2010; Hughes et al., 2012).
Overtopping wave volumes have been
successfully approximated by a Weibull distribution, whose shape factor appears to be
larger for very large overtopping and certainly for wave overtopping combined with
overflow (Hughes and Nadal, 2009; Victor, et al., 2012). The larger shape factor results
in lower maximum overtopping wave volumes.
The percent exceedance distribution of overtopping wave volumes is given by (Hughes et
al., 2012):
%
where
%
[ ( ) ]
(4.15)
is the percentage of wave volumes that will exceed the specified volume ( ).
The two parameters of the Weibull distribution are the non-dimensional shape factor, ,
that helps define the extreme tail of the distribution and the dimensional scale factor, ,
that normalizes the distribution.
Hughes et al. 2012 valid the relationship of the Weibull shape factor
for smooth and
impermeable structures like dikes and levees. The relationship is given as
versus
to describe the distribution of overtopping wave volumes.
Zanuttigh et al. 2013 presents the analysis of the Weibull -value for conventional rubble
mound breakwaters as well as for low crested structures with the crest at or just above the
water level. It is concluded that rubble mound structures show more scatter in the value than smooth impermeable structures and the combined data make even more sense
100
Chapter 4
if the -value is related to relative discharge instead of relative freeboard because the
effects of slope angle and wave steepness are implicitly included.
In this analysis we compared our results with the new trend for the shape factor
found
in Zanuttigh et al. 2013:
(
)
(4.16)
and in the following Figures data for smooth structure, different submergence and
seaward/landward steepness obtained by Hughes and Victor are reported.
Figure 4.33. Comparison numerical results (
) with smooth structures against formula 4.16.
101
Chapter 4
Figure 4.34. Comparison numerical results (
, orange;
, blue) with smooth structures against formula 4.16.
102
, green;
, pink;
CHAPTER 5
TWO-PHASE APPROACH FOR SEDIMENT
TRANSPORT MODELLING.
The present chapter is devoted to the description of the modifications and improvements
that have been made in the IH-2VOF code in order to implement the sediment transport.
Most of these modifications have been carried out based on Hsu et al. 2004.
5.1 Governing equations for fluid and particle phase: RANS
equations
Sediment transport involves a fluid phase and a particle phase. The fluid phase is water
with mass density
diameter
and the particle phase is represented as a identical spheres of
and mass density
.
Assuming that the mixture can be treated as a
continuum, the ensemble averaged two-phase equations of mass and momentum can be
derived readily. In this averaging process, the definition of sediment concentration
is
introduced. Because of the presence of the particle concentration, the two continuum
phases are, essentially, compressible. For this reason, we implement Favre averaging
(Favre 1965). For more details see Hsu et al. 2004.
The fluid and sediment phase continuity equations are:
(5.1)
(5.2)
where:
Chapter 5
(5.3)
and
(5.4)
In these equations
particle velocity and
and
and
represent, respectively, the x-components of the fluid and
the y-component of the fluid and particle velocity.
The x- and y- components of the fluid-phase momentum equations for the uniform flow
can be expressed as:
̅
̅(
)
(5.5)
̅
and
̅
where
̅(
is the fluid pressure,
)
̅
are the fluid phase stresses and
(5.6)
is the
gravitational acceleration. The last two terms in equations (5.5) and (5.6) are the Favre
averaged drag forces, with the drag coefficient
(
In Eq. (5.7),
)
defined as:
̅
is the particle Reynolds number and
velocity between the fluid and sediment phase:
104
(5.7)
is the magnitude of the relative
Chapter 5
(5.8)
√
where
is the fluid viscosity and
(5.9)
is a coefficient
(5.10)
The concentration dependence in equation (5.7) is taken from the experimental results of
Richardson and Zaki (1954). The drag force contribution in (5.5) and (5.6) is composed
of two terms. The first is the averaged drag force due to the relative mean velocity
between the two phases. The second, called fluid turbulent suspension, is the correlation
between the concentration and the large-scale fluid velocity fluctuations. It is modelled
here as a gradient transport (see McTigue, 1981).
The corresponding sediment-phase momentum equations are:
̅
̅
̅(
)
(5.11)
̅
and
̅
where
̅
̅(
)
̅
(5.12)
are the stresses of the sediment phase, including the small-scale
particle (inter-granular) stresses and the Reynolds stresses of the Favre averaged particle
105
Chapter 5
velocities. Closures for fluid turbulence and sediment stresses are major issues of sheet
flow modelling and are detailed in the next paragraph.
5.2 Closure of fluid stresses
Closures for fluid turbulence stresses in two-phase flows are very similar to the case of
clear water, with the exception of the contribution of sediment - in terms of concentration
and correlation between fluid and sediment velocities - in the governing equations for the
turbulent kinetic energy and for the rate of turbulent energy dissipation.
The total stress of the fluid-phase in equation (5.5) and (5.6) can be written as:
(5.13)
(5.14)
(5.15)
(5.16)
where
are the averaged small-scale stresses consisting of the viscous
stress and the small-scale Reynolds stress of the turbulence generated in the fluid between
the sediment particles or induced by fluctuations of the particles. The large-scale fluid
Reynolds stresses, defined as the correlations between the concentration and fluid
velocity fluctuations
and
,
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
(5.17)
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
(5.18)
106
Chapter 5
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
(5.19)
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
(5.20)
result from the Favre averaging process. They represent the transfer of momentum that
occurs on the scale at which the concentration fluctuates.
The turbulent eddy viscosity hypothesis is used here to model the large-scale fluid
Reynolds stresses:
(
)
(5.21)
(
)
(5.22)
and
where
(
)
(5.23)
(
)
(5.24)
is the kinematic viscosity of the fluid and
is the fluid-phase turbulent kinetic
energy, defined as
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
(5.25)
The second term on the right-hand side of equations (5.23) and (5.24) appears because the
divergence of the fluid-phase velocity is not zero. We assume that the fluid phase eddy
viscosity
is given by:
107
Chapter 5
(5.26)
where
is an empirical coefficient and
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
(5.27)
is the fluid-phase turbulent dissipation rate. Because
and
appear in the eddy
viscosity, we need to introduce balance equations for both.
Following Hsu et al. (2004), the fluid phase turbulent kinetic energy equation in the
uniform flow can be written as
(5.28)
[(
̅
)
(
]
[(
)
̅
̅
)
]
(
)
The last term in eq. (5.28), which originally involves correlations between fluctuations of
fluid and sediment velocities, represents a dissipation mechanism for the turbulent
energy, where
is a parameter that measures the degree of correlation between the fluid
and sediment velocity fluctuations. It is determined by the relative magnitudes of a
particle response time
scale
, the time between collisions
, and the fluid turbulence time-
:
(
)
The particle response time is defined as
108
(5.29)
Chapter 5
(5.30)
and it is a measure of the time needed to accelerate a single particle from rest to the
velocity of surrounding fluid (Drew 1976). The time between collisions is estimated
based on the mean free path
of colliding particles and the strength
of sediment
velocity fluctuations:
(5.31)
where
√
(5.32)
The fluid turbulence time-scale is defined as (Elghobashi & Abou-Arab 1983)
(5.33)
The rate of turbulent energy dissipation
is assumed to be governed by an equation
similar to that for a clear fluid (Elghobashi & Abou-Arab 1983):
(
[(
)
)
[
]
[(
̅
)
]
[
]
̅
̅
[
(
)]
109
(
(5.34)
)]
Chapter 5
Due to the lack of information regarding the appropriate values of numerical coefficient
in the present
model, we employ the same coefficients as those implemented in
the standard
model for a clear fluid flow as already done by Hsu (see Hsu et al.
2004).
(5.35)
5.3 Closure of sediment stresses
In the sediment momentum equations, the two-scale averaging process results in a
sediment stress, which can be written as
(5.36)
(5.37)
(5.38)
(5.39)
where
are the mean particle shear and normal stresses due to small scale
interactions, while
are components of the large-scale sediment
Reynolds stress.
The small-scale stresses
are mainly due to granular interactions
resulting from particle collisions or interstitial fluid effects. Here, we adopt the kinetic
theory for collisional granular flow (Jenkins & Hanes 1998) for their closure.
Following this theory, the transport coefficients in the constitutive relations for
are obtained from the kinetic theory of dense gases (Chapman &
Cowling 1970). The particle normal and shear stresses due to collision are represented
as:
110
Chapter 5
(5.40)
(
)
(5.41)
(5.42)
where
, with
the radial distribution function at contact for identical
spheres.
Torquato (1995) provides an accurate expression for this radial distribution function that
is good for concentrations between 0.49, at which a phase transition between random and
hexagonal packing is first possible, and the random close-packed concentration,
, at which the mean distance between the edges’ nearest neighbors is zero:
(5.43)
{
where
The product
.
in eq. (5.40) and (5.42) is the sediment viscosity due to collisions and
we have:
̅ (
)
⁄
(5.44)
⁄
(
)
111
(5.45)
Chapter 5
We model the large-scale sediment Reynolds stresses
using an eddy
viscosity. The shear stresses are written as
(
√
)
(5.46)
and the normal stresses as
√
(5.47)
√
(5.48)
The sediment viscosity
is related to the sediment fluctuation energy through a
sediment mixing length ,
√
where
(5.49)
is a numerical coefficient, assumed to be equal to 0.55 based on the value used
in the one-equation turbulence model for clear fluid. We assume that the sediment
mixing length can be related to the turbulent fluid flow mixing length
through ,
(5.50)
is a parameter that measures the degree of correlation between the fluid and sediment
velocity fluctuations. It is determined by the relative magnitudes of a particle response
time.
The fluid turbulent mixing length is calculated from the fluid turbulent kinetic energy and
its dissipation rate:
(5.51)
112
Chapter 5
For massive particles with a long particle response time,
, the fluid turbulent eddies
cannot induce any sediment velocity fluctuations, and
. For fine particles with
small particle response time,
, the fine particles follow the turbulent eddies, and
.
The kinetic theory of dense gases is based on the fundamental assumption of a significant
number of collisions. Therefore, when the sediment concentration becomes very dilute,
the validity of the collisional grain flow theory becomes questionable. Therefore, we
introduce a damping parameter
the mean free path
for the small-scale sediment stress defined in terms of
of the sediment particles and the fluid turbulent mixing length
,
(5.52)
When the mean free path of collision becomes much larger than the fluid turbulent
mixing length, the small-scale collisional transport is reduced through a diminishing .
The constitutive relations for particle collisions based on the kinetic theory of dense
molecular gases have been successfully implemented to study problems of rapid granular
flow at concentrations smaller than the random loose packing
. The primary reason for
the close similarity between particle and molecular collisions is that they are of relatively
short duration, compared with the time between collisions.
shearing flows at concentrations greater than
However, for granular
, particles are in enduring contact.
Therefore, the analogy between the particles and molecules is no longer valid.
Here, we model the sediment transport above the stationary bed, where the concentration
is near random close packing. Therefore, modifications to the collisional grain flow
theory for the closure of particle stress are needed. The discrete particle simulations of
Zhang & Campbell (1992) indicate that between the random close-packed concentration
and the random loose-packed concentration
, the granular material is in a transitional
state between solid-like and fluid-like behavior. Bocquet et al . (2001) carried out
experiments on the Couette flow of grains in this regime and observed that the viscosity
of the particle shear stress increased dramatically as the concentration approached
They suggested that in the viscosity, the power
113
.
in equation (33) should be changed
Chapter 5
from 1.00 to 1.75. That is, in our numerical implementation,
and 1.75 when
is taken to be 1.00 when
. Therefore, as far as the particle shear stress is concerned,
the region involving enduring contacts is modelled by taking the granular material to be
an extremely viscous fluid. As the concentration increases above
, the collisional
contribution to the particle normal stress diminishes, because the shearing of the particle
phase that is the source of the collisional fluctuations becomes very small. However, in
this range of concentration, the contribution to particle normal stress due to enduring
contacts becomes important. Therefore, we further assume that the small-scale particle
normal stress
normal stress (
and
and
is the sum of the collisional normal stress (
and
) and the
) due to enduring contact:
(5.53)
(5.54)
We model the collisional stress using equations (12) and (14), while for the normal stress
due to enduring contacts we adopt a Hertz contact relation. For a homogeneously packed,
dry granular material consisting of identical spheres in Hertzian contact, the normal stress
is (Jenkins et al . 1989)
(5.55)
( )
where
is the average compressive volume strain,
modulus
e and Poisson’s ratio
is given in terms of the shear
of the material of the particles
(5.56)
√
and
is the average number of contacts per particle or coordination number. We do not
solve Eq. (5.50) for , but assume that ⁄ can be related to the difference between the
local average concentration and that of random loose-packing
114
by
Chapter 5
⁄
in which
(5.57)
is a coefficient. Based on numerical experiments for plastic particles
implement ted by Sumer et al . (1996),
gives a failure concentration of ca. 62%.
Therefore, this value is adopted.
Then
{
(5.58)
where the coordination number
̅
Because
is taken to be a function of concentration.
̅
[ (
)]
̅
appears both in the sediment viscosity
(5.59)
and in the normal-shear stresses,
we need to introduce the transport equation for it (see Hsu 2002).
(5.60)
̅(
with
the flux of the fluctuation energy and
)
the dissipation. The last term in the above
equation describes the interaction between the two phases. Therefore, there is an
additional source term,
̅
, due to the fluid turbulent kinetic energy. This term
models the influence of fluid turbulent eddies on the random motions of sediment
particles and permits turbulent eddies to enhance the sediment fluctuation energy.
Moreover, an additional dissipation mechanism,
̅
, also appears due to the drag of
the interstitial fluid.
The flux of sediment fluctuation energy
and the large-scale
is taken to be the sum of the small-scale
components:
115
Chapter 5
(5.61)
where:
(
)
(5.62)
(
)
(5.63)
and
being
a numerical coefficient (1.0).
Based on the kinetic theory for collisional granular flow, M in eq. (5.57) is:
(
Finally, we take the dissipation rate
)
(5.64)
in equation (5.60) to be the collisional dissipation
associated with the inelasticity of the particles. Based on the analysis of Jenkins &
Savage (1983), and considering
[
as the coefficient of restitution (0.8), we can write:
̅ (
)]
116
(5.65)
Chapter 5
5.4 Model implementation
In order to solve the RANS equations both for fluid and sediment phase a method very
similar to the two-step projection method used in the original IH-2VOF has been
implemented. In the follow the method is described in detail.
1. An intermediate velocity both for fluid and sediment phase is introduced. These
velocities ( ̃ , ̃ , ̃ , ̃ ) does not, in general, satisfy the continuity equation
and derive from the momentum equation without the pressure gradient term.
For the x-direction is possible write
(̃ )
(
)
[(
(
)
)
(
(
)
[
((
(
)
)
]
[
)
)
(
)
(
)
(
)
(
)
(
)
]
(5.66)
]
(5.67)
]
̃
[(
)
]
[
((
[
)
)
and for the y-direction
117
]
Chapter 5
(̃ )
(
)
(
[(
)
)
(
(
)
[
((
(
)
)
]
[
(
)
(
)
(
(
)
)
)
)
]
(5.68)
]
̃
[
]
((
[
[
(
)
)
)
(
)
]
(5.69)
]
2. The next step is to project the intermediate velocity field into a divergence free
plane to obtain the final velocity. In x-direction we have
(
(̃ )
)
(5.70)
and in y- direction
(
(̃ )
)
(5.71)
Combining (5.70) and (5.71) with the continuity equation (5.72)
(
)
(
)
(5.72)
the Poisson Pressure Equation (PPE) for the two-phase approach is obtained:
118
Chapter 5
(
)
[
(̃ )
(
(̃ )
)
]
(̃ )
(̃ )
(5.73)
By solving (5.73) with the appropriate boundary conditions, the correct pressure
information at the n+1-th time step will be obtained.
3. Substituting the updated pressure information into (5.70) and (5.71), the new
velocity field for the fluid at the n+1-th time step, which satisfies the continuity
equation, is obtained. In the same way, by substituting the updated pressure
information into the following equations
̃
(5.74)
̃
(5.75)
also the new velocity field for the sediment at the n+1-th time step is obtained.
4. Finally, the new value of the concentration can be calculated on the base of the
continuity equation of the sediment phase at the n+1-th time step.
(5.76)
(5.77)
119
Chapter 5
5.5 Spatial discretization in finite different form
In the equations presented in the previous paragraph, spatial derivatives need to be
specified. The nonlinear advection terms are discretized by using a combination of the
upwind scheme and the central difference scheme to achieve an accurate numerical
solution. The central difference method is employed to discretize the pressure gradient
terms and stress gradient terms. Introducing the discretization of spatial derivatives into
the Poisson pressure equation yields a set of linear algebraic equations for the pressure
field that is solved using conjugate gradient method with the preconditioned of
incomplete Cholesky decomposition.
Stability of finite difference scheme is performed by Heuristic analysis. Implicit
discretization of the pressure term in the momentum equation leads to a linear system of
equations that needs considerable computational effort to be solved. However this kind
of discretization avoids any stability condition related to pressure term. On the other
hand, the explicit discretization of the advection and diffusion terms in the momentum
equation leading to a time step constraints such that
{
| |
| |
{
}
(5.78)
}
(5.79)
are required.
The scheme implemented in the two-phase approach, as in the original IH-2VOF, is a
finite-different scheme. As shown in Figure 3.1, the scheme calculates the velocity
components of the fluid/sediment,
and
, on the vertical and horizontal cell
faces, respectively, whereas the pressure and other scalars such
fluid function
,
, the volume of
and the sediment concentration , are defined at the cell center.
It is noted that in the finite difference form, some variables are needed at the place where
they are not originally defined, for example, the horizontal velocity at the top face of the
cell or the vertical velocity at the right face of the cell. In such circumstances, the linear
interpolation is used. In the follow the most commonly used interpolation variables are
120
Chapter 5
given and it is not specified the superscript
or
because the interpolation is the same
both for the fluid phase and the sediment phase.
(
)
(5.80)
(
)
(5.81)
(5.82)
(5.83)
(5.84)
(5.85)
(5.86)
(
)
121
(5.87)
Chapter 5
5.5.1 Advection terms
Both for sediment and fluid equations, in equations (5.66), (5.67), (5.68) and (5.69) all
the advection terms will be evaluated at the n-th time step. The advection terms in the xmomentum equation
are evaluated at the right face of the cell. The advection
terms in the y-momentum equation
are evaluated at the top face of the cell.
(
)
(
)
(5.88)
(
)
(
)
(5.89)
As in Lin and Liu (1998), also here to calculate the spatial derivatives of the velocity the
combination of the upwind scheme and the central difference scheme is used. The
upwind scheme is represented by
(
(
)
)
(5.90)
{
(
)
and the center difference scheme is represented by
(
(
)
(
)
In both (5.90) and (5.91)
122
)
(5.91)
Chapter 5
(
)
(
(5.92)
)
(5.93)
are defined. Since the upwind scheme usually introduces significant numerical damping
and the central difference scheme generates numerical instability, a combination of these
two schemes usually yields a more accurate numerical solution.
Thus, the general
formula for the spatial derivative becomes
(
[
(
)] (
)
)
(
[
(
)] (
)
(5.94)
)
and similarly
(
[
(
)] (
)
)
(
[
(
)(
)] (
)
(5.95)
)
where
(
)
(5.96)
and
123
Chapter 5
(
)
(5.97)
For the advection terms in the y-direction we have to define the following derivatives:
(
(
[
)] (
)
)
(
[
(
)] (
)(
)
(5.98)
)
and
(
[
(
)] (
)
)
(
[
(
)(
)] (
)
(5.99)
)
where
(
)
(5.100)
(
)
(5.101)
(
)
(5.102)
124
Chapter 5
(
)
(5.103)
In the above equations, the coefficient
is the weighing factor between the up-wind
method and the central difference method. When
becomes the central difference; while when
upwind difference. In practice,
, the finite difference form
, the finite difference form becomes the
is generally selected in the range of
to
to
produce the stable and accurate results.
In the x-direction is also necessary to define
(
)
(
)
(5.104)
and in the y-direction
(
)
(
(
)
(5.105)
)
5.5.2 New advection terms
Since the sediment presence, the density is not constant. Therefore, in equations (5.66),
(5.67), (5.68) and (5.69) new terms appears. These terms will be evaluated at the n-th
time step. In the x-momentum equation the new term is
(
) and has to be
evaluated at the right face of the cell. In the y-momentum equation the new term is
(
) and has to be evaluated at the top face of the cell.
For the -x and -y direction we have respectively:
125
Chapter 5
[
(
)
(
)
]
(5.106)
[
(
)
(
)
]
(5.107)
where
(5.108)
(5.109)
(
(
)
(5.110)
)
(
(
)(
)
)
(
(
(
)
)
(5.111)
)
(
)
(
(
)
)
126
(5.112)
Chapter 5
(
)
(5.113)
5.5.3 Tangential terms
5.3.3.1 Fluid phase
The gradient of the total stress for the fluid in Equation (5.66) is multiplied to
. This
term has to be defined in the first equation at the right face of the cell and in the second at
the top face (see equations 5.108 and 5.109 respectively).
The gradient of the total stresses in (5.66) can be written as
(5.114)
for the x-momentum equation and
(5.115)
for the y-momentum equation. Once again the stress gradient in the x-direction is
calculated at the right face of the computational cell, while in the y-direction it is
computed at the top face of the cell. Thus, the first term of 5.114 can be written in the
following finite difference from
(
)
(
)
(
)
(5.116)
The second term in 5.114 can be written as
127
Chapter 5
(
(
)
(
)
)
(5.117)
The total stresses (normal and shear) above are the summation of molecular stresses and
Reynolds stresses. The former are the products of the molecular viscosity and the strain
rates of the mean flow, and the latter can be obtained by the nonlinear algebraic Reynolds
stress model (5.118). Both of them involve the evaluation of the strain rates of the mean
flow. The normal stress in the x-direction is evaluated at the center of the cell which
involves the calculation of normal strain rate of the mean flow
(
(
)
(
)
)
(5.118)
The shear stress is evaluated at the vertices of the cell which involves the calculations of
both
and
. The finite difference form of these derivatives can be referred to (5.96)
and (5.97). the similar finite-difference formulas can be obtained for stress gradient terms
in the y-momentum equation (5.117). The shear strain rate of the mean flow can be again
referred to (5.96) and (5.97) and the normal strain rate at the cell
in the y-direction is
expressed as
(
(
)
(
)
(5.119)
)
5.3.3.2 Sediment phase
Also the gradient of the total stress of the sediment phase is multiplied in Equation (5.67)
to
(see equations 5.108 and 5.109 respectively).
The gradients of the total stresses can be written as
128
Chapter 5
(5.120)
for the x-momentum equation and
(5.121)
for the y-momentum equation.
One again the stress gradient in the x-momentum
equation is calculated at the right face of the computational cell, while in the ymomentum equation it is computed at the top face of the cell.
In finite difference form for the x-direction we can write:
(
)
(
(
)
)
(
)
(5.122)
and for the y-direction we have
(
)
The total stresses (
(
(
)
)
(
(
)
(
)
(5.123)
) are the summation of the mean particle shear and
normal stresses due to small scale interactions (
sediment Reynolds stress (
)
) and the large-scale
), see Equations 5.36, 5.37, 5.38, 5.39.
129
Chapter 5
The mean particle shear and normal stresses due to small scale interactions for the xdirection is
[
(
]
)
(
[(
)
(
)
)
]
(5.124)
(5.125)
and for the y-direction is:
(
)
[
(
In which (
(
)
and (
]
)
[(
)
)
(
is defined as in Equation (5.94), (
as in Equation (5.98) and (
)
(
)
)
)
]
)
(5.126)
(5.127)
as in Equation (5.95),
as in Equation (5.99). While (
)
are defined as:
(
)
(5.128)
(
)
(5.129)
130
Chapter 5
In Equations (5.124) and (5.125),
and
depend on
:
(5.130)
and in Equations (5.126) and (5.127),
and
depend on
:
(5.131)
The Reynolds stresses for the x-direction are written as
(
(
)
[(
)
(
)
)
(5.132)
]
(5.133)
and for the y-direction as
(
)
(
(
)
[(
)
131
(
)
)
]
(5.134)
(5.135)
Chapter 5
5.5.4 Drag force terms
As for the others terms, also the drag force terms are defined in the x-direction at the right
face of the numerical cell and in the y-direction at the top face.
(
)
(5.136)
(
)
(
(
)
(
)
)
(5.137)
(
)
(
)
(
)
where
(
)
(
(
)
(
)
)
(5.138)
(5.139)
(
)
(5.140)
(
)
(5.141)
132
Chapter 5
(
(
(
)
(5.142)
)
(
(
)
)
(
)
(5.143)
)
5.5.5 Pressure terms
Pressure is defined at the center of the computational cell, hence the Poisson Equation has
to be discretized calculating all terms at the center of the cell.
If we consider the right terms of Equation (5.73), we have:
[
{
[
(
(
[
)]
)]}
(
[
{
)]
(
)]}
)(
)
(5.144)
{(
)(
{(
)
)(
)
(
(
)(
)
}
}
where
(
)
(5.145)
133
Chapter 5
(
)
(5.146)
(
)
(5.147)
(
)
(5.148)
(5.149)
(5.150)
The left terms of the Equation (5.73) can be discretized as:
[
(̃ )
)
[(
(̃ )
(̃ )
) , (
(̃ )
(̃ )
(
(̃ )
(
(̃ )
(
where (
(̃ )
)
)
(
(̃ )
) , (̃ )
)
(̃ )
(
]
(
)
(5.151)
) ]
)
and ( ̃ )
are defined in Equations
(5.118), (5.119), (5.80) and (5.81), respectively. Whereas the derivatives of the density at
the centre of the computational cells are defined as following:
134
Chapter 5
(
(
)
(
)
(
(
)
(
)
(
(
)
(
)
(
)
(5.152)
)
) (
)
)(
(
)
(5.153)
)
By this way, a set of linear algebraic equations for the pressure field that can be solved by
standard matrix solvers. Here the conjugate gradient method with the preconditioner of
incomplete Cholesky decomposition is used to solve the resulting sparse and symmetric
system of equations.
Solving the PPE equation the values of the pressure at the n+1 time step are determined
and the values of the sediment and fluid velocities and the concentration can be update.
135
CHAPTER 6
WAVE-INDUCED EROSION AND DEPOSITION
PATTERNS: VERIFICATION OF MODEL
RESULTS.
In this chapter a preliminary verification of the two-phase numerical model is presented.
Some simple cases were chosen in order to check all the new subroutines and equations
implemented in the IH-2VOF model. In Table 6.1 the principal characteristic of the cases
tested are summarized. The behavior of the sediment bottom in terms of concentration
and elevation was investigated.
Table 6.1. Characteristics of cases tested.
Test
Wave attack
Water
depth
Hs [m]
Tp [s]
wd [m]
P1
0.1
4.3
2.5
P2
0.1
4.3
2.5
Boundary Condition
Seaward
Mesh resolution
Landward
cell width [m]
cell height [m]
open
0.05
0.05
close
0.05
0.05
Hs , Tp
regular waves
Hs , Tp
regular waves
6.1 Computational set-up
Both tests - P1 and P2 - were performed in a numerical flume 10 m long and 4 m deep
under regular waves with significant wave height
. An homogeneous layer of sand (
𝑆
and peak period
) on the bottom of the channel was
present. The solid phase was characterized by a density
. The water depth
was set equals to
and a diameter
(Figure 6.1).
In the test P1 an open boundary condition was set on the landward (right) boundary,
whereas, in the test P2, the right boundary condition has been changed and a close
boundary was set.
Chapter 6
Figure 6.1. Sketch of the cases tested.
In both tests the same regular wave attack is imposed and in Figure 6.2 the water level at
the wave gouge sets 2
from the beginning of the channel is reported.
Figure 6.2. Water depth trend measured at
from the beginning of the channel.
6.2 Results for test P1
Figure 6.3 reports the water level along the channel at three different time steps:
color),
(green color) and
s (red
(blue color). Because the wave length is about equals
to 20 m and the channel length 10 m, we can observe one wave along the channel for
each time step.
137
Chapter 6
Figure 6.3. Water depth trend along the channel at t = 20 s (red), t = 50 s (green) and t = 80 s (blue).
The evolution of the sand bottom at the same time steps (
color;
s, red color;
, green
, blue color) is reported in Figure 6.4. It is possible to observe that the trend is
the following: erosion in the first half of the channel and deposition in the second half.
Besides, the bottom level along the numerical channel tends to decrease over the time
because the landward boundary condition (right) is open and the sand can get out from
the numerical domain.
Figure 6.4. Bottom level along the channel at t = 20 s (red), t = 50 s (green) and t = 80 s (blue). hs is the
original sediment bottom.
138
Chapter 6
Figures 6.5 and 6.6 report the velocity profiles at different time steps and at different
distances from the beginning of the channel. In particular, in Figure 6.5 the horizontal
velocities at the gauge sets at
from the beginning of the channel are shown
whereas Figure 6.6 presents the velocity profiles at the gauge sets at
. By
comparing these figures with Figure 6.3, it can be observed, as expected, that in
correspondence with the wave crest the velocity values are positive whereas in
correspondence with the wave trough the velocity values are negative.
Figure 6.5. Horizontal velocity at the gauge sets at
from the begin of the channel.
Figure 6.6. Horizontal velocity at the gauge sets at
from the begin of the channel.
139
Chapter 6
Figures 6.7 and 6.8 present calculated values of the concentration. In the lowest portion
of the sheet there is a high concentration (almost equal to close-packed concentration,
) and then the concentration slowly decreases in the vertical direction. In the
same region, the fluid velocity is relatively weak (see Figures 6.5 and 6.6). Such features
are due to the contact stress and the high viscosity implemented in the model for the
region of enduring contacts. Over time the values of the concentration decrease because
the right boundary condition is open.
The position of the bottom (that you can see in Figure 6.4) is determined by considering
that if, in a given cell, the concentration has a value greater than 50% of the close-packed
concentration, then that cell is part of the bottom. However, there is some material in
suspension that is not included in the seabed.
In particular, Figure 6.7 presents the sediment concentration at the gauge sets at
from the beginning of the channel. By observing Figure 6.4, it appears clearly that in this
section the sediment bottom is eroded, hence in Figure 6.7 the diamonds show the
concentration of cells that are included into the seabed, whereas the crosses the
concentration of cells that have less material and then are not included into the seabed.
These are cells with materials in suspension.
As said before, in the second half of the numerical domain an deposition phenomenon is
present. As for the previous figure, the diamonds represent the concentration of the cells
that are included into the seabed and the crosses the concentration of the cells that are not
included.
140
Chapter 6
Figure 6.7. Sediment concentration at the gauge sets at
from the begin of the channel.
Figure 6.8. Sediment concentration at the gauge sets at
from the begin of the channel.
141
Chapter 6
6.3 Results for test P2
The only difference between tests P1 and P2 is that in P1 the landward (right) boundary
was set as an open boundary, whereas, in test P2, the right boundary was set as a close
boundary. The goal in this paragraph is to verify if the numerical model is affected by
changing the boundary condition.
As for test P1, Figure 6.9 reports the water level along the channel at three different time
steps:
s (red color),
(green color) and
(blue color). Also in this case, the
channel length (10 m) is half the wave length (about 20 m), therefore only one wave
along the channel for each time step is observed.
The following figure (Figure 6.10) shows the evolution of the sand bottom at the time
steps:
s (red color),
(green color) and
(blue color). By comparing this figure
with Figure 6.4, it is clear that the numerical model is affected by the change of the right
boundary condition. In fact, the erosion-accumulation pattern change completely. In
particular, the principal area of erosion is set at the middle of the channel in
correspondence with the antinode point. In the second half of the numerical domain is
present a strong deposition of the sediment because sand cannot get out from the channel
(landward boundary close).
Also close to the beginning of the channel, there is a
deposition section. Moreover, over the time, the erosion and the accumulation increase
because the landward boundary is close and the sediment remains inside the channel.
142
Chapter 6
Figure 6.9. Water depth trend along the channel at t = 20 s (red), t = 50 s (green) and t = 80 s (blue).
Figure 6.10. Bottom level along the channel at t = 20 s (red), t = 50 s (green) and t = 80 s (blue).
Figure 6.11, 6.12 and 6.13 present the sediment concentration at the gauges set at
,
and
from the beginning of the channel respectively. As
said before, in the first and second half of the numerical domain an accumulation
phenomenon is present. Hence, Figure 6.11 and 6.13 show that the maximum bottom
level exceed the original value
.
143
Chapter 6
In the section
the sediment bottom is eroded (see Figure 6.10), hence Figure
6.12 shows that the maximum bottom level remains always under
, and in particular
the erosion increase over the time.
Figure 6.11. Sediment concentration at the gauge sets at
Figure 6.12. Sediment concentration at the gauge sets at
144
from the begin of the channel.
from the begin of the channel.
Chapter 6
Figure 6.13. Sediment concentration at the gauge sets at
145
from the begin of the channel.
CHAPTER 7
CONCLUSIONS AND FUTURE WORK.
The main objective of this thesis work was to develop a tool that can represent wave runup and overtopping together with beach reshaping during storms. A two-dimensional
numerical model based on the Reynolds Averaged Navier–Stokes equations (RANS),
called IH-2VOF, was selected. This numerical model was used to study the wavestructure interaction and then the new equations for the representation of the sediment
transport were introduced.
More than 90 numerical simulations with IH-2VOF model were carried out in order to
analyze the flow characteristics (velocities and layer thicknesses) on a dike crest. The
numerical data, derived by this analysis, allowed to perform a systematic investigation,
which may be useful to extend the existing theoretical approach and provide criteria for
design application.
First, in order to verify the numerical simulations, selected numerical results were
compared against experimental data and consolidated theoretical formulae. In particular,
the wave reflection coefficients calculated numerically for each test was compared with
the experimental results obtained by Zanuttigh et al. (2006). The numerical values of
appear to be slightly greater than the experimental values. However, the numerical trends
show two key issues in agreement with the physical process: the greater the submergence
and/or the lower the wave height, the lower the reflection.
As regards the discharge, the numerical values were compared with the theoretical
formulae proposed by Eurotop 2007. Both for emerged and zero freeboard cases, the
theoretical discharge was well represented by the numerical one. Instead, for submerged
cases, the theoretical overflow discharge is well represented by the numerical one,
whereas the theoretical overtopping discharge is overestimated. However, this contribute
is very small and so it does not affect too much the total results.
Chapter 7
The effects of the structure design parameters (i.e. slopes and submergence) on the trend
of both depths and velocities over the crest were in-depth investigated. First it was found
that the seaward/landward slope does not significantly affect the evolution of the flow
depth and velocity over the dike crest whereas the most important parameter is the
relative submergence. Wave heights decrease and flow velocities increase while waves
travel over the crest. In particular, by increasing the submergence, the wave height decay
is less marked and it completely disappears when
, whereas the increase of
the velocity start from about the middle of the crest of the structure and however are very
modest. Besides, an appropriate curve able to fit the variation of the wave height/velocity
over the dike crest were found. Both for the wave height and for the wave velocity
different fitting coefficients were determined on the basis of the submergence and of the
significant wave height. The results show that by keeping in constant the submergence,
the wave height decay tends to decrease, almost halved, when the significant wave height
increases. Similarly, the numerical results show that the wave height decay increases
with increasing the submergence when the significant wave height is constant.
These conclusions can be very important in terms of design criteria. In particular, by
considering that climate change might cause sea level rise and the increase of the
intensity of storms, an increasing of the risk of flooding of low lying areas, an accelerate
erosion of exposed soft beach and a damage to existing coastal protection structures may
occur. The results obtained in this thesis work (in particular reported in Chapter 4) could
be taken into consideration for the upgrade of the structures.
In this context it is also important to predict the dimensionless decay coefficient
in
order to understand the rate of decay of the wave height on the dike crest. Hence the
eventual dependence of the decay coefficient on the characteristics of the structure and/or
wave attack was investigated. In particular, an equation describing the trend of the
dimensionless coefficient
for the wave height was derived. By this way it is possible
to predict the decay coefficient that controls the decrease of the wave height over the dike
crest.
147
Chapter 7
The second part of this thesis work was focused on further developing the numerical
model for sediment transport, aiming at representing beach erosion while waves run-up
and overtop the sea banks during storms. Balance equations for the average mass,
momentum and energy for the two phases are phrased in terms of concentration-weighted
(Favre averaged) velocities.
Closures for the correlations between fluctuations in
concentration and particle velocities are based on those for collisional grain flow.
The new model allows us to calculate sediment fluxes everywhere in the water column
together with the sediment concentration. Moreover it is possible to model the bed
profile evolution.
Two different tests were performed under low-intensity regular waves with an
homogeneous layer of sand on the bottom of a channel whose length is about half the
tested wave length. The only difference between the two tests is that in the first one the
landward (right) boundary was set as an open boundary, whereas, in the second one, the
right boundary was set as a closed boundary. The change of boundary conditions affects
the numerical model response as it can be observed from the erosion-deposition pattern.
When the landward boundary is an open boundary, erosion occurs in the first half of the
channel (within L/4 from the wave-maker) and the deposition in the second half (between
L/4 and L/2). When the landward boundary is set as a closed boundary instead, the
erosion area occurs in the middle of the channel, i.e. in correspondence with the antinode
point (L/4). In the second half of the numerical domain a strong deposition of the
sediment occurs because of the closing wall. Another important difference is that in the
first case, the sediment mass tends to decrease over the time because the landward
boundary is open and the sand can get out from the numerical domain, whereas in the
second case the sediment mass is trapped inside the channel and therefore erosion and
accumulation patterns appear to be more marked in time.
Further work should be done to

validate the sediment transport model on a quantitative basis, considering
experimental and theoretical results available in the literature;
148
Chapter 7

examine the robustness of the model to reproduce different wave attacks and
sediment configurations, including the representation of more complex sediment
geometries inside the numerical code developed so far;

analyse beach reshaping during storms and compare to real prototype data for
assessing beach retreat and increase hydraulic vulnerability.
149
REFERENCES
Alsina, J.M., Baldock, T.E., Hughes, M.G., Weir, F., Sierra, J.P., 2005.
Sediment transport numerical modelling in the swash zone. In: Proceedings of Coastal
Dynamics ‘05, Barcelona, ASCE.
Antuono, M., Brocchini, M., Grosso, G., 2007. Integral properties of the swash
zone and averaging. Part 3. Longshore shoreline boundary conditions for waveaveraged nearshore circulation models. Journal of Fluid Mechanics 573, 399–415.
Archetti, R., Brocchini, M., 2002. An integral swash zone model with friction:
an experimental and numerical investigation. Coastal Engineering 45, 89–110.
Asano, T., 1990. Two-phase flow model on oscillatory sheet-flow. In:
Proceedings of the 22nd ICCE, Delft, Netherlands, pp. 2372–2384.
Asano, T., 1994. Swash motion due to obliquely incident waves. In: Proceedings
of the 24th Coastal Engineering conference, ASCE, October 23–38, Kobe, Japan, pp. 27–
41.
Baba, Y., Camenen, B., 2008. Importance of the swash longshore sediment
transport in morphodynamic models. Coastal Sediments ’07, ASCE, pp. 1938–1946.
Bagnold, R.A., 1963. Mechanics of marine sedimentation. In: Hill, M.N. (Ed.),
The Sea. Wiley-Interscience, New York, pp. 507–528.
Bagnold, R.A., 1966. An Approach to the Sediment Transport Problem from
General Physics. US Geological Survey, Washington, DC, 37pp.
Bailard, J.A., 1981. An energetics total load sediment transport model for a
plane sloping beach. Journal of Geophysical Research 86 (C11), 938–954.
Bakhtyar, R., Yeganeh-Bakhtiary, A., Ghaheri, A., 2007. Numerical
simulation of wave breaking and wave run-up in nearshore. International Conference on
Violent Flow, 20–22 November 2007, Fukuoka, Japan.
Bakhtyar, R., Ghaheri, A., Yeganeh-Bakhtiary, A., Baldock, T.E., 2008.
Longshore sediment transport estimation using fuzzy inference system. Applied Ocean
Research (in press).
Bakhtyar, R., Barry, D.A., Yeganeh-Bakhtiary, A., Ghaheri, A., 2009.
Numerical simulation of surf-swash zone motions and turbulent flow. Advances in Water
Resources 32, 250–263.
Chapter 7
Baldock, T.E., Holmes, P., 1997. Swash hydrodynamics on a steep beach.
Coastal Dynamics Conference, ASCE, Plymouth, UK, pp. 784–793.
Baldock, T.E., Huntley, D.A., 2002. Long wave forcing by the breaking of
random gravity waves on a beach. Proceedings of Royal Society of London A 458, 2177–
2201.
Baldock, T.E., Hughes, M.G., Day, K., Louys, J., 2005. Swash overtopping and
sediment overwash on a truncated beach. Coastal Engineering 52, 633–645.
Battjes, J. A., 1974. Computation of set-up, longshore currents, run-up and
overtopping due to wind –generated waves. Report 74-2. Committee on Hydraulics. Dept.
Of Civil Engineering, Delf University of Technology.
Bayram, A., Larson, M., Miller, H.C., Kraus, N.C., 2001. Cross-shore
distribution of longshore sediment transport: comparison between predictive formulas
and field measurements. Coastal Engineering 44, 79–99.
Bijker, E.W., 1971. Longshore transport computations. Journal of the
Waterways, Harbors and Coastal Engineering Division, ASCE 97 (4), 687–703.
Bocquet, L., Losert, W., Schalk, D., Lubensky, T. C. & Gollub, J. P., 2001
Granular shear flow dynamics and forces: experiment and continuum theory. Phys. Rev.
E65, 011307.
Bosman, G., van der Meer, J. W., Hoffmans, G., Schuttrumpf, H., and
Verhagen, H., 2008. Individual overtopping events at dikes. Proc., 31st Int. Conf.
Coastal Engineering, Vol. 4, ASCE, Reston, Va., 2944–2956.
Bradford, S.F., 2000. Numerical simulation of surf zone dynamics. Journal of
Waterway, Port, Coastal, and Ocean Engineering, ASCE 126 (1), 1–13.
Brocchini, M., Peregrine, D.H., 1995. Flow properties of the swash zone.
Proceedings of Coastal Dynamics’95, ASCE, 221-232.
Brocchini, M., Peregrine, D.H., 1996. Integral flow properties of the swash
zone and averaging. Journal of Fluid Mechanics 317, 241–273.
Calantoni, J., Puleo, J.A., Holland, K.T., 2006. Simulation of sediment motions
using a discrete particle model in the inner surf and swash-zones. Continental Shelf
Research 26, 610–621.
Carrier, G.F., Greenspan, H.P., 1958. Water waves of finite amplitude on a
sloping beach. Journal of Fluid Mechanics 4 (1), 97–109.
151
Chapter 7
Champagne, F.H., Harris, V.G., Coorsin, S., 1970. Experiments on nearly
homogeneous turbulent shear flow. Journal of Fluid Mechanics, 41: 81-139.
Chapman, S. & Cowling, T. G., 1970 The mathematical theory of nonuniform gases. 3rd edn. Cambridge University Press.
Chopakatla1, S.C., Lippmann, T.C., Richardson, J.H., 2008. Field
verification of a computational fluid dynamics model for wave transformation and
breaking in the surf zone. Journal of Water way, Port, Coastal, and Ocean Engineering,
ASCE 134 (2), 71–80.
Christensen, E.D., Deigaard, R., 2001. Large eddy simulation of breaking
waves. Coastal Engineering 42, 53–86.
Christensen, E.D., 2006. Large eddy simulation of spilling and plunging
breakers. Coastal Engineering 53, 463–485.
CLASH, 2004. Crest Level Assessment of coastal Structures by full scale
monitoring, neural network prediction and Hazard analysis on permissible wave
overtopping. EC-contract EVK3-CT-2001-00058. www.clash-eu.org.
Crowe, C.T. (Ed.), 2006. Multiphase Flow Handbook. Taylor and Francis, Boca
Raton.
Cundall, P.A., Strack, O.D., 1979. A discrete numerical model for granular
assembles. Geotechnique 29 (1), 47–65.
Dalrymple, R.A., Gómez-Gesteira M., Rogers B.D., Panizzo A., Zou S.,
Crespo A.J.C., Cuomo G., M. Narayanaswamy, 2009. Smoothed Particle
Hydrodynamics for Water Waves, in Advances in Numerical Simulation of Nonlinear
Waves Shi F., Q.Ma, ed., World Scientific Press, ISBN 981-283-649-7.
Davies, A.G., Ribberink, J.S., Temperville, A., Zyserman, J.A., 1997.
Comparisons between sediment transport models and observations made in wave and
current flows above plane beds. Coastal Engineering 31 (1–4), 163–198.
Dodd, N., 1998. A numerical model of wave run-up, overtopping and
regeneration. Journal of Waterway, Port, Coastal, and Ocean Engineering, ASCE 124
(2), 73–81.
Dong, P., Zhang, K., 1999. Two-phase flow modeling of sediment motions in
oscillatory sheet flow. Coastal Engineering 36, 87–109.
152
Chapter 7
Drake, T.G., Calantoni, J., 2001. Discrete particle model for sheet flow
sediment transport in the nearshore. Journal of Geophysical Research 106 (C9), 19859–
19868.
Drago, M., Iovenitti, L., 1995. The modelling of plunging breakers by the
introduction of a K-l turbulence closure model. In: Proceedings of Coastal Dynamics,
ASCE, pp. 317–328.
Drew, D. A., 1976 Production and dissipation of energy in the turbulent flow of a
particle–fluid mixture, with some results on drag reduction. J. Appl. Mech. 43, 543–547.
Elfrink, B., Baldock, T.E., 2002. Hydrodynamics and sediment transport in the
swash zone: a review and perspectives. Coastal Engineering 45, 149–167.
Elghobashi, S. E. & Abou-Arab, T. W., 1983. A two-equation turbulence model
for two-phase flows. Phys. Fluids 26, 931–938.
EUROSION, European Union Project ENV.B.3-SER-2001-0030, Reports
Online, Living with coastal erosion in Europe: sediment and space for sustainability, Part
1 – Major Findings and Policy Recommendations of the Eurosion project,
www.eurosion.org.
EurOtop, 2007. European Manual for the Assessment of Wave Overtopping. T.
Pullen, N.W.H. Allsop, T. Bruce, A. Kortenhaus, H. Schüttrumpf and J.W. van der Meer.
At: www.overtopping-manual.com.
Favre, A., 1965. Equations des gaz turbulent compressible. J. Méc. 4, 361–390.
Frijlink,H.C., 1952. Discussion des formules de debit solide de Kalinske,
Einstein et Meyer-Peter et Mueller compte tenue des mesures recentes de transport dans
les rivieres Neerlandaises. 2me Journal Hydraulique Societe Hydraulique de France,
Grenoble (in French), pp.98–103.
Fuhrman,D.R., Madsen, P.A.,2008. Simulation of non-linear wave run-up with
a high-order Boussinesq model. Coastal Engineering 55, 139–154.
Garcia, N., Lara, J.L., Losada, I.J., 2004a. 2-D numerical analysis of near-field
flow at low-crested breakwaters. Coastal Engineering 51 (10), 991–1020.
Garcia, N., Lara, J.L., Lomonaco, P., Losada, I.J., 2004b. Flow at low-crested
breakwaters under breaking conditions. Proceedings of the 29th International Conference
on Coastal Engineering. Word Scientific, pp. 4240-4252.
153
Chapter 7
Gotoh,H., Sakai,T., 1997. Numerical simulation of sheet flow as granular
materials. Journal of Waterway, Port, Coastal, and Ocean Engineering 123(6),329–336.
Guanche, R., I.J. Losada and J.L. Lara., 2009. Numerical analysis of wave
loads for coastal structure stability, Coastal Engineering, ELSEVIER, vol. 56, 543-558
Guard, P.A., Baldock,T.E., 2007. The influence of seaward boundary conditions
on swash zone hydrodynamics. Coastal Engineering 54, 321–331.
Higuera, P., Lara, J.L., Losada, I.J., 2013. Realistic wave generation and
active wave absorption for Navier–Stokes models. Application to OpenFOAM. Coastal
Engineering.
Higuera, P., Lara, J.L., Losada, I.J., 2013. Simulating Coastal Engineering
Processes with OpenFOAM. Coastal Engineering.
Hirt, C.W., Nichols, B.D., 1981. Volume of fluid (VOF) method for the dynamics
of free boundaries. Journal of Computational Physics 39, 201–225.
Hoffmann, M. et al., 2008. Conservation planning and the IUCN Red List.
Endangered Species Research 6, 113–125.
Hsu, T.-J., 2002. A two-phase flow approach for sediment transport. PhD thesis,
University, Ithaca, NY, USA.
Hsu, T.-J., Sakakiyama, T., Liu, P.L.-F., 2002. A numerical model for wave
motions and turbulence flows in front of a composite breakwater. Coastal
Engineering, 46: 25-50.
Hsu, T.-J., Jenkins, J.T., Liu, P.L.-F., 2003. On two-phase sediment transport:
dilute flow. Journal of Geophysical Research 108, 3057.
Hsu, T., Jenkins, J.T., Liu, P.L.-F., 2004. On two-phase sediment transport
sheet flow of massive particles. Proceedings of the Royal Society of London, Seriees A
460, 2223–2250.
Hsu, T.-J., and B. Raubenheimer, 2006. .A numerical and field study on innersurf and swash sediment transport, Cont. Shelf. Res., 26, 589-598.
Hu, K., Mingham, C.G., Causon, D.M., 2000. Numerical simulations of wave
overtopping of coastal structures using the non-linear shallow water wave equations.
Coastal Engineering 41, 346–365.
Hughes, M.G., Masselink, G., Brander, R.W., 1997. Flow velocity and
sediment transport in the swash zone of a steep beach. Marine Geology 138, 91–103.
154
Chapter 7
Hughes, M.G., Baldock, T.E., 2004. Eulerian flow velocities in the swash zone:
field data and model predictions. Journal of Geophysical Research 109, C08009.
Hughes, S.A., N.C. Nadal, 2009. Laboratory study of combined wave
overtopping and storm surge overflow of a levee. Coastal Engineering, 56, 244-259.
Hughes, S, C. Thornton, J.W. van der Meer and B. Scholl, 2012.
Improvements in describing wave overtopping processes. ASCE, Proc. ICCE 2012,
Santander, Spain.
Hunt, A., 1959. Design of seawalls and breakwaters. Journal of the Waterways
and Harbors Division. ASCE, pp. 123– 152.
Jenkins, J. T. & Savage, S. B., 1983. A theory for the rapid flow of identical,
smooth, nearly elastic particles. J. Fluid Mech. 370, 29–52.
Jenkins, J. T., Cundall, P. A. & Ishibashi, I., 1989. Micromechanics modeling
of granular material. In Powders and grains (ed. J. Biarez & R. Gourves), pp. 257–264.
Rotterdam: Balkema.
Jenkins, J. T. & Hanes, D. M., 1998 Collisional sheet flows of sediment driven
by a turbulent fluid. J. Fluid Mech. 370, 29–52.
Jensen, A., Pedersen, G.K., Wood, D.J., 2003. An experimental study of wave
run-up at a steep beach. Journal of Fluid Mechanics 486, 161–188.
Kamphuis, J.W., 1991. Alongshore sediment transport rate distribution. In:
Coastal Sediments ‘91, ASCE, pp. 170–183.
Kamphuis, J.W., 2002. Alongshore transport of sand, Proceedings of the 28th
Coastal Engineering Conference, ASCE, in press.
Karambas, T.V., Koutitas, C., 2002. Surf and swash zone morphology evolution
induced by nonlinear waves. Journal of Waterway, Port, Coastal, and Ocean Engineering,
ASCE 128 (3), 101–142.
Karambas, T.V., 2003. Modelling of infiltration–exfiltration effects of crossshore sediment transport in the swash zone. Coastal Engineering Journal 45, 63–82.
Karambas, T.V., 2006. Prediction of sediment transport in the swash zone by
using a nonlinear wave model. Continental Shelf Research 26, 599–609.
Karunarathna, H., Chadwick, A., Lawrence, J., 2005. Numerical experiments
of swash oscillations on steep and gentle beaches. Coastal Engineering 52, 497–511.
155
Chapter 7
Kennedy, A.B., Chen, Q., Kirby, J., Dalrymple, R.A., 2000. Boussinesq
modeling of wave transformation, breaking, and run-up. I: 1D. Journal of Waterway,
Port, Coastal, and Ocean Engineering, ASCE 126 (1), 39–47.
Kirby, J.T., 2003. Boussinesq models and applications to nearshore wave
propagation, surf zone processes and wave-induced currents. In: Lakhan, C. (Ed.),
Advances in Coastal Engineering. Elsevier, Amsterdam.
Kobayashi, N., Seo, S.N., 1985. Fluid and sediment interaction over a plan bed.
Journal of Hydraulic Engineering 111 (6), 903–921.
Kobayashi, N., De Silvia, G.S., Watson, K.D., 1989. Wave transformation and
swash oscillation on gentle and steep slopes. Journal of Geophysical Research 94 (C1),
951–966.
Kobayashi, N., Poff, M.T., 1994. Numerical model RBREAK2 for random waves
on impermeable coastal structures and beaches. Research Report No. CACR-94-12,
Center for Applied Coastal Research, Department of Civil Engineering, University of
Delaware, Newark, 325pp.
Kobayashi, N., Agarwal,A., Johnson,B.D., 2007. Longshore current and
sediment transport on beaches. Journal of Waterway, Port, Coastal, andOcean
Engineering, ASCE133(4),296–304.
Komar, P.D., Inman, D.L., 1970. Longshore sand transport on beaches. Journal
of Geophysical Research 75(30), 5514–5527.
Komar, P.D., 1998. Beach processes and sedimentation (2nd edition). PrenticeHall, New Jersey, 545 p.
Koshizuka, S., Go, S., Oka, Y., Tanizawa, K., 2004. Numerical analysis of a
two-dimensional experiment of shipping water on deck using a particle method.
Proceedings of ASME HTFED’04.
Kothe, D.B., Mjolsness, R.C., Torrey, M.D., 1991. RIPPLE: a computer
program for incompressible flows with free surfaces, Los Alamos National Laboratory.
Kumar, V.S., Anand, N.M., Chandramohan,P., Naik, G.N., 2003. Longshore
sediment transport rate – measurement and estimation, central west coast of India.
Coastal Engineering 48,95–109.
156
Chapter 7
Lara, J.L., Losada, I.J., Liu, P.L.-F., 2006. Breaking waves over a mild gravel
slope: experimental and numerical analysis. Journal of Geophysical Research, AGU,
Vol. 111, C11019; doi: 10-1029/2005 JC003374.
Lara J.L., Losada I.J., R. Guanche, 2008. Wave interaction with low-mound
breakwater using a rans model. Ocean Engineering, 56, 543-558.
Lara, J.L., Ruju, A. and Losada, I.J., 2011. Reynolds Averaged Navier-Stokes
modelling of long waves induced by a transient wave group on a beach. Proceedings of
the Royal Society A, vol. 467, 1215-1242.
Larson, M., Kubota,S., Erikson,L., 2004. Swash-zone sediment transport and
foreshore evolution: field experiment sand mathematical modeling. Marine Geology 212
(1–4), 61–80.
Lemos, C.M., 1992. Wave breaking. Springer-Verlag, New York, USA.
Leont’yev, I.O., 1999. Modelling of morphological changes due to coastal
structures. Coastal Engineering 38,143–166.
Lin, P., 1998. Numerical modeling of breaking waves. Ph.D. Thesis, Cornell
University.
Lin, P., Liu, P.L.-F., 1998a. A numerical study of breaking wave in the surf zone.
Journal of Fluid Mechanics 359, 239–264.
Lin, P., Liu, P.L.-F., 1998b. Turbulence transport, vorticity dynamics, and
solute mixing under plunging breaking waves in surf zone. Journal of Geophysical
Research 103, 15677–15694.
Lin, P., Liu, P.L.-F., 1999. Internal wave maker for Navier-Stokes equations
models. Journal of Waterway, Port, Coastal, and Ocean Engineering, ASCE 125, 207–
215.
Liu, P.L.-F., Lin, P., 1997. A numerical model for breaking waves: the volume of
fluid method. Research Report No. CACR-97-02, Center for Applied Coastal Research,
Ocean Engineering Laboratory, University of Delaware, Newark, DE.
Liu, P.L.-F., Lin, P., Chang, K.-A., Sakakiyama, T., 1999. Numerical
modeling of wave interaction with porous structures. J. Waterw., Port, Coast., and Ocean
Eng., ASCE 125 (6), 322 – 330.
Liu, H., Sato, S., 2006. A two-phase flow model for asymmetric sheet flow
conditions. Coastal Engineering 53, 825–843.
157
Chapter 7
Losada I.J., 2003. Advances in modeling the effects of permeable and reflective
structures on waves and nearshore flows. In: Chris Lakhan, V. (Ed.), Advances in
Coastal Modeling. Elsevier Oceanography Series, vol. 67.
Losada, I.J., Lara, J.L., Christensen, E.D., Garcia, N., 2005. Breaking
process, velocities and turbulence intensity around low crested structures. Coastal
Engineering, “DELOS” Special Issue.
Losada, I.J., J.L. Lara, R. Guanche and J.M. Gonzalez-Ondina, 2008.
Numerical analysis of wave overtopping of rubble mound breakwaters, Coastal
Engineering, ELSEVIER, 55(1), 47-62 (2008)
Lynett, P., Wu, T.-R., Liu, P.L.-F., 2002. Modelling wave run-up with depthintegrated equations. Coastal Engineering 46, 89–107.
Lynett, P., Liu, P.L.-F., 2004. A two-layer approach to water wave modeling.
Proceedings of the Royal Society of London, Series A 460, 2637–2669.
Lynett, P., 2006. Nearshore modeling using high-order Boussinesq equations.
Journal of Waterway, Port, Coastal, and Ocean Engineering, ASCE 132 (5), 348–357.
McTigue, D. F., 1981. Mixture theory for suspended sediment transport. J.
Hydraul. Div. ASCE 107, 659–673.
Madsen, O.S., 1991. Mechanics of cohesionless sediment transport in coastal
waters. Coastal Sediments ‘91. ASCE, Seattle, Washington, pp. 15–27.
Madsen, O.S., 1993. Sediment transport on the shelf. In: Proceedings of the
Sediment Transport Workshop DRP TA1, 8–9 June. CERC, Vicksburg, MS.
Madsen, P.A., Sørensen, O.R., Schaffer, H.A., 1997b. Surf zone dynamics
simulated by a Boussinesq type model. Part II. Surf beat and swash oscillations for wave
groups and irregular waves. Coastal Engineering 32, 289–319.
Masselink, G., Hughes, M., 1998. Field investigation of sediment transport in
the swash zone. Continental Shelf Research 18 (10), 1179–1199.
Masselink, G., Evans, D., Hughes, M.G., Russell, P., 2005. Suspended
sediment transport in the swash zone of a dissipative beach. Marine Geology 216, 169–
189.
Meyer-Peter, E., Muller, R., 1948. Formulas for bed-load transport. In:
Proceedings of the 3rd Meeting of the International Association for Hydraulic Research,
Stockholm, Sweden, pp. 39–64.
158
Chapter 7
Miles, L., Newton, A. C., Defries, R., Ravilious, C., May, I., Blyth, S., Kapos,
V., Gordon, J., 2006. A global overview of the conservation status of tropical dry
forests. Journal of Biogeography 33: 491-505.
Nakayama, A., Kuwahara, F., 1999. A macroscopic turbulence model for
flow in a porous medium. Journal of Fluids Engineering, 121: 427-433.
nd
Nield, D.A., Bejan, A., 1999. Convection in porous media. 2 edition, SpringerVerlag, New York, USA.
Nielsen, P., 1990. Tidal dynamics of the water table in beaches. Water Resources
Research 26 (9), 2127–2134.
Nielsen, P., 2002. Shear stress and sediment transport calculations for swash
zone modeling. Coastal Engineering 45, 53–60.
Nielsen, K.B., Mayer, S., 2004. Numerical prediction of green water incidents.
Ocean Engineering 31 (3–4), 363–399.
Ono, M., Deguchi, I., Sawaragi, T., 1996. Numerical modeling of sediment
transport for various mode. In: Proceedings of 25th International Conference on Coastal
Engineering. ASCE, New York, pp. 3888–3900.
Owen, M.W., 1980. Design of seawalls allowing for wave overtopping.
Technical Report EX-924. H.R. Wallingford, United Kingdom.
Park, J.C., Kim, M.H., Miyata, H., 1999. Fully non-linear free-surface
simulations by a 3D viscous numerical wave tank. International Journal for Numerical
Methods in Fluids 29 (6), 685–703.
Pedrozo-Acuña, A., Simmonds, D.J., Otta, A.K., Chadwick, A.J., 2006. On
the cross- shore profile change of gravel beaches. Coastal Engineering 53(4), 335–347.
Pedrozo-Acuña, A., Simmonds, D.J., Chadwick, A.J., Silva, R., 2007. A
numerical- empirical approach for evaluating morphodynamic processes on gravel and
mixed sand-gravel beaches. Marine Geology 241, 1–18.
Pilkey, O. H., Hume, T., 2001. Coastal research – the shoreline erosion
problem: lessons from the past. Water and Atmosphere, 9(2), National Institute of Water
and Atmosphere, New Zealand. [http://www.niwa.cri.nz]
Pope, S.B., 2000. Turbulent Flows. University Press, Cambridge, UK,771pp.
159
Chapter 7
Puleo, J.A., Slinn, D.N., Holland, K.T., Smith, E., Webb, B.M., 2002.
Numerical modeling of swash zone hydrodynamics. In: Proceedings of the American
Society of Civil Engineers, Cardiff, Wales, UK.
Puleo, J.A., Holland, K.T., Plant, N., Slinn, D.N., Hanes, D.M., 2003. Fluid
acceleration effects on suspended sediment transport in the swash zone. Journal of
Geophysical Research 108 (C11), 3350, 10.1029/2003JC001943.
Rakha, K.A., Deigaard, R., Brøker, I., 1997. A phase-resolving cross shore
sediment transport model for beach profile evolution. Coastal Engineering 31, 231–261.
Raubenheimer, B., 2002. Observations and predictions of fluid velocities in the
surf and swash zones. Journal of Geophysical Research 107 (C11), 3190.
Ribberink, J.S., Al-Salem, A.A., 1995. Sheet flow and suspension of sand in
oscillatory boundary layers. Coastal Engineering 25, 205–225.
Richardson, J. F., Zaki, W. N., 1954. Sedimentation and fluidization. Part 1.
Trans. IChE 32, 35–53.
Rodi, W., 1980. Turbulence models and their application in hydraulics – a stateof-the-art review. IAHR Publication.
Schüttrumpf,
H.,
2001.
Wellenüberlaufströmung
bei
Seedeichen
–
Experimentelle und Theoretische Untersuchungen. PhD-Thesis.
Schüttrumpf, H., van Gent, M.R., 2003. Wave overtopping at seadikes.
Proceedings of Coastal Structures, '03. American Society of Civil Engineers, pp. 431–
443.
Schüttrumpf H., H. Oumeraci, 2005. Layer thicknesses and velocities of wave
overtopping flow at sea dikes. Coastal Engineering, 52 (6), 473-495.
Shen, M.C. and Meyer, R.E., 1963. Climb of a bore on a beach. Part 3. Runup.
J. Fluid Mech., 16: 113-125.
Shiach, J.B., Mingham, C.G., Ingram, D.M., Bruce, T., 2004. The
applicability of the shallow water equations for modelling violent wave overtopping.
Coastal Engineering 51 (1), 1–15.
Shih, T.H., Zhu, J., Lumley, J.L., 1996. Calculation of wall-bounded
complex flows and free shear flows. International Journal for Numerical Methods in
Fluids, 23: 1133-1144.
160
Chapter 7
Shore Protection Manual,1984. USArmy Engineer Waterways Experiment
Station, US Government Printing Office, Washington, DC. Shen, M.C., Meyer, R.E.,
1963. Climb of a bore on a beach: Part3, run-up. Journal of Fluid Mechanics 16(8),113–
125.
Shen, Y.M., Ng, C.O., Zheng, Y.H., 2004. Simulation of wave propagation over
a submerged bar using the VOF method with a two-equation k-eps turbulence modeling.
Ocean Engineering 31(1), 87–95.
Shields, A.,1936. An wending der Ahnlichkeits Mechanik und der Turbulenzforschung auf die Geschiebe Bewegung, 26. Mitteilungen Preuss. Versuchan- stalt fur
Wasserbau and Schiffbau, Berlin, 26pp.
Sitanggang, K., Lynett, P., 2005. Parallel computation of a highly nonlinear
Boussinesq equation model through domain decomposition. International Journal for
Numerical Methods in Fluids 49 (1), 57–74.
Smith, E.R.,Ebersole,B.A.,Wang,P.,2004. Dependence of total longshore
sediment transport rate son incident wave parameter sand breaker type. Technical Report
ERDC/CHLCHETN-IV-62, USArmy Engineer Research and Development Center,
Vicksburg, MS.
Sørensen, O.R., Schaffer, H.A., Sørensen, L.S., 2004. Boussinesq-type
modelling using an unstructured finite element technique. Coastal Engineering 50, 181–
198.
Sumer, B. M., Kozakiewicz, A., Fredsoe, J. & Deigaard, R., 1996. Velocity
and concentration profiles in sheet-flow layer of movable bed. J. Hydraul. Engng 122,
549–558.
Tajima, Y., 2004. Waves, currents, and sediment transport in the surf zone along
long, straight beaches. Ph.D. Thesis, Massachusetts Institute of Technology, 313 pp.
Tautenhain,
E.,
1981.
Der
Wellenu¨berlauf
an
Seedeichen
unter
Beru¨cksichtigung des Wellenauflaufs. Mitteilungen des Franzius- Instituts. H., 53, 1 –
245.
THESEUS project “Innovative technologies for safer European coasts in a
changing climate”, FP7.2009-1, Contract 244104, is gratefully acknowledged.
Torquato, S., 1995. Nearest-neighbor statistics for packings of hard spheres and
disks. Phys. Rev. E51, 3170–3182.
161
Chapter 7
Torres-Freyermuth, A., Lara, J.L. and Losada, I.J., 2010. Numerical
modelling of short and long-wave transformation on a barred beach. Coastal
Engineering, vol. 57, 317-330.
Van der Meer, J.W., Janssen, W., 1995. Wave run-up and wave overtopping at
dikes. In: Kabayashi, Demirbilek (Eds.),Wave Forces on Inclined and VerticalWall
Structures. American Society of Civil Engineers, pp. 1–27.
Van der Meer, J.W., Bernardini, P., Snijders,W., E. Regeling, 2006. The wave
overtopping simulator. Proceedings of the 30th International Conference on Coastal
Engineering, vol. 5. World Scientific, 4654–4666.
Van der Meer, J. W., Steendam, G. J., de Raat, G., and Bernardini, P.,2008.
Further developments on the wave overtopping simulator. Proc., 31st Int. Conf. Coastal
Engineering, Vol. 4, ASCE, Reston, Va., 2957–2969.
Van der Meer, J.W., R. Schrijver, B. Hardeman, A. van Hoven, H. Verheij
and G.J. Steendam, 2009. Guidance on erosion resistance of inner slopes of dikes from
three years of testing with the Wave Overtopping Simulator. Proc. ICE, Breakwaters,
Marine Structures and Coastlines; Edinburgh, UK.
Van der Meer J. W., Hardeman B., Steendam S.J., Schüttrumpf H., H.
Verheij, 2010. Flow depth and velocities at crest and landward slope of a dike, in theory
and with the wave overtopping simulator. ASCE, Proc. ICCE 2010, Shanghai.
Van Wellen, E., Baldock, T., Chadwick, A.J., Simmonds, D., 2000. STRAND:
a model for longshore sediment transport in the swash zone. In: Proceedings of the 27th
International Conference on Coastal Engineering, ASCE, Sydney, pp. 3139–3150.
Van Gent, M.R., 1994. The modelling of wave action on and in coastal
structures. Coastal Engineering, 22: 311-339.
Van Gent M.R., 2002. Wave overtopping events at dikes. Proceedings of the
28th International Coastal Engineering Conference, vol. 2. World Scientific, 2203–2215.
Victor, L., J.W. van der Meer and P. Troch, 2012. Probability distribution of
individual wave overtopping volumes for smooth impermeable steep slopes with low crest
freeboards. Coastal Engineering, 64, pp.87–101.
Wassing, F., Model investigations of wave run-up on dikes carried out in the
Netherlands during the past twenty years, Proc. 6th Conf. Coastal Eng., Gainesville,
1957, p. 700-714.
162
Chapter 7
Walton Jr., T.L., Bruno, R.O., 1989. Longshore transport at a detached break
water, phase II. Journal of Coastal Research 65 (9), 667–668.
Watanabe, A., 1992. Total rate and distribution of longshore sand transport. In:
Proceedings of the 23rd Coastal Engineering Conference, ASCE, New York, pp. 2528–
2541.
Wood, D.J., Pedersen, G.K., Jensen, A., 2003. Modelling of run-up of steep
non- breaking waves. Ocean Engineering 30, 625–644.
Yeganeh-Bakhtiary, A., Gotoh, H., Sakai, T., 2000. Applicability of Euler–
Lagrange coupling multiphase-flow model to bed-load transport under high bottom
shear. Journal of Hydraulic Research 38, 389–398.
Zanuttigh B., J.W. van der Meer, Wave reflection from coastal structures in
design conditions, Coastal Engineering, 2008, 55 (10), pp. 771 – 779.
Zanuttigh, B., J.W. van der Meer, T. Bruce and S. Hughes, 2013. Statistical
characterization of extreme overtopping wave volumes. Proc. ICE 2013, Edinburgh.
Zhang, Y., Campbell, C. S., 1992 The interface between fluid-like and solid-like
behavior in two-dimensional granular flows. J. Fluid Mech. 237, 541–568.
Zhang, Q., Liu, P.L.-F., 2008. A numerical study of swash flows generated by
bores. Coastal Engineering 55 (12), 1113–1134. Zhao, Q., Armfield, S., Tanimoto, K.,
2004. Numerical simulation of breaking waves by a multi-scale turbulence model.
Coastal Engineering 51, 53–80.
Zhao, Q., S. Armfield and K. Tanimoto, 2004: Numerical simulation of
breaking waves by a multi-scale turbulence model. Coastal Engineering 51, pp. 53-80.
Zhou, J.G., Stansby, P.K., 1999. An arbitrary Lagrangian–Eulerian model with
non- hydrostatic pressure for shallow water flows. Computer Methods in Applied
Mechanics and Engineering 178, 199–214.
163
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