\chapter{Literature Review}

In this chapter, literature relevant to the present study will be reviewed.
Three sections will be detailled: the separation of incident and reflected
components from wave measurements, the modelisation of wave impacts on a
rubble-mound breakwater, and the modelisation of block displacement by wave
impacts.

\section{Separating incident and reflected components from wave buoy data}

\subsection{Introduction}

The separation of incident and reflected waves is a crucial step in numerically
modeling a sea state. Using the raw data from a buoy as the input of a wave
model will lead to incorrect results in the domain as the flow velocity at the
boundary will not be correctly generated.

Several methods exist to extract incident and reflected components in measured
sea states, and they can generally be categorised in two types of methods:
array methods and PUV methods \parencite{inch2016accurate}. Array methods rely
on the use of multiple measurement points of water level to extracted the
incident and reflected waves, while PUV methods use co-located pressure and
velocity measurements to separate incident and reflected components of the
signal.

\subsection{Array methods}

\subsubsection{2-point methods}

Array methods were developped as a way to isolate incident and reflected wave
components using multiple wave records.
\textcite{goda1977estimation,morden1977decomposition} used two wave gauges
located along the wave direction, along with spectral analysis, in order to
extract the incident and reflected wave spectra. Their work is based on the
earlier work of \textcite{thornton1972spectral}. \textcite{goda1977estimation}
analyzed the wave spectrum components using the Fast Fourier Transform, and
suggests that this method is adequate for studies in wave flumes. They noted
that this method provides diverging results for gauge spacings that are
multiples of half of the wave length. \textcite{morden1977decomposition}
applies this technique to a field study, where the sea state is wind generated.
\textcite{morden1977decomposition} showed that, using appropriate spectral
analysis methods along with linear wave theory, the decomposition of the sea
state into incident and reflected waves is accurate. A relation between the
maximum obtainable frequency and the distance between the sensors is provided.
According to \textcite{morden1977decomposition}, the only needed knowledge on
the wave environment is that wave frequencies are not modified by the
reflection process.

\subsubsection{3-point methods}

In order to alleviate the limitations from the 2-point methods,
\textcite{mansard1980measurement} introduced a 3-point method. The addition of
a supplementary measurement point along with the use of a least-squares method
most importantly provided less sensitivity to noise, non-linear interactions,
and probe spacing. The admissible frequency range could also be widened. A
similar method was proposed by \textcite{gaillard1980}. The accuracy of the
method for the estimation of incident and reflected wave components was once
again highlighted, while the importance of adequate positioning of the gauges
was still noted.

\subsubsection{Time-domain method}

\textcite{frigaard1995time} presented a time-domain method for reflected and
incident wave separation. This method, called SIRW method, used discrete
filters to extract the incident component of an irregular wave field. The
results were as accurate as with the method proposed by
\cite{goda1977estimation}, while singularity points are better accounted for.
The main advantage of the SIRW method is that it works in the time-domain,
meaning that real time computations can be performed.
\textcite{frigaard1995time} also mentions the possibility of replacing one of
the wave gauges by a velocity meters to prevent singularities.

This method was improved by \textcite{baldock1999separation} in order to
account for arbitrary bathymetry. Linear theory is used to compute shoaling on
the varying bathymetry. Resulting errors in the computed reflection coefficient
are low for large reflection coefficients, but increase with lower
coefficients. The neglect of shoaling can lead to important error in many
cases. The presented method could also be extended to three-dimensionnal waves
and bathymetry by considering the influence of refraction.

\subsubsection{Further improvements}

Further additions were made to array methods. \textcite{suh2001separation}
developped a method taking constant current into account to separate incident
and reflected waves. This method relies on two or more gauges, using a least
squares method. Results are very accurate in the absence of noise, but a small
amount of error appears when noise is added.

\textcite{inch2016accurate} noticed that the presence of noise lead to
overestimation of reflection coefficient. The creation of bias lookup tables is
proposed in order to account for noise-induced error in reflection coefficient
estimations.

\textcite{andersen2017estimation,roge2019estimation} later proposed
improvements to account for highly non-linear regular and irregular waves
respectively. The improved method provides very accurate results for highly
non-linear waves, but are expected to be unreliable in the case of steep
seabeds, as shoaling is not part of the underlying model.

\subsubsection{Conclusion}

Array methods have been developped enough to provide accurate results in a wide
range of situations. Sensibility to noise has been reduced, and the influence
of shoaling has been considered. Those methods can also be applied to irregular
non-linear waves.

However, they require at least two wave gauges to be used. That means that in
some situations such as the Saint-Jean-de-Luz event of 2017, other methods are
needed since only one field measurement location is available.

\subsection{PUV methods}

The goal of PUV methods is to decompose the wave field into incident and
reflected waves using co-located wave elevation and flow velocity measurements
\parencite{tatavarti1989incoming}. \textcite{tatavarti1989incoming} presented a
detailled analysis of separation of incoming and outging waves using co-located
velocity and wave height sensors. Their method allows to obtain the reflection
coefficient relative to frequency, as well as to separate incident and
reflected wave components. Compared to array methods, this method also strongly
reduces the influence of noise.

\textcite{kubota1990} studied the influence of the considered wave theory on
incident and reflected wave separation. Three methods, based on linear
long-wave theory, small-amplitude wave theory and quasi-nonlinear long-wave
theory respectiveley were developped and compared. The results show that the
quasi-nonlinear approach gave the most accurate results.

%\textcite{walton1992} applied a separation method based on co-located pressure
%and velocity measurements on field, studying two natural beaches. This study
%showed that reflection is not significant on natural beaches. Additionnaly,
%the method that is used allowed for larger reflected energy than incident
%energy.

Research by \textcite{hughes1993} showed how co-located horizontal velocity and
vertical velocity (or pressure) sensors can be used to extract incident and
reflected wave spectra. Their method is based on frequency domain linear
theory, and provided accurate results for full reflection of irregular
non-breaking waves. Low-reflection scenarii were evaluated against the results
from \textcite{goda1977estimation}, and showed good agreement between both
methods. \textcite{hughes1993} also highlights that reflection estimates are
unreliable for higher frequency, where coherency between the two measured
series is lower.

Following the work of \textcite{tatavarti1989incoming},
\textcite{huntley1999use} showed how principal component analysis can alleviate
noise-induced bias in reflection coefficient calculations compared to
time-domain analysis. They also stuied the influence of imperfect collocation
of the sensors, showing that the time delay between sensors leads to a peak in
the reflection coefficient at a frequency related to this time delta.

%%% TODO? %%%
% \cite{sheremet2002observations}

\subsection{Conclusion}

Numerous methods have been developped in order to separate incident and
reflected components from wave measurements. Array methods rely on the use of
multiple, generally aligned, wave gauges, while PUV methods rely on the use of
co-located sensors, generally a wave height sensor and a horizontal velocity
sensor. Array methods generally have the advantage of being more cost-effective
to implement, as the cost of reliable velocity measurement devices can be
important \parencite{hughes1993}. Nevertheless, PUV methods are generally more
accurate regarding noise, varying bathymetry, and can be setup closer to
reflective surfaces \parencite{hughes1993,inch2016accurate}.

In the case of the 2017 event on the Artha breakwater, the results from a
single wave gauge are available, which means that the array methods are not
applicable. A PUV method \parencite{tatavarti1989incoming,huntley1999use}
should then be used to evaluate the reflection coefficient of the Artha
breakwater and to separate the incident and reflected wave components from the
measured data.

\section{Modelling wave impact on a breakwater}

\subsection{Introduction}

Modelling rubble-mound breakwaters such as the Artha breakwater requires
complex considerations on several aspects. First of all, an accurate of the
fluid's behavior in the porous armour of the breakwater is necessary. Then,
adequate turbulence models are needed in order to obtain accurate results.
Several types of models have been developped that can be used to study breaking
wave flow on a porous breakwater.

\subsection{SPH models}

\subsubsection{Introduction}

Smoothed-Particle Hydrodynamics (SPH) models rely on a Lagrangian
representation of the fluid \parencite{violeau2012fluid}. These models are
meshless, and work by considering fluids as a collection of particles.

SPH models have been shown to provide satisfactory results for the modeling of
turbulent free surface flows \parencite{violeau2007numerical}. Additionnaly,
\textcite{dalrymple2006numerical} showed that SPH models can be used in small
scale models of water waves. In this part, literature on modeling flow in
porous media and the adequate boundary conditions for wave modeling will be
reviewed.

\subsubsection{Porosity modelling}

Multiple approaches can be used when modeling porous media using SPH models.
The most obvious approach relies on the use of discrete elements in the porous
domain. For instance, \textcite{altomare2014numerical} showed that an SPH model
along with discrete modeling of the blocks composing a breakwater could yield
satisfactory results. The meshless character of SPH models allows for modeling
the large scale outside the porous media and the small scale of the space
between blocks effectively.

Nevertheless, the more common approach is to use a macro-scale model in porous
media, in which the porous domain is considered to have a set of homogeneous
properties.
\textcite{jiang2007mesoscale} used randomly placed fixed particles in the
porous media in order to model porosity at a microscopic scale from mesoscopic
porosity properties. The resulting model showed reliable results in studying
the flow through porous media.

By contrast, \textcite{shao2010} used volume-averaged Navier-Stokes equations
along with an averaged porosity model
\parencite{huang2003structural,burcharth1995one} in an incompressible SPH
(ISPH) model in order to model wave flow in porous media, accounting for a
linear and quadratic term in porosity induced friction. Turbulence was modeled
with a $k-\varepsilon$ volume averaged model. Good agreement was highlighted
between the results from this model and other models, analytical results and
experimental measurements for solitary and regular waves interacting with a
porous breakwater.

Similarly, \textcite{ren2016improved} presents a weakly-compressible SPH
(WCSPH) model using the volume averaged Favre averaged Navier-Stokes (VAFANS)
equations along with a large Eddy simulation (LES, \cite{ren2014numerical})
turbulence model. Interaction between turbulent flows and porous media is
studied and good agreement is shown between model results and experimental
data. Additionnaly, it is highlighted that the addition of the turbulence model
does increase the accuracy of the model. Similar results are found by
\textcite{wen2016sph} when studying wave impact on non-porous structures using
the same model.

The same model was then extended to a three-dimensional model by
\textcite{wen20183d}. The computed free surface and forces on a structure were
shown to be accurately predicted by the 3D model.

%\paragraph{Notes}
%
%\cite{jiang2007mesoscale}: Meso-scale SPH model of flow in isotropic porous
%media; randomly placed particles with repulsive force; reasonable results.
%
%\cite{shao2010}: incompressible flow with porous media; Navier-Stokes,
%Volume-Averaged $k-\varepsilon$; porosity model is same as
%\cite{troch1999development} but without inertia term
%\parencite{huang2003structural}
%
%\cite{altomare2014numerical} "microscopic" model of breakwater
%
%\cite{kunz2016study} comparison of sph model with micro-model experiments; not
%quite applicable
%
%\textbf{\cite{ren2016improved}} VAFANS equations to solve incompressible
%turbulent flow with porous media. Same porosity model as \cite{shao2010}
%
%\cite{wen2016sph}
%
%\cite{pahar2016modeling}
%\cite{peng2017multiphase}
%\cite{wen20183d}: 3D VAFANS
%\cite{kazemi2020sph}

\subsubsection{Wave generation}

One of the more recent research subject with SPH models has been wave
generation. Wave paddles were initially used as a way to generate waves in
numerical basins \parencite{zheng2010numerical}, with the major drawback of
such wave makers begin their high reflectivity.

\textcite{liu2015isph} proposed an improved wave generator using a momentum
source in an ISPH model. The use of a momentum source was a major improvement
as it enabled the use of non-reflective wave generators. The proposed solution
was developed for two-dimensional linear waves, but the same algorithm could be
used for three-dimensional models.

\textcite{altomare2017long} presented a wave generation method for long-crested
(second order) waves in a WCSPH model using a piston wave maker. Although this
method leads to high reflection, but the possibility of generating irregular
waves was highlighted.

Similarly to \textcite{liu2015isph}, \textcite{wen2018non} proposed a wave
generation method using a momentum source to create a non reflective wave
maker. The proposed method was used for generating regular as well as random
waves in a flume, and could be extended to three-dimensional simulations.
Nevertheless, the method proposed was limited to linear wave theory.

%\cite{zheng2010numerical}
%
%\cite{liu2015isph}: 2D non-reflective linear wave generator using a momentum
%source in ISPH
%
%\cite{altomare2017long}: Wave generation and absorption of long-crested waves
%(2nd order) in WCSPH. Generation of monochromatic as well as irregular waves.
%
%\cite{wen2018non}: Non reflective spectral wave maker, using momentum source

\subsubsection{Conclusion}

SPH models have been showed to be extremely powerful tools in modelling
wave-structure interaction, due to their ability to model complex interfaces
and highly dynamic situations \parencite{altomare2017long}.

Modeling wave interaction with porous structures using SPH models has been
widely studied, and generally adequate results are obtained
\parencite{wen20183d}. Nonetheless, SPH models still face some limitations
regarding their ability to represent incompressible flows, leading to high
diffusivity \parencite{higuera2015application}.

Moreover, wave-generation techniques, especially for long simulations, are
still at an early stage of developement \parencite{wen2018non}, limiting the
applicability to such models in studying real cases using in-situ data.

\subsection{VOF models}

\subsubsection{Introduction}

Contrary to SPH models, the volume of fluid (VOF) method relies on a Eulerian
representation of the fluid \parencite{hirt1981volume}. This method uses a
marker function, the value of which represents the fraction of fluid in a cell.

\subsubsection{2D models}

Using the VOF method along with Navier-Stokes equations, several models have
been developed in order to model fluid dynamics around porous structures.
\textcite{van1995wave} first implemented 2D-V incompressible Navier-Stokes
equations using the VOF method while accounting for porous media. The results
of the numerical model were validated with analytical solutions for simple
cases, as well as physical model tests. The model yielded acceptable results,
but the representation of turbulence and air-extrusion still required
improvement.

\textcite{troch1999development} developed the VOFbreak\textsuperscript{2} model
in order to provide improvements to earlier models. The Forchheimer theory
\parencite{burcharth1995one} is used in order to model the behavior of the flow
inside porous media. The hydraulic gradient generated in porous media is
decomposed as a linear term, a quadratic term, and an inertia term. Those terms
are ponderated by three coefficients that need to be calibrated. Several
attempts have been made to obtain analytical formulas for those
\parencite{burcharth1995one,van1995wave}, but no universal result has been
provided for the inertia term in particular. \textcite{vieira2021novel}
additionnaly proposed using artificial neural networks in order to calibrate
those values, which are generally calibrated using experimental results.

Parallely, \textcite{liu1999numerical} created a new model (COBRAS) that used
the VOF method. The model is based on the combination of Reynolds averaged
Navier-Stokes (RANS) equations and a $k-\varepsilon$ turbulence model. The
porous media is modelled similarly to \textcite{troch1999development}. The
offered results were improved compared to earlier models as more a more
accurate consideration of turbulence outside porous media was added. This model
was further improved by \textcite{hsu2002numerical} in order to account for
small scale turbulence inside the porous media thanks to volume averaged RANS
(VARANS) equations.

The COBRAS model was then reworked by
\textcite{losada2008numerical,lara2008wave} to add improvements to wave
generation and usability. The main difference between this new code (COBRAS-UC)
and COBRAS is the addition of irregular waves generation. The code was also
optimized to reduce the number of iterations. The improvements allowed for
longer simulations to be computed. The predictions for free surface elevation
and pressure in front of a porous breakwater were accurate, but improvements
were still needed, in particular considering computation time.

\subsubsection{3D models}

The combination of VARANS equations and the VOF method was then brought to 3D
domains by \textcite{del2011three} in IH3VOF. Specific boundary conditions were
also added for several wave theories. Additionnaly, an improved turbulence
model was used ($\omega$-SST model, \cite{menter1994two}), which provides
strongly improved results in zones where strong pressure gradients appear.
Strong agreement between IH3VOF and experimental results was obtained, but the
need for accurate boundary conditions limited the applicability of the model.

\textcite{higuera2015application} reworked the equations from
\textcite{del2011three} as discrepancies were observed with earlier literature
and added several improvements to the model. Notably, time-varying porosity was
added in order to account for eventual sediment displacement. New boundary
conditions were added, with static and dynamic boundary wave generators as well
as passive and acive wave absorption being implemented. The resulting model
(IHFOAM/olaFlow, \cite{olaFlow}) was implemented in the OpenFOAM toolbox.

\subsubsection{Conclusion}

VOF models have been developped to provide accurate results for the study of
wave impact on porous structures. The validation results from
\textcite{higuera2015application} show the capabilities of such models in
accurately representing rubble-mound breakwaters subject to irregular
three-dimensional wave fields.

Nonetheless, the representation of porosity in those models is still mainly
based on experimental calibration, particularly for the inertia term of
porosity induced friction.

%\subsection{Conclusion}

%\paragraph{Notes}
%
%\cite{van1995wave,troch1999development}
%
%COBRAS \parencite{liu1999numerical}: spatially averaged RANS
%with $k-\varepsilon$ turbulence model. Drag forces modeled by empirical linear
%and non-linear friction terms; \cite{hsu2002numerical}: introduced VARANS in
%order to account for small scale turbulence inside the porous media.
%->
%COBRAS-UC/IH2VOF \parencite{losada2008numerical,lara2008wave}: VOF VARANS (2D);
%refactor of COBRAS code, with improved wave generation, improvement of input
%and output data.
%->
%IH3VOF \parencite{del2011three}: 3D VOF VARANS, updated porous media equations,
%optimization of accuracy vs computation requirements, specific boundary
%conditions, validation. Adding SST model.
%->
%IHFOAM/olaFlow \parencite{higuera2015application}: Rederivation of
%\cite{del2011three}, add time-varying porosity; Improvement to wave generation
%and absorption; implementation in OpenFOAM; extensive validation; application
%to real coastal structures.
%
%\cite{vieira2021novel}: Use of artificial neural networks to determine porosity
%parameter for VOF VARANS model.

%\subsection{Other}
%
%BEM: \cite{hall1994boundary,koley2020numerical}

\section{Block displacement by waves}

\subsection{Introduction}

Displacement of blocks or boulders by waves has been a major topic in
understanding the influence of storm and tsunami waves in coastal regions.
Several approaches have been taken to study this phenomenon.

% In a first part, we will discuss in-situ studies on displaced boulders. In a
% second part, we will review models of block displacements.

\subsection{Models}

The main goal of studying boulder displacement is generally to establish the
cause of boulder deposits in coastal areas. \textcite{nott1997extremely} was
among the first to propose hydrondynamic equations that aimed to calculate the
wave height that would lead to the displacement of a boulder for storm and
tsunami waves. The main difference between storm and tsunami waves in those
equations was the flow velocity relative to wave height. The calculation of the
minimum flow velocity for boulder transport is obtained by calculating an
equilibrium between drag, lift and restraining forces.

Those equations were refined by \textcite{nott2003waves} in order to account
for the pre-transport environment of the boulder. \citeauthor{nott2003waves}
derived equations for submerged, sub-aerial and joint bounder boulders. A new
parameter ($\delta$) was introduced to differentiate between tsunami and storm
waves. This study highlights the importance of the initial environment of a
boulder for wave transport conditions. Compared to the equations from
\textcite{nott1997extremely}, an aditionnal inertia term is added to the
equilibrium equation for sub-aerial boulders, while the drag force is removed
for joint-bounded blocks.

\textcite{nandasena2011reassessment} noted that \citeauthor{nott2003waves}'s
equations could be improved, and proposed a new set of equations correcting
the lift and inertia terms in \citeauthor{nott2003waves}'s equations.
\textcite{nandasena2011reassessment} found that the new equations produced up
to a \SI{65}{\percent} difference with \citeauthor{nott2003waves}'s equations.

\textcite{buckley2012inverse} proposed alternative equations for sliding and
overturning of submerged boulders. An equation for block sliding was introduced
by considering friction on the bed. \textcite{weiss2012mystery} investigated
the influence of bed roughness on block displacement. A new stability criteria
was established, and bed roughness was found to be a major factor in boulder
displacement. In contrast with the findings from \textcite{nott2003waves}, the
threshold wave amplitude for block displacement was found to be similar between
tsunami and storm waves.

\textcite{nandasena2013boulder,liu2014experimental} performed experimental
studies of block displacement using dam break scenarios in a flume. The results
from both studies indicate that the primary mode of boulder motion for large
boulders is sliding, rather than rolling or saltation.

\textcite{weiss2015untangling} highlights inadequacies in the criteria that are
generally used \parencite{nott2003waves,nandasena2011reassessment}. According
to \textcite{weiss2015untangling}, the use of a minimum threshold on block
displacement does not account for the possibility of a block returning to its
initial position after being slightly disloged. A new threshold is proposed on
the minimal movement of a block, while considering the time-dependent nature of
wave-induced flow. \textcite{weiss2015untangling} also shows the importance of
the pre-transport conditions on block displacement.

\textcite{kennedy2017extreme} derived new equations following the approach from
\textcite{nandasena2011numerical} accounting for non-parallelepipedic blocks.
The revised equations lead to a lower velocity threshold for block movement.
This highlights the importance of boulder shape in displacement considerations.

\textcite{lodhi2020role} highlighted the importance of hydrodynamic pressure in
block displacement. A new equation was given for the threshold flow velocity
for block movement. An experimental validation of the models was performed, and
showed the overestimation of the threshold velocity by previous models.

\textcite{oetjen2021experiments} performed a review of boulder displacement
experiments. They found that the initial position of boulders relative to the
wave impact has a major influence on block displacement. Conversely, the
influence of bed roughness seems to have been overestimated in the past.
Similarly to \textcite{lodhi2020role}, \textcite{oetjen2021experiments}
highlights an overestimation of minimum wave height for block displacement by
earlier equations \parencite{nott1997extremely,nandasena2011reassessment}.

\subsection{Conclusion}

%\subsection{In-situ studies}
%
%\cite{barbano2010large}: boulders deposity in Sicily -> probably tsunamis
%
%\cite{paris2011}: 
%
%\cite{nandasena2011numerical}
%\cite{may2015block}
%\cite{biolchi2016}
%\cite{kennedy2016observations}
%\cite{erdmann2018boulder}
%\cite{cox2018extraordinary}
%
%\subsection{Models}
%
%\cite{nott1997extremely}
%
%\cite{nott2003waves} Submerged boulder:
%\begin{equation}
%u^2 \ge \frac{2\left(\frac{\rho_s}{\rho_w}-1\right)ag}
%{C_d\left(\frac{ac}{b^2}\right)+C_l}
%\end{equation}
%
%\cite{imamura2008numerical}
%\cite{barbano2010large}
%\cite{nandasena2011numerical}
%
%\cite{nandasena2011reassessment}
%\begin{equation}
%u^2 \ge \frac{2\left(\frac{\rho_s}{\rho_w}-1\right) gc
%\left(\cos\theta+\frac{c}{b}\sin\theta\right)}
%{C_d\frac{c^2}{b^2}+C_l}
%\end{equation}
%
%\cite{buckley2012inverse}
%\cite{weiss2012mystery}
%\cite{nandasena2013boulder}
%\cite{liu2014experimental}
%\cite{weiss2015untangling}
%
%
%\cite{kennedy2016observations}
%\cite{kennedy2017extreme}
%\cite{weiss2017toward}
%
%\cite{bressan2018laboratory} Partially submerged boulders
%\begin{equation}
%u^2 \ge \frac{2b_wW}{\rho_w\left(b_DC_DA_{wfs}+b_LC_LA_{wbs}\right)}
%\end{equation}
%
%\cite{lodhi2020role}
%\cite{oetjen2020significance}
%
%\cite{oetjen2021experiments}:  Review
%
%---
%\cite{zainali2015boulder}: Numerical model of block displacement