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\usepackage[
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\title{OpenFoam Project\\\huge Simulation of the breaking wave flow at the
Artha breakwater}
\author{Edgar P. Burkhart}

\begin{document}
\maketitle

\tableofcontents

\section{Introduction}
In February 2017, a \SI{50}{\tonne} concrete block was displaced by a wave at
the Artha breakwater, in the entrance of the bay of Saint-Jean-de-Luz. This
event was captured by a photographer, and an initial study (\cite{amir})
allowed to highlight the circumstances which caused the block displacement.

The phenomenon of block displacement by waves has been studied in the past with
multiple approaches (\cite{cox2018extraordinary,shah2013coastal}). In 2014, a
study of displaced blocks on the coast of Ireland was conducted by
\cite{cox2018extraordinary}. This study highlighted a strong correlation
between the mass of displaced boulders and coastal topography. Notably, an
inverse exponential relation between boulder mass and elevation was
established. According to the presentation by \cite{abadie}, the block that was
displaced at the Artha breakwater in 2017 falls in accordance with these
results, as shown in \autoref{fig:compcox}.

\begin{figure}
	\centering
	\begin{tikzpicture}
		\begin{semilogyaxis}[
				xmin=0,
				xmax=25,
				ymin=0.1,
				ymax=1000,
				domain=0:25,
				grid=both,
				legend entries={\cite{cox2018extraordinary}, Artha 2017},
				xlabel={Elevation (\si{\m})},
				ylabel={Mass (\si{\tonne})},
			]
			\addplot[no markers] {exp(5.01-0.15*x)};
			\addplot[only marks] coordinates {
				(8.2,50)
			};
			\draw [dashed,help lines] (axis cs:0,50) -| (axis cs:8.2,0.1);
		\end{semilogyaxis}
	\end{tikzpicture}
	\caption{Comparison between the correlation found by
	\cite{cox2018extraordinary} and the block displaced at the Artha breakwater
	in 2017 (\SI{50}{\tonne}, \SI{8.2}{\m}).}\label{fig:compcox}
\end{figure}

\cite{shah2013coastal} studied coastal boulders in Martigues, on the french
mediterranean coast. Similarly to \cite{cox2018extraordinary}, displaced
boulders were studied regarding their mass and position on the shore.
The study concludes that those blocks are evidence of the risks associated with
high energy waves on the mediterranean coast, and links the displaced boulders
to extreme storms, but does not exclude the possibility of tsunamis.

Other studies focus on the theoretical aspects of bock displacement by water
flow. \cite{nott2003waves} proposed a set of equations for determining the
minimum wave height that would lead to displacement of a boulder in different
scenarios. This study highlights that the environment of the boulder before
transport is a major factor in calculating that minimum wave height, as well
as water depth at the boulder initial location.

The goal of this project will be to perform a sensibility study on the
parameters representing the porosity of the shell of the artha breakwater in a
two-dimensionnal olaFlow (\cite{olaFlow}) simulation of wave impact on the
breakwater.

Studies on the transformation of waves over the Artha breakwater have already
been conducted. \cite{poncet2021characterization} has performed such a study
using the SWASH (\cite{zijlema2011swash}) model. The results from a calibration
study on the porosity of the breakwater shell yielded a porosity of \num{0.25}
for the lowest root mean squared error compared to experimental values.
This value is unexpectedely low, and a porosity of \num{0.40} was used in the
following computations, as it represents the expected porosity of such a
block armour.

\section{Methods}
\subsection{Model}
In this project, we will model the transformation of a storm wave over the
Artha breakwater using the olaFlow (\cite{olaFlow}) model in a two-dimensionnal
domain. This model has several features that will be important for this study.
It provides powerful wave-generation and wave absorption capabilities, as well
as the ability to study two-phase flow through porous media.

The olaFlow model is based on VARANS equations\footnote{Volume-Averaged
Reynolds-Averaged Navier-Stokes equations}. This model is based on a
finite-volume approach, and provides multiple volume-averaging methods for
turbulence: the $k-\varepsilon$ model and the $k-\omega$ sst model. The
$k-\omega$ sst model should provide better results in situations where strong
pressure gradients are present, at the cost of computing power.
For the purposes of this initial sensibility study, the $k-\omega$ model will
be used.

\subsection{Domain}
The studied domain will be a two-dimensionnal vertical slice going through the
Artha breakwater. In order to provide accurate results while keeping
computation times acceptable, a domain length of \SI{150}{\m} was chosen, with
a length of \SI{120}{\m} towards the offshore and \SI{30}{\m} towards the
inside of the Saint-Jean-de-Luz bay, as shown in \autoref{fig:map}.

\begin{figure}
	\centering
	\input{fig/map.pgf}
	\caption{Studied domain.}\label{fig:map}
\end{figure}

The bathymetry was generated using bathymetric data from the SHOM
(\cite{shomsjl}). Although the density of points is fairly low in this data,
it should be precise enough for the purposes of this project.
In addition to this data, the geometry of the superstructure was taken from
\cite{amir}, and the size of the porous armor layer was inspired from
\cite{poncet2021characterization}. The necessary stl files that are used by
snappyHexMesh for the solid boundaries and olaFlow for the porosity
configuration were generated using a custom python script, and the resulting
case is plotted in \autoref{fig:conf}.

\begin{figure}
	\centering
	\input{fig/bathy.pgf}
	\caption{Studied configuration (bathymetry and porosity).}\label{fig:conf}
\end{figure}

\subsection{Mesh}
In order to provide accurate results with an acceptable computation time, a
mesh with \SI{1}{\m} cells was generated on the domain. It seems like olaFlow
cannot use variable mesh sizing, which meant that the mesh was still fairly
coarse near the edges of the domain.

The boundary conditions were set to a wave generator on the offshore boundary,
and a wave absorption boundary on the bay side. The bottom and the caisson were
set to wall boundaries, and the top side of the domain was set to an
atmospheric boundary.

\subsection{Model setup}
In an attempt to model the event of February 2017 in Saint-Jean-de-Luz, the
water level was set to \SI{5}{\m}. The inbound wave on the offshore side of
the domain was set as a solitary Boussinesq wave, with a wave height
$H=\SI{7.5}{\m}$. The wave equation is the following:
\begin{equation}
	\eta=H\left[\sech\sqrt{\frac 34\frac Hh\frac{x-ct}h}\right]^2
\end{equation}

\subsection{Porosity setup}
The goal of the study is to find out the influence of the porosity parameters
on the model results. Porosity in the olaFlow model is goverened by five
parameters: a, b, c are tuning parameters that represent friction inside
the porous material, D50 is the mean nominal diameter of blocks, and porosity.

In this study, we will focus on the D50 and porosity parameters, as a, b and c
should be tuned according to experimental data. The friction parameters will be
set to the default from the breakwater example in the olaFlow model.

Four cases will be run, with all combinations of the parameters in
\autoref{tab:params}. The D50 parameters is based on the smallest and largest
edge size of the blocks that make up the breakwater armour. The porosity
parameters are based on the work of \cite{poncet2021characterization}, with
\num{0.40} being the value that he used for the numerical study, and \num{0.25}
being the value that yielded the lowest error in the model calibration.

\begin{table}
	\centering
	\begin{tabular}{lcc}
		\toprule
		\bfseries Parameters & & \\
		\midrule
		D50 & \SI{4}{\m} & \SI{2}{\m} \\
		Porosity & \num{0.40} & \num{0.25} \\
		\bottomrule
	\end{tabular}
	\caption{Parameter values.}\label{tab:params}
\end{table}

\section{Results}

\section{Conclusion}

\printbibliography
\end{document}
