193 lines
8.0 KiB
TeX
193 lines
8.0 KiB
TeX
\documentclass[english, a4paper, 12pt]{article}
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\usepackage{cours}
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\setmainlanguage{english}
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\usepackage[
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backend=biber,
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sorting=ynt,
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style=authoryear
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]{biblatex}
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\bibliography{library}
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\title{OpenFoam Project\\\huge Simulation of the breaking wave flow at the
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Artha breakwater}
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\author{Edgar P. Burkhart}
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\begin{document}
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\maketitle
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\tableofcontents
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\section{Introduction}
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In February 2017, a \SI{50}{\tonne} concrete block was displaced by a wave at
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the Artha breakwater, in the entrance of the bay of Saint-Jean-de-Luz. This
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event was captured by a photographer, and an initial study (\cite{amir})
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allowed to highlight the circumstances which caused the block displacement.
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The phenomenon of block displacement by waves has been studied in the past with
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multiple approaches (\cite{cox2018extraordinary,shah2013coastal}). In 2014, a
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study of displaced blocks on the coast of Ireland was conducted by
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\cite{cox2018extraordinary}. This study highlighted a strong correlation
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between the mass of displaced boulders and coastal topography. Notably, an
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inverse exponential relation between boulder mass and elevation was
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established. According to the presentation by \cite{abadie}, the block that was
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displaced at the Artha breakwater in 2017 falls in accordance with these
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results, as shown in \autoref{fig:compcox}.
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\begin{figure}
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\centering
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\begin{tikzpicture}
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\begin{semilogyaxis}[
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xmin=0,
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xmax=25,
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ymin=0.1,
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ymax=1000,
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domain=0:25,
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grid=both,
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legend entries={\cite{cox2018extraordinary}, Artha 2017},
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xlabel={Elevation (\si{\m})},
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ylabel={Mass (\si{\tonne})},
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]
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\addplot[no markers] {exp(5.01-0.15*x)};
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\addplot[only marks] coordinates {
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(8.2,50)
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};
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\draw [dashed,help lines] (axis cs:0,50) -| (axis cs:8.2,0.1);
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\end{semilogyaxis}
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\end{tikzpicture}
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\caption{Comparison between the correlation found by
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\cite{cox2018extraordinary} and the block displaced at the Artha breakwater
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in 2017 (\SI{50}{\tonne}, \SI{8.2}{\m}).}\label{fig:compcox}
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\end{figure}
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\cite{shah2013coastal} studied coastal boulders in Martigues, on the french
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mediterranean coast. Similarly to \cite{cox2018extraordinary}, displaced
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boulders were studied regarding their mass and position on the shore.
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The study concludes that those blocks are evidence of the risks associated with
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high energy waves on the mediterranean coast, and links the displaced boulders
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to extreme storms, but does not exclude the possibility of tsunamis.
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Other studies focus on the theoretical aspects of bock displacement by water
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flow. \cite{nott2003waves} proposed a set of equations for determining the
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minimum wave height that would lead to displacement of a boulder in different
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scenarios. This study highlights that the environment of the boulder before
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transport is a major factor in calculating that minimum wave height, as well
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as water depth at the boulder initial location.
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The goal of this project will be to perform a sensibility study on the
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parameters representing the porosity of the shell of the artha breakwater in a
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two-dimensionnal olaFlow (\cite{olaFlow}) simulation of wave impact on the
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breakwater.
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Studies on the transformation of waves over the Artha breakwater have already
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been conducted. \cite{poncet2021characterization} has performed such a study
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using the SWASH (\cite{zijlema2011swash}) model. The results from a calibration
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study on the porosity of the breakwater shell yielded a porosity of \num{0.25}
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for the lowest root mean squared error compared to experimental values.
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This value is unexpectedely low, and a porosity of \num{0.40} was used in the
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following computations, as it represents the expected porosity of such a
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block armour.
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\section{Methods}
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\subsection{Model}
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In this project, we will model the transformation of a storm wave over the
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Artha breakwater using the olaFlow (\cite{olaFlow}) model in a two-dimensionnal
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domain. This model has several features that will be important for this study.
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It provides powerful wave-generation and wave absorption capabilities, as well
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as the ability to study two-phase flow through porous media.
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The olaFlow model is based on VARANS equations\footnote{Volume-Averaged
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Reynolds-Averaged Navier-Stokes equations}. This model is based on a
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finite-volume approach, and provides multiple volume-averaging methods for
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turbulence: the $k-\varepsilon$ model and the $k-\omega$ sst model. The
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$k-\omega$ sst model should provide better results in situations where strong
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pressure gradients are present, at the cost of computing power.
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For the purposes of this initial sensibility study, the $k-\omega$ model will
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be used.
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\subsection{Domain}
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The studied domain will be a two-dimensionnal vertical slice going through the
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Artha breakwater. In order to provide accurate results while keeping
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computation times acceptable, a domain length of \SI{150}{\m} was chosen, with
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a length of \SI{120}{\m} towards the offshore and \SI{30}{\m} towards the
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inside of the Saint-Jean-de-Luz bay, as shown in \autoref{fig:map}.
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\begin{figure}
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\centering
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\input{fig/map.pgf}
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\caption{Studied domain.}\label{fig:map}
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\end{figure}
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The bathymetry was generated using bathymetric data from the SHOM
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(\cite{shomsjl}). Although the density of points is fairly low in this data,
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it should be precise enough for the purposes of this project.
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In addition to this data, the geometry of the superstructure was taken from
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\cite{amir}, and the size of the porous armor layer was inspired from
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\cite{poncet2021characterization}. The necessary stl files that are used by
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snappyHexMesh for the solid boundaries and olaFlow for the porosity
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configuration were generated using a custom python script, and the resulting
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case is plotted in \autoref{fig:conf}.
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\begin{figure}
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\centering
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\input{fig/bathy.pgf}
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\caption{Studied configuration (bathymetry and porosity).}\label{fig:conf}
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\end{figure}
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\subsection{Mesh}
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In order to provide accurate results with an acceptable computation time, a
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mesh with \SI{1}{\m} cells was generated on the domain. It seems like olaFlow
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cannot use variable mesh sizing, which meant that the mesh was still fairly
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coarse near the edges of the domain.
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The boundary conditions were set to a wave generator on the offshore boundary,
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and a wave absorption boundary on the bay side. The bottom and the caisson were
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set to wall boundaries, and the top side of the domain was set to an
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atmospheric boundary.
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\subsection{Model setup}
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In an attempt to model the event of February 2017 in Saint-Jean-de-Luz, the
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water level was set to \SI{5}{\m}. The inbound wave on the offshore side of
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the domain was set as a solitary Boussinesq wave, with a wave height
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$H=\SI{7.5}{\m}$. The wave equation is the following:
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\begin{equation}
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\eta=H\left[\sech\sqrt{\frac 34\frac Hh\frac{x-ct}h}\right]^2
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\end{equation}
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\subsection{Porosity setup}
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The goal of the study is to find out the influence of the porosity parameters
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on the model results. Porosity in the olaFlow model is goverened by five
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parameters: a, b, c are tuning parameters that represent friction inside
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the porous material, D50 is the mean nominal diameter of blocks, and porosity.
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In this study, we will focus on the D50 and porosity parameters, as a, b and c
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should be tuned according to experimental data. The friction parameters will be
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set to the default from the breakwater example in the olaFlow model.
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Four cases will be run, with all combinations of the parameters in
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\autoref{tab:params}. The D50 parameters is based on the smallest and largest
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edge size of the blocks that make up the breakwater armour. The porosity
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parameters are based on the work of \cite{poncet2021characterization}, with
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\num{0.40} being the value that he used for the numerical study, and \num{0.25}
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being the value that yielded the lowest error in the model calibration.
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\begin{table}
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\centering
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\begin{tabular}{lcc}
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\toprule
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\bfseries Parameters & & \\
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\midrule
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D50 & \SI{4}{\m} & \SI{2}{\m} \\
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Porosity & \num{0.40} & \num{0.25} \\
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\bottomrule
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\end{tabular}
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\caption{Parameter values.}\label{tab:params}
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\end{table}
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\section{Results}
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\section{Conclusion}
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\printbibliography
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\end{document}
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