diff --git a/nature/fig/U.pdf b/nature/fig/U.pdf index 33ba638..f7ce199 100644 Binary files a/nature/fig/U.pdf and b/nature/fig/U.pdf differ diff --git a/nature/fig/bathy2d.pdf b/nature/fig/bathy2d.pdf new file mode 100644 index 0000000..370aae8 Binary files /dev/null and b/nature/fig/bathy2d.pdf differ diff --git a/nature/fig/pic1.jpg b/nature/fig/pic1.jpg new file mode 100755 index 0000000..b2167aa Binary files /dev/null and b/nature/fig/pic1.jpg differ diff --git a/nature/fig/pic2.jpg b/nature/fig/pic2.jpg new file mode 100755 index 0000000..a312bfb Binary files /dev/null and b/nature/fig/pic2.jpg differ diff --git a/nature/fig/wavelet.pdf b/nature/fig/wavelet.pdf new file mode 100644 index 0000000..a661e85 Binary files /dev/null and b/nature/fig/wavelet.pdf differ diff --git a/nature/fig/wavelet9312.pdf b/nature/fig/wavelet9312.pdf deleted file mode 100644 index 45f1ac8..0000000 Binary files a/nature/fig/wavelet9312.pdf and /dev/null differ diff --git a/nature/fig/wavelet_sw.pdf b/nature/fig/wavelet_sw.pdf index 89e83d9..a9ce959 100644 Binary files a/nature/fig/wavelet_sw.pdf and b/nature/fig/wavelet_sw.pdf differ diff --git a/nature/library.bib b/nature/library.bib index 40a67c8..0066c63 100644 --- a/nature/library.bib +++ b/nature/library.bib @@ -119,3 +119,19 @@ year={2020}, publisher={Elsevier} } +@book{violeau2012, + title={Fluid mechanics and the SPH method: theory and applications}, + author={Violeau, Damien}, + year={2012}, + publisher={Oxford University Press} +} +@article{violeau2007, + title={Numerical modelling of complex turbulent free-surface flows with the SPH method: an overview}, + author={Violeau, Damien and Issa, Reza}, + journal={International Journal for Numerical Methods in Fluids}, + volume={53}, + number={2}, + pages={277--304}, + year={2007}, + publisher={Wiley Online Library} +} diff --git a/nature/main.tex b/nature/main.tex index 3e2470d..7114d49 100644 --- a/nature/main.tex +++ b/nature/main.tex @@ -56,47 +56,59 @@ initiation. A parametrisation of waves depending on their source is also used to on the type of scenario --- wave or tsunami. Those equations were later revised by \textcite{nandasena2011}, as they were found to be partially incorrect. A revised formulation based on the same considerations was provided. -The assumptions on which \citeauthor{nott2003, nandasena2011} are based were then critisized by \textcite{weiss2015}. In +The assumptions on which \textcite{nott2003, nandasena2011} are based were then critisized by \textcite{weiss2015}. In fact, according to them, the initiation of movement is not sufficient to guarantee block displacement. \textcite{weiss2015} highlights the importance of the time dependency on block displacement. A method is proposed that -allows to find the wave amplitude that lead to block displacement. +allows to find the wave amplitude that lead to block displacement. Additionally, more recent research by +\textcite{lodhi2020} has shown that the equations proposed by \textcite{nott2003, nandasena2011} tend to overestimate +the minimum flow velocity needed to displace a block. % Lack of observations -> observation Whether it is \textcite{nott2003}, \textcite{nandasena2011} or \textcite{weiss2015}, all the proposed analytical -equations suffer from a major flaw; they are all based on simplified analytical models and statistical analysis. -Unfortunately, no block displacement event seems to have been observed directly in the past. +equations suffer from a major flaw: they are all based on very simplified analytical models and statistical analysis. +Unfortunately, no block displacement event seems to have been observed directly in the past, and those events are +difficult to predict. In this paper, we study such an event. On February 28, 2017, a \SI{50}{\tonne} concrete block was dropped by a wave on -the crest of the Artha breakwater. Luckily, the event was captured by a photographer, and a wave buoy located -\SI{1.2}{\km} offshore captured the seastate. Information from the photographer allowed to establish the approximate -time at which the block displacement occured. The goal of this paper is to model the hydrodynamic conditions near the -breakwater that lead to the displacement of the \SI{50}{\tonne} concrete block. +the crest of the Artha breakwater (Figure~\ref{fig:photo}). Luckily, the event was captured by a photographer, and a +wave buoy located \SI{1.2}{\km} offshore captured the seastate. Information from the photographer allowed to establish +the approximate time at which the block displacement occured. The goal of this paper is to model the hydrodynamic +conditions near the breakwater that lead to the displacement of the \SI{50}{\tonne} concrete block. % Modeling flow accounting for porous media Several approaches can be used when modelling flow near a breakwater. Depth-averaged models can be used to study the -transformation of waves on complex bottoms. Studying the hydrodynamic conditions under the surface can be achieved using -smoothed-particles hydrodynamics (SPH) or volume of fluid (VOF) models. SPH models rely on a Lagrangian representation -of the fluid, while VOF models rely on an Eulerian representation. VOF models are generally more mature for the study of -multiphase incompressible flows. +transformation of waves on complex bottoms. Studying the hydrodynamic conditions under the surface can be achieved +using smoothed-particles hydrodynamics (SPH) or volume of fluid (VOF) models. SPH models rely on a Lagrangian +representation of the fluid \parencite{violeau2012}, while VOF models rely on an Eulerian representation. VOF models +are generally more mature for the study of multiphase incompressible flows, while SPH models generally require more +processing power for similar results \parencite{violeau2007}. -In this paper, we first use a one-dimensionnal depth-averaged non-linear non-hydrostatic model to verify that the signal -measured by the wave buoy can be used as an incident wave input for the determination of hydrodynamic conditions near -the breakwater. For this model, we use a SWASH model \parencite{zijlema2011} already calibrated by \textcite{poncet2021} -on a domain reaching \SI{1450}{\m} offshore of the breakwater. +In this paper, we first use a one-dimensionnal depth-averaged non-linear non-hydrostatic model to verify that the +signal measured by the wave buoy can be used as an incident wave input for the determination of hydrodynamic conditions +near the breakwater. For this model, we use a SWASH model \parencite{zijlema2011} already calibrated by +\textcite{poncet2021} on a domain reaching \SI{1450}{\m} offshore of the breakwater. Then, we use a nested VOF model in two vertical dimensions that uses the output from the larger scale SWASH model as -initial and boundary conditions to obtain the hydrodynamic conditions on the breakwater. The models uses olaFlow +initial and boundary conditions to obtain the hydrodynamic conditions on the breakwater. The model uses olaFlow \parencite{higuera2015}, a VOF model based on volume averaged Reynolds averaged Navier-Stokes (VARANS) equations, and which relies on a macroscopic representation of the porous armour of the breakwater. The model is qualitatively -calibrated using photographs from the storm of February 28, 2017. Results from the nested models are finally compared to -the analytical equations provided by \textcite{nandasena2011}. +calibrated using photographs from the storm of February 28, 2017. Results from the nested models are finally compared +to the analytical equations provided by \textcite{nandasena2011}. + +\begin{figure*} + \centering + \includegraphics[height=.4\textwidth]{fig/pic1.jpg} + \includegraphics[height=.4\textwidth]{fig/pic2.jpg} + \caption{Photographs taken during and after the wave that displaced a \SI{50}{\tonne} concrete block onto the Artha + breakwater.}\label{fig:photo} +\end{figure*} \section{Results} \subsection{Identified wave} Preliminary work with the photographer allowed to identify the time at which the block displacement event happened. Using the data from the wave buoy located \SI{1250}{\m} offshore of the Artha breakwater, a seamingly abnormally large -wave of \SI{14}{\m} amplitude was identified that is supposed to have lead to the block displacement. +wave of \SI{14}{\m} amplitude was identified that is supposed to have led to the block displacement. Initial analysis of the buoy data plotted in Figure~\ref{fig:wave} shows that the movement of the buoy follows two orbitals that correspond to an incident wave direction. These results would indicate that the identified wave is @@ -115,14 +127,10 @@ with a period of over \SI{30}{\s}. \subsection{Reflection analysis} -The results from the large scale SWASH model using two configurations --- one of them being the real bathymetry, and the -other being a simplified bathymetry without the breakwater --- are compared in Figure~\ref{fig:swash}. The results -obtained with both simulations show a maximum wave amplitude of \SI{13.9}{\m} for the real bathymetry, and \SI{12.1}{\m} -in the case where the breakwater is removed. - -The 13\% difference between those values highlights the existence of a notable amount of reflection at the buoy. -Nonetheless, the gap between the values is still fairly small and the extreme wave identified on February 28, 2017 at -17:23:08 could still be considered as an incident wave. +The results from the large scale SWASH model using two configurations --- one of them being the real bathymetry, and +the other being a simplified bathymetry without the breakwater --- are compared in Figure~\ref{fig:swash}. The results +obtained with both simulations show a maximum wave amplitude of \SI{13.9}{\m} for the real bathymetry, and +\SI{12.1}{\m} in the case where the breakwater is removed. \begin{figure*} \centering @@ -146,12 +154,10 @@ crest increases, with a zone reaching \SI{400}{\m} long in front of the wave whe qualitatively estimated position of the wave front.}\label{fig:swash_trans} \end{figure*} -\subsection{Wavelet analysis} - In an attempt to understand the identified wave, a wavelet analysis is conducted on raw buoy data as well as at different points along the SWASH model using the method proposed by \textcite{torrence1998}. The results are displayed in Figure~\ref{fig:wavelet} and Figure~\ref{fig:wavelet_sw}. The wavelet power spectrum shows that the major component -in the identified wave is a high energy infragravity wave, with a period of around \SI{60}{\s}. +in identified rogue waves is a high energy infragravity wave, with a period of around \SI{60}{\s}. The SWASH model seems to indicate that the observed transformation of the wave can be characterized by a transfer of energy from the infragravity band to shorter waves from around \SI{600}{\m} to \SI{300}{\m}, and returning to the @@ -159,8 +165,9 @@ infragravity band at \SI{200}{\m}. \begin{figure*} \centering - \includegraphics{fig/wavelet9312.pdf} - \caption{Normalized wavelet power spectrum from the raw buoy timeseries.}\label{fig:wavelet} + \includegraphics{fig/wavelet.pdf} + \caption{Normalized wavelet power spectrum from the raw buoy timeseries for identified rogue waves on february 28, + 2017.}\label{fig:wavelet} \end{figure*} \begin{figure*} \centering @@ -172,16 +179,17 @@ infragravity band at \SI{200}{\m}. The two-dimensionnal olaFlow model near the breakwater allowed to compute the flow velocity near and on the breakwater during the passage of the identified wave. The results displayed in Figure~\ref{fig:U} show that the flow velocity -reaches a maximum of \SI{14.5}{\m\per\s} towards the breakwater during the identified extreme wave. Although the maximum -reached velocity is slightly lower than earlier shorter waves (at $t=\SI{100}{\s}$ and $t=\SI{120}{\s}$, with a maximum -velocity of \SI{17.3}{\m\per\s}), the flow velocity remains high for twice as long as during those earlier waves. The -tail of the identified wave also exhibits a water level over \SI{5}{\m} for over \SI{40}{\s}. +reaches a maximum of \SI{14.5}{\m\per\s} towards the breakwater during the identified extreme wave. Although the +maximum reached velocity is similar to earlier shorter waves, the flow velocity remains high for twice as long as +during those earlier waves. The tail of the identified wave also exhibits a water level over \SI{5}{\m} for over +\SI{40}{\s}. \begin{figure*} \centering \includegraphics{fig/U.pdf} - \caption{Horizontal flow velocity computed with the olaFlow model at $x=\SI{-20}{\m}$ on the breakwater armor. The - identified wave reaches this point around $t=\SI{175}{\s}$.}\label{fig:U} + \caption{Horizontal flow velocity computed with the olaFlow model at $x=\SI{-20}{\m}$ on the breakwater armor. + Bottom: horizontal flow velocity at $z=\SI{5}{\m}$. The identified wave reaches this point around + $t=\SI{175}{\s}$.}\label{fig:U} \end{figure*} \section{Discussion} @@ -194,12 +202,19 @@ twice the significant wave height over a given period. The identified wave fits rogue waves often occur from non-linear superposition of smaller waves. This seems to be what we observe on Figure~\ref{fig:wave}. -The wavelet power spectrum shows that a very prominent infragravity component is present, which usually corresponds to -non-linear interactions of smaller waves. \textcite{dysthe2008} mentions that such waves in coastal waters are often the -result of refractive focusing. On February 28, 2017, the frequency of rogue waves was found to be of 1 wave per 1627, -which is considerably more than the excedance probability of 1 over 10\textsuperscript4 calculated by -\textcite{dysthe2008}. Additionnal studies should be conducted to understand focusing and the formation of rogue waves -in front of the Saint-Jean-de-Luz bay. +As displayed in Figure~\ref{fig:wavelet}, a total of 4 rogue waves were identified on february 28, 2017 in the raw buoy +timeseries using the wave height criteria proposed by \textcite{dysthe2008}. The wavelet power spectrum shows that a +very prominent infragravity component is present, which usually corresponds to non-linear interactions of smaller +waves. \textcite{dysthe2008} mentions that such waves in coastal waters are often the result of refractive focusing. On +February 28, 2017, the frequency of rogue waves was found to be of 1 wave per 1627, which is considerably more than the +excedance probability of 1 over 10\textsuperscript4 calculated by \textcite{dysthe2008}. Additionnal studies should be +conducted to understand focusing and the formation of rogue waves in front of the Saint-Jean-de-Luz bay. + +An important point to note is that rogue waves are often short-lived: their nature means that they often separate into +shorter waves shortly after appearing. A reason for which such rogue waves can be maintained over longer distances can +be a change from a dispersive environment such as deep water to a non-dispersive environment. The bathymetry near the +wave buoy (Figure~\ref{fig:bathy}) shows that this might be what we observe here, as the buoy is located near a step in +the bathymetry, from around \SI{40}{\m} to \SI{20}{\m} depth. \subsection{Reflection analysis} @@ -207,10 +222,11 @@ The 13\% difference between those values highlights the existence of a notable a Nonetheless, the gap between the values is still fairly small and the extreme wave identified on February 28, 2017 at 17:23:08 could still be considered as an incident wave. -Unfortunately, the spectrum wave generation method used by SWASH could not reproduce simlar waves to the one observed at -the buoy. As mentionned by \textcite{dysthe2008}, such rogue waves cannot be deterministicly from the wave spectrum. For -this reason, this study only allows us to observe the influence of reflection on short waves, while mostly ignoring -infragravity waves. +Unfortunately, the spectrum wave generation method used by SWASH could not reproduce simlar waves to the one observed +at the buoy. As mentionned by \textcite{dysthe2008}, such rogue waves cannot be deterministicly from the wave spectrum. +For this reason, this study only allows us to observe the influence of reflection on short waves, while mostly ignoring +infragravity waves. Those results are only useful if we consider that infragravity waves behave similarly to shorter +waves regarding reflection. \subsection{Wave transformation} @@ -236,11 +252,16 @@ Those results tend to confirm recent research by \textcite{lodhi2020}, where it threshold tend to overestimate the minimal flow velocity needed for block movement, although further validation of the model that is used would be needed to confirm those findings. -Additionally, the flow velocity that is reached during the identified wave is not the highest that is reached in the -model. Other shorter waves yield similar flow velocities on the breakwater, but in a smaller timeframe. The importance -of time dependency in studying block displacement would be in accordance with research from \textcite{weiss2015}, who -suggested that the use of time-dependent equations for block displacement would lead to a better understanding of the -phenomenon. +Additionally, similar flow velocities are reached in the model. Other shorter waves yield similar flow velocities on +the breakwater, but in a smaller timeframe. The importance of time dependency in studying block displacement would be +in accordance with research from \textcite{weiss2015}, who suggested that the use of time-dependent equations for block +displacement would lead to a better understanding of the phenomenon. + +Although those results are a major step in a better understanding of block displacement in coastal regions, further +work is needed to understand in more depth the formation and propagation of infragravity waves in the near-shore +region. Furthermore, this study was limited to a single block displacement event, and further work should be done to +obtain more measurements and observations of such events, although their rarity and unpredictability makes this task +difficult. \section{Methods} @@ -261,14 +282,23 @@ over \SI{0.2}{\Hz}. All wavelet analysis in this study is conducted using a continuous wavelet transform over a Morlet window. The wavelet power spectrum is normalized by the variance of the timeseries, following the method proposed by -\textcite{torrence1998}. +\textcite{torrence1998}. This analysis extracts a time-dependent power spectrum and allows to identify the composition +of waves in a time-series. \subsection{SWASH models} \subsubsection{Domain} +\begin{figure} + \centering + \includegraphics{fig/bathy2d.pdf} + \caption{Bathymetry in front of the Artha breakwater. The extremities of the line are the buoy and the + breakwater.}\label{fig:bathy} +\end{figure} + A \SI{1750}{\m} long domain is constructed in order to study wave reflection and wave transformation over the bottom -from the wave buoy to the breakwater. Bathymetry with a resolution of around \SI{1}{\m} was used for most of the domain. +from the wave buoy to the breakwater. Bathymetry with a resolution of around \SI{1}{\m} was used for most of the domain +(Figure~\ref{fig:bathy}). The breakwater model used in the study is taken from \textcite{poncet2021}. A smoothed section is created and considered as a porous media in the model. @@ -322,12 +352,12 @@ the SWASH model, the porous armour was considered at a macroscopic scale. A volume-of-fluid (VOF) model in two-vertical dimensions based on volume-averaged Reynolds-averaged Navier-Stokes (VARANS) equations is used (olaFlow, \cite{higuera2015}). The model was initially setup using generic values for porous -breakwater studies. A sensibility study conducted on the porosity parameters found a minor influence of these values on -the final results. +breakwater studies. A sensibility study conducted on the porosity parameters found the influence of these values on +the final results to be very minor. -The k-ω SST turbulence model was used, as it produced much more realistic results than the default k-ε model, especially -compared to the photographs from the storm of February 28, 2017. The k-ε model yielded very high viscosity and thus -strong dissipation in the entire domain, preventing an accurate wave breaking representation. +The k-ω SST turbulence model was used, as it produced much more realistic results than the default k-ε model, +especially compared to the photographs from the storm of February 28, 2017. The k-ε model yielded very high viscosity +and thus strong dissipation in the entire domain, preventing an accurate wave breaking representation. \subsubsection{Boundary conditions}