613 lines
29 KiB
TeX
613 lines
29 KiB
TeX
\chapter{Literature Review}
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In this chapter, literature relevant to the present study will be reviewed.
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Three sections will be detailled: the separation of incident and reflected
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components from wave measurements, the modelisation of wave impacts on a
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rubble-mound breakwater, and the modelisation of block displacement by wave
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impacts.
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\section{Separating incident and reflected components from wave buoy data}
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\subsection{Introduction}
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The separation of incident and reflected waves is a crucial step in numerically
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modeling a sea state. Using the raw data from a buoy as the input of a wave
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model will lead to incorrect results in the domain as the flow velocity at the
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boundary will not be correctly generated.
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Several methods exist to extract incident and reflected components in measured
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sea states, and they can generally be categorised in two types of methods:
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array methods and PUV methods \parencite{inch2016accurate}. Array methods rely
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on the use of multiple measurement points of water level to extracted the
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incident and reflected waves, while PUV methods use co-located pressure and
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velocity measurements to separate incident and reflected components of the
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signal.
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\subsection{Array methods}
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\subsubsection{2-point methods}
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Array methods were developped as a way to isolate incident and reflected wave
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components using multiple wave records.
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\textcite{goda1977estimation,morden1977decomposition} used two wave gauges
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located along the wave direction, along with spectral analysis, in order to
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extract the incident and reflected wave spectra. Their work is based on the
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earlier work of \textcite{thornton1972spectral}. \textcite{goda1977estimation}
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analyzed the wave spectrum components using the Fast Fourier Transform, and
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suggest that this method is adequate for studies in wave flumes. They noted
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that this method provides diverging results for gauge spacings that are
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multiples of half of the wave length. \textcite{morden1977decomposition}
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applies this technique to a field study, where the sea state is wind generated.
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\textcite{morden1977decomposition} showed that, using appropriate spectral
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analysis methods along with linear wave theory, the decomposition of the sea
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state into incident and reflected waves is accurate. A relationship between the
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maximum obtainable frequency and the distance between the sensors is provided.
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According to \textcite{morden1977decomposition}, the only needed knowledge on
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the wave environment is that wave frequencies are not modified by the
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reflection process.
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\subsubsection{3-point methods}
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In order to alleviate the limitations from the 2-point methods,
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\textcite{mansard1980measurement} introduced a 3-point method. The addition of
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a supplementary measurement point along with the use of a least-squares method
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most importantly provided less sensitivity to noise, non-linear interactions,
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and probe spacing. The admissible frequency range could also be widened. A
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similar method was proposed by \textcite{gaillard1980}. The accuracy of the
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method for the estimation of incident and reflected wave components was once
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again highlighted, while the importance of adequate positioning of the gauges
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was still noted.
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\subsubsection{Time-domain method}
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\textcite{frigaard1995time} presented a time-domain method for reflected and
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incident wave separation. This method, called SIRW method, used discrete
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filters to extract the incident component of an irregular wave field. The
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results were as accurate as with the method proposed by
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\cite{goda1977estimation}, while singularity points are better accounted for.
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The main advantage of the SIRW method is that it works in the time-domain,
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meaning that real time computations can be performed.
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\textcite{frigaard1995time} also mentions the possibility of replacing one of
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the wave gauges by a velocity meters to prevent singularities.
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This method was improved by \textcite{baldock1999separation} in order to
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account for arbitrary bathymetry. Linear theory is used to compute shoaling on
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the varying bathymetry. Resulting errors in the computed reflection coefficient
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are low for large reflection coefficients, but increase with lower
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coefficients. The neglect of shoaling can lead to important error in many
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cases. The presented method could also be extended to three-dimensionnal waves
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and bathymetry by considering the influence of refraction.
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\subsubsection{Further improvements}
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Further additions were made to array methods. \textcite{suh2001separation}
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developped a method taking constant current into account to separate incident
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and reflected waves. This method relies on two or more gauges, using a least
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squares method. Results are very accurate in the absence of noise, but a small
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amount of error appears when noise is added.
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\textcite{inch2016accurate} confirmed that the presence of noise led to
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overestimation of the reflection coefficient. The creation of bias lookup
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tables is proposed in order to account for noise-induced error in reflection
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coefficient estimations.
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\textcite{andersen2017estimation,roge2019estimation} later proposed
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improvements to account for highly non-linear regular and irregular waves
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respectively. The improved method provides very accurate results for highly
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non-linear waves, but are expected to be unreliable in the case of steep
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seabeds, as shoaling is not part of the underlying model.
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\subsubsection{Conclusion}
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Array methods have been developped enough to provide accurate results in a wide
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range of situations. Sensibility to noise has been reduced, and the influence
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of shoaling has been considered. Those methods can also be applied to irregular
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non-linear waves.
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However, they require at least two wave gauges to be used. That means that in
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some situations such as the Saint-Jean-de-Luz event of 2017, other methods are
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needed since only one field measurement location is available.
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\subsection{PUV methods}
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The goal of PUV methods is to decompose the wave field into incident and
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reflected waves using co-located wave elevation and flow velocity measurements
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\parencite{tatavarti1989incoming}. \textcite{tatavarti1989incoming} presented a
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detailled analysis of separation of incoming and outging waves using co-located
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velocity and wave height sensors. Their method allows to obtain the reflection
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coefficient relative to frequency, as well as to separate incident and
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reflected wave components. Compared to array methods, this method also strongly
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reduces the influence of noise.
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\textcite{kubota1990} studied the influence of the considered wave theory on
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incident and reflected wave separation. Three methods, based on linear
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long-wave theory, small-amplitude wave theory and quasi-nonlinear long-wave
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theory respectiveley were developped and compared. The results show that the
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quasi-nonlinear approach gave the most accurate results.
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%\textcite{walton1992} applied a separation method based on co-located pressure
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%and velocity measurements on field, studying two natural beaches. This study
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%showed that reflection is not significant on natural beaches. Additionnaly,
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%the method that is used allowed for larger reflected energy than incident
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%energy.
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Research by \textcite{hughes1993} showed how co-located horizontal velocity and
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vertical velocity (or pressure) sensors can be used to extract incident and
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reflected wave spectra. Their method is based on frequency domain linear
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theory, and provided accurate results for full reflection of irregular
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non-breaking waves. Low-reflection scenarii were evaluated against the results
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from \textcite{goda1977estimation}, and showed good agreement between both
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methods. \textcite{hughes1993} also highlights that reflection estimates are
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unreliable for higher frequencies, where coherency between the two measured
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series is lower.
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Following the work of \textcite{tatavarti1989incoming},
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\textcite{huntley1999use} showed how principal component analysis can alleviate
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noise-induced bias in reflection coefficient calculations compared to
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time-domain analysis. They also stuied the influence of imperfect collocation
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of the sensors, showing that the time delay between sensors leads to a peak in
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the reflection coefficient at a frequency related to this time delta.
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% \cite{sheremet2002observations}
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\subsection{Conclusion}
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Numerous methods have been developped in order to separate incident and
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reflected components from wave measurements. Array methods rely on the use of
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multiple, generally aligned, wave gauges, while PUV methods rely on the use of
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co-located sensors, generally a wave height sensor and a horizontal velocity
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sensor. Array methods generally have the advantage of being more cost-effective
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to implement, as the cost of reliable velocity measurement devices can be
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important \parencite{hughes1993}. Nevertheless, PUV methods are generally more
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accurate regarding noise, varying bathymetry, and can be setup closer to
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reflective surfaces \parencite{hughes1993,inch2016accurate}.
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In the case of the 2017 event on the Artha breakwater, the results from a
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single wave gauge are available, which means that the array methods are not
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applicable. A PUV method \parencite{tatavarti1989incoming,huntley1999use}
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should then be used to evaluate the reflection coefficient of the Artha
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breakwater and to separate the incident and reflected wave components from the
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measured data.
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\section{Modelling wave impact on a breakwater}
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\subsection{Introduction}
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Modelling rubble-mound breakwaters such as the Artha breakwater requires
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complex considerations on several aspects. First of all, an accurate of the
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fluid's behavior in the porous armour of the breakwater is necessary. Then,
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adequate turbulence models are needed in order to obtain accurate results.
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Several types of models have been developped that can be used to study breaking
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wave flow on a porous breakwater.
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\subsection{SPH models}
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\subsubsection{Introduction}
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Smoothed-Particle Hydrodynamics (SPH) models rely on a Lagrangian
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representation of the fluid \parencite{violeau2012fluid}. These models are
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meshless, and work by considering fluids as a collection of particles.
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SPH models have been shown to provide satisfactory results for the modeling of
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turbulent free surface flows \parencite{violeau2007numerical}. Additionnaly,
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\textcite{dalrymple2006numerical} showed that SPH models can be used in small
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scale models of water waves. In this part, literature on modeling flow in
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porous media and the adequate boundary conditions for wave modeling will be
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reviewed.
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\subsubsection{Porosity modelling}
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Multiple approaches can be used when modeling porous media using SPH models.
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The most obvious approach relies on the use of discrete elements in the porous
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domain. For instance, \textcite{altomare2014numerical} showed that an SPH model
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along with discrete modeling of the blocks composing a breakwater could yield
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satisfactory results. The meshless character of SPH models allows for modeling
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the large scale outside the porous media and the small scale of the space
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between blocks effectively.
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Nevertheless, the more common approach is to use a macro-scale model in porous
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media, in which the porous domain is considered to have a set of homogeneous
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properties.
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\textcite{jiang2007mesoscale} used randomly placed fixed particles in the
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porous media in order to model porosity at a microscopic scale from mesoscopic
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porosity properties. The resulting model showed reliable results in studying
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the flow through porous media.
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By contrast, \textcite{shao2010} used volume-averaged Navier-Stokes equations
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along with an averaged porosity model
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\parencite{huang2003structural,burcharth1995one} in an incompressible SPH
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(ISPH) model in order to model wave flow in porous media, accounting for a
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linear and quadratic term in porosity induced friction. Turbulence was modeled
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with a $k-\varepsilon$ volume averaged model. Good agreement was highlighted
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between the results from this model and other models, analytical results and
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experimental measurements for solitary and regular waves interacting with a
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porous breakwater.
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Similarly, \textcite{ren2016improved} presents a weakly-compressible SPH
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(WCSPH) model using the volume averaged Favre averaged Navier-Stokes (VAFANS)
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equations along with a large Eddy simulation (LES, \cite{ren2014numerical})
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turbulence model. Interaction between turbulent flows and porous media is
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studied and good agreement is shown between model results and experimental
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data. Additionnaly, it is highlighted that the addition of the turbulence model
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does increase the accuracy of the model. Similar results are found by
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\textcite{wen2016sph} when studying wave impact on non-porous structures using
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the same model.
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That model was then extended to a three-dimensional model by
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\textcite{wen20183d}. The computed free surface and forces on a structure were
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shown to be accurately predicted by the 3D model.
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%\paragraph{Notes}
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%
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%\cite{jiang2007mesoscale}: Meso-scale SPH model of flow in isotropic porous
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%media; randomly placed particles with repulsive force; reasonable results.
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%
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%\cite{shao2010}: incompressible flow with porous media; Navier-Stokes,
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%Volume-Averaged $k-\varepsilon$; porosity model is same as
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%\cite{troch1999development} but without inertia term
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%\parencite{huang2003structural}
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%
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%\cite{altomare2014numerical} "microscopic" model of breakwater
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%
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%\cite{kunz2016study} comparison of sph model with micro-model experiments; not
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%quite applicable
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%
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%\textbf{\cite{ren2016improved}} VAFANS equations to solve incompressible
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%turbulent flow with porous media. Same porosity model as \cite{shao2010}
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%
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%\cite{wen2016sph}
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%
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%\cite{pahar2016modeling}
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%\cite{peng2017multiphase}
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%\cite{wen20183d}: 3D VAFANS
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%\cite{kazemi2020sph}
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\subsubsection{Wave generation}
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One of the more recent research subject with SPH models has been wave
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generation. Wave paddles were initially used as a way to generate waves in
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numerical basins \parencite{zheng2010numerical}, with the major drawback of
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such wave makers begin their high reflectivity.
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\textcite{liu2015isph} proposed an improved wave generator using a momentum
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source in an ISPH model. The use of a momentum source was a major improvement
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as it enabled the use of non-reflective wave generators. The proposed solution
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was developed for two-dimensional linear waves, but the same algorithm could be
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used for three-dimensional models.
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\textcite{altomare2017long} presented a wave generation method for long-crested
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(second order) waves in a WCSPH model using a piston wave maker. Although this
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method leads to high reflection, the possibility of generating irregular
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waves was highlighted.
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Similarly to \textcite{liu2015isph}, \textcite{wen2018non} proposed a wave
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generation method using a momentum source to create a non reflective wave
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maker. The proposed method was used for generating regular as well as random
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waves in a flume, and could be extended to three-dimensional simulations.
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Nevertheless, the method proposed was limited to linear wave theory.
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%\cite{zheng2010numerical}
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%
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%\cite{liu2015isph}: 2D non-reflective linear wave generator using a momentum
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%source in ISPH
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%
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%\cite{altomare2017long}: Wave generation and absorption of long-crested waves
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%(2nd order) in WCSPH. Generation of monochromatic as well as irregular waves.
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%
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%\cite{wen2018non}: Non reflective spectral wave maker, using momentum source
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\subsubsection{Conclusion}
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SPH models have been showed to be extremely powerful tools in modelling
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wave-structure interaction, due to their ability to model complex interfaces
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and highly dynamic situations \parencite{altomare2017long}.
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Modeling wave interaction with porous structures using SPH models has been
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widely studied, and generally adequate results are obtained
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\parencite{wen20183d}. Nonetheless, SPH models still face some limitations
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regarding their ability to represent incompressible flows, leading to high
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diffusivity \parencite{higuera2015application}.
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Moreover, wave-generation techniques, especially for long simulations, are
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still at an early stage of developement \parencite{wen2018non}, limiting the
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applicability to such models in studying real cases using in-situ data.
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\subsection{VOF models}
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\subsubsection{Introduction}
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Contrary to SPH models, the volume of fluid (VOF) method relies on a Eulerian
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representation of the fluid \parencite{hirt1981volume}. This method uses a
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marker function, the value of which represents the fraction of fluid in a mesh
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cell.
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\subsubsection{2D models}
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Using the VOF method along with Navier-Stokes equations, several models have
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been developed in order to model fluid dynamics around porous structures.
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\textcite{van1995wave} first implemented 2D-V incompressible Navier-Stokes
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equations using the VOF method while accounting for porous media. The results
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of the numerical model were validated with analytical solutions for simple
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cases, as well as physical model tests. The model yielded acceptable results,
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but the representation of turbulence and air-extrusion still required
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improvement.
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\textcite{troch1999development} developed the VOFbreak\textsuperscript{2} model
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in order to provide improvements to earlier models. The Forchheimer theory
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\parencite{burcharth1995one} is used in order to model the behavior of the flow
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inside porous media. The hydraulic gradient generated in porous media is
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decomposed as a linear term, a quadratic term, and an inertia term. Those terms
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are ponderated by three coefficients that need to be calibrated. Several
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attempts have been made to obtain analytical formulas for those
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\parencite{burcharth1995one,van1995wave}, but no universal result has been
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provided for the inertia term in particular. \textcite{vieira2021novel}
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additionnaly proposed using artificial neural networks in order to calibrate
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those values, which are generally calibrated using experimental results.
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Parallely, \textcite{liu1999numerical} created a new model (COBRAS) that used
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the VOF method. The model is based on the combination of Reynolds averaged
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Navier-Stokes (RANS) equations and a $k-\varepsilon$ turbulence model. The
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porous media is modelled similarly to \textcite{troch1999development}. The
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offered results were improved compared to earlier models as more a more
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accurate consideration of turbulence outside porous media was added. This model
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was further improved by \textcite{hsu2002numerical} in order to account for
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small scale turbulence inside the porous media thanks to volume averaged RANS
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(VARANS) equations.
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The COBRAS model was then reworked by
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\textcite{losada2008numerical,lara2008wave} to add improvements to wave
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generation and usability. The main difference between this new code (COBRAS-UC)
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and COBRAS is the addition of irregular waves generation. The code was also
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optimized to reduce the number of iterations. The improvements allowed for
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longer simulations to be computed. The predictions for free surface elevation
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and pressure in front of a porous breakwater were accurate, but improvements
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were still needed, in particular considering computation time.
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\subsubsection{3D models}
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The combination of VARANS equations and the VOF method was then brought to 3D
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domains by \textcite{del2011three} in IH3VOF. Specific boundary conditions were
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also added for several wave theories. Additionnaly, an improved turbulence
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model was used ($\omega$-SST model, \cite{menter1994two}), which provides
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strongly improved results in zones where strong pressure gradients appear.
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Strong agreement between IH3VOF and experimental results was obtained, but the
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need for accurate boundary conditions limited the applicability of the model.
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\textcite{higuera2015application} reworked the equations from
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\textcite{del2011three} as discrepancies were observed with earlier literature
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and added several improvements to the model. Notably, time-varying porosity was
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added in order to account for eventual sediment displacement. New boundary
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conditions were added, with static and dynamic boundary wave generators as well
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as passive and acive wave absorption being implemented. The resulting model
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(IHFOAM/olaFlow, \cite{olaFlow}) was implemented in the OpenFOAM toolbox.
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\subsubsection{Conclusion}
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VOF models have been developped to provide accurate results for the study of
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wave impact on porous structures. The validation results from
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\textcite{higuera2015application} show the capabilities of such models in
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accurately representing rubble-mound breakwaters subject to irregular
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three-dimensional wave fields.
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Nonetheless, the representation of porosity in those models is still mainly
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based on experimental calibration, particularly for the inertia term of
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porosity induced friction.
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%\subsection{Conclusion}
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%\paragraph{Notes}
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%
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%\cite{van1995wave,troch1999development}
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%
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%COBRAS \parencite{liu1999numerical}: spatially averaged RANS
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%with $k-\varepsilon$ turbulence model. Drag forces modeled by empirical linear
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%and non-linear friction terms; \cite{hsu2002numerical}: introduced VARANS in
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%order to account for small scale turbulence inside the porous media.
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%->
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%COBRAS-UC/IH2VOF \parencite{losada2008numerical,lara2008wave}: VOF VARANS (2D);
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%refactor of COBRAS code, with improved wave generation, improvement of input
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%and output data.
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%->
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%IH3VOF \parencite{del2011three}: 3D VOF VARANS, updated porous media equations,
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%optimization of accuracy vs computation requirements, specific boundary
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%conditions, validation. Adding SST model.
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%->
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%IHFOAM/olaFlow \parencite{higuera2015application}: Rederivation of
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%\cite{del2011three}, add time-varying porosity; Improvement to wave generation
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%and absorption; implementation in OpenFOAM; extensive validation; application
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%to real coastal structures.
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%
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%\cite{vieira2021novel}: Use of artificial neural networks to determine porosity
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%parameter for VOF VARANS model.
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%\subsection{Other}
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%
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%BEM: \cite{hall1994boundary,koley2020numerical}
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\section{Block displacement by waves}
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\subsection{Introduction}
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Displacement of blocks or boulders by waves has been a major topic in
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understanding the influence of storm and tsunami waves in coastal regions.
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Several approaches have been taken to study this phenomenon.
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% In a first part, we will discuss in-situ studies on displaced boulders. In a
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% second part, we will review models of block displacements.
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\subsection{Block displacement models}
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The main goal of studying boulder displacement is generally to establish the
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cause of boulder deposits in coastal areas. \textcite{nott1997extremely} was
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among the first to propose hydrondynamic equations that aimed to calculate the
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wave height that would lead to the displacement of a boulder for storm and
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tsunami waves. The main difference between storm and tsunami waves in those
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equations was the flow velocity relative to wave height. The calculation of the
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minimum flow velocity for boulder transport is obtained by calculating an
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equilibrium between drag, lift and restraining forces.
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Those equations were refined by \textcite{nott2003waves} in order to account
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for the pre-transport environment of the boulder. \citeauthor{nott2003waves}
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derived equations for submerged, sub-aerial and joint bounder boulders. A new
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parameter ($\delta$) was introduced to differentiate between tsunami and storm
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waves. This study highlights the importance of the initial environment of a
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boulder for wave transport conditions. Compared to the equations from
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\textcite{nott1997extremely}, an aditionnal inertia term is added to the
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equilibrium equation for sub-aerial boulders, while the drag force is removed
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for joint-bounded blocks.
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\textcite{nandasena2011reassessment} noted that \citeauthor{nott2003waves}'s
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equations could be improved, and proposed a new set of equations correcting
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the lift and inertia terms in \citeauthor{nott2003waves}'s equations.
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\textcite{nandasena2011reassessment} found that the new equations produced up
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to a \SI{65}{\percent} difference with \citeauthor{nott2003waves}'s equations.
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\textcite{buckley2012inverse} proposed alternative equations for sliding and
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overturning of submerged boulders. An equation for block sliding was introduced
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by considering friction on the bed. \textcite{weiss2012mystery} investigated
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the influence of bed roughness on block displacement. A new stability criteria
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was established, and bed roughness was found to be a major factor in boulder
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displacement. In contrast with the findings from \textcite{nott2003waves}, the
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threshold wave amplitude for block displacement was found to be similar between
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tsunami and storm waves.
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\textcite{nandasena2013boulder,liu2014experimental} performed experimental
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studies of block displacement using dam break scenarios in a flume. The results
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from both studies indicate that the primary mode of boulder motion for large
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boulders is sliding, rather than rolling or saltation.
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\textcite{weiss2015untangling} highlights inadequacies in the criteria that are
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generally used \parencite{nott2003waves,nandasena2011reassessment}. According
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to \textcite{weiss2015untangling}, the use of a minimum threshold on block
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movement does not account for the possibility of a block returning to its
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initial position after being slightly disloged. A new threshold is proposed on
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the minimal movement of a block, while considering the time-dependent nature of
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wave-induced flow. \textcite{weiss2015untangling} also shows the importance of
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the pre-transport conditions on block displacement.
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\textcite{kennedy2017extreme} derived new equations following the approach from
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\textcite{nandasena2011reassessment} accounting for non-parallelepipedic
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blocks. The revised equations led to a lower velocity threshold for block
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movement. This highlights the importance of boulder shape in displacement
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considerations.
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\textcite{lodhi2020role} highlighted the importance of hydrodynamic pressure in
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block displacement. A new equation was given for the threshold flow velocity
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for block movement. An experimental validation of the models was performed, and
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showed the overestimation of the threshold velocity by previous models.
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\textcite{oetjen2021experiments} performed a review of boulder displacement
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experiments. They found that the initial position of boulders relative to the
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wave impact has a major influence on block displacement. Conversely, the
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influence of bed roughness seems to have been overestimated in the past.
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Similarly to \textcite{lodhi2020role}, \textcite{oetjen2021experiments}
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highlights an overestimation of minimum wave height for block displacement by
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earlier equations \parencite{nott1997extremely,nandasena2011reassessment}.
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\subsection{Breakwater stability}
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Breakwater stability has been a central issue in breakwater design.
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\textcite{hudson1959laboratory} showed that Iribarren's formula could be used
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to design breakwater when combined with an experimental parameter depending on
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the shape of armour blocks and the geometry of the armour layer. A no-damage
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and no-overtopping criteria is provided and validate using experimental
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results. \textcite{losada1979joint} noted that Iribarren's criterion was fairly
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accurate, but the results from \textcite{hudson1959laboratory} are found to be
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too conservative.
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\textcite{van1987stability} proposed an new criterion regarding breakwater
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stability under random wave action. Extensive experimental validation using
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both small-scale and large-scale models was conducted, and seem to provide
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coherent results. Similarly to earlier work, the criterion is provided as an
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dimensionless form.
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\textcite{galland1995rubble} observed that armour stability relative to oblique
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waves was increased. \textcite{van2014oblique} confirmed the influence of wave
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direction on stability, and provided a way of estimating breakwater stability
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depending on wave orientation.
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%\cite{hudson1959laboratory}
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%
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%%\cite{hudson1975reliability}
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%
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%\cite{losada1979joint}
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%
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%\cite{van1987stability}
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%
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%\cite{van1995conceptual}
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%
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%\cite{galland1995rubble}
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%
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%\cite{iglesias2008virtual}
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%
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%\cite{etemad2012stability}
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%
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%\cite{van2014oblique}
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\subsection{Conclusion}
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Block displacement by waves has been widely studied in the literature.
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Nevertheless, most validation has been conducted using laboratory experiments,
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and reliable real-world data on that subject is scarce. This highlights the
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opportunity provided by the 2017 Saint-Jean-de-Luz event, as the availability
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of in-situ data allows for real-world validation of the results from earlier
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research.
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%\subsection{In-situ studies}
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%
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%\cite{barbano2010large}: boulders deposity in Sicily -> probably tsunamis
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%
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%\cite{paris2011}:
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%
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%\cite{nandasena2011numerical}
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%\cite{may2015block}
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%\cite{biolchi2016}
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%\cite{kennedy2016observations}
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%\cite{erdmann2018boulder}
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%\cite{cox2018extraordinary}
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%
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%\subsection{Models}
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%
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%\cite{nott1997extremely}
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%
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%\cite{nott2003waves} Submerged boulder:
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%\begin{equation}
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%u^2 \ge \frac{2\left(\frac{\rho_s}{\rho_w}-1\right)ag}
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%{C_d\left(\frac{ac}{b^2}\right)+C_l}
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%\end{equation}
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%
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%\cite{imamura2008numerical}
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%\cite{barbano2010large}
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%\cite{nandasena2011numerical}
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%
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%\cite{nandasena2011reassessment}
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%\begin{equation}
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%u^2 \ge \frac{2\left(\frac{\rho_s}{\rho_w}-1\right) gc
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%\left(\cos\theta+\frac{c}{b}\sin\theta\right)}
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%{C_d\frac{c^2}{b^2}+C_l}
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%\end{equation}
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%
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%\cite{buckley2012inverse}
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%\cite{weiss2012mystery}
|
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%\cite{nandasena2013boulder}
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|
%\cite{liu2014experimental}
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|
%\cite{weiss2015untangling}
|
|
%
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|
%
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|
%\cite{kennedy2016observations}
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%\cite{kennedy2017extreme}
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%\cite{weiss2017toward}
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%
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%\cite{bressan2018laboratory} Partially submerged boulders
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%\begin{equation}
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%u^2 \ge \frac{2b_wW}{\rho_w\left(b_DC_DA_{wfs}+b_LC_LA_{wbs}\right)}
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%\end{equation}
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%
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%\cite{lodhi2020role}
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%\cite{oetjen2020significance}
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%
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|
%\cite{oetjen2021experiments}: Review
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|
%
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|
%---
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|
%\cite{zainali2015boulder}: Numerical model of block displacement
|