commit 5598887c04305be89324783534aac03f2bf7c43f
parent 73dc0fe0eab52c827609e23c30326510c4c27d1b
Author: Ivan Gankevich <i.gankevich@spbu.ru>
Date: Thu, 14 Nov 2019 13:37:36 +0300
Update the paper after the review.
Diffstat:
8 files changed, 315 insertions(+), 276 deletions(-)
diff --git a/Makefile b/Makefile
@@ -11,6 +11,7 @@ FLAGS = \
NAME = mmcp-19-gerstner
all: build/$(NAME).pdf
+all: build/$(NAME)-grayscale.pdf
all: build/$(NAME).zip
build/$(NAME).pdf: build
@@ -19,6 +20,18 @@ build/$(NAME).pdf: main.tex
@-$(LATEXMK) $(FLAGS) -f $<
@cp build/main.pdf $@
+build/$(NAME)-grayscale.pdf: build
+build/$(NAME)-grayscale.pdf: build/$(NAME).pdf
+ gs \
+ -sOutputFile=$@ \
+ -sDEVICE=pdfwrite \
+ -sColorConversionStrategy=Gray \
+ -dProcessColorModel=/DeviceGray \
+ -dCompatibilityLevel=1.4 \
+ -dNOPAUSE \
+ -dBATCH \
+ $<
+
build/$(NAME).zip: Makefile
build/$(NAME).zip: main.tex
build/$(NAME).zip: refs.bib
diff --git a/graphics/01-gerstner.png b/graphics/01-gerstner.png
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diff --git a/graphics/02-gerstner.png b/graphics/02-gerstner.png
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diff --git a/graphics/03-wind-wave.png b/graphics/03-wind-wave.png
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diff --git a/graphics/05-wind-wave.png b/graphics/05-wind-wave.png
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diff --git a/graphics/06-group-wave.png b/graphics/06-group-wave.png
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diff --git a/main.tex b/main.tex
@@ -1,275 +1,283 @@
-\documentclass{webofc}
-
-\usepackage[varg]{txfonts}
-
-\begin{document}
-
-\title{Computational model of unsteady hydromechanics\\of large amplitude Gerstner waves}
-%\subtitle{Do you have a subtitle?\\ If so, write it here}
-
-\author{%
-\firstname{Alexander} \lastname{Degtyarev}\inst{1}%
-\and%
-\firstname{Ivan} \lastname{Gankevich}\inst{1}%
-\fnsep\thanks{\email{i.gankevich@spbu.ru}}%
-\and%
-\firstname{Nataliia} \lastname{Kulabukhova}\inst{1}%
-\and%
-\firstname{Vasily} \lastname{Khramushin}\inst{1,2}%
-}
-
-\institute{%
-Saint Petersburg State University, Russia%
-\and%
-Scientific Society of Shipbuilders named after Alexey Krylov, Russia%
-}
-
-\abstract{%
-Numerical experiments in ship hydromechanics involve non-stationary interaction
-of a ship hull and wavy surface that include formation of vortices, surfaces of
-jet discontinuities, and discontinuities in fluid under influence of negative
-pressure. These physical phenomena occur not only near ship hull, but also at
-a distance where waves break as a result of interference of sea waves and waves
-reflected from the hull.
-In the study reported here we simulate wave breaking and reflection near the
-ship hull. We use explicit numerical schemes to simulate propagation of
-large-amplitude sea waves and their transformation after the impact with a
-ship. The problem reduces to determining wave kinematics on a moving boundary
-of a ship hull and a free boundary of a computational domain. We build a grid
-of large particles having a form of a parallelepiped, and in wave equation in
-place of velocity field we integrate streams of fluid represented by functions
-as smooth as wavy surface elevation field. We assume that within boundaries of
-computational domain waves do not disperse, i.e.~their length and period stays
-the same. Under this assumption we simulate trochoidal Gerstner
-waves of a particular period. Wavy surface boundary have to
-satisfy Bernoulli equation: pressure on the surface of the wave becomes
-non-constant, fluid particles drift in the upper layers of a fluid in the
-direction of wave propagation, and vortices form as a result.
-The drift is simulated by changing curvatures of particles trajectories based
-on the instantaneous change of wavy surface elevation.
-This approach allows to simulate wave breaking and reflection near ship hull.
-The goal of the research is to develop a new method of taking wave reflection
-into account in ship motion simulations as an alternative to the classic method
-that uses added masses.
-}
-
-\maketitle
-
-\section{Introduction}
-
-Ship and sea wavy surface motion do not have an abundance of geometric forms
-and physical phenomena. Waves formation due to ship motion and interaction
-between ocean and atmosphere are governed by continuity condition for heavy
-fluid and the law of conservation of energy.
-
-Strict theoretical solution (and in fact the only solution) for large-amplitude
-wind waves on the surface of heavy fluid was obtained in 1802 by Franz Josef
-von Gerstner~--- a professor from a university in Prague~\cite{gerstner1809}.
-Generic trochoidal wave mathematical model has large
-dispersion~\cite{sommerfeld1945mechanik}, the dependence of the speed of wave
-propagation on their length and period. As a result
-\begin{itemize}
- \item wave energy propagation speed becomes half the visible phase velocity
- of wave crests,
- \item wave front constantly changes its phase, and
- \item wave are quantised into packets and wave transformation
- and propagation phenomena become nonstationary.
-\end{itemize}
-
-\section{Computational model of intense sea waves}
-
-\begin{figure}
- \centering
- \includegraphics[width=\textwidth]{graphics/01-gerstner.png}
- \includegraphics[width=\textwidth]{graphics/02-gerstner.png}
- \caption{Analytic solutions: progressive Gerstner wave (top),
- a wave with critical height~--- a standing wave
- (bottom).\label{fig-gerstner}}
-\end{figure}
-
-Gerstner wave (fig.~\ref{fig-gerstner}) is a cycloid, fluid particle trajectory
-radius \(r_W=1.134\lambda_Wh_W/4\pi\text{ }\left[\text{m}\right]\) of which is fixed
-relative to flat wavy surface level \(z_W\), hence \(z\)-coordinates of the
-crest and trough are the same. Here \(\lambda_W\) is the wave length,
-\(h_W\)~--- relative wave height defined on the interval \([0..1]\) with
-\(h_W=1\) being the maximum wave height for which the crest does not
-break (fig.~\ref{fig-gerstner}). Vertical displacement of a fluid particle
-is given by
-\begin{equation*}
- \zeta_Z = r_W \cos x_W \exp\left(-2\pi z_W / \lambda_W\right)
- \qquad \left[\text{m}\right].
-\end{equation*}
-Horizontal displacement of the same fluid particle with respect to its initial
-position for progressive wave is given by analogous equation, but with
-a shift by one fourth of the phase:
-\begin{equation*}
- \zeta_X = -r_W \sin x_W \exp\left(-2\pi z_W / \lambda_W\right)
- \qquad \left[\text{m}\right].
-\end{equation*}
-Critical wave height of Gerstner waves (fig.~\ref{fig-gerstner}) gives the correct
-ratio of wave height to wave length, but 60 degree slope limit for standing
-wave with steepness \(\approx{}1/4\) as well as 30 degree slope limit for
-progressive (traveling) wave with steepness \(\approx{}1/7\) are not correctly
-captured by the model.
-
-State of the art mathematical and computational models do not simulate wave
-groups that are integral part of ocean wavy surface motion. Our model, which is
-a modified version of Gertner wave, includes wave groups. They are described as
-a dependency between fluid particle trajectory radius and instantaneous
-displacement of the particle with respect to calm sea level
-(fig.~\ref{fig-trochoidal}).
-
-\begin{figure}
- \centering
- \includegraphics[width=\textwidth]{graphics/03-wind-wave.png}
- \includegraphics[width=\textwidth]{graphics/05-wind-wave.png}
- \caption{Simulation result: regular trochoidal waves with vertical
- displacement of sea level and wind stress on the wave sea
- surface. Propagating wind waves (top), extremely high wind waves
- (bottom).\label{fig-trochoidal}}
-\end{figure}
-
-We write adjusted radius as \(^{A}R_W={}^{A}K{}r_W\left(\cos{}x_W-1\right)\), where
-\(^{A}K=[1,0..\sqrt{2}]\) is radius coefficient that makes wave crests cnoidal
-and raises mean sea level. We choose \(^{A}K\) to be slightly less than 1 to
-reduce the effect of gusty winds on the wave form and prevent forming of
-cycloidal loops in wave crests, that appear for waves with overly large
-amplitude, that may occur as a result of the interference with waves
-heading from the opposite direction.
-
-The pressure on windward slope of the wave is smaller, because wind slides on
-the surface of the wave at a high speed and makes the slope more flat, while on
-the leeward slope of the wave wind speed drops significantly or even goes to
-nought and creates vorticity.
-
-The coefficient of wind stress \(^{W}K\) (the parameter that was shifted by one
-fourth of the phase) determines the assymmetry of steepness of windward and
-leeward slopes of the wave (fig.~\ref{fig-trochoidal}):
-\(^{W}R_W={}^{W}Kr_W\left(\sin{}x_W-1\right)\), where \(^{W}K=[0..1]\). The
-coefficient is close to unity for fresh wind waves and close to nought for
-swell. For two-dimensional sea surface \(^{W}K\) is used in dot product
-between wind and wave direction vectors:
-\begin{equation*}
- \begin{aligned}
- \zeta_Z &= r_W \cos x_W \exp\left(
- 2\pi \left[
- -z_W + r_W ^{A} K \left( \cos x_W - 1\right) + r_W ^{W} K\sin x_W
- \right] / \lambda_W
- \right) \\
- \zeta_X &= -r_W \sin x_W \exp\left(
- 2\pi \left[
- -z_W + r_W ^{A} K \left( \cos x_W - 1\right) + r_W ^{W} K\sin x_W
- \right] / \lambda_W
- \right).
- \end{aligned}
-\end{equation*}
-Then energy conservation is defined by Bernoulli's principle
-\begin{equation*}
- \frac{\rho V^2}{2} + \rho g \zeta_W = \text{const},
- \qquad \left[\text{N}/\text{m}^2\right]
-\end{equation*}
-where particle velocity \(V\) contributes the most to balancing
-the pressure \(\rho{}g\zeta_W\) on the wavy surface down to nought
-for breaking waves.
-
-\section{Trochoidal wave groups}
-
-In our modified model we simulate two wavy surfaces simultaneously: one for
-regular waves with normal length and one for waves with nine times higher
-length, that propagate under the same laws but with two times less speed
-(fig.~\ref{fig-group}). The product of these surfaces allows to simulate wave
-groups.
-
-\begin{figure}
- \centering
- \includegraphics[width=\textwidth]{graphics/06-group-wave.png}
- \caption{Trochoidal wave groups.\label{fig-group}}
-\end{figure}
-
-On the first entry the profile of the long wave is given by specific smoothing
-function, the form of which is close to phase wave profile. This function
-defines continuous change of wave front phase, which is needed to simulate
-waves produced by the ship.
-
-There is also a simpler approach to simulate wave groups: a superposition of
-regular waves with slightly different periods propagating in the opposite
-directions. Interference of waves of comparable lengths produces beats, in
-which nineth waves has double height and are standing waves. This approach
-generally gives satisfactory wavy surface, but does not work for waves produced
-by the ship, because they have complex wave front.
-
-\section{Direct numerical simulation of sea waves}
-
-We use explicit numerical scheme to simulate sea wavy surface that satisfies
-continuity equation; we call it direct numerical simulation
-(fig.~\ref{fig-waves-1}). We use the following definitions for three sea wave
-systems, that are used in the scheme.
-
-\begin{figure}
- \centering
- \includegraphics[width=\textwidth]{graphics/waves-01.png}
- \caption{Large-amplitude trochoidal waves.\label{fig-waves-1}}
-\end{figure}
-
-\begin{itemize}
-
- \item Fresh wind waves have a period of 6-8 seconds near the shore
- and up to 10-12 seconds in the ocean. The height of the wave is close
- to critical, that typically corresponds to 6 on Beaufort scale with
- wave crests greater than 5-6 metres.
-
- \item Fresh swell waves skew from mean wind direction by \(\approx{}30\)
- degrees. When the storm in northern hemisphere increases, wind
- direction goes counterclockwise and vice versa, i.e. the swell is
- always present in the ocean. Swell waves are comparable to wind waves:
- their height is two times smaller than critical wave height, and their
- length is 1.5-2 times greater.
-
- \item Old swell waves are long waves that come from higher latitudes. Their
- height is two times smaller than than of wind waves and fresh swell,
- their length is two times greater, and their direction is close to
- meridional (i.e.~south in northern latitudes and vice
- versa\footnote{The wind blows into the compass rose, the waves
- propagate in the direction of the rose.}).
-
-\end{itemize}
-
-These waves may add up in unfavourable way to a wave with the height of 13-15
-metres, however, in real world mean wave height will be 8-10 metres. Wave
-groups have nineth wave with double height, breaking crest and wave slope
-greater than 45 degrees.
-
-\begin{figure}
- \centering
- \includegraphics[width=0.9\textwidth]{graphics/07-exp-1.png}
- \caption{In the course of the simulation we visualise all three
- wave systems and create a view of ship hull dynamics
- and sea wave profiles in a different convenient scale.\label{fig-waves-2}}
-\end{figure}
-
-Oceanographers use well-established solutions~\cite{poplavskii1997} for regular
-progressive waves of arbitrary shape. Using trochoidal waves as a source, we
-fix wave periods and speeds in time to satisfy continuity equation and
-energy conservation law. We simulate all three wave systems (described above)
-simultaneously and indepedently (fig.~\ref{fig-waves-2}) and add individual
-wavy surfaces together to produce the resulting wavy surface.
-
-\section{Conclusion}
-
-We use explicit numerical schemes to simulate modified version of Gerstner
-waves. We simulate particle drift in the upper fluid layers by changing the
-curvature of the trajectory depending on the instantaneous change of wavy
-surface elevation. Our model is nonstationary, hence ship motions can also be
-nonstationary. Computational power of a desktop computer is enough for
-performing such simulations in real-time, and these types of simulations can
-even be performed on the board of the ship to chose optimal and efficient mode
-of ship operation.
-
-\begin{acknowledgement}
-Research work is supported by Saint Petersburg State University (grant
-no.~26520170 and~39417213).
-\end{acknowledgement}
-
-\bibliography{refs.bib}
-
-\end{document}
+\documentclass{webofc}
+
+\usepackage[varg]{txfonts}
+\usepackage{wrapfig}
+
+\begin{document}
+
+\title{Computational Model of Unsteady Hydromechanics\\of Large Amplitude Gerstner Waves}
+%\subtitle{Do you have a subtitle?\\ If so, write it here}
+
+\author{%
+\firstname{Alexander} \lastname{Degtyarev}\inst{1}%
+\and%
+\firstname{Ivan} \lastname{Gankevich}\inst{1}%
+\fnsep\thanks{\email{i.gankevich@spbu.ru}}%
+\and%
+\firstname{Nataliia} \lastname{Kulabukhova}\inst{1}%
+\and%
+\firstname{Vasily} \lastname{Khramushin}\inst{1,2}%
+}
+
+\institute{%
+Saint Petersburg State University, Universitetskaya
+Emb. 7-9 , 199034 St.~Petersburg, Russia%
+\and%
+Alexey Krylov All-Russian Scientific Shipbuilder Society, %with headquarters in
+Saint Petersburg, % Scientific Society of Shipbuilders named after Alexey Krylov,
+Russia%
+}
+
+\abstract{%
+The computational experiments in the ship fluid mechanics involve the
+non-stationary interaction of a ship hull with wave surfaces that include the
+formation of vortices, surfaces of jet discontinuities, and discontinuities in
+the fluid under the influence of negative pressure. These physical phenomena
+occur not only near the ship hull, but also at a distance where the waves break
+as a result of the interference of the sea waves with waves reflected from the
+hull. In the study reported here we simulate the wave breaking and reflection
+near the ship hull. The problem reduces to determining the wave kinematics on
+the moving boundary of a ship hull and the free boundary of the computational
+domain. We build a grid of large particles having the form of a parallelepiped
+and, in the wave equation instead of the velocity field we integrate streams of
+fluid represented by functions as smooth as the wave surface elevation field.
+We assume that within the boundaries of the computational domain the waves do
+not disperse, i.e.~their length and period stay the same. Under this
+assumption, we simulate trochoidal Gerstner waves of a particular period.
+This approach allows to simulate the wave breaking and reflection near the ship
+hull. The goal of the research is to develop a new method of taking the wave
+reflection into account in the ship motion simulations as an alternative to the
+classic method which uses added masses.}
+
+\maketitle
+
+
+\section{Introduction}
+
+The ship and sea wave surface motions do not involve an abundance of geometric
+forms and physical phenomena. The formation of waves due to the ship motion and
+the interaction between the ocean and the atmosphere are governed only by the
+continuity condition for a heavy fluid and the law of conservation of the
+energy.
+
+A strict theoretical solution (and in fact the only solution) for
+large-amplitude wind waves at the surface of a heavy fluid was obtained in 1802
+by Franz Josef von Gerstner~\cite{gerstner1809}. The generic trochoidal wave
+mathematical model has large dispersion~\cite{sommerfeld1945mechanik}, the
+speed of the wave propagation depends on their length and period. As a result
+\begin{itemize}\addtolength\itemsep{-1mm}
+ \item wave energy propagation speed becomes half the visible phase velocity
+ of the wave crests,
+ \item wave front constantly changes its phase, and
+ \item waves are quantised into packets and the phenomena of wave
+ transformation and propagation become nonstationary.
+\end{itemize}
+
+
+\section{Computational model of intense sea waves}
+
+\begin{wrapfigure}{l}{0.5\textwidth}
+ \centering
+ \includegraphics[width=0.5\textwidth]{graphics/01-gerstner.png}
+ \includegraphics[width=0.5\textwidth]{graphics/02-gerstner.png}
+ \caption{Analytic solutions: progressive Gerstner wave (top),
+ a wave of critical height, as a standing wave (bottom).\label{fig-gerstner}}
+\end{wrapfigure}
+
+The Gerstner wave (Fig.~\ref{fig-gerstner}) is a cycloid with the radius
+$r_W=1.134\lambda_Wh_W/4\pi$ of the partile trajectory being fixed relative to
+the flat wavy surface level \(z_W\), hence \(z\)-coordinates of the crest and
+trough are the same. Here \(\lambda_W\) is the wave length, \(h_W\) is the
+relative wave height defined on the interval \([0,1]\) with \(h_W=1\) being the
+maximum wave height for which the crest does not break
+(Fig.~\ref{fig-gerstner}). The vertical displacement of a fluid particle is
+given by
+
+\begin{equation*}
+ \zeta_Z = r_W \cos x_W \exp\left(-2\pi z_W / \lambda_W\right).
+\end{equation*}
+The horizontal displacement of the same fluid particle with respect to its
+initial position for progressive wave is given by an analogous equation, but
+with a shift by one fourth of the phase:
+\begin{equation*}
+ \zeta_X = -r_W \sin x_W \exp\left(-2\pi z_W / \lambda_W\right).
+\end{equation*}
+The critical wave height of the Gerstner waves (fig.~\ref{fig-gerstner}) gives
+the correct ratio of the wave height to the wave length, but 60 degree slope
+limit for standing wave with steepness \(\approx{}1/4\) as well as 30 degree
+slope limit for progressive (traveling) wave with steepness \(\approx{}1/7\)
+are not correctly captured by the model.
+
+\begin{wrapfigure}{r}{0.5\textwidth}
+ \centering
+ \includegraphics[width=0.5\textwidth]{graphics/03-wind-wave.png}
+ \includegraphics[width=0.5\textwidth]{graphics/05-wind-wave.png}
+ \caption{Simulation result: regular trochoidal waves with vertical
+ displacement of sea level and wind stress on the wave sea
+ surface. Propagating wind waves (top), extremely high wind waves
+ (bottom).\label{fig-trochoidal}}
+\end{wrapfigure}
+
+The main focus of state of the art mathematical and computational models that
+simulate wave groups is on stochastic properties rather than fluid
+mechanics~\cite{anastopoulos2016}. Our model is a modified version of the
+Gerstner wave which includes wave groups. They are described as a dependency
+between fluid particle trajectory radius and instantaneous displacement of the
+particle with respect to the calm sea level (Fig.~\ref{fig-trochoidal}).
+
+We write the adjusted radius as \(R^A_W={}K^A{}r_W\left(\cos{}x_W-1\right)\), where
+\(K^A\in[1,\sqrt{2}]\) is the radius coefficient that makes the wave crests cnoidal
+and raises the mean sea level. We choose \(K^A\) to be slightly less than 1 to
+reduce the effect of gusty winds on the wave form and to prevent the formation
+of cycloidal loops in the wave crests, that appear for waves with overly large
+amplitude, which may occur as a result of the interference with waves
+heading from the opposite direction.
+
+The pressure on the windward slope of the wave is smaller, because the wind
+slides on the surface of the wave at a high speed makes the slope flatter,
+while on the leeward slope of the wave the wind speed drops significantly or
+even comes to nought and creates vorticity.
+
+The coefficient of wind stress \(K^W\) (the parameter that was shifted by one
+fourth of the phase) determines the asymmetry of steepness of windward and
+leeward slopes of the wave (Fig.~\ref{fig-trochoidal}):
+\(R^W_W=K^Wr_W\left(\sin{}x_W-1\right)\), where \(K^W\in[0,1]\). The
+coefficient is close to unity for fresh wind waves and close to nought for
+swell. For a two-dimensional sea surface, \(K^W\) is used in dot product
+between the wind and wave direction vectors:
+\begin{equation*}
+ \begin{aligned}
+ \zeta_Z &= r_W \cos x_W \exp\left(
+ 2\pi \left[
+ -z_W + r_W K^A \left( \cos x_W - 1\right) + r_W K^W\sin x_W
+ \right] / \lambda_W
+ \right) \\
+ \zeta_X &= -r_W \sin x_W \exp\left(
+ 2\pi \left[
+ -z_W + r_W K^A \left( \cos x_W - 1\right) + r_W K^W\sin x_W
+ \right] / \lambda_W
+ \right).
+ \end{aligned}
+\end{equation*}
+
+The energy conservation is defined by the Bernoulli's principle:
+\(\rho V^2/2+\rho g \zeta_W=\text{const}\), [N/m$^{2}$],
+where the particle velocity \(V\) brings the largest contribution to balancing
+the pressure \(\rho{}g\zeta_W\) on the wave surface down to nought for breaking
+waves.
+
+\section{Trochoidal wave groups}
+
+\begin{wrapfigure}{l}{0.5\textwidth}
+ \centering
+ \includegraphics[width=0.5\textwidth]{graphics/06-group-wave.png}
+ \caption{Trochoidal wave groups.
+ \label{fig-group}}
+\end{wrapfigure}
+
+In our modified model we simulate two wavy surfaces simultaneously: one for
+regular waves with normal length and one for waves with nine times higher
+length, that propagate under the same laws but with half speed
+(Fig.~\ref{fig-group}). The product of these surfaces allows to simulate wave
+groups.
+
+On the first entry the profile of the long wave is given by a specific
+smoothing function the form of which is close to phase the wave profile. This
+function defines continuous changes of the wave front phase which is needed to
+simulate waves produced by the ship.
+
+\begin{wrapfigure}{r}{0.6\textwidth}
+ \centering
+ \includegraphics[width=0.6\textwidth]{graphics/waves-01.png}
+ \caption{Large-amplitude trochoidal waves.
+ \label{fig-waves-1}}
+\end{wrapfigure}
+
+There is also a simpler approach to simulate wave groups: a superposition of
+regular waves with slightly different periods propagating in opposite
+directions. The interference of waves of comparable lengths produces beats, in
+which ninth wave have double height and is standing wave. This approach
+generally gives satisfactory wavy surfaces, but does not work for waves
+produced by the ship, because these have intricate wave fronts.
+
+
+\section{Direct numerical simulation of sea waves}
+
+We use an explicit numerical scheme to simulate a sea wave surface that
+satisfies the continuity equation; we call it direct numerical simulation
+(Fig.~\ref{fig-waves-1}). We use the following definitions for three sea wave
+systems, that are used in the scheme.
+%\vspace{-0.5\baselineskip}
+\begin{itemize} \addtolength\itemsep{-1mm}
+ \item Fresh wind waves have a period of 6--8 seconds near the shore and up
+ to 10--12 seconds in the open ocean. The height of the wave is close to
+ critical, that typically corresponds to 6 on the Beaufort scale with
+ wave crests larger than 5--6 metres.
+
+ \item Fresh swell waves skew from mean wind direction by \(\approx{}30\)
+ degrees. When the storm in northern hemisphere increases, wind
+ direction goes counterclockwise and vice versa, i.e. the swell is
+ always present in the ocean. Swell waves are comparable to wind waves:
+ their height is two times smaller than the critical wave height, and
+ their length is 1.5--2 times larger.
+
+ \item Old swell waves are long waves that come from higher latitudes. Their
+ height is two times smaller than than of the wind waves and fresh
+ swell, their length is two times larger, and their direction is close
+ to meridional (i.e.~south in northern latitudes and vice
+ versa\footnote{The wind blows into the compass rose, the waves
+ propagate in the direction of the rose.}).
+
+\end{itemize}
+
+\begin{wrapfigure}{l}{0.55\textwidth}
+ \centering
+ \includegraphics[width=0.55\textwidth]{graphics/07-exp-1.png}
+ \caption{In the course of the simulation we visualise all three
+ wave systems and create a view of ship hull dynamics
+ and sea wave profiles in a different convenient scale.
+ \label{fig-waves-2}}
+\end{wrapfigure}
+
+These waves may add up in unfavourable way to a wave with the height of 13--15
+metres, however, in real world mean wave height will be 8--10 metres. Wave
+groups have ninth wave with double height, breaking crest and wave slope larger
+than 45 degrees.
+
+Oceanographers use well-established solutions~\cite{poplavskii1997} for regular
+progressive waves of arbitrary shape. Using trochoidal waves as a source, we
+fix wave periods and speeds in time to satisfy continuity equation and
+energy conservation law. We simulate all three wave systems (described above)
+simultaneously and indepedently (Fig.~\ref{fig-waves-2}) and add individual
+wave surfaces together to produce the resulting wave surface.
+
+\vspace{-0.5\baselineskip}
+
+\section{Conclusion}
+
+We use explicit numerical schemes to simulate a modified version of the
+Gerstner waves. We simulate particle drift in the upper fluid layers by
+changing the curvature of the trajectory depending on the instantaneous change
+of the wave surface elevation. Our model is nonstationary, hence ship motions
+can also be nonstationary. The computational power of a desktop computer is
+enough for performing such simulations in real-time, and these types of
+simulations can even be performed on the board of the ship to chose efficiently
+the optimal mode of ship operation.
+
+%\begin{acknowledgement}
+
+\vspace{-0.5\baselineskip}
+
+\subsection*{Acknowledgement}
+
+\vspace{-0.5\baselineskip}
+
+Research work is supported by Saint Petersburg State University (grant
+no.~26520170 and~39417213).
+%\end{acknowledgement}
+
+\vspace{-0.5\baselineskip}
+
+\bibliography{refs.bib}
+
+\end{document}
diff --git a/refs.bib b/refs.bib
@@ -8,7 +8,8 @@
pages = {412--445},
doi = {10.1002/andp.18090320808},
url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/andp.18090320808},
- year = {1809}
+ year = {1809},
+ note = {Republished from an 1802 paper.}
}
@Book{ sommerfeld1945mechanik,
@@ -36,3 +37,20 @@
year = {1997},
note = {in Russian}
}
+
+@Article{ anastopoulos2016,
+ title = {Towards an improved critical wave groups method for the
+ probabilistic assessment of large ship motions in irregular
+ seas},
+ journal = {Probabilistic Engineering Mechanics},
+ volume = {44},
+ pages = {18--27},
+ year = {2016},
+ note = {Special Issue Based on Papers Presented at the 7th
+ International Conference on Computational Stochastic Mechanics
+ (CSM7)},
+ issn = {0266-8920},
+ doi = {10.1016/j.probengmech.2015.12.009},
+ author = {Panayiotis A. Anastopoulos and Kostas J. Spyrou and
+ Christopher C. Bassler and Vadim Belenky},
+}
