mmcp-19-gerstner

main.tex (12748B)

---

 \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{(Color online) 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 particle trajectory being fixed relative to
 the flat wave 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{(Color online) 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:
 \(\rho V^2/2+\rho g \zeta_W=\text{const}\),
 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{(Color online) Trochoidal wave groups.
 	\label{fig-group}}
 \end{wrapfigure}
 
 In our modified model we simulate two wave 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 the phase 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{(Color online) 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 the ninth wave has double height and is standing wave. This approach
 generally gives satisfactory wave 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{(Color online) 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 the ninth wave with double height, breaking crest and wave slope larger
 than 45 degrees.
 
 Oceanographers use well-estab\-lished 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}