commit bffadcad85a45364afd0783ca1c8b65607a15fd6
parent c6c94b450d0058d102b792c10f06c6f98e8a2c9e
Author: Ivan Gankevich <igankevich@ya.ru>
Date: Sun, 21 Jul 2019 11:32:25 +0300
finish
Diffstat:
| main.tex | | | 53 | ++++++++++++++++++++++++++++++++++++++++++++++++++--- |
1 file changed, 50 insertions(+), 3 deletions(-)
diff --git a/main.tex b/main.tex
@@ -200,7 +200,12 @@ which ninth 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 for the simulation of sea waves}
+\section{Direct numerical simulation of sea waves}
+
+We use explicit numerical schemes 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 sea wave
+systems.
\begin{figure}
\centering
@@ -208,19 +213,61 @@ by the ship, because they have complex wave front.
\caption{Contours of high-amplitude trochoidal wave profiles.\label{fig-waves-1}}
\end{figure}
+\begin{itemize}
+
+ \item Fresh wind waves have a period of 6-8 seconds for 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 tow times greater, and their direction is close
+ 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 ninth wave with double height, breaking crest and wave slope
+greater than 45 degrees.
+
\begin{figure}
\centering
- \includegraphics[width=\textwidth]{graphics/07-exp-1.png}
+ \includegraphics[width=0.9\textwidth]{graphics/07-exp-1.png}
\caption{In the course of the simulation we visualise all three
monochromatic fields and their overall representatation in
interaction with ship hull 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 wave speed 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).
