commit bb82b11862e21441f807aa4eb6aa38aa9c710815
parent bffadcad85a45364afd0783ca1c8b65607a15fd6
Author: Ivan Gankevich <igankevich@ya.ru>
Date: Sun, 21 Jul 2019 21:14:20 +0300
proof-read
Diffstat:
| main.tex | | | 49 | +++++++++++++++++++++++-------------------------- |
1 file changed, 23 insertions(+), 26 deletions(-)
diff --git a/main.tex b/main.tex
@@ -60,15 +60,14 @@ that uses added masses.
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: wind waves have uniform energy
-distribution.
+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 high
+Generic trochoidal wave mathematical model has large
dispersion~\cite{sommerfeld1945mechanik}, the dependence of the speed of wave
-crest propagation on their length and period. As a result
+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,
@@ -88,19 +87,19 @@ crest propagation on their length and period. As a result
(bottom).\label{fig-gerstner}}
\end{figure}
-Gerstner wave (fig.~\ref{fig-gerstner}) is a cycloid, fluid parcel trajectory
+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 \(z_W\)~--- flat wavy surface level, hence \(z\)-coordinates of the
+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 parcel
+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 parcel with respect to its initial
+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*}
@@ -116,8 +115,8 @@ 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 parcel trajectory radius and instantaneous
-displacement of the parcel with respect to calm sea level
+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}
@@ -125,7 +124,7 @@ displacement of the parcel with respect to calm sea level
\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 tension on the wave sea
+ 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}
@@ -139,7 +138,7 @@ 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 high speed and makes the slope more flat, while 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.
@@ -148,7 +147,7 @@ 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. In two-dimensional sea surface \(^{W}K\) is used in dot product
+swell. For two-dimensional sea surface \(^{W}K\) is used in dot product
between wind and wave direction vectors:
\begin{equation*}
\begin{aligned}
@@ -184,8 +183,7 @@ groups.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{graphics/06-group-wave.png}
- \caption{Phase wave product with packet groups contours of wave
- field.\label{fig-group}}
+ \caption{Trochoidal wave groups.\label{fig-group}}
\end{figure}
On the first entry the profile of the long wave is given by specific smoothing
@@ -196,26 +194,26 @@ 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 ninth waves has double height and are standing waves. This approach
+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 schemes to simulate sea wavy surface that satisfies
+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 sea wave
-systems.
+(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{Contours of high-amplitude trochoidal wave profiles.\label{fig-waves-1}}
+ \caption{Large-amplitude trochoidal waves.\label{fig-waves-1}}
\end{figure}
\begin{itemize}
- \item Fresh wind waves have a period of 6-8 seconds for near the shore
+ \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.
@@ -229,7 +227,7 @@ systems.
\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
+ 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.}).
@@ -238,21 +236,20 @@ systems.
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
+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
- monochromatic fields and their overall representatation in
- interaction with ship hull and create a view of ship hull dynamics
+ 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 wave speed in time to satisfy continuity equation and
+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.
