commit c91e6b4e53f1293fbcfa40e52848c2ac30690116
parent 5ea1f8d33c5786f213a03c5a5bb5e304dcb36ea1
Author: Ivan Gankevich <i.gankevich@spbu.ru>
Date: Fri, 13 Mar 2020 16:12:16 +0300
Solutions.
Diffstat:
| main.tex | | | 161 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
1 file changed, 161 insertions(+), 0 deletions(-)
diff --git a/main.tex b/main.tex
@@ -5,6 +5,8 @@
\usepackage{graphicx}
\usepackage{url}
+\newcommand{\Length}[1]{\big|#1\big|}
+
\begin{document}
\title{TODO\thanks{Supported by Saint Petersburg State University (grants
@@ -94,10 +96,169 @@ wind velocity on the ship roll angle and carry out computational performance
analysis of our programme.
\section{Methods}
+
+\subsection{Analytic representation of wind velocity field}
+
+Air motion without turbulence can be decomposed into two components:
+translational motion~--- air particles travel in the same direction with
+constant velocity, circular motion~--- air particles travel on a circle.
+Translational motion describe sea breeze, that occures on the shore on the
+sunrise and after the sunset. Rotational motion describe storms suchs as
+typhoons and hurricane. Translational motion is a particular case of circular
+motion when the radius of the circle is infinite. Given the scale of circular
+motion relative to the scale of translational motion, and the size of a typical
+ship hull we consider only translational motion in this paper.
+
+Since there is no rotational component, air flow is described by equations for
+irrotational inviscid incompressible fluid. In this context fluid velocity
+\(\vec\upsilon\) is determined as a vector gradient \(\vec\nabla\) of scalar
+velocity potential \(\phi\) and continuity equation and equation of motion are
+written as
+\begin{equation}
+ \label{eq-governing}
+ \begin{aligned}
+ & \Delta\phi = 0; \qquad \vec\upsilon=\vec\nabla\phi; \\
+ & \rho\frac{\partial\phi}{\partial{}t} +
+ \frac{1}{2}\rho\Length{\vec\nabla\phi}^2 +
+ p + \rho g z = p_0.
+ \end{aligned}
+\end{equation}
+Here \(p_0\) is atmospheric pressure, \(g\) is gravitational acceleration,
+\(\rho\) is air density, \(p\) is pressure. We seek solution to this system
+of equations for velocity potential \(\phi\). Continuity equation restricts
+the type of the function that can be used as the solution, and equation of
+motion gives the pressure for a particular velocity potential value.
+
+Ship hull boundary is defined by a parametric surface \(\vec{S}\) and surface
+normals \(\vec{n}\):
+\begin{equation*}
+\vec{S}=\vec{S}\left(a,b,t\right)
+\qquad
+a,b\in{}A=[0,1];
+\qquad
+\vec{n}=\frac{\partial\vec S}{\partial a} \times \frac{\partial\vec S}{\partial b}
+\end{equation*}
+The simplest parametric surface is infinite plane which has the same normal in
+each point. The computer model of a real ship hull is composed of many panels
+with different areas and different orientations that approximate continuous
+surface. On the boundary the projection of wind velocity on the surface normal
+is nought:
+\begin{equation}
+ \label{eq-boundary}
+ \vec\nabla\phi\cdot\vec{n} = 0;
+ \qquad
+ \vec{r} = \vec{S}.
+\end{equation}
+
+The solutions to the governing system of equations differ in how boundary is
+incorporated in them: in our model the boundary is taken into account by adding
+velocity of a reflected air particle in the solution. Velocity
+\(\vec\upsilon_r\) of the particle that is reflected from the surface with
+surface normal \(\vec{n}\) is given by the law of reflection
+(fig.~\ref{fig-law-of-reflection}):
+\begin{equation}
+ \label{eq-reflected}
+ \vec\upsilon_r = \vec\upsilon - 2\left(\vec\upsilon\cdot\vec{n}\right)\vec{n}.
+\end{equation}
+When we add velocity of incident and reflected air particles we get a vector
+that is parallel to the boundary. As we move away from the boundary its impact
+on the velocity decays quadratically with the distance. The known analytic
+solution for the potential flow around a cylinder contains similar term:
+\begin{equation*}
+ \phi\left(r,\theta\right) = U r \left( 1 + \frac{R^2}{r^2} \right) \cos\theta.
+\end{equation*}
+Here \(r\) and \(\theta\) are polar coordinates, \(R\) is cylinder radius and \(U\)
+is velocity magnitude.
+
+\begin{figure}
+ \centering
+ %\includegraphics{}
+ \caption{\label{fig-law-of-reflection}}
+\end{figure}
+
+In the following subsections we describe the solution that we obtained for the
+velocity field \emph{on} the boundary and \emph{near} the boundary.
+
+\subsection{Uniform translational motion on the static body surface}
+
+On the surface we neglect the impact of neighbouring panels on the velocity
+field on the ground that the real ship hull surface is smooth,
+i.e.~neighbouring panels have approximately the same normals. This assumption
+does not hold for aft and bow of some ships, and, as a result, velocity field
+near these features has stream lines with sharp edges. We consider this effect
+negligible for the determination of roll angle caused by the wind, since the
+area of panels that distort wind field is small compared to the area of all
+other panels.
+
+We seek solution to the governing system of equations~\eqref{eq-governing} with
+boundary condition~\eqref{eq-boundary} of the form
+\begin{equation*}
+\phi = \vec\upsilon\cdot\vec{r}
++ C \left(\vec\upsilon_r\cdot\vec{r}\right);
+\qquad
+\vec{r}=\left(x,y,z\right),
+\end{equation*}
+Here \(\vec{r}\) is spatial coordinate, \(C\) is the coefficient, and
+\(\vec\upsilon_r\) is velocity of reflected air particle defined
+in~\eqref{eq-reflected}. This solution is independent for each panel.
+Plugging the solution into boundary condition~\eqref{eq-boundary} gives
+\begin{equation*}
+\left(\vec\upsilon + C\vec\upsilon_r\right)\cdot\vec{n} = 0,
+\end{equation*}
+hence
+\begin{equation*}
+C = -\frac{ \vec\upsilon\cdot\vec{n} }{ \vec\upsilon_r\cdot\vec{n} } = 1
+\end{equation*}
+and velocity is written simply as
+\begin{equation}
+\vec\nabla\phi = \vec\upsilon + \vec\upsilon_r.
+\end{equation}
+
+This solution satisfies continuity equation. It gives velocity only at the
+centre of each ship hull panel, but this is sufficient to calculate pressure
+and force moments acting on the ship hull.
+
+\subsection{Uniform translational motion near the static body surface}
+
+Near the surface there are no neighbouring panels, the impact of which we can
+neglect, instead we add reflected particle velocities for all the panels and
+decay the velocity quadratically with the distance to the panel. Here we can
+neglect panels surface normals of which has large angles with the wind
+direction for efficiency, but they do not blow up the solution.
+
+We seek solution of the form
+\begin{equation*}
+\phi = \vec\upsilon\cdot\vec{r}
++ C \frac{\vec\upsilon_r\cdot\vec{r}}{1+\Length{\vec{r}-\vec{S}}^2};
+\qquad
+\vec\upsilon_r = \vec\upsilon - 2\left(\vec\upsilon\cdot\vec{n}\right)\vec{n};
+\qquad
+\vec{r}=\left(x,y,z\right),
+\end{equation*}
+where \(\Length{\cdot}\) is vector length. Plugging the solution into boundary
+condition gives the same coefficient \(C=1\), but velocity vector is written differently
+as
+\begin{equation*}
+\vec\nabla\phi =
+\vec\upsilon +
+\left(
+ \frac{1}{s} \vec\upsilon_r
+ - \frac{2}{s^2} \left(\vec\upsilon_r\cdot\vec{r}\right) \left(\vec{r}-\vec{S}\right)
+\right);
+\qquad
+s = 1+\Length{\vec{r}-\vec{S}}^2.
+\end{equation*}
+Besides the term for reflected air particle velocity that decays quadratically with
+the distance to the panel, there is a term that decays quaternary with the distance and
+that can be neglected because of this.
+
+
\section{Results}
\section{Discussion}
\section{Conclusion}
+Future work is to include circular motion in the model.
+
\subsubsection*{Acknowledgements.}
Research work is supported by Saint Petersburg State University (grants
no.~TODO) and Council for grants of the President of the Russian Federation
