\documentclass{article}
\usepackage{fullpage}
\usepackage{parskip}
\usepackage{titlesec}
\usepackage{xcolor}
\usepackage[colorlinks = true,
linkcolor = blue,
urlcolor = blue,
citecolor = blue,
anchorcolor = blue]{hyperref}
\usepackage[natbibapa]{apacite}
\usepackage{eso-pic}
\AddToShipoutPictureBG{\AtPageLowerLeft{\includegraphics[scale=0.7]{powered-by-Authorea-watermark.png}}}
\renewenvironment{abstract}
{{\bfseries\noindent{\abstractname}\par\nobreak}\footnotesize}
{\bigskip}
\titlespacing{\section}{0pt}{*3}{*1}
\titlespacing{\subsection}{0pt}{*2}{*0.5}
\titlespacing{\subsubsection}{0pt}{*1.5}{0pt}
\usepackage{authblk}
\usepackage{graphicx}
\usepackage[space]{grffile}
\usepackage{latexsym}
\usepackage{textcomp}
\usepackage{longtable}
\usepackage{tabulary}
\usepackage{booktabs,array,multirow}
\usepackage{amsfonts,amsmath,amssymb}
\providecommand\citet{\cite}
\providecommand\citep{\cite}
\providecommand\citealt{\cite}
% You can conditionalize code for latexml or normal latex using this.
\newif\iflatexml\latexmlfalse
\AtBeginDocument{\DeclareGraphicsExtensions{.pdf,.PDF,.eps,.EPS,.png,.PNG,.tif,.TIF,.jpg,.JPG,.jpeg,.JPEG}}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\begin{document}
\title{Numerical Flow Simulation\\
Initial Report}
\author[ ]{Son Pham-Ba}
\author[ ]{Tristan Revaz}
\affil[ ]{}
\vspace{-1em}
\date{}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\emph{Abstract}\\
This initial report serves as a background structure for the later final report. It does not claim to be complete, as it only demonstrate that the chosen mesh and physical models are verified, and just need to be refined before the post-processing.
\section{Introduction}
When a boat hull is set into motion on still water, it generates a wave pattern, resulting in the creation of a wave drag opposing the boat motion. The Wigley hull is a simplified parabolic boat geometry, defined analytically. It has been studied a lot, both experimentally and numerically, in various flow conditions. That's why it is used to provide verification and validation when testing a new numerical simulation algorithm for example.
\section{State of the art}
\subsection{Experiments}
\cite{kajitani} gives measured physical quantities obtained during various experiments on different sizes of the Wigley hull, especially on one of the same size as the one studied in this report. This paper will be useful to validate our computed data.
\subsection{Numerical simulations}
\subsubsection{Computationnal mesh}
A lot of types of meshes can be found in the literature. \cite{ciortan} and \cite{beddhu} both use a structured mesh refined around the hull and the water free surface.
\section{Problem definition}
In this mini-project, we will do the analysis of the flow around a Wigley hull, shown in Figure \ref{fig:wigley}. Its shape is defined by the equation:
\begin{equation*}
y(x, z) = \frac{B}{2} \left\{ 1 - \left( \frac{2x}{L} \right)^2 \right\} \left\{ 1 - \left( \frac{z}{T} \right)^2 \right\}
\label{eqn:wigley}
\end{equation*}
with the hull length $L = 2.5\text{m}$, the hull beam $B = 0.25\text{m}$ and the hull draught $T = 0.15625\text{m}$. The equation describes the wetted section of the hull where $z<0$, while the above water geometry is obtained by extending the shape vertically. The wetted surface area is $S = 0.929\text{m}^2$\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.7\columnwidth]{figures/wigley/wigley}
\caption{{Four views of a Wigley hull geometry
\label{fig:wigley}%
}}
\end{center}
\end{figure}
The origin of the boat coordinate system is located at the water surface ($z=0$). The boat moves in the positive $x$ direction, so the flow is in the negative $x$ direction in the boat frame of reference.\\
The domain of computation correspond to the towing tank of following dimensions:
\begin{align*}
-6.25\text{m} & < x < 3.75\text{m}\\
0 & < y < 2.5\text{m}\\
-2.5\text{m} & < z < 0.15625\text{m}
\end{align*}
The tank walls are considered to be stationary, with the top surface open, and a symmetry about the $y=0$ plane is assumed.\\
The boat is subject to two freestream flow, which have the following experimental values:
\begin{itemize}
\item inflow velocity: $U_0 = 1.322\text{m/s}$ and $U_0 = 1.753\text{m/s}$ (i.e. $Fr = 0.267$ and $Fr = 0.354$),
\item density of water: $\rho_w = 998.2 \text{kg/m}^3$,
\item density of air: $\rho_a = 1.225 \text{kg/m}^3$,
\item pressure: $p_w(z) = p_{w0} - \rho_w g z$, where $p_{w0} = 1.013 \cdot 10^5\text{Pa}$,
\item kinematic viscosity of water: $\nu_w = 1.00 \cdot 10^{-6}\text{m}^2/\text{s}$,
\item kinematic viscosity of air: $\nu_a = 1.46 \cdot 10^{-5} \text{m}^2/\text{s}$,
\item turbulence intensity: $1\%$,
\item gravity: $g_z = -9.81 \text{m/s}^2$.
\end{itemize}
The flow is comprised of two immiscible phases: a water region and an air region. The Volume of Fluid (VOF) method will be applied to determine the interface between the two phases.
Our goal is to compute the physical properties of the flow around the boat, like the pressure and velocity fields, the height of the water, the drag coefficient and the pressure coefficient and compare them with experimental data available.
\section{Techniques employed}
\subsection{Computationnal mesh}
Multiple meshes will be used throughout this mini-project. The first mesh is represented in the figure \ref{fig:mesh1}. It is a structured mesh with $80 \times 10 \times 16 = 12800$ elements. It is refined near the hull and the water free surface. The cell size near the walls has not yet been adapted to correspond to the turbulence model.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.7\columnwidth]{figures/mesh1/mesh1}
\caption{{Initial mesh
\label{fig:mesh1}%
}}
\end{center}
\end{figure}
\subsection{Physical model}
A Volume of Fluid (VOF) method is used to simulate the two phase flow. As stated in the Problem definition, the two phases are the air and the water. We define $\alpha$ as the volume fraction of water, so that a value of $\alpha = 1$ correspond to the water phase and a value of $\alpha = 0$ to the air phase. The interface between the phases is the surface where $\alpha = 0.5$.\\
The turbulence model is a standard k-$\epsilon$. The near-wall treatment of the flow has not yet been studied in our firsts simulations. We then plan to use a shear-stress transport k-$\omega$ model.
\subsection{Numerical method}
We used two approaches to solve the problem. We first tried to solve it with a direct computation of the steady state, with a SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) pressure based solver. We then considered a transient flow and computed it until a steady state was reached. In that case, we used a fractional time-step solver.\\
In both cases, we impose an open channel boundary condition at the inlet with a fixed mean velocity of the flow. On the hull, a no-slip condition is imposed. On the walls, we impose a no-shear condition so that they do not oppose to the flow motion by inducing drag. Finally, a symmetry is forced at $y=0$.
\section{Results and discussion}
The figures \ref{fig:water_steady} and \ref{fig:water_transient} show the wave height for both the steady and the transient solutions for an inflow velocity of $U_0 = 1.322\text{m/s}$.\\
We see that in the steady case, large waves are generated behind the hull. The result seems to be more realistic in the case given by the unsteady calculation.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.7\columnwidth]{figures/Steady-OpenChan-Pressure/Steady-OpenChan-Pressure}
\caption{{Wave height obtained from the steady solution
\label{fig:water_steady}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.7\columnwidth]{figures/Transient-OpenChan-Pressure/Transient-OpenChan-Pressure}
\caption{{Wave height obtained form the transient solution
\label{fig:water_transient}%
}}
\end{center}
\end{figure}
The figure \ref{fig:graph} compare the wave height along the hull of both solutions with the experimental data from \cite{kajitani}.\\
We see that the wave locations are the same as the experimental data for both cases. However their amplitude is not the same, as they appear to be smaller.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.7\columnwidth]{figures/Initial/Initial}
\caption{{Normalized wave heights along the hull for $U_0 = 1.322\text{m/s}$
\label{fig:graph}%
}}
\end{center}
\end{figure}
\section{Proposed improvements}
The initial mesh is very coarse. It may be the reason of the inaccuracies found when comparing our results with the experimental data. A general refinement of the mesh will be done.\\
The cell size at the boundary of the hull does not has the right size according to the turbulence model. This value will be extracted from the solutions to be used in the next mesh.
\section{Conclusion}
For the moment, our results show that we are on a good way to obtain data in accordance with the experimental data. We still have to go on with the simulations and then verify and validate our final results.
\section*{References}
\section*{Appendix}
\section*{Time expenditure}
\selectlanguage{english}
\FloatBarrier
\bibliographystyle{apacite}
\bibliography{bibliography/converted_to_latex.bib%
}
\end{document}