56 lines
1.9 KiB
TeX
56 lines
1.9 KiB
TeX
\chapter{Introduction}
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\label{s:introduction}
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\section{Objective}
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The objective of this document is introduce briefly how the waves of the
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seakeeping simulator are propagated.\rc
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In the seakeeping simulator Boundary Elements Method (BEM) will be used,
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that is detailed described in several books, like \citet{bem_2007}.
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\citet{vinayan2007} gives a detailed description of the propagation
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of waves in a 2D case, and is a good starting point.\rc
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We will start briefly describing the governing equations in order to can
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start working with the 2D problem. First the incident waves over our
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computational domain will be described, introducing also the potential,
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discussing then the BEM applied to this case. As we will see the Laplace
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problem in the 2D case will not be really useful for us.\rc
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After that we can start working in the 3D case, that is our real objective.
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The incident waves will be rewritten, and the Laplace problem and the BEM
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application purposed again.
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\chapter{Governing equations}
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\label{s:governing_equations}
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Assuming no viscous fluid (that allows to transform Navier-Stokes
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equations into Euler ones), and imposing an initial condition such
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that\footnote{With no viscous fluid this condition is preserved along the
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time}:
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\begin{eqnarray}
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\rotational \bs{u} = 0
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\end{eqnarray}
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The fluid velocity derives from a scalar function potential $\phi$
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\begin{eqnarray}
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\label{eq:governing_equations:v_potential}
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\gradient \phi = \bs{u}
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\end{eqnarray}
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Then the Navier-Stokes equations can be rewriten as a Laplacian problem
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and Bernoulli equation:
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\begin{eqnarray}
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\label{eq:governing_equations:laplace}
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\laplacian \phi = & 0
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\\
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\label{eq:governing_equations:bernoulli}
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p = & - \rho \left( \vert \bs{u} \vert^2 + g \bs{z} \right)
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\end{eqnarray}
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And in order to solve the Laplace problem \ref{eq:governing_equations:laplace}
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we will use the BEM as described by \citet{bem_2007}.
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