commit 7b849fc4d1adb299062925df54b23c1b5c0ca74d
parent 43dd6794db49b1181b0691fc1232fc3b029b31b4
Author: Ivan Gankevich <igankevich@ya.ru>
Date: Sat, 23 Mar 2019 11:57:26 +0300
Pressure.
Diffstat:
| main.tex | | | 77 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
| references.bib | | | 10 | ++++++++++ |
2 files changed, 87 insertions(+), 0 deletions(-)
diff --git a/main.tex b/main.tex
@@ -1,6 +1,16 @@
\documentclass[runningheads]{llncs}
\usepackage{graphicx}
+\usepackage{amsmath}
+
+\newcommand{\WaveVector}{\vec{k}}
+\newcommand{\WaveNumber}{\lvert\smash[b]{\WaveVector}\rvert}
+\newcommand{\Cosh}[1]{\cosh\left(#1\right)}
+\newcommand{\Fourier}[2]{\mathcal{F}_{#2}\!\left\{#1\right\}}
+\newcommand{\InverseFourier}[2]{\mathcal{F}^{-1}_{#2}\!\left\{#1\right\}}
+% properly aligned version of sqrt for \zeta_y^2
+\newcommand{\SqrtZeta}[1]{\sqrt{\vphantom{\zeta_x^2}\smash[b]{#1}}}
+\newcommand{\SqrtZetaXY}{\SqrtZeta{1 + \zeta_x^2 + \zeta_y^2}}
\begin{document}
@@ -176,7 +186,74 @@ generate it using graphical accelerator, and for other models it is to trivial
to discuss. This field is an input for velocity potential solver.
\subsection{Velocity potential computation}
+
+Since wavy surface generator produces discretely given elevation field we may
+not use formula from linear wave theory to compute velocity potential; instead,
+we derived a formula for arbitrary surface for inviscid incompressible fluid:
+\begin{equation}
+ \label{eq:phi}
+ \phi(x,y,z,t) = \InverseFourier{
+ \frac{ \Cosh{\smash{2\pi \WaveNumber (z+h)}} }{ 2\pi\WaveNumber }
+ \frac{ \Fourier{ f(x,y,t) }{u,v} }
+ { \Fourier{\mathcal{D}_3\left( x,y,\zeta\left(x,y\right) \right)}{u,v} }
+ }{x,y},
+\end{equation}
+where
+\begin{align*}
+ f(x,y,t)&=\zeta_t(x,y,t) / \left( i f_1(x,y) + i f_2(x,y) - f_3(x,y) \right),\\
+ f_1(x,y)&={\zeta_x}/{\SqrtZetaXY}-\zeta_x,
+ \quad
+ \Fourier{\mathcal{D}_3\left(x,y,z\right)}{u,v}=\Cosh{\smash{2\pi\WaveNumber{}z}},\\
+ f_2(x,y)&={\zeta_y}/{\SqrtZetaXY}-\zeta_y,
+ \quad
+ \WaveNumber = \sqrt{u^2+v^2},\\
+ f_3(x,y)&=1/\SqrtZetaXY.
+\end{align*}
+Here \(\WaveVector\) is wave number, \(\zeta\)~--- wavy surface elevation,
+\(h\)~--- water depth, \(\mathcal{F}\)~--- Fourier transform, \(\phi\)~---
+velocity potential. The formula is derived as a solution for continuity
+equation with kinematic boundary condition
+\begin{align}
+ & \nabla^2\phi = 0,\nonumber\\
+ & \phi_t+\frac{1}{2} |\vec{\upsilon}|^2 + g\zeta=-\frac{p}{\rho}, & \text{at }z=\zeta(x,y,t),\label{eq:problem}\\
+ & D\zeta = \nabla \phi \cdot \vec{n}, & \text{at }z=\zeta(x,y,t),\nonumber
+\end{align}
+without assumptions of linear wave theory (wave length is much larger than wave
+height). Hence it can be used for arbitrary-amplitude ocean waves. Here the
+first equation is continuity equation, the second is dynamic boundary
+condition, and the last one is kinematic boundary condition; \(p\)~---
+pressure, \(\rho\)~--- fluid density,
+\(\vec{\upsilon}=(\phi_x,\phi_y,\phi_z)\)~--- velocity vector, \(g\)~---
+acceleration of gravity, and \(D\)~--- substantial (Lagrange) derivative. Since
+we solve for \(\phi\), dynamic boundary condition becomes explicit formula for
+pressure and is used to compute pressure force acting on a ship hull
+(see~sec.~\ref{sec:pressure-force}).
+
+Formula~\eqref{eq:phi} converges when summation goes over a range of wave
+numbers that are actually present in discretely given wavy surface. This range
+is determined numerically by finding crests and troughs for each spatial
+dimension of the wavy surface with polynomial interpolation and using these
+values to determine wave length. For small-amplitude waves this approach gives
+the same values of velocity potential field as direct application of the
+formula from linear wave theory.
+
+Formula~\eqref{eq:phi} is particularly suitable for computation on a graphical
+acelerator: it contains transcendental mathematical functions (complex
+exponents) that help offset slow global memory loads and stores, it is explicit
+which makes it easy to compute in parallel and it is written using Fourier
+transforms that are efficient to compute on a graphical
+accelerator~\cite{volkov2008fft}.
+
+This paper gives only a short description of the method, please refer
+to~\cite{gankevich2018thesis,gankevich2018ocean} for in-depth study.
+
\subsection{Pressure force computation}
+\label{sec:pressure-force}
+
+Wave pressure at any point under wavy surface is computed using dynamic
+boundary condition from~\eqref{eq:problem} as an explicit formula.
+
+\subsection{Translational and angular motion computation}
\section{Results}
diff --git a/references.bib b/references.bib
@@ -44,6 +44,16 @@
doi = {10.1007/s10055-008-0088-8}
}
+@TechReport{ volkov2008fft,
+ title = {Fitting {FFT} onto the {G80} architecture},
+ author = {Volkov, Vasily and Kazian, Brian},
+ institution = {University of California},
+ year = {2008},
+ month = {May},
+ address = {Berkeley},
+ number = {6}
+}
+
@InProceedings{ varela2011interactive,
title = {Interactive Simulation of Ship Motions in Random Seas based
on Real Wave Spectra.},
