commit 53199fcfd1024b62726397f0b2e27499cb5a052b
parent c27b0ff48c020883d523f5fc25857f2ee8cfd60d
Author: Ivan Gankevich <igankevich@ya.ru>
Date: Mon, 29 May 2017 17:02:25 +0300
Edit introduction.
Diffstat:
| arma.org | | | 149 | +++++++++++++++++++++++++++++++++---------------------------------------------- |
1 file changed, 62 insertions(+), 87 deletions(-)
diff --git a/arma.org b/arma.org
@@ -25,17 +25,18 @@ instance of which is included in the paper.
#+end_abstract
* Introduction
-Any research related to the study of the behaviour of a ship at sea, requires a
-description of wind waves, which are the major disturbance that displaces the
-vessel from equilibrium. Currently, the most popular models for describing wind
-waves, are models based on the linear expansion of a stochastic moving surface
-as a system of independent random variables. These include models by St. Denis &
-Pearson\nbsp{}cite:st1953motions, Rosenblatt\nbsp{}cite:rosenblatt1956random,
+Studying behaviour of a ship at sea is often based on some model of external
+excitations\nbsp{}--- any disturbance that displaces the vessel from
+equilibrium\nbsp{}--- major component of which is wind waves. Currently, the
+most popular sea wave simulation models are based on the linear expansion of a
+stochastic moving surface as a system of independent random variables. Such
+models were studied by St. Denis & Pearson\nbsp{}cite:st1953motions,
+Rosenblatt\nbsp{}cite:rosenblatt1956random,
Sveshnikov\nbsp{}cite:sveshnikov1959determination, and
Longuet---Higgins\nbsp{}cite:longuet1957statistical. The most popular model is
-that of Longuet---Higgins, which is based on a stochastic approximation of the
-moving wave front as a superposition of elementary harmonic waves with random
-phases \(\epsilon_n\) and random amplitudes \(c_n\):
+that of Longuet---Higgins (LH), which approximates propagating sea waves as a
+superposition of elementary harmonic waves with random phases \(\epsilon_n\) and
+random amplitudes \(c_n\):
\begin{equation}
\label{eq-lhmodel}
\zeta(x,y,t) = \sum\limits_n c_n \cos(u_n x + v_n y - \omega_n t + \epsilon_n),
@@ -45,7 +46,7 @@ plane, i.e. the unit area \(du \times dv\) contains an infinite number of wave
numbers. The frequency \(\omega_n\) associated with wave numbers \((u_n,v_n)\)
is given by a dispersion relation
\begin{equation*}
- \omega_n = \omega(u_n,v_n).
+ \omega_n = \omega(u_n,v_n).
\end{equation*}
The phase \(\epsilon_n\) are jointly independent random variables uniformly
distributed in the interval \([0,2\pi]\).
@@ -54,99 +55,69 @@ Longuet---Higgins showed that under the above conditions, the function
\(\zeta(x,y,t)\) is a three-dimensional steady-state homogeneous ergodic
Gaussian field, defined by
\begin{equation*}
- 2 E_{\zeta}(u,v) du dv = \sum\limits_{n} c_n^2,
+ 2 E_{\zeta}(u,v) du dv = \sum\limits_{n} c_n^2,
\end{equation*}
-where \(E_{\zeta}(u,v)\) is two dimensional spectral density of wave energy.
-
-Longuet---Higgins model is simple and easily computed. It incorporates the
-physical fundamentals of the process of wind waves and is fully consistent with
-the task of modelling ocean waves.
-
-Indeed, consider one of the simplest general conservation laws, the law of
-continuity:
-\begin{equation}
- \label{eq-continuity}
- \frac{\partial{\rho}}{\partial{t}} = \vec{\nabla} \cdot \left(\rho\vec{V}\right) = 0,
-\end{equation}
-where \(\rho\) is the density of the liquid, and \(V\) the fluid velocity.
-
-In relation to ocean waves, we can make the assumptions of incompressibility and
-isotropy within the marine environment. In this case eq.\nbsp{}eqref:eq-continuity
-reduces to
-\begin{equation}
- \label{eq-continuity-2}
- \text{div}\vec{V} =
- \frac{\partial{V_x}}{\partial{x}} +
- \frac{\partial{V_y}}{\partial{y}} +
- \frac{\partial{V_z}}{\partial{z}} = 0
-\end{equation}
-The physics of wind waves is defined primarily by the action of gravitational
-forces, which simplifies nature of the phenomenon under investigation. This
-approach allows us to consider the irrotational motion of the fluid and
-introduce the velocity potential \(\phi\). Then eq.\nbsp{}eqref:eq-continuity &
-eqref:eq-continuity-2 reduce to Laplace's equation, which is the most general
-field equation for the problem of wave motions of a liquid:
+where \(E_{\zeta}(u,v)\) is two-dimensional spectral density of wave energy.
+
+Formula\nbsp{}eqref:eq-lhmodel is derived from equation of continuity and
+equation of motion for incompressible inviscid fluid. For ocean waves
+incompressibility and isotropy of a fluid is assumed; since the motion of ocean
+waves is due to gravitational forces, irrotational motion of the fluid is
+assumed which let us introduce the velocity potential \(\phi\). Under these
+assumptions the equation of continuity reduces to Laplace equation:
\begin{equation*}
- \Delta\phi =
- \frac{\partial^2{\phi_x}}{\partial{x^2}} +
- \frac{\partial^2{\phi_y}}{\partial{y^2}} +
- \frac{\partial^2{\phi_z}}{\partial{z^2}} = 0.
+ \label{eq-continuity}
+ \Delta\phi =
+ \frac{\partial^2{\phi_x}}{\partial{x^2}} +
+ \frac{\partial^2{\phi_y}}{\partial{y^2}} +
+ \frac{\partial^2{\phi_z}}{\partial{z^2}} = 0.
\end{equation*}
-
-The difference between one formulation or another of the wave problem will be in
-the nature of the boundary conditions satisfied at the surface of the air-water
-interface. In our case, it is important to remember that the linear formulation
-of the boundary conditions is (a) applied on the surface of the undisturbed
-fluid \((z=0)\) and (b) all nonlinear terms in the boundary conditions are
-ignored. The Laplace equation is linear and its solution can be found using
-Fourier transforms. Thus, for plane waves a well-known solution is given in the
-form of a definite integral cite:kochin1966theoretical:
+The Laplace equation is linear and its solution can be found using Fourier
+transforms. Thus, for plane waves a well-known solution is given in the form of
+a definite integral\nbsp{}cite:kochin1966theoretical:
\begin{equation*}
\phi(x,z,t) =
\int\limits_{0}^{\infty}
e^{kz} \left[ A(k,t) \cos{kx} + B(k,t) \sin{kx} \right] dk.
\end{equation*}
+
A similar, but slightly more complicated solution is obtained for the
-three-dimensional case. The constants A and B are determined from the boundary
+three-dimensional case. The constants \(A\) and \(B\) are determined from the boundary
conditions on the surface. In the linear formulation the equation of the wave
-profile is
+profile (which is derived from linearised kinematic boundary condition and
+equation of motion, see sec.\nbsp{}[[#sec-pressures]]) is
\begin{align}
\zeta(x,t) &= -\frac{1}{g} \frac{\partial{\phi(x,0,t)}}{\partial{t}} \label{eq-integ-zeta}\\
&= \int\limits_{0}^{\infty}
\left[ \frac{\partial{A(k,t)}}{\partial{t}} \cos{kx} + \frac{\partial{B(k,t)}}{\partial{t}} \sin{kx} \right] dk \nonumber \\
&= \int\limits_{0}^{\infty} C_t(k,t) \cos\left(kx + \epsilon(k,t)\right).\nonumber
\end{align}
-If we set \(c_n = C_t(k_n, t) dk\), then wave model eqref:eq-lhmodel may be
-associated with an approximation of integral eqref:eq-integ-zeta.
-
-Thus, Longuet---Higgins' model is distinguished by its considerable clarity and
-the simplicity of the computational algorithm. However, it is not without some
-serious shortcomings inherent in models of this class:
-- The Longuet---Higgins model is only designed to represent a stationary
- Gaussian field. Normal distribution of the simulated process eqref:eq-lhmodel is
- a consequence of the central limit theorem. Its application to the analysis of
- more general problems (e.g., the evolution of ocean waves in a storm, or the
- study of ocean waves distorted in shallow water) represents a significant
- challenge.
-- Models of this class are periodic and need a very large set of frequencies to
- re-creating long-term simulation.
-- In the numerical implementation of the Longuet---Higgins model, it appears
- that the rate of convergence of eqref:eq-lhmodel is very slow. This is seen as
- a distortion of the energy spectrum of the simulated process (i.e. not
- provided by the convergence of \((u, v)\)), and the laws for the distribution
- of elementary waves, especially in terms of extreme events (not ensured by the
- convergence of \(\epsilon\)). This problem becomes especially significant when
- simulating complex waves that have a broad spectrum with many peaks.
-
-The latter point becomes particularly critical in numerical simulation. In a
-time domain computation of the responses of a vessel in a random seaway, the
-repeated evaluation of the apparently simple equation, eqref:eq-lhmodel at
-hundreds of points on the hull for thousands of time steps can becomes a major
-factor determining the execution speed of the code cite:beck2001modern. This
-becomes an even more significant issue in a nonlinear computation where the wave
-model is even more complex. Thus identifying a significantly less time intensive
-method for modelling the ambient ocean-wave environment has the potential for
-significantly speeding the total simulation process.
+If we set \(c_n = C_t(k_n, t) dk\), then wave model\nbsp{}eqref:eq-lhmodel may
+be associated with an approximation of integral\nbsp{}eqref:eq-integ-zeta.
+
+Although, LH model is based on simple linear wave theory and has straightforward
+computational algorithm, it has some serious shortcomings.
+- LH model is designed to represent a stationary Gaussian field. Normal
+ distribution of the simulated process\nbsp{}eqref:eq-lhmodel is a consequence
+ of the central limit theorem: its application to the analysis of
+ storm or shallow water waves represents a significant challenge.
+- LH model is periodic and need a large set of frequencies to perform long-term
+ simulation.
+- In the numerical implementation of the LH model, it appears that convergence
+ rate of\nbsp{}eqref:eq-lhmodel is slow. This leads to a skewed simulated wave
+ energy spectrum and skewed cumulative distribution functions of various wave
+ parameters (heights, lengths, etc.). This problem becomes especially
+ significant when simulating complex sea waves that have a wide spectrum with
+ multiple peaks.
+
+The latter point becomes particularly critical in long-term numerical
+simulation. In a time domain computation of the responses of a vessel in a
+random seaway, the repeated evaluation of the apparently simple
+eq.\nbsp{}eqref:eq-lhmodel at hundreds of points on the hull for thousands of
+time steps becomes a major factor determining the execution speed of the
+code\nbsp{}cite:beck2001modern. So, finding a less computationally intensive
+method for modelling ocean waves has the potential to increase performance of
+long-term simulation.
* Related work
** Ocean wave modelling
@@ -751,6 +722,10 @@ to generate waves of arbitrary profiles, and is one of the directions of future
work.
* Determining wave pressures for discretely given wavy surface
+:PROPERTIES:
+:CUSTOM_ID: sec-pressures
+:END:
+
Analytic solutions to boundary problems in classical equations are often used to
study different properties of the solution, and for that purpose general
solution formula is too difficult to study, as it contains integrals of unknown
