commit 653e87bd77303501e41dabc8f3cb087d9f0d4f08
parent e73981355377723279d67208de34fe1ff563f7f2
Author: Ivan Gankevich <igankevich@ya.ru>
Date: Mon, 5 Jun 2017 13:20:41 +0300
Add directional derivative derivation.
Diffstat:
2 files changed, 59 insertions(+), 19 deletions(-)
diff --git a/arma-thesis-ru.org b/arma-thesis-ru.org
@@ -436,7 +436,8 @@ Motion Programme (LAMP), программе для моделирования к
**** Формула для поля давлений.
Задача определения поля давлений под взволнованной морской поверхностью
представляет собой обратную задачу гидродинамики для несжимаемой невязкой
-жидкости. Система уравнений для нее в общем виде записывается как\nbsp{}cite:kochin1966theoretical
+жидкости. Система уравнений для нее в общем виде записывается
+как\nbsp{}cite:kochin1966theoretical
\begin{align}
& \nabla^2\phi = 0,\nonumber\\
& \phi_t+\frac{1}{2} |\vec{\upsilon}|^2 + g\zeta=-\frac{p}{\rho}, & \text{на }z=\zeta(x,y,t),\label{eq-problem}\\
@@ -450,7 +451,7 @@ Motion Programme (LAMP), программе для моделирования к
называют динамическим граничным условием); третье уравнение\nbsp{}--- кинематическое
граничное условие, которое сводится к равенству скорости перемещения этой
поверхности (\(D\zeta\)) нормальной составляющей скорости жидкости
-(\(\nabla\phi\cdot\vec{n}\)).
+(\(\nabla\phi\cdot\vec{n}\), см.\nbsp{}разд.\nbsp{}[[#directional-derivative]]).
Обратная задача гидродинамики заключается в решении этой системы уравнений
относительно \(\phi\). В такой постановке динамическое ГУ становится явной
@@ -3503,3 +3504,21 @@ bibliography:bib/refs.bib
Здесь \(\epsilon\)\nbsp{}--- белый шум, а \(C_t\) включает в себя значение \(dk\).
Подставляя бесконечную сумму вместо интеграла, получаем двухмерную форму
ур.\nbsp{}[[eq-longuet-higgins]].
+** Производная в направлении нормали к поверхности
+:PROPERTIES:
+:CUSTOM_ID: directional-derivative
+:END:
+Производная от \(\phi\) в направлении вектора \(\vec{n}\) определяется как
+\(\nabla_n\phi=\nabla\phi\cdot\frac{\vec{n}}{|\vec{n}|}\). Вектор \(\vec{n}\),
+направленный по нормали к поверхности \(z=\zeta(x,y)\) в точке \((x_0,y_0)\)
+определяется как
+\begin{equation*}
+ \vec{n} = \begin{bmatrix}\zeta_x(x_0,y_0)\\\zeta_y(x_0,y_0)\\-1\end{bmatrix}.
+\end{equation*}
+Отсюда производная в направлении нормали к поверхности определяется
+\begin{equation*}
+\nabla_n \phi = \phi_x \frac{\zeta_x}{\sqrt{\zeta_x^2+\zeta_y^2+1}}
+ + \phi_y \frac{\zeta_y}{\sqrt{\zeta_x^2+\zeta_y^2+1}}
+ + \phi_z \frac{-1}{\sqrt{\zeta_x^2+\zeta_y^2+1}},
+\end{equation*}
+где производные \(\zeta\) вычисляются в \((x_0,y_0)\).
diff --git a/arma-thesis.org b/arma-thesis.org
@@ -696,16 +696,18 @@ for it in general case is written as\nbsp{}cite:kochin1966theoretical
& \phi_t+\frac{1}{2} |\vec{\upsilon}|^2 + g\zeta=-\frac{p}{\rho}, & \text{на }z=\zeta(x,y,t),\label{eq-problem}\\
& D\zeta = \nabla \phi \cdot \vec{n}, & \text{на }z=\zeta(x,y,t),\nonumber
\end{align}
-where \(\phi\)\nbsp{}--- velocity potential, \(\zeta\)\nbsp{}--- elevation (\(z\) coordinate)
-of wavy surface, \(p\)\nbsp{}--- wave pressure, \(\rho\)\nbsp{}--- fluid density,
-\(\vec{\upsilon}=(\phi_x,\phi_y,\phi_z)\)\nbsp{}--- velocity vector, \(g\)\nbsp{}---
-acceleration of gravity, and \(D\)\nbsp{}--- substantial (Lagrange) derivative. The
-first equation is called continuity (Laplace) equation, the second one is the
-conservation of momentum law (the so called dynamic boundary condition); the
-third one is kinematic boundary condition for free wavy surface, which states
-that rate of change of wavy surface elevation (\(D\zeta\)) equals to the change of
-velocity potential derivative along the wavy surface normal
-(\(\nabla\phi\cdot\vec{n}\)).
+where \(\phi\)\nbsp{}--- velocity potential, \(\zeta\)\nbsp{}--- elevation
+(\(z\) coordinate) of wavy surface, \(p\)\nbsp{}--- wave pressure,
+\(\rho\)\nbsp{}--- fluid density,
+\(\vec{\upsilon}=(\phi_x,\phi_y,\phi_z)\)\nbsp{}--- velocity vector,
+\(g\)\nbsp{}--- acceleration of gravity, and \(D\)\nbsp{}--- substantial
+(Lagrange) derivative. The first equation is called continuity (Laplace)
+equation, the second one is the conservation of momentum law (the so called
+dynamic boundary condition); the third one is kinematic boundary condition for
+free wavy surface, which states that rate of change of wavy surface elevation
+(\(D\zeta\)) equals to the change of velocity potential derivative along the
+wavy surface normal (\(\nabla\phi\cdot\vec{n}\), see
+section\nbsp{}[[#directional-derivative]]).
Inverse problem of hydrodynamics consists in solving this system of equations
for \(\phi\). In this formulation dynamic boundary condition becomes explicit
@@ -1921,13 +1923,13 @@ they may (and often) overlap.
#+caption: Values of Weibull shape parameter for different wave characteristics.
#+attr_latex: :booktabs t
| Characteristic | Weibull shape (\(k\)) |
-|----------------------+---------------------|
-| Wave height | 2 |
-| Wave length | 2.3 |
-| Crest length | 2.3 |
-| Wave period | 3 |
-| Wave slope | 2.5 |
-| Three-dimensionality | 2.5 |
+|----------------------+-----------------------|
+| Wave height | 2 |
+| Wave length | 2.3 |
+| Crest length | 2.3 |
+| Wave period | 3 |
+| Wave slope | 2.5 |
+| Three-dimensionality | 2.5 |
Verification was performed for standing and propagating waves. The corresponding
ACFs and quantile-quantile plots of wave characteristics distributions are shown
@@ -3559,3 +3561,22 @@ Plugging it in the boundary condition yields
Here \(\epsilon\) is white noise and \(C_t\) includes \(dk\). Substituting
integral with infinite sum yields two-dimensional form of
eq.\nbsp{}[[eq-longuet-higgins]].
+
+** Derivative in the direction of the surface normal
+:PROPERTIES:
+:CUSTOM_ID: directional-derivative
+:END:
+Directional derivative of \(\phi\) in the direction of vector \(\vec{n}\) is
+given by \(\nabla_n\phi=\nabla\phi\cdot\frac{\vec{n}}{|\vec{n}|}\). Normal
+vector \(\vec{n}\) to the surface \(z=\zeta(x,y)\) at point \((x_0,y_0)\) is
+given by
+\begin{equation*}
+ \vec{n} = \begin{bmatrix}\zeta_x(x_0,y_0)\\\zeta_y(x_0,y_0)\\-1\end{bmatrix}.
+\end{equation*}
+Hence, derivative in the direction of the surface normal is given by
+\begin{equation*}
+\nabla_n \phi = \phi_x \frac{\zeta_x}{\sqrt{\zeta_x^2+\zeta_y^2+1}}
+ + \phi_y \frac{\zeta_y}{\sqrt{\zeta_x^2+\zeta_y^2+1}}
+ + \phi_z \frac{-1}{\sqrt{\zeta_x^2+\zeta_y^2+1}},
+\end{equation*}
+where \(\zeta\) derivatives are calculated at \((x_0,y_0)\).
