The Ocean Environment
Non Linear Free-surface Condition
[math]\displaystyle{
\begin{matrix}
&\bullet(X,Y,Z) &: &\mbox{Earth Fixed Coordinate System} \\
&\vec X &: &\mbox{Fixed Eulerian Vector} \\
&\vec V &: &\mbox{Flow Velocity Vector at} \ \vec X \\
&\zeta &: &\mbox{Free Surface Elevation}
\end{matrix} }[/math]
[math]\displaystyle{ \bullet }[/math] Assume ideal fluid (No shear stresses) and irrotational flow:
[math]\displaystyle{ \nabla \times \vec V = 0 }[/math]
Let:
[math]\displaystyle{
\vec V = \nabla \Phi \Rightarrow \nabla \times \nabla \Phi \equiv 0
}[/math]
Where [math]\displaystyle{ \Phi(\vec{X},t) }[/math] is the velocity potential assumed sufficiently continuously differentiable.
[math]\displaystyle{ \bullet }[/math] Potential flow model of surface wave propagation and wave-body interactions very accurate. Few important exceptions will be noted.
[math]\displaystyle{ \bullet }[/math] Conservation of mass:
[math]\displaystyle{ \nabla \cdot \vec V = 0 \Rightarrow }[/math]
[math]\displaystyle{ \nabla \cdot \nabla \Phi = 0 \Rightarrow \nabla^2 \Phi = 0 }[/math] or
[math]\displaystyle{ \frac{\partial^2 \Phi}{\partial X^2} + \frac{\partial^2\Phi}{\partial Y^2} + \frac{\partial^2\Phi}{\partial Z^2} = 0, \quad \mbox{Laplace Equation} }[/math]
[math]\displaystyle{ \bullet }[/math] Conservation of Linear momentum. Euler's Equation in the Absence of Viscosity.
[math]\displaystyle{
\frac{\partial \vec V}{\partial t} + (\vec V \cdot \nabla) \vec V = - \frac1{\rho} \nabla P + \vec g
}[/math]
[math]\displaystyle{ P(\vec X, t) : \mbox{Fluid Pressure at} (\vec X, t) }[/math]
[math]\displaystyle{ \vec g = - \vec k g : \mbox{Acceleration of Gravity} }[/math]
[math]\displaystyle{ \vec k : \mbox{unit vector pointing in the positive z-direction} }[/math]
[math]\displaystyle{ \rho : \mbox{water density} \, }[/math]
[math]\displaystyle{ \bullet }[/math] Vector Identity:
[math]\displaystyle{ (\vec V \cdot \nabla) \vec V = \frac 1{2} \nabla (\vec V \cdot \vec V) - \vec V \times ( \nabla \times \vec V) }[/math]
in irrotational flow: [math]\displaystyle{ \nabla \times \vec V = 0 }[/math], thus Euler's equations become:
[math]\displaystyle{ \frac{\partial \vec V}{\partial t} + \frac 1{2} \nabla (\vec V \cdot \vec V) = - \frac 1{\rho} \nabla P - \nabla (g Z) }[/math]
[math]\displaystyle{ \mbox{Note} : \quad \nabla Z = \vec K, \vec V = \nabla \Phi }[/math]
Upon substitution:
[math]\displaystyle{ \nabla \underbrace{(\frac{\partial \Phi}{\partial t} + \frac 1{2} \nabla \Phi \cdot \nabla \Phi + \frac {P}{e} + g Z )} = 0 }[/math]
[math]\displaystyle{ F ( \vec X, t) }[/math]
[math]\displaystyle{ \nabla F( \vec X, t) =0 \quad \Longrightarrow \quad F (\vec X, t) = \mathbb{C} }[/math]
where [math]\displaystyle{ \mathbb{C} = \mbox{constant} }[/math]
Bernovlli's equation follows:
[math]\displaystyle{ \frac{\partial \Phi}{\partial t} + \frac 1{2} \nabla \Phi \cdot \nabla \Phi + \frac {P}{\rho} + g Z = \mathbb{C} }[/math]
or
