Conservation Laws and Boundary Conditions

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]