Conservation Laws and Boundary Conditions

==The Ocean Environment

===Non Linear Free-surface Condition

[math]\displaystyle{ \bullet (X,Y,Z) }[/math]: Earth Fixed Coordinate System \br [math]\displaystyle{ \vec X }[/math]: Fixed Eulerian Vector [math]\displaystyle{ \vec V }[/math]: Flow Velocity Vector at [math]\displaystyle{ \vec X }[/math] [math]\displaystyle{ \zeta }[/math]: Free Surface Elevation

Assume ideal fluid (No shear stresses) and irrotational flow:

[math]\displaystyle{ \nabla \times \overrightarrow{V} = 0 }[/math]

Let:

[math]\displaystyle{ \overrightarrow{V} = \nabla \Phi \Rightarrow \nabla \times \nabla \Phi \equiv 0 }[/math]

Where [math]\displaystyle{ \Phi(\overrightarrow{X},t) }[/math] is the velocity potential assumed sufficiently continuously differentiable.

Potential flow model of surface wave propagation and wave-body interactions very accurate. Few important exceptions will be noted.

Conservation of mass:

[math]\displaystyle{ \nabla \cdot \overrightarrow{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.