Difference between revisions of "Conservation Laws and Boundary Conditions"

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<math>\bullet</math> Conservation of Linear momentum. Euler's Equation in the Absence of Viscosity.
 
<math>\bullet</math> Conservation of Linear momentum. Euler's Equation in the Absence of Viscosity.
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\frac{\partial \vec V}{\partial t} + (\vec V \cdot \nabla) \vec V = - \frac1{\rho} \nabla P + \vec g
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</math> </center>

Revision as of 05:49, 17 January 2007

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]