Difference between revisions of "Conservation Laws and Boundary Conditions"

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Conservation of mass:
 
Conservation of mass:
  
<center><math> \Delta \dot V = 0 </math></center>
+
<center><math> \nabla \cdot \overrightarrow{V} = 0 \Rightarrow</math></center>
 +
 
 +
 
 +
<center><math> \nabla \cdot \nabla \Phi = 0 \Rightarrow \sqrt{\nabla} \Phi = 0 </math></cente> or

Revision as of 11:28, 16 January 2007

The Ocean Environment

Non Linear Free-surface Condition

(X,Y,Z): Earth Fixed Coordinate System X: Fixed Eulerian Vector v: Flow Velocity Vector At X

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(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 \sqrt{\nabla} \Phi = 0 }[/math]</cente> or