Template:Equations for fixed bodies in the time domain: Difference between revisions

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pointing in the horizontal direction and <math>z</math> is pointing in the
pointing in the horizontal direction and <math>z</math> is pointing in the
vertical upwards direction (we denote <math>\mathbf{x}=\left( x,z\right) ).</math> The
vertical upwards direction (we denote <math>\mathbf{x}=\left( x,z\right) ).</math> The
free surface is at <math>z=0</math> and the sea floor is at <math>z=-h</math> (the theory
free surface is at <math>z=0</math> and the sea floor is at <math>z=-h</math> (the equations
would be almost identical if the sea floor depth varied within some
would be almost identical if the sea floor depth varied). The equations
finite region and was at <math>z=-h</math> outside this region). The equations
of motion in the time domain are
of motion in the time domain are
<center><math>
<center><math>
\Delta\Phi\left(  \mathbf{x,}t\right)  =0,\ \ \mathbf{x}\in\Omega,
\Delta\Phi\left(  \mathbf{x,}t\right)  =0,\ \ \mathbf{x}\in\Omega,
(1)
</math></center>
</math></center>
<center><math>
<center><math>
Line 30: Line 27:
and the dynamic condition (the linearized Bernoulli equation)
and the dynamic condition (the linearized Bernoulli equation)
<center><math>
<center><math>
\zeta = -\partial_{t}\Phi,\ \ z=0,\ x\in F,(2)
\zeta = -\partial_{t}\Phi,\ \ z=0,\ x\in F,
</math></center>
</math></center>
 
where <math>\zeta</math> is the free-surface elevation.  These equations are in non-dimensional
where <math>\zeta</math> is the free-surface elevation.  Equations
\eqref{laplace_time} to \eqref{dynamic_time} are in non-dimensional
form (so that the fluid density and gravity are both unity).  They are
form (so that the fluid density and gravity are both unity).  They are
also subject to initial conditions
also subject to initial conditions
<center><math>
<center><math>
  (3)
   \left.\zeta\right|_{t=0} = \zeta_0(x)\,\,\,\mathrm{and}\,\,\,
   \left.\zeta\right|_{t=0} = \zeta_0(x)\,\,\,\mathrm{and}\,\,\,
\left.\partial_t\zeta\right|_{t=0} = v_0(x).
\left.\partial_t\zeta\right|_{t=0} = \partial_t\zeta_0(x).
</math></center>
</math></center>
Figure~4 is a schematic
diagram of the problem.
\begin{figure}
\begin{center}
\begin{pspicture}(0,0)(8,6)
\psline[linewidth=2pt](0,1)(7.5,1)
\psline[linewidth=2pt](0,5)(7.5,5)
\rput[l](6,3){\Large<math>\Delta\Phi = 0</math>}
\rput[1](6.5,5.3){<math>\partial_t \zeta = \partial_n\Phi</math>}
\rput[l](0,5.3){<math>\partial_t \Phi = - \zeta</math>}
\rput[l](5,1.5){<math>\partial_n \Phi =0 </math>}
\rput[l](3.35,2.5){<math>\partial_n \Phi =0 </math>}
\rput[l](2,4){<math>\partial_n \Phi =0</math>}
\rput[l](4.5,4){<math>\partial\Omega</math>}
\rput[l](2,1.5){<math>\partial\Omega</math>}
\rput[l](1,3){\Large <math>\Omega</math>}
\rput[l](7.7,5.1){<math>z=0</math>}
\rput[l](7.7,1.1){<math>z=-h</math>}
\pscurve[linewidth=2pt, fillstyle=solid, fillcolor=lightgray,
    showpoints=false](5,5)(4,4)(2,5)
\psccurve[linewidth=2pt, fillstyle=solid, fillcolor=lightgray,
    showpoints=false](3,3)(2,2)(3,2)
\end{pspicture}
\end{center}
\caption{Schematic diagram showing the time-dependent equations}
(4)
\end{figure}

Revision as of 04:13, 12 August 2009

Equations for fixed bodies in the time domain

We consider a two-dimensional fluid domain of constant depth, which contains a finite number of fixed bodies of arbitrary geometry. We denote the fluid domain by [math]\displaystyle{ \Omega }[/math], the boundary of the fluid domain which touches the fixed bodies by [math]\displaystyle{ \partial\Omega }[/math], and the free surface by [math]\displaystyle{ F. }[/math] The [math]\displaystyle{ x }[/math] and [math]\displaystyle{ z }[/math] coordinates are such that [math]\displaystyle{ x }[/math] is pointing in the horizontal direction and [math]\displaystyle{ z }[/math] is pointing in the vertical upwards direction (we denote [math]\displaystyle{ \mathbf{x}=\left( x,z\right) ). }[/math] The free surface is at [math]\displaystyle{ z=0 }[/math] and the sea floor is at [math]\displaystyle{ z=-h }[/math] (the equations would be almost identical if the sea floor depth varied). The equations of motion in the time domain are

[math]\displaystyle{ \Delta\Phi\left( \mathbf{x,}t\right) =0,\ \ \mathbf{x}\in\Omega, }[/math]
[math]\displaystyle{ \partial_{n}\Phi=0,\ \ \mathbf{x}\in\partial\Omega, }[/math]
[math]\displaystyle{ \partial_{n}\Phi=0,\ \ z=-h, }[/math]

where [math]\displaystyle{ \Phi }[/math] is the velocity potential for the fluid. At the free surface we have the kinematic condition

[math]\displaystyle{ \partial_{t}\zeta=\partial_{n}\Phi,\ \ z=0,\ x\in F, }[/math]

and the dynamic condition (the linearized Bernoulli equation)

[math]\displaystyle{ \zeta = -\partial_{t}\Phi,\ \ z=0,\ x\in F, }[/math]

where [math]\displaystyle{ \zeta }[/math] is the free-surface elevation. These equations are in non-dimensional form (so that the fluid density and gravity are both unity). They are also subject to initial conditions

[math]\displaystyle{ \left.\zeta\right|_{t=0} = \zeta_0(x)\,\,\,\mathrm{and}\,\,\, \left.\partial_t\zeta\right|_{t=0} = \partial_t\zeta_0(x). }[/math]
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