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| − | =Equations for fixed bodies in the time domain=
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| − | We consider a two-dimensional fluid domain of constant depth, which
| + | {{coordinate definition in two dimension}} |
| − | contains a finite number of fixed bodies of arbitrary geometry. We
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| − | denote the fluid domain by <math>\Omega</math>, the boundary of the fluid domain
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| − | which touches the fixed bodies by <math>\partial\Omega</math>, and the free
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| − | surface by <math>F.</math> The <math>x</math> and <math>z</math> coordinates are such that <math>x</math> is
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| − | pointing in the horizontal direction and <math>z</math> is pointing in the
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| − | vertical upwards direction (we denote <math>\mathbf{x}=\left( x,z\right) ).</math> The
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| − | free surface is at <math>z=0</math> and the sea floor is at <math>z=-h</math> (the theory
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| − | would be almost identical if the sea floor depth varied within some
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| − | finite region and was at <math>z=-h</math> outside this region). The equations
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| − | of motion in the time domain are
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| − | <center><math>
| + | {{equations of motion time domain without body condition}} |
| − | \Delta\Phi\left( \mathbf{x,}t\right) =0,\ \ \mathbf{x}\in\Omega,
| + | |
| − | (1)
| + | The body boundary condition for a fixed body is |
| − | </math></center>
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| | <center><math> | | <center><math> |
| | \partial_{n}\Phi=0,\ \ \mathbf{x}\in\partial\Omega, | | \partial_{n}\Phi=0,\ \ \mathbf{x}\in\partial\Omega, |
| | </math></center> | | </math></center> |
| − | <center><math>
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| − | \partial_{n}\Phi=0,\ \ z=-h,
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| − | </math></center>
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| − | where <math>\Phi</math> is the velocity potential for the fluid. At the free
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| − | surface we have the kinematic condition
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| − | <center><math>
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| − | \partial_{t}\zeta=\partial_{n}\Phi,\ \ z=0,\ x\in F,
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| − | </math></center>
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| − | and the dynamic condition (the linearized Bernoulli equation)
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| − | <center><math>
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| − | \zeta = -\partial_{t}\Phi,\ \ z=0,\ x\in F,(2)
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| − | </math></center>
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| − |
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| − | where <math>\zeta</math> is the free-surface elevation. Equations
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| − | \eqref{laplace_time} to \eqref{dynamic_time} are in non-dimensional
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| − | form (so that the fluid density and gravity are both unity). They are
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| − | also subject to initial conditions
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| − | <center><math>
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| − | (3)
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| − | \left.\zeta\right|_{t=0} = \zeta_0(x)\,\,\,\mathrm{and}\,\,\,
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| − | \left.\partial_t\zeta\right|_{t=0} = v_0(x).
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| − | </math></center>
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| − | Figure~4 is a schematic
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| − | diagram of the problem.
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| | | | |
| − | \begin{figure}
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| − | \begin{center}
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| − | \begin{pspicture}(0,0)(8,6)
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| | | | |
| − | \psline[linewidth=2pt](0,1)(7.5,1)
| + | The initial conditions are |
| − | \psline[linewidth=2pt](0,5)(7.5,5)
| + | {{initial free surface time domain}} |
| − | \rput[l](6,3){\Large<math>\Delta\Phi = 0</math>}
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| − | \rput[1](6.5,5.3){<math>\partial_t \zeta = \partial_n\Phi</math>}
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| − | \rput[l](0,5.3){<math>\partial_t \Phi = - \zeta</math>}
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| − | \rput[l](5,1.5){<math>\partial_n \Phi =0 </math>}
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| − | \rput[l](3.35,2.5){<math>\partial_n \Phi =0 </math>}
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| − | \rput[l](2,4){<math>\partial_n \Phi =0</math>}
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| − | \rput[l](4.5,4){<math>\partial\Omega</math>}
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| − | \rput[l](2,1.5){<math>\partial\Omega</math>}
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| − | \rput[l](1,3){\Large <math>\Omega</math>}
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| − | \rput[l](7.7,5.1){<math>z=0</math>}
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| − | \rput[l](7.7,1.1){<math>z=-h</math>}
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| − | \pscurve[linewidth=2pt, fillstyle=solid, fillcolor=lightgray,
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| − | showpoints=false](5,5)(4,4)(2,5)
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| − | \psccurve[linewidth=2pt, fillstyle=solid, fillcolor=lightgray,
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| − | showpoints=false](3,3)(2,2)(3,2)
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| − | \end{pspicture}
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| − | \end{center}
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| − | \caption{Schematic diagram showing the time-dependent equations}
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| − | (4)
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| − | \end{figure}
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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
fluid motion is described by a velocity potential [math]\displaystyle{ \Phi }[/math] and free surface by
[math]\displaystyle{ \zeta }[/math].
The equations
of motion in the time domain are
Laplace's equation through out the fluid
[math]\displaystyle{
\Delta\Phi\left( \mathbf{x,}t\right) =0,\ \ \mathbf{x}\in\Omega.
}[/math]
At the bottom surface we have no flow
[math]\displaystyle{
\partial_{n}\Phi=0,\ \ z=-h.
}[/math]
At the free
surface we have the kinematic condition
[math]\displaystyle{
\partial_{t}\zeta=\partial_{n}\Phi,\ \ z=0,\ x\in \partial\Omega_{F},
}[/math]
and the dynamic condition (the linearized Bernoulli equation)
[math]\displaystyle{
\partial_{t}\Phi = -g\zeta ,\ \ z=0,\ x\in \partial\Omega_{F}.
}[/math]
The body boundary condition for a fixed body is
[math]\displaystyle{
\partial_{n}\Phi=0,\ \ \mathbf{x}\in\partial\Omega,
}[/math]
The initial conditions are
[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]
