Template:Equations for fixed bodies in the time domain

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 theory would be almost identical if the sea floor depth varied within some finite region and was at [math]\displaystyle{ z=-h }[/math] outside this region). The equations of motion in the time domain are

[math]\displaystyle{ \Delta\Phi\left( \mathbf{x,}t\right) =0,\ \ \mathbf{x}\in\Omega, (1) }[/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,(2) }[/math]

where [math]\displaystyle{ \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 also subject to initial conditions

[math]\displaystyle{ (3) \left.\zeta\right|_{t=0} = \zeta_0(x)\,\,\,\mathrm{and}\,\,\, \left.\partial_t\zeta\right|_{t=0} = v_0(x). }[/math]

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]\displaystyle{ \Delta\Phi = 0 }[/math]} \rput[1](6.5,5.3){[math]\displaystyle{ \partial_t \zeta = \partial_n\Phi }[/math]} \rput[l](0,5.3){[math]\displaystyle{ \partial_t \Phi = - \zeta }[/math]} \rput[l](5,1.5){[math]\displaystyle{ \partial_n \Phi =0 }[/math]} \rput[l](3.35,2.5){[math]\displaystyle{ \partial_n \Phi =0 }[/math]} \rput[l](2,4){[math]\displaystyle{ \partial_n \Phi =0 }[/math]} \rput[l](4.5,4){[math]\displaystyle{ \partial\Omega }[/math]} \rput[l](2,1.5){[math]\displaystyle{ \partial\Omega }[/math]} \rput[l](1,3){\Large [math]\displaystyle{ \Omega }[/math]} \rput[l](7.7,5.1){[math]\displaystyle{ z=0 }[/math]} \rput[l](7.7,1.1){[math]\displaystyle{ 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}