Difference between revisions of "Template:Equations for fixed bodies in the time domain"
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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 | + | 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 | + | would be almost identical if the sea floor depth varied). The equations |
− | |||
of motion in the time domain are | of motion in the time domain are | ||
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<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, | ||
− | |||
</math></center> | </math></center> | ||
<center><math> | <center><math> | ||
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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, | + | \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. | ||
− | |||
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> | ||
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\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} = | + | \left.\partial_t\zeta\right|_{t=0} = \partial_t\zeta_0(x). |
</math></center> | </math></center> | ||
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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
where [math]\displaystyle{ \Phi }[/math] is the velocity potential for the fluid. At the free surface we have the kinematic condition
and the dynamic condition (the linearized Bernoulli equation)
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