Template:Equations for fixed bodies in the time domain - Revision history

Meylan: /* Two Dimensional Equations for fixed bodies in the time domain */

2009-08-21T10:46:33Z

Two Dimensional Equations for fixed bodies in the time domain

← Older revision		Revision as of 10:46, 21 August 2009
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	~~== Two Dimensional Equations for fixed bodies in the time domain ==~~

	{{coordinate definition in two dimension}}		{{coordinate definition in two dimension}}

Meylan: /* Two Dimensional Equations for fixed bodies in the time domain */

2009-08-21T10:44:43Z

Two Dimensional Equations for fixed bodies in the time domain

← Older revision		Revision as of 10:44, 21 August 2009
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	The initial conditions are		The initial conditions are
	{{~~intial~~ free surface time domain}}		{{initial free surface time domain}}

Meylan: /* Two Dimensional Equations for fixed bodies in the time domain */

2009-08-21T10:44:30Z

Two Dimensional Equations for fixed bodies in the time domain

← Older revision		Revision as of 10:44, 21 August 2009
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	{{equations of motion time domain without body condition}}		{{equations of motion time domain without body condition}}

			The body boundary condition for a fixed body is

	The ~~initial conditions~~
	<center><math>		<center><math>
	\~~left.\zeta~~\~~right\|_{t~~=0~~} = \zeta_0(x)\~~,\,\,\~~mathrm~~{~~and~~}\,\,\,		\partial_{n}\Phi=0,\ \ \mathbf{x}\in\partial\Omega,
	~~\left.\partial_t\zeta\right\|_{t=0} = \partial_t\zeta_0(x).~~
	</math></center>		</math></center>


			The initial conditions are
			{{intial free surface time domain}}

Meylan: /* Two Dimensional Equations for fixed bodies in the time domain */

2009-08-21T10:37:28Z

Two Dimensional Equations for fixed bodies in the time domain

← Older revision		Revision as of 10:37, 21 August 2009
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	{{coordinate definition in two dimension}}		{{coordinate definition in two dimension}}

			{{equations of motion time domain without body condition}}

	~~The equations~~
	~~of motion in the time domain, in non-dimensional~~
	~~form (so that the fluid density and gravity are both unity are~~
	~~Laplace's equation through out the fluid~~
	~~<center><math>~~
	~~\Delta\Phi\left( \mathbf{x,}t\right) =0,\ \ \mathbf{x}\in\Omega,~~
	~~</math></center>~~
	~~At the bottom surface we have no flow~~
	~~<center><math>~~
	~~\partial_{n}\Phi=0,\ \ z=-h.~~
	~~</math></center>~~
	~~At the free~~
	~~surface we have the kinematic condition~~
	~~<center><math>~~
	~~\partial_{t}\zeta=\partial_{n}\Phi,\ \ z=0,\ x\in F,~~
	~~</math></center>~~
	~~and the dynamic condition (the linearized Bernoulli equation)~~
	~~<center><math>~~
	~~\zeta = -\partial_{t}\Phi,\ \ z=0,\ x\in F,~~
	~~</math></center>~~

Meylan: /* Two Dimensional Equations for fixed bodies in the time domain */

2009-08-21T10:35:49Z

Two Dimensional Equations for fixed bodies in the time domain

← Older revision		Revision as of 10:35, 21 August 2009
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	The equations		The equations
	of motion in the time domain are		of motion in the time domain, in non-dimensional
			form (so that the fluid density and gravity are both unity are
			Laplace's equation through out the fluid
	<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>
			At the bottom surface we have no flow
	<center><math>		<center><math>
	\partial_{n}\Phi=0,\ \ z=-h,		\partial_{n}\Phi=0,\ \ z=-h.
	</math></center>		</math></center>
	~~<center><math>~~		At the free
	~~\partial_{n}\Phi=0,\ \ \mathbf{x}\in\partial\Omega,~~
	~~</math></center>~~
	~~where <math>\Phi</math> is the velocity potential for the fluid.~~ At the free
	surface we have the kinematic condition		surface we have the kinematic condition
	<center><math>		<center><math>
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	\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~~
	~~form (so that the fluid density and gravity are both unity). They are~~
	~~also subject to~~ initial conditions		The initial conditions
	<center><math>		<center><math>
	\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} = \partial_t\zeta_0(x).		\left.\partial_t\zeta\right\|_{t=0} = \partial_t\zeta_0(x).
	</math></center>		</math></center>

Meylan: /* Two Dimensional Equations for fixed bodies in the time domain */

2009-08-21T10:32:17Z

Two Dimensional Equations for fixed bodies in the time domain

← Older revision		Revision as of 10:32, 21 August 2009
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	== Two Dimensional Equations for fixed bodies in the time domain ==		== Two Dimensional Equations for fixed bodies in the time domain ==

	~~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~~
	~~denote the fluid domain by <math>\Omega</math>, the boundary of the fluid domain~~
	~~which touches the fixed bodies by <math>\partial\Omega</math>, and the free~~		The equations
	~~surface by <math>F.</math> The <math>x</math> and <math>z</math> coordinates are such that <math>x</math> is~~
	~~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~~
	~~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).~~ The equations
	of motion in the time domain are		of motion in the time domain are
	<center><math>		<center><math>

Meylan: /* Two Dimensional Equations for fixed bodies in the time domain */

2009-08-20T04:39:41Z

Two Dimensional Equations for fixed bodies in the time domain

← Older revision		Revision as of 04:39, 20 August 2009
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	</math></center>		</math></center>
	<center><math>		<center><math>
	\partial_{n}\Phi=0,\ \ ~~\mathbf{x}\in\partial\Omega~~,		\partial_{n}\Phi=0,\ \ z=-h,
	</math></center>		</math></center>
	<center><math>		<center><math>
	\partial_{n}\Phi=0,\ \ ~~z=-h~~,		\partial_{n}\Phi=0,\ \ \mathbf{x}\in\partial\Omega,
	</math></center>		</math></center>
	where <math>\Phi</math> is the velocity potential for the fluid. At the free		where <math>\Phi</math> is the velocity potential for the fluid. At the free

Meylan: /* Equations for fixed bodies in the time domain */

2009-08-19T08:54:45Z

Equations for fixed bodies in the time domain

← Older revision		Revision as of 08:54, 19 August 2009
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	== Equations for fixed bodies in the time domain ==		== Two Dimensional Equations for fixed bodies in the time domain ==

	We consider a two-dimensional fluid domain of constant depth, which		We consider a two-dimensional fluid domain of constant depth, which

Meylan: /* Equations for fixed bodies in the time domain */

2009-08-19T08:53:28Z

Equations for fixed bodies in the time domain

← Older revision		Revision as of 08:53, 19 August 2009
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	\left.\partial_t\zeta\right\|_{t=0} = \partial_t\zeta_0(x).		\left.\partial_t\zeta\right\|_{t=0} = \partial_t\zeta_0(x).
	</math></center>		</math></center>

	~~{{standard linear wave scattering equations}}~~

Meylan: /* Equations for fixed bodies in the time domain */

2009-08-18T01:35:11Z

Equations for fixed bodies in the time domain

← Older revision		Revision as of 01:35, 18 August 2009
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	\left.\partial_t\zeta\right\|_{t=0} = \partial_t\zeta_0(x).		\left.\partial_t\zeta\right\|_{t=0} = \partial_t\zeta_0(x).
	</math></center>		</math></center>

			{{standard linear wave scattering equations}}