Conservation Laws and Boundary Conditions - Revision history

Adi Kurniawan: /* Gauss theorem */

2010-11-06T11:06:17Z

Gauss theorem

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	Note the three scalar identities that follow:		Note the three scalar identities that follow:
	<center><math> \iiint_{{\Omega}} \partial_x f \mathrm{d}v = - \iint_{\partial\Omega} f n_1 \mathrm{d}s ~~</math></center> <br>~~		<center><math>
	~~<center><math>~~ \iiint_{{\Omega}} \partial_y f \mathrm{d}v = - \iint_{\partial\Omega} f n_2 \mathrm{d}s ~~</math></center> <br>~~		\begin{align}
	~~<center><math>~~ \iiint_{{\Omega}} \partial_z f \mathrm{d}v = - \iint_{\partial\Omega} f n_3 \mathrm{d}s </math></center>		\iiint_{{\Omega}} \partial_x f \mathrm{d}v &= - \iint_{\partial\Omega} f n_1 \mathrm{d}s \\
			\iiint_{{\Omega}} \partial_y f \mathrm{d}v &= - \iint_{\partial\Omega} f n_2 \mathrm{d}s \\
			\iiint_{{\Omega}} \partial_z f \mathrm{d}v &= - \iint_{\partial\Omega} f n_3 \mathrm{d}s .
			\end{align}
			</math></center>

	The scalar version is as follows where		The scalar version is as follows where

Adi Kurniawan: /* Definition of force and moment in terms of fluid pressure */

2010-09-02T14:08:34Z

Definition of force and moment in terms of fluid pressure

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	</math></center>		</math></center>

	It follows from the Gauss theorem that if <math> \P = C </math> the force and moment over a closed boundary S vanish identically. Hence without loss of generality in the context of wave body interactions we will set <math> C=0 </math>. It follows that the dynamic free surface condition takes the form		It follows from the Gauss theorem that if <math> P = C </math> the force and moment over a closed boundary S vanish identically. Hence without loss of generality in the context of wave body interactions we will set <math> C=0 </math>. It follows that the dynamic free surface condition takes the form

	<center><math> \zeta (x,y,t) = - \frac{1}{g} \left \{ \partial_t\Phi + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi \right \}, \qquad z=\zeta </math></center>		<center><math> \zeta (x,y,t) = - \frac{1}{g} \left \{ \partial_t\Phi + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi \right \}, \qquad z=\zeta </math></center>

Adi Kurniawan: /* Definition of force and moment in terms of fluid pressure */

2010-09-02T14:07:41Z

Definition of force and moment in terms of fluid pressure

← Older revision		Revision as of 14:07, 2 September 2010
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	where <math>P</math> is the pressure and the moment <math>\mathbf{M}</math> is given by		where <math>P</math> is the pressure and the moment <math>\mathbf{M}</math> is given by
	<center><math>		<center><math>
	\mathbf{M} = \iint_{\partial\Omega} P(\mathbf{x}\times\mathbf{v})\mathrm{d}s		\mathbf{M} = \iint_{\partial\Omega} P(\mathbf{x}\times\mathbf{n})\mathrm{d}s
	</math></center>		</math></center>

	It follows from the Gauss theorem that if <math> \~~rho~~ = C </math> the force and moment over a closed boundary S vanish identically. Hence without loss of generality in the context of wave body interactions we will set <math> C=0 </math>. It follows that the dynamic free surface condition takes the form		It follows from the Gauss theorem that if <math> \P = C </math> the force and moment over a closed boundary S vanish identically. Hence without loss of generality in the context of wave body interactions we will set <math> C=0 </math>. It follows that the dynamic free surface condition takes the form

	<center><math> \zeta (x,y,t) = - \frac{1}{g} \left \{ \partial_t\Phi + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi \right \}, \qquad z=\zeta </math></center>		<center><math> \zeta (x,y,t) = - \frac{1}{g} \left \{ \partial_t\Phi + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi \right \}, \qquad z=\zeta </math></center>

Adi Kurniawan: /* Method I */

2010-08-28T07:11:52Z

Method I

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	<center><math> \left (\partial_t + \mathbf{v} \cdot \nabla \right ) (z-\zeta) =0, \qquad \mbox{on} \ z=\zeta </math></center>		<center><math> \left (\partial_t + \mathbf{v} \cdot \nabla \right ) (z-\zeta) =0, \qquad \mbox{on} \ z=\zeta </math></center>
	which in turn implies that		which in turn implies that
	<center><math> \partial_t\zeta + \partial_x\Phi \partial_x\zeta + \partial_y\Phi \partial_y\zeta = \partial_z\Phi, \, z=\zeta </math></center>		<center><math> \partial_t\zeta + \partial_x\Phi \partial_x\zeta + \partial_y\Phi \partial_y\zeta = \partial_z\Phi, \qquad \mbox{on} \ z=\zeta </math></center>
	which is the Kinematic free-surface condition.		which is the Kinematic free-surface condition.

	We have already derived the dynamic condtion from Bernoulli's equation		We have already derived the dynamic condtion from Bernoulli's equation
	<center>		<center>
	<math> \partial_t\Phi + \frac{1}{2} \nabla \Phi \cdot \nabla \Phi + g \zeta = ~~\mathbb{~~C} - \frac{P_a}{\rho}, \ z=\zeta </math></center>		<math> \partial_t\Phi + \frac{1}{2} \nabla \Phi \cdot \nabla \Phi + g \zeta = C - \frac{P_a}{\rho}, \qquad \mbox{on} \ z=\zeta </math></center>

	Constants in Bernoulli's equation may be set equal to zero when we are eventually interested in integrating pressures over closed or open boundaries (floating or submerged bodies) to obtain forces & moments. This follows from a simple application of one of the two Gauss vector theorems.		Constants in Bernoulli's equation may be set equal to zero when we are eventually interested in integrating pressures over closed or open boundaries (floating or submerged bodies) to obtain forces & moments. This follows from a simple application of one of the two Gauss vector theorems.

Adi Kurniawan: /* Method I */

2010-08-28T07:03:26Z

Method I

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	<center><math> \frac{D\tilde{f}}{Dt}=0=\left (\partial_t + \mathbf{v} \cdot \nabla \right ) \tilde{f}=0, \qquad \mbox{on} \ z=\zeta </math></center>		<center><math> \frac{D\tilde{f}}{Dt}=0=\left (\partial_t + \mathbf{v} \cdot \nabla \right ) \tilde{f}=0, \qquad \mbox{on} \ z=\zeta </math></center>

	Expanding we obtain:		Expanding, we obtain
	<center><math> \left (\partial_t + \mathbf{v} \cdot \nabla \right ) (z-\zeta) =0, \qquad \mbox{on} \ z=\zeta </math></center>		<center><math> \left (\partial_t + \mathbf{v} \cdot \nabla \right ) (z-\zeta) =0, \qquad \mbox{on} \ z=\zeta </math></center>
	which in turn implies that		which in turn implies that

Adi Kurniawan: /* Method I */

2010-08-28T07:01:39Z

Method I

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	On <math> z=\zeta; \ P=P_a \equiv \mbox{Atmospheric Pressure} </math>.		On <math> z=\zeta; \ P=P_a \equiv \mbox{Atmospheric Pressure} </math>.
	This allows us to rewrite Bernoulli's equation as		This allows us to rewrite Bernoulli's equation as
	<center><math> \frac{P_a}{\rho}=-\frac{\partial\Phi}{\partial t}-\frac{1}{2}\nabla\Phi\cdot\nabla\Phi-g\zeta+~~\mathbb{~~C} \qquad \mbox{on} \ z=\zeta(x,y,t) </math></center>		<center><math> \frac{P_a}{\rho}=-\frac{\partial\Phi}{\partial t}-\frac{1}{2}\nabla\Phi\cdot\nabla\Phi-g\zeta+C \qquad \mbox{on} \ z=\zeta(x,y,t) </math></center>
	We will simplify this equation by showing that we are free to set the pressure to any value.		We will simplify this equation by showing that we are free to set the pressure to any value.

Adi Kurniawan: /* Conservation of linear momentum */

2010-08-28T06:58:25Z

Conservation of linear momentum

← Older revision		Revision as of 06:58, 28 August 2010
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	== Conservation of linear momentum ==		== Conservation of linear momentum ==

	We begin with [http://en.wikipedia.org/wiki/~~Euler_equations~~ Euler's equation] in the absence of viscosity		We begin with [http://en.wikipedia.org/wiki/Euler_equations_%28fluid_dynamics%29 Euler's equation] in the absence of viscosity
	<center> <math>		<center> <math>
	\partial_t \mathbf{v} + (\mathbf{v}\cdot \nabla)\mathbf{v}= - \frac1{\rho} \nabla P + \mathbf{g}		\partial_t \mathbf{v} + (\mathbf{v}\cdot \nabla)\mathbf{v}= - \frac1{\rho} \nabla P + \mathbf{g}
	</math> </center>		</math> </center>
	where		where
	<math> P(\mathbf{v}, t) </math> is the fluid Pressure at <math>(\mathbf{v}, t)</math> and		<math> P(\mathbf{x}, t) </math> is the fluid Pressure at <math>(\mathbf{x}, t)</math> and
	<math> \mathbf{g}= - \mathbf{k} g </math> is the acceleration due to gravity where		<math> \mathbf{g}= - \mathbf{k} g </math> is the acceleration due to gravity where
	<math> \mathbf{k} </math> is the unit vector pointing in the positive <math>z</math>-direction (so we are now setting the <math>z</math> coordinate to point in the vertical direction). Finally <math> \rho </math> is the water density.		<math> \mathbf{k} </math> is the unit vector pointing in the positive <math>z</math>-direction (so we are now setting the <math>z</math> coordinate to point in the vertical direction). Finally <math> \rho </math> is the water density.

Meylan at 10:35, 30 August 2009

2009-08-30T10:35:39Z

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	\| previous chapter =		\| previous chapter =
	}}		}}

			{{complete pages}}

	We begin by deriving the equations of motion used to model ocean waves. The equations of motion of a general fluid are given by the celebrated [http://en.wikipedia.org/wiki/Navier_Stokes Navier Stokes equations]. However, for the large scale processes that occur in ocean waves many simplifications are possible.		We begin by deriving the equations of motion used to model ocean waves. The equations of motion of a general fluid are given by the celebrated [http://en.wikipedia.org/wiki/Navier_Stokes Navier Stokes equations]. However, for the large scale processes that occur in ocean waves many simplifications are possible.

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	[[Category:Linear Water-Wave Theory]]		[[Category:Linear Water-Wave Theory]]
	~~[[Category:Complete Pages]]~~

Adi Kurniawan at 11:29, 11 April 2009

2009-04-11T11:29:26Z

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	We begin by deriving the equations of motion used to model ocean waves. The equations of motion of a general fluid are given by the celebrated [http://en.wikipedia.org/wiki/Navier_Stokes Navier Stokes equations]. However, for the large scale processes that occur in ocean waves many simplifications are possible.		We begin by deriving the equations of motion used to model ocean waves. The equations of motion of a general fluid are given by the celebrated [http://en.wikipedia.org/wiki/Navier_Stokes Navier Stokes equations]. However, for the large scale processes that occur in ocean waves many simplifications are possible.

	== Coordinate ~~System~~ ==		== Coordinate system and velocity potential ==

	[[Image:Coordinate_system.png\|right\|thumb\|500px\|Coordinate System]]		[[Image:Coordinate_system.png\|right\|thumb\|500px\|Coordinate System]]
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	At the moment we have not yet defined the region of space which is occupied by the fluid. However, the free surface plays a very important role in the propagation of ocean waves.		At the moment we have not yet defined the region of space which is occupied by the fluid. However, the free surface plays a very important role in the propagation of ocean waves.

	The most important assumption we make is that the fluid is an [http://en.wikipedia.org/wiki/Viscosity ideal fluid], i.e. there are no shear stresses due to viscosity and that the flow is [http://en.wikipedia.org/wiki/Irrotational irrotational]. This means that		The most important assumption we make is that the fluid is an [http://en.wikipedia.org/wiki/Viscosity ideal fluid], i.e. there are no shear stresses due to viscosity and that the flow is [http://en.wikipedia.org/wiki/Irrotational irrotational]. This means that

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	It turns out that the potential flow model of surface wave propagation and wave-body interactions is very accurate for the kind of length and time scales which are important in the ocean. It is important to realise, however, that we have made considerable simplifications and that certain processes, most notably wave breaking, are in no way covered by this theory. In fact, the process of wave breaking is extremely complicated and is much less well understood than the potential flow model.		It turns out that the potential flow model of surface wave propagation and wave-body interactions is very accurate for the kind of length and time scales which are important in the ocean. It is important to realise, however, that we have made considerable simplifications and that certain processes, most notably wave breaking, are in no way covered by this theory. In fact, the process of wave breaking is extremely complicated and is much less well understood than the potential flow model.

	=== Conservation of mass ===		== Conservation of mass ==

	The key equation we will solve to understand ocean waves is [http://en.wikipedia.org/wiki/Laplaces_equation Laplace's equation] which is derived very simply from the expression for the velocity potential. We assume that the fluid is incompressible so that conservation of mass gives us the following condition		The key equation we will solve to understand ocean waves is [http://en.wikipedia.org/wiki/Laplaces_equation Laplace's equation] which is derived very simply from the expression for the velocity potential. We assume that the fluid is incompressible so that conservation of mass gives us the following condition
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	In particular, if the particles are modelled as spheres, this equation implies no angular velocity at all times.		In particular, if the particles are modelled as spheres, this equation implies no angular velocity at all times.

	== Derivation of ~~Nonlinear Free~~-surface ~~Condition~~ ==		== Derivation of nonlinear free-surface condition ==

	A very important result is the boundary condition at the free surface of the fluid and air. There are two conditions which relate the free surface displacement <math>\zeta(x,y,t)</math> and the velocity potential <math>\Phi(x,y,z,t)</math> at the free surface. The dynamic condition is derived from the Bernoulli's equation and the kinematic condition is derived from the equations linking the fact that the velocity vector at the surface is given by the gradient of the potential. We will present two methods to derive these equations.		A very important result is the boundary condition at the free surface of the fluid and air. There are two conditions which relate the free surface displacement <math>\zeta(x,y,t)</math> and the velocity potential <math>\Phi(x,y,z,t)</math> at the free surface. The dynamic condition is derived from the Bernoulli's equation and the kinematic condition is derived from the equations linking the fact that the velocity vector at the surface is given by the gradient of the potential. We will present two methods to derive these equations.
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	The scalar identity is often used to prove mass conservation principle.		The scalar identity is often used to prove mass conservation principle.

	==== Definition of force and moment in terms of fluid pressure. ====		==== Definition of force and moment in terms of fluid pressure ====

	The force <math>\mathbf{F}</math> is given by		The force <math>\mathbf{F}</math> is given by

Adi Kurniawan: /* Coordinate System */

2009-04-11T11:25:45Z

Coordinate System

← Older revision		Revision as of 11:25, 11 April 2009
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	At the moment we have not yet defined the region of space which is occupied by the fluid. However, the free surface plays a very important role in the propagation of ocean waves.		At the moment we have not yet defined the region of space which is occupied by the fluid. However, the free surface plays a very important role in the propagation of ocean waves.
	The most important assumption we make is that the fluid is an [http://en.wikipedia.org/wiki/Viscosity ideal fluid]. ~~This means that~~ there are no shear stresses due to viscosity and that the flow is [http://en.wikipedia.org/wiki/Irrotational irrotational]. This means that		The most important assumption we make is that the fluid is an [http://en.wikipedia.org/wiki/Viscosity ideal fluid], i.e. there are no shear stresses due to viscosity and that the flow is [http://en.wikipedia.org/wiki/Irrotational irrotational]. This means that

	<center><math>\nabla \times \mathbf{v} = 0</math></center>		<center><math>\nabla \times \mathbf{v} = 0</math></center>

	We now introduce the very important concept of the velocity potential. Essentially this allows us to express the solution as a function of a scalar (a function which has a single value as opposed to a vector function which has multiple values) rather than a vector throughout the fluid domain. There is an important theorem in vector calculus [http://en.wikipedia.org/wiki/Irrotational_vector_field] that if <math>\nabla \times \mathbf{v} = 0</math> then we can express the irrotational vector as the ~~gradiant~~ of a scalar function, i.e.		We now introduce the very important concept of the velocity potential. Essentially this allows us to express the solution as a function of a scalar (a function which has a single value as opposed to a vector function which has multiple values) rather than a vector throughout the fluid domain. There is an important theorem in vector calculus [http://en.wikipedia.org/wiki/Irrotational_vector_field] that if <math>\nabla \times \mathbf{v} = 0</math> then we can express the irrotational vector as the gradient of a scalar function, i.e.
	<center><math>		<center><math>
	\mathbf{v} = \nabla \Phi		\mathbf{v} = \nabla \Phi