WikiWaves - User contributions [en]

Linear Wave-Body Interaction

2008-04-07T16:48:46Z

Len: typo fix

= Introduction =

We derive here the equations of motion for a floating body in [[Linear Plane Progressive Regular Waves]] in two dimensions.

= Linear wave-body interactions =

[[Image:Rigid_body.jpg|thumb|right|600px|Rigid body motions]]

Consider a [[Linear Plane Progressive Regular Waves|Linear Plane Progressive Regular Wave]] interacting with a floating body in two dimensions (the main concepts survive almost with no change in the more practical three-dimensional problem). We begin by defining the following,

<center><math> \zeta(t): \quad \mbox{ambient wave elevation. Regular or random with definitions to be given below}. \,</math></center>

<center><math> \xi_1(t): \quad \mbox{Body surge displacement} \,</math></center>

<center><math> \xi_3(t): \quad \mbox{Body heave displacement} \,</math></center>

<center><math> \xi_4(t): \quad \mbox{Body roll displacement} \,</math></center>

where the surge, heave and roll are the three [http://en.wikipedia.org/wiki/Rigid_body_dynamics rigid body] motions which are possible in two dimensions.

= Linear theory =

We assume that we can apply linear theory to the motions, which are an extension of the [[Linear and Second-Order Wave Theory| linear equations]] for a free-surface. We assume that
<center><math> \left| \frac{\partial\zeta}{\partial x} \right| = O(\varepsilon) \ll 1 \, </math></center>
This is an assumption of small wave steepness which is a reasonable assumption for gravity waves in most cases, except when waves are near breaking conditions. Further more we assume
<center><math> \left| \frac{\xi_1}{A} \right| = O(\varepsilon) \ll 1 \, </math></center>
<center><math> \left| \frac{\xi_3}{A} \right| = O(\varepsilon) \ll 1 \, </math></center>
<center><math> \left| \xi_4 \right| = O(\varepsilon) \ll 1 \, </math></center>
These assumptions are valid in most cases and most bodies of practical interest, unless the vessel response at resonance is highly tuned or lightly damped. This is often the case for roll when a small amplitude wave interacts with a vessel weakly damped in roll.

= Linear Systems theory =

[[Image:Linear_systems_theory.jpg|thumb|right|600px|Linear systems theory]]

The vessel dynamic responses in waves may be modelled according to linear system theory. By virtue of linearity, a random seastate may be represented as the linear super position of [[Linear Plane Progressive Regular Waves]] (see [[Waves and the Concept of a Wave Spectrum]])
<center><math> \zeta(x,t) = \sum_j A_j \cos ( K_j x - \omega_j t + \epsilon_j ) \,</math></center>
where in deep water: <math> K_j = \frac{\omega_j^2}{g} \,</math>. Note that the sum here can be replace by an integral in many formulations. According to the theory of St. Denis and Pierson, the phases <math> \epsilon\, </math>, are random and uniformly distributed between <math> ( - \pi, \pi ] \, </math>. For now we assume them known constants:

At <math> X=0\,</math>:

<center><math> \zeta(t) = \sum_j A_j \cos ( \omega_j t - \epsilon_j ) \,</math></center>

<center><math> = \mathfrak{Re} \sum_j A_j e^{i\omega_j t - i \epsilon_j} </math></center>

The corresponding vessel responses follow from linearity in the form:

<center><math> \xi_K (t) = \mathfrak{Re} \sum_j \Pi_K (\omega_j) e^{i\omega_j t - i\epsilon_j},

\qquad K = 1, 3, 4 </math></center>

Where <math> \Pi_K (\omega) \, </math> is the complex RAO ([[Response Amplitude Operator]]) for mode <math> K\,</math>. It is the object of linear seakeeping theory to derive equations for <math>\Pi\omega)\,</math> the frequency domain. The treatment in the stochastic case is then a simple exercise in linear systems.

= Calculation of the RAO =

The equations of motion for <math> \xi_K(t)\,</math> follow from Newton's law applied to each mode in two dimensions. The same principles apply with very minor changes in three dimensions

We begin by considering the equation in surge.
<center><math> \mathbf{M} \frac{d^2\xi_1}{dt^2} = F_{1\omega} ( \xi_1, \dot\xi_1, \ddot\xi_1, t) </math></center>
where <math> \frac{d\xi_1}{dt} = \dot\xi_1 \, </math> and <math> F_{1\omega} \, </math> is the force on the body due to the fluid pressures, by virtue of linearity, <math> F_{1\omega} \,</math> will be assumed to be a linear functional of <math> \xi_1, \dot\xi_1, \ddot\xi_1 \, </math>. [[Memory effects]] exist when surface waves are generated on the free surface, so <math> F_{1\omega} \,</math> depends in principle on the entire history of the vessel displacement. We adopt here the [[Frequency Domain Problem|frequency domain]] formulation where the vessel motion has been going on over an infinite time interval, <math> (-\infty, t)\,</math> with <math> e^{i\omega t}\,</math> dependence.

We will therefore set:
<center><math> \xi_K(t) = \mathfrak{Re} \left\{ \Pi_K e^{i\omega t} \right\}, \qquad K=1,3,4 </math></center>
In this case we can linearize the water induced force on the body as follows:
<center><math> F_{1\omega}(t) = X_1(t) - A_{11} \ddot \xi_1 - A_{13} \ddot \xi_3 - A_{14} \ddot \xi_4 </math></center>

<center><math> - B_{11} \dot \xi_1 - B_{13} \dot \xi_3 - B_{14} \dot \xi_4 </math></center>

<center><math> - C_{11} \xi_1 - C_{13} \xi_3 - C_{14} \xi_4 \,</math></center>

<center><math> = X_1(t) - \sum_j \left[ A_{1j} \ddot \xi_j + B_{1j} \dot \xi_j + C_{1j} \xi_j \right] </math></center>

The same expansion applies for other modes, namely Heave (<math> K = 3 \, </math>) and Roll (<math> K=4 \, </math>). In sum:

<center><math> F_{K\omega} (t) = X_K - \sum_j \left[ A_{Kj} \ddot \xi_j + B_{Kj} \dot \xi_j + C_{Kj} \xi_j \right], \qquad K = 1,3,4 </math></center>

* The added-mass matrix <math> A_{Kj} \,</math> represents the added inertia due to the acceleration of the body in water with acceleration <math>\ddot\xi_j\,</math>.

* The damping matrix <math> B_{Kj}\,</math> governs the energy dissipation into the fluid domain in the form of surface waves.

* The hydrostatic restoring matrix <math> C_{Kj} \, </math> represents the system stifness due to the hydrostatic restoring forces and moments.

For harmonic motions, the matrices <math> A_{Kj} \, </math> and <math> B_{Kj} \, </math> are functions of <math> \omega\,</math>, so we write <math> A_{Kj} (\omega), \ B_{Kj} (\omega)\, </math>. This functional form will be discussed below. The hydrostatic matrix <math> C_{Kj} \, </math> is independent of <math>\omega\,</math> and many of its elements are identically equal to zero. Collecting terms in the left-hand side and denoting by <math> M_{Kj}\,</math> the body inertia matrix:

== Surge ==

<center><math> \sum_j \left[ -\omega^2 \left( M_{1j} + A_{1j} \right) + i\omega B_{1j} + C_{1j} \right] \Pi_j = \mathbf{X}_1 (\omega), \quad j=1,3,4 </math></center>

==Heave ==

<center><math> \sum_j \left[ -\omega^2 \left( M_{3j} + A_{3j} \right) + i\omega B_{3j} + C_{3j} \right] \Pi_j = \mathbf{X}_3 (\omega), \quad j=1,3,4 </math></center>

== Roll ==

<center><math> \sum_j \left[ -\omega^2 \left( I_G + A_{4j} \right) + i\omega B_{4j} + C_{4j} \right] \Pi_j = \mathbf{X}_4 (\omega)</math></center>

The extension of these equations to six degrees of freedom is straightforward. However before discussing the general case we will study specific properties of the 2D Problem for the sake of clarity.

= Symmetric body =

Consider a body symmetric about the <math> X = 0\,</math> axis.

[[Image:Symmetric.jpg|thumb|right|600px|Symmetric body]]

For a body symmetric port/starboard:

* Verify that Heave is decoupled from Surge and Roll. In other words the Surge and Roll motions do not influence Heave and vice versa:

<center><math> \left[ -\omega^2 \left( M + A_{33} \right) + i\omega B_{33} + C_{33} \right] \Pi_3 = A </math></center>

* The only nonzero hydrostatic coefficients are <math> C_{33} \, </math> and <math> C_{44} \, </math>. Verify that this is the case even for non-symmetric sections.

* Surge and Roll are coupled for symmetric and non-symmetric bodies. The coupled equation of motion becomes:

Surge-Roll

<center><math> \sum_{j=1,4} \left[ -\omega^2 \left( M_{ij} + A_{ij} \right) + i\omega B_{ij} + C_{ij} \right] \Pi_j = \mathbf{X}_i, \quad i,j = 1,4 </math></center>

* When Newton's law is expressed about the center of gravity:

<center><math> M_{14} = M_{41} = 0, \ M_{11} = M, \ M_{44} = I_G </math></center>

where <math> I_G\,</math> is the body moment of inertia about the center of gravity. If the equations are to be expressed about the origin of the coordinate system, then the formulation must start with respect to <math> G\,</math> and expressions derived w.r.t. <math> O \,</math>, via a coordinate transformation.

* The exciting forces <math> \mathbf{X}_1, \mathbf{X}_3 \,</math> are defined in an obvious manner along the X- and Z-axis. The Roll moment <math> \mathbf{X}_4 \, </math> is defined initially about <math> G\,</math>.

Need to derive definitions for the coefficients that enter the Heave & Surge-Roll equations of motion:

<center><math> M = \rho \forall, \qquad \forall = \ </math> volume of water displaced by body (archimedian principle of buoyancy) </center>

<center><math> C_{33} = \rho g A_\omega = \rho g B, </math></center>

<center><math> A_\omega = </math> body waterplane area = <math>B</math> (Beam in two dimensions)</center>

<center><math> \left( C_{44} \right)_G = \rho g \frac{B^3}{12} = \ </math> Roll restoring moment due to a small angular displacement </center>
<center> about the center of gravity. Verify for all wall and non-wall sided sections </center>

<center><math> C_{44} \equiv \left( C_{44} \right)+G \ne \left( C_{44} \right)_O; \quad </math> Derive an expression for <math> \ \left( C_{44} \right)_O \ </math> in terms of <math> \ \left( C_{44} \right)_G. </math></center>

-----

This article is based on the MIT open course notes and the original article can be found
[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/666E84F4-5679-47FD-BD7B-9D39877DE5A1/0/lecture9.pdf here]

[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]

Conservation Laws and Boundary Conditions

2008-04-07T13:34:14Z

Len: typo fix

We begin by derving 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.

= Non-Linear Free-surface Condition =

[[Image:Non_linear_free_surface.jpg|center|frame|Co-ordinate System]]

We begin by defining the coordinate system.

<center><math>
\begin{matrix}
&(X,Y,Z) &: &\mbox{Earth Fixed Coordinate System} \\
&\vec X &: &\mbox{Fixed Eulerian Vector} \\
&\vec V &: &\mbox{Flow Velocity Vector at} \ \vec X \\
&\zeta &: &\mbox{Free Surface Elevation}
\end{matrix} </math></center>

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

<center><math>\nabla \times \vec 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 through out the fluid domain. There is an important theorem in vector calculus [http://en.wikipedia.org/wiki/Irrotational_vector_field] that if <math>\nabla \times \vec V = 0</math> then we can express the irrotational vector as the gradiant of a scalar function, i.e.
<center><math>
\vec V = \nabla \Phi
</math></center>
where <math>\Phi(\vec{X},t)</math> is called the [http://en.wikipedia.org/wiki/Velocity_potential velocity potential].

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 that the potential flow model.

== 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
<center><math> \nabla \cdot \vec V = 0 </math></center>
This condition in turn implies, using the definition of the velocity potential that
<center><math> \nabla \cdot \nabla \Phi = 0 \Rightarrow \nabla^2 \Phi = 0 </math></center> or

<center><math> \frac{\partial^2 \Phi}{\partial X^2} + \frac{\partial^2\Phi}{\partial Y^2} + \frac{\partial^2\Phi}{\partial Z^2} = 0, \quad \mbox{Laplace Equation} </math></center>.

== Conservation of linear momentum ==

We begin with [http://en.wikipedia.org/wiki/Euler_equations Euler's equation] in the absence of viscosity
<center> <math>
\frac{\partial \vec V}{\partial t} + (\vec V \cdot \nabla) \vec V = - \frac1{\rho} \nabla P + \vec g
</math> </center>
where
<math> P(\vec X, t) </math> is the fluid Pressure at <math>(\vec X, t)</math> and
<math> \vec g = - \vec k g </math> is the acceleration due to gravity where
<math> \vec k </math> is the unit vector pointing in the positive z-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.

We then use the folliwng vector identity
<center><math> (\vec V \cdot \nabla) \vec V = \frac 1{2} \nabla (\vec V \cdot \vec V) - \vec V \times ( \nabla \times \vec V) </math></center>
and since we have irrotational flow (i.e. <math> \nabla \times \vec V = 0 </math>) Euler's equation becomes
<center><math> \frac{\partial \vec V}{\partial t} + \frac 1{2} \nabla (\vec V \cdot \vec V) = - \frac 1{\rho} \nabla P - \nabla (g Z) </math></center>
where we have used <math> \nabla Z = \vec K </math>.

We now substitute <math> \vec V = \nabla \Phi </math> and we obtain
<center><math> \nabla (\frac{\partial \Phi}{\partial t} + \frac 1{2} \nabla \Phi \cdot \nabla \Phi + \frac {P}{e} + g Z ) = 0 </math></center>
We now observe that if
<center><math> \nabla F( \vec X, t) =0 \quad \Longrightarrow \quad F (\vec X, t) = C </math></center>
where <math> C </math> is an arbitrary constant.

=== Bernoulli's equation ===

[http://en.wikipedia.org/wiki/Bernoulli%27s_equation Bernoulli's equation] follows

<center><math> \frac{\partial \Phi}{\partial t} + \frac 1{2} \nabla \Phi \cdot \nabla \Phi + \frac {P}{\rho} + g Z = C </math></center>
or
<center><math> \frac{P}{\rho} = - \frac{\partial\Phi}{\partial t}-\frac{1}{2}\nabla\Phi \cdot \nabla \Phi - g Z + C </math></center>

The value of the constant <math> C </math> is immaterial.
It is also woth noting that the
angular momentum conservation principle is contained in
<math> \nabla \times \vec V = 0 </math>
In particualr, if the particles are modelled as spheres, this equation implies no angular velocity at all times.

= Derivation of Nonlinear Free-surface Condition =

[[Image:Free_surface.jpg|center|Figure]]

A very important result is the boundary condition at the free surface of the fluid and air. There are two condions which relate the free surface displacement <math>\zeta(X,Y,t)</math> and the velocity potential <math>\Phi(X,Y,Z)</math> at the free surface. The dynamic condition is derived from the Bernouilli 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.

== Method I ==

We derive the dynamic condition directly from Bernouilli's equation.
On <math> Z=\zeta; \ P=P_a \equiv \mbox{Atmospheric Pressure} </math>.
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>
We will simplify this equation by showing that we are free to set the pressure to any value.

The kinematic condition is derived as follows.
On <math>Z=\zeta</math> The mathematical function
<center><math>Z-\zeta(X,Y,t)\equiv\tilde{f}(X,Y,Z,t)</math></center>

is always zero when tracing a fluid particle on the free surface. So the [http://en.wikipedia.org/wiki/Total_derivative substantial or total derivative] of <math>\tilde{f}</math> must vanish, thus

<center><math> \frac{D\tilde{f}}{Dt}=0=\left ( \frac{\partial}{\partial t} + \vec V \cdot \nabla \right ) \tilde{f}=0, \qquad \mbox{on} \ Z=\zeta </math></center>

Expanding we obtain:
<center><math> \left (\frac{\partial}{\partial t} + \vec V \cdot \nabla \right ) (Z-\zeta) =0, \qquad \mbox{on} \ Z=\zeta </math></center>
which in turn implies that
<center><math> \frac{\partial\zeta}{\partial t}+\frac{\partial\Phi}{\partial X} \frac{\partial\zeta}{\partial X}+\frac{\partial\Phi}{\partial Y}\frac{\partial\zeta}{\partial Y}=\frac{\partial\Phi}{\partial Z}, \ Z=\zeta </math></center>
which is the Kinematic free-surface condition.

We have already derived the dynamic condtion from Bernoulli's equation
<center>
<math> \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla \Phi \cdot \nabla \Phi + g \zeta = \mathbb{C} - \frac{P_a}{\rho}, \ 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

=== Gauss theorem ===

We need to use the following theorems often called [http://en.wikipedia.org/wiki/Gauss_theorem Gauss theorem] although more properly known as the divergence theorem.
We begin with the vector version. If
<math> \vec n </math> is the unit normal vector pointing inside the volume <math> V </math> and
<math> f(\bar{X})</math> is an arbitrary sufficiently differentiable scalar function, then
<center>
<math> \iiint_{V} \nabla f dv = -\iint_{S} f_{s} \vec n ds </math>
</center>

Note the three scalar identities that follow:
<center><math> \iiint_{{V}} \frac{\partial f}{\partial x} dv = - \iint_{S} f n_1 ds </math></center> 
<center><math> \iiint_{{V}} \frac{\partial f}{\partial y} dv = - \iint_{S} f n_2 ds </math></center> 
<center><math> \iiint_{{V}} \frac{\partial f}{\partial z} dv = - \iint_{S} f n_3 ds </math></center>

The scalar version is as follows where
<math> \vec{V}</math> is an arbitrary sufficiently differentiable vector function
<center>
<math> \iiint_{\bar{V}} \nabla \cdot \vec V = - \iint_{s} \vec V \cdot \vec n ds </math></center>
The scalar identity ofter used to prove mass conservation principle.

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

[[Image:Force_coordinates.jpg|right|frame|Force coordinates]]

The force <math>\vec{F}</math> is given by
<center><math>
\vec{F} = \iint_{S} P\vec{n}ds
</math></center>
where <math>P</math> is the pressure and the moment <math>\vec{M}</math> is given by
<center><math>
\vec{F} = \iint_{S} P(\vec{x}\times\vec{n})ds
</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

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

== Method II ==

When tracing a fluid particle on the free surface the hydrodynamic pressure given by Bernoulli (after the constant <math>C</math> has been set equal to zero) must vanish as we follow the particle:

<center><math> \frac{D}{Dt} \left \{ \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi+gZ \right \} =0, \qquad Z=\zeta </math></center>
or
<center><math> \left ( \frac{\partial}{\partial t} + \vec V \cdot \nabla \right ) \left ( \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi +gZ \right ) =0, \qquad Z=\zeta </math></center>

This condition also follows upon elimination of <math>\zeta</math> from the kinematic & dynamic conditions derived under method I.

-----

This article is based on the MIT open course notes and the original article can be found
[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/1814747D-3A05-45A1-BDE5-2CEF40DEA25F/0/lecture1.pdf here]

[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]

Conservation Laws and Boundary Conditions

2008-04-07T13:33:01Z

Len: typo fix

We begin by derving 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.

= Non-Linear Free-surface Condition =

[[Image:Non_linear_free_surface.jpg|center|frame|Co-ordinate System]]

We begin by defining the coordinate system.

<center><math>
\begin{matrix}
&(X,Y,Z) &: &\mbox{Earth Fixed Coordinate System} \\
&\vec X &: &\mbox{Fixed Eulerian Vector} \\
&\vec V &: &\mbox{Flow Velocity Vector at} \ \vec X \\
&\zeta &: &\mbox{Free Surface Elevation}
\end{matrix} </math></center>

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

<center><math>\nabla \times \vec 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 through out the fluid domain. There is an important theorem in vector calculus [http://en.wikipedia.org/wiki/Irrotational_vector_field] that if <math>\nabla \times \vec V = 0</math> then we can express the irrotational vector as the gradiant of a scalar function, i.e.
<center><math>
\vec V = \nabla \Phi
</math></center>
where <math>\Phi(\vec{X},t)</math> is called the [http://en.wikipedia.org/wiki/Velocity_potential velocity potential].

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 theorey. In fact, the process of wave breaking is extremely complicated and is much less well understood that the potential flow model.

== 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
<center><math> \nabla \cdot \vec V = 0 </math></center>
This condition in turn implies, using the definition of the velocity potential that
<center><math> \nabla \cdot \nabla \Phi = 0 \Rightarrow \nabla^2 \Phi = 0 </math></center> or

<center><math> \frac{\partial^2 \Phi}{\partial X^2} + \frac{\partial^2\Phi}{\partial Y^2} + \frac{\partial^2\Phi}{\partial Z^2} = 0, \quad \mbox{Laplace Equation} </math></center>.

== Conservation of linear momentum ==

We begin with [http://en.wikipedia.org/wiki/Euler_equations Euler's equation] in the absence of viscosity
<center> <math>
\frac{\partial \vec V}{\partial t} + (\vec V \cdot \nabla) \vec V = - \frac1{\rho} \nabla P + \vec g
</math> </center>
where
<math> P(\vec X, t) </math> is the fluid Pressure at <math>(\vec X, t)</math> and
<math> \vec g = - \vec k g </math> is the acceleration due to gravity where
<math> \vec k </math> is the unit vector pointing in the positive z-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.

We then use the folliwng vector identity
<center><math> (\vec V \cdot \nabla) \vec V = \frac 1{2} \nabla (\vec V \cdot \vec V) - \vec V \times ( \nabla \times \vec V) </math></center>
and since we have irrotational flow (i.e. <math> \nabla \times \vec V = 0 </math>) Euler's equation becomes
<center><math> \frac{\partial \vec V}{\partial t} + \frac 1{2} \nabla (\vec V \cdot \vec V) = - \frac 1{\rho} \nabla P - \nabla (g Z) </math></center>
where we have used <math> \nabla Z = \vec K </math>.

We now substitute <math> \vec V = \nabla \Phi </math> and we obtain
<center><math> \nabla (\frac{\partial \Phi}{\partial t} + \frac 1{2} \nabla \Phi \cdot \nabla \Phi + \frac {P}{e} + g Z ) = 0 </math></center>
We now observe that if
<center><math> \nabla F( \vec X, t) =0 \quad \Longrightarrow \quad F (\vec X, t) = C </math></center>
where <math> C </math> is an arbitrary constant.

=== Bernoulli's equation ===

[http://en.wikipedia.org/wiki/Bernoulli%27s_equation Bernoulli's equation] follows

<center><math> \frac{\partial \Phi}{\partial t} + \frac 1{2} \nabla \Phi \cdot \nabla \Phi + \frac {P}{\rho} + g Z = C </math></center>
or
<center><math> \frac{P}{\rho} = - \frac{\partial\Phi}{\partial t}-\frac{1}{2}\nabla\Phi \cdot \nabla \Phi - g Z + C </math></center>

The value of the constant <math> C </math> is immaterial.
It is also woth noting that the
angular momentum conservation principle is contained in
<math> \nabla \times \vec V = 0 </math>
In particualr, if the particles are modelled as spheres, this equation implies no angular velocity at all times.

= Derivation of Nonlinear Free-surface Condition =

[[Image:Free_surface.jpg|center|Figure]]

A very important result is the boundary condition at the free surface of the fluid and air. There are two condions which relate the free surface displacement <math>\zeta(X,Y,t)</math> and the velocity potential <math>\Phi(X,Y,Z)</math> at the free surface. The dynamic condition is derived from the Bernouilli 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.

== Method I ==

We derive the dynamic condition directly from Bernouilli's equation.
On <math> Z=\zeta; \ P=P_a \equiv \mbox{Atmospheric Pressure} </math>.
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>
We will simplify this equation by showing that we are free to set the pressure to any value.

The kinematic condition is derived as follows.
On <math>Z=\zeta</math> The mathematical function
<center><math>Z-\zeta(X,Y,t)\equiv\tilde{f}(X,Y,Z,t)</math></center>

is always zero when tracing a fluid particle on the free surface. So the [http://en.wikipedia.org/wiki/Total_derivative substantial or total derivative] of <math>\tilde{f}</math> must vanish, thus

<center><math> \frac{D\tilde{f}}{Dt}=0=\left ( \frac{\partial}{\partial t} + \vec V \cdot \nabla \right ) \tilde{f}=0, \qquad \mbox{on} \ Z=\zeta </math></center>

Expanding we obtain:
<center><math> \left (\frac{\partial}{\partial t} + \vec V \cdot \nabla \right ) (Z-\zeta) =0, \qquad \mbox{on} \ Z=\zeta </math></center>
which in turn implies that
<center><math> \frac{\partial\zeta}{\partial t}+\frac{\partial\Phi}{\partial X} \frac{\partial\zeta}{\partial X}+\frac{\partial\Phi}{\partial Y}\frac{\partial\zeta}{\partial Y}=\frac{\partial\Phi}{\partial Z}, \ Z=\zeta </math></center>
which is the Kinematic free-surface condition.

We have already derived the dynamic condtion from Bernoulli's equation
<center>
<math> \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla \Phi \cdot \nabla \Phi + g \zeta = \mathbb{C} - \frac{P_a}{\rho}, \ 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

=== Gauss theorem ===

We need to use the following theorems often called [http://en.wikipedia.org/wiki/Gauss_theorem Gauss theorem] although more properly known as the divergence theorem.
We begin with the vector version. If
<math> \vec n </math> is the unit normal vector pointing inside the volume <math> V </math> and
<math> f(\bar{X})</math> is an arbitrary sufficiently differentiable scalar function, then
<center>
<math> \iiint_{V} \nabla f dv = -\iint_{S} f_{s} \vec n ds </math>
</center>

Note the three scalar identities that follow:
<center><math> \iiint_{{V}} \frac{\partial f}{\partial x} dv = - \iint_{S} f n_1 ds </math></center> 
<center><math> \iiint_{{V}} \frac{\partial f}{\partial y} dv = - \iint_{S} f n_2 ds </math></center> 
<center><math> \iiint_{{V}} \frac{\partial f}{\partial z} dv = - \iint_{S} f n_3 ds </math></center>

The scalar version is as follows where
<math> \vec{V}</math> is an arbitrary sufficiently differentiable vector function
<center>
<math> \iiint_{\bar{V}} \nabla \cdot \vec V = - \iint_{s} \vec V \cdot \vec n ds </math></center>
The scalar identity ofter used to prove mass conservation principle.

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

[[Image:Force_coordinates.jpg|right|frame|Force coordinates]]

The force <math>\vec{F}</math> is given by
<center><math>
\vec{F} = \iint_{S} P\vec{n}ds
</math></center>
where <math>P</math> is the pressure and the moment <math>\vec{M}</math> is given by
<center><math>
\vec{F} = \iint_{S} P(\vec{x}\times\vec{n})ds
</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

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

== Method II ==

When tracing a fluid particle on the free surface the hydrodynamic pressure given by Bernoulli (after the constant <math>C</math> has been set equal to zero) must vanish as we follow the particle:

<center><math> \frac{D}{Dt} \left \{ \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi+gZ \right \} =0, \qquad Z=\zeta </math></center>
or
<center><math> \left ( \frac{\partial}{\partial t} + \vec V \cdot \nabla \right ) \left ( \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi +gZ \right ) =0, \qquad Z=\zeta </math></center>

This condition also follows upon elimination of <math>\zeta</math> from the kinematic & dynamic conditions derived under method I.

-----

This article is based on the MIT open course notes and the original article can be found
[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/1814747D-3A05-45A1-BDE5-2CEF40DEA25F/0/lecture1.pdf here]

[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]

Conservation Laws and Boundary Conditions

2008-04-07T13:22:30Z

Len: typo fix

We begin by derving 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.

= Non-Linear Free-surface Condition =

[[Image:Non_linear_free_surface.jpg|center|frame|Co-ordinate System]]

We begin by defining the coordinate system.

<center><math>
\begin{matrix}
&(X,Y,Z) &: &\mbox{Earth Fixed Coordinate System} \\
&\vec X &: &\mbox{Fixed Eulerian Vector} \\
&\vec V &: &\mbox{Flow Velocity Vector at} \ \vec X \\
&\zeta &: &\mbox{Free Surface Elevation}
\end{matrix} </math></center>

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

<center><math>\nabla \times \vec 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 through out the fluid domain. There is an important theorem in vector calculus [http://en.wikipedia.org/wiki/Irrotational_vector_field] that if <math>\nabla \times \vec V = 0</math> then we can express the irrotational vector as the gradiant of a scalar function, i.e.
<center><math>
\vec V = \nabla \Phi
</math></center>
where <math>\Phi(\vec{X},t)</math> is called the [http://en.wikipedia.org/wiki/Velocity_potential velocity potential].

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 considerably simplifications and that certain processes, most notably wave breaking are in no way covered by this theorey. In fact, the process of wave breaking is extremely complicated and is much less well understood that the potential flow model.

== 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
<center><math> \nabla \cdot \vec V = 0 </math></center>
This condition in turn implies, using the definition of the velocity potential that
<center><math> \nabla \cdot \nabla \Phi = 0 \Rightarrow \nabla^2 \Phi = 0 </math></center> or

<center><math> \frac{\partial^2 \Phi}{\partial X^2} + \frac{\partial^2\Phi}{\partial Y^2} + \frac{\partial^2\Phi}{\partial Z^2} = 0, \quad \mbox{Laplace Equation} </math></center>.

== Conservation of linear momentum ==

We begin with [http://en.wikipedia.org/wiki/Euler_equations Euler's equation] in the absence of viscosity
<center> <math>
\frac{\partial \vec V}{\partial t} + (\vec V \cdot \nabla) \vec V = - \frac1{\rho} \nabla P + \vec g
</math> </center>
where
<math> P(\vec X, t) </math> is the fluid Pressure at <math>(\vec X, t)</math> and
<math> \vec g = - \vec k g </math> is the acceleration due to gravity where
<math> \vec k </math> is the unit vector pointing in the positive z-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.

We then use the folliwng vector identity
<center><math> (\vec V \cdot \nabla) \vec V = \frac 1{2} \nabla (\vec V \cdot \vec V) - \vec V \times ( \nabla \times \vec V) </math></center>
and since we have irrotational flow (i.e. <math> \nabla \times \vec V = 0 </math>) Euler's equation becomes
<center><math> \frac{\partial \vec V}{\partial t} + \frac 1{2} \nabla (\vec V \cdot \vec V) = - \frac 1{\rho} \nabla P - \nabla (g Z) </math></center>
where we have used <math> \nabla Z = \vec K </math>.

We now substitute <math> \vec V = \nabla \Phi </math> and we obtain
<center><math> \nabla (\frac{\partial \Phi}{\partial t} + \frac 1{2} \nabla \Phi \cdot \nabla \Phi + \frac {P}{e} + g Z ) = 0 </math></center>
We now observe that if
<center><math> \nabla F( \vec X, t) =0 \quad \Longrightarrow \quad F (\vec X, t) = C </math></center>
where <math> C </math> is an arbitrary constant.

=== Bernoulli's equation ===

[http://en.wikipedia.org/wiki/Bernoulli%27s_equation Bernoulli's equation] follows

<center><math> \frac{\partial \Phi}{\partial t} + \frac 1{2} \nabla \Phi \cdot \nabla \Phi + \frac {P}{\rho} + g Z = C </math></center>
or
<center><math> \frac{P}{\rho} = - \frac{\partial\Phi}{\partial t}-\frac{1}{2}\nabla\Phi \cdot \nabla \Phi - g Z + C </math></center>

The value of the constant <math> C </math> is immaterial.
It is also woth noting that the
angular momentum conservation principle is contained in
<math> \nabla \times \vec V = 0 </math>
In particualr, if the particles are modelled as spheres, this equation implies no angular velocity at all times.

= Derivation of Nonlinear Free-surface Condition =

[[Image:Free_surface.jpg|center|Figure]]

A very important result is the boundary condition at the free surface of the fluid and air. There are two condions which relate the free surface displacement <math>\zeta(X,Y,t)</math> and the velocity potential <math>\Phi(X,Y,Z)</math> at the free surface. The dynamic condition is derived from the Bernouilli 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.

== Method I ==

We derive the dynamic condition directly from Bernouilli's equation.
On <math> Z=\zeta; \ P=P_a \equiv \mbox{Atmospheric Pressure} </math>.
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>
We will simplify this equation by showing that we are free to set the pressure to any value.

The kinematic condition is derived as follows.
On <math>Z=\zeta</math> The mathematical function
<center><math>Z-\zeta(X,Y,t)\equiv\tilde{f}(X,Y,Z,t)</math></center>

is always zero when tracing a fluid particle on the free surface. So the [http://en.wikipedia.org/wiki/Total_derivative substantial or total derivative] of <math>\tilde{f}</math> must vanish, thus

<center><math> \frac{D\tilde{f}}{Dt}=0=\left ( \frac{\partial}{\partial t} + \vec V \cdot \nabla \right ) \tilde{f}=0, \qquad \mbox{on} \ Z=\zeta </math></center>

Expanding we obtain:
<center><math> \left (\frac{\partial}{\partial t} + \vec V \cdot \nabla \right ) (Z-\zeta) =0, \qquad \mbox{on} \ Z=\zeta </math></center>
which in turn implies that
<center><math> \frac{\partial\zeta}{\partial t}+\frac{\partial\Phi}{\partial X} \frac{\partial\zeta}{\partial X}+\frac{\partial\Phi}{\partial Y}\frac{\partial\zeta}{\partial Y}=\frac{\partial\Phi}{\partial Z}, \ Z=\zeta </math></center>
which is the Kinematic free-surface condition.

We have already derived the dynamic condtion from Bernoulli's equation
<center>
<math> \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla \Phi \cdot \nabla \Phi + g \zeta = \mathbb{C} - \frac{P_a}{\rho}, \ 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

=== Gauss theorem ===

We need to use the following theorems often called [http://en.wikipedia.org/wiki/Gauss_theorem Gauss theorem] although more properly known as the divergence theorem.
We begin with the vector version. If
<math> \vec n </math> is the unit normal vector pointing inside the volume <math> V </math> and
<math> f(\bar{X})</math> is an arbitrary sufficiently differentiable scalar function, then
<center>
<math> \iiint_{V} \nabla f dv = -\iint_{S} f_{s} \vec n ds </math>
</center>

Note the three scalar identities that follow:
<center><math> \iiint_{{V}} \frac{\partial f}{\partial x} dv = - \iint_{S} f n_1 ds </math></center> 
<center><math> \iiint_{{V}} \frac{\partial f}{\partial y} dv = - \iint_{S} f n_2 ds </math></center> 
<center><math> \iiint_{{V}} \frac{\partial f}{\partial z} dv = - \iint_{S} f n_3 ds </math></center>

The scalar version is as follows where
<math> \vec{V}</math> is an arbitrary sufficiently differentiable vector function
<center>
<math> \iiint_{\bar{V}} \nabla \cdot \vec V = - \iint_{s} \vec V \cdot \vec n ds </math></center>
The scalar identity ofter used to prove mass conservation principle.

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

[[Image:Force_coordinates.jpg|right|frame|Force coordinates]]

The force <math>\vec{F}</math> is given by
<center><math>
\vec{F} = \iint_{S} P\vec{n}ds
</math></center>
where <math>P</math> is the pressure and the moment <math>\vec{M}</math> is given by
<center><math>
\vec{F} = \iint_{S} P(\vec{x}\times\vec{n})ds
</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

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

== Method II ==

When tracing a fluid particle on the free surface the hydrodynamic pressure given by Bernoulli (after the constant <math>C</math> has been set equal to zero) must vanish as we follow the particle:

<center><math> \frac{D}{Dt} \left \{ \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi+gZ \right \} =0, \qquad Z=\zeta </math></center>
or
<center><math> \left ( \frac{\partial}{\partial t} + \vec V \cdot \nabla \right ) \left ( \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi +gZ \right ) =0, \qquad Z=\zeta </math></center>

This condition also follows upon elimination of <math>\zeta</math> from the kinematic & dynamic conditions derived under method I.

-----

This article is based on the MIT open course notes and the original article can be found
[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/1814747D-3A05-45A1-BDE5-2CEF40DEA25F/0/lecture1.pdf here]

[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]