
English: 
Hello welcome to my talk, All about Fluids.
This talk is the first part of the boundary element method
for wave-structure interactions on the fundamentals of the velocity potentials and
the boundary integral equation.
in this talk, I will first show the difference between the body flying in the air and
the structures on the seas,
in terms of constructing the boundary element method,
and then how we can use the Green's theorem and the de-singularisation method
to establish the boundary integral equation for the wave structure interaction,

English: 
Hello welcome to my talk, All about Fluids.
This talk is the first part of the boundary element method
for wave-structure interactions on the fundamentals of the velocity potentials and
the boundary integral equation.
in this talk, I will first show the difference between the body flying in the air and
the structures on the seas,
in terms of constructing the boundary element method,
and then how we can use the Green's theorem
and the de-singularisation method
to establish the boundary integral equation for the wave structure interaction,

English: 
and why the special Green function is chosen for simplifying the boundary integral equation,
so that we can implement the numerical scheme for the wave structure interaction.
As it has been shown the wave velocity potential function
can be expressed as this in deep water
with the fluid particle velocity component expressed as this.
in this plot, we can see the fluid velocities
are changing with the position and with the time,
The structure on the sea could move freely
under the wave actions in six degrees of freedom motion.
If we compare the structure on the sea with the body flying in the air

English: 
and why the special Green function is chosen for simplifying the boundary integral equation,
so that we can implement the numerical scheme for the wave structure interaction.
As it has been shown the wave velocity potential function
can be expressed as this in deep water
with the fluid particle velocity component expressed as this.
in this plot, we can see the fluid velocities
are changing with the position and with the time,
The structure on the sea could move freely
under the wave actions in six degrees of freedom motion.
If we compare the structure on the sea with the body flying in the air

English: 
We can see the differences:
For the flying body the problem would be same as the fixed body in a uniform flow
while the structure on the sea can move under the wave actions
the flow domain is simply
bounded by the body surface and the control surface for the body flying in the air
while the structure on the sea would be bounded by the body surface,
free surface; seabed and the control surface.
The flow is steady and uniform for the flying body
But the flow is changing
with both the time and the position for the structure on the sea.

English: 
We can see the differences:
For the flying body the problem would be same as the fixed body in a uniform flow
while the structure on the sea can move under the wave actions
the flow domain is simply
bounded by the body surface and the control surface for the body flying in the air
while the structure on the sea would be bounded by the body surface,
free surface; seabed and the control surface.
The flow is steady and uniform for the flying body
But the flow is changing
with both the time and the position for the structure on the sea.

English: 
For the wave-structure interaction
the flow can be regarded as irrotational and incompressible flow,
and the dynamic system is normally linearized.
As such, the real velocity
potential, capital PHI, would be expressed as a time independent
potential function Phi * the time factor as this, or simply expressed as this.
the time independent potential function Phi is
actually the complex amplitude of the potential function
capital PHI, and both satisfy the Laplace equation as this.

English: 
For the wave-structure interaction
the flow can be regarded as irrotational and incompressible flow,
and the dynamic system is normally linearized.
As such, the real velocity
potential, capital PHI, would be expressed as a time independent
potential function Phi
times the time factor as this, or simply
expressed as this.
the time independent potential function Phi is
actually the complex amplitude of the potential function
capital PHI,
and both satisfy the Laplace equation as this.

English: 
Our purpose for the boundary element method is to solve the potential function Phi,
which is a frequency domain function,
and for simplifying the potential problem for the marine structure and for a better physical understanding,
We can separate the potential into three parts:
the potential of the incoming flow, Phi_0
and this potential function is a known potential if the wave amplitude
frequency and the water depth are given;
the potential of the scattered wave, Phi_S,
meaning that the incoming wave would be scattered
due to the existence of the marine structure, and in this case, the structure is fixed;

English: 
Our purpose for the boundary element method is to solve the potential function Phi,
which is a frequency domain function,
and for simplifying the potential problem for the marine structure and for a better physical understanding,
We can separate the potential into three parts:
the potential of the incoming flow, Phi_0
and this potential function is a known potential if the wave amplitude
frequency and the water depth are given;
the potential of the scattered wave, Phi_S,
meaning that the incoming wave would be scattered
due to the existence of the marine structure, and in this case, the structure is fixed;

English: 
The potential of the radiated wave, Phi_R,
the waves are radiated to the structure motions
which can be superimposed on the incoming and the scattered waves
Such a separation is made
mainly for easing the boundary conditions for the boundary element method.
we will see this later in the talk.
As a linear superimposition
the three potentials could form a perfect dynamic system
for the wave-structure interactions
that is, the structure scatters the incoming wave,
together with the wave radiated away
due to the structure motion

English: 
The potential of the radiated wave, Phi_R,
the waves are radiated to the structure motions
which can be superimposed on the incoming and the scattered waves
Such a separation is made
mainly for easing the boundary conditions for the boundary element method.
we will see this later in the talk.
as a linear superimposition
the three potentials could form a perfect dynamic system
for the wave-structure interactions
that is, the structure scatters the incoming wave,
together with the wave radiated away
due to the structure motion

English: 
For the wave-structure interaction problems,
The fluid domain of interest is enclosed by the body surface, Sb
free surface, Sf, the seabed, Sz and the control surface, Sc
as seen in this figure.
The free surface condition on Sf are
given at z=0, for all three potential functions.
the free surface boundary conditions can be derived from the linearised free surface equation
given by this.
Here the free surface boundary conditions are given in frequency domain,

English: 
For the wave-structure interaction problems,
The fluid domain of interest is enclosed by the body surface, Sb
free surface, Sf,
the seabed, Sz and the control surface, Sc
as seen in this figure
The free surface condition on Sf are
given at z=0, for all three potential functions.
the free surface
boundary conditions can be derived from the linearised free surface equation
given by this
Here the free surface boundary conditions are given in frequency domain,

English: 
with the wave number capital K, defined with the wave frequency as this
The seabed conditions on Sz are very simple as this
at z = -h, in a water depth h
Basically, these conditions are the no-penetration boundary conditions
meaning that the fluid cannot penetrate the seabed
into the soil of the seabed
Under the free surface and the seabed conditions
and by solving the Laplace equation
we can obtain the potential function for the incoming wave.
in deep water

English: 
with the wave number capital K, defined with the wave frequency as this
The seabed conditions on Sz are very simple as this
at z = -h, in a water depth h
Basically, these conditions are the no-penetration boundary conditions
meaning that the fluid cannot penetrate the seabed
into the soil of the seabed
Under the free surface and the seabed conditions
and by solving the Laplace equation
we can obtain the potential function for the incoming wave.
in deep water

English: 
where the water depth h is larger than or equal to half of the wave length,
the potential function for the incoming wave in deep water is given by this, Phi_0
where A is the amplitude of the incoming wave;
Omega is the wave frequency;
capital K is the wave number in deep water;
BETA is the incident wave angle, defined with the x-axis,
see here in the plot.
In a finite water depth,
the potential function of the incoming flow is given by this,
and here h is the water depth;
k is the wave number,

English: 
where the water depth h is larger than or equal to half of the wave length,
the potential function
for the incoming wave in deep water is given by this, Phi_0
where A is the amplitude of the incoming wave,
Omega is the wave frequency,
capital K is the wave number in deep water
BETA is the incident wave angle
defined with the x-axis,
see here in the plot
in a finite water depth,
the potential function of the incoming flow is given by this ,
and here h is the water depth
k is the wave number

English: 
(which) can be solved from the dispersion relation given by this.
For the potentials of the scattered and the radiated waves,
these potentials are also subject to the body conditions and the far field condition
On the body surface Sb, the boundary condition for the scattered wave is given by this
or simply as this.
This means the no-penetration condition for the scattered wave and the incoming wave
For the radiated wave, the body boundary condition is given by this
here V is the velocity of the structure motion. This body boundary condition means

English: 
can be solved from the dispersion relation given by this
For the potentials of the scattered and the radiated waves,
these potentials are also subject to the body conditions and the far field condition
On the body surface Sb, the boundary condition for the scattered wave is given by this
or simply as this
This means the no-penetration condition for the scattered wave and the incoming wave
For the radiated wave, the body boundary condition is given by this
here V is the velocity of the structure motion. This body boundary condition means

English: 
due to the motion of the structure, the fluid has same normal velocity as on the body surface.
The far field conditions are given for the potential of the scattered and radiated wave
both disturbance potentials to that of the incoming wave
This boundary conditions mean that the disturbance potential Phi_S and Phi_R
and the water particle velocities both vanish in distance
due to the conservation of the mass and energy.
In the fluid domain of
interest, we suppose two velocity potential functions: the target potential function

English: 
due to the motion of the structure, the fluid has same normal velocity as on the body surface.
The far field conditions are given for the potential of the scattered and radiated wave
both disturbance potentials to that of the incoming wave
This boundary conditions
mean that the disturbance potential Phi_S and Phi_R
and the water particle velocities
both vanish in distance
due to the conservation of the mass and energy
In the fluid domain of
interest, we suppose two velocity potential functions: the target potential function

English: 
Phi and a known potential function Phi_0
for instance, the potential for the Green function
or from the Ranking source and both potential functions
satisfy the Laplace equation.
Using the Gauss divergence theorem, we have this equation
so the volume V is enclosed by the control surface, Sc
body surface Sb; free surface Sf and
the seabed surface Sz,
And some mathematical manipulations
as well as the application of the Laplace equation

English: 
Phi and a known potential function Phi_0
for instance,
the potential for the Green function
or from the Ranking source
and both potential functions
satisfy the Laplace equation
Using the Gauss divergence theorem, we have this equation
so the volume V is enclosed by the
control surface, Sc
body surface Sb;
free surface Sf and
the seabed surface Sz
And some mathematical manipulations
as well as the application of the Laplace equation

English: 
so we can obtain the Green's theorem as this
This Green's Theorem is a very important
and the fundamental equation for the boundary element method
However, this Green's Theorem is only valid in 3D, not in 2D,
Due to the existence of the free surface and seabed boundaries,
to simplify the problem, the known potential Phi_0 must be carefully chosen.
Following the suggestions in the famous boundary element method, WAMIT,  'Wave Analysis MIT'

English: 
so we can obtain the Green's theorem as this
This Green's Theorem is a very important
and the fundamental equation for the boundary element method
However, this Green's Theorem is only valid in 3D
not in 2D,
Due to the existence of the free surface and seabed boundaries,
to simplify the problem
the known potential Phi_0 must be carefully chosen.
Following the suggestions in the famous boundary element method, WAMIT,  'Wave Analysis MIT'
the special Green function, G

English: 
can be chosen in deep water as this,
Here r, r' and capital R
are defined in here
And in a finite water depth, h
The Green function is given by this form
here r double prime is given in this formula
The reason for choosing this Green function
it is because this Green function could satisfy
the free surface and the seabed conditions automatically,
thus, it could simplify the boundary element method a great deal
The Green's Theorem is only connect

English: 
the special Green function, G, can be chosen in deep water as this,
Here r, r' and capital R are defined in here.
And in a finite water depth, h, the Green function is given by this form
here r double prime is given in this formula.
The reason for choosing this Green function
it is because this Green function could satisfy
the free surface and the seabed conditions automatically,
thus, it could simplify the boundary element method a great deal
The Green's Theorem is only connect

English: 
if there are no singularities on the boundaries or in the fluid domain
however, when we choose the Green function G
there will be singularities on the boundary when the target point P and the
source point are coincident,that is, r = 0
Therefore we must de-singularise the singularities on the boundaries,
for the point P, we can construct a small semi-sphere as EPSILON
to exclude the point P,
so the volume would be enclosed by the control surface, the body surface,
free surface and seabed, as well as the semi-sphere, S_Epsilon

English: 
if there are no singularities on the boundaries or in the fluid domain
however, when we choose the Green function G
there will be singularities on the boundary when the target point P and the
source point are coincident,
That is, r = 0
Therefore we must de-singularise the singularities on the boundaries,
for the point P
we can construct a small semi-sphere as EPSILON,
to exclude the point P,
so the volume would be enclosed by the control surface, the body surface,
free surface and seabed
as well as the semi-sphere, S_Epsilon

English: 
So the Green's theorem is a given by this
Thus we separate the integration for surfaces
We have this
Now we will see how we can get the integration on this small semi-sphere
since on the semi-sphere, S_epsilon
the potential and its normal gradient would be taken as constants
Thus we can write the Greens
theorem as this here.
Phi is a constant for the S_epsilon
and the normal gradient of Phi is also a constant for the small semi-sphere, S_epsilon
So we calculate this integration and

English: 
So the Green's theorem is a given by this.
Thus we separate the integration for surfaces
We have this.
Now we will see how we can get the integration on this small semi-sphere
since on the semi-sphere, S_epsilon
the potential and its normal gradient would be taken as constants
Thus we can write the Green's theorem as this here.
Phi is a constant for the S_epsilon
and the normal gradient of Phi is also a constant for the small semi-sphere, S_epsilon
So we calculate this integration and

English: 
we can see here the normal gradient of the Green function
is in opposite of differentiation with r,
so we can have this, and on the semi-sphere
r is a constant, so we have this and finally we can get the integration as 2*pi.
and for the second time, the similar method can be used
and this term for a very small semi-sphere is zero
Therefore the de-singularisation on the boundaries would lead to
an equation as this.
This equation is called the boundary integral equation.

English: 
we can see here the normal gradient of the Green function
is in opposite of differentiation with r,
so we can have this, and on the semi-sphere
r is a constant, so we have this and finally we can get the integration
as 2*pi.
and for the second time, the similar method can be used
and this term for a very small semi-sphere is zero
Therefore the de-singularisation on the boundaries would lead to
an equation as this.
This equation is called the boundary integral equation.

English: 
this equation means the potential function on the boundaries
can be expressed by the potential function and the sources
distributed on the boundaries
this equation provides an equation for solving the potential function or the strength of the distributed sources
it should be noted the integration on the surface
should exclude the small area around the point P on the boundaries,
see the figure here.
Here we further separate the integration on the surfaces
we can have this
This term is the integration on the free surface and this term is the integration
on the seabed.

English: 
this equation means the potential function on the boundaries
can be expressed by the potential function and the sources
distributed on the boundaries.
this equation provides an equation for solving the potential function or the strength of the distributed sources
it should be noted the integration on the surface
should exclude the small area around the point P on the boundaries,
see the figure here.
Here we further separate the integration on the surfaces
we can have this.
This term is the integration on the free surface and this term is the integration
on the seabed.

English: 
in the next two slides, we will see what we can get
for the integration on free surface and on seabed,
because of the special Green function.
In this slide, the surface integration on the free surface is examined.
Since the chosen Green function G
satisfies the free surface boundary condition automatically
which is given by this in frequency domain
or we can write it as this.
Now, we assume the potential can be
expressed as this surface integration of

English: 
In the next two slides, we will see what we can get
for the integration on free surface and on seabed,
because of the special Green function.
In this slide, the surface integration on the free surface is examined.
Since the chosen Green function G
satisfies the free surface boundary condition automatically
which is given by this in frequency domain
or we can write it as this.
Now, we assume the potential can be expressed as this surface integration of

English: 
the Green function, with the strength  SIGMA,  this surface
is the total surface of the fluid domain
So we can calculate the normal gradient of the potential, it's given by this
So we look at the integration on the free surface
and substitute the boundary condition for the Green function,
so we have this and this.
Then we substitute
the potential and the potential normal gradient into the
surface integration we can have this and this.
So if we change the order of the
integrations on S and Sf, we have the expression as this

English: 
the Green function, with the strength SIGMA, this surface
is the total surface of the fluid domain.
So we can calculate the normal gradient of the potential, it's given by this
So we look at the integration on the free surface
and substitute the boundary condition for the Green function,
so we have this and this.
Then we substitute the potential and the potential normal gradient into the
surface integration we can have this and this.
So if we change the order of the
integrations on S and Sf, we have the expression as this

English: 
So in here we can see the integration on the free surface
the integrand would become zero if we apply the
free surface condition for the Green function
thus we will obtain the integration on the free surface this as zero
that is, the term of
the surface integration on free surface is
zero, because of the Green function
Here we are going to examine the
integration on seabed boundary Sz.
on seabed Sz at z=-h, the Green function

English: 
So in here we can see the integration on the free surface
the integrand would become zero if we apply the
free surface condition for the Green function
thus we will obtain the integration on the free surface this as zero
that is, the term of the surface integration on free surface is
zero, because of the Green function.
Here we are going to examine the
integration on seabed boundary Sz.
on seabed Sz at z=-h, the Green function

English: 
satisfies the no-penetration condition as this
We assume the potential
can be expressed as a result of the distributed sources on the entire boundary S
here GAMMA is the strengths of the
distributed sources
and the normal gradient of the potential can be expressed as this
So for the integration on the surface, Sz
by applying the seabed condition
The integration would become as this
So we can substitute
the normal gradient of the potential
into this integration,

English: 
satisfies the no-penetration condition as this
We assume the potential
can be expressed as a result of the distributed sources on the entire boundary S
here GAMMA is the strengths of the
distributed sources
and the normal gradient of the potential can be expressed as this
So for the integration on the surface, Sz
by applying the seabed condition
The integration would become as this
So we can substitute the normal gradient of the potential
into this integration,

English: 
and if we have exchanged the order of the integration on the surface, Sz and S
We have this equation.
so for the integration on Sz,
we can see here because of the Green function satisfying the seabed condition
Thus this term would become zero,
so the integration on the surface Sz would be zero
thus the term of the integration on Sz would be zero because of the
special Green function.
This is the conventional boundary integral equation on the surfaces
Sc and Sb, Sf and Sz
Because we have chosen the special Green function

English: 
and if we have exchanged the order of the integration on the surface, Sz and S
We have this equation
so for the integration on Sz,
we can see here because of the Green function satisfying the seabed condition
Thus this term would become zero,
so the integration on the surface Sz would be zero
thus the term of the integration on Sz would be zero because of the
special Green function
This is the conventional boundary integral equation on the surface
Sc and Sb, Sf and Sz
Because we have chosen the special Green function

English: 
so the integration on Sf and on Sz will vanish
thus the boundary integral equation for the wave-
structure interaction problem becomes as this,
we achieve this because we choose the special Green function,
if we separate the integration based on the surface, we have this equation
So this boundary integral equation is correct for any potential function Phi
for instance, we could have the equation for
the scattered potentials Phi_S as this,
or for the potential of the radiated wave as this.

English: 
so the integration on Sf and on Sz will vanish
thus the boundary integral equation for the wave-
structure interaction problem becomes as this,
we achieve this because we choose the special Green function,
if we separate the integration based on the surface, we have this equation
So this boundary integral equation is correct for any potential function Phi
for instance, we could have the equation for
the scattered potentials Phi_S as this,
or for the potential of the radiated wave as this.

English: 
these are for the wave-structure interaction for the problem of the scattered wave and
radiated wave.

English: 
these are for the wave-structure interaction for the problem of the scattered wave and
radiated wave.
