
English: 
Hello welcome to my talk, All about Fluids
it is the third part of the talk on the topic of
the boundary element method for wave-structure interaction. This talk
focuses on how the hydrostatic and hydrodynamic forces and the moments can
be calculated once the relevant potential functions have been solved
which would include the hydrostatic forces, wave exciting forces, as well as
the radiation forces in terms of added mass and the radiation damping coefficient
the flow pressure, or more specifically, the gauge pressure in fluid

English: 
Hello welcome to my talk, All about Fluids.
It is the third part of the talk on the topic of
the boundary element method for wave-structure interaction. This talk
focuses on how the hydrostatic and hydrodynamic forces and the moments can
be calculated once the relevant potential functions have been solved
which would include the hydrostatic forces, wave exciting forces, as well as
the radiation forces in terms of added mass and the radiation damping coefficient
the flow pressure, or more specifically, the gauge pressure in fluid

English: 
can be obtained from the Bernoulli's equation, expressed as this. Here the capital
PHI is the time dependent total velocity potential
for many applications in wave-structure interactions, the high order terms in the
pressure is generally omitted as a result of a linearisation to the problem
and this had been employed in the analysis for the first-order
forces and motions as we have this. in frequency domain, the pressure is
expressed as this, here the potential Phi is the complex amplitude of the

English: 
can be obtained from the Bernoulli's equation, expressed as this. Here the capital
PHI is the time dependent total velocity potential
for many applications in wave-structure interactions, the high order terms in the
pressure is generally omitted as a result of a linearisation to the problem
and this had been employed in the analysis for the first-order
forces and motions as we have this. in frequency domain, the pressure is
expressed as this, here the potential Phi is the complex amplitude of the

English: 
potential function. Substitute the diffraction and
radiation potentials into the linearised pressure, we have the equation as this.
Here we can see the radiation potential is the sum of radiation
potentials of unit amplitude six-degrees of freedom motion: surge, sway,
heave as the translational motions and roll, pitch and yaw as the rotational
motion as seen in this figure.
the forces and the moments acting on the marine structure are calculated as the
integrals of the fluid pressure on the body surface Sb.

English: 
potential function. Substitute the diffraction and
radiation potentials into the linearised pressure, we have the equation as this.
Here we can see the radiation potential is the sum of radiation
potentials of unit amplitude six-degrees of freedom motion: surge, sway,
heave as the translational motions and roll, pitch and yaw as the rotational
motion as seen in this figure.
the forces and the moments acting on the marine structure are calculated as the
integrals of the fluid pressure on the body surface Sb.

English: 
here normal vector n can be seen in here, while the vector r is the vector from
the origin of the coordinate to the body surface. For a convenience, we can define
the new vector component, the conventional vector components: n1, n2 and n3,
for the translational motion and the vector components: n4, n5 and n6,
for the rotational motion. this vector component as same as that we
have used for defining the body boundary condition for the radiation potentials
as such the forces and the moments can
be expressed in a uniform format as this.

English: 
here normal vector n can be seen in here, while the vector r is the vector from
the origin of the coordinate to the body surface. For a convenience, we can define
the new vector component, the conventional vector components: n1, n2 and n3,
for the translational motion and the vector components: n4, n5 and n6,
for the rotational motion. this vector component as same as that we
have used for defining the body boundary condition for the radiation potentials
as such the forces and the moments can be expressed in a uniform format as this.

English: 
here the subscript i equals to 1 to 6 corresponding to the six degrees
freedom motion: surge, sway, heave, roll, pitch and yaw,
so substitute the fluid pressure into the integral for the first component,
we have the force fi as this, and further separate the total force
we have the expression for the hydrodynamic and hydrostatic forces as this,
this time is the wave excitation, and this represents the radiation force
and this is the hydrostatic force.
for formulating the hydrodynamic and hydrostatic forces for the structure

English: 
here the subscript i equals to 1 to 6 corresponding to the six degrees
freedom motion: surge, sway, heave, roll, pitch and yaw,
so substitute the fluid pressure into the integral for the first component,
we have the force fi as this, and further separate the total force
we have the expression for the hydrodynamic and hydrostatic forces as this,
this time is the wave excitation, and this represents the radiation force
and this is the hydrostatic force
for formulating the hydrodynamic and hydrostatic forces for the structure

English: 
motion, here we defined the parameters for the hydrostatics
the volume displaced by the structure can be calculated in three directions,
on x-axis, on y-axis and on z-axis, respectively, and all three volumes
should be same if the calculation is accurate enough. This information can be
very useful in wave-structure interaction analysis when we need to check whether
all generated panels are orientated correctly. If all panels are all orientated
correctly, these three volumes should be very close if they are not identical.

English: 
motion, here we defined the parameters for the hydrostatics
the volume displaced by the structure can be calculated in three directions,
on x-axis, on y-axis and on z-axis, respectively, and all three volumes
should be same if the calculation is accurate enough. This information can be
very useful in wave-structure interaction analysis when we need to check whether
all generated panels are orientated correctly. If all panels are all orientated
correctly, these three volumes should be very close if they are not identical.

English: 
and the coordinates of the buoyancy center, xb, yb and zb are given by these
formulas.
The coordinates of the buoyancy center are needed when we calculate the
hydrostatic restoring coefficient for the structure
the hydrostatic forces can be calculated as this, for all 6 components for the six
degrees of freedom motion, and in principle, this hydrostatic forces are
the so-called zero-order forces, these zero-order forces would be the largest
for wave-structure interaction, but they would be balanced out with the structure

English: 
and the coordinates of the buoyancy center, xb, yb and zb are given by these formulas.
The coordinates of the buoyancy center are needed when we calculate the
hydrostatic restoring coefficient for the structure.
the hydrostatic forces can be calculated as this, for all 6 components for the six
degrees of freedom motion, and in principle, these hydrostatic forces are
the so-called zero-order forces, these zero-order forces would be the largest
for wave-structure interaction, but they would be balanced out with the structure

English: 
mass in equilibrium. however in the analysis of wave-structure interaction,
due to the motion of the structure there will be the net hydrostatic forces, given as this,
the equation 1 here, which can be regarded as the restoring forces and
this hydrostatic forces can be regarded as the first-order hydrostatic forces,
with the non-zero c_ij, given as this. Some of these may vanish depending on the
symmetry of the structure
for the radiation force if we exchange the order of the integral and the summation,

English: 
mass in equilibrium. However, in the analysis of wave-structure interaction,
due to the motion of the structure there will be the net hydrostatic forces, given as this,
the equation 1 here, which can be regarded as the restoring forces and
these hydrostatic forces can be regarded as the first-order hydrostatic forces,
with the non-zero c_ij, given as this. Some of these may vanish depending on the
symmetry of the structure
for the radiation force if we exchange the order of the integral and the summation,

English: 
we have this. Substitute the radiation body boundary condition into the
equation, and then we have this expression.
Now we can define the coefficient f_ij by this. This coefficient can be
understood as the complex force in the direction i due to the motion of unit
amplitude in the direction j, as such, we have the expression as this for the
total radiation force.
now if we enclose the fluid domain using the body surface Sb, the larger control
surface Sc, the free surface Sf, and seabed Sz. We can formulate the

English: 
we have this. Substitute the radiation body boundary condition into the
equation, and then we have this expression.
Now we can define the coefficient f_ij by this. This coefficient can be
understood as the complex force in the direction i due to the motion of unit
amplitude in the direction j, as such  we have the expression as this for the
total radiation force
now if we enclose the fluid domain using the body surface Sb, the larger control
surface Sc, the free surface Sf, and seabed Sz. We can formulate the

English: 
equation as this. using the Green's theorem, for the radiation potential Phi_i
and Phi_j, and for the radiation potentials, we have shown the integral on
the control surface, a very large surface, Sc and the integrals on free
surface Sf and seabed Sz would be all 0, for the details of this
please see my talks on the boundary element method for wave-structure
interaction: part 1 and part 2. So as such we can have the integral
equation for the radiation potentials on body surface, Sb only, and this

English: 
equation as this. using the Green's theorem, for the radiation potential Phi_i
and Phi_j, and for the radiation potentials, we have shown the integral on
the control surface, a very large surface, Sc and the integrals on free
surface Sf and seabed Sz would be all 0, for the details of this
please see my talks on the boundary element method for wave-structure
interaction: part 1 and part 2. So as such we can have the integral
equation for the radiation potentials on body surface, Sb only, and this

English: 
equation shows the coefficient f_ij is the symmetric, means f_ij equals to f_ji.
if the structure moves on the sea or near the free surface, the coefficient f_ij
would contain both real and imaginary parts, expressed as this, and a_ij is
called as the added mass, and b_ij is the radiation damping due to the wave radiation.
so we can write the total radiation force as this, the  equation 2.
the wave exciting forces are the forces due to the incoming wave and the

English: 
equation shows the coefficient f_ij is the symmetric, means f_ij equals to f_ji.
if the structure moves on the sea or near the free surface, the coefficient f_ij
would contain both real and imaginary parts, expressed as this, and a_ij is
called as the added mass, and b_ij is the radiation damping due to the wave radiation.
so we can write the total radiation force as this, the equation 2.
the wave exciting forces are the forces due to the incoming wave and the

English: 
scattered wave on the structure. the incoming wave could cause the fluid
Krylov force on a fixed structure. so the wave excitation can be expressed as this.
this is the director integral for the wave excitation if the scattering
potential Phi_S and the radiation potential Phi_I are both solved. and here
the incoming wave potential Phi_0 is a known function, if the wave amplitude and
the wave frequency and the water depth are given,
based on the Green's theorem, again, we can construct an equation for the scattering
and the radiation potentials on an enclosed the boundary Sb + Sc + Sf

English: 
scattered wave on the structure. the incoming wave could cause the fluid
Krylov force on a fixed structure. so the wave excitation can be expressed as this.
this is the director integral for the wave excitation if the scattering
potential Phi_S and the radiation potential Phi_I are both solved. and here
the incoming wave potential Phi_0 is a known function, if the wave amplitude and
the wave frequency and the water depth are given,
based on the Green's theorem, again, we can construct an equation for the scattering
and the radiation potentials on an enclosed the boundary Sb + Sc + Sf

English: 
+Sz as this. for a similar reason, the integral for the scattering
and radiation potentials on the very large control surface Sc and on the free
surface Sf and on the seabed Sz, would be all 0, hence we have equation
as this, and put these together, we have the integral on Sb only for the
scattering and radiation potentials, as this.
Therefore, the wave excitation now can be expressed after using the relation,
We have the expression as this. Now we apply the body boundary condition for the
scattering potential, so we have the expression for the wave excitation as this,

English: 
+Sz as this. for a similar reason, the integral for the scattering
and radiation potatios on the very large control surface Sc and on the free
surface Sf and on the seabed Sz, would be all 0, hence we have equation
as this, and put these together, we have the integral on Sb only for the
scattering and radiation potentials, as this
therefore the wave excitation now can be expressed after using the relation,
We have the expression as this. Now we apply the body boundary condition for the
scattering potential, so we have the expression for the wave excitation as this,

English: 
the equation 3. This equation is called the Haskind relation.
Using the Haskind relation means we can use the radiation potential to calculate
the exciting forces, even if the scattering potential is not solved.
once the forces and the moments are calculated, we can establish the dynamic
equation for the structure motion, based on the Newton's second law of
motion as this. For that translational motion and for the rotational motion,
if we put this in the component form, we can write the dynamic equation as this,
here fi is the total hydrostatic and hydrodynamic forces, expressed as the

English: 
the equation 3. This equation is called the Haskind relation.
using the Haskind relation means we can use the radiation potential to calculate
the exciting forces, even if the scattering potential is not solved.
once the forces and the moments are calculated, we can establish the dynamic
equation for the structure motion, based on the Newton's second law of
motion as this. For that translational motion and for the rotational motion,
if we put this in the component form, we can write the dynamic equation as this,
here fi is the total hydrostatic and hydrodynamic forces, expressed as the

English: 
sum of hydrostatic force, radiation force and exciting force as this, and M_ij
is called mass matrix, given as this. here m is the mass of the structure, and I_ij
is calculated as this, and the coordinates xg, yg and zg, are
the coordinates of the center of gravity

English: 
sum of hydrostatic force, radiation force and exciting force as this, and M_ij
is called mass matrix, given as this. here m is the mass of the structure, and I_ij
is calculated as this, and the coordinates xg, yg and zg, are
the coordinates of the center of gravity.
