From potential to forces

Calculating the velocity potential is necessary for then being able to calculate the forces and pressures acting on the body. To get there, BEMUse uses the equations written in Chapter 2. In particular, the total pressure working on the surface of the body is calculated through the Bernoulli equation

𝑝(𝑥 , 𝑡) = −𝜌 (𝜕𝜙

𝜕𝑡 +1

2∇𝜙 ∙ ∇𝜙 + 𝑔𝑧) (3.32)

There are although numerous other parameters of great interest when dealing with floating platforms and hydrodynamics in general.

3.9.1 Added mass and damping

First there is added mass, which is defined as the inertia added to a system because a body moves or deflects some volume of the fluid surrounding it when subjected to forces, thus accelerating and decelerating. Since it identifies inertia, it's expressed in 𝑘𝑔 in the case of translational motions and 𝑘𝑔 ∙ 𝑚² for rotational ones. In alternative added mass can also be expressed in dimensionless form, becoming the added mass coefficient. To obtain it, it’s sufficient to divide the added mass times the displaced fluid mass.

Then there is damping, which in physics is generally described as the influence on an oscillatory system which reduces, restricts or prevents its oscillation. Damping is directly proportional to the relative velocity between the two bodies (platform and water, in this case). In the case of fluids, we refer to radiation damping which is mostly due to the viscosity of the water. It is expressed in 𝑘𝑔/𝑠 for translational motions and in 𝑘𝑔 ∙ 𝑚²/𝑠for rotational ones. Also in this case, a dimensionless damping coefficient can be expressed as the ratio of the calculated damping and the mass of displaced water.

The added mass and damping are obtained integrating the value of the radiation potential over the body:

𝐴_𝑖𝑗 − 𝑖

𝜔𝐵_𝑖𝑗 = ∬ 𝑛_𝑖𝜙_𝑟,𝑗𝑑𝑆

𝑆

(3.33) Where the subscript i indicates the particular direction of the force and j the coordinate in which the surface oscillation occurs. A11, therefore, designates the x-forces caused by an oscillation in the x-direction. This kind of motion is commonly referred to as surge motion. In the y-direction it's called sway, and in z-direction it's heave. For rotational motions, rotation around the x-, y- and z-axis are called roll, pitch and yaw, respectively.

In WAMIT the non-dimensional added mass and damping coefficients are obtained as follows 𝐴̅_𝑖𝑗 = 𝐴_𝑖𝑗

𝜌𝐿^𝑘 𝐵̅_𝑖𝑗 = 𝐵_𝑖𝑗

𝜌𝐿^𝑘𝜔 (3.34 𝑎 𝑎𝑛𝑑 𝑏)

Where k = 3 for both i, j being equal to 1, 2 or 3; k = 4 for i = 1, 2, 3 and j = 4, 5, 6 or i = 4, 5, 6 and j = 1, 2, 3; k = 5 for both i, j = 4, 5, 6.

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3.9.2 Exciting forces

The periodic forces on the body which arise due to the diffraction of the incoming harmonic wave are called exciting forces. To calculate them, the potential of the incoming wave is needed.

In the infinite-depth case, it’s expressed as 𝜙_𝐼 = 𝑖𝑔𝐴

𝜔 𝑒^𝜈𝑧𝑒𝑖𝜈(𝑥 𝑐𝑜𝑠𝛽+𝑥𝑦 𝑠𝑖𝑛𝛽) (3.35)

Where A is the wave’s amplitude, and β is its heading. Making use of Haskind relations for the diffraction potential, the forces can be expressed as a function of the radiation potential defined previously for the oscillating body:

𝑋_𝐼 = −𝑖𝜔𝜌 ∬ (𝑛_𝑖𝜙_𝑖 −𝜕𝜙_𝑖

𝜕𝑛 𝜙_𝑗) 𝑑𝑆

𝑆

(3.36) This allows to avoid the calculation of the diffraction potential on the surface, but simply to manipulate the results of the radiation potential. The exciting forces differ according to the direction of the incoming waves. Therefore, the results obtained must always be expressed indicating the angle which they refer to. This angle is usually measured starting from the positive x-axis. In this text, an angle of 45° was always used.

Exciting forces are expressed in 𝑁/𝑚 for translational motions and in 𝑁𝑚/𝑚 for rotational ones, but they are also found in their non-dimensional form

𝑋̅_𝑖 = 𝑋_𝑖

𝜌𝑔𝐴𝐿^𝑚 (3.37)

Where m = 2 for i = 1, 2, 3 and m = 3 for i = 4, 5, 6.

3.9.3 Response Amplitude Operators

The Response Amplitude Operator, or RAO, is an indication of the degree of movement induced in a floating body due to a passing hydrodynamic wave. RAOs can be calculated for each degree of freedom of the body, both translational and rotational. Like the exciting force, the direction of the incoming wave must be defined. As before, 45° was always the choice. The formula used to calculate each RAO is taken from Baghfalaki, Das, and Das (2012) [28]:

𝑍_𝑖(𝜔) =𝑋_𝑖(𝜔)

𝐷_𝑖 = 𝐹_𝑖(𝜔)

−𝜔²(𝐴_𝑖𝑖(𝜔) + 𝑀_𝑖𝑖) + 𝑖𝜔𝐵_𝑖𝑖(𝜔) + 𝐶_𝑖𝑖 (3.38) Where Xk (ω) is the motion in the frequency domain, and Dk is the corresponding wave amplitude. Fk is the exciting force, Akk is the value of added mass, Mkk the mass and inertia matrix, Bkk is the damping coefficient, Ckk is the value of the hydrostatic restoring matrix and ω is the frequency of the waves.

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The mass and inertia matrix contains the value of mass and rotational inertia of the body under inspection and is defined as

𝑀_𝑖𝑖 =

[

𝑚 0 0 0 𝑚𝑧_𝑔 −𝑚𝑦_𝑔

0 𝑚 0 −𝑚𝑧_𝑔 0 𝑚𝑥_𝑔

0 0 𝑚 𝑚𝑦_𝑔 −𝑚𝑥_𝑔 0

0 −𝑚𝑧_𝑔 𝑚𝑦_𝑔 𝐼_𝑥 −𝐼_𝑥𝑦 −𝐼_𝑥𝑧

−𝑚𝑧_𝑔 0 −𝑚𝑥_𝑔 −𝐼_𝑦𝑥 𝐼_𝑦 −𝐼_𝑦𝑧

−𝑚𝑦_𝑔 𝑚𝑥_𝑔 0 −𝐼_𝑧𝑥 −𝐼_𝑧𝑦 𝐼_𝑧 ]

(3.39)

On the other hand, the hydrostatic restoring matrix is defined as follows in the WAMIT User Manual [29]

𝐶_𝑖𝑖 = [

0 0 0 0 0 0

0 0 0 0 0 0

0 0 𝐶_3,3 𝐶_3,4 𝐶_3,5 0 0 0 0 𝐶_4,4 𝐶_4,5 𝐶_4,6

0 0 0 0 𝐶_5,5 𝐶_5,6

0 0 0 0 0 0 ]

(3.40)

Where

𝐶(3, 3) = 𝜌𝑔 ∬ 𝑛₃𝑑𝑆

𝑆𝑏

(3.41𝑎)

𝐶(3, 4) = 𝜌𝑔 ∬ 𝑦𝑛₃𝑑𝑆

𝑆_𝑏

(3.41𝑏)

𝐶(3, 5) = −𝜌𝑔 ∬ 𝑥𝑛₃𝑑𝑆

𝑆𝑏

(3.41𝑐)

𝐶(4, 4) = 𝜌𝑔 ∬ 𝑦²𝑛₃𝑑𝑆

𝑆_𝑏

+ 𝜌𝑔𝑉𝑧_𝑏− 𝑚𝑔𝑧_𝑔 (3.41𝑑)

𝐶(4, 5) = −𝜌𝑔 ∬ 𝑥𝑦𝑛₃𝑑𝑆

𝑆𝑏

(3.41𝑒)

𝐶(4, 6) = −𝜌𝑔𝑉𝑥_𝑏+ 𝑚𝑔𝑥_𝑔 (3.41𝑓)

𝐶(5, 5) = 𝜌𝑔 ∬ 𝑥²𝑛₃𝑑𝑆 + 𝜌𝑔𝑉𝑧_𝑏− 𝑚𝑔𝑧_𝑔

𝑆_𝑏

(3.41𝑔)

𝐶(5, 6) = −𝜌𝑔𝑉𝑦_𝑏+ 𝑚𝑔𝑦_𝑔 (3.41ℎ)

V is the volume of the body, ρ is the density of the water, and m is the mass of the body. The coordinates with subscript b refer to the center of buoyancy of the body, the ones with subscript g to its center of mass.

To improve the computational performance of the software, the surface integrals are calculated as the discrete sum of the values on each panel. x and y are thus the coordinates of each centroid, and n is the vector normal to the surface, with its three components.

44 These values are then normalized as follows:

𝐶̅(3, 3) = 𝐶(3, 3) 𝜌𝑔𝐿⁄ ² (3.42𝑎)

𝐶̅(3, 4) = 𝐶(3, 4) 𝜌𝑔𝐿⁄ ³ (3.42𝑏)

𝐶̅(3, 5) = 𝐶(3, 5) 𝜌𝑔𝐿⁄ ³ (3.42𝑐)

𝐶̅(4, 4) = 𝐶(4, 4) 𝜌𝑔𝐿⁄ ⁴ (3.42𝑑)

𝐶̅(4, 5) = 𝐶(4, 5) 𝜌𝑔𝐿⁄ ⁴ (3.42𝑒)

𝐶̅(4, 6) = 𝐶(4, 6) 𝜌𝑔𝐿⁄ ⁴ (3.42𝑓)

𝐶̅(5, 5) = 𝐶(5, 5) 𝜌𝑔𝐿⁄ ⁴ (3.42𝑔)

𝐶̅(5, 6) = 𝐶(5, 6) 𝜌𝑔𝐿⁄ ⁴ (3.42ℎ)

Where L is the characteristic length of the body.

RAOs have the dimension of 𝑚/𝑚 for translational motions because they measure the ratio between the motion of the body and the amplitude of the wave, while rotational ones are expressed in °/𝑚, as they evaluate the ratio between the angular motion of the body and, again, the amplitude of the wave. As for the other parameters calculated, also RAOs have their non-dimensional form

𝑍̅_𝑖 = 𝑍_𝑖 𝐴 𝐿^𝑛

(3.43) Where n = 0 for i = 1, 2, 3 and n = 1 for i = 4, 5, 6.

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