Chapter 3 Details of the numerical approach- es in FD

25

Φ = Φ_!+ Φ_!+ Φ_!+ ⋯ (2.39)

Replacing 2.39 in 2.37 and developing in series through 2.38 the following expression is ob- tained:

𝑝 𝜂 = −𝜌 Φ_! + Φ_!+ Φ_{! !}1

2∇ Φ_!+ Φ_! + Φ_!

∙ ∇ Φ_! + Φ_!+ Φ_! + 𝑔𝑧 −1 2𝑈^!

− 𝜌∇ Φ_!+ Φ_!+ Φ_{! !}1

2∇ Φ_!+ Φ_! + Φ_!

∙ ∇ Φ_! + Φ_!+ Φ_! + 𝑔𝑧 −1

2𝑈^! 𝜂 +1 2⋯

(2.40)

𝑝 𝜂 = −𝜌 Φ_!!+ Φ_!!+1

2∇Φ_!∙ ∇Φ_! +1

2∇Φ_! ∙ ∇Φ_! +1

2∇Φ_!∙ ∇Φ_! +1

2∇Φ_!∙ ∇Φ_!+1

2∇Φ_!∙ ∇Φ_!+1

2∇Φ_!∙ ∇Φ_!+ 𝑔𝑧

−1 2𝑈^! 𝜂

(2.41)

By the equation 2.40 this following equation is obtained:

𝑝_! = −𝜌 Φ_!!+1

2∇Φ_!∙ ∇Φ_!+1

2∇Φ_!∙ ∇Φ_! (2.42)

𝑝_! = −𝜌 Φ_!!+1

2∇Φ_!∙ ∇Φ_! (2.43)

By the equation 2.41:

∇𝑝_! = −𝜌∇ Φ_!!+1

2∇Φ_!∙ ∇Φ_! (2.44)

∇𝑝_! = −𝜌∇ 𝑔𝑧 (2.45)

26 Then this equation can be substituted in 2.42 and 2.43.

In order to numerically solve the integral the surface of integration is replaced with a double line integration. Then a cylinder with director the contour of the stationary waterplane and as generators the sides considered locally vertical is constructed.

The terms of pressure that are part of the integrand are those ensure the second order when integrated on the curvilinear abscissa of the water line and the instantaneous wave elevation (which is a quantity of the first order of infinitesimal).

𝐹_! = ∫_!∫_!^!𝑝𝑛 + 𝑝 𝛼×𝑛 + ∇𝑝 𝛼×𝑥 + 𝑢 𝑛 + ∇𝑝 𝛼×𝑥 + 𝑢 𝛼×𝑛 𝑑𝑍𝑑𝐶

! (2.46) Since only the terms of second order have to be retain, the expression becomes:

𝐹_! = ∫_!∫_!^!𝑝_!𝑛 + 𝑝_! 𝛼×𝑛 + ∇𝑝_! 𝛼×𝑥 + 𝑢 𝑛𝑑𝑍𝑑𝐶

! (2.47)

The added resistance can be defined as the component of the forces of the second order in the longitudinal direction.

The formulas previously identified are then rewritten considering only the longitudinal com- ponent of the normal.

𝐼 = ∫_!∫_!^!𝑝_!𝑛_!+ 𝑝_! 𝛼×𝑛 _! + ∇𝑝_! 𝛼×𝑥 + 𝑢 𝑛_!𝑑𝑍𝑑𝐶 (2.47)

𝐼𝐼 = 𝑝_!+ ∇𝑝_! 𝛼×𝑥 + 𝑢 𝑛_!𝑑𝑆

+ 𝑝_!+ ∇𝑝_! 𝛼×𝑥 + 𝑢 𝛼×𝑛 _!𝑑𝑆

!

(2.48)

the term 𝐼 in particular the second term 𝐼_! = ∫_!∫_!^!𝑝_! 𝛼×𝑛 _!𝑑𝑍𝑑𝐶 is now considered Regarding the term of pressure −𝜌 ^!_!∇Φ ∙ ∇Φ + gz −^!_!𝑈^! for integration only −𝜌𝑔𝑧 inter- ests.

27 The development of 𝛼×𝑛 _! leads to the expression 𝛼^!𝑛_!− 𝛼_!𝑛_{! !} and if only heave and pitch motion in head sea are considered, only the term 𝛼^!𝑛_! holds.

𝐼_! = ∫_!∫_!^!− 𝜚𝑔𝑧𝛼_!𝑛_!𝑑𝑍𝑑𝐶 (2.49)

with integration a term of degree higher than the second is obtained, therefore the term in its totality can be neglected.

Now considering:

𝑝_! = −𝜌𝑔𝜁_! (2.50)

𝜁_! = 𝑍 − 𝑢_!− 𝑢_!𝑥 + 𝑢_!𝑦 (2.51)

𝜁_!=instantaneous wave profile The first term become:

𝐼_! = ∫_!∫_!^!− 𝜚𝑔𝜁_!𝑛𝑑𝑍𝑑𝐶

= ∫_!∫_!^!− 𝜚𝑔 𝑍 − 𝑢_!− 𝑢_!𝑥 + 𝑢_!𝑦 𝑛𝑑𝑍𝑑𝐶 (2.52)

Now the third term is:

𝐼_! = ∫_!∫_!^!∇𝑝_! 𝛼×𝑥 + 𝑢 𝑛_!𝑑𝑍𝑑𝐶 (2.53)

the development of the term in parentheses in 2.53 leads to:

𝐼_! = ∫_!∫_!^!− 𝜌𝑔 𝛼_!𝑥_! − 𝛼_!𝑥_!+ 𝑢_! 𝑛_!𝑑𝑍𝑑𝐶 (2.54)

comparing the 2.52 and the 2.54 remains only:

𝐼 = ∫_!∫_!^!− 𝜌𝑔𝑍𝑑𝑍𝑑𝐶 (2.55)

Considering the second term:

28 𝐼𝐼_! = ∫_! 𝑝_!+ ∇𝑝_! 𝛼×𝑥 + 𝑢 𝑛_!𝑑𝑆

! (2.56)

𝑝_! = −𝜚 Φ_!!+ ∇Φ_!∇Φ_!+1

2∇Φ_!∇Φ_! (2.57)

The terms that contain the derivatives of the second order disappear with the time average.

Only the following term remains:

𝑝_! = −𝜌 1

2∇Φ_!∇Φ_! (2.58)

then developing the term ∇𝑝^! 𝛼×𝑥 + 𝑢 the following expression can be obtained:

𝜕

𝜕𝑥 Φ_!!− 𝑈Φ_{!! !}+ 𝜕

𝜕𝑦 Φ_!!− 𝑈Φ_{!! !} + 𝜕

𝜕𝑧 Φ_!!− 𝑈Φ_{!! !} 𝑢_!+ 𝑧𝛼_!− 𝑦𝛼_{! !} + 𝑢_!+ 𝑧𝛼_!− 𝑦𝛼_{! !}+ 𝑢_! + 𝑧𝛼_!− 𝑦𝛼_{! !}

(2.59)

Then the components of added resistance are:

𝐼 = ∫_!∫_!^!− 𝜌𝑔𝑍^!𝑑𝑍𝑑𝐶 (2.60)

𝐼𝐼 = −𝜚∫_!1

2∇Φ_!∇Φ_! ∙ 𝑛_!𝑑𝑆 (2.62)

𝐼𝐼𝐼 = ∫_! 𝑝_!+ ∇𝑝_! 𝛼×𝑥 + 𝑢 𝛼×𝑛 _!𝑑𝑆

! (2.63)

𝐼𝑉 = ∫_! 𝜕

𝜕𝑥 Φ_!!− 𝑈Φ_!! 𝑧𝛼_!+ 𝜕

𝜕𝑧 Φ_!!− 𝑈Φ_!! 𝑢_!− 𝑥𝛼_! 𝑛_!𝑑𝑆 (2.64)

29

2.2.3 Analysis of non-stationary wave field

This is an experimental method, which allows to determine, through tank test, the characteris- tics of the non-stationary wave field generated by ship which advances in a regular incident wave field.

Ohkusu (1980) proposed and applied this technique first. Subsequently an alternative method has been proposed by Naito (1992).

The quantities that are measured are the amplitude and phase of the wave motions; from the- se, through mathematical analysis, Kochin functions are obtained, which represent the energy of non-stationary waves in function of their angle of propagation 𝜃.

By obtaining the Kochin functions 𝐻_! 𝜃 and 𝐻_! 𝜃 it is possible to calculate the added re- sistance by the formula 2.65 proposed by Maruo, which is reported here with the notation used by Ohkusu and Naito.

𝑅_!" = 2𝜋𝜌 ∫_{!! !}^!!! +∫_!

!

! !−∫_{! !}^{!! !} 𝐻_! 𝜃 ^!𝑘_! 𝑘_!cos 𝜃 − 𝑘 cos 𝛼 1 − 4Ω cos 𝜃 𝑑𝜃 + ^!!!!! 𝐻_! 𝜃 ^!

!!_!

𝑘_! 𝑘_!cos 𝜃 − 𝑘 cos 𝛼

1 − 4Ω cos 𝜃 𝑑𝜃 (2.65)

It should be noted that, having the possibility to carry out tank tests, the added resistance can be easily obtained detecting the average resistance in the wave field and subtracting from this the resistance in still water.

30

Chapter 3 Details of the numerical approach- es in FD

The added resistance due to ship motion-Near Field 3.1

methods

The evaluation of added resistance began from the analysis of the 3D BEM (FD) program available in house (Bruzzone,2003).

This program is a seakeeping software that calculates the added resistance through the direct integration of pressure using a method in 3D Rankine panels that has been developed at DITEN.

In this case the velocity potential is approximated by a free stream potential with a series of perturbation potentials due to ship motions and waves. These perturbation potential are de- termined through a BEM method based on Rankine Sources. These elements are special pan- els of the first order evaluated through numerical integration and positioned on the surface of the ship and in part on the free surface.

The panel system of the models and the free surface has been developed through another software, also developed at DITEN.

The input data needed to create the panels model are the geometry of the hull; it consists of a sufficient number of cross-sections and possibly a longitudinal section in the plane of sym- metry to represent the line of bow.

In the case of complex geometries the shell at the points where the most important variations in the form occurs must be divide into different zones and for each area the panelling must be make.

Finally, an appropriate number of panels for the hull and the free surface must be chosen.

31

Figure 3.1: Example of Kriso’ s panel system

After creating the panels model with the free surface 3D BEM can be used, including other data input required:

- Wave amplitude (generally we consider a unitary amplitude) - Position of gravity and radius of inertia

- Froude number

- Angle of encounter of the wave field incident - Wave frequency of the incident wave field

In addition to the added resistance as a function of wave frequency you obtain also as output the values of the transfer function, amplitude and phase, of the motions of surge, heave and pitch, sway, roll, yaw.

Thereafter, using a post-processor, part of the program that calculates the added resistance us- ing the near field method is analysed.

In particular two additional terms of resistance that were not present in the program are added in order to improve the result. Then this post-processor has been extended to oblique angle of encounter.

Subsequently then the calculation of added resistance in short waves is investigated. In fact, the discrepancy with the experimental data in the short waves may be induce by the nonlinear effects induced by the forward speed, such as nonlinear bow waves.

So to improve the calculated results in shortwave a program that calculates, using the method of Kuroda, a semi-empirical correction to be applied to the added resistance calculated has been implemented.

Y X

Z

32 The input data required for this program are:

- Waterline

- Geometric data of the hull - Froude number

- Wave frequencies - Wave lengths

Figure 3.2: Summary of necessary steps

33

The added resistance due to reflected wave at the 3.2

bow

Comparing the measured and the computed added resistance, fairly good agreement is shown as a whole except for the range of shorter wave length where the measured value larger than the computed one. It is known that there is a remarkable discrepancy particularly in the case of high-speed ship such as a container ship.

The calculation methods for added resistance due to wave reflection are based on an equation for drift force acting on an object in short waves.

An asymptotic formula for the added resistance in short waves have been obtained by Faltinsen et al. (1980) with hypothesis of the perfect reflection of the incident waves.

The formula was:

𝐹_! = ∫_!𝐹_!sin 𝜑𝑑𝑙 (3.1)

Where:

𝐹_! =1

2𝜚𝑔𝜁_!^! sin^! 𝜑 − 𝜒 2𝜔_!𝑉

𝑔 1 + cos 𝜑 cos 𝜑 − 𝜒 (3.2)

𝜁_! and 𝜔_! are wave amplitude and frequency respectively.

The domain of integration of Eq. (3.2) is showed in Fig. 3.3.

Figure 3.3: Coordinate system for the short waves range added resistance calculation methods (from Papaniko- laou, 2009)

34 This formulation generates good results for quite full bodies; however this expression ob- tained not satisfactory results for fine hull as container ships.

In order to improve this drawback Fuji and Takahashi (1975) have considered an approxi- mate calculation method of the resistance increase due to the wave reflection at the bow on the basis of the formula of wave drifting force. Thereafter Kuroda et al. (2008) further exam- ined Fuji and Takahashi semi-empirical method and considered a better expression which gives good correspondence with experimental data, in particular for fine and high-speed ship such as container ship. Then this last semi-empirical method is applied.

𝑅_!"# = 1

2𝜚𝑔𝜁_!^!𝐵 ∙ 𝐵_!𝛼_! 1 + 𝛼_! (3.3)

Figure 3.4 Definition of the bluntness coordinate [adapted from Kuroda et al., 2008]

Eq. (3.3) is composed of effect of draft and frequency ( ), effect of advance speed ( ), and bluntness factor .

𝛼_! 𝑘_!𝑑 = 𝜋^!𝐼_! 𝑘_!𝑑

𝜋^!𝐼_! 𝑘_!𝑑 + 𝐾_!^! 𝑘_!𝑑 (3.4)

𝐼_! is modified Bessel function of first kind of order 1; 𝐾_! is modified function of the second kind of order 1; 𝑘_! = 𝜔_!^! 𝑔 ; d is draft.

α_d 1+α_u

B_f

35 𝐵_!= 1

𝐵 ∫_!sin^! 𝛼 + 𝛽_! sin 𝛽_!𝑑𝑙 + ∫_!!sin^! 𝛼 − 𝛽_! sin 𝛽_! (3.5)

I and II are domains of integration in Fig. 3.4 ; dl is line element; is slope of line element.

1 + 𝛼_! = 1 + 𝐶_!𝐹_! (3.6)

𝐶_! = 𝑚𝑎𝑥 10.0, −310𝐵_!+ 68 (3.7)

𝐹_! is the Froude number

Then the total resistance increment of a ship in waves can be evaluated approximately as the sum of the resistance enhancement due to ship motions and the component due to wave re- flection at the bow.

𝑅_!"!"! = 𝑅_!"+ 𝑅_!"! (3.8) βw

36

Chapter 4 Mathematical Formulation and out- lines of numerical method in TD

In order to calculate ship motions and added resistance the same approach is presented in time domain.

In fact in the previous chapter a frequency-domain approach using a Fourier decomposition of the problem to predict the added resistance and consequently the ship motions has been presented. This approach considers unit wave amplitude. Then the results derived from this method are only valid for waves of small amplitude.

Instead, by using a time domain approach, it is easier to introduce the non-linear aspects, thus obtaining more accurate results.

A quick outline of the Boundary Value Problem (BVP) is presented for the study of a ship advancing in sea waves. The numerical solutions of this approach are here described both for the two-dimensional and for the three-dimensional case.

It is assumed the fluid is inviscid, incompressible and the flow irrotational.

Considering this hypothesis the hydrodynamic problem may be represented using a velocity potential Φ, with the fluid velocity u (x, t) = ∇Φ.

Another simplified formulations considered in this approach are the Laplace’s equation and the Bernoulli’s equation.

Boundary value problem and equation of motion 4.1

The complexity of the motions problem for a ship advancing in waves is mainly caused by non-linear effects. Then, in order to simplify the mathematical description of the problem, different hypothesis are considered. In particular considering small forces, small inertial terms of the body and also small wave amplitude, only the linear terms are retained.

37 A ship advancing and oscillating in the free surface of a fluid domain with a defined forward speed is considered. Two reference systems are identified: the inertial and the body-fixed frame, indicated by 𝑋 and 𝑥 respectively .The BVP is solved in the body-fixed coordinates.

The ship position at each time is individuated from the orientation of the body-axes in rela- tion to the inertial reference. In Fig. 4.1 the reference system of the general problem and the six degree of freedom of a ship model are showed.

Figure 4.1: The two different reference systems.

The fluid domain D is composed by the free-surface 𝑆_!" and the body surface 𝑆_!, the depth is considered infinite.

With the hypothesis of perfect fluid and irrotational flow, the velocity potential Φ that sa- tisfies the Laplace equation can be considered.

∇^!Φ = 0 𝑥 𝜖 𝐷 (4.1)

The boundary condition applied on the hull surface 𝑆_! is expressed in the following form:

U

38

∇Φ ∙ 𝑛 = 𝑉_!∙ 𝑛 𝑜𝑛 𝑆_! (4.2)

Where 𝑛 = 𝑛_!, 𝑛_!, 𝑛_! is the unit normal vector at a point P on the body and 𝑉_! is the com- ponent of the hull velocity at point P.

Then regarding free surface two boundary conditions 𝑧 = 𝜂 𝑥, 𝑦, 𝑧, 𝑡 are imposed:

• A kinematic condition (a point on the free-surface remains on the free-surface):

𝜕𝜂

𝜕𝑡 −𝜕Φ

𝜕𝑧 + ∇Φ∇𝜂 = 0 𝑜𝑛 𝑆_!" (4.3)

• A dynamic condition (the pressure on the free surface is equal to the atmospheric pressure):

−𝜕Φ

𝜕𝑡 −1

2∇Φ ∙ ∇Φ − 𝑔𝜂 =𝑝_!

𝜌 𝑜𝑛 𝑆_!" (4.4)

Neglecting the higher order terms and considering Φ = Φ_!+ 𝜙 the linearized boundary con- ditions are obtained:

𝜕𝜂

𝜕𝑡 − 𝜙_!+ Φ_!∙ ∇𝜂 = 0 𝑜𝑛 𝑆_!" (4.5) 𝑝_!

𝜌 = −𝜙_!− ∇Φ_! ∙ ∇ϕ − 𝑔𝜂 𝑜𝑛 𝑆_!" (4.6)

Where Φ_! is the steady base potential and 𝜙 is a perturbation potential φ. The latter is com- posed by the sum of the incident wave potential 𝜙_!, the diffraction potential 𝜙_! and the six radiation potentials 𝜙_! relating to the six body motions.

Φ 𝑥, 𝑡 = −Φ_!+ 𝜙_! + 𝜙_!+ 𝜙_!

!

!!!

(4.8)

39 Then with this decomposition of the total velocity potential Φ and considering Φ_! as a zero order basic potential (Neumann-Kelvin linearization) the boundary conditions, expressed in 4.5 and 4.6, evolve into the following form:

𝜕𝜂_!

𝜕𝑡 −𝜕𝜙_!

𝜕𝑧 + ∇Φ_!∙ ∇𝜂_! = 0 𝑜𝑛 𝑆_!" (4.9)

𝜕𝜙_!

𝜕𝑡 = −𝑔𝜂_!+ ∇Φ_! ∙ ∇ϕ_! − 𝑝_!

𝜌 𝑜𝑛 𝑆_!" (4.10)

The body boundary conditions take the following forms:

𝜕𝜙_!

𝜕𝑛 = 𝜂_!𝑛_!+ 𝑚_! 𝑜𝑛 𝑆_! (4.11)

𝜕𝜙_!

𝜕𝑛 +𝜕𝜙_!

𝜕𝑛 = 0 𝑜𝑛 𝑆_! (4.12)

where 𝑛 is the generalized normal vector and 𝑚 is expressed as:

𝑚_! = 𝑛 ∙ ∇ ∇Φ_! 𝑓𝑜𝑟 𝑗 = 1, 2, 3 (4.13)

𝑚_! = 𝑛 ∙ ∇ 𝑥×∇Φ_! 𝑓𝑜𝑟 𝑗 = 4, 5, 6 (4.14)

The m-terms represent the interaction between the steady and unsteady problem.

Considering Φ_! = −𝑈𝑥 the free-surface conditions evolve into this formulation:

𝜕𝜂

𝜕𝑡+ 𝑈𝜕𝜂

𝜕𝑥−𝜕ϕ

𝜕𝑧 = 0 𝑜𝑛 𝑆_!" (4.15)

𝜕𝜙

𝜕𝑡 + 𝑈𝜕𝜙

𝜕𝑥+ 𝑔𝜂 −𝑝_!

𝜌 = 0 𝑜𝑛 𝑆_!" (4.16)

and the 𝑚 term is simplified in:

40 𝑚_! = (0, 0, 0, 0, 𝑈𝑛_!, −𝑈𝑛_!) (4.17)

Using Bernoulli’s equation the pressure can be evaluated to compute the forces acting on the body surface:

𝑝 = −𝜌 𝜕𝜙

𝜕𝑡 − 𝑈𝜕𝜙

𝜕𝑥+ 𝑔𝑧 (4.18)

in 4.18 the time derivative of the potential is calculated using this formulation:

𝜕𝜙

𝜕𝑡

!

= 𝜙^!− 𝜙^!!∆!

∆𝑡 (4.19)

Obtaining the total pressure surface, the force acting on the mean body surface (for the linear problem) or on the instantaneous immerged body surface (for the ”body exact” approach) can be derived by integrating the 4.18:

𝐹 = ∫_!

!"𝑝𝑛 𝑑𝑆 (4.20)

where 𝑛 is the generalized normal of the body.

Then solving an equation of motion the ship motion can be obtained .

The Euler equation is simplified as a linear equation for the six variables 𝜂_!, where k = 1, 2, 3 represent the translation indexes and k = 4, 5, 6 the rotation indexes respect to an inertial ref- erence system advancing at the ship speed U. This simplified rigid body equation is showed in eq. 4.21 considering a generalized 6 x 1 vector.

𝑀_!"

!

!!!

𝜂_! 𝑡 = 𝐹_!^!"# = 𝐹_!^!+ 𝐹_!^!!+ 𝐹_!^!! (4.21)

41 where M is the mass matrix of the ship and the 𝐹_!^!, 𝐹_!^!! and 𝐹_!^!! are gravitational, hydrostatic and hydrodynamic components respectively.

The mathematical formulation for the 3D linear prob- 4.2

lem

As set in the previous section, the linearized boundary value problem is described by the sys- tem of equations:

∇^!Φ = 0 𝑥 𝜖 𝐷 (4.22a)

∇Φ ∙ 𝑛 = 𝑉_!∙ 𝑛 𝑜𝑛 𝑆_! (4.22b)

𝜕𝜂

𝜕𝑡 + 𝑈∂η

𝜕𝑥−𝜕ϕ

𝜕𝑧 = 0 𝑜𝑛 𝑆_!" (4.22c)

𝜕ϕ

𝜕𝑡 = −𝑔𝜂 + 𝑈 ∙ ∇ϕ −𝑝_!

𝜌 𝑜𝑛 𝑆_!" (4.22d)

Where:

Φ = 𝑡𝑜𝑡𝑎𝑙 𝑝𝑒𝑟𝑡𝑢𝑟𝑏𝑎𝑡𝑖𝑜𝑛 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝜙 = 𝑝𝑒𝑟𝑡𝑢𝑟𝑏𝑎𝑡𝑖𝑜𝑛 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙

𝑛 = 𝑡ℎ𝑒 𝑢𝑛𝑖𝑡 𝑛𝑜𝑟𝑚𝑎𝑙 𝑣𝑒𝑐𝑡𝑜𝑟 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑡𝑜𝑤𝑎𝑟𝑑 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑 𝑉_! = 𝑡ℎ𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑎 𝑝𝑜𝑖𝑛𝑡 𝑃 𝑜𝑓 𝑡ℎ𝑒 ℎ𝑢𝑙𝑙 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑆_! = 𝑡ℎ𝑒 ℎ𝑢𝑙𝑙 𝑠𝑢𝑟𝑓𝑎𝑐𝑒

𝑆_!" = 𝑡ℎ𝑒 𝑓𝑟𝑒𝑒 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝜂 = 𝑡ℎ𝑒 𝑤𝑎𝑣𝑒 𝑒𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛

42 𝑈 = 𝑡ℎ𝑒 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦

𝑝_! = 𝑡ℎ𝑒 𝑎𝑡𝑚𝑜𝑠𝑝ℎ𝑒𝑟𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝜌 = 𝑡ℎ𝑒 𝑤𝑎𝑡𝑒𝑟 𝑑𝑒𝑛𝑠𝑖𝑡𝑦

𝑔 = 𝑡ℎ𝑒 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛

Two different time stepping schemes can be used to the solution of the free surface boundary conditions.

In general the fluid flow may be represented in two different ways: the Lagrangian and the Eulerian.

In the Lagrangian scheme the fluid motions are obtained following an individual fluid particle through space and time. In the Eulerian scheme a specific position in the space that fluid crosses is considered. In this last case, the flow velocity is obtained at position x at time t.

In view of these different schemes there are two kind of solutions: the mixed Eulerian- Lagrange and the fully Eulerian solutions

Zhang et al. (2010b) used the first solution that rewrites the boundary conditions using the re- lation between the Eulerian and Lagrangian coordinates systems:

𝛿 𝛿𝑡= 𝜕

𝜕𝑡+ 𝑣 ∙ ∇ (4.23)

Consequently the free surface boundary conditions begin:

𝛿𝜂 𝛿𝑡 =𝜕𝜙

𝜕𝑧 + 𝑈𝜕𝜂

𝜕𝑥+ 𝑣 ∙ ∇η (4.24)

𝛿𝜙

𝛿𝑡 = −𝑔𝜂 + 𝑈𝜕𝜙

𝜕𝑥−𝑝_!

𝜌 + 𝑣 ∙ ∇ϕ (4.25)

where 𝑣= (u, v, 0) is the velocity of the collocation points; in particular the horizontal transla- tion is represented by the velocity of advance of the ship u = −U and v is provided in such a way that the collocation points follow the paths around the body.

In the second solution the velocity of the collocation points is considered as 𝑣 = (0, 0, 0). In this case, the Lagrangian and the Eulerian derivatives coincide and the collocation points are fixed in the translating coordinate system. The advantage of this scheme is that no-regridding

43 is necessary but the disadvantage is that the free-surface slope ∇η must be numerically com- puted.

In this thesis the last solution was chosen due to the linearity of the problem and the related unnecessary regridding and the wave slope was calculated using numerical differentiation furnished by Fortran Libraries. This faster scheme was preferred as a first approach to this problem since the regridding is complex and time consuming.

Another important aspect to solve this BVP regards the evaluation of the influence coeffi- cients for the body panels, necessary to calculate the source strengths. Hess and Smith (1964) proposed a mathematical technique to evaluate the incompressible potential flow about an ar- bitrary body shape.

The sources are located on the center of each body panel and the velocity potential satisfies the Laplace equation with the boundary conditions, see 4.22, in all points of the fluid domain except when source point and collocation point coincide, 𝑥_! ≅ 𝑥_!.

Hess and Smith (1964) proposed different methods based on the distance between x^S𝑒 xC to evaluate the influence coefficients. In fact on the base of this distance R there are different methods to calculate the potential. R is defined in the following equation:

𝑅 = (𝑥_!− 𝑥_!)^!+ (𝑦_!− 𝑦_!)^!+ (𝑧_!− 𝑧_!)^! (4.26)

and it is compared with the max length of the quadrilateral edge.

If 𝑅 > 4𝑆_!, an exact method is used; if 2𝑆_! < 𝑅 < 4𝑆_! a quadrupole pole is used while, if a point closer to the body surface is considered and 𝑅 < 2𝑆_!, a monopole method obtains the velocity coefficients.

In this thesis the solution characterized by a non-desingularized body Bandyk (2009) was chosen, as detailed in chapter 5.

This choice is due to the fact that the body desingularization may induce a sort of instability in the final solution, when the hull shapes are particularly thin. Then with this particular ap- proach a code without limitation about hull shapes is obtained.

44

The mathematical formulation for the added re- 4.3

sistance in TD

After the BVP in time domain were solved, then the force and the motions were obtained, it is possible to evaluate the added resistance with near field method, as detailed in section 2.2.2, but in time domain Kim & Kim (2011).

In the near field method added resistance is the averaged second order force in the longitudi- nal direction. It can be evaluated integrating the second-order pressure on the body surface that is obtained using Bernoulli’s equation and Taylor’s expansion. As seen in the section 2.2.2, the added resistance is consisted of only linear terms then the second-order boundary value problem is not to be solved. The difference respect to the frequency approach is the fact that in domain approach added resistance is obtained averaging the second-order signal.

The components of added resistance in time domain are:

𝐼 = ∫_!"1

2𝜌𝜚 𝜁 − 𝑧_! ^!𝑛𝑑𝐿 (4.27)

𝐼𝐼 = −𝜌∬_!

!𝑔𝑧_!∙ 𝑛_!𝑑𝑆 (4.28)

𝐼𝐼𝐼 = −𝜌∬_!

! 𝑔𝑧_!+𝜕𝜙_!

𝜕𝑡 − 𝑈𝜕𝜙_!

𝜕𝑡 ∙ 𝑛_!𝑑𝑆 (4.29)

𝐼𝑉 = −𝜌∬_!

!

1

2∇𝜙_!∙ ∇𝜙_! ∙ 𝑛𝑑𝑆 (4.30)

𝑉 = −𝜌∬_!

!𝑥_!∙ ∇ 𝜕𝜙_!

𝜕𝑡 − 𝑈𝜕𝜙_!

𝜕𝑡 𝑛𝑑𝑆 (4.31)

Where:

𝜙_! = 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙

𝑛_! = 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒𝑎𝑟 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑜𝑓 𝑛𝑜𝑟𝑚𝑎𝑙 𝑣𝑒𝑐𝑡𝑜𝑟 𝑜𝑛 𝑡ℎ𝑒 𝑏𝑜𝑑𝑦 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑛_! = 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛 𝑜𝑟𝑑𝑒𝑟 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑜𝑓 𝑛𝑜𝑟𝑚𝑎𝑙 𝑣𝑒𝑐𝑡𝑜𝑟 𝑜𝑛 𝑡ℎ𝑒 𝑏𝑜𝑑𝑦 𝑠𝑢𝑟𝑓𝑎𝑐𝑒

45 𝑥_! = 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒𝑎𝑟 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑜𝑑𝑦 𝑚𝑜𝑡𝑖𝑜𝑛

𝑆_! = 𝑡ℎ𝑒 ℎ𝑢𝑙𝑙 𝑠𝑢𝑟𝑓𝑎𝑐𝑒

Figure 4.2: Components of added resistance on S175 containership: Fn=0.20,wave heading angle=180° (from Kim, 2010)

To understand the importance of the various terms of the added resistance the fig. 4.2 shows the components of added resistance on S175 containership. In this thesis the term (II) was not calculated because is very small and then negligible.

It can be notated that the most principal component of added resistance is the waterline- integral term ( ). Also terms ( ) and represent a significant part of added resistance.

In the case of barge with zero speed it can be observed that the principal part is the same of the forward speed problem but the term (IV) is predominant on the other terms (Pinkster and Van Oortmerssen (1977).

The fig. 4.3 shows how this difference is due of the radial component that is more significant in the forward speed problem.

In fact in the fig. 4.3 the diffraction component is much smaller than the radiation component and it doesn’t change considerably regarding the wave length.

I III (V )

46

Figure 4.3: Radiation and diffraction components of added resistance on S175 containership (from Kim, 2010)

The mathematical formulation for the 2D body exact 4.4

problem

This methodology considers the exact body wetted-surface at each time step then it is a more accurate method than the two-dimensional linear approach. This solution represents a first in- troduction of the non-linear effects. It was decided to develop this methodology for two- dimensional case as a basis for further developments in three-dimensional version. In fact in 2D the re-pannellization of the body surface at each time-step and the related new description of the free-surface are less complicated.

In this approach the partial desingularization, detailed in the next chapter, is used to avoid the numerical singularity presented in the calculation of the influence of a panel on itself. Then the body surface is not desingularized and a distribution of constant sources is applied.

A general two-dimensional section floating on the free-surface with three degree of freedom is considered. In fig. 4.4 an earth-fixed Cartesian coordinate system is shown. The y-axis co- incides with the calm water level and the z-axis is positive upward.

47

Figure 4.4 Reference system

As seen in the previous section, considering perfect, homogeneous, incompressible, inviscid fluid, the irrotational flow and the depth infinite, the fluid motions can be described using the velocity potential Φ that satisfy Laplace’s equation:

∇^!Φ = 0 (4.32)

The free-surface boundary conditions can be simplified as follow:

𝜁_!− Φ_!= 0 𝑜𝑛 𝑧 = 0 (4.33)

Φ_!− 𝑔𝜁 = 0 𝑜𝑛 𝑧 = 0 (4.34)

where z represents the free-surface deformation ζ(y, t).

The body boundary condition is applied on the mean hull surface:

𝜕Φ

𝜕𝑛 = 𝑉_! 𝑜𝑛 𝑆_! (4.35)

Where 𝑉_! is the body normal velocity and n is the normal vector pointing to the fluid.

48 In time domain, initial conditions at t = 0 is imposed in the whole fluid domain as follow:

Φ = Φ_! = 0 (4.36)

After solving this BVP, the potential and its derivatives are computed; so using linearized Bernoulli’s equation the total pressure acting on the body can be evaluated as follow:

𝑝 = −𝜌 𝜕Φ

𝜕𝑡 − 𝑈𝜕Φ

𝜕𝑥 + 𝑔𝑧 (4.37)

In this equation the time derivative of the potential on the body is obtained directly by incre- mental ratio.

𝜕Φ

𝜕𝑡 =Φ^!− Φ^!!!

Δ𝑡 (4.38)

Integrating 4.37 on the main hull surface, the forces acting on the body can be obtained.

49

Chapter 5 Details of the numerical approach- es in TD

In this chapter the numerical solution of linear BVP is detailed. In particular, the partial des- ingularized approach is presented.

The advantages of this solution procedure respect the conventional methods are underlined.

Moreover other relevant aspects in time formulation are analysed as the mesh size, the values of the distance of desingularization, the time stepping.

The indirect method (with partial desigularization) in 5.1

3D

In order to solve the BVP in the conventional methods the differential equation with the boundary conditions are generally satisfied with a fundamental solution. Considering the col- location point of each panel a sources distribution is chosen, the number of sources and the number of the panels are the same.

Since the source points overlap the collocation points, an accurate evaluation of the singular integrands is required; in the time domain approach this solution increase the computational time to solve the boundary equation. Therefore the conventional approach might be unsuita- ble in the time procedure.

Considering the singularity placed slightly away from the centroid of the panels, an easier approach can be obtain, called desingularized method . There are two different versions of the desingularized approach: the direct method or the indirect method.

In the direct method, considering Green’s second identity the boundary integral equation is derived, and the solution is obtained directly by solving the boundary integral equation.

50 In the indirect method, a singularity distribution is considered and the boundary conditions are satisfied using the strength of the singularities in the boundary integral equation. After ob- taining the strength, the potential can be evaluated.

The difference between the two methods is due to the fact that the direct method considers the integration surface as the boundary of the problem, while in the indirect method the control surface is considered (Cao, 1991). In this study this method was chosen for the solution of the problem, then a brief analysis of this approach is presented in the next section.

In this work the indirect method is applied with a partial application of the desingularized sources. In particular in the case of the hull is characterized by thin shapes the desingulariza- tion of the body surface may induce some instability problems.

In fact using the desingularization on the body surface an over-position of the source points may occur and cause instability in the time-histories of the forces.

In this case, considering constant source strength panels on the body surface (a partial des- ingularization) this problem can be avoid.

Then the first step to solve the three-dimensional problem is to discretize the free-surface and the body surface. A grid of flat quadrilateral panels is used for this discretization.

In the body surface the panel dimensions decrease in the part of the hull where the geometry shapes are complex, as in the zone of the bow.

In the zone of free surface closed to the hull the dimension of the free surface panels is simi- lar to that of the near body panels, so a geometrical adherence is maintained. While in the zone far from the body the size of the panels are increased to avoid reflection of waves gen- erated by body.

Desingularized sources are distributed only above the calm water surface, so the collocation points and the source points never coincide and a source constant distribution for the body panels is considered . The distance of desingularization depends on the local mesh size Lee (1992):

𝐷_! = 𝑎 ∙ 𝑆_! (5.1)

Where 𝑆_! is the area of the panel and 𝑎 is the desingularized parameter considered as a = 1.

51

Figure 5.1: Desingularized sources (only free-surface).

For the numerical solution of the BVP the velocity potential at any collocation point can be formulated as:

Φ 𝑥_!! = 𝜎 𝑥_!!

!!"

!!!

𝐺 𝑥_!!; 𝑥_!!

!"#$_!

𝑑𝑆_!+ 𝜎 𝑥_!!

!!"

!!!

𝐺 𝑥_!!; 𝑥_!!

(5.2)

where 𝐺 𝑥_!!; 𝑥_!! is the Rankine source Green functions for the 3D case:

𝐺 𝑥_!!; 𝑥_!! = 1

𝑟 (5.3)

52 where:

𝑅 = (𝑥_!!− 𝑥_!!)^!+ (𝑦_!!− 𝑦_!!)^!+ (𝑧_!!− 𝑧_!!)^! (5.4)

The G function must satisfy the boundary conditions both on the hull and on the free surfac- es, considering the infinite water depth assumption, and the integral equations for the un- known strength become:

𝜎 𝑥_!!

!!"

!!!

𝐺 𝑥_!!; 𝑥_!!

!"#$_!

𝑑𝑆_!+ 𝜎 𝑥_!!

!!"

!!!

𝐺 𝑥_!!; 𝑥_!! = Φ 𝑥_!! 𝑜𝑛 𝑆_!" (5.5)

𝜎 𝑥_!!

!_!"

!!!

𝑛_!∇_!!𝐺 𝑥_!!; 𝑥_!!

!"#$_!

𝑑𝑆_!+ 𝜎 𝑥_!!

!_!"

!!!

𝑛_!∇_!!𝐺 𝑥_!!; 𝑥_!!

= 𝑛_!∙ 𝑉 𝑜𝑛 𝑆_!

(5.6)

Using this partial desingularization the calculation of the influence matrix composed by the influence coefficients and, the evaluation of the integral equation, are simplified with a reduc- tion of the computational time.

The equations 5.5 and 5.6 are applied at all the collocation point 𝑥_! and an 𝑁 × 𝑁 linear sys- tem is derived:

𝐴_!" ∙ 𝜎_! = 𝑏_! (5.7)

The term 𝐴_!" is the matrix where all the influence coefficients are located, 𝜎_! is the vector of unknown source strength and 𝑏_! represents the vector of the boundary conditions.

Considering a linear approach, the inverse of the influence matrix is not re-calculated at every time-step. After finding the source strengths, the potential and its derivatives can be evaluated for every panel of the fluid domain.

Once the BVP is solved, the pressure can be evaluated considering Bernoulli’s equation:

𝑝 = −𝜌 𝜕𝜙_!

𝜕𝑡 − 𝑈𝜕𝜙_!

𝜕𝑥 (5.8)

53 where 𝜙_! is the radiation velocity potential in the kth mode.

The force and moment can be calculated integrating 5.8 on the body surface, for each time step. Then the hydrodynamic coefficients can be derived.

Time-stepping in 3D 5.2

Applying a 3rd-order Adams-Bashforth scheme the free-surface elevation and the potential are updated. Time-stepping is obtained with the follow expressions:

𝜁^!!!! = 𝜁^!+ ∆𝑡

12 23 𝜕𝜙_!

𝜕𝑧

!

− 16 𝜕𝜙_!

𝜕𝑧

!!∆!

+ 5 𝜕𝜙_!

𝜕𝑧

!!!∆!

(5.9)

𝜙_!^!!!! = 𝜙_!^!− 𝑔∆𝑡

12 23 𝜁 ^!− 16 𝜁 ^!!∆!+ 5 𝜁 ^!!!∆! (5.10) The time-step size is chosen with respect to period T in this range:

𝑇 100 < ∆𝑇 < 𝑇 200 (5.10)

If the time-step is too large, the spatial resolution of the free-surface can resulted in stability problems. Park and Troesch (1992) and Wang and Troesch (1997) introduced a free-surface stability index by relating the panel dimensions with the time-step size:

𝐹𝑆𝑆 = 𝜋𝑔 ∆𝑡^! Δ𝑥 (5.11)

To obtain a stable solution in this work FSS < 1 is considered as conservative value.

The indirect method (with partial desigularization) in 5.3

2D Body exact

For the numerical solution the free-surface is divided into an inner and an outer zone (Zhang

54 (2007) and Bandyk (2009). This approach provides to reduce the wave reflection using the outer zone as a sort of numerical beach.

By considering the deep water dispersion relation:

𝜆 =2𝜋𝑔

𝜔^! (5.12)

the dimension of the domain can be evaluated. The number of wave-lengths and nodes per wave-length can be chosen as appropriate. In the zone of the intersection between the free surface e the body surface the size of the panels must be chosen so that the two boundaries connect properly.

Regarding the outer region distribution, Lee (1992) proposed the following expression:

𝑑𝑦_!"#$%! = 𝑑𝑦 ∙ 1.0378^!(!!!)! (5.13)

Where 𝑑𝑦 represents the spacing in the inner domain.

In this work a partial desingularization approach is considered. Then Desingularized sources are distributed only above the calm water surface.

The desingularized distance, as in Bandyk (2009), is obtained by:

𝐷_! = 𝑎 ∙ 𝑙_! (5.14)

where a represents parameter that in this case is considered unitary and 𝑙_! is the dimension of the panel.

As described in the linear case 3D, the expression of the potential at any point in the fluid domain is expressed in terms of Rankine singularities, considered as source points over the free-surface and as source constant distribution for the body panels (partial desingulariza- tion):

Φ 𝑥_!! = 𝜎 𝑥_!!

!_!"

𝐺 𝑥_!!; 𝑥_!! 𝑑𝑙 + 𝜎 𝑥_!!

!_!"

!!!

𝐺 𝑥_!!; 𝑥_!! (5.16)