Numerical and experimental study on the wave-body interaction problem with the effects of forward speed and finite water depth in regular waves

Ocean Engineering xxx (xxxx) xxx

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Numerical and experimental study on the wave-body interaction problem with the effects of forward speed and finite water depth in regular waves Tian-Long Mei a, b, Guillaume Delefortrie b, c, Manas�es Tello Ruiz b, Evert Lataire b, Marc Vantorre b, Changyuan Chen b, Zao-Jian Zou a, d, * a

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China Maritime Technology Division, Ghent University, Belgium Flanders Hydraulics Research, Antwerp, Belgium d State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China b c

A R T I C L E I N F O

A B S T R A C T

Keywords: Time domain higher-order Rankine panel method wave-body interaction finite water depth DTC

In this study, a time domain higher-order Rankine panel method is developed and applied to evaluate the wavebody interaction problem in shallow water and regular waves. The hydrodynamic effects of forward speed and finite water depth are accounted for. In order to verify the proposed numerical method, the Duisburg Test Case (DTC) container ship is chosen as a study object. The numerical results of the wave-induced ship motions and the added resistance are validated against the data of model tests, which were carried out in the Towing Tank for Manoeuvres in Confined Water at Flanders Hydraulics Research (FHR) in cooperation with Ghent University (UGent) as part of the SHOPERA project.

1. Introduction With more and more deep-drafted ships navigating or manoeuvring near shore or in coastal areas, it is desired to have a better insight into the effects of shallow water. The influence of shallow water leads to a more complex hydrodynamic situation compared to open sea. In addi­ tion, these areas are often subjected to wave conditions. Under the presence of waves in shallow water areas, not only the safety-related issues of the manoeuvring ships should arouse concern, but also the seakeeping behaviour, such as wave-induced motions and added resis­ tance, needs to be taken into account to avoid grounding and to quantify speed reduction. Hence, it is essential to investigate the wave-induced motions and second-order wave forces in finite water depth. For this reason, many researchers have developed numerical methods to solve the wave-body interaction problems in finite water depth. In an early stage, the 2D strip theory was used for its practica­ bility and high efficiency in the initial design (Kim, 1968; Hwang and Lee, 1975; Takaki, 1977; Tasai et al., 1978; Andersen, 1979; Kim, 1999; Perunovic and Jensen, 2003; Vantorre and Journ�ee, 2003; Kim and Kim, 2012). The mentioned method, however, is only suitable for relatively high frequencies and a slender ship hull. Because of this limitation, re­ searchers turned their attention to the matched asymptotic expansion

method (MAEM), where the solutions of the far field is matched with the solution of the near field on the interface boundary by using different numerical techniques on the divided field zones (Tuck, 1970; Vorobyov and Stasenko, 2010). The main problem with this method is that because of the long wave assumption, large deviations occur when the wave length is close to the ship length. More advanced techniques are available nowadays to address this problem in shallow water, such as 3D panel methods based on Green function or Rankine source. The free surface Green function panel method has been initially applied by researchers (Oortmerssen, 1976; Chan, 1990; Li, 2001). Though the accuracy seems to be better than the 2D theory or MAEM, the form and evaluation of Green function is relatively complex and difficult. Furthermore, the Neumann-Kelvin linearization of the method cannot consider the interaction between the steady and unsteady flows, while it can be accounted for by using double-body linearization method. In comparison, the Rankine source based 3D panel method has been widely used for its flexible treatment of the free surface as well as the simpler kernel form. Moreover, the Rankine source method can more easily be extended to include nonlinear factors when addressing the ship motion in waves, such as nonlinear free surface conditions and dynamic squat, nonlinear Froude-Krylov and hydrostatic forces, see the works of Riesner et al. (2016), Riesner and el Moctar (2018). Among others, Gao et al. (2008),

* Corresponding author. School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China. E-mail address: [email protected] (Z.-J. Zou). https://doi.org/10.1016/j.oceaneng.2019.106366 Received 19 February 2019; Received in revised form 4 August 2019; Accepted 23 August 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Tian-Long Mei, Ocean Engineering, https://doi.org/10.1016/j.oceaneng.2019.106366

T.-L. Mei et al.

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Nomenclature ½Aij � ½Bij � Bj ½Cij � Fr Fi Fij F ⇀ð2Þ

Sb Sf t T TS

Added mass matrix Damping coefficient matrix B-spline basis function Restoring coefficient matrix Froude number Wave exciting force Radiation force Sum of wave exciting force and radiation force

⇀

U ¼ ðU0 ; 0; 0Þ Velocity vector r Volume of displacement β Wave angle ⇀ ξS

λ ζ ζA ζd ζI

F g G h H k L; B; d mj ½Mij �

Second-order wave force vector Gravitational acceleration Rankine source Water depth Second order transformation matrix Wave number Ship length, breadth and draft m terms Mass matrix

n Nb ; N f o xyz O XYZ

Unit normal vector Number of panels on hull surface and free surface Ship-fixed coordinate system Earth-fixed coordinate system

⇀

x; x Pi;j ; Qi;j

⇀ ⇀ ’

Field point and source point Qi;j ðshoulb be deletedÞInfluence coefficients

r

Position vector in the ship-fixed frame

⇀

Mean wetted hull surface Mean free surface Time Wave period Period of the artificial spring

υ

⇀ ⇀ ξT ; ξR

ξj ξ ϕd ϕI ϕl ϕm φs Ψ

ω0 ωe

Yao et al. (2017) applied the frequency domain Rankine panel method for ship hydrodynamic analysis in shallow water conditions. In their frequency domain method, transient dynamic effects such as the coupling effect with the nonlinear external force cannot be considered (Kim and Kim, 2012). The time domain Rankine panel method was applied for the wave-induced motion problems in shallow water by Kim and Kim (2012), Kim and Kim (2013), Feng et al. (2016), Wang et al. (2016), but their studies only focused on the zero forward speed cases. Another point of attentions is the added wave resistance. When a ship navigates in a seaway, this added resistance will lead to power loss and speed reduction. Moreover, the arising greenhouse gas emissions are now constrained by the International Maritime Organization (IMO) and the directives have already been established by the Marine Envi­ ronment Protection Committee (MEPC) from the perspective of an En­ ergy Efficiency Design Index (EEDI). Recent research projects, such as Energy Efficient Safe Ship Operation (SHOPERA) (Sprenger et al., 2017) have been carried out to examine the safety and performance of ships in €ding et al. (2014) calculated the added adverse weather conditions. So resistance by a frequency domain Rankine panel method which accounts for nonlinear steady flow. The added wave resistance has also been studied using potential flow theory by many researchers, such as Seo et al. (2014), Lyu and el Moctar (2017), Kim et al. (2017), Zhang and el Moctar (2019), Park et al. (2019). In order to fully investigate the

Wave-induced ship motion Wave length Total wave elevation Incident wave amplitude Unsteady disturbance wave elevation Incident wave elevation Newtonian cooling strength Translation and rotation vectors of ship motion Motion amplitude in j-th mode

Harmonic motion amplitude Unsteady disturbance velocity potential Incident wave velocity potential Local flow velocity potential Memory velocity potential Steady velocity potential Total velocity potential Incident wave frequency Encounter frequency

viscous effects (braking waves, etc.) on added resistance in waves, Computational Fluid Dynamics (CFD) methods are also widely applied, see for instance Riesner and el Moctar (2018), Sigmund and el Moctar (2018). However, the mentioned studies on the added resistance in waves are mostly analysed in deep water cases and investigations in view of shallow water cases are scarce. In the present study, a time domain higher-order Rankine panel method is developed to investigate the motion responses and added resistance of a ship moving in shallow water of various water depths in regular waves. The sea bottom is assumed to be flat and the image method is used to satisfy the bottom boundary condition. A good agreement is achieved when comparing the numerical results with the data of model tests which were executed in the Towing Tank for Ma­ noeuvres in Confined Water at Flanders Hydraulics Research in coop­ eration with Ghent University as part of the SHOPERA project. 2. Mathematical formulation 2.1. Boundary value problem In the frame of potential flow theory, an inertial coordinate system o xyz fixed to a ship moving with the forward speed U0 in the earthfixed coordinate system O XYZ is introduced as shown in Fig. 1. The

Fig. 1. Coordinate systems. 2

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frame 0 xyz is on the mean water surface with its origin amidships, the x-axis positive to the bow, the y-axis positive to the port side and the

∂ ∂t

z-axis pointing upwards. The total velocity potential Ψ ðx; tÞ can be written as: ⇀

�

6 � ∂ϕd X ξ_ j nj þ ξj mj ¼ ∂n j¼1

(2)

ζA g cosh½kðz þ hÞ� sin½kðx cos β þ y sin βÞ cosh kh ω0

⇀

⇀

⇀

⇀

⇀

⇀

⇀

(15)

∂ϕd ¼0 ∂n

(16)

The initial condition at t ¼ 0:

(5)

The velocity potential Ψ ðx; y; z; tÞ satisfies the following boundary value problem (BVP):

ϕd ¼ 0;

∂ϕd ¼0 ∂t

(17)

2.2. Hydrodynamic forces and motion equations

(6)

r2 Ψ ¼ 0; in ​ the ​ fluid ​ domain

(14)

The mj terms represent the interaction between the steady and unsteady flows. Since the second-order derivatives of basic velocity potential can be considered as solutions of the Dirichlet type problem when putting the first-order derivatives of φs as the right-hand-side terms, the mj terms can be evaluated directly. The detailed evaluation of the mj terms can be found in the authors’ previous work (Mei et al., 2019). The sea bottom condition on z ¼ h is given as:

(4)

kU0 cosβ

∂ϕI ∂n

ðm4 ; m5 ; m6 Þ ¼ ðn ⋅rÞ½x� ðU rφs Þ �

The encounter frequency ωe is defined as:

ωe ¼ ω0

(13)

ðm1 ; m2 ; m3 Þ ¼ ðn ⋅rÞðU rφs Þ;

where ζA is the wave amplitude, g is the gravitational acceleration, ω0 is the wave frequency, k is the wave number, h is the water depth and β is the wave angle. The dispersion relation in finite water depth is:

ω20 ¼ kg tanhðkhÞ

1 rφ ⋅rφs 2 s

ðn4 ; n5 ; n6 Þ ¼ x� n;

(3)

ωe t�

⇀

rφs ⋅rϕI þ U ⋅rφs

where

where ζI ðx; y; tÞ and ζd ðx; y; tÞ denote the incident wave elevation and the unsteady disturbance wave elevation, respectively. The velocity potential ϕI ðx; y; z; tÞ of a regular incident wave is given as: ϕI ðx; y; z; tÞ ¼

gζd

(12)

rφs ⋅rζI

The boundary condition on the mean wetted hull surface Sb :

where φs ðx; y; zÞ, ϕI ðx; y; z; tÞ and ϕd ðx; y; z; tÞ are the steady velocity potential, the incident velocity potential and the unsteady disturbance velocity potential, respectively. The unsteady wave elevation ζðx; y ; tÞ can be written as: ζðx; y ; tÞ ¼ ζI ðx; y; tÞ þ ζd ðx; y; tÞ

� ⇀ ðU rφs Þ⋅r ϕd ¼

∂ ∂t

(1)

Ψ ðx; y; z; tÞ ¼ φs ðx; y; zÞ þ ϕI ðx; y; z; tÞ þ ϕd ðx; y; z; tÞ

� ⇀ ∂ϕ ∂2 φ ðU rφs Þ⋅r ζd ¼ d þ 2s ζ ∂z ∂z

�

The kinematic and dynamic boundary conditions on the free surface ​ z ¼ ζðx; y; tÞ: � � ⇀ ∂ (7) ðU rΨÞ ⋅ r ðz ζÞ ¼ 0 ∂t

Using the method proposed by Kring (1994), the unsteady distur­ bance velocity potential ϕd ðx; y; z; tÞ can be further subdivided as the sum of a local flow potential ϕl ðx; y; z; tÞ and a memory potential ϕm ðx;y; z; tÞ:

�

ϕd ðx; y; z; tÞ ¼ ϕl ðx; y; z; tÞ þ ϕm ðx; y; z; tÞ

∂ ∂t

� rΨÞ ⋅ r Ψ ¼

⇀

ðU

1 gζ þ rΨ⋅rΨ 2

(8)

Then the diffraction potential is attributed to the memory potential, which accounts for the effect on ship motion and wave elevation of the flow field at next moment, while the radiation potential is attributed to the local potential and the memory potential. The reason of taking out the local flow from the unsteady disturbance velocity potential is because of the instability of the numerical solution. The local potential can be further decomposed as:

⇀

where U ¼ ðU0 ; 0; 0Þ. The boundary condition on the instantaneous wetted hull surface Sb : ⇀

∂Ψ ∂ξ ⇀ ¼ U0 n1 þ S ⋅n ∂n ∂t

(9) ⇀

where n ¼ ðn1 ; n2 ; n3 Þ is the unit normal vector on Sb ; ξ S is the wave⇀

induced motion vector and can be written as ⇀

where ξ

T

⇀ ξS ⇀

¼ ðξ1 ; ξ2 ; ξ3 Þ is the translation vector, ξ

⇀

¼ ξ Tþ R

⇀

ξ

R

�

⇀

ϕl ðx ; tÞ ¼

∂Ψ ¼0 ∂n

r,

¼ ðξ4 ; ξ5 ; ξ6 Þ is the

(19)

where Nj ðxÞ and Mj ðxÞ satisfy the following boundary conditions: ⇀

⇀

Nj ðx Þ¼ 0; ⇀

∂Nj ðx Þ ¼ nj ; ∂n ⇀ ∂Nj ðx Þ ¼ 0; ∂n

(10)

In the present study, the double-body linearization is applied. By substituting Eq. (1) into Eqs. (6)–(10), the linearized BVP of ϕd ðx; y; z; tÞ can be derived as follows: r2 ϕd ¼ 0; in ​ the ​ fluid ​ domain

6 X � ⇀ � ⇀ Nj ðx Þξ_ j ðtÞ þ Mj ðx Þξj ðtÞ j¼1

⇀

rotation vector and r is the position vector in the ship-fixed frame. The sea bottom condition on z ¼ h is given as: ⇀

(18)

⇀

⇀

Mj ðx Þ¼ 0;

j ¼ 1; 2; ⋯; 6

on z ¼ 0

⇀

∂Mj ðx Þ j ¼ 1; 2; ⋯; 6 on Sb ¼ mj ; ∂n ⇀ ∂Mj ðx Þ ¼ 0; j ¼ 1; 2; ⋯; 6 on z ¼ h ∂n

The BVP for the memory potential ϕm ðx; y; z; tÞ is given as:

(11)

The kinematic and dynamic boundary conditions on the undisturbed free surface z ¼ 0:

3

(20)

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8 2 r ϕ ¼ 0; in ​ the ​ fluid ​ domain > > >� m > � > > ⇀ > ∂ðϕl þ ϕm Þ ∂2 φs > ∂ ðU > rφs Þ⋅r ζd ¼ þ 2 ζ rφs ⋅rζI ; on z ¼ 0 > > ∂t ∂z > ∂z > > > � > >� ∂ ⇀ ⇀ 1 > > > rφ ⋅rφs ; on z ¼ 0 ðU rφs Þ⋅r ϕm ¼ gζd rφs ⋅rϕI þ U ⋅rφs > < ∂t 2 s > ∂ϕm ∂ϕI > > ¼ ; on Sb > > ∂ n ∂n > > > > > ∂ϕm > > > ¼ 0; on z ¼ h > > ∂n > > > > > > ϕ ¼ 0; ∂ϕm ¼ 0; t ¼ 0 : m ∂t

ρ

υðyÞ ¼ 3Cs

ðx

x0 Þ

x0 Þ3

ðy

y0 Þ2

ðyr

�

ρ

∂ ∂t

ρ

⇀

U ⋅rϕ þ rφs ⋅rϕ þ gðξ3 þ ξ4 y

ZZ � � r ZZ �

(21)

x0 j � jxr

x0 j

; y0 Þ3

0 � jy

y0 j � jyr

y0 j

1 ⇀ ðrϕ⋅rϕÞnds 2

⇀ 1 U ⋅rφs þ rφs ⋅rφs 2

��

�� � ⇀ ⇀ ξ5 xÞ ⋅ ξ R � n ds

� � ⇀ ⇀ ξ S ⋅ ξ R � n ds

⇀

� ⇀ 1 ⇀ U ⋅rφs þ rφs ⋅rφs þ gz H n ds 2

ρ Sb

Z ​ ⇀ 1 n þ ρg ½ζ ðξ3 þ ξ4 y ξ5 xÞ �2 dl 2 sin α wl � � Z ​ ⇀ 1 U ⋅rφs þ rφs ⋅rφs ½ζ ðξ3 þ ξ4 y ρ 2 wl

(22)

� rφs Þ⋅r ϕd ni ds

ZZ

ρ ⇀ Sb

Sb

0 � jx

⇀

� ⇀ 1 ⇀ U⋅rφs þ rφs ⋅rφs þ gz n ds 2

Sb

;

ðU

� ⇀ H r ⋅r

∂ϕ ∂t

ρ

⇀

⇀

ξ5 xÞ � ξ R �

n dl sin α (28)

where ϕ ¼ ϕI ðx; tÞ þ ϕd ðx; tÞ, H is the second-order transformation matrix. α represents the angle of the hull flare at free surface. 3 2 � 0 0 ξ25 þ ξ26 7 � 16 7 H¼ 6 (29) 0 2ξ4 ξ5 ξ24 þ ξ26 5 24 � 2 2 2ξ4 ξ6 2ξ5 ξ6 ξ4 þ ξ5 ⇀

⇀

2.3. Numerical implementation According to Green’s second theorem, the boundary integral equa­ tion (BIE) can be derived as: � � � � ZZ ZZ ⇀ ⇀ ⇀ ∂ϕd ðx ’ ; tÞ ∂Gðx; x ’ Þ ⇀ ⇀ ⇀ ⇀ 2π ϕd ðx; tÞ ¼ ϕd x ’ ; t G x; x ’ ds ds (30) ∂n ∂n

Sb

ZZ

Sb

� ⇀ ⇀ ⇀ ðU rφs Þ⋅rϕ ⋅ξS ​ nds

Sb

where Cs is the coefficient of cooling strength, x0 ; y0 and xr ; yr represent the coordinates of x axis and y axis of the start and end positions in the free surface damping region, respectively. Once the unsteady velocity potentials ϕl and ϕm are obtained, the wave exciting forces Fi and radiation force Fij can be determined by integrating the pressure, which is evaluated from Bernoulli’s equation, along the mean wetted hull surface Sb : � � ZZ ⇀ ∂ Fi ¼ ρ (23) ðU rφs Þ⋅r ðϕI þ ϕm Þni ds ∂t

Fij ¼

∂ϕ ∂t

r

ZZ �

2

ðxr

ρ ZZ

In addition, an artificial damping term (Huang, 1997), defined as 2υζd þ υ2 ϕd =g, should be added in the kinematic free surface condition to satisfy the radiation condition, where υ is the Newtonian cooling strength and is determined by:

υðxÞ ¼ 3Cs

�

ZZ

⇀

F ð2Þ ¼

(24)

S

S

where x ¼ ðx; y; zÞ and x ¼ ðx ; y ; z Þ are the field point and the source point, respectively; the boundary surfaces S include the mean free sur­ face Sf , the mean wetted hull surface Sb and the sea bottom surface Sd . ⇀

Sb

For a harmonically oscillating ship, the radiation force Fij​ can be expressed as: h �i � �� F ij​ ¼ Aij​ ðωe Þ€ξj ðtÞ þ Bij​ ðωe Þξ_ j ðt ¼ ω2e Aij ðωe Þ iωe Bij ðωe ξj eiωe t (25)

⇀

0

0

0

By using the image method, the Green function Gðx; x’ Þ can be expressed as: � � 1 ⇀ ⇀ G x ; x ’ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ’ ðx x Þ þ ðy y’ Þ2 þ ðz z’ Þ2 ⇀⇀

where ωe and ξj are the encounter frequency and the amplitude of harmonic motion, respectively. Then the added mass and damping coefficient can be obtained as: � � Re Fij Im Fij ​ ​ Aij ¼ 2 ; Bij ¼ (26) ωe ξj ωe ξj

1 þqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ’ ðx x Þ þ ðy y’ Þ2 þ ðz þ z’ þ 2hÞ2

(31)

In order to avoid numerical errors due to a direct difference method, the related physical variables are approximately described by a quadratic B-spline function:

The 6-DOF ship motion equations can be obtained based on New­ ton’s Second Law: � Mij €ξj ðtÞ þ Cij ξj ðtÞ ¼ Fi ξ_ j ; ξj ; t ; i ¼ 1; 2; ⋯; 6 (27)

8 9 X ⇀ ⇀ > > ϕd ðx ; tÞ � ðϕd Þj ðtÞBj ðx Þ > > > > j¼1 > > > > > 9 < X ⇀ ⇀ ζd ðx ; tÞ � ðζd Þj ðtÞBj ðx Þ > > j¼1 > > > > > � ⇀ > 9 � > x ; tÞ X ∂ϕd > ⇀ > ðtÞBj ðx Þ � : ∂ϕd ð ∂z ∂z j j¼1

where Mij and Cij represent the mass matrix and restoring coefficient matrix, respectively. The hydrodynamic forces Fi ðξ_ j ; ξj ; tÞ include wave exciting force and radiation force. In this study, the second-order force is evaluated by applying the pressure integration method (also known as near field method) proposed by Joncquez (2009), and it is given by:

(32)

Substituting Eq. (32) into Eq. (12) and Eq. (13), the temporal term in the kinematic boundary condition is discretized by an explicit Euler scheme to update the wave elevation, while an implicit Euler scheme is used for the dynamic boundary condition to obtain the velocity potential 4

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Table 1 Benchmark tests: Captive tests with the bare hull in waves. Test ID

Velocity

CW1; CW2; CW3 CW4; CW5

Table 3 Model tests of the DTC container ship in deep water.

Environment

Model scale (m/s)

Full scale (knots)

Fr ( )

UKC( )

λ=L( )

0; 0.327; 0.872 0; 0.327

0; 6; 16

0; 0.052; 0.139 0; 0.052

100%

0.55

20%

0.55

0; 6

Table 2 Benchmark tests: Free running tests in waves. Test ID

Velocity

FW1 FW2

Environment

Model scale (m/s)

Full scale (knots)

Fr ( )

UKC( )

λ=L( )

0.327 0.872

6 16

0.052 0.139

100% 100%

0.55 0.40

�

⇀

ðζd Þðnþ1Þ ðU rφs Þ⋅rBij ¼ j

∂ðϕd ÞðnÞ ∂2 φ j Bij þ 2s ðζd ÞðnÞ j Bij þ ζI ∂z ∂z

� rφs ⋅rζI (33)

where Bij with a bar is the B-spline function at the i-th collocation point; and the superscript (n) denotes the time t¼tn. ðϕd Þðnþ1Þ ðϕd ÞðnÞ j j Bij Δt

⇀ ðϕd Þðnþ1Þ ðU j

⇀

rφs ⋅rϕI þ U⋅rφs

1 rφ ⋅rφs 2 s

(34)

X

Qi;j ðϕd Þj

Pi;j

j¼1

�

Nb X

Pi;j

¼ j¼1

∂ϕd ∂n

�

j¼Nb þ1

X

Qi;j ðϕd Þj ; j

i ¼ 1; 2; ⋯; Nb

Pi;j j¼1

∂ϕd ∂n

�

X

Pi;j j¼Nb þ1

∂ϕd ∂n

X

2πðϕd Þi ;

βðÞ

CE4000 CE4010 CE4090 CE4100 CE4241 CE4251 CE4260 CE4020 CE4030 CE4040 CE4050 CE4060 CE4070 CE4080

0.139 0.139 0.139 0.139 0.139 0.139 0.139 0.139 0.139 0.139 0.139 0.139 0.139 0.139

11.3236 12.5025 9.463 6.2325 9.0174 6.1649 7.6006 13.0552 9.9967 7.2495 4.448 1.5009 0.9966 0.65

15.7628 15.0429 15.0119 14.9913 14.3478 14.312 14.2829 9.9868 9.0284 8.0136 7.0066 5.008 5.0049 5.0033

180 180 180 180 180 180 180 180 180 180 180 180 180 180

Designation

Full scale

Length (L) (m) Breadth (B) (m) Draft (d) (m) Longitudinal centre of gravity (m) Vertical centre of gravity (m) Displacement volume (m3) Block coefficient (CB) ( ) Froude number (Fr) ( )

355.0 51.0 14.5 2.941 19.851 173467 0.661 0.139

3. Experimental benchmark tests

j

The tests in shallow water were executed using a 1:89.11 scale model of the Duisburg Test Case (DTC) container ship in the Towing Tank for Manoeuvres in Confined Water (cooperation FHR and UGent). Eight captive model tests and two tests in free running mode are selected. The details of the experimental setup can be found in Van Zwijnsvoorde et al. (2019), here only the selected benchmark test conditions are listed in Table 1 and Table 2. For the experimental setup of the DTC container ship in deep water, tests in regular waves have been conducted by MARINTEK (Now SINTEF Ocean) (el Moctar et al., 2015). The presented model test results for deep water in this paper are from Lyu and el Moctar (2017). Here, the selected test conditions are given in Table 3, where Hw is the wave height and T is the wave period.

(35)

� j

Nb þNf

Qi;j ðϕd Þj j

�

Nb þNf

j¼1

¼

�

j¼Nb þ1

Qi;j ðϕd Þj �

∂ϕd ∂n

Nb þNf

Nb X

Nb X

�

Nb þNf

Nb X

T(s)

where mii is the added mass of iith mode; mii;∞ represents the infinitefrequency added mass; Ti is the period of artificial spring model which is used to the strength of the spring model.

By substituting Eq. (32) into Eq. (30), a set of linear equations can be obtained according to the distribution of source points on Sb and Sf : 2πðϕd Þi þ

Hw(m)

modes are zero in the restoring coefficients matrix Cij , an artificial spring model by Kim and Kim (2011) should be applied for evaluation of horizontal motion and resistance to avoid numerical divergence: � �2 2π Cii ¼ ðmii þ mii;∞ Þ ; i ¼ 1; 2; 6 (38) Ti

gðζd Þðnþ1Þ Bij j

rφs Þ⋅rBij ¼

Fr

Table 4 Main particulars of DTC.

at the free surface: ðζd Þðnþ1Þ ðζd ÞðnÞ j j Bij Δt

Run

i ¼ Nb þ 1; ⋯; Nb þ Nf

j¼Nb þ1

(36) where Nb and Nf are the numbers of discretized panels on Sb and Sf , respectively; the influence coefficients Pi;j and Qi;j are given by: � � ZZ 8 ⇀ m n P ds ¼ ξ η G x ;ξ; η > i;j i > > > < PanelðjÞ (37) � � ZZ > ∂ > ⇀ > > Qi;j ¼ ξm ηn G xi ;ξ; η ds; m; n ¼ 0;1; 2 : ∂n

4. Results and discussions The main particulars of the DTC container ship are listed in Table 4. Due to the symmetry of the flow field, only half of the computational domain is considered. Fig. 2 shows the adopted discretized panels on the boundaries after convergence analysis, where the truncated free surface of the computational domain is 1.5L upstream, 2L downstream and 0.8L half width. The total number of discretized panels is 4950, with 1200 on half-ship hull and 3750 on half-free surface.

PanelðjÞ

where ξ and η denote the coordinates in the panel local coordinate system. The numerical evaluation method for Pi;j and Qi;j can be found in Dai (1998). In addition, since the restoring coefficients of surge, sway and yaw 5

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Fig. 2. Discretized panels on hull surface and free surface.

Fig. 3. Time histories of wave exciting force and moment at four water depths. Heave force (top), Pitch moment (bottom), ω0

Fig. 4. Wave exciting heave force (left) and pitch moment (right). Fr ¼ 0:139.

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pffiffiffiffiffiffiffi L=g ¼ 3:0, Fr ¼ 0:139.

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Fig. 5. Added mass and damping coefficient. (a), (b) added mass and damping coefficient due to heave motion; (c), (d) added mass and damping coefficient due to pitch motion. Fr ¼ 0:139.

4.1. Wave exciting forces

20% UKC. As can be seen, all of the curves are smooth, which demon­ strates the good numerical stability of the present program. It is worth noting that, even for the same wave frequency of the incident wave pffiffiffiffiffiffiffi (ω0 L=g ¼ 3:0), the time history curves at four water depths show

Fig. 3 shows the time histories of wave exciting force and moment at four water depths, i.e., infinite water depth, 100% UKC, 50% UKC and

Fig. 6. Time histories of heave and pitch motions at four water depths. (a) Heave motion; (b) Pitch motion. ω0 7

pffiffiffiffiffiffiffi L=g ¼ 3:0, Fr ¼ 0:139.

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Fig. 7. Comparison of numerical and experimental time history results. (a) Heave motion; (b) Pitch motion. λ=L ¼ 0:55, Fr ¼ 0:139.

different encounter wave frequencies due to the dispersion relation in finite water depth. The non-dimensional wave exciting heave force and pitch moment with respect to frequency at different water depths are plotted in Fig. 4. The amplitudes of the results are obtained by using Fourier series expansion from the time history results corresponding to the same fre­ quency. From Fig. 4, it can be observed that the obvious difference mainly occurs at low frequencies for four water depths, whereas the values at high frequencies do not show obvious differences except for phase shift persistence. In low frequency range, the most distinct feature is that the peak values of wave exciting force and moment RAO move towards lower frequencies with decreasing water depth, the same conclusion was also found in Kim and Kim (2012) for the zero forward speed case by Rankine panel method and in Perunovic and Jensen (2003) for the nonzero forward speed case by strip method. The reason is that the peak value is closely related to the ratio of wave length to the

projected length of the ship on the direction of incident wave (i.e., λ=Lcosβ). 4.2. Hydrodynamic coefficients Fig. 5 presents the added masses and damping coefficients due to wave-induced heave and pitch motions at different water depths. From Fig. 5 (a), (c), it can be observed that the added masses decline sharply in low frequencies, while almost tend to flat with the increase of the fre­ quency; as for the damping coefficients in Fig. 5 (b), (d), they first in­ crease in relative low frequency range and then decrease as the frequency increases. Generally speaking, shallower water induces larger hydrodynamic coefficients; low frequency makes enormous contribu­ tions to the hydrodynamic coefficients in both deep and shallow waters, especially for the added mass (see Fig. 5 (a), (c)). The reason is that the radiation wave length does not change dramatically at high wave

Fig. 8. Heave and pitch motion RAOs at four water depths. Heave motion RAO (left), Pitch motion RAO (right). Fr ¼ 0:139. 8

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Fig. 9. Wave contours around DTC container ship at different water depths. Infinite water (left), 100% UKC (middle) and 20% UKC (right). ω0 0:139.

Fig. 10. Time histories of added resistance at three water depths. (a) Infinite water, (b) 100% UKC and (c) 20% UKC. ω0

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pffiffiffiffiffiffiffi L=g ¼ 1:98, Fr ¼

pffiffiffiffiffiffiffi L=g ¼ 3:0, Fr ¼ 0:139.

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Fig. 11. Comparisons of added resistance at infinite and 100% UKC water depths. Fr ¼ 0:139

Fig. 12. Comparisons of added resistance at 20% UKC water depth. Fr ¼ 0:139.

frequency according to the dispersion relation. However, the shallow water effect is much more significant in relative low frequency ranges. This is because the radiation waves are limited by the sea bottom for such low frequency, thus more energy is needed to accelerate the fluid movement around the ship in the low frequency range in shallow water.

nonlinear one when the forward speed problem in shallow water is studied. Fig. 9 shows the wave contours around DTC container ship at different water depths in regular waves at normalized wave frequency pffiffiffiffiffiffiffi ω0 L=g ¼ 1:98. It can be seen that the smaller the water depth, the shorter the wavelength at the same wave frequency.

4.3. Motions in regular waves

4.4. Added resistance

Fig. 6 shows the time histories of heave and pitch motions under four pffiffiffiffiffiffiffi water depths at ω0 L=g ¼ 3:0, Fr ¼ 0:139. Similar to the time histories of wave exciting force in Fig. 3, the curves in Fig. 6 also show good smoothness and stability. Note that the wave-induced motions show different encounter wave periods even for the same incident wave fre­ quency. The reason is that the orbit of a fluid particle in waves is strongly dependent on water depth, hence in shallow water, the flow is obstructed by the keel clearance which is different from that in deep water. The upper plots in Fig. 7 (a), (b) present the comparison between the numerical and experimental time histories of heave and pitch motions at � pffiffiffiffiffiffiffi � λ=L ¼ 0:55 ω0 L=g � 2:89 and 100% UKC. As can be seen from

Fig. 10 presents the time series of the first-order and second-order surge forces at three water depths (i.e., infinite water depth, 100% UKC and 20%UKC). From the figures, it can be observed that the oscillation period of the second-order surge force is almost twice as large as the first-order surge force. In addition, the speed at which the secondorder force converges is lower than that for the first-order force, which implies that a longer simulation time is needed. Fig. 11 shows the comparisons between the numerical and experi­ mental results of added resistance obtained at infinite and 100%UKC water depths. As can be seen from the figures, the normalized added resistance values in infinite water show fair agreement with the exper­ imental results by Sprenger et al. (2017), but the values are over­ estimated in the small wavelength ranges. Though some deviations exist between the numerical and experimental results, the order of deviation magnitudes for 100%UKC is acceptable. In general, the peak values of numerical added resistance RAO shift towards smaller wavelengths with the decrease of water depth. In addition, although the numerical values for 100% UKC(Fig. 11) and 20% UKC (Fig. 12) present a similar trend as that in infinite water, it is not possible to find a consistent trend compared with experiment re­ sults, especially for the case of 20% UKC. The reasons can be explained as follows: On one hand, as described in Sprenger et al. (2017), the 20% UKC in the test is no more a static condition because the squat effects lead to dynamic UKC of merely 0.9% in the presence of waves. More­ over, an issue worth noting is that in the test the incident wave ampli­ tudes are very small (in order to avoid the bottom contact problem because of very small UKC), which could have introduced inaccuracies in the measurements. A detailed explanation can be found in Sprenger et al. (2017). On the other hand, other sources of discrepancies can be associated to the numerical analysis based on the linear potential flow method, which does not account for the squat effects and viscous effects in shallow water. Thus the numerical accuracy is also questionable. Therefore, the effects of water depth on added resistance in waves need to be further investigated both numerically and experimentally. Nevertheless, to an extent, the results can still be used for a qualitative analysis at the initial stages of design.

Fig. 7, during the experimental process, the acceleration and transition zones exist before the ship reaches a stable state (between two dotted lines); after that, the intended regular wave pattern is disturbed because of reflections by the beach and the wave maker, see the detailed description in Van Zwijnsvoorde et al. (2019). In addition, due to the significant squat behaviour in shallow water, the mean values of experimental heave and pitch motion time samples (red line) are not zero anymore, which are different from the values calculated by the linear potential flow method (blue line). However, when the mean values of the experimental time histories in steady state (between two dotted lines) are set to zero (see the lower plots in Fig. 7 (a), (b)), though some deviations of phase angle exist, a good agreement can be found by comparing with the numerical results. Fig. 8 depicts the present numerical results of heave and pitch motion RAOs at different water depths in comparison with the experimental data, where the numerical amplitudes of motion responses are obtained from the time series of corresponding wave frequency by Fourier series expansion. Here the test data and CFD results for deep water are taken from Lyu and el Moctar (2017). As can be seen in Fig. 8, generally the numerical results are in fair agreement with experimental data and CFD results, however some deviations can also be observed, especially for heave motion, even for deep water case. Similar discrepancies were reported in Lyu and el Moctar (2017). One reason, as explained in Lyu and el Moctar (2017), might be that the pitch resonance has an influence on heave motion; another reason might be the linear potential flow method used in this study, which does not consider the transient wetted hull surface or ship squat. Therefore, it is likely that the linear potential flow theory based Rankine panel method should be extended to a

5. Conclusions In the present study, the wave-body interaction problems in finite 10

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water depth are investigated for the DTC container ship by using a time domain Rankine panel method; and the hydrodynamic forces, motion response and added resistance are numerically predicted. The developed numerical scheme and program are validated by comparing the nu­ merical results with the corresponding model test data. From this study, the following conclusions can be drawn:

Gao, Z.L., Zou, Z.J., Wang, H.M., 2008. Numerical solution of the ship wave-making problem in shallow water by using a high order panel method. J. Ship Mech. 12 (6), 858–869. Huang, Y., 1997. Nonlinear Ship Motions by a Rankine Panel Method.. Ph.D. thesis Massachusetts Institute of Technology, Cambridge, USA. Hwang, J.H., Lee, S.J., 1975. The effect of forebody forms on the ship motion in water of finite depth. Bulletin of the Society of Naval Architects of Korea 2 (12), 59–66. Joncquez, S.A.G., 2009. Second-order Forces and Moments Acting on Ships in Waves.. Ph.D. thesis Technical University of Denmark, Kgs. Lyngby, Denmark. Kim, C.H., 1968. The influence of water depth on the heaving and pitching motions of a ship moving in longitudinal regular head waves. Schiffstechnik 15 (79), 127–132. Kim, K.H., Kim, Y., 2011. Numerical study on added resistance of ships by using a timedomain Rankine panel method. Ocean Eng. 38 (13), 1357–1367. 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1) Generally, in shallow water larger hydrodynamic coefficients are obtained. Lower frequency radiation waves induce larger added mass in both deep and shallow waters, and the results will converge to those in the deep-water case with the increase of wave frequency. 2) The smaller the water depth, the larger the peak value of the waveinduced force and moment is. Peak values of excitation RAO shift toward to lower frequencies with the decrease of water depth; similar results can also be found for the peak values of heave and pitch motion responses. From the comparison of numerical results with model test data for motion responses at different water depths, it can be concluded that the present linear approach is applicable for the prediction of wave-body interaction problems related to motion dynamics in shallow water. 3) As for the added resistance, the peak values of added resistance RAO shift toward to smaller wavelengths with the decrease of water depth. Though the numerical results in deep water agree well with model test data, it is not possible to find a consistent trend for cases of finite water depths, especially for the extreme test condition (20% UKC). In order to estimate the second-order force in shallow water accurately, it is necessary to extend the present potential flow method to consider the squat and viscous effects. Acknowledgments The first author gratefully acknowledges the financial support from China Scholarship Council (CSC), and the Lloyd’s Register Foundation (LRF) through the joint centre involving University College London, Shanghai Jiao Tong University, and Harbin Engineering University. LRF help protect life and property by supporting engineering-related edu­ cation, public engagement, and the application of research. The authors acknowledge the contributions of MARINTEK and the Towing Tank for Manoeuvres in Confined Water (cooperation FHR and UGent) with respect to the experimental data for the Duisburg Test Case (DTC) container ship. The authors would also like to thank the SHOPERA (Energy Efficient Safe SHip OPERAtion) consortium and the 5th MASHCON International Conference for making the model test available. References Andersen, P., 1979. Ship motions and sea loads in restricted water depth. Ocean Eng. 6 (6), 557–569. Chan, H.S., 1990. A Three-Dimensional Technique for Predicting First and Second Order Hydrodynamic Forces on a Marine Vehicle Advancing in Waves.. Ph.D. thesis University of Glasgow, Glasgow, UK. Dai, Y.S., 1998. Potential Flow Theory of Ship Motions in Waves in Frequency and Time Domain (in Chinese). National Defense Industry Press. el Moctar, O., Sigmund, S., Schellin, T.E, 2015. Numerical and experimental analysis of added resistance of ship in waves. In: Proceedings of the ASME 2015 34th International Conference on Ocean, Offshore and Arctic Engineering (OMAE 2015), St. John’s, NL, Canada. Feng, A.C., Bai, W., You, Y.X., Chen, Z.M., Price, W.G., 2016. A Rankine source method solution of a finite depth, wave–body interaction problem. J. Fluids Struct. 62, 14–32.

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