Numerical analysis on floating-body motion responses in arbitrary bathymetry

Ocean Engineering 62 (2013) 123–139

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Numerical analysis on floating-body motion responses in arbitrary bathymetry Taeyoung Kim, Yonghwan Kim n Department of Naval Architecture and Ocean Engineering, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 151-744, Republic of Korea

a r t i c l e i n f o

abstract

Article history: Received 14 August 2012 Accepted 12 January 2013 Available online 28 February 2013

To simulate the incident waves over a varying bottom of the ocean floor, a Rankine panel method is utilized with numerical wave tank scheme. Nonlinear incident Boussinesq-type waves are also applied to verify the shallow depth effect. The effect of shallow water depth is especially observed in moderate depth around the draft of a ship. The numerical results do not show significant difference between linear and nonlinear waves and also induced load, which supports the plausibility of the linear regime. By using the linear Rankine panel method, a stepwise analysis is performed for an LNG carrier, with focus on the hydrodynamic coefficients, wave loads and motion responses. In addition, the influence of varying bathymetry is carefully observed for various sloped bottoms. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Floating-body motion Finite-depth effects Rankine panel method Varying bathymetry Numerical wave generation

1. Introduction Recently, the installation of floating-type offshore structures in coastal areas instead of inland facilities has seriously been considered. The targets mainly considered are platforms related to liquefied natural gas (LNG), which have abilities to store, convert and transfer LNG. In general, these offshore structures have drafts over 10 m, and thus the shallowest conditions for operation are nearly 15–30 m depth. Since the restricted water depth changes the characteristics of waves like direction of propagation and celerity, the accurate estimation of wave loads is a key issue to evaluate the seakeeping performance of floating structures. The basic step to include the bottom effect is assuming the bottom to be flat. Several methods have been introduced and have provided satisfactory results to this simplified problem. For instance, the strip method (Tuck, 1970; Tasai et al. 1978; Andersen, 1979; Perunovic and Jensen, 2003) is still widely used nowadays for practical purposes. However, this method has limitations as a two-dimensional theory and thus it is not suitable for accurate analysis. To address such limitations, Kim (1999) introduced a new unified theory considering the finite-depth effect, showing much improved level of accuracy. Nowadays, three-dimensional panel methods are widely used, such as WAMIT (Lee, 1995), which provides a solver for constant depth. Many researchers have recently moved their focus to floatingbody motion over an actual coastal seabed. Sloped bottom is a simple but quite realistic description of near-shore area. Kyoung et al. (2005), Belibassakis and Athanassoulis (2009) solved the

n

Corresponding author. Tel.: þ82 2 880 1543; fax: þ82 2 876 9226. E-mail addresses: [email protected] (T. Kim), [email protected] (Y. Kim).

0029-8018/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2013.01.012

elastic body response over sloped bathymetry and showed different responses with respect to specific location. For the problem of rigid ships, there have also been several trials with the sloped bottom (Buchner, 2006; Ferreira and Newman, 2009; Hauteclocque et al., 2009). The researchers have used the panel method by assuming a sloped bottom with an additional fixed body. However, this second-body approach is not a complete formulation in diffraction and radiation problems. In this study, the Rankine panel method is applied in the time domain to account for the bottom effects on a floating body. In the case of linear problem, a frequency-domain formulation is applicable. However, this study adopts a time-domain formulation since it has an advantage when the problem is extended to nonlinear wave and motion dynamics (Kim, 1999; Song et al., 2011). Also the coupling with nonlinear external mechanisms, e.g. mooring cable and nonlinear viscous force, is possible in the time-domain approach. To satisfy the exact bottom boundary conditions, a Rankine source is distributed over the bottom surface. The present method applies a B-spline basis function to present physical parameters such as velocity potential and wave elevation. In order to simulate wave propagation over a varying bottom more accurately, the numerical wave tank concept is applied. Numerical wave tank can be easily added to the Rankine panel method, enabling the computation efficient. The detailed numerical schemes and their effectiveness are well reviewed by Kim et al. (1999) and Tanizawa (2000). More precisely, two specific methods are wrapped onto this mixture of the panel method and numerical wave tank. At first, following Boo et al. (1994), the solution of linear incident waves is fed at an inflow boundary, which can act like a wave-making wall. Furthermore, a numerical damping zone is also utilized for wave absorption and generation. To this end, adopting the idea of Ferrent (1998), the free

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favorable agreements with the results from unified theory. To see the effect of varying bathymetry, a floating body is assumed to be installed over a sloped bottom. In the case of sloped bottom with a mild variation of 1/20, motion responses and other hydrodynamic coefficients show similar tendencies with the result of constant water depth, particularly the water depth in the location of midship. This means mild slope does not cause significant difference in the wave propagation and induced wave loads. To observe the increase of nonlinear effects, two different tests are carried out with focusing on the depth of the adjacent bottom and the steepness of slope. However, a remarkable difference is only found for the extremely steep bottom. Those results imply that the nonlinear effect due to a varying bottom is not apparent near the moderate depth where a floating body can be installed. Therefore, the linear approach can be effectively utilized for observing the hydrodynamic characteristics of a floating body over finite and varying depth.

surface boundary condition is modified to pass the targeting wave component along this layer. The combined method shows favorable results on the generation of targeting waves in numerical tests. For practical purposes, our approach is restricted to the linear regime for boundary value problems and wave loads. In general, the linear approach has given favorable results to many problems regarding floating-body dynamics. However, it requires careful caution to apply linear approaches to problems of finite depth. In extremely shallow water depth, it has been well known that nonlinear effects are too significant to be ignored. In the case of coastal structures, such as seawalls or other fixed platforms, which can be installed in shallow regions, the nonlinear effect needs to be considered carefully. However, it is not easy to estimate the significance of nonlinear effects when the target is a floating structure. A floating body cannot be installed in extremely shallow regions because they have certain values of draft. However, it still remains a question whether nonlinear effects might be strong even in intermediate depths larger than the draft of the floating body. Therefore, the magnitude of the nonlinear effect due to shallow-water waves needs to be checked to achieve realistic evaluation of a floating-body’s motion. To observe the effect of shallow water waves, the representative shallow wave theory—the Boussinesq equation—is introduced to solve incident waves. Recently, the focus on this problem has been on enlarging the restricted range of its applicability to more intermediate depths (Agnon et al., 1999; Madsen et al., 2002, 2006). The extension has made it possible for the method to be applied to floating-body dynamics because the floating body is located not in shallow but in intermediate depths. One recent method (Madsen et al., 2006) including rapidly varying bathymetry has been adopted in the present study to calculate nonlinear waves for various bottom conditions. In the present study, two sets of comparisons are carried out by using two different methods: wave propagation and induced wave loads on a floating body. Tests are carried out for constant and varying water depths. From the test in constant depth, the effect induced by depth itself can be observed. In contrast, varying bathymetry can provide the effect of refraction, reflection and shoaling. In both cases, however, the Boussinesq model does not show significant difference from linear theory, except for very steep waves. Even though nonlinear effects are expected to be stronger in shallower depths, it is beyond the normal conditions where floating bodies can operate. Therefore, these results support the usefulness of the linear approach for evaluating seakeeping performance of floating bodies in finite depth. Based on the linear regime, hydrodynamic coefficients, wave excitations and motions responses of an LNG carrier are compared in different constant water depths, and all the results show

a

2. Background 2.1. Boundary value problem in linear Rankine panel method Consider a freely floating-body under 6-degrees-of-freedom (DOF) motion in the presence of water wave, as shown in Fig. 1(a). From Newton’s second law, the linear equation of motion can be written as Eq. (1). {x} means the displacement vector of 6-DOF ship motion. [M] is a mass matrix and [C] is a restoring coefficients matrix. {F} is all hydrodynamic forces induced from wave and body motions. ½Mfx€ g þ½Cfxg ¼ fFg

ð1Þ

The boundary value problem can be set up by adopting the velocity potential, f, which satisfies the following boundary value problem:

r2 fðx,y,z,tÞ ¼ 0 in fluid domain

ð2Þ

@z @f ¼ @t @z

ð3Þ

on z ¼ 0

@f ¼ g z @t

on z ¼ 0

 6  @xj @f X ¼ nj @n j ¼ 1 @t @f ¼0 @n

ð4Þ

on the mean body ðSB Þ

ð5Þ

on z ¼ hðx,yÞ

ð6Þ

b d dt

x, y

2

z

2

g

d dt

2

z

2

I

I

g

x y

I

n

h

z

n

h x, y

xs xe

Fig. 1. Coordinate system and computational domain with damping zone. (a) Coordinate system and (b) computational domain with damping zone.

T. Kim, Y. Kim / Ocean Engineering 62 (2013) 123–139

125

Table 1 Principal dimensions of hull model.

Fig. 2. Evaluation points to calculate fifth order differentiation in the finite difference method.

Capacity

LNG carrier

LBP (m) Beam (m) Draft (m) Displacement (m3) Longitudinal center of gravity (m) Vertical center of gravity (m) Roll radius of gyration (m) Pitch radius of gyration (m)

274 44.2 11 97,120  1.06 from midship 16.3 from keel line 15.2 68.5

damping zone is made by modifying the kinematic free surface boundary condition as written in Eq. (9), which was originally suggested by Nakos (1993). The other type of zone lies near the wave-making wall. Adopting the idea of Ferrent (1998), the free surface boundary condition is modified as in Eq. (10). Therefore, only the disturbed wave component is absorbed and the targeting solution of the incident wave can pass this layer. This damping zone can also eliminate the evanescent mode and absorb the reflected wave from the downstream region. Two parameters should be determined: the damping intensity, n, and the zone size. In this study, the damping intensity is defined as Eq. (11), where a is a constant. Also, it is assumed that the damping zone lies from xs to xe. The length between two values is usually 1–2 times of wave length. @f @fI ¼ @n @n

ð8Þ

dz @f n2 ¼ 2nz þ f dt @z g

ð9Þ

 dz @f n2  ¼ 2nðzzI Þ þ ffI dt @z g Fig. 3. Panel model of test hull.

n¼ where xj and nj denote the j-th motion and normal vector among 6-DOF, respectively. z means the wave elevation. It should be noted that the water depth hðx,yÞ is a function of x and y, so that it can vary arbitrary. The integration of hydrodynamic pressure on a mean wetted surface gives the wave force as Eq. (7). ZZ ZZ ! @f ! ! F ¼ n dS ð7Þ p n dS ¼ r SB SB @t If water depth is constant, an analytic solution of the velocity potential for an incident wave, fI, is available in the boundary value problems. Then the velocity potential can be divided into two components such as f ¼ fI þ fd . The remaining problem is calculating the disturbed velocity potential, fd, which satisfies the boundary conditions described above. When water depth is not constant, solving the boundary value problem becomes so difficult that analytic approaches are restricted. Mild-slope equation (Takagi and Naito, 1994) or step model (Liu et al., 2011) could be a usable approximation method, but it could not be used in the general bottom conditions. Another possible approach is generating an incident wave numerically. Because this scheme can be easily added to the Rankine panel method, the mixture of incident and disturbed potential can be solved simultaneously. To this end, the concept of a numerical wave tank including a wave-making layer is considered (see Fig. 1(b)). At the wave-making wall, a linear solution of an incident wave, which is a locally linear solution at the inlet boundary, is given by Neumanntype boundary conditions (Eq. (8)). Two types of artificial damping zones are used for wave generation and absorption. One type of zone lies at the rear side of the back stream flow for wave absorption. This

h i 8 eÞ < asin2 p ðxðxx x Þ

xs rx o ðxs þ2 xe Þ

:

ðxs þ xe Þ 2

s

a

e

r x oxe

ð10Þ

ð11Þ

2.2. Numerical method in linear Rankine panel method A time-domain Rankine panel method is applied to solve the boundary value problem described above. In this method, a higherorder B-spline basis function is used to represent the physical variables such as the velocity potential, wave elevation, and normal velocity on the fluid boundary as written in Eq. (12). Then an integral equation can be formulated by using Green’s 2nd identity. In the integral equation in Eq. (13), SF, SB and SBT denote the boundaries of the free surface, body, and bottom, respectively. Discretization of this equation leads to the algebraic form, and which can be solved by popular inverse operation. 2 3 2 3 ðfÞj f     X 6 @f 7 ! 6 @f 7 ! 6 7 ð12Þ 4 @n 5 x ,t ¼ 6 @n j 7ðt ÞBj x 4 5 j z ðzÞj     ZZ ZZ ZZ @ 1 1 @f @ 1 1 @f f ds þ ds ds ds ¼ @n r @n r SB SF r @n SBT SB r @n   ZZ ZZ @ 1 1 @f ds þ ds ð13Þ  f @n r SF SBT r @n

2pf þ

ZZ

f

Time integration is carried out using a mixed implicit–explicit method for the free-surface boundary conditions of Eqs. (3) and (4). The fourth-order predictor–corrector method is applied to the equation of motion written in Eq. (1). The radiation condition at the

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T. Kim, Y. Kim / Ocean Engineering 62 (2013) 123–139

far field can be implemented by adopting the damping zone described in Eq. (9). Rankine sources are distributed on the entire boundary surfaces. The number of panels depends on the computational condition. It is usually about 5000–7000. Maximum domain size is determined at least 6 times larger than the wave length. Time step is selected smaller than 1/100 times of wave period.

A/A incident (m/m)

4

Linear analytic Linear (Rankine panel method) Nonlinear (A/h = 0.067) Nonlinear (A/h = 0.133)

3 2 1 0 -1 -2

-4

-2

0

2

4

x/ (m/m)

A/A incident (m/m)

4

Linear analytic Linear (Rankine panel method) Nonlinear (A/h = 0.033) Nonlinear (A/h = 0.067)

3 2 1 0 -1 -2

-4

-2

0

2

4

x/ (m/m)

A/A incident (m/m)

4

Linear analytic Linear (Rankine panel method) Nonlinear (A/h = 0.067) Nonlinear (A/h = 0.133)

3 2 1 0

Fig. 5. Comparison of linear and nonlinear waves over a varying bottom. (a) Varying bottom (o(l0/g)1/2 ¼1.773, h0/l0 ¼ 0.088), (b) Varying bottom (o(l0/ g)1/2 ¼1.939, h0/l0 ¼0.110) and (c) Varying bottom (o(l0/g)1/2 ¼ 2.433, h0/l0 ¼ 0.279).

-1 -2

-4

-2

0

2

4

x/ (m/m)

A/A incident (m/m)

4

Linear analytic Linear (Rankine panel method) Nonlinear (A/h = 0.033) Nonlinear (A/h = 0.067)

3 2 1 0 -1 -2

Fig. 6. The notation of lengths and depths in the test bathymetry with an LNG carrier. -4

-2

0

2

4

x/ (m/m) Fig. 4. Comparison of linear and nonlinear waves over a constant bottom. (a) Constant depth (o(l/g)1/2 ¼ 1.509, h/l ¼ 0.060), (b) Constant depth (o(l/g)1/2 ¼ 1.773, h/l ¼0.088), (c) Constant depth (o(l/g)1/2 ¼ 2.057, h/l ¼ 0.130) and (d) Constant depth (o(l/g)1/2 ¼ 2.321, h/l ¼ 0.204).

2.3. Nonlinear shallow water wave as Boussinesq equation Boussinesq’s equation reduces the three-dimensional equation into two dimensions by eliminating the vertical gradient. The

T. Kim, Y. Kim / Ocean Engineering 62 (2013) 123–139

detailed procedure can be found in Madsen et al. (2006), therefore only the essence of the final formulation is stated here. The horizontal and vertical velocity components in an arbitrary location,

127

u and w, within fluid domain can be derived as Eq. (14). n denotes ˆ n vertical velocity which are not physical horizontal velocities and w but pseudo velocities defined at arbitrary location. The infinite series

Fig. 7. Froude–Krylov force and moment on LNG carrier over sloped bottom: head sea, o(l0/g)1/2 ¼ 1.939, h0/l0 ¼ 0.110, the same bottom condition to Fig. 5(b): Ls ¼ 300 m, L0 ¼  13 m, L1 ¼  13 m, h0 ¼ 45 m and h1 ¼15 m.

Fig. 8. Added mass and damping coefficient of LNG carrier for three different water depths. (a) Added mass of heave–heave, (b) Added moment of inertia of pitch–pitch, (c) Damping coefficient of heave–heave and (d) Damping coefficient of pitch–pitch.

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a

b

Fig. 9. Wave exciting force RAO of LNG carrier for three different water depths: head sea. (a) Heave force, (b) Pitch moment.

Fig. 10. Motion RAO of LNG carrier for three different water depths: head sea. (a) Heave motion and (b) Pitch motion.

of partial differential operators, JI and JII, include the differentiation up to fifth order. n

2.4. Numerical method in Boussinesq equation

n

^ uðx,y,z,tÞ ¼ J I u^ þJ II w ^n ^ n JII u wðx,y,z,tÞ ¼ JI w

ð14Þ

Hydrodynamic pressure at any location can be calculated from the Bernoulli equation in Eq. (15). However, it requires calculating the time derivative of the velocity potential, which is not defined in this Boussinesq equation based on velocity formulation. Bingham (2000) approximates it from the dynamic free surface boundary conditions and linear velocity potential of an incident wave. The derivate can be approximated as Eq. (16), where k means the local wave number which is defined at the evaluation point. p

r

¼

 @f 1  2  u þ v2 þw2 @t 2

 @f coshfkðz þ hÞg C g z coshkh @t

ð15Þ

ð16Þ

In order to treat the differentiations up to fifth order, the finite difference method is applied in the rectangular domain. As shown in Fig. 2, 25 points are used to evaluate the differentiation at a central point. At the end of the boundary, the mixed Dirichlet and Neumanntypes boundary conditions are applied according to the velocity components and their direction. The number of evaluating points is reduced when lower-order differential terms are required. "

n

n

^ þ J II w ^ uðx,y,z,tÞ ¼ JI u wþ rhUu ¼ 0

9 8 9 8 #> u^ n > > U~ > = < = < n ¼ V~ v^ on z ¼ h > ; : > ; > : n> ^ 0 w

on z ¼ z

ð17Þ

The systematic differential equations can be aligned in matrix form as in Eq. (17). Because velocities in wave elevation are known values at the initial stage and following each time step, two equations

T. Kim, Y. Kim / Ocean Engineering 62 (2013) 123–139

can be utilized for both of the horizontal velocities on the exact location of the wave surface. Furthermore, the bottom boundary conditions give one more relation between the horizontal and vertical velocities on the bottom surface. Then, pseudo-velocities can be calculated by solving the algebraic equation. One practical concern is the computation time to solve the algebraic equation in Eq. (17). The components in this matrix include the exact location of wave elevation, which requires updating the components in the matrix at each of new time step. An iterative scheme, such as the Gauss–Seidel method, is used to make the computation fast. When pseudo-velocities are obtained, the vertical velocity at the free surface can be calculated using the second relation in Eq. (14). Then, new wave elevation and horizontal velocities can be computed from the time-marching method using kinematic and dynamic boundary conditions. These time marching methods are similar to those in the panel method. The fourth-order RungeKutta method is applied for the time integration, as in Eq. (18). " # " # (" # " # " # " #) zt þ Dt k11 k21 k31 k41 zt Dt ¼ þ þ2 þ2 þ U t þ Dt k12 k22 k32 k42 Ut 6

"

k21

#

k22

"

k31

#

k32

"

k41 k42

129

3 2  f zt þ ðDt=2Þk11 ,U t þ ðDt=2Þk11 ,wt þ ðDt=2Þk11 , t þ D2t 6 7 ¼4  5 g zt þ ðDt=2Þk12 ,U t þ ðDt=2Þk12 ,wt þ ðDt=2Þk12 , t þ D2t 3 2  f zt þ ðDt=2Þk21 ,U t þ ðDt=2Þk21 ,wt þ ðDt=2Þk21 , t þ D2t 6 7 ¼4  5 g zt þ ðDt=2Þk22 ,U t þ ðDt=2Þk22 ,wt þ ðDt=2Þk22 , t þ D2t "

# ¼

f ðzt þ Dtk31 ,U t þ Dtk31 ,wt þ Dtk31 ,t þ DtÞ

#

gðzt þ Dtk32 ,U t þ Dtk32 ,wt þ Dtk32 ,t þ DtÞ

@z ¼ UUrz þ ð1 þ rzUrzÞw @t  @U 1 1 ¼ r UUU w2 ð1þ rzUrzÞ g rz g ðz,U,w,t Þ : @t 2 2 f ðz,U,w,t Þ :

ð18Þ 3. Numerical results

where

3.1. Test model

U ¼ ðU,VÞ ¼ u þ wrz "

k11

#

k12

" ¼

f ðzt ,U t ,wt ,tÞ

#

gðzt ,U t ,wt ,tÞ

Table 2 Dimension of sloped bathymetry (notions follow the definition in Fig. 6). Case Slope Slope Slope Slope Slope Slope

I II III IV V VI

h0

h1

h2

Ls

L0

L1

Slope (h2/Ls)

30 45 30 30 30 30

15 15 15 15 15 15

15 30 15 15 15 15

300 300 100 30 10 0

 13  13 87 122 132 137

 13  13 87 122 132 137

1/20 1/20 3/20 1/2 3/2 Infinite

An LNG carrier is considered for the present numerical computation because it can operate near coastal areas to provide LNG to terminals. Table 1 shows the principal dimensions of the hull, and Fig. 3 shows the corresponding panel model. This LNG carrier has a draft of 11 m (m), and this length can be a guideline to decide the range of depths for normal operation. 3.2. Comparison of linear and nonlinear waves In the present study, a part of the focus is given to verify the magnitude of nonlinear effects on wave loads. This problem can be divided into two categories. The first concern is the effect of shallow depth itself, because the nonlinear effect definitely becomes stronger as water depth becomes shallower. The other issue is the effect of

h/L=0.0625

h/L=0.0625

h/L=infinite

h/L=infinite

Fig. 11. Comparison of instantaneous disturbed wave between h/L¼ 0.0625 and infinite depth: head sea. (a) Heave forced motion with a period of 31.42 s and (b) Fixed motion with a wave period of 31.42 s.

130

T. Kim, Y. Kim / Ocean Engineering 62 (2013) 123–139

Fig. 12. Wave contour around LNG carrier over sloped bottom: head sea, o(L/g)1/2 ¼1.586 (Slope I defined in Table 2). (a) t(g/L)1/2 ¼9.46, (b) t(g/L)1/2 ¼18.92, (c) t(g/L)1/2 ¼28.38 and (d) t(g/L)1/2 ¼ 37.84.

Fig. 13. Comparison of wave elevations with and without side damping layer: o(L/g)1/2 ¼ 1.586 (Slope I defined in Table1). (a) Calculated wave at FP position (if body is located) and (b) Calculated wave at AP position (if body is located).

T. Kim, Y. Kim / Ocean Engineering 62 (2013) 123–139

varying bathymetry, which causes different features on the reflection and refraction of waves. It is very meaningful to define the range of two factors used in the numerical computation. In the case of depth, the meaningful range is quite clear. The depth should be larger than the draft of modern floating structures. Because the test model has a draft of 11 m, a reasonable range would not be smaller than 15 m because the freely floating body requires a marginal depth for safety of under keel clearance. In contrast to depth, the effect of varying bathymetry is not easily restricted. According to specific geometric conditions, any kind of varying bathymetry can be a concern regarding seakeeping performance of a floating body. However, for general and practical purposes, the commonly possible problem might be a sloped bottom, because it is a realistic description of near shore. Therefore, the tests are restricted to sloped bottom with different

131

slopes and depths. In the tests of varying bathymetry, the shallowest depth is restricted to 15 m.

3.2.1. Comparison of linear and nonlinear waves over constant water depths To verify the effect of depths, a regular wave is simulated over constant water depths. Fig. 4 compares linear and nonlinear waves with different water depths and frequencies. More precisely, Fig. 4(a) and (c) have the same depths of 15 m, and Fig. 4(b) and (d) have depths of 30 m. The same wave frequency (o) of 0.3 rad/ sec is given to Fig. 4(a) and (b), 0.6 rad/sec to Fig. 4(c) and (d). The length scale is normalized by the wave length of Stokes’ linear wave (l). In four tests, the wave amplitude is increased to see the enhancement of nonlinear effects. However, significant nonlinear

Fig. 14. Motion RAO and time histories of LNG carrier over sloped (Slope I defined in Table 2) and constant bottom: head sea. (a) Surge motion, (b) Heave motion, (c) Pitch motion and (d) Pitch time histories at o(L/g)1/2 ¼ 1.057.

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T. Kim, Y. Kim / Ocean Engineering 62 (2013) 123–139

effect is not obviously observed in all tests. Linear and nonlinear waves show similar shapes, and closely overlap with the analytical solution of Stokes’ linear wave. A little distortion of shape is only found in the result of the highest wave in Fig. 4(a), which corresponds to the most extreme cases because the wave length is longest and depth is shallowest among the tests. A phase difference is also observed, especially near the damping layer, which might have happened due to the limitation of the finite difference scheme. From the tests, it can be concluded that the nonlinear effect is not stronger in those test conditions. Because a little distortion is observed in the most extreme condition in Fig. 4(a), it can be guessed that nonlinear effects might be apparent in shallower depth and with longer waves than in Fig. 4(a). Despite the expectation for increased nonlinear effects, the shallower depth does not belong to standard conditions for evaluating seakeeping performance of this ship.

3.2.2. Comparison of linear and nonlinear waves over varying water depths Another test is planned to observe the nonlinear effect due to varying bottom. The problem is designed such that a sloped bottom is located between two flat bottoms. The length of the slope is 300 m. Two sets of depths are planned to observe the different effects of the steepness. Fig. 5 shows linear and nonlinear waves over these bottom models. The subscript, 0, denotes the local values at the depth of h0. It can be found that wave length becomes shorter and wave amplitude grows in the shallower region. These effects can be explained by the simple theory of wave refraction and shoaling. However, it seems obvious that the nonlinear wave shows slightly different behavior of the increased amplitude in the shallower region. Because the slope is not very steep, reflection of the incident wave is hard to be found.

0.6

10

0.5 8

0.4 6 0.3 4 0.2

2

0

0.1

0

1

2

3

4

5

0

0

1

2

3

4

5

3

4

5

(L/g)1/2

(L/g)1/2 0.5

14

12 0.4 10 0.3

8

6

0.2

4 0.1 2

0

0

1

2

3

(L/g)1/2

4

5

0

0

1

2

(L/g)1/2

Fig. 15. Added mass and damping coefficient of LNG carrier over sloped (Slope I defined in Table 2) and constant bottom. (a) Added mass of heave–heave, (b) Added moment of inertia of pitch–pitch, (c) Damping coefficient of heave–heave and (d) Damping coefficient of pitch–pitch.

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Fig. 5(a) and (b) compare the different features of waves due to the slope; the incident wave has the same wave frequency, but the slope and water depth of the incoming region are different. Because the condition in Fig. 5(b) has a more rapidly changing slope than that in Fig. 5(a), slightly higher increase in wave amplitude is found. Regarding nonlinear effects, the differences between the two conditions are not visible. Fig. 5(b) and (c) have the same bottom conditions but different wave frequencies. More specifically, Fig. 5(c) has a larger wave frequency, thus wave length is shorter than in Fig. 5(b). Because of the shortened wave length, the effect of refraction and shoaling is reduced. As such, the expected conclusion is confirmed, that long waves are more sensitive to variation of the bottom. The special feature of these bottom models is that they cause refraction and shoaling of incoming waves. As a result, induced wave loads might be different between linear and nonlinear waves. To confirm the effect, it is necessary to calculate the wave loads on an LNG carrier. In the Boussinesq model, however it is difficult to include the body in the computation domain because the numerical method uses a rectangular grid to apply the finite difference scheme. Due to this restriction on numerical treatment, only the Froude– Krylov force is calculated. Calculation of the Froude–Krylov force just requires the pressure on the body surface due to an undisturbed wave, which leads to the body not needing to be included in the computation domain. Despite this restriction, meaningful information is expected because the Froude–Krylov force is a dominant factor determining the characteristics of wave loads. The general definition of test bathymetry is presented in Fig. 6. L0 and L1 are defined as positive toward the ship center. For all test conditions described in Fig. 6, the ship is assumed to be located at the center of the slope and with the bow toward the deeper region, which corresponds to head sea condition for seakeeping problems. This location is quite meaningful because the different wave length and amplitude features coexist along the slope. One representative result of nonlinear Froude–Krylov force and moment is plotted in Fig. 7. This bathymetry condition corresponds to that in Fig. 5(b). Linear Froude–Krylov force in the Rankine panel method is not evaluated because decomposing the velocity potential of an incident wave from the total velocity potential is not easy. Therefore, the trend of nonlinear wave load is observed as wave amplitude grows. The time scale is normalized with the combination of ship length (L) and the gravitational constant (g). In addition, the force and moment are normalized by a combination of fluid density (r), gravitational constant, ship length and wave amplitude (A). According to the observation in Fig. 5, the waves show similar shape in the region of the slope. Similarly, the time histories of the Froude–Krylov forces on heave and pitch nearly coincide for different wave amplitudes. These results indicate that nonlinear effects are not remarkable in these bathymetry even though wave amplitude grows. Hence, it can be reasonably concluded that the main portion of wave load originates from the linear wave load. Moreover, the linear regime has another advantage with respect to computation time. Despite the fact that the Boussinesq theory requires much time to solve the algebraic equations and to update the matrix at every time step, linear theory shows fast computation because the value of matrix in the algebraic equation is fixed over the entire computation. Therefore, the linear regime can be thought of as an effective method for the practical estimation of wave loads. Based on this conclusion, the motion and hydrodynamic properties of the LNG carrier are numerically estimated using only the linear regime.

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3.3. Constant depth problem Finite-depth effect is observed by comparing hydrodynamic coefficients, exciting force and motion response over various constant water depths. In the time-domain analysis, added mass and damping coefficient can be obtained from the Fourier analysis of hydrodynamic force on a body under a forced harmonic motion. Fig. 8 shows the added mass and damping coefficient of an LNG carrier for three different water depths. The present results using the Rankine panel method show similar tendencies with those from the unified theory (Kim, 1999), despite slight discrepancy, which is inevitable due to differences in numerical treatment between two methods. The added masses from the unified theory tend to slightly increase at high frequency. The unified theory presents the kernel of integral equation as infinite series in finite depth. At high frequencies, the convergence of series is very slow since it becomes deep water case. In this computation, the number of series truncated in the program might be not sufficient to make converged results in high frequency.

Fig. 16. Exciting force RAO of LNG carrier over sloped (Slope I defined in Table 2) and constant bottom: head sea. (a) Heave force and (b) Pitch moment.

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However, the correspondence of overall trend is obvious. A significant change in hydrodynamic coefficients is observed at low frequencies for the two results of finite depths. In both computations of the Rankine panel method and the unified theory, the added mass and damping coefficients of heave–heave component increase dramatically as the water depth decreases. This is caused by the increased movement of water particles around the whole area of the restricted water depth. The wave exciting force is the sum of the Froude–Krylov and diffraction forces, and the both forces are highly related to incident waves. In the computation, the head sea condition is assumed because it is the usual operating condition of a floating body in the normal weather conditions. Fig. 9 compares the response amplitude operators (RAOs) of wave excitation in infinite and finite depths. The overall trend shows good accordance between the Rankine panel method and unified theory. In the high frequency region where the

change of dispersion relation is small, the wave excitation force RAOs are very similar to those in deep water. However, at low frequencies, the peak RAO is shifted to lower frequencies as water depth becomes shallower. The motion response is the combined result of hydrodynamic coefficients and wave excitation. Similar to the trend observed in the hydrodynamic coefficients and wave excitation, the motion RAOs in Fig. 10 show a variation due to finite depth, especially in lower frequencies. The variation becomes more apparent as water depth becomes shallower. The shift of the peak to lower frequency is an important feature, and it mainly follows the change of dispersion relation, because the length scale between the ship and the wave length is an important factor for determining properties of the motion response. Fig. 11 compares the instantaneous wave contours in a forced heaved motion and in fixed motion. In fixed motion, the incident

Fig. 17. Added mass and damping coefficient of LNG carrier over two different slopes (Slopes I and II defined in Table 2) and constant bottoms. (a) Added mass of heave– heave, (b) Added moment of inertia of pitch–pitch, (c) Damping coefficient of heave–heave and (d) Damping coefficient of pitch–pitch.

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Fig. 18. Exciting force RAO of LNG carrier over two different slopes (Slopes I and II defined in Table 2) and constant bottoms: head sea. (a) Heave force and (b) Pitch moment.

wave is removed and only a disturbed wave is plotted. Shorten wave length is an apparent feature in a shallower depth, which is caused by the change of the dispersion relation. Increased wave amplitude, which is especially observed in a heave-forced motion, contributes to the different behavior of hydrodynamic coefficients and force RAOs. 3.4. Uneven bottom problem As is already stated above, the problem of varying bathymetry is restricted to a sloped bottom. All definitions of lengths and depths follow the notations defined in Fig. 6. The LNG carrier is assumed to be located at the center of the sloped bottom. In the problem with sloped bottom, the reflection and change of local wave length can cause remarkable effect on floating-body motion. These properties about waves can be different according to the bottom conditions, precisely due to the depths of adjacent flat bottoms and length of slope. To verify the effect induced by each factor, two kinds of tests are considered. In the first case, the

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Fig. 19. Motion RAO of LNG carrier over two different slopes (Slopes I and II defined in Table 2) and constant bottoms: head sea. (a) Heave motion and (b) Pitch motion.

depth of flat bottom in incoming region is increased to make the slope steeper. Therefore, the mean depth changes according to the increased depth. In the other case, the depths of the two flat bottoms are fixed and only the length of the slope is reduced. In this case, however, the mean depth remains fixed. The dimensions of sloped bathymetries for both tests are presented in Table 2. Before the computations, the mildest slope condition (Slope I in Table 2) is chosen and several tests are carried out to verify the reliability of numerical results. As described in Table 2, the length of slope and depths of two adjacent flat bottoms are designed as Ls ¼300, h0 ¼30 and h1 ¼15 m. The incident wave flows from the deeper region to shallower region. The slope is 1/20, which can be considered a mild slope. The same test condition was applied to the problem of wave propagation in Fig. 5(a). As verified earlier, the nonlinear effects on the wave load are not severe in this kind of model.

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One numerical type of error can happen because distortion of the progressive wave due to damping zone at the side wall causes unwanted effects on the ship motion. Therefore, it should be confirmed that the disturbed component by the damping zone does not exist near the position of the floating body. This is examined by the following procedure. The floating body is removed to observe the incident wave only. Then the incident wave is simulated without the side damping zone because the zone is not necessary in this head sea condition. The same test is carried out with the side damping zone applied in the absence of the body. In this computation, though body is absent, the shape and location of damping zone are designed as the same to the problem with body. Both wave elevations from two different tests

Fig. 12 shows the series of wave contours around the floating body for one representative case. On the stern side, the wave length is shorter and the wave amplitude is larger than on the bow side due to shallower depth. Interference by wave reflection is not significant because the slope is mild. Wave distortion near the side wall in the shallower domain is due to a damping zone. The damping zone on the side boundary starts from nearly the same location as the ship, enabling it to absorb the disturbed waves from the body. Most of incident and disturbed waves along longitudinal direction are effectively absorbed by damping layer in the behind of domain. The purpose of side damping zone is absorbing disturbed waves from body. As observed in Fig. 12(d), the surrounding damping zone effectively absorbs the disturbed waves.

a

b

c

d

Fig. 20. Added mass and damping coefficient of LNG carrier over five different slopes (Slopes I, III–VI defined in Table 2). (a) Added mass of heave–heave, (b) Added moment of inertia of pitch–pitch, (c) Damping coefficient of heave–heave and (d) Damping coefficient of pitch–pitch.

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are compared at specified points around floating body. Fig. 13 shows the wave elevation at the two points, where are correspond to AP and FP position if floating-body would be located. In this panel model, wave interference by the side damping zone is small. Therefore, the present combination of panel modeling and damping zone can be applied to the motion analysis of the floating body. The motion RAOs of surge, heave and pitch are presented in Fig. 14. The surge and heave motion RAOs are compared with the numerical results from Hauteclocque et al. (2009). The overall trend shows good agreement except for a small region in heave. To verify the relation of varying and constant depths, the RAOs are also compared to the results from constant water depths. Those results are calculated by assuming the water depth is constant at 15, 22.5 and 30 m. In the results, the motion responses in the sloped bottom region are quite similar to those from constant water depth corresponding to 22.5 m, which is the local depth at the position of midship. That means mild slope does not significantly influence the floating-body motion. A small difference is observed in the pitch response at low frequencies, as plotted in Fig. 14(d). The response becomes larger compared to the other results from constant depths. The reason can be found from the change of wave amplitude aft of the body. As observed in Fig. 13, the wave amplitude on the stern side increases compared to that off the bow due to the shoaling effect. This variation is especially significant in the lower frequencies because the effect of refraction and shoaling becomes stronger. This increased difference in wave amplitude on both sides increases. Other hydrodynamic properties are also evaluated for the sloped bottom. Added mass and damping coefficients are presented in Fig. 15. The overall trend is quite similar to those from constant depth in midship. This trend is also observed at the force RAOs in Fig. 16. In Figs. 14–16, the computation is conducted at the non-dimensional frequency larger than 1.0. The nondimensional wave frequency less than 1.0 means that wave length is greater than 6.28, i.e. more than 1.9 km in real scale. Therefore, there is no strong need to carry out the computation longer than that. Furthermore, in the problem with varying bottom, the generation of long wave requires larger domain, while higher resolution is needed around the body and varying bottom surface. It is not easy to satisfy the both conditions simultaneously, especially at long wave length. Some fluctuation of the pitch excitation moment in Fig. 16(a) seems due to the numerical error caused by such domain and resolution problem. In the region of frequencies higher than 2.0, the exciting force RAOs are similar to the results of 22.5 m. Those results indicate that the basic properties of disturbed wave around ship are quite similar to those from constant depth when the frequency is high. Particularly, the force RAO grows slightly as frequency becomes smaller. As shown earlier in Figs. 13 and 14, this is the combined effect of refraction and shoaling which becomes apparent in long waves. The growth is remarkably significant in pitch, and it brings larger pitch response as shown earlier in Fig. 14(c). In the case of mild slope, the motion response and other hydrodynamic coefficients show similar tendencies to those at constant depth at midship. As such, more concern should naturally be given to steeper slopes. In this problem, the stronger reflection and rapid change of local wave length can cause serious effect on floating-body motion. As explained earlier, two kinds of tests are designed to verify the possible strong effect by steeper slope. In the first case, two different depths are assumed at the incoming region. According to the notation defined in Fig. 6, the designed depths of deeper region (h0) are 30 and 45 m (Slopes I and II in Table 2). Lengths of slope and depth of shallower region are defined as Ls ¼300 and h1 ¼15 m, respectively. This model was used earlier to observe nonlinear effects as shown in Fig. 5. A

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strong nonlinear effect is not observed in the result of nonlinear incident waves. Based on this conclusion, added mass and damping coefficients are calculated using the linear Rankine panel method. Fig. 17 shows the numerical results for two bottom models, which are compared to the results from each mean depth. The overall trend is quite similar to the results of each mean depth. Slightly increased values are observed in the damping coefficient of pitch–pitch. This is reasonable since the sloped bottom causes different behavior in the propagation of dissipated waves along each direction. Hence, the most significant difference occurs in the pitch response, especially for the steeper slope. The exciting force and motion response are also plotted in Figs. 18 and 19. Similar to the results of added mass and damping coefficient, the heave does not show remarkable differences between sloped and constant bottoms. The increase in exciting force and the motion response of pitch are both noticeable. The other test is designed to have steeper slope with the same mean depth. The depths of two different flat regions are fixed as h0 ¼30 and h1 ¼15 m. Five different lengths of slopes (Ls) are

a

b

Fig. 21. Exciting force RAOs of LNG carrier over five different slopes: head sea (Slopes I, III–VI defined in Table 2). (a) Heave force and (b) Pitch moment.

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Based on the linear regime, the hydrodynamic properties of a floating body over various bathymetries can be effectively obtained using the Rankine panel method. Important information can be obtained from the comparison of the results obtained for the two different tests above. In the case of sloped bottom, the depths of the adjacent bottoms are important factors for determining the hydrodynamic characteristics of a floating body. In addition, the results show similar tendencies to those from the problem of constant depth corresponding to mean depth of sloped bottom. When the mean depths of sloped bottoms are the same, there exists little variation in the responses for bottoms below a certain value of slopes. The observation of significant difference is restricted only to very steep slope (3/2 among the conditions tested here).

4. Conclusions

Fig. 22. Motion RAOs of LNG carrier over five different slopes: head sea (Slopes I, III–VI defined in Table 2). (a) Heave motion and (b) Pitch motion.

assumed as 0, 10, 30, 100 and 300 m (Slopes I, III–VI in Table 2). Then, hydrodynamic coefficients are computed for those five slopes. As observed in Fig. 20, the added mass and damping coefficient computed from bottoms with different slopes are located closely within a narrow region. Moreover, as clearly seen in the added mass of heave–heave, the hydrodynamic coefficients slightly increase as the slope becomes steeper. The exciting force and motion RAOs are also computed by applying the same bottom models. In Figs. 21 and 22, two results related to pitch show remarkable increase in lower frequencies. The increase is apparent in both slopes of infinite and 3/2, but the force and motion responses rapidly decrease below those slopes. The responses show small changes in the milder slopes below 1/2. Because the influence of varying bathymetry is also affected by the value of depth itself, it is not easy to point out the general criteria remarkable changes for different slopes. Among these test conditions, the slope of 3/2 is regarded as important criteria for remarkable changes.

Hydrodynamic features of a floating body have been observed, especially for operation near coastal area. Because nonlinear effects are important concerns in this shallow region, effects are compared by applying the linear Rankine panel method and the nonlinear Boussinesq equation. The incident waves and induced Froude–Krylov force are compared between the two methods. In the moderate depths where floating bodies usually operate, the nonlinear effect can be ignored because the main portion of wave loads is contributed by linear wave forces. Therefore, the linear approach is expected to be a reasonable and favorable choice for the motion analysis even near coastal areas. Based on the linear approach, the hydrodynamic properties of an LNG carrier were observed for various depth conditions. In the case of constant depth, the hydrodynamic coefficients and motion responses showed favorable agreement with the responses from unified theory. As water depth becomes shallower, dramatic change of the hydrodynamic properties is observed, especially at lower frequencies. Shift in the exciting forces and motion responses were noticeable. In the problem with sloped bottom, one important feature was the changes in the wave length and amplitude as waves passed the sloped layer. Reflection was not severe in the mildly varying model. Furthermore, the hydrodynamic coefficients, exciting force and motion response showed similar tendencies with the results from the problem of constant depth corresponding to the mean depth of slopes. The pitch response was slightly increased in the lower frequency range. The changed wave length and amplitude off the bow and stern regions increased the pitch moment and response. A similar tendency was also observed even in the cases of steeper slopes. In the two different tests focusing on adjacent depth and length of slope, the results were similar to those for each of the mean depths over a wide range of frequencies. When the mean depth is fixed, remarkable differences in the hydrodynamic properties were not observed below the slope of 3/2. This implies that nonlinear effects might be not stronger unless the slope becomes remarkably steep. Therefore, the present linear Rankine panel method can be utilized to evaluate the hydrodynamic properties of floating bodies over various bathymetries.

Acknowledgements This study has been carried out as a part of WISH-OFFSHORE JIP which is funded by ABS, ClassNK, DSME, HHI, KR, SHI, and STX Offshore and Shipbuilding. This work was also partly supported by the Global Leading Technology Program of the Office of Strategic R&D Planning(OSP) funded by the Ministry of

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