Modelling of multi-bodies in close proximity under water waves—Fluid forces on floating bodies

Ocean Engineering 38 (2011) 1403–1416

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Modelling of multi-bodies in close proximity under water waves—Fluid forces on floating bodies Lin Lu a,b,n, Bin Teng b, Liang Sun c, Bing Chen d a

Center for Deepwater Engineering, Dalian University of Technology, Dalian 116024, China State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China c Centre for Offshore Research and Engineering, Department of Civil Engineering, National University of Singapore, 117576 Singapore, Singapore d School of Hydraulic Engineering, Dalian University of Technology, Dalian 116024, China b

a r t i c l e i n f o

abstract

Article history: Received 11 October 2010 Accepted 26 June 2011 Editor-in-Chief: A.I. Incecik Available online 19 July 2011

This work presents two-dimensional numerical results of the dependence of wave forces of multiple floating bodies in close proximity on the incident wave frequency, gap width, body draft, body breadth and body number based on both viscous fluid and potential flow models. The numerical models were validated by the available experimental data of fluid oscillation in narrow gaps. Numerical investigations show that the large amplitude responses of horizontal and vertical wave forces appear around the fluid resonant frequencies. The convectional potential flow model is observed to un-physically overestimate the magnitudes of wave forces as the fluid resonance takes place. By introducing artificial damping term with appropriate damping coefficients mA[0.4, 0.5], the potential flow model may work as well as the viscous fluid model, which agree with the damping coefficients used in our previous work for the predication of wave height under gap resonance. In addition, the numerical results of viscous fluid model suggest that the horizontal wave force is highly dependent on the water level difference between the opposite sides of an individual body and the overall horizontal wave force on the floating system is generally smaller than the summation of wave force on each body. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Multiple floating bodies Narrow gaps Fluid resonance Wave forces Viscous fluid model Potential flow model Artificial damping term

1. Introduction The topic of complex hydrodynamics of multiple floating bodies in close proximity under water waves has received increasing attentions during the last two decades, for example Kagemoto and Yue (1993), Koo and Kim, (2005), Koutandos et al. (2005), Kristiansen and Faltinsen (2010) and Sauder et al. (2010) among others. This is mainly attributed to the great demands for oil and gas, which leads to the flourishing applications of multiple floating facilities for the extraction of ocean resources. The typical examples are the Floating Production Storage and Offloading (FPSO) operation to shuttle tanker, Liquified Natural Gas (LNG) carriers in the proximity of the terminals, the assembly of Very Large Floating Structures (VLFS) involving a great number of individual modules and the appearance of catamarans. Under these circumstances, the effects of hydrodynamic interactions have to be carefully taken into account for the safe operations in the practical engineering. For the purpose of simplification but not loss generality, the multiple floating bodies are often simplified to fixed rectangular

n Corresponding author at: Center for Deepwater Engineering, Dalian University of Technology, Dalian 116024, China. Tel.: þ86 411 84708481; fax: þ 86 411 84708526. E-mail address: [email protected] (L. Lu).

0029-8018/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2011.06.008

boxes. It has been shown that the extreme large amplitude wave oscillations may take place in the narrow gaps formed by the multiple boxes with narrow gaps under specific incident wave frequencies (Saitoh et al., 2006; Iwata et al., 2007; Lu et al., 2011), and accordingly giving rise to the increase of wave forces on the floating bodies (Miao et al., 2001; Zhu et al., 2005, 2008; Sun et al., 2010). Three issues are of concern, i.e., what is the appearance condition of the fluid resonance in the narrow gaps? What is the exact amplitude of the resonant wave height in the narrow gaps? And how does the wave force respond near the resonant frequency? The appearance conditions of fluid resonance in narrow gaps can be estimated roughly using the natural frequency of fluid bulk involving the oscillation. That means the fluid resonance takes place as the incident wave frequency matches the natural frequency of oscillating fluid. The previous theoretical analysis (Miao et al., 2001; Saitoh et al., 2006) show that, for example, the resonant condition can be approximated by kls ¼np or kls tanh (kh)¼1, where k is the wave number, ls is the characteristic geometrical parameter but with different definitions in the literatures, h is the water depth and n¼ 1, 2, y. The above formulas have been confirmed by the numerical results (Miao et al., 2001; Zhu et al., 2005) and the experimental data (Saitoh et al., 2006; Iwata et al., 2007). As for the second question

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mentioned above, many efforts including experimental investigations and numerical simulations based upon both potential flow model and viscous fluid model have been made. The experimental investigations (Saitoh et al., 2006; Iwata et al., 2007) demonstrated that the maximal wave height in the narrow gap under resonant motion may reach up to 5 times of the incident wave height. The numerical investigations conducted so far are mainly based on the potential flow theory (Miao et al., 2000, 2001; Li et al., 2005; Zhu et al., 2008; Sun et al., 2010) employing the boundary element method and scaled boundary finite element method. It was understood that the conventional potential flow model trends to over-predict the resonant wave heights in the narrow gaps and consequently leads to the over-estimation of the wave forces on the floating bodies because the physical energy dissipations due to the fluid viscosity, vortex shedding and even turbulence cannot be taken into account in the context of potential flow theory. On the other hand, the Computational Fluid Dynamics (CFD) methods were also utilized to investigate the fluid resonance in the narrow gaps induced by the incident waves (Lu et al., 2008, 2010a,b, 2011; Sauder et al., 2010). The numerical results indicated that the viscous fluid model performs well in predicting the violent free surface oscillation at the fluid resonance and leads to good predictions compared with the experimental observations. However, the computational efforts are considerably excessive. In order to save computer time and improve the accuracy of potential flow model at resonant condition, some particular numerical techniques were proposed, such as the rigid lid concept (Huijsmans et al., 2001), linear damping term as body force (Newman, 2003) and linear dissipative term in free surface boundary condition (Chen, 2005; Pauw et al., 2007; Bunnik et al., 2009). But the introduction of artificial damping term seems somewhat arbitrary for the rigorous potential theory, and hence the careful calibrations are necessary. As for the wave forces on the multiple floating bodies in close proximity, most of the available examinations were conducted by means of numerical investigations employing the conventional potential flow model. It was demonstrated by Miao et al. (2001), for example, that the horizontal wave force on the twin bodies with narrow gap is about ten times of that for an isolated floating body. It was believed that the wave forces were also overestimated by the conventional potential flow model, just as its over-prediction for the resonant wave heights in the narrow gaps. From the viewpoint of practical applications, the potential flow model has more attractions due to its low computer demands. Although the method of introducing calibrated damping term into the conventional two-dimensional potential flow model can obtain reasonable predictions on the resonant wave heights (Lu et al., 2011), it needs to re-examine whether the calibrated damping coefficients are applied to evaluate the wave forces on the floating bodies with acceptable accuracy, especially for the resonant condition. In addition, the influences of gap width, body draft, body breadth ratio and body number on the variation of wave forces (both in horizontal and vertical) with the incident wave frequency remain unclear nowadays. These issues will be addressed in this work. This article is organized as follows. In Section 2, the twodimensional numerical models used in this work will be described briefly including the viscous fluid model, the conventional linear potential flow model and the potential flow model with artificial damping term. The numerical models are validated in Section 3 using the available experimental data considering the wave heights in the narrow gaps at various wave frequencies. The numerical results of the wave forces and the corresponding response frequencies are presented in Section 4. The dependence of the horizontal and vertical wave forces on the incident wave frequency, body configuration and spatial arrangement is

examined. The damping coefficients used for the potential flow model in order to obtain the reasonable predictions of the amplitudes of wave forces are also calibrated in this work. In addition, the mechanism of wave forces is discussed in this section. Finally, the main concluding remarks are given in Section 5.

2. Numerical models 2.1. Viscous fluid model The governing equations for the incompressible Newton viscous fluid flows read ( qðx,tÞ x A Os @ui ð1Þ ¼ 0 x2 = Os @xi   @ui @u 1 @p @ @ui u @q þ uj i ¼  þu þfi þ @xj @xj 3 @xi @t @xj r @xi

ð2Þ

where, ui and fi are the velocity and body force components in the ith direction and p, r, n denote the pressure, fluid density and kinematic viscosity, respectively. In Eq. (1), a source term q(x, t) is activated in the source region Os in order to generate desirable target waves. For the linear monochromatic waves, the source function reads (Lin and Liu, 1999) qðtÞ ¼ CH0 sinðotÞ=S

ð3Þ

where C is the phase velocity of water waves, H0 is the incident wave height with H0 ¼2A and A is the wave amplitude, o denotes the wave frequency with o ¼2p/T, T is the wave period and S is the area of source region. In the present viscous fluid model, the reflected wave is eliminated by means of the artificial spongy layers. Therefore, the body force term in Eq. (2) is devised as fi ¼ gi þ Ri

ð4Þ

where, fi denotes the total forces in xi direction, gi is the gravitational force, Ri is the artificial force used in the spongy layers (Kim et al., 2001; Lu et al., 2010a,b). The free surface is captured using the Computational Lagrangian–Eulerian Advection Remap Volume of Fluid method (CLEAR-VOF, Ashgriz et al., 2004). For the present water wave problem, the void cells are ignored due to the rather small density of air and the full elements are processed as usual, whereas the partial elements are treated specially by means of averaging the density and viscosity

r ¼ jrw þð1jÞra , u ¼ juw þ ð1jÞua

ð5Þ

where the subscript w and a represent the water and air phases, respectively, and j denotes the volume fraction of water phase. The governing equations are solved by using the three-step Taylor–Galerkin finite element method (Jiang and Kawahara, 1993; Zhao et al., 2004), which shows good performance in dealing with the advection-diffusion problems. When the pressure and velocities are obtained, the fluid forces F on the bodies resulted from the pressure and shear stress components are computed by the integral operations along the solid wall Z Z F ¼  pUn ds þ mð@ut =@nÞds ð6Þ s

s

where n is the unit normal vectors, ut is the tangent velocity component and m is the dynamic viscosity. The initial conditions for the numerical computations are u(x, 0)¼ 0 and p¼ps, where ps is the static water pressure. At the ends of sponge layers and the solid walls, the non-slip boundary condition is imposed. According to the requirements of stress balance,

L. Lu et al. / Ocean Engineering 38 (2011) 1403–1416

but neglecting the viscous effects, the simple normal dynamic free surface condition p¼0 is implemented while the velocity extrapolations (Yang et al., 2006) on the free surface is supplemented. 2.2. Potential flow model The governing equation for the potential flows can be described by the Laplace equation of velocity potential ð7Þ

where Fðx,y,tÞ is the velocity potential. For the harmonic wave motion, the time-dependent part of velocity potential can be separated out as Fðx,y,tÞ ¼ Re½fðx,yÞeiot , where fðx,yÞ is the complex potential and it can be further divided into the incident and scattering parts

fðx,yÞ ¼ fI ðx,yÞ þ fS ðx,yÞ

ð8Þ

The incident potential fI ðx,yÞ for the present linear water wave reads igA

fI ðx,yÞ ¼ 

cosh kðyþ hÞeikx

ð9Þ

o cosh kh where k¼2p/L is the wave number, L is the wave length and h is

the water depth. In order to account for the flow resistance in the narrow gaps an artificial damping term is introduced into the potential flow model (Chen, 2005; Pauw et al., 2007) f ¼ mðxÞV ¼ mðxÞrF

ð10Þ

where m is an artificial damping coefficient and V is the averaging flow velocity in the narrow gaps. By using the first order approximation of wave profile x, we have 1 g

zðx,y,tÞ ¼  Ft ðx,y,tÞmðxÞ=g Fðx,y,tÞ

ð11Þ

Considering the kinematic condition on the free surface, @z=@t ¼ @F=@y, Eq. (10) can be re-written as @F2 ðx,y,tÞ @Fðx,y,tÞ @Fðx,y,tÞ ¼ g þ mðxÞ @t @y @t 2

of the corresponding experimental and numerical setup is shown in Fig. 1. Two or three identical boxes with the same breadth B¼0.5 m are fixed in a wave flume with water depth h¼ 0.5 m. The draft D and gap width Bg are 0.252 m and 0.05 m, respectively. The incident wave heights range from 0.023 m to 0.025 m. The fluid resonance in single narrow gap is firstly simulated using the present viscous fluid model and potential flow model with different damping coefficients. Note that, the Body C shown in Fig. 1 is absent for the present case. The numerical results of the dimensionless mean wave height Hg/H0 in the narrow gap at various wave number kh are plotted in Fig. 2, together with the experimental data for the purpose of comparison. It can be seen from Fig. 2 that both the viscous fluid model and the potential flow model can predict the resonant frequency with accuracy. The predicted resonant frequency, kh¼1.556, is almost identical to that observed in the laboratory tests (Saitoh et al., 2006). The variation of Hg/H0 with kh obtained from the viscous fluid model is seen to agree well with the experimental data. However, the conventional potential flow model (that is, m ¼0) over-predicts the wave heights in the narrow gap at the frequencies near the resonant frequency while it works as well as the viscous fluid model at the other wave frequencies. The accuracy of the predicted wave heights around the resonant frequency is improved substantially when the non-zero damping coefficients are used. It is found that the damping coefficient m ¼0.4 leads to the potential numerical results consistent well with the experimental data. It is also observed that the predicted wave heights by the potential flow model are hardly affected by the values of the damping coefficient at the frequencies outside a frequency range of 10% at either side of the resonant frequency. The numerical results shown in Fig. 2 appear to suggest that the potential flow model can be used in predicting the resonant wave height in the narrow gap if a properly calibrated damping coefficient is used. The fluid resonance in two narrow gaps between three identical fixed boxes is also used to validate the present numerical models. The dimensions of water depth, body draft, body breadth and gap width are the same as those used in the previous twin-body case, but with the intrusion of the third box of Body C and consequently

ð12Þ

Accordingly, the time-independent complex potential reads @f 1 ¼ ðo2 f þ iomfÞ ð13Þ g @y The scattering potential satisfies the Laplace Equation as well and is subjected to the following boundary conditions, @fS =@y ¼ ðo2 fS þ iomfS Þ=g at the still water level of y¼0; @fS =@n ¼ @fI =@n along the solid walls; @fS =@y ¼ 0 at the sea bottom y¼  h; @fS =@x ¼ 7ikfS at the far field downstream and upstream, respectively. The governing equations of potential flow together with the boundary conditions are solved using a standard boundary element method (Li and Teng, 2002) in the frequency domain. Note that, only the first order results are concerned in this work. The wave forces on the fixed floating bodies are evaluated as Z F ¼ rðiomÞf ds ð14Þ

3. Numerical validations The available laboratory tests (Saitoh et al., 2006; Iwata et al., 2007) are adopted to validate the present viscous fluid model and potential flow model (with and without the damping term), in which the variation of the mean wave height in the narrow gap with the incident wave frequency is considered. Two benchmark cases are used, i.e., single narrow gap formed by twin boxes and two narrow gaps separated by three identical boxes. A sketch definition

Incident wave

y Body A

Body B

Body C

Transmitted wave x D

Reflected wave B

h

B

Fig. 1. Sketch definition of experimental and numerical setup.

10

Hg / H0

@2 Fðx,y,tÞ @2 Fðx,y,tÞ þ ¼0 @x2 @y2

1405

Exp. Saitoh et al (2006) Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

5

0 0.5

1.0

1.5

2.0 kh

2.5

3.0

3.5

Fig. 2. Comparison of non-dimensional wave height with respect to incident wave frequency for twin bodies with h ¼ 0.5 m, B¼ 0.5 m, D¼ 0.252 m, Bg ¼ 0.05 m and H0 ¼ 0.024 m.

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two narrow gaps, referring to Fig. 1. Note that, the Gap 1 is defined as between Body A and Body B while the Gap 2 is that formed by Body B and Body C. The variations of Hg/H0 with kh from the viscous flow model and the potential flow models are compared with experimental data of Iwata et al. (2007), as shown in Fig. 3. It can be seen from Fig. 3(a) that the variation of Hg/H0 with kh in Gap 1 from viscous flow model agrees well with the experiment data. Both the viscous numerical results and the experimental data show that the variation of wave height in Gap 1 has two peak values. This suggests that the fluid resonance in Gap 1 may take place at two distinct frequencies in contrast to the single resonant frequency in the previous case of twin bodies with single narrow gap. The double-peak variation of Hg/H0 with kh is also observed in Gap 2, as shown in Fig. 3(b), although the second fluid resonant response is not as significant as that observed in Gap 1. It is believed that the existence of two resonant frequencies is mainly attributed to the presence of the second narrow gap or the intrusion of the third box. The numerical results from the conventional potential flow model (m ¼0.0) show that the overall characteristics of the variation of Hg/H0 with kh in Gap 1 and Gap 2 are in general accordance to the experimental data, but with over-predictions over the frequency band covering the two resonant frequencies. By introducing artificial damping term, the numerical discrepancies from the potential flow model near the resonant frequencies are reduced significantly for the both narrow gaps. Particularly, the value of m ¼0.4 can lead to relatively accurate predictions of wave heights around the resonant frequencies. It is worth noting that the present damping coefficient m ¼0.4 is valid not only for the Gap 1 but also for the Gap 2. It was also observed that the using of damping force in the potential flow model does not degrade the accuracy of potential flow model in predicting the wave height at the frequencies far from the resonant frequencies. In addition, it was confirmed again that the potential flow model with and without damping term are capable of predicting the resonant frequencies as accurately as the viscous fluid model and the laboratory tests.

Hg / H0

10 Exp. Iwata et al (2007) Present Viscous Model Potential Model, μ=0.0 Potential Model, μ=0.3 Potential Model, μ=0.4 Potential Model, μ=0.5

5

0

0.5

1.0

1.5

2.0 kh

2.5

3.0

3.5

Hg / H 0

10 Exp. Iwata et al (2007) Present Viscous Model Potential Model, μ=0.0 Potential Model, μ=0.3 Potential Model, μ=0.4 Potential Model, μ=0.5

5

0

0.5

1.0

1.5

2.0 kh

2.5

3.0

The above numerical simulations and comparisons confirmed that the present viscous fluid model has good performance in simulating the fluid resonance in narrow gaps. The conventional potential flow model can also produce satisfying predictions for the resonant frequencies and wave heights outside the near region of resonant frequencies. As far as the limitation of conventional potential flow model in prediction the wave height around the resonant frequencies is concerned, it can be improved by introducing appropriate damping term. However the damping coefficient used for the potential flow model has to be verified by available experimental data or reliable viscous fluid model. The more details on the numerical results of resonant wave height in the narrow gaps and the calibrations of damping coefficient are referred to Lu et al. (2011). However, our previous numerical investigations were mainly concerned with the fluid oscillation in the narrow gaps. The dependence of the horizontal and vertical wave forces on the floating bodies and the validation of the appropriate damping coefficient for wave forces are remained, which will be examined in this work.

4. Numerical results The variation of horizontal and vertical wave forces on the multiple floating bodies with the incident wave frequencies under different gap widths Bg, body drafts D, body breadths B and body numbers are examined in this section. Referring to Fig. 1 again, two or three rectangular boxes are fixed in a wave flume with a constant water depth h¼ 0.5 m. A uniform incident wave height H0 ¼ 0.024 m is used throughout this work. Considering the unavailable experimental data of wave forces, the damping coefficients of the potential flow model, mainly used to evaluate the wave forces in this work, are tuned by the CFD results since the viscous fluid model has shown good performance in simulating the problem of gap resonance. For the purpose of clarity and simplification, we define herein the dimensionless amplitude of the horizontal wave forces as FAx , while the amplitude of vertical wave forces is denoted by FAy . They are both normalized by rghA, where A is the incident wave amplitude. The wave forces are evaluated by Eqs. (6) and (14). The characteristic response frequency of the wave forces, corresponding to the counterpart of resonant frequency of fluid oscillation in narrow gaps, is denoted by fr. In the following Sections 4.1–4.4, the dependence of wave forces on gap width, body draft, body breadth ratio and body number will be presented together with the comparisons between the numerical results of viscous fluid model and potential flow model with different artificial damping coefficients. As for the correlation of response frequencies of wave forces and fluid resonance, it will be investigated separately in Section 4.5 and compared with the analytical formula of Molin (2001). The mechanism of wave forces on the bodies, related closely with the water level difference, and the overall wave forces on the floating system are discussed in Section 4.6. 4.1. Influence of gap width

3.5

Fig. 3. Comparison of non-dimensional wave height with respect to incident wave frequency in Gap 1 (a) and Gap 2 (b) for three identical boxes with h ¼ 0.5 m, B¼ 0.5 m, D¼ 0.252 m, Bg ¼ 0.05 m and H0 ¼0.024 m.

In order to investigate the influence of gap width on the wave forces, three different gap widths Bg ¼0.03, 0.05 and 0.07 m are considered for the situation of two identical boxes with the same body breadth 0.5 m, draft 0.252 m and water depth 0.5 m. The numerical results of the variation of horizontal wave forces on Body A and Body B with the incident wave frequencies are shown in Figs. 4 and 5, respectively. It can be seen from Fig. 4 that, according to the numerical results of viscous fluid model, the horizontal wave forces on Body A increase gradually with kh at low frequencies and reach the maximums around some particular

L. Lu et al. / Ocean Engineering 38 (2011) 1403–1416

3

3 Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

2 1

1

1.5

2.0 kh

2.5

0 1.0

3.0

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

2.0 kh

2.5

3.0

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

2 FxA

FxA

2

1

1

1.5

2.0 kh

2.5

0 1.0

3.0

3

1.5

2.0 kh

2.5

3.0

3 Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

2 FxA

FxA

2

1

1 0 1.0

1.5

3

3

0 1.0

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

FxA

FxA

2

0 1.0

1407

1.5

2.0 kh

2.5

3.0

0 1.0

1.5

2.0 kh

2.5

3.0

Fig. 4. Influence of gap width on the horizontal wave forces on Body A of twin bodies at h ¼ 0.5 m, B¼ 0.5 m, D ¼0.252 m and H0 ¼ 0.024 m: (a) horizontal wave force on Body A at Bg ¼ 0.03 m, (b) horizontal wave force on Body A at Bg ¼ 0.05 m and (c) horizontal wave force on Body A at Bg ¼0.07 m.

Fig. 5. Influence of gap width on the horizontal wave forces on Body B of twin bodies at h ¼0.5 m, B ¼0.5 m, D ¼ 0.252 m and H0 ¼0.024 m: (a) horizontal wave force on Body B at Bg ¼ 0.03 m, (b) horizontal wave force on Body B at Bg ¼0.05 m and (c) horizontal wave force on Body B at Bg ¼ 0.07 m.

wave frequencies. However, the maximal amplitudes of horizontal wave force on Body A are definitely smaller than two times of rghA and it has little dependence on the gap width. On the other hand, the corresponding response frequencies of the maximal horizontal wave forces fr trend to decrease with the increase of gap width, which is similar to the resonant frequency of fluid oscillation in the narrow gap (Lu et al., 2011). The comparisons between potential flow model and viscous fluid model show that the conventional potential flow model predicts very large horizontal wave force on Body A in the narrow band of frequency around fr, although it works as well as the viscous fluid model at the other wave frequencies. However, the over-predicted amplitudes of horizontal wave force on Body A can be reduced significantly by introducing artificial damping force. It can be seen that the potential flow model with m ¼0.4–0.5 can produce reasonable predictions for the maximal horizontal wave forces comparing with the viscous fluid model. Fig. 4(a)–(c) indicate also that the potential flow model is able to predict the response frequency fr as same as the viscous fluid model. The numerical results of horizontal wave forces on Body B are presented in Fig. 5. The similar variation of FAx with kh as Fig. 4 can be observed in these figures. The peak values of horizontal wave forces on Body B are almost in the same magnitude regardless of the increase of gap width. On the other hand, the response frequencies of horizontal wave force decrease as the increase of gap width. In generally, the numerical results from potential flow model with m ¼0.5 are in good agreement with those obtained by the viscous fluid model. Different from what was observed in Fig. 4, where the amplitudes of horizontal wave force on Body A at the lower frequencies are generally smaller than those at the

higher wave frequencies, the numerical results of Fig. 5 show that the larger horizontal wave force on Body B appears at the lower frequencies. Meanwhile, the very small amplitudes of horizontal wave forces are observed at the higher frequencies, even approaching to zero as kh42.0. The influence of gap width on the vertical wave forces on Body A and Body B are further investigated in Figs. 6 and 7, respectively. According to the numerical results of viscous fluid model, the amplitudes of vertical wave force on Body A trend to decrease in general with the increase of wave frequency, but three phases can be identified. In the first phase, covering the lower frequency region, FAy is observed to decrease slowly with kh. Then, it decreases quickly at the medium frequencies, that is, the second phase. Finally, the very slow variation is observed in the higher frequency band, that is, the third phase. Distinct to the results of viscous fluid model, the conventional potential flow model leads to two obvious peak values of the vertical wave forces on Body A. It can be found that the lower frequency, corresponding to the largest vertical wave forces, is almost identical to the boundary between the above mentioned phase 1 and phase 2, while the second one corresponding to the lowest vertical wave force locates at the frequency bound between phase 2 and phase 3. The former is referred to the response frequency of vertical wave forces since it also appears in the CFD results. The comparisons between the viscous and potential models confirmed again that the conventional potential flow model gives rise to unreasonable over-prediction of vertical wave forces around the near region of response frequency fr. Moreover, a fictional response frequency is even produced and the vertical wave force is under-estimated there. However, the potential flow

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2 Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

1

0 1.0

1.5

2.0 kh

2.5

F yA

F yA

2

1

1.5

2.0 kh

2.5

2.5

3.0

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

1.5

2.0 kh

2.5

3.0

2 Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

1

1.5

2.0 kh

2.5

FyA

FyA

2.0 kh

1

0 1.0

3.0

2

0 1.0

1.5

2 Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

FyA

FyA

2

0 1.0

1

0 1.0

3.0

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

3.0

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

1

0 1.0

1.5

2.0 kh

2.5

3.0

Fig. 6. Influence of gap width on the vertical wave forces on Body A of twin bodies at h ¼ 0.5 m, B¼ 0.5 m, D ¼0.252 m and H0 ¼ 0.024 m: (a) vertical wave force on Body A at Bg ¼ 0.03 m, (b) vertical wave force on Body A at Bg ¼ 0.05 m and (c) vertical wave force on Body A at Bg ¼ 0.07 m.

Fig. 7. Influence of gap width on the vertical wave forces on Body B of twin bodies at h ¼ 0.5 m, B¼ 0.5 m, D ¼0.252 m and H0 ¼ 0.024 m: (a) vertical wave force on Body B at Bg ¼0.03 m, (b) vertical wave force on Body B at Bg ¼ 0.05 m and (c) vertical wave force on Body B at Bg ¼ 0.07 m.

model may work as well as the viscous fluid model in the frequency regions with certain extend smaller or greater than the response frequency. In addition, the characteristic response frequency is observed to decrees with the increase of gap width, which is confirmed by both the viscous fluid model and the potential flow model. It can be seen from Fig. 6 that the potential flow model is also capable of producing reasonable predictions for the amplitude of the vertical wave forces on Body A when the damping coefficients of m ¼0.3–0.5 are introduced. This is valid throughout the wave frequency band considered here. As far as the vertical wave forces on Body B is concerned, the similar three-phase variation of FAy with kh is observed according to the numerical results of viscous fluid model, as shown in Fig. 7. However, it was found that the magnitude of vertical wave forces on Body B is overall smaller than those on Body A, especially in the higher wave frequency region of kh42.0, where the FAy approaches almost to zero. The conventional potential flow seems to over-predict again the vertical wave forces on Body B near the characteristic response frequency fr. Moreover, the response frequency fr trends to decrease with the increase of gap width, which is consistent with the results of the vertical wave forces on Body A. It was demonstrated again by Fig. 7 that the damping coefficients of m ¼0.3–0.5 can guarantee the potential flow model to work as well as the viscous fluid model.

respectively. Then the numerical results of vertical wave forces on the two bodies are presented in Figs. 10 and 11, respectively. We consider here three different drafts, i.e., D¼ 0.105, 0.152 and 0.252 m for the twin bodies with the fixed h¼0.5 m, B¼0.5 m and Bg ¼0.05 m. As for the last case with D¼0.252 m, the corresponding numerical results have been discussed before but the figures are included again for the conveniences of comparisons. It can be seen from Fig. 8(a)–(c) that the horizontal wave forces on Body A are rather limited at the lower frequencies. They increase gradually to their maximums as the incident wave frequency kh approaches to a particular wave frequency fr, and then the FAx trends to decrease. In general, the horizontal wave forces on Body A at higher frequencies are greater than those observed at the lower frequencies. The numerical results from viscous fluid model indicate that the peak values of horizontal wave forces on Body A trend to increase significantly with the increase of body draft. In addition, it is demonstrated obviously by Fig. 8 that the lager draft leads to the smaller characteristic response frequency fr. Similar to the previous findings, the conventional potential flow model produces very large horizontal wave forces on Body A. However, it can be reduced to the satisfying magnitudes if an appropriate artificial damping coefficient m ¼0.5 is adopted. The horizontal wave forces on Body B, as shown in Fig. 9, are also observed to increase gradually with the incident wave frequency at the lower frequencies and attain their maximums at the particular response frequency fr. After that, the FAx on Body B decreases with the incident wave frequency and finally approaches to zero at the extreme high wave frequencies. As far as the same body draft and geometric dimension are concerned here, it can be

4.2. Influence of body draft As before, we first investigate here the variation of horizontal wave force with the incident wave frequency, concerning with the influence of draft, as shown in Figs. 8 and 9 for Body A and Body B,

L. Lu et al. / Ocean Engineering 38 (2011) 1403–1416

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3 Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

2 1

1

1.5

2.0

2.5 kh

3.0

3.5

0

4.0

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

2.0

2.5 kh

3.0

3.5

4.0

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

2 FxA

FxA

2

1

1

1.5

2.0

2.5 kh

3.0

3.5

0

4.0

1.5

2.0

2.5 kh

3.0

3.5

4.0

3

3 Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

2 FxA

FxA

2

1

1 0

1.5

3

3

0

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

FxA

FxA

2

0

1409

0

1.5

2.0

2.5 kh

3.0

3.5

4.0

1.5

2.0

2.5 kh

3.0

3.5

4.0

Fig. 8. Influence of draft on the horizontal wave forces on Body A of twin bodies at h ¼ 0.5 m, B¼ 0.5 m, Bg ¼0.05 m and H0 ¼ 0.024 m: (a) horizontal wave force on Body A at D ¼ 0.103 m, (b) horizontal wave force on Body A at D ¼0.153 m and (c) horizontal wave force on Body A at D ¼0.252 m.

Fig. 9. Influence of draft on the horizontal wave forces on Body B of twin bodies at h ¼0.5 m, B¼0.5 m, Bg ¼0.05 m and H0 ¼0.024 m: (a) horizontal wave force on Body B at D ¼ 0.103 m, (b) horizontal wave force on Body B at D ¼ 0.153 m and (c) horizontal wave force on Body B at D ¼ 0.252 m.

found that the maximum of horizontal wave forces on Body B are smaller than that observed for Body A. However, unlike the previously described horizontal wave forces on Body A, the FAx on Body B are generally with larger values at lower frequencies while the rather limited amplitudes appear at the higher frequencies. In addition, with the increasing of body draft, the characteristic response frequencies fr shift to the lower frequencies. This is similar to what was found in Fig. 8. It is observed again in Fig. 9 that the promising results of horizontal wave forces on Body B from potential flow model are obtained by using m ¼0.5. Focusing on the vertical wave force on Body A, referring to Fig. 10, the previously mentioned three-phase variation of FAy with kh from the viscous fluid model appears again. The amplitudes of vertical wave forces are observed to decrease in general with the increase of incident wave frequency and drops rapidly at the medium frequencies. It seems that the increase of body draft has limited influence on the magnitude of the vertical wave force on Body A, which is different from the horizontal wave forces. But the characteristic frequencies fr shares the similar decreasing tendency with the increase of draft as the horizontal wave forces. It can be seen that the conventional potential flow model produces satisfying results of the vertical wave forces on Body A except at the frequencies around fr. However, the accuracy of the maximal vertical wave forces predicted by the potential flow model are improved substantially if the damping coefficient m ¼0.3–0.5 are used. The variation of vertical wave forces on Body B with incident wave frequency, as depicted in Fig. 11, seems the transition from the pattern of three-phase to that of the regular single peak mode.

From Fig. 11, the magnitude of vertical wave forces on Body B is observed to be generally smaller than that on Body A, especially at the higher wave frequencies where the FAy on Body B are nearly zero. Moreover, the damping coefficients m ¼0.3–0.5 used for the potential flow model are confirmed again to produce satisfying results comparing with the viscous fluid model. 4.3. Influence of body breadth The influence of breadth ratio between Body A and Body B is examined here. We examined three different cases, in which the Body B located downstream takes three different breadths, i.e., BB ¼0.2, 0.3 and 0.5 m, while the breadth of Body A upstream is set to be a constant of BA ¼ 0.5 m. The corresponding breadth ratios BB/BA are of 0.4, 0.6 and 1.0. The numerical results associated with the horizontal and vertical wave forces on Body A and Body B are illustrated in Figs. 12–15. According to the numerical results of viscous fluid model, Fig. 12(a)–(c) show that the variations of horizontal wave force on Body A with the incident wave frequency attain their maximums at the particular response frequencies fr as usual. Generally speaking, at the lower wave frequencies, the amplitudes of horizontal wave forces on Body A are smaller than those at the higher frequencies. The comparisons under different breadths of Body B suggest that the maximum of horizontal wave force on Body A trends to increase slightly with the increase of BB. In addition, it was found that the breadth ratio has little influence on the response frequencies for the appearance of the maximal horizontal wave force on Body A.

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1

0

2

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

1.5

2.0

2.5 kh

FyA

F yA

2

3.0

3.5

1.5

2.0

2.5 kh

3.0

3.5

FyA

FyA

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

1

2.5 kh

3.0

3.5

4.0

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

1.5

2.0

2.5 kh

3.0

3.5

4.0

2 Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

1

FyA

FyA

2.0

1

0

4.0

2

0

1.5

2

2

0

1

0

4.0

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

1

0

1.5

2.0

2.5 kh

3.0

3.5

4.0

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

1.5

2.0

2.5 kh

3.0

3.5

4.0

Fig. 10. Influence of draft on the vertical wave forces on Body A of twin bodies at h ¼0.5 m, B¼ 0.5 m, Bg ¼0.05 m and H0 ¼ 0.024 m: (a) vertical wave force on Body A at D ¼ 0.103 m, (b) vertical wave force on Body A at D ¼0.153 m and (c) vertical wave force on Body A at D ¼ 0.252 m.

Fig. 11. Influence of draft on the vertical wave forces on Body B of twin bodies at h ¼0.5 m, B¼ 0.5 m, Bg ¼0.05 m and H0 ¼ 0.024 m: (a) vertical wave force on Body B at D ¼0.103 m, (b) vertical wave force on Body B at D ¼ 0.153 m and (c) vertical wave force on Body B at D ¼0.252 m.

It can be seen from Fig. 13(a)–(c) that the horizontal wave force on Body B behaves the similar variation for Body A, but with smaller magnitude in general. In contrast to Fig. 12, the relative larger horizontal wave forces on Body B are observed at the lower frequencies, while it shows very small magnitudes at the higher wave frequencies. Figs. 12 and 13 show clearly that the conventional potential flow model over-predicts the maximal horizontal wave forces on both Body A and Body B although it works well in the spans far from the characteristic response frequencies fr despite the different body breadth of BB are considered here. For the present three different breadth ratios, the potential flow model with m ¼0.4–0.5 can produce satisfying horizontal wave forces on both Body A and Body B, throughout the frequency band considered. The numerical results of vertical wave force on Body A at various incident wave frequencies are presented in Fig. 14(a)–(c). It can be seen that the variation of FAy with kh can also be described roughly by the three-phase pattern. However, different from the previously discussed horizontal wave forces, the influence of breadth ratio on the vertical wave forces becomes more significantly, leading to the gradual increase of the maximal FAy at fr with the increase in BB. Moreover, the smaller breadth ratio BB/BA (with constant BA) seems to result in the more obviously decreasing of FAy with kh in the lower frequency band. It was found that the dimensionless vertical wave forces on Body A are generally smaller than 1.0 under the three typical breadth ratios. On the other hand, Fig. 14 suggests that the over-prediction of vertical wave forces by potential flow model is not as severe as that of the horizontal wave force shown in Figs. 12 and 13. The numerical results of potential flow model indicate that the

characteristic response frequencies fr for the vertical wave forces are not sensitive to the variation of breadth ratio. Fig. 14 suggests also that the potential flow model may produce reasonable predictions for the vertical wave forces on Body A by adopting the damping coefficients ranging from 0.3 to 0.5. As far as the vertical wave force on Body B is concerned, as illustrated in Fig. 15, the overall characteristic of the variation of FAy with kh behaves as similar as that on Body A. The numerical results from both viscous fluid model and potential flow model confirm that the vertical wave force on Body B is rather limited, especially in the higher frequency range of kh41.9. Fig. 15 indicates that the wider breadth of Body B leads to the larger vertical wave forces as kho1.9. Again, the damping coefficients used as m ¼0.4–0.5 may give rise to satisfying results of the vertical wave forces from the potential flow model. 4.4. Influence of body number In addition to the previously described twin bodies with single narrow gap, this work examines further the situation involving three identical rectangular boxes with double narrow gaps. We intend to demonstrate the influence of the additional third body (or the second narrow gap) on the response of wave forces. As before, the horizontal wave forces are discussed first and then followed by the vertical wave forces. Note that the three identical boxes are of 0.5 m in breadth and 0.252 m in draft with two narrow gaps in the same widths of 0.05 m. Such an assembly can be thought of as the counterpart of the case of twin body in Figs. 4(b) and 5(b), but with an additional intrusion of Body C at the rear of Body B.

L. Lu et al. / Ocean Engineering 38 (2011) 1403–1416

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3 Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

2 FxA

FxA

2

1

1 0 1.0

1411

1.5

2.0

2.5

3.0

0 1.0

3.5

1.5

2.0

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

3.5

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

2 FxA

FxA

2

1

1

1.5

2.0

2.5

3.0

0 1.0

3.5

1.5

2.0

2.5

3.0

3.5

kh

kh 3

3 Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

2 FxA

Fx A

2

1

1 0 1.0

3.0

3

3

0 1.0

2.5 kh

kh

1.5

2.0

2.5

3.0

3.5

kh

0 1.0

1.5

2.0

2.5

3.0

3.5

kh

Fig. 12. Influence of breadth ratio on the horizontal wave forces on Body A at h ¼ 0.5 m, BA ¼0.5 m, D ¼ 0.252 m and H0 ¼ 0.024 m: (a) horizontal wave force on Body A at BB ¼ 0.2 m (BB/BA ¼ 0.4), (b) horizontal wave force on Body A at BB ¼ 0.3 m (BB/BA ¼0.6) and (c) horizontal wave force on Body A at BB ¼0.5 m (BB/BA ¼ 1.0).

Fig. 13. Influence of breadth ratio on the horizontal wave forces on Body B at h ¼0.5 m, BA ¼0.5 m, D ¼ 0.252 m and H0 ¼ 0.024 m: (a) horizontal wave force on Body B at BB ¼ 0.2 m (BB/BA ¼0.4), (b) horizontal wave force on Body B at BB ¼0.3 m (BB/BA ¼ 0.6) and (c) horizontal wave force on Body B at BB ¼ 0.5 m (BB/BA ¼1.0).

Looking at Fig. 16(a), it can be found that the variation of FAx with kh appears double peak values for Body A. This is distinct to the previously investigated two boxes with single narrow gap. This is mainly induced by the presence of the second narrow gap in the floating system. The double-peak phenomenon was also observed for the problem of fluid oscillation in the narrow gaps, from both numerical simulations and experimental investigations. It can be seen that the first peak value of the horizontal wave force on Body A is larger than the second one. In addition, the comparisons between Figs. 16(a) and 4(b) shown that the first response frequency is smaller than that observed in the case of twin bodies, while the second response frequency is found to be larger than that of twin boxes. The more obvious double peak variation of FAx with kh can be observed from the conventional potential flow model with m ¼0.0, where the amplitude of horizontal wave forces is seriously over-predicted. Fig. 16(a) shows that the numerical results of potential flow model can be improved significantly by introducing an appropriate damping term with m ¼ 0.4, although the small discrepancy with respect to the viscous fluid model is observed in the higher frequencies. The double peak variation of FAx with kh appears also on the Body C, as shown in Fig. 16(c). According to the numerical results of viscous fluid model, the second response frequency associated with Body C is not as significant as that observed on Body A. However, it can be distinguished by the conventional potential flow model although the corresponding peak value of FAx at the second frequency is unphysical. As far as the middle Body B is concerned, as shown in Fig. 16(b), there exists only one peak value for the variation of horizontal wave forces, which is similar to what was observed for the cases of twin bodies. It is noted that

the characteristic response frequency in Fig. 16(b) is slightly greater than that of twin body, as shown in Fig. 4(b). The comparisons of the horizontal wave forces on the three bodies indicate that the Body A is subjected to the large FAx at the higher frequencies, while the Body A seems to experience a relative small horizontal wave force as kho1.0. Generally speaking, the maximal dimensionless horizontal wave forces on the three bodies are totally smaller than 2.0. The conventional potential flow model over-predicts again the wave forces as for the twin bodies. Fig. 16 shows that the using of damping coefficient m ¼0.4 can result in satisfying results of the horizontal wave forces. The variation of vertical wave force FAy with the incident wave frequency kh are shown in Fig. 17. It can be observed from Fig. 17(a) that the vertical wave force on Body A remains a high level at the lower frequencies. And then it decreases rapidly within a narrow band of frequency. Two distinct response frequencies fr1 and fr2 can be identified. This is more obvious according to the numerical results from the conventional potential flow model. At the higher frequencies, the vertical wave force on Body A is found to be in rather limited amplitude. The similar variation of FAy with kh associated with Body C can be observed in Fig. 17(c). However, the magnitude of vertical wave forces on Body C is much smaller that on Body A. Similar to Fig. 16(b), the vertical wave force on Body B behaves only one peak value. In general, the over-prediction of vertical wave forces on the three floating bodies is also produced by the conventional potential flow model. According to the numerical comparisons, it seems that m ¼0.4 used for the potential flow model may give rise to fairly good results of the vertical wave forces on the three floating bodies, which is consistent well with the tuned damping coefficient for the horizontal wave forces.

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2 Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

1

0 1.0

1.5

2.0 kh

2.5

FyA

FyA

2

1.5

2.0 kh

2.5

FyA

FyA

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

1

2.5

3.0

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

1.5

2.0 kh

2.5

3.0

2 Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

1

1.5

2.0 kh

2.5

FyA

FyA

2.0 kh

1

0 1.0

3.0

2

0 1.0

1.5

2

2

0 1.0

1

0 1.0

3.0

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

1

0 1.0

3.0

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

1.5

2.0 kh

2.5

3.0

Fig. 14. Influence of breadth ratio on the vertical wave forces on Body A at h ¼0.5 m, BA ¼ 0.5 m, D ¼ 0.252 m and H0 ¼ 0.024 m: (a) vertical wave force on Body A at BB ¼0.2 m (BB/BA ¼0.4), (b) vertical wave force on Body A at BB ¼0.3 m (BB/BA ¼0.6) and (c) vertical wave force on Body A at BB ¼0.5 m (BB/BA ¼ 1.0).

Fig. 15. Influence of breadth ratio on the vertical wave forces on Body B at h ¼0.5 m, BA ¼ 0.5 m, D ¼0.252 m and H0 ¼ 0.024 m: (a) vertical wave force on Body B at BB ¼ 0.2 m (BB/BA ¼ 0.4), (b) vertical wave force on Body B at BB ¼ 0.3 m (BB/BA ¼ 0.6) and (c) vertical wave force on Body B at BB ¼ 0.5 m (BB/BA ¼1.0).

4.5. Response frequency of wave force and resonant frequency of fluid oscillation

mode of fluid oscillation inside the moonpool, denoted by o0, is derived as

In the previous sections, we presented the numerical results of the variation of wave forces with the incident wave frequency, where the magnitude of wave forces on the multiple floating bodies and the corresponding response frequencies are discussed. Considering the available numerical results associated with the fluid oscillation in the narrow gaps in our previous work (Lu et al., 2011), therefore it is worthy to exploring the correlation between the resonant frequency of fluid oscillation in narrow gaps and the response frequency of wave forces on the floating bodies. Tables 1–4 summarized the comparisons of the characteristic frequencies (kh) of fluid resonance and wave forces at varied gap widths, body drafts, body breadths and body numbers. In these tables, the rows denoted by fR represent the resonant frequency of fluid oscillation while the fr is used to denote the response frequency of wave forces. It should be noted that the characteristic frequencies (fR and fr) in terms of kh listed in Tables 1–4 are entirely determined by the potential flow model. This is because that the potential flow model requires small computational efforts and hence allows us to use very fine frequency increment of Dkh ¼ 0:025 in the numerical calculations. This enables us to identify the frequency response as accurately as possible. In addition, comparisons between the viscous fluid model and potential flow model have confirmed that they can produce the identical predictions on the characteristic response frequency. The numerical results of this work associated with the response frequencies are also compared with the analytical solutions of Molin (2001), where the natural frequency of the piston

o0 ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g D þ ðBg =pÞ½1:5 þlnðHB =2Bg Þ

ð15Þ

where HB ¼1.5B is used according to the suggestion of Molin (2001). The natural frequency o0 is further related to the wave number k according to the dispersion formula and is normalized by water depth h. This allows the response frequencies to be compared by using the unified non-dimensional parameter kh. It can be seen from Table 1, where the influence of gap width of twin bodies is concerned, that the response frequencies of horizontal and vertical wave forces on Body A and Body B are nearly identical under the same conditions according to the numerical results of this work, at most with a frequency discrepancy of 0.025 for the special case of vertical wave forces on Body A. In addition, the response frequencies of wave forces collide generally well with the resonant wave frequency in the narrow gap. It is found that the response frequencies predicted by the formula of Molin (2001) are slightly greater than the numerical results of this work. This is because that the Eq. (15) is restricted to the fluid bulk oscillation in the moonpool, corresponding to the narrow gap in this work. The fluid motion under the structures is not considered well, leading to the less fluid mass involving the fluid oscillation and consequently the increase of resonant frequency. As mentioned before, the data shown in Table 1 demonstrate again that the characteristic frequency associated with the fluid resonance trend to decrease with the increase of gap width.

L. Lu et al. / Ocean Engineering 38 (2011) 1403–1416

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3

FxA

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

2 FyA

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

2

1

1 0 0.5

1413

1.0

1.5

2.0

2.5

0 0.5

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1.0

1.5

3.0

FxA

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

2 FyA

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

2

1

1

1.0

1.5

2.0

2.5

0 0.5

3.0

1.0

1.5

2.0

2.5

3.0

kh

kh 3

3

FxA

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

2 FyA

Present Viscous Model Potential Model, µ=0.0 Potential Model, µ=0.3 Potential Model, µ=0.4 Potential Model, µ=0.5

2

1

1 0 0.5

2.5

3

3

0 0.5

2.0 kh

kh

1.0

1.5

2.0

2.5

0 0.5

3.0

1.0

1.5

Fig. 16. Horizontal wave forces on three identical bodies at h ¼0.5 m, Bg ¼ 0.05 m, B¼ 0.5 m, D ¼0.252 m and H0 ¼ 0.024 m: (a) horizontal wave force on Body A, (b) horizontal wave force on Body B and (c) horizontal wave force on Body C.

As shown in Table 2, it was found that the response frequencies of wave force for different drafts agree generally well with the fluid resonant frequencies in the narrow gap. It was noted also that the characteristic frequencies fF and fr decrease dramatically with the increase of body draft, which is much more significant than what was observed in Table 1. Again, the larger response frequencies of Molin (2001) are found in Table 2, but one may observe the same decrease in the response frequency as the draft increases. The influence of body breadth on the frequency response is summarized in Table 3. The numerical results of this work suggest that the characteristic frequencies of fluid resonance and wave forces decrease slightly as the body breadth ratio of BB/BA increases from 0.4 to 1.0. Good agreement between fF and fr is observed but the response frequency of FAy on Body A at the three different breadths of Body B are slightly smaller than the other cases. Due to the present unequal breadths of Body A and Body B, the formula of Molin (2001) cannot be applied and no comparison is carried out in Table 3. As discussed in Section 4.4, there are generally two distinct response frequencies of wave forces under the situation of three identical bodies with double narrow gaps. We qualified the corresponding numerical results in Table 4 and compared with the cases of twin body with single narrow gap. It can be seen from this table that the first resonant frequencies in the two narrow gaps of three bodies are smaller than that the resonant frequency of twin-body system, while the second frequency of three identical bodies is greater than the resonant frequency of twin bodies. It is interesting to find that the response frequency of twin body is almost of the mean values of the first and second response frequencies of three bodies. The frequency response of wave

2.0

2.5

3.0

kh

kh

Fig. 17. Vertical wave forces on three identical bodies at h ¼0.5 m, Bg ¼ 0.05 m, B¼0.5 m, D ¼0.252 m and H0 ¼ 0.024 m: (a) vertical wave force on Body A, (b) vertical wave force on Body B and (c) vertical wave force on Body C.

Table 1 Comparisons of the characteristic frequencies at different gap widths for twin bodies with D ¼0.252 m, B¼0.5 m, h ¼ 0.5 m and H0 ¼0.024 m. Gap width (Bg)

0.03 m

0.05 m

0.07 m

fR in narrow gap fr of FAx on Body A fr of FAx on Body B fr of FAy on Body A fr of FAy on Body B fR of Molin (2001)

1.700 1.700 1.700 1.700 1.700 1.815

1.575 1.575 1.575 1.550 1.575 1.729

1.475 1.475 1.475 1.450 1.475 1.664

Table 2 Comparisons of the characteristic frequencies at different draft for twin bodies with Bg ¼0.05 m, B¼ 0.5 m, h ¼0.5 m and H0 ¼ 0.024 m. Draft (D)

0.103 m

0.153 m

0.252 m

fR in narrow gap fr of FAx on Body A fr of FAx on Body B fr of FAy on Body A fr of FAy on Body B fR of Molin (2001)

2.750 2.750 2.750 2.725 2.750 3.157

2.175 2.200 2.175 2.175 2.175 2.430

1.575 1.575 1.575 1.550 1.575 1.729

forces on the leading Body A and the rear Body C are generally consistent with the fluid resonant frequency in narrow gaps. However, the response frequency of wave forces on the middle Body B appears only one value. It was found that the fr of FAx on Body B happens at the second resonant frequency, while the response frequency of vertical wave forces on Body B corresponds to the first resonant frequency of fluid oscillation in the narrow gaps.

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Breadth ratio (BB/BA)

0.4

0.6

1.0

fR in narrow gap fr of FAx on Body A fr of FAx on Body B fr of FAy on Body A fr of FAy on Body B

1.650 1.650 1.650 1.625 1.650

1.625 1.625 1.625 1.600 1.600

1.575 1.575 1.575 1.550 1.575

Fx / ρghA

Table 3 Comparisons of the characteristic frequencies at different breadth ratio for twin bodies with Bg ¼ 0.05 m, D¼ 0.252 m, h ¼ 0.5 m, and H0 ¼0.024 m.

3

F

2

F

1 0 -1 -2 -3 40

41

42

43

44

45

43

44

45

T Table 4 Comparisons of the characteristic frequencies involving different body number with Bg ¼0.05 m, D ¼ 0.252 m, B¼ 0.5 m, h ¼0.5 m and H0 ¼ 0.024 m.

fR in Gap 1 fR in Gap 2 fr of FAx on Body fr of FAy on Body fr of FAx on Body fr of FAy on Body fr of FxA on Body fr of FAy on Body

A A C C B B

Three identical bodies 1st freq.

2nd freq.

1.375 1.400 1.400 1.350 1.375 1.375

1.675 1.675 1.675 1.650 1.675 1.675 1.675 1.375

Body A h

y

Twin bodies

0 -1 -3 40

1.575 1.550 – – 1.575 1.575

41

42 T

Fig. 19. Comparisons of total horizontal wave forces on individual floating body and horizontal fluid forces due to water elevation difference: (a) for Body A upstream and (b) for Body B downstream.

Body B

h

Fig. 18. Illustration of horizontal fluid forces due to water level difference.

4.6. Mechanism of wave forces The mechanism of wave forces is examined based on the numerical results of viscous fluid model. Considering an individual body, for example Body A, as shown in Fig. 18, the timedependent wave elevation at the left side of Body A is denoted by hL(t) in this work, which may lead to the positive horizontal fluid force, i.e., Z hL FxL ¼ rgðhL yÞdy ¼ 0:5rgh2L ð16Þ 0

Analogously, the negative horizontal fluid force due to the water level hR(t) along the right side of Body A can be evaluated as ð17Þ

The horizontal fluid force induced by the water elevation difference between the opposite sides of Body A is, FxED ¼ FxR þ FxL ¼ 0:5rgðh2L h2R Þ

1

-2

1.575

x

FxR ¼ 0:5rgh2R

2 Fx / ρghA

Number of bodies

3

ð18Þ

Fig. 19(a) and (b) demonstrate the stable time series of the total horizontal wave forces FFB x (from 40T to 45T) on the individual floating bodies of Body A and Body B, respectively, for the typical example of D¼0.252 m, B¼0.5 m, Bg ¼0.07 m, h¼0.5 m, H0 ¼0.024 m and kh¼ 1.469, which is around the resonant frequency of fluid oscillation in the narrow gap. The time dependent FED x on the two bodies are also ED included in this figure. Note that, both FFB x and Fx are normalized by rghA. It can be seen clearly for Fig. 19 that the total horizontal wave ED force FFB x is dominated by the static pressure force Fx . They vary with the same phase. That means the horizontal wave force

depends greatly on the water elevation difference between the opposite sides of the individual floating body. Since the conventional potential flow model produces very larger resonant wave height in the narrow gap, it accordingly results in much overpredication of horizontal wave force on the individual body. On the other hand when the incident wave frequency is out of the resonant band the conventional potential flow model can work as well as the viscous fluid model in predicting the wave height in the narrow gap. Therefore, the wave forces on the floating bodies can also be predicted accurately. For the potential flow model with appropriate artificial damping term, it may even work well under the resonant conditions. This is the reason why the potential flow model with artificial damping term is able to predict the wave forces in good accordance with the viscous fluid model. From the above analysis, it can be seen that the numerical model has to predict accurately the wave height in the narrow gap if one expects that it can give satisfying results of the wave forces on the floating bodies with narrow gaps. Fig. 19 shows also two interesting phenomenon: (1) the ED amplitude of FFB x is smaller than that of Fx . In the frame of linear water wave theory, the total wave force is composed of the zero and first order components in which the FED x can be interpreted as the zero order pressure contribution. This implies that the dynamic pressure has an anti-phase variation with respect to the total horizontal wave force which cancels the over amount of static pressure force; (2) the mean value of FED on Body A is x slightly smaller than zero, while it is slightly greater than zero for Body B. This is induced by the fact that the fluid oscillation in the narrow gap is not exactly symmetric with respect to the initial still water level, but with a small rising due to the non-linear effect (Lu et al., 2010a,b). The existence of the small increase of mean water level leads to a net static pressure force trending to push the bodies away. The time series of non-dimensional overall horizontal wave force on the floating system is further examined in Fig. 20. The horizontal wave force of Body A and that of Body B are also included in this figure for the purpose of comparison. Three typical incident wave frequencies kh¼1.187, 1.469 and 1.893 are considered in Fig. 20(a)–(c), respectively, but with the same D ¼0.252 m, B ¼0.5 m, Bg ¼ 0.07 m, h¼0.5 m, H0 ¼0.024 m. It can

L. Lu et al. / Ocean Engineering 38 (2011) 1403–1416

Fx / ρghA

2 1

F on Body A F on Body B F on Floating System

0 -1 -2 40

42

44

46

48

50

46

48

50

46

48

50

t

Fx / ρghA

2 1 0 -1 -2 40

42

44 t

Fx / ρghA

2 1 0 -1 -2 40

42

44 t

Fig. 20. Time series of horizontal wave forces on Body A, Body B and floating system at typical incident wave frequencies for D ¼0.252 m, B¼ 0.5 m, Bg ¼ 0.07 m, h ¼ 0.5 m and H0 ¼0.024 m: (a) kh ¼ 1.187, T¼1.430 s, (b) kh ¼1.469, T¼ 1.234 s and (c) kh ¼ 1.893, T ¼1.055 s.

be observed from Fig. 20 that the horizontal wave forces on the individual Body A and Body B generally vary with different phases. The phase lag between them leads to the smaller amplitude of the overall horizontal wave force on the floating system compared with the summation of horizontal wave forces on Body A and Body B. At the lower frequencies, for example Fig. 20(a), the overall horizontal wave forces are dominated by the horizontal wave forces on Body B, while it is governed by the horizontal wave force on Body A at the higher frequencies, as shown in Fig. 20(c). When the fluid resonance in narrow gap takes place, it can be found in Fig. 20(b) that the horizontal wave forces on Body A and Body B have almost the same magnitude, but still with significant phase-lag. It indicates that, under fluid resonance, the horizontal wave forces on Body A and Body B have the same order influence on the overall horizontal wave force on the floating system.

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observed, while there is only one response frequency for the middle Body. The numerical results of viscous numerical fluid model suggest that the dimensionless amplitude of horizontal wave forces on the floating bodies are generally smaller than 2.0 under the various body geometric dimensions and spatial arrangements considered in this work, while the vertical wave forces are smaller than the horizontal wave forces. The conventional potential flow model is found to over-predict the actual wave forces on the floating bodies as the incident wave frequencies are close to the resonant frequency. But it can estimate the wave forces as well as the viscous fluid model at the frequencies far from the resonant frequency. Moreover, the potential flow model is capable of predicting the response frequency of wave forces on each floating body. Based on the numerical results of the conventional potential flow model in frequency domain, it was confirmed that the response frequencies of wave forces on the floating bodies and the resonant frequencies of fluid oscillation in the narrow gaps are generally in good agreements. The numerical investigations of this work indicate that the accuracy of potential flow model in predicting wave forces can be improved promisingly by introducing appropriate damping force. It was suggested that the damping coefficient mA[0.4, 0.5] may guarantee the potential flow model to work even as well as the viscous fluid model. The significances are that: (1) the calibrated damping coefficient is observed not sensitive to the variation of gap width, body draft, breadth ratio and body number; (2) the tuned damping coefficient mA[0.4, 0.5] for the wave forces in the present studies are consistent well with what were obtained in our previous work (Lu et al., 2011) concerning with the fluid resonance in narrow gaps. The mechanism of wave force on the floating body is also investigated in this work. It was found that the horizontal wave force is greatly dependent on the water level difference between the opposite sides of an individual body, which is the origin of the over-predicated amplitude of wave forces from conventional potential flow model. It was also identified that the overall horizontal wave force on the floating system is generally smaller than the summation of horizontal wave forces on the individual body due to the phase-lag between them.

Acknowledgment Grateful acknowledges are given to the financial supports from the Natural Science Foundation of China with Grant nos. 50909016, 50921001 and 10802014. This work was also partially supported by the Open Fund from the State Key Laboratory of Structural Analysis for Industrial Equipment with Grant no. GZ0909.

5. Concluding remarks References The wave forces on multi-floating bodies with narrow gaps are investigated in this work employing both viscous fluid model and potential flow model with and without artificial damping term. The numerical results of the variation of horizontal and vertical wave forces on the individual floating body with the incident wave frequency are presented considering various gap widths, body drafts, body breadth ratios and body numbers. The numerical examinations of this work show that the characteristic response frequencies of wave forces trend to decrease with the increase of gap width, body breadth ratio and body draft. When the third identical body intrudes, two distinct response frequencies of wave forces on the leading and rear bodies are

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