Numerical and Experimental Investigation of Hydrodynamic Interactions of Two VLFS Modules Deployed in Tandem

This paper numerically and experimentally investigates the hydrodynamic interaction between two semi-submersible type VLFS modules in the frequency domain. Model tests were conducted to investigate the relationship between interactions and wave headings. Numerical studies were performed by solving the radiation-diffraction problem and were validated against the experimental results. Motion Response Amplitude Operators (RAOs) were obtained from numerical and experimental studies. The dependency of the hydrodynamic interaction effect on wave headings is clarified. The influence of different wave periods on the motion responses of two-module VLFS and wave elevations in the gap is studied. The results indicate that the hydrodynamic interactions of the two modules are directly related to the wave headings and the periods of the incident wave. The shielding effect plays an important role in short wave, and the influence decreases with the increase of the incident wavelength. The numerical results based on the diffraction-radiation code can give a relatively good estimation to the responses in short wave while for long wave, it would over-predict the response.


Introduction
Land pressure of some coastal cities motivates the authorities to extend their urban space into the sea. Compared with the land reclamation, very large floating structure (VLFS) is an alternative solution which is environmentalfriendly, mobile, and sustainable (Wang and Wang, 2015). The design concepts of VLFS can be divided into two categories: pontoon type and semi-submersible type (Lamas-Pardo et al., 2015). For pontoon type VLFS, the estimation of the hydroelastic response is an essential concern for the researchers and engineers; while for multi-modular semisubmersible type VLFS, the entire system can be viewed as several rigid modules connected by flexible connectors and the hydrodynamic interaction between modules needs to be taken care of when adjacent two modules are in proximity.
It can be easily found that similar narrow-deployed scenes exist in the field of ocean engineering, such as LNG FPSO and LNG carrier offloading operation. The hydrodynamic interaction between bodies in wave has been one of the research focuses in the field of ocean engineering for decades. Therefore, lots of works have been carried out on the problems of hydrodynamic interaction. Ohkusu (1976) used the two-dimensional (2-D) strip theory to investigate the interactions between two side-by-side vessels. Van Oortmerssen (1979) used three-dimensional linear diffraction theory to extend the cognition of the problem. Maniar (1995) discussed the trapped-wave mode in an array of cylinders, and the relationship between the trapped wave in the channel and the large diffraction load was presented. Chakrabarti (2000) developed a numerical solution coupling linear radiation-diffraction method with scattering analytical method to predict the response of multiple floating cylindrical structures in waves. The results showed that the interaction is evident between two semi-submersible modules. Riggs et al. (2000) compared the responses of a 5module Mobile Offshore Base (MOB) between rigid modules flexible connector model (RMFC) and the finite element model. They concluded that the RMFC model could predict the motion responses well, and this assumption was suitable for the preliminary stage. Huijsmans et al. (2001) found that numerical errors would occur in a very close situation. Unrealistically high wave elevations were observed in the gap between two floating bodies because of the neglect of the viscous terms and energy dissipation. Some attempts were made to reduce the extremely considerable wave elevation, such as rigid lid method (Huijsmans et al., 2001), damping lid method (Chen, 2005;Jean-Robert et al., 2006) and so on. Hong et al. (2005) used a high order boundary element method to compute the interaction of side-by-side moored vessels and compared with experimental data. The above-mentioned works were all based on some forms or some reasonable improvements of the frequency-domain free-surface Green function drawn from linear potential theory. In addition, some researchers adopted the time-domain method based on the Cummins formulation (Cummins, 1962) to investigate the two-body problem. Through this method, Bunnik et al. (2009) computed the motion response of a side-by-side deployed LNG offloading system in the time domain. Zhao et al. (2014) also investigated the side-by-side LNG offloading system using the time-domain simulation code SIMO (MARINTEK, 2009). Xu et al. (2016) investigated the hydrodynamic interactions between three barges in a float-over installation. Wu et al. (2017a) developed a simplified algorithm for calculating the hydrodynamics in long waves by treating all the parts of the MOB as small components. The shielding effects are also considered to evaluate the wave diffraction loads. Wu et al. (2018) took the effect of the hydrodynamic interaction, variable bottom, and connector system into consideration. The hydroelastic responses of a three-module VLFS over varying bottom were investigated and discussed.
Different from the aforementioned time-domain method, which is called indirect time-domain method depending on the frequency-domain Green function calculation, the direct time-domain method is another solution for the multi-body problem. Kim et al. (2009) developed a 3D time-domain Rankine panel method to solve the two-body problem. Two cases were studied and had a good agreement with the results in the literature. Watai et al. (2015) adopted 3D rankine panel method with application to the side-by-side two-body problem. Wave resonance in the gap between the two ships was observed. The numerical results were improved and had a better agreement with experimental results using dampinglid methods. Computational Fluid Dynamic (CFD) method is also one of the direct time-domain simulation methods, which can also be used to investigate the issue of 3D multiple floating structures considering the viscous effect. Lu et al. (2010) studied the wave resonance in the gap using both potential flow method and CFD method. With the viscous CFD results, the damping-lid coefficient used in potential flow method can be correctly defined, and the influence of the damping coefficient adopted in the gap was studied. Although CFD has an advantage of considering the viscous influence, using such a method may not be a practical solution to the multi-body problem because of its expensive computational cost.
At the preliminary design stage, a reasonable estimation of the motion responses of two modules with limited resources is essential for the safety of the connector system. Thus, following the previous work (Wang et al., 2018), this paper focuses on the hydrodynamic interaction between two semi-submersible modules in the tandem configuration in different wave headings. A low-order panel model was adopted to perform frequency-domain analyses of two modules and validated against the model tests. A series of experiments were conducted to clarify the relationship of the motion responses of two-module VLFS with different wave headings. Some observations and discussions were presented based on the numerical and experimental results.

Description of the semi-submersible type module
The semi-submersible type VLFS discussed in the present paper consists of several modules connected by the connector system. Different conceptual designs were carried out by the cooperation institution: China Ship Science and Research Center (CSSRC) (Wu et al., 2017b) and the geometry presented in this paper was proved to have a good hydrodynamic performance. Yoshida et al. (2001) also concluded that the semi-submersible type Mega-Float (SSMF) (Suzuki, 2005) had a better performance than the pontoon type Mega-Float because of the small waterplane. In this paper, the major particulars of the module are listed in Table 1. The coordinate systems and notations of the two-adjacentmodule VLFS in tandem operation are shown in Fig. 1, where M i (i = 1, 2) denotes the module i. Two reference systems, as can be seen in Fig. 2, are established: one is the earth-fixed reference system (o e -x e y e z e ), and the other is the body- . It should be noted that all the data presented in this paper are in prototype value if there are no specific annotations.

Mathematical formulation
3.1 Diffraction-radiation problem Φ Based on some basic assumptions: fluid is ideal, irrotational and incompressible, the fluid velocity can be described by the velocity potential , and the governing equation is simplified as Laplace equation (Newman, 1977): (1) The time-dependent part of the velocity potential is separated out [ (x, y, z, t) = ] due to the harmonic wave motion. Here, refers to the circular frequency of the incident wave. Based on the linear assumption, (x, y, z) can be decomposed into three parts: incident potential (x, y, z), diffraction potential (x, y, z) and radiation potential (x, y, z): ϕ N is the 6×N B degrees of freedom, and N B refers to the number of bodies. Therefore, the solution to the whole velocity potential (x, y, z) is separated into the solutions to different parts of the potential. ϕ I (x, y, z) can be expressed based on the linear periodic wave: ω ω where A is the wave amplitude, g the acceleration of gravity, k the wave number, the wave frequency and h the water depth. The wave number k is decided by the dispersion relation with and h: Diffraction and radiation potential can be obtained by solving the classical boundary-value problems (Lee, 1995). Based on the perturbation for small amplitude waves, the potentials are expended corresponding to the mean position. The boundary conditions, satisfied by the diffraction and radiation potentials, are formulated as follows: where U n represents the normal velocity of the body surface. The velocity potential can be solved by free-surface Green function (Lee and Newman, 2005). Once the potential is obtained, the hydrodynamic pressure, added mass and damping coefficients can be derived. The first-order wave excitation forces f ω composed of so-called Froude-Krylov and diffraction forces and moments are expressed as follows: For detailed explanations, formulations and solutions, there are lots of relevant researches and works available in the literature (Faltinsen, 1993).

Motion equation
The motion equations for the multi-body system are similar to the single body motion equation, and expressed as follows: ηηη where M and M a are respectively the mass matrix and added mass matrix. C is the damping matrix and K the restoring force matrix. F represents the wave excitation force matrix. , and are respectively the displacements, velocity and acceleration vectors of the multi-body system. Viscous damping is ignored in the diffraction-radiation problem, which has a significant influence on the resonance. Based on the results of free decay tests, the viscous damping term can be linearized and included in the linear theory, which can be expressed as follows:   where is the non-dimensional damping ratio and c crit represents the critical damping. K( ) is the hydrostatic restoring stiffness matrix of the specific motion. In this paper, the low-order boundary element method was used for calculating the hydrodynamic performance of the two modules. The wetted surface of the semi-submersible type module was modeled by several panel elements. The state-of-art diffraction-radiation code Wadam (DNV, 2011) was adopted for frequency-domain calculations of the two-adjacent-module VLFS.

Experimental setup
Model tests, including free decay tests, regular wave tests, and white-noise wave tests, were performed with a scale of 1:100 in Ocean Engineering Basin at Shanghai Jiao Tong University. The scaling ratio is selected with the consideration of the dimensions (50 m×30 m×6 m) of the basin. The water depth in model-scale was set at 50 cm corresponding to the prototype water depth of 50 m. These model tests were designed to clarify the hydrodynamic interaction between two modules in tandem configuration and its relationship against wave headings. The experimental results also provided support for the numerical validation works. A multi-flap wave-maker was used for modeling a large frequency range of wave spectrum. The wave energy was absorbed, and reflected wave was minimized by an array of absorbing beach at the boundary. Meanwhile, the model tests were carried out at the center of the basin in order to avoid the boundary effects. The six degrees-of-freedom (DOFs) motions of two modules were captured by the noncontact optical motion tracking system. The experimental configuration is illustrated in Fig. 2. A snapshot of the model test is shown in Fig. 3.
As can be seen in Fig. 2, a horizontal mooring system was adopted to avoid the modules being drifted away. The stiffness of the mooring system is designed to be small, and therefore, has limited influence on the hydrodynamic performance of the two modules. The effects of the connector system are not the focus of this research. In addition, the connector system will make it hard to explain the interaction phenomenon if the connector system is included. The spacing of two modules is 10 cm in model-scale without specific connector system.
Free decay tests are conducted to ensure that two modules have the same moment of inertia and hydrostatic resilience. It also provided the viscous damping ratio so that the numerical model can be improved. Table 2 listed the results measured in free decay tests. One can see that two modules have the same natural periods and damping ratios, confirming that the two modules are well modeled and physical similarity was satisfied.
Motion Response Amplitude Operators (RAOs) of the two disconnected modules are obtained from regular wave tests and white-noise wave tests in three wave headings: 180° head wave, 225° oblique wave and 270° beam wave. The wave conditions are summarized in Table 3. Wave elevations in time series were captured by three wave probes in the gap between two modules. The arrangements of the wave probes can also be observed in Fig. 2. The wave calibration test at wave probe 2 was conducted before the experiments. Because of the inevitable reflection wave caused by boundary, the best group of wave periods is selected as the calibration reference. The motion RAOs are calculated based on the calibrated wave amplitude. The white-noise wave calibration results are shown in Fig. 4. The results show that the obtained wave data have a good agreement with the target.

Mesh convergence test
Mesh convergence test was carried out first in order to have an accurate cognition of the effect of the different  mesh sizes. The number of panels allowed in the radiationdiffraction code is limited; besides, the computational cost would be expensive due to the excessive meshes. Therefore, appropriate mesh size and sufficient computational precision are important for the numerical analysis. Three different low-order panel meshes were adopted, as shown in Fig. 5. The mesh size decreased from Mesh-1 to Mesh-3. In this paper, all the cases, including single-module and twomodule cases with 41 periods and 7 headings, have been computed. The element number of the meshes and the calculation time are summarized in Table 4. A horizontalmoored single-module white-noise wave test was conducted to validate the accuracy of these calculations.
It can be seen that the panel number has an essential effect on the calculation time. With the increase of the panel number by about 2.5 times, the calculation time has increased more than 10 times. In single-module case, sway, heave and roll motion RAOs of the single module in beam sea are given by both experimental results and numerical results for different meshes (see Fig. 6). It can be observed that a good agreement of the single-module RAOs of different meshes is obtained except for some difference in the range of low frequency. The discrepancy can be caused by two main reasons. First, the experimental errors caused by the restriction of the experimental equipment may contribute to the difference. Because of the constant slope of the absorbing beach, the low-frequency part of the incident, diffracted and radiated wave may not be fully absorbed by the beach. The lateral boundary effect can also reflect some waves, especially for the low-frequency wave. Therefore, the reflecting low-frequency wave introduced by lateral and backside boundaries may also cause some differences. Second, the ignorance of the viscous term in the numerical study can lead to some errors compared with physical model tests. Overall, results indicate that meshes used in frequency-domain calculations are almost independent. Thus, Mesh-2 was chosen for the rest two-module case calculation by a compromise of computational efficiency and accuracy.

Wave elevations in the gap
Wave elevations in the gap were captured by the wave probes deployed as Fig. 2. Figs. 7a and 7b present two selected time trains of the regular wave respectively with a period of 8 s and 25 s in calibration test and corresponding regular wave train during the 270° two-module test. It can be found that for both long and short waves, the wave elevations in the gap are depressed because of the two-module interaction. A non-linear phenomenon in the crest of wave train is observed in Fig. 7a in wave tests. The non-linear wave induced by relative motions of two modules would significantly affect the incident wave of period 8 s, and eventually, change the crest of wave elevation at the position of wave probe 2. Evidently, the linearity of long waves is maintained well as can be seen in Fig. 7b, which means that the non-linear influence is relatively small in long wave. These observations are opposed to those from Jean-Robert et al. (2006) where relative motions between two vessels amplify the wave in the gap. It may be explained as follows: the geometry is different between two studies; the closed waterplane curve of the vessel may contribute to the amplification of the wave; while for semi-submersible floating structures, the wave in the gap is dominated by the incident wave, and the scattering wave induced by columns may reduce the amplitude of the incident wave. The effect of the interaction between two modules on the amplitude of a large-period incident wave is relatively small. In short, the short wave is more easily influenced by the multi-column type floating structure than the long wave.

Motion RAOs of two-module VLFS
RAOs of two-adjacent-module VLFS in different wave headings were obtained using the diffraction-radiation code Wadam. These numerical results were compared with corresponding experimental results (Fig. 8 for head wave, Fig. 9 for oblique wave, Fig. 10 for beam wave). It should be noted that the surge, pitch, and yaw motions are relatively small in beam wave (90°) compared with the sway, heave, and roll motions. Therefore, for beam wave, only sway, heave and roll motions are studied, while for head wave (0°), surge, heave and pitch motions are investigated.
Overall, the numerical simulation predicts the motion responses of the two modules deployed in tandem configuration well, and it gives a good agreement with experimental results by including the experimental errors, especially for short wave with small periods. The experimental data from regular wave tests are consistent with those from whitenoise wave tests. It shows that white-noise wave tests can present a reliable result. Therefore, the subsequent discussions are all based on the data of white-noise wave tests.  WANG Yi-ting et al. China Ocean Eng., 2020, Vol. 34, No. 1, P. 46-55 51 One can see that for head wave, the motion response and hydrodynamic interaction are different for the incident wave of different wave periods. The surge RAO of M-2 module in short wave whose period is smaller than 10 s is more significant than that of the M-1 module. From the configuration of the 2-module VLFS, M-2 module can give a significant shielding effect to M-1. It reduces the amplitude of the incident wave and dissipates the wave energy. Therefore, the surge response of M-1 module is small. This phenomenon can be seen both in numerical and experimental results and also in heave and roll motions. From this point of view, the numerical results obtained from diffraction-radiation code can present a reasonable result and predict the interaction between two modules well for small period wave. However, when it comes to long wave, the difference between numer-ical and experimental data is increased. In the wave range of 10−25 s, the M-2 roll motion is smaller than that of M-1 module for numerical results, while for experimental results, roll motion of M-2 module shows no significant difference with that of M-1. Thus, it can be concluded that from the experimental data, the interactions and shielding effects decreased for long wave conditions. Similar observations and conclusions can also be drawn from the oblique wave results. The phenomenon obtained in the oblique wave is not as clear as that in the head wave. For short wave, the motion responses of the M-1 module are slightly affected by the shielding impact of the M-2 module. The motion RAOs of the M-2 module is a little larger than those of the M-1 module. The errors between numerical and experimental results increase with the increase of wave peri-

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WANG Yi-ting et al. China Ocean Eng., 2020, Vol. 34, No. 1, P. 46-55 δ δ od. In beam sea, the interactions between the two modules are minimal. The RAOs curves of the two modules are very close to each other. The above qualitative observations are verified by quantitative researches. The relative error ( ) is defined by the equation =(RAO M-2 −RAO M-1 )/RAO M-1 and can be calculated based on the experimental data. Table 5 lists the mean relative errors of the wave with different headings before the period of 10s. In the head sea, surge and roll RAO of M-1 is more than one time larger than those of M-2. Heave RAO of M-1 is also 58.61% larger than that of M-2. When it comes to oblique wave, the shielding effect is reduced dramatically. By taking the surge motion as an example, the difference between the two modules only is 18.22%. For beam sea, the error between two modules is within 4 % in all the modes, confirming the qualitative conclusion.
In conclusion, the hydrodynamic interactions between two modules and the shielding effects are strongly related to the wave headings and wave periods. For head wave, the multi-module interaction and shielding are apparent for short wave, while for long wave, this phenomenon is not obvious as observed. For oblique and beam wave, the interactions between the two modules are small.

Motion RAOs with different spacings
Because of the limited experimental resources, not all parameters of interest can be investigated by the model tests. The validation works performed in Subsection 5.3 indicate that the numerical calculation can give a reliable prediction. With these conclusions, a sensitivity analysis was conducted to study the spacings influence on the motion RAOs of the two-module VLFS. Three more spacings between two modules are considered: 5 m, 15 m and 20 m. Based on the numerical and experimental results in Subsection 5.3, only head wave condition is considered to show the shielding effects on the surge, heave, and pitch direction. Figs. 11 and 12 respectively show the motion RAOs of M-1 and M-2 modules in the head wave.
It can be found that the shielding effects on different spacing conditions are related to the wave period. When the wave period is larger than 12 s, the motion RAO curves of   M-1 are close to those of M-2, especially for the surge direction, and the differences between two modules in heave and pitch direction are smaller than 2%. However, when the period is smaller than 12 s, the motion RAOs change with the spacings. These phenomena may be explained that the short incident wave is dramatically affected by the multicolumn structure compared with the long incident wave. This point is confirmed by the observation presented in Subsection 5.2. What is more, the smaller spacing would lead to larger peak response. These observations are even more pronounced for M-1 module while the difference between different spacings is relatively insignificant. That is because M-2 on the incoming wave side gives an interaction effect on M-1, which is related to the spacings. The increase of the spacing would reduce the interaction effect.

Conclusions
In this paper, frequency-domain analyses on the twomodule hydrodynamic interactions were conducted by use of a 3-D diffraction-radiation code. Well-designed model tests were performed and compared with the numerical results. Based on the numerical and experimental data, free surface and motion RAOs in the head, oblique and beam wave are obtained. The relationships among wave periods, wave headings, and motion responses are clarified. Some concluding remarks are drawn based on the numerical and experimental results.
(1) Overall, numerical and experimental results suggest that the numerical calculation can give a good estimation on the motion responses and wave elevations of the two modules in the short wave; while in the long wave, the numerical results would over-predict the motions, especially for the head wave.
(2) The hydrodynamic interactions between two modules are related to the period of the incident wave. The amplitude of the incident wave at the center of the gap reduces because of the interaction influence. When the period increases to 25 s, the influence is decreased. It implies that the two-module VLFS gives a clearer interaction effect in the short wave than that in the long wave.
(3) Both numerical and experimental results show that there is a strong relationship between hydrodynamic interactions and wave headings. According to the cases described in this paper, the existence of the interaction between two modules in the head wave can easily be indicated. When the wave heading increases from 0 deg to 90 deg, the interaction effect is reduced. In beam wave, motion responses of the two modules are almost the same (no more than 4 % difference between two modules). It can be explained that one of the two modules provides a shielding effect to the other module in the head wave, and the shielding effect reduces with the increase of the wave heading.
(4) The dependency of the motion RAOs on spacing is further investigated. The change of the spacing has an insig-nificant influence on the motion RAOs of the two modules in the large period wave. When the period is smaller than 12 s, the larger spacing will present a smaller peak response, especially for surge direction. Because the interaction effect decreases with the increase of the spacing, M-2 has a small response in a large spacing of the two modules deployed in tandem.