Experimental, Numerical and Simplified Theoretical Model Study for Internal Solitary Wave Load on FPSO with Emphasis on Scale Effect

Scale effect of ISWs loads on Floating Production Storage and Offloading (FPSO) is studied in this paper. The application conditions of KdV, eKdV and MCC ISWs theories are used in the numerical method. The depthaveraged velocities induced by ISWs are used for the velocity-inlet boundary. Three scale ratio numerical models λ=1, 20 and 300 were selected, which the scale ratio is the size ratio of numerical models to the experimental model. The comparisons between the numerical and former experimental results are performed to verify the feasibility of numerical method. The comparisons between the numerical and simplified theoretical results are performed to discuss the applicability of the simplified theoretical model summarized from the load experiments. Firstly, the numerical results of λ=1 numerical model showed a good agreement with former experimental and simplified theoretical results. It is feasible to simulate the ISWs loads on FPSO by the numerical method. Secondly, the comparisons between the results of three scale ratio numerical models and experimental results indicated that the scale ratios have more significant influence on the experimental horizontal forces than the vertical forces. The scale effect of horizontal forces mainly results from the different viscosity effects associated with the model’s dimension. Finally, through the comparisons between the numerical and simplified theoretical results for three scale ratio models, the simplified theoretical model of the pressure difference and friction forces exerted by ISWs on FPSO is applied for large-scale or full-scale FPSO.


Introduction
In recent decades, the internal solitary waves (ISWs), as a unique phenomenon in the stratified ocean, are frequently and widely (Cai et al., 2012;Guo and Chen, 2014;Huang et al., 2017) observed in the South China Sea with the exploration and exploitation of oil and gas. According to the reports, the ISWs have posed severe security risks to ocean engineering structures. In 1990, the mooring cables were found cracked, the ships crashed and the floating flexible pipes were broken in Liuhua Oilfield (Chen, 1996) when ISWs propagated past the platform. ISWs therefore became an important factor that should be considered in the hydrodynamics study and safety evaluation of an ocean engineering structure.
In practice, the loads induced by an ISW on the vertical cylinder are studied by many scholars based on Morison's empirical formula by adopting different ISW theories. The large-amplitude ISWs load on vertical circular cylinder with Miyata-Choi-Camassa (MCC) equation (Choi and Camassa, 1996) was investigated by Xie et al. (2010). The load and the dynamic response of tension leg platform (TLP), semi-submersible platform and FPSO under ISWs with extended Korteweg-de Vries (eKdV) equation (Helfrich and Melville, 2006) were investigated by You et al. (2010You et al. ( , 2012 and Zhang et al. (2017). The load and dynamic response of Spar platform exerted by ISWs with Korteweg-de Vries (KdV) equation were studied by Song et al. (2010). Cai et al. (2003Cai et al. ( , 2006Cai et al. ( , 2008Cai et al. ( , 2014 calculated the forces on piles with and without seasonal water stratification variation. It should be noted that the accuracy of the Morison's empirical formula may highly depend on the coefficients C d and C m that are considered in the calculation. The two coef-ficients are commonly determined by experiences or based on surface wave cases. In recent years, You's group in Shanghai Jiao Tong University has performed a series of experiments and simplified theoretical models on the coefficients C d and C m (Huang et al., 2013a(Huang et al., , 2013b(Huang et al., , 2013cXu et al., 2014aXu et al., , 2014bWang et al., 2016) including circular cylinder, semi-submersible platform, TLP and FPSO in a stratified fluid tank. The feasibility of different ISWs theories and the coefficient of Morison's empirical formula were summarized in their studies. However, the maximum Reynolds number in the tank experiments only reaches 10 4 , thus the scale-effect cannot be ignored. According to the surrounding ocean environment in the South China Sea, ISWs phase velocity is 1.6-2.0 m/s and the maximum amplitude is larger than 100 m (Fang and Du, 2005). The Reynolds number in reality may reach 10 8 . Apparently, such a large Reynolds number is impossible to be realized in a tank experiment.
Another way to evaluate the ISW loads is the numerical simulation. For wave-body interaction, many approaches, such as the mixed Euler-Lagrange method, the continuous Rankine source method, the desingularized integral method and so on, were developed by Feng et al. (2014Feng et al. ( , 2015Feng et al. ( , 2017. The study on ISWs-body interaction has just started. Xie et al. (2011) and Wang et al. (2017aWang et al. ( , 2017b investigated the ISWs load on a cylindrical pile using simulation. The ISWs load with background parabolic current on smalldiameter cylindrical tender leg was studied by Lv et al. (2016). A numerical flume considering the applicability conditions of KdV, eKdV and MCC theory was established by Wang et al. (2014Wang et al. ( , 2015aWang et al. ( , 2015bWang et al. ( , 2015cWang et al. ( , 2015dWang et al. ( , 2017c. Using that numerical flume, the ISW load on the circular cylinder, TLP and semi-submersible platform can be analyzed. Compared with the experiments, numerical method is applicable to the cases within a broader Reynolds number range, but its accuracy and reliability may still be controversial unless some practical measures can be taken to verify the results firstly.
In this paper, the authors would like to propose the work on combining the experimental results and the numerical simulation, and consider the FPSO (floating production storage and offloading) as an example. The horizontal forces, vertical forces and their load components exerted by ISWs are investigated with numerical FPSO models with different scale ratios. The smallest model is exactly the same as the experimental one formerly presented by Xu et al. (2014aXu et al. ( , 2014b, and their results are compared to verify the numerical method. The results of different scale ratios are further analyzed to reveal the scale effect of the experiments. Besides, the feasibility of the simplified theoretical model under the full-scale situation is analyzed.
In Section 2, the model, the parameters and the design of scale ratio values are described. The results are shown and discussed in Section 3. The conclusions are drawn in Section 4.

Numerical methods and simplified model
F x For FPSO with hull forms, the horizontal forces exerted by ISWs mainly consist of the frictional force and the pressure-difference forces.
where the first term represents the frictional force at the side and bottom surfaces, the second one represents the pressuredifference forces at the side surface. F z Similarly, vertical forces follows where the first term represents the frictional force at the side surface, the second one represents the pressure-difference forces at the side surface.
In Eqs. (1) and (2), is the wetted area of FPSO; is the tangential velocity along the wetted surface of FPSO; is the instantaneous dynamic pressure induced by ISWs on FPSO; is the internal normal vector of the wetted surface of FPSO.
According to Bernoulli equation, the instantaneous dynamic pressure induced by ISWs is expressed as: , where and are the horizontal and vertical instantaneous velocity of water particles induced by ISWs. They are defined as follows (Camassa et al., 2006): Considering the applicability of ISWs theory, the nonlinear parameter is defined as , the dispersion parameter is defined as (where is the characteristics width, is the total depth of calculation flow field, is the amplitude of ISWs). For a given ISW, the nonlinear parameter and dispersion parameter for the three ISW equations are calculated respectively. The KdV equation is selected to calculate the inlet velocity for and , where is the critical dispersion parameter summarized by laboratory experiments. The eKdV equation is selected to calculate the inlet velocity for and . The MCC equation is selected to calculate the inlet velocity for or . The series of experiments for the ISW load on FPSO scale model have been carried out by You's group (Xu et al., 2014a(Xu et al., , 2014b. The simplified theoretical model is sum-marized. They found that the ISWs friction force is integrated with the tangential velocity induced by ISWs along FPSO wetted surface: where is the tangential unit vector of the wetted area and rear toward, and is the tangential velocity on the wetted area. The friction coefficient is totally related to Reynolds number. The simplified formula (Xu et al., 2014b) is proposed as: where Reynolds number is defined as Re=U max L wl /ν, and U max represents the maximum horizontal velocity induced by ISWs. The pressure-difference force is calculated with Froude-Krylov formula, written as: n where is the normal unit vector of the wetted area and inward.

Results and discussions
Denote scale ratio , which is the size ratio of numerical models to the experimental model. In order to analyze the scale effect, three scale ratio numerical models including 1:1, 20:1 and 300:1 are selected. The sizes of the numerical models are shown in Table 1. During the simulation, the density of upper and lower layer fluid, and their layer thickness ratio are consistent with the experimental conditions. The upper layer fluid density is 998 kg/m 3 while the lower 1025 kg/m 3 . Three ratios of the upper and lower layer thickness, 10:90, 15:85 and 20:80, are considered. The draft diagram of the numerical flume is shown in Fig. 1. The inlet boundary adopts the depth-averaged velocities. The velocity induced by ISWs is defined as: where represents phase velocity of ISWs, ( ) is inlet velocity of the upper (lower) layer fluid on the inlet boundary.
The VOF (volume of fluid) method is employed for tracking the two-layer fluid interface during the generating and propagating of ISWs. Meanwhile, the sponge layer technique (Han et al., 2009) is adopted to absorb ISWs at the end of numerical flume.
In this paper, the dimensionless horizontal and vertical forces exerted by ISWs on FPSO are defined as , , where ( ) represents the projected area that the wetted surface area of FPSO are projected to x-direction (z-direction) of the coordinate system (the coordinate systems are defined with the horizontal plane on the undisturbed interface and the -axis pointing upwards and passing through the gravity center of FPSO.). Characteristic time of numerical simulation is defined as , where t is the characteristic time of the previous laboratory experiments.
Considering the viscosity effect of fluid, Reynolds number is defined as Re=U max L wl /ν, in which U max represents the maximum horizontal velocity induced by ISWs.
The experimental ISWs loads on FPSO for comparison were derived from Xu et al. (2014b).

Scale effect on the ISWs horizontal forces
The comparisons of horizontal ISWs force for numerical models, experiments and simplified model are listed in Fig. 2. Fig. 2a shows the experimental and simplified model results agree well with the λ=1 numerical simulations both on the amplitudes and trends. The maximum dimensionless horizontal ISWs forces for experiments, λ=1 numerical and simplified model are 28.3×10 -3 , 27.1×10 -3 and 27.5×10 -3 , respectively. The relative error of the maximum horizontal forces is merely 4%. Figs. 2b and 2c show, with the increase of scale ratio, the amplitude of dimensionless horizontal forces for λ=20 and λ=300 numerical models has noticeably decreased to 25.5×10 -3 and 24.3×10 -3 , respectively. The relative errors reach 10% and 14%, respectively. It means that the scale ratio has significant influence on the experimental ISWs horizontal force on FPSO.
Comparing the numerical and simplified model results in Figs. 2a-2c, with the increase of scale ratio, the maximum dimensionless horizontal ISWs forces for λ=20 and λ=300 simplified models has decreased to 20.2×10 -3 and 18.5×10 -3 , respectively. According to the results of λ=20 and λ=300 numerical models, the relative errors are 21% and 24%, respectively. It means that the scale ratio has significant influence on the ISWs horizontal force calculated by the simplified theoretical model on FPSO.
According to the series of the load experiments on FPSO, the ISWs load can be decomposed into the friction and pressure-difference forces. The pressure-difference forces can be further decomposed into the wave and viscous pressure-difference forces. Using the ISW numerical flume, one can calculate the pressure-difference forces and the wave pressure-difference force in viscous fluid (ν=1.003×10 -6 m 2 /s) and inviscid fluid (ν=0), respectively. The viscous pressure-difference force equals the pressuredifference force minus the wave pressure-difference force. Fig. 3 shows the time histories of ISW load components for the three numerical models. Table 2 lists the maximum dimensionless horizontal ISWs load components for the three numerical models. The wave pressure difference force plays the main role in the horizontal forces exerted by ISWs. The friction force and viscous pressure difference force are significantly small in the horizontal forces. However, the scale ratio shows more influence on the friction force and viscous pressure difference force . Figs. 3a-3c and Table 2 show that Reynolds number increases with the scale ratio, and the effect of viscosity consequently vary. The value of friction force decreases two or three orders of magnitude, and the value of viscous pressure difference force decreases one or two orders of magnitude. The scale effect emerging in the horizontal loads  ZHANG Rui-rui et al. China Ocean Eng., 2019, Vol. 33, No. 1, P. 26-33 29 mainly results from the different viscosity effects associated with the model's dimension. As for the scale effect of simplified model, Fig. 4 shows the time histories of the pressure difference force from the numerical model and the simplified model. The curves of the pressure difference forces from the simplified model always agree well with those from the numerical model both on trends and amplitudes. It means that the simplified theoretical model of the pressure difference force can be applied for the large-scale or full-scale ocean engineering structure. But the pressure difference forces with numerical and simplified model are large when time t is about 65 s. It is due to the trailing wave in the numerical simulation. Fig. 5 shows the time histories of the friction force from the numerical simulation and the simplified model. The peak friction force from the simplified model is one or two order smaller when the scale ratio is 20:1 or 300:1, respectively. Comparing Figs. 4a-4c, the curves of simplified and the numerical models generally share the same trend. But the amplitude of friction force from simplified models is 10 to 100 times of those in numerical models under λ=20 and λ=300. It means that the simplified model can be only applied for the small-scale ocean engineering structures like the experimental model. But, considering that the proportion of the friction force in the horizontal forces decreases with the increasing scale ratio, it is reasonable that this simplified theoretical model of the friction force is applicable to large-scale or full-scale structures.
Through the above analysis, the horizontal forces, the wave pressure difference force, the friction force and the viscous pressure difference force are affected by the scale ratio in various degrees. The scale effect on the horizontal forces is mainly caused by the discrepancy of the friction force. The horizontal forces exerted by ISWs on FPSO will be overestimated if the experimental results are directly converted to the loads on practical FPSO ignoring the scale effect. As far as the simplified theoretical model is concerned, the proportion of the friction force is significantly small in the horizontal forces on large-scale structure. Therefore, it is reasonable that the simplified theoretical model based on the ISWs load experiments is applies to practical FPSO.

Scale effect on the ISWs vertical forces
The comparisons of ISWs vertical forces between experimental results, the numerical models and simplified models are plotted in Fig. 6 with different scale ratios. Fig. 6a shows the experimental and simplified model results match well with the λ=1 numerical simulation both on amplitudes and trends. The amplitude of dimensionless vertical ISWs forces for experiments, λ=1 numerical and simplified model  Table 3 lists the amplitude of the vertical ISWs load components. The wave pressure difference force dominates the ISWs vertical forces. From Table 3, for λ=1, λ=20 and λ=300 numerical model, the amplitudes of wave pressure difference force are 10 4 , 10 6 and 10 7 times of those of the friction force, respectively. These amplitudes are 10 or 100 times of the amplitude of viscous pressure difference force. Comparing Figs. 7a-7c, the friction force and viscous pressure difference force decrease three and one orders of magnitude when the scale ratio increase from 1:1 to 300:1. These mean that the friction force occupies a negligible portion of vertical forces. The scale effect has less influence on the vertical ISWs forces. The vertical forces of large-scale FPSO can be reasonably converted from the experimental results while ignoring the scale effect.
The comparisons of the time histories of vertical pressure difference force by using the numerical and simplified models are plotted in Fig. 8 under different scale ratios. The ZHANG Rui-rui et al. China Ocean Eng., 2019, Vol. 33, No. 1, P. 26-33 31 amplitudes and trends of the vertical pressure difference forces from the two methods agree well with each other. The relative error of vertical pressure difference force is smaller than 2%. The vertical pressure difference forces for numerical and simplified models are slightly off when time t is larger than 67 s. It is also due to the trailing waves in the numerical simulation. Therefore, the simplified model of vertical pressure difference force may also be applicable to the large-scale or full-scale engineering structures. As a result, the scale effect has slight influence on the vertical ISWs forces. The simplified theoretical model for the vertical ISWs forces can be applied to any scale engineering structures.

Conclusions
In this paper, the scale effect on experimental ISWs loads on FPSO and applicability of simplified theoretical model have been analyzed through comparisons of the experimental, numerical and theoretical results under three scale ratio models. The parameters of numerical method are consistent with the former ISWs load experiments. The applicability conditions of KdV, eKdV and MCC theories are considered in the inlet velocity of numerical method. Main conclusions are drawn as follows.
(1) The numerical results of horizontal and vertical ISWs forces on model FPSO are in good agreement with the results by former experiments. It is feasible using the numerical method to simulate the strongly nonlinear interaction between ISWs and FPSO.
(2) Comparing the results of experiments and numerical simulations, the scale ratios have significant influence on the experimental horizontal ISWs load. That mainly results from the different viscosity effects associated with the model's dimension. The scale ratios have less influence on the experimental vertical load. Converting the vertical ISWs forces from the experiments for the large-scale engineering is reasonable.
(3) Comparing the results of numerical simulations and simplified theoretical model, the scale ratios have significant influence on ISWs friction force. However, the friction force is a negligible portion of the horizontal or vertical ISWs load. The scale ratios' influence on pressure difference force is very small. Therefore, the simplified theoretical model of the ISWs load is applied to large-scale or fullscale engineering. But the simplified theoretical model is just applicable to FPSO.
The scale effect on the ISWs load of FPSO is analyzed quantitatively in this paper. The time history of the ISWs load components on FPSO with the increase of the scale ratio is clarified. The applicability of the simplified theoretical model for the ISWs loads is discussed for the large-scale FPSO. The paper provides a more feasible basis for forecasting the ISWs load on the full-scale FPSO in actual marine environment.