Modelling of the Vortex-Induced Motion of A Single-Column Platform with Non-Linear Mooring Stiffness Using the Coupled Wake Oscillators

A phenomenological model for predicting the vortex-induced motion (VIM) of a single-column platform with nonlinear stiffness has been proposed. The VIM model is based on the couple of the Duffing-van der Pol oscillators and the motion equations with non-linear terms. The model with liner stiffness is presented for comparison and their results are compared with the experiments in order to calibrate the model. The computed results show that the predicted VIM amplitudes and periods of oscillation are in qualitative agreements with the experimental data. Compared with the results with linear stiffness, it is found that the application of non-linear stiffness causes the significant reductions in the in-line and transverse motion amplitudes. Under the non-linear stiffness constraint, the lock-in behavior is still identified at 8<Ur<15, and the trajectories of the VIM on the xy plane with eight-figure patterns are maintained. The results with different non-linear geometrically parameters show that both in-line and transverse non-linear characteristics can significantly affect the predict in-line and transverse motion amplitudes. Furthermore, the computed results for different aspect ratios indicate that the in-line and transverse motion amplitudes increase with the growth of aspect ratio, and the range of lock-in region is enlarged for the large aspect ratio.


Introduction
The vortex-induced motion (VIM) of the offshore structures such as Spars and multi-column deep draft floaters has been an important issue in the offshore and ocean engineering. The VIM caused by the vortex shedding behind the column in the current can not only impact the fatigue life of mooring and riser systems, but also cause destructive collision between a floater and supporting vessels.
The studies on the VIM for Spar have revealed that the VIM amplitude could reach the same order of magnitude as its diameter such that helical strakes must be adopted to mitigate the VIM (Finnigan and Roddier, 2007). Beyond that, Fujarra et al. (2009) conducted model tests in a towing tank to study the VIM responses of a monocolumn platform, and check the effect of the current incidences, drafts and spoiler plates on the VIM responses. The VIM of the multi-column platform also has been widely concerned due to the development of deep draft floater concepts (Zhang et al., 2017;Hu et al., 2019). Gonçalves et al. (2012) investigated experimentally on the VIM responses of a semi-submersible platform with four square columns. They found that the largest transverse VIM responses of the semi-submersible platform were around 40% of the column width, and this behavior occurred in a range of reduced velocities between 4.0 and 14.0. This range was also called the lock-in region.
In recent years, the numerical simulations based on the Computational Fluid Dynamics (CFD) have been developed to predict the VIM of the offshore platform (Lefevre et al., 2013;Zhang et al., 2017;Gu et al., 2018). Lefevre et al. (2013) applied the CFD simulations on the VIM of a Spar hard tank with appurtenances, and compared with the model test data. They also provided the guidelines for mesh and time step setting in the numerical simulations. Chen and Chen (2016) developed a Finite-Analytic Navier-Stokes (FANS) code to simulate the VIM of a semisubmersible in both model scale and full scale, and investig-ated the lock-in behavior. Zhang et al. (2017) used the improved delayed detached eddy simulation (IDDES) to investigate the VIM characteristics for a multi-column platform under different current incidences and draft conditions. They found that the largest transverse motion amplitude of the platform occurred for 22.5° current incidence in the range of current incidences from 0° up to 45° current incidences.
Despite the CFD numerical method shows the great potential for simulating the VIM, it is not a common tool for analyzing the practical systems yet due to the high cost of time and the limit of computational power. Therefore, it is essential to find a more efficient solution with a reasonable accuracy. Phenomenological models based on wake oscillators have been applied to describe the vortex-induced vibration (VIV) of the circular cylinder and understand the underlying physics (Facchinetti et al., 2004;Rosetti et al., 2009Rosetti et al., , 2011Stappenbelt, 2011;Srinil and Zanganeh, 2012). In the earlier studies, most of the numerical models used in the engineering industry are limited to the analysis of one degree-of-freedom (DOF) VIV (Facchinetti et al., 2004). Facchinetti et al. (2004) analyzed the cross-flow VIV excited by the lift force using the phenomenological model based on the wake oscillators. They carried out a classical van der Pol equation to model the near wake dynamics, and the wake oscillator interacted with the motion equations by the linear coupling terms. Rosetti et al. (2009) developed a phenomenological model based on van der Pol equations to predict the VIM behavior of a single-column platform. However, the effect of non-linear stiffness was disregarded in their study. Later, Srinil and Zanganeh (2012) proposed an advanced model to predict two-dimensional coupled in-line/transverse VIV of a circular cylinder.
For the offshore structures with a catenary mooring system, the non-linear compliance is commonly encountered. The VIM of Spar platform with linear springs restraint has also been shown significant deviation from that with catenary restraint (van Dijk et al., 2003). Stappenbelt (2011) experimentally investigated the VIM of nonlinearly compliant elastically mounted cylinder with the varying cubic compliance components. They revealed that the linear system responses are conservative in the in-line and transverse motion responses.
The present study aims to develop a phenomenological model based on the wake oscillators with non-linear stiffness, and investigates the influence of the non-linear stiffness on the VIM behaviors of a single-column platform with low aspect ratio. The initial phenomenological model with non-linear stiffness, which was proposed by Srinil and Zanganeh (2012), was applied to simulate the two-dimensional vortex-induced motion of a circular cylinder. Based on their work, we develop the model to simulate the three-dimensional vortex-induced motion of a single-column platform, and the values of these parameters have been re-calculated.
The in-line and transverse motion amplitudes, periods of the oscillations, hydrodynamic coefficients and trajectories are analyzed. The effect of the aspect ratio on the VIM has also been discussed.

VIM model
2.1 Formulation of the VIM model D To model the in-line and transverse VIM of a singlecolumn platform with non-linear stiffness, the mathematical model of an elastically supported circular cylinder with low aspect ratio is developed. As shown in Fig. 1, the cylinder of the diameter is constrained by a two-directional nonlinear spring system, and it can oscillate in both in-line (xaxis) and transverse (y-axis) directions in a uniform steady U flow of the free stream velocity .
As studied by Srinil and Zanganeh (2012), the motion equations of a single-column platform with non-linear stiffness characteristics may be expressed as: T X Y k where (˙) denotes the differentiation with respect to the dimensional time . and are the in-line and transverse motion displacements, respectively. is the spring stiffness.
where is the fluid density, is the column diameter, is the draft of the platform, ( ) is the potential added mass coefficient for a circular cylinder. The damping coefficient ( ) includes the viscous damping coefficient ( ) and the fluid-added damping coefficient ( ), namely where ( ) denotes the structural angular frequency, and is the structural damping coefficient. is related to the average drag coefficient and assumed to be a constant, as proposed by Facchinetti et al. (2004). ( ) is the vortex-shedding angular frequency, and is the Strouhal number. , , and are the geometrical parameters and set to the constant. It should be noted that Eqs. (1) and (2) are the Duffing equations, in which the cubic non-linear terms ( , ) are described to capture the axial stretching feature, and the quadratic terms ( , ) present a physical coupling of in-line/transverse motions (Srinil and Zanganeh, 2012).
In the VIM of the single-column platform, the effects of vortex shedding are modelled by and in Eqs. (1) and (2). and denote the in-line and transverse forces, respectively. They are related to the drag force ( ) and lift force ( ), as seen in Fig. 2, and may be expressed as: θ where is the attack angle of the flow relative to the cylinder (Srinil and Zanganeh, 2012), and is assumed to be small. The drag and life forces are given by and are the drag and lift coefficients, respectively.
The oscillating nature of the fluid caused by the vortex shedding is modelled by the non-linear wake oscillators satisfying the van der Pol equations (Facchinetti et al., 2004) where and are the dimensionless wake variables, and associated to the drag and lift coefficients on the platform; and are the wake empirical coefficients. The terms on the right-hand side of Eqs. (12) and (13) ( and ) model the effects of the platform motion on the near wake, and are assumed to be linearly proportional to the accelera-tion in the in-line/transverse directions, as suggested by Facchinetti et al. (2004). The coupling and interaction between the fluid and the structure are modelled through the drag and lift coefficients (Furnes and Sørensen, 2007) and are the vortex shedding drag and lift coefficients for a stationary cylinder. It should be noted that the formulation for the drag coefficient is different from that in the studies of Facchinetti et al. (2004) and Srinil and Zanganeh (2012) and it is a function of and . This enables the simulation of dynamic amplification, which has been found in experiments (Sarpkaya, 1987;Vandiver, 1983). is the mean drag coefficient of a stationary cylinder. is a constant determined by experimental data. Therefore, and can be written as: The coupled Eqs.

Parameter values
The values of these parameters of the VIM model presented above are estimated through the model test and some benchmark results, as studied by Rosetti et al. (2009).
The dimensions of a single-column platform are similar to those in the model test . In the VIM model, and . The structural damping coefficient is taken as 0.044, and the spring stiffness is . The Reynolds numbers occurred in the model test are in the sub-critical range, and hence the vortex shedding lift coefficient is usually 0.3, as taken in the references (Facchinetti et al., 2004;Rosetti et al., 2009;Srinil and Zanganeh, 2012). Notice that due to the low aspect ratio of the column ( ), the mean drag coefficient for a stationary cylinder ( ) is different from those for 2D profile and the cylinder with large aspect ratio (Schewe, 1983;Gonçalves et al., 2015). According to the experiment data of the circular cylinder with low aspect ratio (Gonçalves et al., 2015), is taken as 0.7 in the present model. Similarly, the Strouhal number is modified as 0.078 because of the low aspect ratio. The parameter is assumed to be a constant equal to 2.14. The wake empirical coefficients , , and are obtained, which are based on the guidelines that or should keep equal to 40 (Facchinetti et al., 2004). The non-linear parameters are given, as Srinil and Zanganeh (2012) suggested.

Results and discussion
3.1 Prediction of the coupled in-line/transverse VIM The coupled in-line and transverse VIM responses of a single-column platform are investigated based on the VIM model. The results with linear stiffness are used to calibrate the parameters in the coupled wake oscillators by comparing with the experimental data of Fujarra et al. (2009). In their experiment, the mooring system had been simplified as the four linear springs, and the stiffness of constraint was linear. The results of a phenomenological model used by Rosetti et al. (2009) are also presented for comparison. ) is the reduced velocity. It is used to establish a relationship between the flow velocity and the frequency of motion, and has been widely used in the VIV or VIM studies (Facchinetti et al., 2004;Rosetti et al., 2009;Fujarra et al., 2009;Gonçalves et al., 2012). and are the non-dimensional in-line (x-axis) and transverse (y-axis) motion amplitudes, respectively. The definition of dimensionless motion amplitude ( ) refers to that in the expe- riment by Fujarra et al. (2009). It was proposed by Gonçalves et al. (2009) to characterize the phenomenon when there is a high motion signal modulation. As shown in Fig. 3a, compared with the experimental data, the VIM model provides larger in-line motion amplitudes at . This overprediction is similar to that using a model based on wake oscillators by Rosetti et al. (2009). The difference can be explained by the fact that it is difficult to predict the in-line motion amplitude in this range of reduced velocities due to the disturbances in the incoming flow at low reduced velocities in the experiments . Whereas, at , the computed increases with the growth of , and this change trend of with is similar to that of experiment. The computed results with linear stiffness are close to the experimental data at . On the other hand, as presented in Fig. 3b, the computed results of transverse motion amplitudes have good agreements with the experimental data at . presents a linear increase with the increase of at but this amplification weakens at . Besides, as illustrated in Figs. 3a and 3b, it should be noted that the application of non-linear stiffness causes the reduction of the VIM amplitudes of a single-column platform, mainly at the high reduced velocities ( ). This observation is consistent with the experimental study of Stappenbelt (2011) with cubic non-linear compliance. Fig. 4 shows the non-dimensional periods for in-line ( , where ) and transverse ( ) oscillation. The computed results are very similar to those from the experiments, except for the cases at the low reduced velocities ( ). It can be seen that the non-linear behavior leads to a slight decrease of the non-dimensional periods. One can notice that the values of are around 1.0 at , which indicates that the period of transverse oscillation ( ) locks in the natural period of the transverse motion. This behavior of lock-in will give rise to the large transverse motion amplitude in this range of (see Fig. 3b).
It is also observed that the periods of transverse oscillation ( ) are always the double of the periods of in-line oscillation ( ).
U r Fig. 5 shows the trajectories of the VIM on the xy plane at different reduced velocities. It is observed that the motion displacement in the transverse direction increases with the growing . In general, the trajectories show so-called eight-figure patterns which is found in the studies on the VIV of a cylinder (Srinil and Zanganeh, 2012). This behavior is associated with the ratio of the frequency of in-line oscillation to that of transverse oscillation (2:1), as found in Fig. 4. Compared with the results using the model with linear stiffness, the model with non-linear stiffness yields a

Analysis of the hydrodynamic behaviors
In this section, in order to further investigate the effect of the non-linear stiffness on the hydrodynamic behaviors in the VIM of the single-column platform, the hydrodynamic coefficients are presented. Figs. 6a and 6b show the variations of drag coefficients of the single-column platform with respect to . As shown in Fig. 6a, increases as grows at , which corresponds to the increase of the VIM amplitude in in-line and transverse directions. According to the results of fluctuating drag coefficient ( ), the dynamic amplification at is verified (see Fig. 6b). This behavior can give rise to the larger in-line motion amplitude (see Fig. 3a). The curve shapes for the fluctuating drag coefficient are similar to those for the in-line motion amplitude. It can be seen that compared with the results using linear-stiffness, the non-linear stiffness causes the reduction in both of mean drag coefficients ( ) and fluctuating drag coefficients ( ). The differences between the results using linear-stiffness and non-linear stiffness magnify with the increase of at . The reduced drag coefficients with non-linear stiffness at these reduced velocities can lead to the reduction of the in-line and transverse motion amplitudes (see Figs. 3a and 3b). Fig. 7a presents the fluctuating lift coefficient ( ) at different reduced velocities. It can be seen that increases with the rise of until , and changes slightly above this reduced velocity. This change trend is qualitatively similar with the experimental data (Gonçalves et al., 2010). The results indicate that the large transverse motion is not only driven by the large lift force, but also associated with the coupled in-line and transverse motions. In comparison with the results of linear stiffness, those of non-linear stiffness have a similar change trend with , even though the values of are much lower. In addition, the lower with nonlinear stiffness can be the reason to cause the decrease of transverse motion amplitudes (see Fig. 3b).
C F a The results of added mass coefficient ( ) are presented in Fig. 7b. The added mass coefficient is obtained by an analysis in frequency domain, as proposed by Fujarra and Pesce (2002) and can be written as: where denotes the total fluid force including the potential added mass and fluid damping. As shown in Fig. 7b, drops quickly with the increase of . The change trend is similar to that of an elastically mounted cylinder (Fujarra and Pesce, 2002) and the experimental data (Gonçalves et al., 2010). It should be noted that of non-linear stiffness is larger than that of linear stiffness. It is obvious that the larger added mass with the non-linear stiffness leads to the lower fluctuating lift force and transverse motion amplitude.

Effects of non-linear geometrical parameters on the VIM
The effects of non-linear geometrical parameters on the VIM are discussed in this section. In these computations, four different cases with non-linear geometrical parameters ( = =0, = =0.7; = =0, = =0.7; = = = = 0.3; = = = =0.7) and the case with linear stiffness ( ) are performed. The case with denotes that the in-line non-linear characteristic is neglected, and the case with 0.7 means that the transverse non-linear characteristic is neglected.
The computed results of in-line and transverse motion amplitudes with different non-linear stiffness characteristics are presented in Figs. 8a and 8b. It can be seen that, compared with the results with linear term ( ), the in-line non-linear characteristic ( ) causes an increase in and , and it has a greater effect on than , However, the transverse non-linear characteristic ( ) gives rise to a reduction in and . These observations are consistent with those found in Srinil and Zanganeh (2012). It is also observed that in the case with , has a reduction at .
By further computing the case with = = = =0.3, and comparing with the results of = = = =0 and , it is found that both and drop as the values of and increase from 0 to 0.7, as shown in Figs. 8a and 8b. By the comparisons, it suggests that the transverse non-linear characteristic has a more significant effect on the in-line and transverse motion amplitudes than the in-line non-linear characteristic does.
Since the platform excursion can affect the in-line and transverse oscillations (Stappenbelt and Thiagarajan, 2004), the mean and max in-line displacements are also presented in Figs. 9a and 9b. As shown in Fig. 9a, compared with the results with linear stiffness ( ), the all imposing non-linear terms always lead to much decrease in the mean in-line displacement ( ) at the high reduced velocities ( ). In comparison with the mean in-line The effects of aspect ratio ( ) on the VIM are investigated in this section. The non-dimensional in-line motion amplitudes using the model with the non-linear stiffness characteristics for different aspect ratios ( ) are presented in Fig. 10a. It is seen that slightly changes with the increase of for . However, for =0.4, 0.6 and 0.8, jumps at . A significant increase of with the growth of from 0.2 to 0.4 is observed. The tendency is consistent with the experimental data with linear spring system. It is also observed that the increase of in-line motion amplitude is insignificant for 0.4. The non-dimensional transverse motion amplitudes for different aspect ratios ( ) are shown in Fig. 10b. Similar to , increases as grows. Figs. 11a and 11b present the non-dimensional periods for in-line and transverse oscillations for different aspect ratios ( ), respectively. It can be seen that the tendencies of the curves for these aspect ratios are similar. Whereas, the values of are closer to 1.0 for larger . As shown in Fig. 11b, for , the vortex shedding frequency is synchronized with the natural frequency of the transverse motion at which is called lock-in region. It should be noted that the range of lock-in region enlarges with the growth of .
The effects of on the trajectories are illustrated in Fig. 12. The computed results at and are presented in Figs. 12a and 12b, respectively, with four aspect ratios ( , 0.4, 0.6 and 0.8). It is observed that most of the trajectories on the xy plane entail the eight-figure pattern and the symmetries of them are maintained. However, for the low aspect ratio ( ), the eight-figure pattern is not yet well-formed. It is also noted that the low aspect ratio effectively mitigates the VIM.

Conclusions
In the present study, a phenomenological model for pre-  HU Xiao-feng et al. China Ocean Eng., 2019, Vol. 33, No. 4, P. 459-467 dicting the VIM of a single-column platform with non-linear stiffness characteristic has been developed. The VIM model is built based on the couple of the Duffing-van der Pol oscillators and the motion equations with non-linear stiffness characteristic. The non-linear stiffness characteristic includes the cubic and quadratic nonlinear terms (Srinil and Zanganeh, 2012). Before analyzing the VIM behaviors with non-linear stiffness, the VIM model with linear stiffness is first performed and calibrated by comparison with the experiments. The comparison shows that the most of results are in qualitative agreement with the experimental data, and the predicted transverse motion amplitude quantitatively agree well with the experimental results.
U r > 10 8 < U r < 15 Compared with the results with linear stiffness, it is found that the application of non-linear stiffness causes the significant reduction of the VIM amplitudes of the singlecolumn platform at , even though the change trends of the results with the reduced velocities are similar. However, the use of non-linear stiffness slightly changes the periods for in-line and transverse oscillation. The lock-in behavior is still identified at . The results with different non-linear geometrically parameters indicate that the Fig. 10. Non-dimensional in-line and transverse motion amplitudes with non-linear stiffness characteristic for different aspect ratios. Experimental data with linear spring system was taken from the reference .  transverse non-linear characteristic has a more significant effect on the in-line and transverse motion amplitudes than the in-line non-linear characteristic does. Besides, the results of hydrodynamic coefficients show that the drag and lift coefficients reduce greatly in the cases with non-linear stiffness. It can be attributed to the larger added mass in the non-linear system. L/D

L/D
In addition, the effects of on the VIM are investigated based on the model. It is found that the in-line and transverse motion amplitude increases with the growth of aspect ratio. The range of lock-in region is enlarged for the large aspect ratio. It is also found that for different , most of the trajectories on the xy plane entail the eight-figure pattern, but for the lower aspect ratio, the trajectories of eight-figure pattern can be not well-formed.