Roundness and slenderness e ﬀ ects on the dynamic characteristics of spar-type ﬂ oating o ﬀ shore wind turbine

: Spar-type ﬂ oating o ﬀ shore wind turbine has been massively developed considering its design simplicity and stability to withstand the wave-induced motion. However, the variation of the local sea level and the readiness of supporting production facilities demand the spar design to adapt in a viable way. Considering this, the present article investigated how the slenderness (length over diameter ratio) and the roundness of cross section in ﬂ uence the hydrodynamic characteristics, which are the crucial parameters of ﬂ oater performances. The OC3-Hywind spar-type ﬂ oating platform was adapted as the reference model. The length of the reference ﬂ oater was then varied with a ratio of 1.5, 2, 2.5, and 3 and the diameter was proportionally scaled to obtain constant buoyancy. The number of the sides which indicated the roundness of the cross section was varied to be 4, 6, 8, 10, 12, 14, and in ﬁ nity (cylindrical shape). The analysis was conducted using potential ﬂ ow theory in a boundary element method solver through an open-source code NEMOH. Initially, panel convergence was conducted and compared with the experimental results of the reference model to obtain the appropriate simulation settings before being used for the case con ﬁ guration analysis. Results stated that the roundness e ﬀ ect with sides greater than 16 had little e ﬀ ect on dynamic characteristics. Meanwhile, the spar with the largest diameter was more stable against the translational motion.


Introduction
In recent years, climate change has been brought on by greenhouse gases on a global scale [1]. The increase in green energy demand indicates that renewable energy plays an essential role in the future sustainable development. Ocean energy resources, especially offshore wind energy, gain more attention and concerns due to the convincing energy potential. To ensure the availability of stable wind sources, maximise wind potential, and reduce noise and visual impact issues, the offshore wind turbine goes further from the shore and at deeper water depth. In this condition, floating-type offshore wind turbines (FOWTs) are suitable for viable deployments.
Various types of FOWTs with different support platforms have been developed and investigated in recent years. The designs are adaptations of floating support structure concepts employed by the oil and gas offshore industry. The technologies that have been developed provide much valuable information, although there are some differences due to differences in aero and hydrodynamic characteristics [2]. FOWTs can be divided into four main categories according to their floating structure [3], i.e., semi-submersible [4,5], tension leg platform (TLP) [6,7], barge [8,9], and spar [10], as shown in Figure 1 [3].
Each design of FOWT has advantages and disadvantages that should be considered during installation parameters, such as proximity to shore, water depth, environmental factors, and seabed characteristics. Considering the simplicity and the dynamic stability, spar-type floating platform is preferable. In spar-type platforms, the structural principle is a vertical cylinder. Ballast is used at the bottom to provide stability [10]. Substantial ballast at the bottom of the structure   contributes significantly to roll, pitch, and displacement motions [11]. Meanwhile, mooring lines prevent drifting and limit surge and sway motions. This type of structure is intended for offshore wind farms with water depths between 100 and 300 m [12,13].
The performance of an FOWT is affected by wind, waves, and currents in the marine environment, resulting in 6 degrees of freedom (6-DOF) motions. Three translational motions (surge X, sway Y, and heave Z) and three rotational motions (roll, pitch, and yaw) are listed in Figure 2 [14]. Therefore, it is necessary to investigate the effects of the overall motion of the structure, focusing on the relevant phenomena of FOWTs operating in natural sea environments and wind conditions [15].
Several researchers have studied the dynamic response of spar FOWT with various methods, which is summarized in Table 1. Utsunomiya et al. [16] studied the motion dynamics of a scaled model of a spar-type offshore wind turbine under regular and irregular waves. The experimental results on the wave tank were compared with the numerical simulation results. Xu and Day [17] conducted a dynamic response study with experimental methods on a tank tester for spar-type FOWT. They explained in detail the experimental setup and the limitations of their method. Chen et al. [18], using the experimental method, explained that mooring cable tension is not only affected by waves, but also by surge and heave motion. Using in-house code, Liu and Yu [19] calculated the FOWT motion under the wave group scenario, and the joint north sea wave project spectrum generated the wave group series. The results show that the variation of surge motion increases slowly within a certain period. Meng et al. [20] analytically calculated the aerodynamic and hydrodynamic damping, which was validated using numerical analysis by FAST and AQWA. Hydrodynamic dampings such as radiational damping and viscous drag effect provide damping values in surge and sway motions (up to 50% and 30%, respectively).
Using the boundary element method (BEM) in ANSYS AQWA, Ma et al. [21] innovated the mooring system to inhibit horizontal and pitch motions because conventional mooring generally only inhibits horizontal motions. The innovative mooring system decreased the average value of surge motions by 37.97%, and the average pitch motions decreased by 17.87%. Yue et al. [22] investigated the effect of heave damping on FOWT spars. Variations were made to the position of the heave plate (a bottom, middle, top) on the spar. The application of the heave plate affects the significant increase of heave-added mass and has little effect on surge-added mass, while pitch-added mass is reduced. In the  effect of radiational damping, the heave plate only gives a slight effect on the bottom, middle, and top configurations. In addition, the heave plate can also significantly reduce the mooring tension in extreme conditions.
Jeon et al. [23] and Prastioanto et al. [24] examined the effect of mooring cables on dynamic responses. Subbulakshmi and Sundaravadivelu [25,26] and Seebai and Sundaravadivelu [27] investigated the effect of damping plates on dynamic responses. ANSYS-AQWA, FAST, and WAMIT are commonly used for numerical methods. Open-source software NEMOH has been used sparingly for numerical methods.
From the literature survey, it can be concluded that the research on spar-type floating offshore wind turbine mainly focused on the motion response comparison obtained from different methods, including the installation of plate damping and the anchoring system. The present article focuses on the effects of the fundamental floater parameters including the  [17] Experimental investigation on dynamic responses of a spar-type offshore floating wind turbine and its mooring system behaviour Experimental in tank test • Scaling parameters • Tank layout (spar only and spar with realistic mooring lines • Wave frequency Prastianto et al. [24] Mooring analysis of SPAR-type floating offshore wind turbine in operating condition due to heave, roll, and pitch motions length over diameter ratio (L/D) which relates to the required sea water depth and number of sides of the cross section which affect the fabrication simplicity on the hydrodynamic characteristics of the floater (added mass and radiation damping).

Methodology
In this section, case configurations are discussed to determine the effects of roundness and slenderness on the dynamic characteristics of the spar-type FOWT. In addition, the equations related to the BEM are also explained in detail.

Case configuration
Before conducting the parametric study to assess the effects of roundness and slenderness on the hydrodynamic characteristics, panel convergence should be done to ensure the validation of the analysis procedure. The benchmarking model used was the spar-type FOWT OC3 Hywind [28]. The panels were varied from 468 to 2,700 panels. The results compared (surge, heave, and pitch) added mass with the number of panels and (surge, heave, and pitch) radiational damping with the number of panels. In addition, the maximum value of the dynamic response at each number of panels was also displayed for consideration of panel convergence. After obtaining the range of panel values that produces convergence results, detailed case configurations are performed numerically for the spar-type FOWT.

Roundness
The number of sides (roundness) of the spar-type FOWT model was varied to simplify the fabrication process. Only the number of sides varies, while the values of r (radius of spar) and z (spar length) were unchanged. The variation in the number of sides was compared with the benchmarking results set as the baseline. The variations in the number of sides used were 4 (tetragon), 6 (hexagon), 8 (octagon), 10 (decagon), 12 (duodecagon), and 14 (tetradecagon) sides on the FOWT spar-type ( Figure 3). The results compared the value of (surge, heave, and pitch) added mass at each variation of the number of sides and (surge, heave, and pitch) radiational damping at each variation of the number of sides. The peak value of each dynamic response will also be shown.

Slenderness
In this section, the effect of floater geometry on the dynamic parameters of the spar-type FOWT was studied. The geometry changes are aimed at design optimization regarding the volume of the OC3 Hywind spar-type FOWT [28]. Changes were made to the floating platform's length (L) and diameter (D). The variations of the platform length were L/1.5, L/2, L/2.5, and L/3. Then the diameter size for each variation of the platform length followed the volume reference of OC3-Hywind. The principal dimensions for all variations and main characteristics for spar are shown in Figure 4 and Table 2. The centre of gravity (CoG) value had the same ratio in each variation. The dynamic response for each variation will be shown to determine the characteristics of each design. All the described case configurations were considered under free-floating conditions.

BEM
Linear potential flow theory can be solved by numerical methods known as the panel method or BEM [29]. Approximations of free surface green function are important in BEM, especially for open-source software NEMOH. In another application in engineering, Lei et al. [30] applied the BEM to calculate the fracture parameters. Then they compared the results with the extended finite element method. The results obtained showed that the two methods have identical values. According to previous studies [31][32][33][34], there are several advantages of BEM compared to the other numerical methods: smaller amount of data required in BEMs for a program to run effectively; the discretization process in BEMs only affects the body's surface (resulting in a significantly smaller system of equations); it takes less time to solve the problem due to the small number of equations; and the approach of BEM is inexpensive because little data are needed. In addition, in BEM, designing boundary techniques that automatically handle singularities, shifting bounds, and infinite boundaries seems simpler.

Airy wave theory
For wave modelling, the assumptions used include incompressible flow, irrational flow, and inviscid fluid [35]. The continuity of incompressible flow for 3D bodies follows Eq. (1).  where u is the velocity component for the x direction, v is the velocity component for the y direction, and w is the velocity component for the z direction. For irrational flow, the vorticity (curl of the velocity) becomes zero, which is stated in the following equation: where ω x , ω y , and ω z are rotational components along x, y, and z directions. Wave flow analysis around the spar-type platform and hydrodynamic force can be described by velocity potential. Velocity potential (ϕ) as a function of x, y, and z is defined as follows [36]: If the potential function exists, the continuity equation of incompressible flow for 3D bodies can be reformed as Laplace's equation by substituting Eqs (5)-(7) in Eq. (1).
Solving Laplace's equations required several boundary conditions. Summarized equations for boundary conditions are as follows: -Seabed boundary condition, for = − z d, -Body surface boundary condition -Fluid domain boundary condition -Free surface kinematic condition, for ( ) = z η x y t , , -Free surface dynamic condition, for where d is the seabed, n is the normal unit vector on body surface, v n is the normal velocity, and η is the free surface elevation. Due to the nonlinear boundary conditions at the free surface following Eqs (12) and (13), to simplify the solution, it can be assumed that ≈ η 0. Thus, the equations at free surface boundary conditions can be linear.
1 for 0, free surface boundary conditions.
The solution for the velocity potential that satisfies Laplace's equation is given in Eq. (8) by applying all boundary conditions.
where g is the gravity, η a is the wave amplitude, ω is the frequency, t is the time, and k is the number of waves. Based on Eq. (15), the free surface elevation η is calculated according to Eq. (16). Meanwhile, the value of k can be found using the dispersion relationship in Eq. (17).

Linear diffraction theory
Potential function ϕ is a combination of diffraction potential and radiation potential that follows the following equation: where ϕ 0 is the velocity potential of incident waves, ϕ 7 is the velocity potential of scattered waves, and φ j is the velocity potential of radiated waves generated by Ṡj (6-DOF motions). -Seabed As for radiation potentials by following j = (1, 2, …, 6), the boundary conditions follow the following equation: The wave exiting force for j = (1, 2, …, 6) due to diffraction and radiation potential is given by the following equations: where S B is the surface body, ρ is the density, F k is the wave forces, and p dyn is the dynamic pressure. The radiation potential is a complex number in general; hence for an oscillating body with angular frequency = − ω s iωS ,̈j j (time-dependent of − e iωt ), Eq. (28) where A jk is the added mass, B jk is the radiational damping, and jk = 1,2, …, 6.

Green function and boundary integral equation
According to Xie et al. [37], the free surface green function in finite water depth that satisfies the boundary conditions is shown as follows: where ( ) → → G X X ; 0 is the free surface green function, → = X ( ) x y z , , is the field point vector position, → = X x y z , , 0 0 0 0 is the source point vector position on the body surface, r is the distance between field point and source point, r 1 is the distance between field point and mirror source point, PV is the Cauchy principal value of the integral, and = μ k is the number of waves [38]. K is the wave number in-depth water, R is the horizontal coordinate for field point and source point, and J 0 is the zeroth-order Bessel function of the first kind defined [39]. Boundary integral equation for radiation velocity potential φ j ( = j 1, 2, …, 6) is given as follows: Meanwhile, boundary integral equation for diffraction potential φ 7 is shown as the following equation.

Equation of motions
The motion of the rigid structure in 6-DOF using the panel method was carried out based on the frequency domain following Eq. (35) [40].
where M jk is the mass matrices, C jk is the restoring coefficient matrix of structure, and ξ k is the 6-DOF motion frequency-dependent amplitude. In application, Eq. (35) can be defined as the response amplitude operator χ k (RAO) in Eq. (36) which is affected by the value ξ k . Meanwhile, to plot the actual response of RAO is calculated by Eq. (37) [40].

Results and discussion
This section contains the results of numerical analysis using the BEM-NEMOH method. Following the case configurations discussed in the previous section, the results of the analysis of the effects of roundness and slenderness on the dynamic characteristics of the spar-type FOWT are displayed in the form of graphs explained one by one briefly and thoroughly. In addition, the RAO is discussed.

Benchmarking and panel convergence
Benchmarking or validation was done using the model from OC3-Hywind [28], and the results referred to the numerical analysis from Ma et al. [21]. The input geometry in NEMOH was based on data in Figure 7(a). Moreover, other parameters followed Table 2 on the spar-type baseline platform.
The number of sides was set at 36 sides. This number was sufficient to generate a cylindrical shape (infinity of sides). Following the case configurations discussed in the previous section, the panels were set from 468 to 2,700 panels. The numerical analysis results for each panel produced six results, as shown in Figure 5. Surge added mass ( ) A 11 in Figure 5(a) shows that almost all panel variations had results and patterns close to the numerical analysis conducted by Ma et al. [21]; however, the smallest number of panels had the most significant error. Similar to surge added mass, the results of surge radiational damping ( ) B 11 in Figure 5(b) show that all panels had results close to the reference value. However, at the maximum value of each panel, results with tolerable errors were obtained. Heave added mass ( ) A 33 in Figure 5 From the results presented earlier, it can be concluded that the value converged and matched the reference value in the number of panels from 2,000 to 2,700. It can also be seen in Figure 6, which was the normalized maximum value for each panel. The target value for normalization was based on the highest maximum value (spar with 2,700 panels), where A 11 was × 8.308 10 6 kg, B 11 was × 3.836 10 5 kg/s, A 33 was × 2.552 10 5 kg, B 33 was × 1.236 10 4 kg/s, A 55 was × 1.606 10 10 kg m 2 , and B 55 was × 2.730 10 10 kg m 2 /s. The normalization result with a value close to 1 occurred in panels 2,000 to 2,700. Then a panel value of 2,484 panels was selected, which will be used to determine the effect of roundness and slenderness. The mesh results from NEMOH for OC3-hywind with 36 sides using 2,484 panels can be seen in Figure 7(b) and used as a baseline.

Roundness (number of sides)
Per the case configurations described in the previous section, the number of sides or roundness of the cross section was varied to be 4, 6, 8, 10, 12, 14, and infinity (cylindrical shape). In Section 3.1, the cylindrical shape was represented with sides of 36. The results of numerical analysis from the effect of the roundness on the surge added mass    Normalized Maximum Value   and radiational damping are shown in Figure 8. In the surge added mass in Figure 8(a), the tetragon provided the smallest value with a peak of × 6.174 10 6 kg at 0.5 rad/s of frequency. At the same time, the tetradecagon with 14 sides provided the closest value to the baseline with a peak value of × 8.044 10 6 kg. In Figure 8(b), the surge radiational damping of the tetragon provided the lowest damping value of × 2.286 10 5 kg/s at a frequency of 1.5 rad/s. The significant increase occurred from tetragon to octagon, while from octagon to baseline, the increase was insignificant. Tetradecagon produced the closest damping value to the baseline with a value of × 3.689 10 5 kg/s at 1.4 rad/s. The effects of roundness on heave added mass and radiational damping are shown in Figure 8(c) and (d). Like the surge characteristics, the tetragon produced the smallest value in heave added mass and radiational damping, and a significantly increased value occurs from tetragon to octagon. In contrast, the value close to the baseline was found in the tetradecagon. The peak value of heave added mass in tetragon and tetradecagon at a frequency of 1.4 rad/s was × 1.329 10 5 kg and × 2.460 10 5 kg. In heave radiational damping, the tetragon had a peak value of × 0.512 10 4 kg/s at a frequency of 0.9 rad/s, and the tetradecagon gave a peak value of × 1.165 10 4 kg/s at a frequency of 0.9 rad/s. The same trend also happened in rotational motions, namely pitch (added mass and radiational damping), which can be seen in Figure 8(e) and (f). The lowest maximum value of pitch added mass was produced by a tetragon with a value of × 1.190 10 10 kg m 2 at a frequency of 0.9 rad/s. The highest maximum value of pitch added mass was obtained from a tetradecagon with a value of × 1.557 10 10 kg m 2 at 0.9 rad/s. The pitch radiational damping of the tetragon gave the lowest maximum value of × 1.648 10 9 kg m 2 /s at 1.5 rad/s. Tetradecagon gave the maximum value of pitch radiational damping closest to the baseline or the highest of all side variations with a value of × 2.624 10 9 kg m 2 /s at 1.5 rad/s. This tendency was due to the mass difference on each number of sides. In NEMOH, fewer sides would result in a smaller mass. In general, if the value of added mass and radiational damping was small, the stability would decrease, and the platform would be more sensitive to waves. In terms of the normalized maximum value for each side variation ( Figure 9) with the target value for normalization was based on the highest maximum value (baseline spar), where A 11 was × 8.308 10 6 kg, B 11 was × 3.834 10 5 kg/s, A 33 was × 2.560 10 5 kg, B 33 was × 1.236 10 4 kg/s, A 55 was × 1.606 10 10 kg m 2 , and B 55 was × 2.727 10 10 kg m 2 /s. It can be concluded that the number of sides more outstanding than 16 (tetradecagon) did not significantly affect surge, heave, and pitch (added mass and radiational damping). Significant values occurred in tetragon (4 sides) to octagon (8 sides) variations.

Slenderness (L/D ratio)
The slenderness effect in the dynamic characteristic of spar-type FOWT can be seen in Figure 10. Spars 1 to 4 had configurations of L/1.5, L/2, L/2.5, and L/3, respectively, according to Figure 4. Figure 10(a) and (b) shows the surge motion numerical analysis results. It showed that the baseline (OC3-Hywind) with the smallest diameter and longest floater geometry had the best surge added mass value with a peak of × 8.308 10 6 kg. Conversely, spar 4, with the largest diameter and smallest floater length, produced the slightest surge added mass value with a maximum value of × 7.282 10 6 kg. The peak value of all spars in surge added mass was obtained at a frequency of 0.5 rad/s and decreased  after that. In radiational damping for surge motions, the spar with the best radiational damping value was spar 4, while the spar with the lowest radiational damping was the baseline spar. The maximum value of surge radiational damping for spar 4 was × 8.715 10 6 kg/s at 1 rad/s. In addition, the surge radiational damping had the same value at high frequencies.
The heave motion for added mass and radiational damping can be seen in Figure 10(c) and (d). The spar with the best value for these two parameters was spar 4, with the widest diameter and the longest floater geometry. The most unstable spar against heave motion was the baseline spar with the lowest value for added mass and radiational damping. The maximum value of heave added mass in spar 4 and baseline provided a significant difference of × 15.703 10 5 kg for spar 4 and × 2.561 10 5 kg for baseline. For all geometries, the peak value of heave added mass occurred at 0.4 rad/s. For radiational damping, the maximum values for spar 4 and baseline were × 26.046 10 4 kg/s and × 1.236 10 4 kg/s, respectively. The peak value for all geometries in heave radiational damping occurred at 0.9 rad/s. Similar to the surge motion, heave radiational damping had the same value at high frequencies.
The rotational motion in pitch (Figure 10(e) and (f)), spar with the longest floater and smallest diameter, which was baseline spar, gave the best value in added mass and radiational damping. Hence, it indicated that the spar had the best stability in pitch motion. Meanwhile, the least stable spar in pitch motion was the spar with the shortest floater and largest diameter, spar 4, with the smallest value of added mass and radiational damping. The maximum values for pitch added mass of the baseline spar and spar 4 were × 1.606 10 10 kg m 2 and × 0.149 10 10 kg m 2 at 0.7 rad/s. Meanwhile, the maximum values of pitch radiational damping in spar baseline and spar 4 were × 2.727 10 9 kg m 2 /s and × 0.523 10 9 kg m 2 /s at 1.5 rad/s and 1.3 rad/s, respectively.

RAO
The numerical results of RAO surge, heave, and pitch motions are shown in Figure 11. For RAO surge in Figure 11(a), the peak response was generated at low frequency. The baseline spar generated the largest peak response value with a value of 2.029 m/m at a frequency of 0.3 rad/s. In comparison, the lowest peak response was generated on spar 1 with a value of 0.966 m/m at a frequency of 0.1 rad/s. For RAO heave in Figure  11(b), the baseline spar generated the largest response with a peak value of 12 m/m at 0.2 rad/s. Furthermore, the smallest peak response value was generated by spar 4 with a value of 1.011 m/m at a frequency of 0.1 rad/s. This was because the value of heave radiational damping (Figure 10(d)) on the baseline spar had the lowest value. For RAO pitch in Figure 11(c), the platform with the shortest floater and the largest diameter, spar 4, had the largest response compared to other spars. The peak value of spar 4 in RAO pitch was 23.560 deg/m at a frequency of 0.5 rad/s. The large RAO values for surge, heave, and pitch indicated that the platform had low stability compared to others. This also aligned with the radiational damping values shown in Figure 10. Good stability occurred when the radiational damping value was more excellent. All motion responses from the baseline, spar 1, spar 2, spar 3, and spar 4 had different peak frequency values, indicating each spar's natural frequency. Based on these findings, this work can be potentially continued to application in other cylindrical shell structures in marine regions, such as cold water pipe for underwater situation [41][42][43][44][45][46][47][48]. Collaboration between BEM and finite element method to forecast temperature and critical situation effects on the pipe-based structure can be considered as a potential future research topic [49][50][51][52][53][54][55].

Conclusions
A numerical analysis of the dynamic characteristics of spar-type FOWT using the open-source BEM NEMOH had been completed to determine the effect of roundness and slenderness. The model configuration was validated using existing research on OC3-Hywind to ensure the accuracy of the research methodology. The results obtained showed a good compatibility between different BEM solver codes. The variation of roundness (number of sides) showed that the increase in added mass and radiational damping occurred significantly from the tetragon (4 sides) to the octagon (8 sides). At the same time, the number of sides which is more than the tetradecagon (16 sides) results in an insignificant increase in added mass and radiational damping.
The slenderness variation showed that the spar with good stability against the translational motion (surge and heave) had the largest diameter and shortest floater geometry. As for the rotational motion (pitch), the spar with the longest floater geometry and the smallest diameter had a good stability against pitch motion. It can be seen from the value of added mass and radiational damping. The peak value for RAO happened at a low frequency, and each spar had a different peak value. This indicated that the natural frequency of each spar was also different.