Numerical Study on Hydrodynamic Performances of Novel Dual-Layer Flower-Shaped Heave Plates of a Floating Offshore Wind Turbine

Abstract

This paper proposes novel designs of dual-layer flower-shaped heave plates, featuring both aligned and staggered configurations with three, six, and nine petals. Numerical simulations were conducted to study the hydrodynamic effects of these various heave plate designs integrated with the OC4 DeepCwind semisubmersible floating offshore wind turbine platform under prescribed heave oscillations. The overset mesh technique was employed to treat the floating platform’s motions. Comprehensive assessments of vertical force, radiated wave patterns, vorticity fields, added mass, and damping coefficients were conducted. The results revealed that the novel flower-shaped staggered heave plates significantly outperformed conventional circular plates in terms of damping coefficients. Specifically, the damping coefficient of flower-shaped staggered heave plates was greater than that of circular heave plates, while the aligned configuration exhibited a lower damping coefficient. The damping coefficient increased with a reduction in the number of petals for the staggered heave plates. Among the evaluated designs, the dual-layer flower-shaped staggered heave plates with three petals demonstrated the highest effectiveness in attenuating heave motion of the floating platform. The utilization of novel dual-layer flower-shaped staggered heave plates is therefore a promising practice aimed at damping the heave motion of platforms in rough seas.

1. Introduction

Heave plates are commonly incorporated into the design of Floating Offshore Wind Turbines (FOWTs) and other marine structures to mitigate the effects of vertical motion induced by waves, wind, and currents. These motions can lead to structural fatigue, increased maintenance costs, and reduced operational efficiency, hindering the development of renewable and sustainable energy sources [1]. In recent years, green energy represented by the offshore wind energy industry has been thriving both domestically and internationally [2]. In exploiting the vast wind energy resources in deep water, the floating offshore wind turbine (FOWT) is more competitive than the fixed offshore wind turbine [3]. FOWTs are mainly divided into four types, including semisubmersible FOWTs [4], tension leg platform (TLP) FOWTs [5], spar FOWTs [6], and barge FOWTs [7]. However, FOWTs are susceptible to significant oscillatory motions due to environmental loads, particularly in deep seas [8]. In response, the incorporation of heave plates at the base of FOWTs has been identified as an effective passive mechanism for augmenting added mass and damping, thereby mitigating the platform’s oscillatory motion [9].

Research into heave plates includes both model tests and numerical simulations. Extensive model testing has been conducted to elucidate the hydrodynamic performance of these devices. Tao et al. [10] conducted forced oscillation tests on perforated heave plates and demonstrated that that perforated heave plates can enhance damping at lower Keulegan–Carpenter (KC) numbers. Li et al. [11] investigated the hydrodynamic coefficients of heave plates through forced oscillation tests and found that with an increase in the KC number, the added mass coefficient increases. Moreover, greater plate spacing led to higher coefficients for both added mass and damping. Thiagarajan et al. [12] observed discrepancies in the added mass and damping coefficient of heave plates under ambient wave motion versus that under forced oscillation in still water and proposed a modified KC number to obtain reliable added mass and damping coefficients. Ezoji et al. [13] introduced the plate edges area-to-plate area ratio (Re/a) as a coefficient to study the hydrodynamic performance of heave plates. It was found that when decreasing Re/a to slightly above 0.1 (Re/a > 0.1), there was a noticeable enhancement in the damping coefficient. However, if the ratio was lowered further to below 0.1 (Re/a < 0.1), a reduction in the damping coefficients was observed in the majority of scenarios.

With the advancements in computer technology and computational resources, numerical analyses of heave plates have been facilitated in recent years. For example, Tao et al. [14] conducted numerical simulations on the forced oscillation of a cylinder with dual-layer heave plates. The results indicated that dual-layer heave plates with minimal spacing exhibit intense vortex shedding interactions, leading to reduced damping and added mass. Mendoza et al. [15] numerically simulated the forced oscillation of a heave plate at different heights above the impermeable seabed boundary and found that the proximity of a heave plate to the seabed augments both added mass and damping coefficients. Wang et al. [16] proposed perforated fractal heave plates and conducted forced oscillation simulations, revealing that with an increase in the fractal order, the added mass decreased while the damping coefficient initially increased and then decreased. The term ‘fractal order’ refers to the hierarchical structural complexity of the flower-shaped heave plates, where the number of petals corresponds to the degree of geometric self-similarity and spatial distribution patterns inherent to fractal geometry. This parameter quantifies the iterative branching or scaling behavior of the plate design, which directly influences hydrodynamic performance through surface area amplification and flow interaction mechanisms. Zhang et al. [17] investigated the hydrodynamic loads on circular heave plates by applying large-eddy simulations and found that the maximum dynamic pressure appeared at the center of the plate, gradually decreasing towards the outer regions. The hydrodynamic performance of heave plates with different chamfer angles of perforation was numerically analyzed by Wang et al. It is found that as the chamfer angle increased, the damping coefficient and added mass coefficient first decreased and then increased, with the optimal chamfer angle being 35 degrees [18].

Structural geometry variations in heave plates, i.e., thickness, spacing, and shape, have been the subject of significant experimental and numerical studies. Shen et al. [19] studied the hydrodynamic characteristics of heave plates with various form edges of the Truss Spar platform, and results showed that tapered heave plates performed better in hydrodynamic performance. Tian et al. [20] studied the hydrodynamic characteristics of different-shaped heave plates through forced oscillation tests and found that heave plates with sharp edges have greater added mass and damping compared to those with round corners. Tian et al. [21] concluded that nearly fully filled shapes like circles, octagons, and hexagons could provide more added mass than the less-filled shapes by comparing the results of circular and polygonal heave plates. Rusch et al. conducted forced oscillation experiments on hexagonal flat plates and hexagonal conic plates, indicating that the fluid reaction force produced by the flat plate was greater than that produced by the conic plate [22]. Zhang et al. studied damping coefficient of heave plates with different thicknesses through experiments and found that under the same KC number, thinner heave plates exhibit greater damping coefficients [23]. Chen et al. proposed a novel flower-shaped heave plate, and numerical results indicated that its damping coefficient was higher than that of a circular heave plate [24].

Heave plates serve as an effective tool in enhancing the damping of FOWTs. Recently, many researchers have studied the hydrodynamics of FOWTs with heave plates. Manuel et al. conducted forced oscillation tests and free decay tests on the column of a floating wind turbine platform with a heave plate and found that there existed a fair matching of results in the hydrodynamic coefficients obtained by these two methods [25]. Saettone et al. investigated a floating wind turbine platform with a heave plate at three different model scales [26]. Their findings revealed similarities in the velocity and vorticity fields, regardless of scale. This confirmed that the impact of the KC number on hydrodynamic coefficients was greater than the scale factor. Hegde et al. studied the hydrodynamic characteristics of heave plates on spar platforms, conducting free decay tests with a model scaled at 1:100. Their research demonstrated that the placement of a heave plate at the base of the spar platform significantly outperforms a configuration with the plate positioned near the free surface in terms of hydrodynamic efficiency [27]. Philip et al. expanded the understanding of heave plate dynamics by simulating the behavior of a spar platform with heave plates under wave conditions. Their simulations revealed that heave damping exhibits a nonlinear relationship with wave period and a linear correlation with wave height [28]. Jiang et al. introduced an innovative semisubmersible platform design, the HexaSemi, featuring heave plates. Under the coupled wind and wave conditions, the mooring tension of HexaSemi was reduced by 50% compared to the WindFloat platform, with heave and pitch motion responses decreased by over 30% [29].

The current heave plate designs of floating structures used in offshore wind turbines face several challenges and limitations. For example, the effectiveness heave plates in mitigating the motions induced by environmental loads varies under different sea conditions. Heave plates are effective in minimizing heave motion, but they may not be as effective in reducing surge and pitch motions. In addition, finding the optimal design to avoid resonance remains a challenge, which can trigger large amplitudes of motions. Based on the burgeoning interest in heave plates for optimizing the performance of FOWTs, there is a growing need in innovative heave plate designs to further improve their damping capabilities. This paper proposed novel types of cc-layer flower-shaped heave plates, including the aligned and staggered forms with different petals attached to the OC4 DeepCwind semisubmersible floating platform. Forced oscillation simulations of the OC4 platform with various types of heave plates under still water conditions were carried out to study hydrodynamic coefficients, including added mass and damping coefficient.

This paper is organized as follows: Section 1 is a detailed introduction to the present studies on heave plate designs and findings. Section 2 describes the methodologies employed in this study. Section 3 details the computational domain and boundary conditions, and mesh generation. Section 4 describes the verification and validation studies, and Section 5 presents the numerical research results with a detailed analysis. Finally, Section 6 summarizes the conclusions drawn from this investigation.

2. Methodologies

In this paper, the OC4 platform was selected for evaluating the hydrodynamic performance of a FOWT equipped with various heave plates. Detailed information regarding the OC4 platform [30] can be found in the work by Robertson et al. A sketch of the computational domain and geometry of the OC4 DeepCwind semisubmersible floating platform is shown in Figure 1 and Figure 2, respectively. Considering the trade-off between computational cost and the prevention of boundary effects, the length, width, and height of the computational domain were set to 6D, 6D, and 9D, where D is the diameter of the heave plate, based on experience [31].

The detailed dimensions of heave plates are presented in Figure 3. To the underside of the platform, dual-layer configurations of both circular and flower-shaped heave plates were incorporated. Dual-layer flower-shaped heave plates included both aligned and staggered forms. Taking the dual-layer flower-shaped aligned heave plates with three petals as an example, the prototype design was based on a circular shape with a diameter of D = 24.3 m. Subsequently, the center of each petal was positioned relative to the circular coordinates, specifically at x = −6.08 m and y = 10.52 m. Following this, a semi-circular segment with a radius of 10.52 m was circumscribed around each petal center. The final step involved the symmetrical arrangement of three petals, each rotated by 120 degrees, culminating in the completion of the heave plate’s schematic. This methodology was replicated to construct heave plates with six and nine petals.

Dual-layer flower-shaped staggered heave plates are designed by offsetting two layers of heave plates at a certain angular displacement. For both flower-shaped and circular heave plates, the thickness of one layer of heave plates was 0.1 m. The dual-layer heave plates were equipped to the bottom of the platform with the spacing between two layers of heave plates of 0.3 m. It should be noted that the spacing of dual-layer heave plates has been identified as a critical parameter affecting the damping of heave plates [14]. Therefore, a comparison analysis was conducted on the dual-layer flower-shaped staggered heave plates with three petals with another interlayer spacing of 5.9 m. The surface area and volume of dual-layer flower-shaped heave plates are the same as that of dual-layer circular heave plates.

2.1. Governing Equations

The Reynolds-averaged Navier–Stokes (RANS) equations for viscous incompressible fluids were employed as the governing equations in the present simulations. Assuming that flow variables consist of a time-averaged component and a fluctuating component, the incompressible RANS equations can be written as follows:

$${\displaystyle \frac{\partial \rho }{\partial t}}+\nabla ·\left(\rho \overline{V}\right)=0$$

(1)

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho \overline{V}\right)+\nabla ·\left(\rho \overline{V}⨂\overline{V}\right)=-\nabla ·\overline{p}I+\nabla ·\left(\overline{\tau }+{\tau }_{RANS}\right)+f_b$$

(2)

where $\rho$ is the density of fluid, $\overline{V}$ is the average velocity, $\overline{p}$ is the average pressure, $I$ is the unit tensor, $f_b$ is the resultant body force, and $\overline{\tau }$ is the average viscous stress tensor. The additional stress tensor ${\tau }_{RANS}$ is calculated by ${\tau }_{RANS}=2{\mu }_{t}S-2/3\left({\mu }_t\nabla ·\overline{V}\right)I$, in which $S$ is the mean strain rate tensor. The eddy viscosity, ${\mu }_t$, is defined as ${\mu }_t=\rho kT$, where $k$ is the kinetic energy and T is equal to ${\alpha }^{*}/\omega$ (${\alpha }^{*}$ is a coefficient and $\omega$ is the specific dissipation rate).

In this paper, the SST $k-\omega$ model was employed for turbulence modelling due to its ability to combine the strengths of both the $k-\epsilon$ and $k-\omega$ models. The SST $k-\omega$ model enhanced the solution accuracy, particularly for the fluids in the boundary layer region. The equations for the SST $k-\omega$ model are formulated as follows [32]:

$${\displaystyle \frac{\partial \left(\rho k\right)}{\partial t}}+\nabla ·\left(\rho k\overline{V}\right)=P_k+\nabla \left[\left(\mu +{\sigma }_k{\mu }_t\right)\nabla \mathrm{k}\right]-{\beta }^{\mathrm{*}}\rho \omega$$

(3)

$$\begin{array}{c}{\displaystyle \frac{\partial \left(\rho \omega \right)}{\partial t}}+\nabla ·\left(\rho \omega \overline{V}\right)=P_{\omega }-\beta \rho {\omega }^2+\nabla \left[\left(\mu +{\sigma }_{\omega }{\mu }_t\right)\nabla \omega \right]\\ +2\left(1-F_1\right){\displaystyle \frac{\rho {\sigma }_d}{\omega }}\nabla ·k\nabla ·\omega \end{array}$$

(4)

where $\rho$ is the density of fluid, $k$ is the kinetic energy, $\beta$ is the thermal expansion coefficient, $\mu$ is the dynamic viscosity, ${\mu }_t$ is turbulent eddy viscosity, ${\sigma }_k$, ${\sigma }_{\omega }$ and ${\sigma }_d$ are coefficients, ${\beta }^{\mathrm{*}}$ is the model coefficient, $F_1$ is a blend function, and $P_k$ and $P_{\omega }$ are the production terms calculated by the equation $P_k=P_{\omega }k/\gamma \omega$, where $\gamma$ is a coefficient.

These equations are discretized using the finite volume method (FVM), which is based on the Euler grids. The two-phase fluids are simulated and the volume of fluid (VOF) method is applied to capture the free surface. To ensure the accuracy of numerical computations, the spatial discretization employs the second-order upwind scheme. For the temporal discretization, the second-order implicit backward Euler is applied to time marching.

2.2. Forced Oscillation and Overset Mesh Technique

The prescribed motion of forced oscillation is imposed on the OC4 platform with heave plates. The equation of heave motion is simplified as follows:

$$a=A\sin \left(wt\right)$$

(5)

where $a$ is the displacement of the heave motion, $t$ is the movement time, $A$ is motion amplitude, and $w=2\pi f$ is the circular frequency of motion. $f$ is the frequency in Hz. In this study, the values of the amplitude, $A$, and the angular frequency, w, correspond to the power-generation operating conditions of floating offshore wind turbines reported in the previous work [24]. Specifically, the spectral peak period of the wave spectrum under rated-power conditions was adopted as the prescribed period of motion for the OC4 platform, and the heave amplitude, $A$, was chosen as the significant response amplitude of the wind-turbine foundation in heave motion. Consequently, the amplitude of vertical displacement (denoted as $A$) is set to 1.75 m, and the angular frequency ($w$) is assigned a value of 0.8378 rad/s, corresponding to a motion period (T) of 7.5 s [24].

The body motions with moving boundaries can be commonly simulated based on the dynamic mesh techniques. However, there still remains challenges in treating the large amplitude body motions when the dynamic mesh is deformed to be distorted. The dynamic overset mesh technique is a promising tool for heave plate simulations since this technique does not require mesh deformation or remeshing for large body motions or complex geometries [33]. The concept of the overset mesh technique involves partitioning each component of the computational model into individual parts, which are then overlapped together to form the computational domain. The data exchange in the overset mesh region can be performed using interpolation schemes, thereby achieving the solution of moving boundaries.

2.3. Hydrodynamic Coefficients

The expression of the vertical forces on the floating platform can be fitted and formulated as follows:

$$F=F_0\sin (wt+\phi )$$

(6)

where $F$ is the vertical force, $t$ is the movement time, $F_0$ is the force amplitude, and $\phi$ is the phase difference, respectively. $F_0$ can be obtained based on the vertical force curve. Matlab’s Curve Fitting Tool, cftool, is employed to fit the force curve and obtain $\phi$. Equations (5) and (6) can be rewritten as follows:

$$F_0\sin (wt+\phi )=-{\displaystyle \frac{F_0\cos \left(\phi \right)}{A{w}^2}}\ddot{a}+{\displaystyle \frac{F_0\sin \left(\phi \right)}{Aw}}\dot{a}$$

(7)

In addition, the free motion equation can be written in a form as follows:

$$A_m\ddot{a}+B_d\dot{a}+F=0$$

(8)

where $A_m$ represents the added mass and $B_d$ stands for the damping coefficient. $F$ represents the vertical force. Finally, the added mass and damping coefficient can be obtained as follows:

$$B_d=-{\displaystyle \frac{F_0\sin \left(\phi \right)}{Aw}}$$

(9)

$$A_m={\displaystyle \frac{F_0\cos \left(\phi \right)}{A{w}^2}}$$

(10)

3. Computational Settings

3.1. Computational Domain and Boundary Conditions

The size of the computational domain is 6D in length, 6D in width, and 9D in height, where D equals 33 m, corresponding to the diameter of the flower-shaped heave plates, each with six petals. The domain involves a water depth of approximately 200 m and an air height of 100 m. A dedicated CFD sensitivity study confirmed that further enlargement of the domain produces negligible variations in the predicted quantities. Consequently, the forced-motion problem does not necessitate an extended domain. Nestled within this domain is an overset region, which can cover the OC4 platform equipped with heave plates, spanning 2.52D in length, 2.82D in width, and 1.15D in height, centrally located within the background domain.

The boundary conditions of the computational domain are defined as follows. The left side and top of the background are set as fixed velocity inlets with an inlet velocity of 0 m/s. The right side of the background is defined as a pressure outlet and the relative pressure is 0 Pa. Both the bottom of the background domain and the OC4 platform, including the heave plates, are treated as wall boundaries. It should be noted that the OC4 platform with heave plates can generate radiated waves during forced oscillation. To prevent wave reflection at the boundaries of the computational domain, 6 m dissipative regions are strategically placed along the left, right, front, and back sides of the domain for wave damping.

The wave probe, the pressure probe, and the velocity probe are shown in Figure 4. A radiated wave probe was positioned on the free surface adjacent to a column of the OC4 platform to monitor the instantaneous wave elevation of radiated waves. Furthermore, a velocity probe is placed between two layers of heave plates to measure the velocity of the flow field and the pressure probe is set on the surface of the heave plates to measure the pressure exerted on the surfaces of the heave plates.

3.2. Mesh Generation

Hexahedral cells were generated by the trimmed mesher tool in STAR-CCM+ for both the background region and the overset region. The mesh generation and detailed settings are shown in Figure 5. The mesh in the background was refined in three levels, with a cell size of 5 m for the outermost layer. The cell size in the overset region was set as 0.65 m. Additionally, a refinement zone within the background, 6 m in height and with a cell size of 0.65 m, is designed to accommodate the forced oscillation of the OC4 platform equipped with heave plates. To ensure the accuracy of simulation, the cell size on the body surface was 0.4 m. The dimensionless distance from the wall, represented by y⁺, was set to approximately 30, and the thickness of boundary layers was estimated as 0.147 m. The value was obtained from the classical solution for a flat plate (Blasius solution), in which the thickness δ is defined as the wall-normal distance where the local stream-wise velocity reaches 99% of the free-stream value. Oscillatory effects were neglected because the estimation employed the maximum free-stream velocity as a conservative upper bound, effectively treating the flow as quasi-steady.

To precisely capture the radiated waves, both the stationary and overset regions incorporate free surface mesh refinement areas with a cell size of 0.1 m.

4. Verification and Validation Studies

4.1. Mesh and Time Step Sensitivity Studies

The case of the dual-layer flower-shaped staggered heave plates with six petals was chosen to conduct mesh sensitivity and time step sensitivity studies. The maximum heave amplitude ($A$) is set to be 1.75 m and the forced oscillation period ($T$) is 7.5 s. Three levels of meshes are utilized: fine (12.26 million cells), medium (9.24 million cells), and coarse (5.81 million cells), all with a time step of 0.01 s. The cell size in the free surface mesh refinement region remains consistent across these meshes, while the background and overset regions vary, adjusted by a ratio of $\sqrt{2}$.

Comparisons of the radiated waves and the vertical forces for these three different meshes are shown in Figure 6. It can be observed that there is no significant difference in the vertical forces and the radiated waves among three meshes, with a discrepancy in the amplitude within 1%, which indicates that the simulation was converged. Considering the computational accuracy, the medium mesh was adopted in the following studies.

For the time step sensitivity analysis, the same mesh with four different time steps, i.e., $\Delta t$ = 0.005 s, 0.01 s, 0.015 s, and 0.02 s, was applied. The corresponding Courant numbers, defined as $C=V\Delta t/\Delta x_{min}(\Delta x_{min}=0.1$ is the minimum cell size), were 0.07, 0.15, 0.22, and 0.30, respectively. Comparisons of the vertical forces and the radiated waves for four different time steps are shown in Figure 7.

The amplitude errors of the radiated waves at $t$ $t$ = 7.1 s are presented in

Table 1. The errors of the radiated wave amplitude ($H_i$) for using different time steps.

Time Step	${\mathit{H}}_{\mathit{i}}$ (m)	$\left|{(\mathit{H}}_{\mathit{i}}-{\mathit{H}}_{\mathit{i}+1})\right|/{\mathit{H}}_{\mathit{i}+1}$
TS1	0.6223	-
TS2	0.5741	7.735%
TS3	0.5486	4.450%
TS4	0.5471	0.003%

. Note that TS1, TS2, TS3, and TS4 represented $\Delta t=$0.02 s, 0.015 s, 0.01 s, and 0.005 s, respectively. The simulation results revealed that the vertical forces in the time step sensitivity studies show good agreement, with errors not exceeding 1%. However, there was some discrepancy in the radiated wave amplitudes. Upon examining the amplitude at 7.1 s, it can be concluded that the error decreased as the time step decreased and the result converged at TS3 = 0.01 s. Therefore, a time step of 0.01 s was selected for subsequent studies.

4.2. Validation Studies

Due to the limited availability of forced oscillation experiments on the OC4 platform, a numerical simulation [34] of free decay motion of 2 m on the OC4 platform was chosen for validation. The results were obtained by the inhouse code DARwind based on potential flow theory. The OC4 platform was initially displaced vertically by 2 m from its equilibrium position in still water and then released to undergo free decay motion in the heave direction with a mooring system.

The calculation method for damping coefficient in free decay motion $\left(B_{df}\right)$ differed from that applied in forced oscillation, which was determined by the logarithmic decrement method [35]. The equations of $B_{df}$ in the free decay motion are written as follows:

$$\delta ={\displaystyle \frac{1}{n}}\ln {\displaystyle \frac{x_0}{x_n}}$$

(11)

$$B_{df}={\displaystyle \frac{1}{\sqrt{1+{\left(\frac{2\pi }{\delta }\right)}^2}}}$$

(12)

where $x_0$ and $x_n$ represent the initial peak amplitude and the peak amplitude after n cycles. To compare damping coefficients of free decay motion and forced oscillation, it is essential to determine the undamped natural period ($T_n$) of the OC4 platform. $T_n$ can be obtained from the damped natural frequency (${\omega }_d$), derived as follows.

$$w_d={\displaystyle \frac{2\pi }{T_d}}$$

(13)

$$w_n={\displaystyle \frac{w_d}{\sqrt{1-B_{df}^2}}}$$

(14)

$$T_n={\displaystyle \frac{2\pi }{w_n}}$$

(15)

where $T_d$ refers to the damped natural period, while $w_n$ is the undamped natural frequency.

In this paper, a simulation of a free decay test on the OC4 platform was conducted by using STAR-CCM+ (17.04.008-R8) software with a physical time span of 104.75 s. The $T_n$ obtained from the free decay simulation result was utilized as the forced oscillation period (T), and the average amplitude of free decay was used as the maximum heave amplitude ($A$) to perform forced oscillation simulations. It should be noted that $B_d$ of forced oscillation should be non-dimensionalized as $B_{dn}$ ($B_{dn}=B_d/2m{w}_n)$. Figure 8 shows the simulation results of the free decay test. The comparison of the $B_{df}$ obtained from STAR-CCM+ and DARwind and the $B_{dn}$ calculated by STAR-CCM+ are presented in

Table 2. Comparison of damping coefficients.

Type of the Motion	Damping Coefficient
Free decay (STAR-CCM+)	0.0440
Free decay (DARwind)	0.0389
Forced oscillation (STAR-CCM+)	0.0400

. From Figure 8, it can be observed that the simulation results of the free decay motion given by STAR-CCM+ agreed well with the values given by DARwind. From Table 2, it can be observed that the discrepancy between the $B_{df}$ and $B_{dn}$ can be acceptable. This suggests that the damping coefficients for both free decay and forced oscillation were similar, indicating that they can both accurately represent the hydrodynamic performance. As presented in Table 2, quantitative comparison between STAR-CCM+ simulations and DARwind reference data demonstrates a high degree of consistency, with relative error remaining within 2.8% across all evaluated parameters. Thus, the simulation of forced oscillation was validated and deemed reliable.

5. Results and Discussions

In this section, the hydrodynamic performance of the OC4 platform equipped with various types of heave plates is presented and analyzed. One degree-of-freedom translation was focused on; therefore, the body motions were fixed except the heave motion. The simulations were conducted under several conditions: (1) the environment was calm water, with no wind and waves; (2) the OC4 platform fitted with heave plates was initially positioned at its balance position; (3) the simulations were conducted for a duration of over six motion cycles, allowing for the full evolution of the flow field. The heave amplitude ($A$) is set as 1.75 m and the period (T) of the oscillation is 7.5 s. The time series of the displacement and velocity of the forced oscillation are shown in Figure 9.

5.1. Vertical Force and Hydrodynamic Coefficients

Damping of the OC4 platform with heave plates is mainly attributed to three factors involving the frictional damping due to water viscosity, wave-induced damping resulting from the radiated waves generated by the heave motion, and vortex damping caused by the formation of vortices near the heave plates which dissipates energy. Based on Equations (6)–(10), the added mas, $A_m$, and damping coefficient, $B_d$, are significantly associated with the vertical force during forced oscillations. To understand the damping characteristics, the time series of the vertical forces of the OC4 platform with various types of heave plates are compared and shown in Figure 10.

As shown in Figure 10, the vertical force curves appear as period change featured with non-linear characteristics, which could be caused by the wave–body interactions during the forced oscillation of the OC4 platform with heave plates. This could suggest that the addition of heave plates to the OC4 platform led to a significant increase in $F_0$, consequently resulting in increased $B_d$ and $A_m$. It can be observed that $F_0$ was greater for flower-shaped staggered heave plates compared to flower-shaped aligned and traditional circular heave plates. For those flower-shaped staggered heave plates, as the number of petals decreased, $F_0$ also increased. For the aligned configuration, $F_0$ for the flower-shaped aligned heave plates with six petals was very close to that with nine petals, differing by only 0.87%. However, $F_0$ for the flower-shaped aligned heave plates with three petals was larger than that with six petals and nine petals. For the heave plates with a spacing of 0.3 m, it can be observed that $F_0$ was maximized for the flower-shaped staggered heave plates with three petals. In addition, as the spacing between the two-layer heave plates increased to 5.9 m, $F_0$ for the flower-shaped staggered heave plates with three petals also increased.

Table 3. Comparison of the phase differences of vertical force for the OC4 platform.

Types of Heave Plates	Number of Petals	Spacing of the Two Layers (m)	$\mathit{\phi }(°)$
Without heave plates	-	-	20.69°
Circular type	-	0.3	28.63°
Flower-shaped Aligned type	3	0.3	26.56°
Flower-shaped Aligned type	6	0.3	25.21°
Flower-shaped Aligned type	9	0.3	26.00°
Flower-shaped Staggered type	3	0.3	29.82°
Flower-shaped Staggered type	6	0.3	31.43°
Flower-shaped Staggered type	9	0.3	31.19°
Flower-shaped Staggered type	3	5.9	29.82°

presents the phase, $\phi$, of the vertical force curves for the OC4 platform with various types of heave plates. From Table 3, it is observed that the addition of heave plates results in smaller $\phi$ compared to the case without heave plates. Heave plates with a spacing of 0.3 m and 5.9 m have the same $\phi$. Once $F_0$ and $\phi$ are obtained, the $B_d$ and $A_m$ can be calculated using Equations (9) and (10). Comparisons of the $A_m$ and $B_d$ of the OC4 platform with various types of heave plates are presented in

Table 4. Comparison of the damping coefficient and added mass of the OC4 platform.

Types of Heave Plates	Number of Petals	Spacing of the Two Layers (m)	${\mathit{B}}_{\mathit{d}}(\mathit{N}·\mathit{s}·{\mathit{m}}^{-1})$	${\mathit{B}}_{\mathit{d}}\left(\mathit{k}\mathit{g}\right)$
Without heave plates	-	-	3.5027 × 106	1.1074 × 107
Circular type	-	0.3	1.0388 × 107	2.2719 × 107
Flower-shaped Aligned type	3	0.3	9.3816 × 106	2.2405 × 107
Flower-shaped Aligned type	6	0.3	8.6621 × 106	2.1974 × 107
Flower-shaped Aligned type	9	0.3	8.8397 × 106	2.1639 × 107
Flower-shaped Staggered type	3	0.3	1.3557 × 107	2.8231 × 107
Flower-shaped Staggered type	6	0.3	1.3156 × 107	2.5699 × 107
Flower-shaped Staggered type	9	0.3	1.2498 × 107	2.4646 × 107
Flower-shaped Staggered type	3	5.9	1.3796 × 107	3.1857 × 107

.

As shown in Table 4, the results of the damping coefficient and added mass increased when the OC4 platform was equipped with any heave plates. Flower-shaped staggered heave plates exhibited larger $B_d$ and $A_m$ compared to flower-shaped aligned and traditional circular heave plates. For the heave plates with a spacing of 0.3 m, the $B_d$ of the flower-shaped staggered heave plates with three, six, and nine petals was increased by 30.51%, 26.64%, and 20.31%, respectively, compared to the case of circular heave plates. Additionally, the corresponding $A_m$ was increased by 24.26%, 13.12%, and 8.48%, respectively. This can be attributed to the larger projected area in the heave direction for flower-shaped staggered heave plates compared to flower-shaped aligned and circular heave plates. In terms of the flower-shaped staggered heave plates types, as the number of petals decreased, the results of $B_d$ and $A_m$ increased. From the results of the flower-shaped staggered heave plates with three petals in Table 4, it can be seen that for two-layer heave plates with a spacing of 5.9 m compared to a spacing of 0.3 m, $B_d$ and $A_m$ increased by 1.76% and 12.84%, respectively.

For the proposed flower-shaped aligned heave plates, despite having the same projected area in the heave direction as circular heave plates, their more complex structure led to more intricate interactions between the two heave plates, reducing energy dissipation and consequently diminishing $B_d$ and $A_m$.

5.2. Radiated Waves Due to Heave Motion

The OC4 platform with various types of heave plates generates radiated waves due to heave motion, resulting in energy dissipation and damping generation. The time series of the radiated wave elevations for the OC4 platform with various types of heave plates are shown in Figure 11. The addition of heave plates to the OC4 platform resulted in a higher amplitude of the radiated waves compared to the case without heave plates. With the continuous progression of forced oscillation, the amplitude of the radiated waves exhibited an increasing trend. In addition, it can be observed that the average amplitudes of flower-shaped staggered heave plates were greater than those of flower-shaped aligned and traditional circular heave plates. However, the average amplitudes of flower-shaped aligned heave plates were smaller than those of traditional circular heave plates. This could be attributed to the complex interactions between the aligned flower-shaped heave plates and the waves and subsequent reduced energy dissipation.

It can be found that the radiated wave amplitude of the heave plates with a spacing of 5.9 m was greater than that of heave plates with a spacing of 0.3 m. It also made sense that the $B_d$ of heave plates with a spacing of 5.9 m was greater than that of heave plates with a spacing of 0.3 m.

5.3. Local Flow Fields Adjacent to Heave Plates

The evolution of the local flow fields was considered as an important factor which was associated with the dissipation of energy. The time series of the velocity at the velocity probe between two layers of heave plates (spacing = 0.3 m) are shown in Figure 12, and the amplitude results of the FFT analysis are presented in

Table 5. FFT analysis results of the velocity probe.

Types of Heave Plates (Spacing = 0.3 m)	Number of Petals	Amplitude (m/s)
Circular type	-	1.3232
Flower-shaped Aligned type	3	1.3222
Flower-shaped Aligned type	6	1.3155
Flower-shaped Aligned type	9	1.3174
Flower-shaped Staggered type	3	1.3948
Flower-shaped Staggered type	6	1.4101
Flower-shaped Staggered type	9	1.4282

. It should be noted that the velocity probe recorded the velocity in the heave direction. Figure 12 reveals that velocities obtained from different velocity measurement points show minimal differences. It is shown that the amplitudes of the velocity for flower-shaped staggered heave plates were larger than those of circular heave plates and flower-shaped aligned heave plates. Moreover, the amplitudes of the velocity for flower-shaped aligned heave plates were very close to those of circular heave plates, with a difference of less than 2%.

According to Table 5, the energy dissipation of flower-shaped staggered heave plates were higher than that of flower-shaped aligned heave plates and circular heave plates. Moreover, the energy dissipation of flower-shaped aligned heave plates was lower compared to circular heave plates. The numerical results indicate a stronger damping for flower-shaped staggered heave plates and relatively weaker damping for flower-shaped aligned heave plates compared to circular heave plates, which was consistent with the conclusion presented in Section 5.1. In terms of this velocity probe, the energy dissipation of various flower-shaped staggered heave plates was very close, differing by only 2.4% among them.

Another essential factor for energy dissipation and hydrodynamic damping is the shedding vortices created by heave plates. A previous study [34] indicated that for a platform with heave plates for forced oscillation, the effect of radiated wave damping was relatively small and the system’s damping was primarily due to the vortex system, which was generated by heave plates during its motion. The more vortices were generated, the greater the energy dissipation and the better the damping performance of the system that will be obtained. The study systematically examined four configurations: the conventional circular heave plates, the baseline case without plates, the three-petal flower-shaped plates in an aligned arrangement, and the three-petal flower-shaped plates in a staggered arrangement, all set at a uniform spacing of 0.3 m. The vorticity field of the flower-shaped staggered heave plates with three petals with a spacing of 5.9 m was also analyzed. Vorticity magnitudes around the OC4 platform with various types of heave plates (spacing = 0.3 m) for one cycle of forced oscillation (35.6 s–43.1 s) are shown in Figure 13. The snapshots, taken at t = 35.6 s, 37.5 s, 39.4 s, 41.3 s, and 43.1 s, capture the object at the start of the fifth forced-oscillation cycle, at mid-height while ascending, at the upper turning point, at mid-height while descending, and at the lower turning point, respectively. Among the figures, the red arrow represents the direction of the body motion.

As shown in Figure 13a–e, for the case without heave plates, vortices are generated at the two sharp corners of a column bottom. Based on Figure 13f–t, it can be observed that with various types of heave plates, vortices primarily are generated along the edges of the heave plates and in the gaps between two heave plates. The vortex generation during one forced oscillation cycle followed a certain pattern. For instance, with circular heave plates, it can be observed that at t = 35.6 s, vortices started to form upwards along the edges of heave plates, coexisting with vortices from the previous cycle above the plates. At t = 37.5 s, the vortices grew in size and took an oval shape. At t = 39.4 s, when heave plates reached their highest position, the previously formed oval vortices detached, and new vortices formed on the downward side. The vortex generation at t = 41.3 s was similar to that at t = 37.5 s, with the direction of motion reversed. In addition, the vortex generation at t = 43.1 s is similar to that at t = 35.6 s.

Compared to circular heave plates, the flower-shaped aligned heave plates with three petals exhibited smaller oval vortices above it at t = 35.6 s, and thus its $B_d$ is lower than that of circular heave plates. From Figure 13p–t, it can be observed that compared to the flower-shaped aligned heave plates with three petals and circular heave plates, the flower-shaped staggered heave plates with three petals generated more vortices in the spacing of two heave plates. Additionally, there were also more detached vortices on the left side. This indicates that $B_d$ of the flower-shaped staggered heave plates with three petals was higher than that of the flower-shaped aligned heave plates with three petals and circular heave plates. The vorticity fields around the OC4 platform with the dual-layer flower-shaped staggered heave plates with three petals (spacing = 5.9 m) for one cycle of forced oscillation (35.6 s–43.1 s) are shown in Figure 14.

It can be observed that the main vortices were generated and shed from the edges of the two layers of heave plates as well as from the spacing between the heave plates. When the spacing of heave plates increased to 5.9 m, the interaction between two heave plates was smaller during the shedding of vortices. This resulted in more vortices being generated and shed by heave plates with a spacing of 5.9 m compared to heave plates with a spacing of 0.3 m, leading to a greater energy dissipation and subsequently resulting in a larger value of $B_d$.

Figure 15 presents the time series of the pressure probe detected on the OC4 platform with various types of heave plates. The pressure probe was initially located underwater at a depth of 20.25 m, with a water pressure of 198,450 Pa. The cases of the flower-shaped aligned heave plates with three petals and circular heave plates exhibited two peak values in their curves, while the curves of other heave plates existed only one peak value. The case of flower-shaped staggered heave plates indicated higher pressure amplitudes compared to both flower-shaped aligned and circular heave plates, necessitating further consideration of their structural strength.

Figure 16 shows the pressure contours of the OC4 platforms with various types of heave plates at t = 46.88 s. It can be observed that the pressure on OC4 platforms with various types of heave plates increased from the top to the bottom. Without heave plates, the maximum pressure was primarily obtained at the bottoms of the columns of the OC4 platform. However, with the addition of heave plates, the maximum pressure was transferred to these plates. Additionally, there was a high-pressure area between the two heave plates of the flower-shaped staggered heave plates with three petals (spacing = 0.3 m). After adding the two-layer heave plates with a spacing of 5.9 m, the maximum pressure occurred on the lower heave plate.

6. Conclusions

In this paper, novel types of dual-layer flower-shaped aligned and staggered heave plates were proposed and their hydrodynamic performances were comparatively studied through comprehensive numerical simulations. Forced oscillation simulations were performed for the OC4 DeepCwind semisubmersible platform under three conditions: (1) baseline without heave plates, (2) equipped with dual-layer circular heave plates, and (3) integrated with dual-layer flower-shaped heave plates (both aligned and staggered configurations). The principal findings can be summarized as follows:

The addition of heave plates, especially the flower-shaped staggered configurations, significantly increased the added mass $A_m$ and damping coefficient $B_d$ of the OC4 platform. This enhancement is attributed to the larger projected heave-plane area of the staggered configuration, which yielded increases of up to 30.5% in $B_d$ and 24.3% in $A_m$ relative to conventional circular plates and substantially outperformed the aligned variant. However, flower-shaped aligned configurations have lower $B_d$ and $A_m$ compared to circular heave plates. It was primarily due to the flow interactions between the two layers of heave plates, which suppressed vortex shedding and thereby diminished energy dissipation efficiency. Additionally, increasing the spacing between the two layers from 0.3 m to 5.9 m can result in an increased vertical force.

The petal configuration and spacing between dual-layer heave plates were found to be two key factors affecting the hydrodynamic performance. For flower-shaped staggered heave plates, a reduction in the number of petals results in higher added mass and damping coefficients. Additionally, increasing the spacing between the dual-layer heave plates improves damping performance. Specifically, the flower-shaped staggered heave plates with three petals and a spacing of 5.9 m demonstrated the most effective damping, indicating that larger spacing allows for greater vortex shedding and energy dissipation.

The results show the importance of vortex interactions in enhancing the damping characteristics of heave plates. Flower-shaped staggered heave plates generated more vortices and shed them more effectively than other configurations, leading to higher energy dissipation and improved damping performance.

The average amplitude of radiated waves of flower-shaped staggered heave plates were larger than that of traditional circular heave plates and flower-shaped aligned heave plates. As the spacing between two layers of heave plates increased, the amplitude of the radiated wave also increased. Flower-shaped staggered heave plates exhibited higher pressure amplitudes compared to aligned and circular plates, suggesting that structural strength should be considered in their design.

In summary, this study demonstrates that dual-layer flower-shaped heave plates—particularly staggered configurations—significantly enhance the added mass and damping coefficients of FOWT platforms. Among the various configurations, the flower-shaped staggered heave plate with three petals (spacing = 5.9 m) demonstrated the optimum performance improvements. However, this study focuses on the platform’s dynamic response under heave motion and does not account for the effects of the other five degrees of freedom (DoF). Future work should focus on the coupled six-DoF motions including the pitch and heave of FOWTs equipped with the proposed heave plates to provide a more comprehensive understanding of their hydrodynamic characteristics. Additionally, while the current work prioritizes comparative analysis of the hydrodynamic performance between the designed flower-shaped heave plate and the conventional circular heave plate to evaluate potential performance improvements, manufacturing costs are not included in the present scope. A comprehensive assessment incorporating economic factors will be addressed in subsequent studies.