Assessment of Hydrodynamic Performance and Motion Suppression of Tension Leg Floating Platform Based on Tuned Liquid Multi-Column Damper

Abstract

To address the unstable motion of a tension leg platform (TLP) for floating wind turbines in various sea conditions, an improved method of incorporating a tuned liquid multi-column damper (TLMCD) into the TLP foundation is proposed. In order to evaluate the control effect of TLMCD on the motion response of the floating foundation, a multiphase flow solver based on a viscous flow CFD method and overlapping grid technique is applied to model the coupled multi-body dynamics interaction problem involving liquid tanks, waves, and a spring mooring system. This method has been proven to accurately capture the high-frequency motions of the structure and account for complex viscous interferences affecting the geometric motions. Additionally, the volume of fluid (VOF) method and the first-order linear superposition method are used to model the focused wave, enabling a simulation of the effects of transient wave loads on the floating foundation. The results show that the tuned damping effect of TLMCD on the TLP is mainly in the pitch motion, with the maximum pitch amplitude control volume ratio of TLMCD reaching up to 86% and the maximum surge amplitude control volume ratio of TLMCD reaching up to 25.2% under the operating conditions. These findings highlight the potential for additional research on and implementation of TLMCD technology.

1. Introduction

In recent years, wind power has emerged as the fastest-growing type of renewable energy globally due to its significant energy storage capacity, environmental cleanliness, and numerous additional advantages, positioning it as an essential element in the pursuit of new energy sources. While the exploitation of shallow sea resources approaches saturation, deep sea wind energy resources are plentiful. The deployment of floating offshore wind turbines as a vital technical solution for the development of wind energy in deep and remote seas has become a focal point in current offshore wind energy research [1,2]. Currently, the floating foundations in floating wind turbine design can be categorized into three types: semi-submersible (SEMI), tension leg platform (TLP), and Spar. Compared to the semi-submersible and Spar types, the large tension force in the tension leg imposes strong constraints on platform displacement, thereby enhancing the potential of the TLP floating foundation for deep-water offshore wind energy development [3,4].

Suzuki et al. [5] carried out a conceptual design of TLP for offshore wind turbines and performed a numerical analysis of the dynamic response and tension deviation of the tension leg in waves, the natural frequency of vibration, and the dynamic response to seismic loading. The results revealed that under the wind, current, and wave conditions, the maximum tendon tension was significantly lower than the tendon breaking strength, providing an adequate safety margin. Further, Matha [6] presented the development process of the TLP model and validated the MIT/NREL TLP design concept by comparing it with the frequency domain calculation results. Comprehensive analyses were conducted for ultimate loads and fatigue loads. Furthermore, Wu et al. [7] predicted the complex short-period motion response characteristics of a tension leg platform through experiments and numerical modeling. They analyzed the effects of platform dimensions, weight, damping and loading conditions, boundary layer separation, and other influencing factors. The study established platform parameter constraint domains based on the influencing law and the short-period motion sensitive region to guide the design of the platform.

In the exploration of deep-water wind energy, floating offshore wind turbines (FOWTs) encounter a harsh and complex external environment. The combined effects of structural resonance response and extreme wave conditions can result in severe, high-frequency translational and rotational motions in FOWTs [8], leading to increased system downtime, turbine performance degradation, and potential damage to system components, including moorings and anchors. To mitigate the risk of high-frequency motions, the application of passive motion control devices such as tuned mass dampers (TMDs), tuned liquid dampers (TLDs), and tuned liquid column dampers (TLCDs) has been expanded from the traditional civil engineering to the offshore structures [9,10].

In the realm of coupling TMD with FOWT, the focus lies in controlling high-frequency, small-amplitude vibrations. Murtagh et al. [11] proposed a tuned mass damper (TMD) placed at the top of the tower to mitigate the along-wind forced vibration response of wind turbines. Similarly, Si [12] installed a TMD in the Spar platform and conducted a fully coupled aero-hydraulic servo elastic analysis, demonstrating that positioning the TMD on the higher side of the Spar platform exerted a more significant damping effect. Furthermore, Yang et al. [13] employed a frequency tuning method and genetic algorithm to adjust the TMD of the same wind turbine model, achieving a better motion suppression rate through evolutionary techniques. Additionally, Verma et al. [14] proposed a framework for optimizing the TMD, where the stiffness and damping parameters of the TMD were optimally adjusted. The study analyzed the nonlinear response of the FOWT to assess the effectiveness of the TMD in response to various loads.

Compared to TMDs, TLDs have a simpler design process, offering the advantages of low maintenance, easy installation, and zero energy input [15], and are highly effective in suppressing small amplitude vibrations in slender wind turbine structures [16]. Ha et al. [17] found that installing a TLD on the top of the Spar-type FOWT can improve the pitch control effect of the floating foundation and reduce the vibration response of the FOWT. Coudurier et al. [18] incorporated the water contained in TLCD into the dynamics, considering the interaction between floating (no fixed point) and TLCD. In addition, the application of TLD on the TLP has been investigated. For example, V. Jaksic et al. [19] conducted an experimental study comparing the effects of three combinations of multi-tuned liquid column dampers (MTLCDs) on the dynamic performance of TLP structures, demonstrating the effectiveness of the MTLCDs in reducing the motion of the TLP-type FOWT platforms and positively affecting the tensions sustained by the mooring cables. Furthermore, Park et al. [20] developed an orthogonal type TLCD(s) for a tension leg platform, with the simulation results confirming the effectiveness of orthogonal TLCDs in reducing front and side fatigue and ultimate loads.

In order to enhance the damping performance of the TLD, it was applied to the motion suppression in FOWT with a semi-submersible support platform. Coudurier et al. [21] proposed a damping and motion suppression device based on TLMCD interconnection, installed at the bottom of the FOWT. By establishing equations based on the theory of coupled motion of the floater, they found that TLMCD was more effective in controlling both pitch and roll motions of the floating body compared to traditional TLCD. Specifically, Zhou [22] validated an OpenFOAM-based CFD model and applied it to the internal sway of a TLMCD system under prescribed pitch motion, coupled with a semi-submersible FOWT in waves. The initial findings indicated that TLMCD can significantly reduce the pitch motion of the FOWT when tuned to the pitch resonance frequency of the float. Moreover, Xue et al. [23] applied a similar numerical model to verify the anti-pitch motion performance of a substructure-scale model of a semi-submersible FOWT equipped with TLMCD in regular waves, demonstrating that a TLMCD with a mass ratio of 2.0% and a tuning ratio of 1.0 effectively controls pitch motion, particularly near the resonance period, reducing the maximum pitch motion amplitude by approximately 18%.

In consideration of the above, there are studies that can prove the effectiveness of TLMCD in suppressing the motion of floating structures, which mainly focus on large semi-submersible platforms, such as OC4, and there is a lack of TLMCD studies applied to TLPs. However, compared with large semi-submersible platforms, such as OC4, TLPs are very different in terms of the natural period characteristics, especially their pitch motions, with a short period of time. Moreover, the stability of the coupled system in real sea conditions has rarely been considered in previous TLMCD studies. In this paper, four types of TLMCD with different natural periods are designed by combining the spatial distribution characteristics of the TLP. A physical model of TLMCD coupled with a floating body is constructed, and a three-dimensional model capturing the motion response of the floating body to liquid tanks, waves, and spring moorings is established using the Siemens software STAR-CCM+ (Version 15.06). This model aims to examine the impact of TLMCD on the motion response of a floating foundation under different wave excitations. Furthermore, the motion suppression performance of TLMCD on the coupling of the floating foundation is assessed with regard to hydrodynamic load, structural motion response, and energy dissipation. This analysis provides important data support and a theoretical foundation for optimizing the damping and vibration reduction equipment.

2. Numerical Method

The incompressible fluid motion is modeled using the incompressible Navier–Stokes (N-S) equations in the Cartesian coordinate system. The continuity equation and the N-S equation for the continuous, incompressible, unsteady fluid are as follows:

$$\frac{𝜕u_i}{𝜕x_i}=0$$

(1)

$$\frac{𝜕u_i}{𝜕t}+\frac{𝜕}{𝜕x_j}(u_i{u}_j)=-\frac{1}{\rho }\frac{𝜕p}{𝜕x_i}+v\frac{𝜕}{𝜕x_j}\left(\frac{𝜕u_i}{𝜕x_j}+\frac{𝜕u_j}{𝜕x_i}\right)$$

(2)

where $u_i$ represents the components of velocity in the x, y, and z directions, respectively; $x_i$ indicates the coordinates in the x, y, and z directions, respectively; $v$ is the kinematic viscosity coefficient; ρ is the density of fluid; and p is the pressure.

To effectively close the differential control equation and facilitate an efficient solution, it is essential to introduce a turbulence model. Specifically, this study adopts the SST k-ω turbulence model developed by Menter [24], which combines the advantages of the k-ω model and the k-ε model. The k-ε model is employed in the free shear regions, while the k-ω model is utilized in the complex near-wall regions. The SST k-ω turbulence model can be expressed as

$$\frac{𝜕\left(\rho k\right)}{𝜕t}+\frac{𝜕\left(\rho u_{i}k\right)}{𝜕x_i}=P-{\beta }^{\ast }\rho k\omega +\frac{𝜕}{𝜕x_i}\left[\left(\mu +{\sigma }_k{\mu }_t\right)\frac{𝜕k}{𝜕x_i}\right]$$

(3)

$$\frac{𝜕\left(\rho \omega \right)}{𝜕t}+\frac{𝜕\left(\rho u_i\omega \right)}{𝜕x_i}=\frac{\gamma }{{\nu }_t}P-\beta \rho {\omega }^2+\frac{𝜕}{𝜕x_i}\left[\left(\mu +{\sigma }_{\omega }{\mu }_t\right)\frac{𝜕\omega }{𝜕x_i}\right]+2\left(1-F_1\right)\frac{\rho {\sigma }_{\omega 2}}{\omega }\frac{𝜕k}{𝜕x_i}\frac{𝜕\omega }{𝜕x_i}$$

(4)

where k is the turbulence kinetic energy; ω is the specific dissipation rate; μ is the dynamic viscosity; ${\nu }_t$ is the eddy kinematic viscosity; and σ, β, and γ are model coefficients, while the blending function $F_1$ and the production term P are defined by

$$F_1={\tanh }^{\hspace{0.33em}}\left\{{\left\{\min \left[\max \left(\frac{\sqrt{k}}{{\beta }^{\ast }\omega d},\frac{500\nu }{d^2\omega }\right),\frac{4\rho {\sigma }_{\omega 2}k}{C{D}_{k\omega }{d}^2}\right]\right\}}^4\right\}\hspace{0.33em}$$

(5)

$$P=\min \left[{\mu }_t\frac{𝜕u_i}{𝜕x_j}\left(\frac{𝜕u_i}{𝜕x_j}+\frac{𝜕u_j}{𝜕x_i}\right),10\cdot {\beta }^{\ast }\rho k\omega \right]$$

(6)

where CD_kω is the cross-diffusion coefficient and d is the distance to the nearest wall.

The turbulent eddy viscosity equation ${\mu }_t$ is defined as follows:

$${\mu }_t=\frac{\rho a_{1}k}{\max \left(a_1\omega ,\Omega F_2\right)}$$

(7)

where Ω is the vorticity, $a_1$ is constant, and $F_2$ is a second blending function defined by

$$F_2={\tanh }^{\hspace{0.33em}}\left\{{\left\{\max \left(2\frac{\sqrt{k}}{{\beta }^{\ast }\omega d},\frac{500\nu }{d^2\omega }\right)\right\}}^2\right\}\hspace{0.33em}$$

(8)

In addressing the multiphase flow problem, the volume of fluid (VOF) method was employed to accurately represent the free interface between air and water. The VOF method, proposed by Hirt and Nichols [25], is an efficient and straightforward approach to handling complex free surfaces with minimal stored information and automatic management of intersecting free boundaries. However, a primary concern arises from the method’s requirement for precise meshing at the multiphase flow interface to ensure accurate capture of the position and shape of the interface. The fluid phase distribution and interface position are described by the phase volume fraction ${\alpha }_i$, as depicted in Equation (9):

$${\alpha }_i=V_i/V$$

(9)

where ${\alpha }_i$ is a variable, $V_i$ is the volume of the specified fluid, and V is the volume of the cell.

The sum of the volume fractions of all phases in the grid is 1, and there are three different cases of phase volume fractions, as shown in Equations (10) and (11):

$${\displaystyle {\sum }_{i=1}^N{\alpha }_i=1}$$

(10)

$$\left\{\begin{array}{l}\begin{array}{ccc}\alpha =0& ,& air\end{array}\\ \begin{array}{ccc}\alpha =1& ,& water\end{array}\\ \begin{array}{ccc}0<\alpha <1& ,& free surface\end{array}\end{array}\right.$$

(11)

where N is the total number of fluid phases. Fluids are defined a primary fluid and a secondary fluid, which are indexed as 1 and 0. Thus, when a cell is occupied only by the primary fluid, in this case only water, α = 1; when only air is present, α = 0; and finally, when the cell has two types of fluid, the fraction between 0 and 1 determines the volume ratio between them, and the position of the free surface in the usual case corresponds to the value α = 0.5.

The fluid calculation area is discretized using the finite volume method. Subsequently, the coupling of pressure and velocity is solved using the SIMPLE algorithm.

In the linear spring mooring model, the elastic force is used to connect the body to the environment, functioning as a restoring force to return the spring to its equilibrium position when it is stretched. The force of the linear spring mooring model can be applied between the rigid body and a fixed point in the environment. According to Hooke’s theorem, the force on the endpoint can be expressed as

$$f_1(x)=k_{eff}(x-x_0), f_2(x)=-f_1(x)$$

(12)

The variables in the equation are explained as follows: f₁ denotes the force acting on the first endpoint position vector X₁, f₂ represents the reaction force acting on the second endpoint position vector X₂, x signifies the distance between the two endpoints (scalar), x₀ stands for the distance at which the elastic force disappears (relaxation length of spring), and k_eff denotes the effective elastic coefficient.

3. Validation of Numerical Model

3.1. Model Descriptions

The DeepCwind Tensioned Leg Floating Wind Turbine System (referred to as the TLP) [26] designed by the U.S. National Renewable Energy Laboratory (NREL), was selected as the basic platform. A validation exercise with the numerical model in STAR-CCM+ was carried out to ensure the correctness of the modeling. For the purpose of this paper, it was assumed that the platform (including the wind turbine strut and pontoon structure) was rigid, and the only elastic component was the tendon. Despite wind not being a part of the research and the absence of wind turbines in the calculation model, properties such as the mass, center of gravity, and moment of inertia of the whole system were taken into account. Moreover, seismic effects were neglected in this analysis. The main parameters of the TLP system are shown in

Table 1. Basic parameters of the TLP system.

Item	Model Values
Overall system quality	1,361,000 kg
Overall displacement	2,840,000 kg
Overall center of gravity height	64.06 m
Radius of inertia	(52.61, 52.69, 9.40) m
Freeboard to tower base	10 m
Draft	30 m

and Figure 1.

The tension leg mooring system of TLP utilizes gravitational force and elastic tendon deformation to generate platform-restoring forces. This system comprises three tendons with 120° angles between adjacent tension tendons. Each tendon is connected to the upper part of the floater and fixed to the seabed in the lower part. The characteristic parameters of the mooring system are shown in

Table 2. Basic parameters of mooring structure.

Item	Model Values
Legs configuration	3 radially at 120°
Anchor radius	30 m
Anchor depth	200 m
Radius of fairlead	30 m
Tendon porch depth	28.5 m
Unelongated length of tension leg	171.39 m
Tendon diameter	0.6 m
Mass per length (dry)	289.8 kg/m
Axial stiffness	7430 MN/m

.

3.2. Validation of Pitch Free Decay

The grid model for numerical simulation employs the overlapping grid technique using a cut body grid type. This approach involves dividing the complex computational watershed into several geometrically simple sub-regions, each independently divided into grids, with overlapping or nesting between them. The overlapping grid boundaries enable data exchange through interpolation methods, allowing computations across the entire watershed. To reduce the number of grids in the computational domain, only the TLP floating platform is included.

The platform grid division includes the free surface area, background area, and overlapping grid area. The boundary of the numerical tank is defined with the length direction (x-direction) as the velocity inlet and pressure outlet, and the width direction (y-direction) of the numerical tank as the symmetry plane. The bottom boundary condition of the numerical tank in the height direction (z-direction) is set as a non-slip wall boundary, with the platform located 300 m away from the left velocity inlet. Moreover, the turbulence intensity and turbulent viscosity ratio at the inlet and outlet are 0.01 and 10, respectively. The computational domain is depicted in Figure 2, and the grid division, whether global or local, is illustrated in Figure 3.

Before the numerical calculations, the pitch free decay with different grid schemes was tested, and then the natural periods were compared with the experimental value [26]. The employed grids were named fine (Case 1: 730,000 cells), medium (Case 2: 315,000 cells), and coarse (Case 3: 152,000 cells). The time history curve depicting the free decay motion of the test platform is presented in Figure 4. The natural period was obtained by combining Equations (13)–(16), which is the average of multiple sets of T-values, as demonstrated below:

$$T_d=t_3-t_1$$

(13)

$$\delta =\frac{2}{T_d}\ln \frac{\left|A_1\right|+\left|A_2\right|}{\left|A_2\right|+\left|A_3\right|}$$

(14)

$${\omega }_i=\sqrt{{\delta }^2+{(2\pi /T_d)}^2}$$

(15)

$$\overline{T}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{T}_i}=\frac{1}{n}{\displaystyle \sum _{i=1}^n\frac{2\pi }{{\omega }_i}}$$

(16)

where t₁ and t₃ represent the time points corresponding to the adjacent peaks in the free decay curve, and t₂ represents the time point corresponding to the trough between these two peaks. A₁, A₂, and A₃ denote the motion amplitudes at t₁, t₂, and t₃, respectively; $\delta$ is the logarithmic decrement; and n is less than or equal to the number of pitch cycles in the simulation duration.

The results are shown in

Table 3. Comparison of calculated errors of pitch natural period of TLP under different grid schemes.

Item	Natural Period of Pitch (s)	Error (%)
Test Data	3.74
Case 1	3.82	2.14
Case 2	3.85	2.94
Case 3	3.89	4.01

, and it can be concluded that the numerical simulation results for the pitch natural period of the TLP were largely consistent with the experimental results. The numerical model scheme met the required calculation accuracy, allowing for the continuation of subsequent calculations.

3.3. Validation of Numerical Waves

The study employed three regular waves and one focused wave as excitation loads. The wave simulation utilized the VOF wave module of STAR-CCM+. In the regular wave conditions, the fifth-order Stokes wave generated using the VOF method was used as the incident wave [27]. The corresponding wave surface equation and velocity potential equation were as follows:

$$\begin{array}{c}k\eta (x)=\epsilon \cos (kx)+{\epsilon }^2{B}_{22}\cos (2kx)+{\epsilon }^3{B}_{31}[\cos (kx)-\\ \cos (3kx)]+{\epsilon }^4[B_{42}\cos (2kx)+B_{44}\cos (4kx)]+{\epsilon }^5[-(B_{53}+\\ B_{55})\cos (kx)+B_{53}\cos (3kx)+B_{55}\cos (5kx)]\end{array}$$

(17)

$$\begin{array}{c}\Phi (x,z)=-C_{PE}x+\frac{C_0}{k}\sqrt{\frac{g}{k}}{\displaystyle \sum _{i=1}^5}{\epsilon }^i{\displaystyle \sum _{j=1}^i}{A}_{ij}\cosh (jk(z+h))\sin (jkx)\\ \end{array}$$

(18)

where k is the wave number; ε is the parameter determined by the relative water depth and the relative wave height; and the coefficients $A_{ij}$, $B_{ij}$, and C are the fifth-order wave coefficients.

The focused wave was generated through the first-order linear superposition of multiple sets of regular waves calculated by the Jonswap spectrum [28]. This spectrum was analyzed from measurements obtained from the Joint North Sea Wave Project and is suitable for the North Sea, where the wind range is limited.

$$S(f)={\beta }_J{H}_{1/3}^2{T}_P^{-4}{f}^{-5}\exp \left[-\frac{5}{4}{(T_{P}f)}^{-4}\right]\cdot {\gamma }^{\exp \left[-\frac{{(f/f_p-1)}^2}{2{\sigma }^2}\right]}$$

(19)

where $\gamma$ is the spectral peak elevation factor, and the value is 3.3; ${\beta }_J$ is a constant; $H_{1/3}$ is the effective wave height; $T_P$ is the peak period; $\sigma$ is a shape parameter; and $f_p$ is the peak frequency.

Then, the amplitude of each component in the focused wave can be expressed as

$$a_n=\frac{A_{N}S(f_n)\Delta f_n}{{\displaystyle {\sum }_{N}S(f_n)\Delta f_n}}$$

(20)

where A_N is the maximum value of the wavefront amplitude of the focused wave; $a_n$ and $f_n$ are the wave amplitude and frequency of the nth component wave; $\Delta f_n$ is an equidistant frequency interval between successive frequencies; and N is the number of constituent waves, with a value is 66.

$$\eta (x,t)={\displaystyle {\sum }_{n=1}^N{a}_n\cos [k_n(x-x_0)-{\omega }_n(t-t_0)]}$$

(21)

where η(x, t) represents the wave amplitude at the position x and time t, while $x_0$ is the wave focusing position (At the origin of the geodetic coordinate system, see Figure 2); $t_{0 }$ is the wave focusing position and time; $k_n$ is the wave number of each component wave; and ${\omega }_n$ is the circular frequency. The fundamental design parameters of the focused wave are detailed in

Table 4. Theoretical wave parameters for each of the wave load cases.

Item	Focused Wave
Water depth (m)	200
Theoretical peak period, Tp (s)	15.4
Theoretical significant wave height, Hs (m)	9.5
Theoretical focus location, xfocus (m)	0
Theoretical focus time, tfocus (s)	95
Theoretical focus phase, φfocus (°)	0

.

To address the grid validity issue, a quasi-two-dimensional (with narrow width) numerical tank was utilized in this study to validate the wave model for three sets of wave conditions. For instance, in the case of the focused wave, three different grid schemes (referred to as Case 4, Case 5, and Case 6) were employed to encrypt the free surface, with Δx and Δz representing the grid sizes in the x and z directions, as presented in

Table 5. Grid configuration of three sets of grids.

Item	Δx (m)	Δz (m)
Case 4	4.80	0.60
Case 5	6.80	0.85
Case 6	9.60	1.20

.

The wave test results of the three grid schemes (Case 4, Case 5, and Case 6) were compared with the theoretical values. Figure 5 shows the time history curve of wave elevation. Among these schemes, Case 4 and Case 5 exhibited the maximum wave heights closer to the theoretical value. For a comprehensive capture of the liquid surface around the platform, and considering factors such as computational efficiency, the second (Case 5) free surface encryption mesh scheme was utilized as an alternative.

In this section, three time steps (0.006 s, 0.01 s, and 0.02 s) were utilized with a consistent grid. The wave elevation of the focused wave for each time step condition is shown in Figure 6. The comparison results show that the wave height at the focusing time (Δt = 0.02 s) was greatly attenuated compared to implying inadequate calculation accuracy. An abnormal vibration phenomenon occurred when Δt = 0.006 at t > 110 s due to the differing wave period and wavelength of each component of the focused wave. For components of the focused wave with larger wavelengths, the time step was insufficient, leading to instability of the numerical solution, thereby affecting the accuracy of the wave simulation. Additionally, at Δt = 0.01 s, the maximum wave height was closer to the theoretical result, and the time step was selected as 0.01 s, considering the wave period and sufficient sampling frequency within one period.

During wave propagation, the increase in distance from the inlet resulted in energy loss due to the viscosity and wave–boundary interaction, which slightly reduced the height of the focused wave. However, the extent of wave height attenuation was minimal, and the simulation results satisfied the requirements for calculation accuracy.

4. Design of TLMCD

To reduce the motion response of FOWT, a novel TLMCD was designed and applied to the support structure of FOWT, exemplified by the TLP. The TLMCD was designed according to the structural configuration of the TLP, including the structural dimensions of the tension leg and the overall mass of the platform. Specifically, it consisted of three columns of vertical liquid columns and a bottom connected liquid column, intersecting at a 120° angle, and all additional tubes were rigid. The detailed structure of the TLMCD and the structural parameters are shown in Figure 7.

In the previous design of the TLCD system, the natural angular frequency closely aligned with or mirrored the pitch natural frequency of the floating support structure. Taking into account previous research and the short-period motion characteristics of the TLP, as well as the constraints posed by the tension leg structure on the radius of the connected liquid columns, two key aspects emerged. Firstly, the height (L_V) of the TLMCD three-column liquid column had a minimal impact on the design period; secondly, the influence of the connecting rod at the bottom of the platform restricted the change in the cross-sectional radius (R_h) of the bottom liquid column. Consequently, this study focused on modifying the cross-sectional radius (R_v) of the vertical liquid column to achieve the TLMCD’s natural period design objective, with reference to specific benchmark North Sea state wave periods, and TLMCDs had the same bottom connection condition. The primary parameters are detailed in

Table 6. Basic parameters of TLMCD.

Parameter	TLMCD1	TLMCD2	TLMCD3	TLMCD4
Rv (m)	0.57	1.38	2.46	3.9
Rh (m)	1.4	1.4	1.4	1.4
Lv (m)	30	30	30	30
Lh (m)	60	60	60	60
$T_0$ (s)	11.8	15.4	22.2	32.5
${\omega }_0$ (rad/s)	0.53	0.41	0.28	0.19

.

Considering that the natural frequency of TLMCD is similar to that of traditional TLCD [29,30], based on the Lagrange equation, the free surface motion in TLMCD can be expressed as

$${\ddot{x}}_i(t)+\frac{1}{2}\frac{\zeta }{L}\frac{A_{\nu }}{A_h}|{\dot{x}}_i(t)|{\dot{x}}_i(t)+{\omega }_0^2{x}_i(t)=-\frac{L_{\nu }}{L}{\ddot{y}}_g(t)$$

(22)

where ${\ddot{y}}_g(t)$ is the motion acceleration of TLMCD, $\zeta$ is the damping ratio, and ${\omega }_0$ is the natural frequency of the damper. ${\ddot{x}}_i(t)$, ${\dot{x}}_i(t)$, and $x_i(t)$ are the acceleration, velocity, and displacement of liquid in the damper, respectively. L_h is the total length of the bottom connected liquid, with A_h representing its cross-sectional area calculated by the cross-sectional radius R_h of the bottom liquid column; L_v is the height of the three-column liquid column, with A_v representing its cross-sectional area calculated by the cross-sectional radius R_v of the vertical liquid column. $L$ is the total length of liquid in TLMCD, and is written as

$$L=2L_v+\frac{A_v}{A_h}{L}_h$$

(23)

The natural frequency of the TLMCD was calculated according to the following equation:

$${\omega }_0=\sqrt{2g/L}$$

(24)

where the natural frequency of TLMCD liquid oscillation, donated as ${\omega }_0$, is used to calculate the natural period. Since the accuracy of the formula has been verified in many previous studies, the natural period of TLMCD will not be re-verified in this paper [31].

5. Hydrodynamic Performance of TLP-TLMCD

To investigate the hydrodynamic performance of the coupled TLP-TLMCD system, the free decay of TLP with different TLMCDs was simulated firstly to identify the suitable TLMCDs. Subsequently, the motion of TLP with the selected TLMCDs was simulated in regular and focused waves to enhance the overall understanding of the potential impact of TLMCD integration on the TLP.

5.1. Free Decay

In the pitch-free decay test, the simulation of hydrostatic conditions relied on the flat VOF wave module of STAR-CCM+, and the initial conditions and boundary conditions were defined by adding field functions (including hydrostatic pressure of flat, velocity of flat, etc.). The natural period of TLP-TLMCD could be obtained through simulating free decay motion, whereby the platform was initially displaced from its hydrostatic equilibrium and then released to move freely. This study focused on examining the free decay of pitch motion. The TLP, alone and equipped with the TLMCDs, listed in Table 6 was simulated, and the corresponding free decay results of pitch motion are presented in Figure 8. According to Equations (13)–(16), the natural periods of TLP and TLP-TLMCDs could be achieved, which is shown in

Table 7. Natural period and hydrostatic damping of platform.

Parameter	TLP	TLP-TLMCD1	TLP-TLMCD2	TLP-TLMCD3	TLP-TLMCD4
$T_1$ (s)	3.85	4.32	4.61	5.27	6.69
${\omega }_1$ (rad/s)	1.63	1.45	1.36	1.19	0.94
$\overline{\varsigma }$	0.025	0.038	0.040	0.020	0.016

. Furthermore, the hydrostatic damping could also be obtained from the free decay curves using Equation (25), as demonstrated below:

$$\overline{\varsigma }=\frac{{\displaystyle \sum _{i=1}^n(\frac{1}{2\pi }\ln \frac{a_i}{a_{i+1}})}}{n}$$

(25)

where a_i and a_i+1 are the adjacent amplitudes of the i th and (i + 1) th peak or trough of the free decay curve; and $\overline{\varsigma }$ is the average damping value.

According to Table 7, the application of TLMCD led to a natural frequency alteration in the coupled system, resulting in decreases in the pitch natural frequency of TLP-TLMCD1, TLP-TLMCD2, TLP-TLMCD3, and TLP-TLMCD4 by 11.0%, 16.6%, 27.0%, and 42.3%, respectively, compared to the original TLP, with TLMCD4 demonstrating optimal frequency tuning. Moreover, in terms of hydrostatic damping, TLP-TLMCD 2 exhibited the largest damping, increased by 37.9% compared to that of the original platform. The increased damping facilitated rapid reduction in the amplitude of pitch motion over a short time.

Considering the aforementioned factors, especially the tuning effect of TLMCD in reducing the system’s natural frequency and increasing the hydrostatic damping, the TLP-TLMCD2 and TLP-TLMCD4 coupling platforms were selected for wave condition verification. This aimed to further investigate the impact of TLMCD parameters on the system’s hydrodynamic performance.

5.2. Motion Response in Regular Waves

The motion of TLP and the selected TLP-TLMCDs were firstly simulated in regular waves to ascertain the impact of TLMCD on the floating body structure in typical sea conditions. The waves were specifically chosen to mirror key North Sea state wave periods, i.e., wave amplitude A = 3.79 m and wave periods T = 11.7 s, 15.4 s, and 19.8 s. The surge, heave, and pitch response motions were simulated in different waves, and the results are shown in Figure 9, Figure 10 and Figure 11.

It can be seen that heave motion response (Figure 9b, Figure 10b and Figure 11b) was influenced to a small extent by TLMCD. The surge motion amplitude (Figure 9a, Figure 10a and Figure 11a) of the TLP-TLMCD2 decreased by 2–5% relative to the TLP, while that of the TLP-TLMCD4 decreased by 14.8–25.2%. Furthermore, the motion amplitude control ratio (Figure 9a, Figure 10a and Figure 11a) of TLMCD in the surge degree of freedom tended to increase with the wave period. The pitch motion response (Figure 9c, Figure 10c and Figure 11c) was effectively controlled by TLMCD, resulting in a reduction of more than 47% in the maximum pitch amplitude in all cases. Additionally, the results of the Fourier analysis for the platform’s pitch motion (Figure 9d, Figure 10d and Figure 11d) demonstrated the substantial influence of TLMCD in attenuating wave frequency motion across all three wave excitations (0.085 Hz, 0.065 Hz, and 0.05 Hz).

To further analyze the impact of TLMCD properties on the anti-pitch motion effect, the variation in maximum pitch amplitude control of various TLMCDs with the tuning ratio (T/T₀, Ratio of wave period to TLMCD natural period) is depicted in Figure 12, derived from the numerical results of the platform’s pitch motion in Figure 9c, Figure 10c and Figure 11c). Based on the Figure, the following conclusions can be drawn:

(1)

In the range of 0.8 < T/T₀ < 1.3, the maximum pitch amplitude of the TLP-TLMCD2 platform was reduced by 47–54% compared with that of the TLP, and the ratio of pitch amplitude control volume was at its peak at a tuning ratio of 1.0. When the external excitation load frequency was close to the TLMCD tuning frequency, the maximum spectral peak occurred at 1.0, resulting in stronger liquid fluctuations in the cabin and improved damping performance.

(2)

In the range of 1.6 < T/T₀ < 2.8, the maximum pitch amplitude of the TLP-TLMCD4 platform decreased by 81–86% compared to that of the TLP. Furthermore, within this tuning ratio range, the pitch amplitude control volume ratio was reduce as the coordination ratio increased.

(3)

For TLMCDs with the same bottom connection condition, a larger natural period indicated a larger liquid mass in the TLMCD cabin. In fact, TLMCD4 demonstrated a superior damping performance compared to TLMCD2 under various working conditions. This seems to indicate that the TLMCD damping effect increases with the increase in liquid mass in the TLMCD cabin.

Figure 10. Under the wave condition of A = 3.79 m and T = 15.4 s, the time history curve of platform motion and the spectrum distribution map of pitch motion: (a) surge motion of the platform; (b) heave motion of the platform; (c) pitch motion of the platform; and (d) Fourier analysis.

Figure 10. Under the wave condition of A = 3.79 m and T = 15.4 s, the time history curve of platform motion and the spectrum distribution map of pitch motion: (a) surge motion of the platform; (b) heave motion of the platform; (c) pitch motion of the platform; and (d) Fourier analysis.

Figure 11. Under the wave condition of A = 3.79 m and T = 19.8 s, the time history curve of platform motion and the spectrum distribution map of pitch motion: (a) surge motion of the platform; (b) heave motion of the platform; (c) pitch motion of the platform; and (d) Fourier analysis.

Figure 11. Under the wave condition of A = 3.79 m and T = 19.8 s, the time history curve of platform motion and the spectrum distribution map of pitch motion: (a) surge motion of the platform; (b) heave motion of the platform; (c) pitch motion of the platform; and (d) Fourier analysis.

Figure 12. The maximum pitch amplitude control ratio of TLMCD with different masses under different tuning ratios (T/T₀, ratio of wave period to TLMCD natural period).

Figure 12. The maximum pitch amplitude control ratio of TLMCD with different masses under different tuning ratios (T/T₀, ratio of wave period to TLMCD natural period).

5.3. Motion Response in Focused Wave

The numerical simulation results under free decay and regular wave conditions suggest that the TLMCD structure effectively dampened the motion of the TLP. To further explore the nonlinear damping characteristics of TLMCD, this study introduced a focused wave condition (refer to Table 4 for specific parameters) to simulate the motion response of both the TLP and the coupled TLP-TLMCD system under extreme loads. The time history curves illustrating the platform’s surge, pitch, and heave degrees of freedom motion under the focused wave condition are presented in Figure 13.

Figure 13 shows that the TLMCD damping effect aligned with the findings from the previous regular wave motion, which seems to have manifested itself as a more pronounced suppression of wave frequency motion. Additionally, the nonlinear characteristics of the damping process were more prominently evident. Consequently, for further analysis of the nonlinear interactions of the TLP-TLMCD system, the wavelet transform was utilized to represent the frequency components of the amplitude of the pitch motion.

The wavelet analysis of the platform pitch motion in Figure 14 revealed that the focused wave’s significant excitation frequency fell within the range of 0.05–0.12 Hz. Within these excitation ranges, the damping effect of TLMCD on the platform’s pitch motion was most notable. Moreover, for TLMCDs with the same bottom connection condition, a larger natural period of the TLMCD indicated that the mass of the liquid in the cabin of the TLMCD was larger, and the frequency bandwidth of its suppression became larger. In comparison to TLMCD2, the tuning range of TLMCD4 expanded from 0.06–0.10 Hz to 0.04–0.12 Hz.

The Fourier analysis results depicted in Figure 15 indicate that the pitch motion response amplitude in the TLP-TLMCD2 and TLP-TLMCD4 platforms was notably lower than that in the TLP. Furthermore, TLMCD4 demonstrates a 1.8-times-improved tuning efficiency for significant excitation frequency center energy compared to TLMCD2.

The liquid in the TLMCD was disconnected from the external wave. The wave transferred kinetic energy to the float, which in turn transferred kinetic energy to the liquid in the damper, and the height of the liquid column varied with the movement of the float and its own inertia. Figure 16 illustrates the phase relationship between the liquid level change in the liquid column and the response of the platform pitch motion. It was observed that the phase difference between the liquid column’s liquid level height and the platform pitch motion angle was close to π. This indicates that at this phase, the fluid sloshing in the TLMCD liquid tank absorbed and releases mechanical energy, providing damping torque for the floating structure, which effectively suppressed motion. Essentially, this enabled the full utilization of the coupling between the wave force, the sloshing force, and the structural motion.

6. Conclusions

A three-dimensional tuned liquid multi-column damper (TLMCD) was designed for TLPs, and its impact on the TLP under various excitation loads was investigated. Initially, the natural period of the TLP itself was confirmed to validate the numerical tank’s accuracy. Building on this, the physical model of the TLMCD coupled to a floating body was established to expand the study to the modeling of the wave–platform–spring mooring coupled system. Finally, the motion suppression effect of TLMCD was assessed in terms of hydrodynamic loading, structural motion response, and energy dissipation.

The integration of TLMCD fundamentally altered the platform’s natural frequency, gradually shifting the coupled system’s natural frequency to lower levels. Under regular wave conditions, TLMCD effectively controlled the platform’s pitch and surge motion. Moreover, as the mass of liquid in the TLMCD cabin increased and the tuning ratio approached 1.0, the anti-pitch motion effect became more pronounced. However, comparison between the presence and absence of TLMCD revealed that its effect on heave motion was limited. Additionally, under extreme load excitation, the damping effect of TLMCD on the platform’s pitch motion primarily manifested in the frequency range of the focused wave, with the frequency bandwidth of the damping expanding as the mass of liquid in the TLMCD cabin became greater.

In the simulation results presented in this paper, the maximum pitch amplitude control volume ratio of TLMCD was shown to reach up to 86%, while the maximum surge amplitude control volume ratio attained 25.2%, indicating the strong applicability of TLMCD. The paper approximates real sea states through regular and focused wave modeling; however, it is noteworthy that the wind load effect has yet to be examined. Hence, future plans include conducting a coupling analysis of liquid sloshing within the liquid tank and platform motion under the combined excitation of wind and waves.