1. Introduction
Against the backdrop of the accelerating global energy transition, the limitations of traditional fossil energy are becoming increasingly prominent. The International Energy Agency (IEA) predicts that, by 2040, renewable energy will need to meet 65% of the global energy increment demand. In the diversified renewable energy system, ocean energy has become a strategic focus, with its theoretical reserves of
TWh/year (equivalent to twice the current global electricity consumption) [
1]. Tidal current energy has been listed by the European Union as a core direction of the 2050 Marine Energy strategy due to its unique physical properties [
2]. The trend can take into account energy intensity, stability, and predictability. O. Roberts [
3], studying energy density, reported that the tidal current energy density at a flow velocity of 3 m/s reaches 14.5 kW/m
2, which is four times that of wind energy at the same wind speed. Data from the International Renewable Energy Agency (IRENA) shows that the global installed capacity of tidal current energy exceeded 200 MW in 2023. Dai [
4] evaluated the potential for developing tidal flow energy in the Zhoushan area of China using a numerical model. By comparing the maximum average tidal flow power density of different channels, it was shown that the tidal current energy in this area has high development potential and economic feasibility. Spicer [
5] focused on the Aleutian Islands region of Alaska, conducting an in-depth analysis of how mixed tidal regimes influence the spatial distribution of tidal current energy potential and revealing its seasonally modulated variability patterns.
The motion response problem has always been the focus of research on oceanic floats and has been studied by many scholars in conjunction with waves and currents. Sun [
6] introduced a Laplace domain-based method for estimating wave heights from the transient motion response of a floating body. The method provides an effective way to invert the wave characteristics from the float motion data by establishing a mathematical relationship between the motion response of the float and the wave excitation. Zang [
7] conducted an in-depth investigation of the motion response characteristics of a floating wind turbine with a semi-submersible foundation by experimental means, aiming to obtain a comprehensive understanding of its dynamical behavior under different sea states. In recent years, significant progress has been made in the research of carriers for tidal current power generation devices. Wang [
8] designed a submersible carrier for tidal generators and used computational fluid dynamics methods to study the smooth flow distribution of the turbine during operation. It was determined that this design improved the energy acquisition and efficiency of the turbine for tidal energy. Sheng [
9] analyzed the hydrodynamic loads of the double-blade fixed-pitch vertical-axis turbine of the Haineng I
kW floating tidal power station in the same direction of waves and water flow and verified that, under the same wave height conditions, the ultimate load of the main shaft increases with the increase in wave height.
The economic development of shipping has led to increasingly frequent maritime traffic, and the ship waves caused by ship navigation can impact not only other ships but also fixed and floating platforms at sea. Some studies on fixed and floating carriers have considered the effects of ship waves and incorporated them into their research. Huang [
10] employed a three-dimensional numerical simulation approach based on the resolution of the Navier–Stokes equations to ascertain the causal link between ship waves and the erosion phenomenon of river channel banks, as well as to offer critical conditions for the protection of these banks. By examining the decrease in wave elevation near the banks of the waterway, it was demonstrated that the normal impact forces and tangential shear stresses caused by ship waves are the primary culprits in the erodibility of riverbanks. Zhang [
11], taking the MASHCON2022 benchmark model test cases as the research subject, employed the Unsteady Reynolds-Averaged Navier–Stokes (URANS) method coupled with dynamic overlapping grid technology to simulate the entire process of a ship passing by a moored vessel. By simulating four acceleration patterns of the navigating ship, the research validated the impact of the navigating ship on the moored ship during the initial acceleration phase. Wu [
12] focused on dynamic response and vibration control problems of deep-water bridge piers under ship wave excitation. By deeply analyzing the action mechanism of ship waves on deep-water bridge piers, we aim to reveal their dynamic response characteristics and explore effective vibration control strategies to ensure the safety and stability of deep-water bridge structures.
At present, academics mostly focus on the structural research of tidal buoy carriers under the action of regular waves [
13,
14], while there is a relative lack of investigative research on their stability under the influence of ship traveling waves. In fact, ship waves generated by different ship speeds, ship spacing parameters, and draught conditions during ship navigation can significantly perturb the stability of the carrier, which, in turn, reduces the power generation efficiency of the tidal energy generator. The research will be presented across five sections: 1.
Section 1 is responsible for introducing the established research results on ship traveling waves and carrier motion; 2.
Section 2 is responsible for demonstrating the theoretical framework applied in the research; 3.
Section 3 provides a finely detailed introduction of the experimental equipment and model; 4.
Section 4 is responsible for the visual presentation and in-depth analysis of the experimental data; and, finally, 5.
Section 5 presents the conclusions drawn from the thesis and points out the shortcomings of the research. In this research, a catamaran configuration was innovatively introduced into the field of floating tidal energy carriers, and the 3600 TEU KCS container ship designed by KRISO Research Institute of Korea was used as the sailing vessel to form a multi-body coupled experimental whole. Through the combination of theoretical modeling and tank experiments, the motion response law of the six-degree-of-freedom catamaran was systematically analyzed under variable environmental parameters, providing a novel technical idea and experimental basis for the cross-field integration of ship and tidal energy development and utilization and expanding the design boundaries and application scenarios of floating tidal energy carriers.
2. Methods
2.1. Ship Wave Theory
As shown in
Figure 1, the main characteristics of ship waves are as follows: The size of the Kelvin angle is independent of the ship’s speed, and it is invariant to the ship’s speed. The ship speed primarily modulates wave amplitude while preserving the Kelvin angle. When the ship speed is low, the divergent wave is the main wave system. With the increase in the ship speed, the transverse wave gradually strengthens, while the stray wave gradually weakens. The magnitude and direction of the scattered waves are related to the ship type, speed, etc. The scattered waves at the bow and stern of the ship, as well as the scattered waves and transverse waves, do not interfere with each other [
15].
In the above formula, is the wave number, g is the acceleration due to gravity, is the Kelvin angle, is the wavelength of the transverse wave, is the wavelength of the scattered wave, and is the velocity of the ship.
The advancing speed of the transverse wave of the traveling wave of a ship is the same as the moving speed of the origin of the disturbed pressure. With the gradual deepening of the research on the Kelvin pressure disturbance theory by scholars from various countries, the theoretical research on the traveling wave of a ship is also constantly improving. The change in the wave height of the traveling wave of a ship is mainly affected by the shape and speed of the ship.
The phase velocity of transverse waves in a ship’s wake matches the propagation speed of the pressure disturbance source. As research on Kelvin wake theory has progressed, the understanding of ship waves has significantly advanced. The amplitude of ship waves is primarily determined by hull geometry and vessel speed.
Considering the relationship between the ship wave and the type and speed of the ship, the formula for calculating the ship wave is
In the formula,
is the correction coefficient, which is generally taken as 0.2–0.5. In this research, the correction coefficient for the selected KCS ship is
.
is the width of the ship; L is the captain.
is the sailing speed of the ship;
is the acceleration due to gravity [
16].
2.2. Motion Response Motion Equation
The expression of the motion equation of a floating body under the combined action of wind, waves, and currents is as follows [
17]:
In the formula, represents the mass matrix of the floating body; represents the damping matrix of the floating body; represents the stiffness matrix of the floating body; represents the wind load; represents the flow load; represents the wave load; and represents the mooring force provided by the mooring system.
The added mass matrix is determined in this research using thin-body approximation within potential flow theory, accounting for the hydrodynamic characteristics of the catamaran configuration. Following Faltinsen’s theoretical framework, the added mass coefficients demonstrate distinct dependencies on (1) hull separation distance for transverse motions and (2) demi-hull cross-sectional geometry for longitudinal motions. The slender-body formulation yields the following expressions for added mass estimation:
where
is the water density;
is the length of the hull;
is the spacing between the centers of the two hulls; and
,
, and
are the coefficients of the added mass in the surge, sway, and roll directions, respectively. These coefficients are determined according to the geometrical parameters and kinematic state of the catamaran, with reference to relevant literature and experimental data.
For the damping matrix, the effects of viscous damping and rising wave damping are considered together. The viscous damping is obtained by calculating the frictional resistance on the hull surface, while the rising wave damping is estimated based on the wave generation characteristics of the catamaran in different motion states. The following empirical equations were used to calculate the damping coefficients for each degree of freedom:
where
,
, and
are the damping coefficients in the surge, sway, and longitudinal rocking directions, respectively.
The restoring force matrix is derived from the hydrostatic properties of the catamaran, incorporating both buoyancy and stability considerations. The diagonal elements characterize the direct restoring forces and moments, governed by the equilibrium between buoyancy and gravitational forces. The off-diagonal elements account for coupling effects between different motion degrees of freedom, reflecting the interconnected nature of the catamaran’s hydrostatic response. The matrix formulation is expressed as
where
is the volume of discharged water, and
is the initial stability height.
2.3. The Principle of the Push-Type Regular Wave Generator
The form of wave generation used for regular wave generation in this research is mainly push wave generation; the offshore wave tanks of the Danish Hydraulic Institute (DHI) and the Norwegian Marintek wave tank use this form. Its working principle is described below [
18].
① Control system parameter input: Set the target wave parameters—wave height H, period T, and water depth h—and calculate the push plate movement parameters (amplitude S, frequency ).
② Generate push plate movement: The servo motor/hydraulic cylinder drives the pusher plate to perform simple harmonic motion; the specific formula is
where
represents the instantaneous displacement of the pusher plate.
③ Wave formation: The push plate pushes the water body to generate traveling waves, and the waveform satisfies the linear wave theory:
where
represents the wave surface elevation,
represents the wave number
, and
represents the angular frequency
.
④ Wave dissipation treatment: A porous wave-absorbing beach is installed at the downstream end to suppress reflected waves, achieving a reflection coefficient of less than 5% (R < 0.05).
4. Results and Analysis
This experiment was designed to focus on key parameters such as regular waves, ship distance, ship speed, the regular wave period, and draft of the navigating vessel, and conduct systematic control experiments. The experiment also focused on analyzing and comparing data under the following three working conditions: the action of regular waves alone, the action of traveling waves alone, and the superimposed action of traveling waves and regular waves. Through in-depth analysis of the experimental data of different working conditions and parameter combinations, this research will systematically explore the influence laws of various factors on the motion response of the floating carriers of mooring catamarans and ultimately draw corresponding conclusions.
In this experiment, we adopted an infrared motion capture system to monitor the motion response of the floating body in real time. This system has high measurement accuracy and anti-interference ability and is not affected by external electromagnetic noise. However, under different environmental conditions, changes in temperature, humidity, and light intensity may cause a certain degree of interference to the transmission and reception of infrared signals, thereby introducing certain measurement errors. To effectively reduce the impact of such errors on the experimental results and ensure the reliability and accuracy of the data, we conducted at least three repeated measurements for each group of experiments. Through the statistical analysis of the repeated experimental data, it was found that the experimental error was controlled within an acceptable range, and the data had good consistency and repeatability. Therefore, this paper selects one group of representative experimental results for detailed analysis and discussion to verify the errors of the experiments and ensure the scientificity and rigor of the research conclusions at the same time.
As shown in
Figure 10, two experiments of six-degree-of-freedom kinematic response curves are demonstrated for a ship speed of 1.478 m/s and a ship spacing of 1.5 m. The response curves of the six-degree-of-freedom kinematic response are shown in
Figure 10. Neglecting the problem of acquisition start time, the two experiments have the same experimental data pattern, and at the same time, by calculating the peak value of each motion response, the error of the peak value of each motion response is less than 5%, so the experimental setup produces less error and meets the rigor requirements of experimental papers.
By comparing and analyzing the maximum values of the variation amplitude of the six degrees of freedom of the two groups of data, As shown in
Table 5, the results show that the errors were controlled to be around 10%. Specific quantitative assessments showed that the difference in means was less than 5 per cent, the standard deviation was maintained at a low level, the root mean square error (RMSE) was less than 0.15, and the overlap of the 95 per cent confidence intervals was high. All these indicators confirm the reliability and reproducibility of the experimental results. The trends of the two groups of data are highly consistent, and the corresponding error values are within acceptable limits, which indicates that the experimental results have good reliability and reproducibility.
4.1. The Influence of Regular Waves
Before conducting the subsequent control experiments, this research first carried out a systematic verification experiment on the motion response characteristics of the floating carrier of the mooring catamaran under the action of regular waves. The experiment adopted the control variable method and focused on investigating the influence of two key parameters, namely wave height (H = 0.02 m, 0.04 m, 0.06 m) and wave period (T = 1.2 s, 1.5 s), on the motion response of the carrier.
As shown in
Figure 11, the experimental results show that the amplitude of the carrier’s motion shows a significant increasing trend with the increase in wave height, and its variation law is in line with the prediction of the floating body response by the linear wave theory. The motion response shows obvious periodic characteristics with the variation in the wave period, but the influence amplitude is relatively small, which is consistent with the frequency-domain response characteristics of the mooring float. The motion response curves collected in the steady-state stage have good repeatability and regularity. The consistency between the experimental data and the classical mooring float motion theory further verifies the scientific nature of the experimental method.
As shown in
Figure 12, the change curves of the longitudinal and vertical oscillations of the regular wave with 0.02 m wave height and 1.5 s wave period were selected for comparison of the regular wave data, and the curves in the figure were extracted from the data after the motion of the carrier became relatively stable. It can be seen that the change rule of the curves is consistent, but due to the problem of simulation grid setting, there is regular wave attenuation, and the motion response appears to be reduced. At the same time, in the experiment, due to the influence of external factors, the curve of the transverse rocking value change is not smooth, but the change interval is consistent with the simulation.
The peak values of longitudinal rocking and peak values of pendulum oscillations in the experiment and simulation under regular waves are shown in
Table 6. By comparing the curve peaks, we found that the error of longitudinal rocking is about 5.9%, and the error value of drooping oscillation is about 9.6%, and the error values are all within 10%.
4.2. The Influence of Ship Speeds
This research explored the effect of sailboat speed variation on the motion response of a moored catamaran through a controlled experiment. In the experiment, the distance between the sailboat and the catamaran was fixed at 1.5 m. The aim of the experiment was to investigate the effect of traveling waves on the six-degree-of-freedom motion of the catamaran at different boat speeds (0.3, 0.6, 0.9, 1.2, and 1.478 m/s). Given the limited width of the experimental tank, reflections from the tank wall observed during the experiment may have interfered with the results. To minimize the effect of this interference, experimental data collection was initiated when the sailboat started its motion and continued until the catamaran was most affected by the traveling wave, at which point data collection was stopped to ensure the accuracy of the data.
The time curve of the catamaran carrier model with the change in ship speed is shown in
Figure 13. At low speed (0.3–0.9 m/s), the motion response of the catamaran carrier changes less; as the ship speed increases, the motion response changes show a dramatic change. When the ship speed exceeds 1.2 m/s, the motion response shows a significant jump. The peak of pitch ranges from 1.62° to 4.47°, the peak of roll ranges from 0.76° to 2.44°, the peak of yaw ranges from 0.23° to 1.22°, the peak of sway ranges from 35.37 mm to 81.74 mm, and the peak of heave ranges from 3.97 mm to 13.94 mm. By analyzing the variations, it can be seen that the magnitude of the influence of the ship’s traveling wave on the motion response of each degree of freedom of the carrier is small in the low-speed range (0.3–0.9 m/s). This indicates that, at low speeds, the fluid perturbation energy is low, resulting in the relatively small kinematic response of the carrier. The kinetic energy of the fluid at low speeds is not sufficient to cause significant fluctuations or disturbances, making the flow field around the carrier relatively smooth. The significant jump in the high-speed interval (1.2–1.478 m/s) reflects the nonlinear nature of the hydrodynamic forces. The complexity and dynamics of the hydrodynamic forces increase as the vessel speed increases, leading to a significant increase in the carrier motion response. This nonlinear effect is particularly evident at high speeds, suggesting that the influence of the ship wave on the stability of the carrier is more significant under high-speed conditions.
4.3. The Influence of the Distance Between Two Ships
Figure 14 reveals the significant effect of boat pitch on the six-degree-of-freedom kinematic response of the catamaran carrier. When the ship speeds are all 1.478 m/s, the roll remains stable (2.62° ± 0.03°), with the increase in ship distance being from 1.5 m to 2.5 m, indicating that it is less affected by the ship distance; the pitch decreases sharply by 40% (4.48°→2.69°), reflecting that the longitudinal perturbation of the ship’s traveling wave attenuates rapidly with the distance; the yaw decreases by 64% (3.10°→1.13°), confirming the dominant effect of ship track flow on the steering moment of the carrier; surge decreases by 76% (72.3 mm→17.5 mm), and sway decreases by 55% (81.7 mm→36.8 mm), reflecting the exponential decay property of the wave force with the distance; and heave decreases by 27% (13.9 mm→10.2 mm), showing that the pendant movement has the lowest sensitivity to the distance of the ship.
Overall, during the navigation of a ship, the distance between it and the carrier has a significant impact on the propagation of the ship’s traveling waves and the motion response of the carrier. When the distance between the ship and the carrier gradually increases, the time for the ship’s traveling wave to diffuse from the ship to the carrier will also be delayed accordingly. Meanwhile, the wave energy will gradually be consumed and attenuated during the long propagation path, which causes the wave energy received by the carrier to continuously weaken. Therefore, the motion response of the carrier will also decrease accordingly, and the peak value of its motion response will gradually decrease with the increase in the ship distance. The influence of this distance on the propagation of ship traveling waves and the response of the carrier fully demonstrates the important role of the distance factor in the interaction between ships and carriers and also reveals the intrinsic connection between the propagation of ship traveling waves and the response of the carrier.
4.4. The Influence of Ship Draft
The experimental results in
Figure 15 show that, under the conditions of a fixed ship distance (1.5 m) and ship speed (1.478 m/s), the change in ship load has a significant influence on the motion response of mooring catamarans. This research finds that, when the ship is in an empty load state (draft of 0.1 m), due to the shallower draft of the hull, the wave formation resistance increases, and the ship’s traveling wave energy is concentrated and distributed in the horizontal plane, resulting in a significant increase in the horizontal plane motion responses, such as the pitch, yaw, sway, and surge of the floating body, compared with the full-load condition (draft of 0.2 m). When the ship is in the unloaded state (draft 0.1 m), due to the shallow draft of the hull, the formation of wave resistance increases; the ship’s traveling wave energy is centrally distributed in the horizontal plane, resulting in pitch, yaw, sway, and surge waves that are larger; the transverse rocking moment mainly originates from the ship’s transverse displacement, which is weakly correlated to the draught, meaning the increase or decrease in the draught has a small impact on it; and the value of the swaying motion is positively correlated to the draught, so the increase in the ship’s draught is not significant. Therefore, the increase in the ship’s draft will increase the influence on the carrier’s heave motion.
4.5. The Superimposed Influence of Ship Waves and Regular Waves
Considering that the occurrence of ship traveling waves at sea is often accompanied by the influence of other waves, in the experiment, for this problem, conditions of superimposing ship traveling waves and regular waves were applied. As shown in
Figure 16, the sailing direction of the KCS ship and the propagation direction of the rule wave are opposite to each other, and the ship-generated traveling wave and the rule wave act together on the catamaran model.
Here, the six-degree-of-freedom motion response of the catamaran floating body was compared under the conditions of a single 1.5 m/s ship speed regular wave, three different regular waves, and the superimposed wave generated by the superposition of the two.
According to the analysis results in
Figure 17, it can be known that the influence of the regular wave on the motion response of the carrier manifests as periodic changes in all six degrees of freedom, but its fluctuation amplitude is relatively small. In contrast, under still-water conditions, the carrier is basically in a static state. However, as the sailing ships pass by, all six degrees of freedom of the carrier change significantly. Further analysis of the motion response curves after the superposition of regular waves and ship traveling waves revealed that, compared with the action of a single regular wave, the motion responses of the carrier in all six degrees of freedom under the superposition condition increased. Compared with the traveling wave effect of a single ship, under the superimposed condition, the longitudinal sway, lateral sway, bow sway, and lateral sway responses of the carrier are reduced, while the longitudinal sway and vertical sway responses are increased. This phenomenon can be attributed to the relative interaction between the direction of the traveling wave and the direction of the incoming regular wave: Since the ship begins to sail only after the regular wave reaches the carrier, the motion response of the carrier is first affected by the regular wave and then by the traveling wave. The existence of regular waves has a certain inhibitory effect on the influence of ship waves, thereby leading to the variation characteristics of the above-mentioned motion response.
In order to verify whether the changes in the wave height and wave period of the regular wave have a certain inhibitory effect on the influence of the ship wave, the superposition conditions of 0.02 m wave height and 1.5 s wave period, as well as 0.04 m wave height and 1.2 s wave period with the ship wave, were applied. The variation curves of the six degrees of freedom over time are shown in
Figure 18. It can be seen from the analysis of the curve that, under the condition of a single ship traveling wave, with the decrease in the period of the regular wave and the increase in the height of the regular wave, the changes in the six degrees of freedom all increase. Pitch, yaw, and heave are greatly affected by the changes in the regular wave. The change in roll is still dominated by the ship traveling wave, and the addition of the regular wave will only slightly reduce the amplitude of the change in roll. Due to the consideration of the influence of the tank wall effect, the data collection for the longitudinal swing curve and the transverse swing is relatively short. However, it can be seen that, in terms of amplitude, the traveling wave still dominates.