Hydrodynamic Response of a Large-Scale Mariculture Ship Based on Potential Flow Theory

Abstract

The marine fishery will be the main form of the marine economy in the future. Simulating a hydrodynamic response under normal and extreme working conditions is the main means of structural analysis and design of a mariculture ship. In this paper, a simulation methodology is proposed based on potential flow theory, focusing on a semi-submersible large-scale mariculture ship with a rigid frame. Abaqus/Aqua 2020 software is used to establish a full-scale dynamic analysis model of the fishery. In the simulation, a nonlinear implicit integration method is applied, and the non-deterministic boundary conditions of the floating body are optimized using dynamic equilibrium principles. By varying the wave and flow conditions, the variations in mooring forces, vibration amplitudes, and average vibration values are analyzed. Furthermore, the dynamic changes in the overall spatial displacements of the fishery, characteristics of longitudinal and vertical oscillations, and mid-span deflections are analyzed. It is concluded that the mooring force is linearly correlated with the flow velocity, that a higher wave increases the longitudinal oscillation amplitude, and that a longer wave period leads to higher mooring forces and longitudinal heaving amplitude. These dynamic response and displacement results of the mariculture ship are expected to provide a basis for its design and safety assessment.

1. Introduction

Mariculture products account for more than 80 percent of all seafood products in China [1]. Research works on various mariculture platforms have been conducted in many countries, including the United States and Norway, since 1970 [2,3]. With the introduction of large-scale, wind-resistant, deep-sea fish farms (HDPE type) in Norway [4], significant progress has been made in China’s fishery equipment construction in terms of system safety, intelligent management, and mariculture models [5]. Since 2017, large-scale fishery equipment, such as fully submersible, intelligent bottom-located, and single-column semi-submersible mariculture cages, have been developed successfully [6]. Deep-sea mariculture is evolving to become large-scale, automated, intelligent, and eco-friendly [7]. ‘Dehai-1’, the prototype of this research, is located in the sea 32 km away from Zhuhai, China. It is a semi-submersible large-scale intelligent mariculture ship, and its main structure is a combination of box and truss structures [8]. It survived the working condition test of a strong typhoon. Therefore, it is selected as the design prototype for this study. It has a slender rigid frame structure, with low material consumption and strong wind and wave resistance. It is a typical single-point anchoring mariculture ship, as depicted in Figure 1.

The hydro-elastic response and interaction between floating fuel storage facilities and breakwaters have been studied by many researchers. For example, a plate model structure was used to simulate both floating storage facilities and the breakwaters [9], and the linear wave theory was utilized to simulate water waves in order to correctly predict the hydrodynamic response and hydrodynamic interactions among floating storage modules placed side-by-side. Similarly, in 2011, Gao, R.P. studied a very large floating structure (VLFS) by using a single-plate model based on the Mindlin plate theory, in which the hydrodynamic response of the VLFS was investigated by using the potential flow theory [10]. The boundary element method is used to solve the Laplace equation together with the fluid boundary conditions for velocity potential, whereas the finite element method is used for solving the deflection of the floating plate. Moreover, Karperaki, A.E. (2016) conducted a time-domain, shallow-water, hydro-elastic analysis of a VLFS elastically connected to the seabed [11]. The boundary conditions were discussed in detail. The structure was approximated as a plate and modeled as a Euler–Bernoulli beam. The hydrodynamic model was built based on linear shallow-water wave theory, while the boundary conditions were assumed as elastic connectors, and the adjacent cells were assumed to be connected through a series of simple spring-damping systems.

Deep-sea cage-like floating structures exhibit complexity and diverse forms. It is suggested in the literature that mass-spring models are suitable for the simulation of traditional large flexible fish cages to predict their dynamic response [12]. A net cage was modeled using truss elements that represented several parallel lines. Sub-elements allowed the trusses to buckle in compression, and negligible compressive forces were observed in the numerical results [13]. A plate unit is commonly used to simulate large, non-permeable platforms in the sea. Beam elements offer effective alternatives for simulating elongated slender members, where the computational complexity could be reduced significantly. A spring-damping system was proven to be excellent for the configuration of mooring chains [14]. In the numerical simulation of a floating structure, the boundary conditions had to evolve from deterministic to non-deterministic, and the constraints transform from a semi-fixed to a dynamic equilibrium condition by using the buoyancy distribution method. Inertial release techniques were subsequently applied to achieve dynamic equilibrium. It was proven that the dynamic equilibrium method is effective in setting the boundaries of floating structures. Based on existing research works, this paper proposes a numerical simulation method for large-scale mariculture ships, which might provide help for the design and safety assessment of deep-sea fishery equipment.

2. Theoretical Foundation

2.1. Potential Flow Theory

For structures floating in fluid, subjected to a relatively minor adhesive force, smaller surface tension, and Coriolis forces, the potential flow theory is commonly used to describe waves under the assumption of an ideal fluid [4]. In the case of a flat seabed with a net water depth of d (m), considering a wave propagates along the X-axis and the vertical direction extending along the Z-axis, the coordinate origin is situated at the still water surface, where t (s) represents the wave period, L represents the wavelength, and C = L/t represents the wave speed.

The velocity potential of the fluid can be defined as $\mathrm{\Phi }(x,z,t)$, a function of position and time. It satisfies the Laplace equation:

$$\frac{{\partial }^2\mathrm{\Phi }}{\partial x^2}+\frac{{\partial }^2\mathrm{\Phi }}{\partial z^2}=0$$

(1)

Due to the fact that the vertical flow velocity at the seabed is zero, it adheres to the boundary condition:

$${\left.\frac{\partial \mathrm{\Phi }}{\partial z}\right|}_{z=-d}=0$$

(2)

For water mass points on the fluid surface, their vertical velocity equals the surface’s motion velocity in that direction [15]. Thus:

$$\frac{\partial \eta }{\partial t}+\frac{\partial \mathrm{\Phi }}{\partial x}\cdot \frac{\partial \eta }{\partial x}-\frac{\partial \mathrm{\Phi }}{\partial z}=0(z=\eta )$$

(3)

It is supposed that the water mass points on the wave surface always remain on the wave surface. The pressure on the wave surface is equal to the atmospheric pressure. Hence:

$$\frac{\partial \mathrm{\Phi }}{\partial t}+\frac{1}{2}\left[{\left(\frac{\partial \mathrm{\Phi }}{\partial x}\right)}^2+{\left(\frac{\partial \mathrm{\Phi }}{\partial z}\right)}^2\right]+g\eta =0(z=\eta )$$

(4)

The periodic condition of the waves is expressed as:

$$\mathrm{\Phi }\left(x,z,t\right)=\mathrm{\Phi }\left(x-ct,z,t\right)$$

(5)

The wave velocity potential’s boundary conditions are nonlinear. However, when solving this nonlinear problem, perturbation methods are often used to transform the nonlinear problem into a linear problem and obtain approximate solutions of different orders [16].

2.2. Stokes Wave Theory

To solve the problem that the boundary conditions for a free surface are nonlinear, the velocity potential and wave surface are expanded using a perturbation parameter ${\tau }^n$:

$$\left\{\begin{array}{c}\mathrm{\Phi }={\displaystyle \sum _{n=1}{\tau }^n{\mathrm{\Phi }}_n}\\ \eta ={\displaystyle \sum _{n=1}{\tau }^n{\eta }_n}\end{array}\right.$$

(6)

where $\mathrm{\Phi }$ is the potential function and $\eta$ is wave surface.

Substitute Φ_n into the Laplace equation and boundary conditions. The specific approach involves expanding the potential function $\mathrm{\Phi }$ and its derivatives at the free surface position of $z=\eta$ using a Taylor series, and substituting them into the boundary conditions for the free surface. A series of independent partial differential equations (PDEs) can be obtained by setting the coefficients to zero. The first-order PDEs are shown in Equations (7) and (8), and the second-order PDEs are shown in Equations (9) and (10):

$$\frac{\partial {\mathrm{\Phi }}_1}{\partial z}-\frac{\partial {\eta }_1}{\partial t}=0$$

(7)

$$\frac{\partial {\mathrm{\Phi }}_1}{\partial t}+g{\eta }_1=0$$

(8)

$$\frac{\partial {\mathrm{\Phi }}_2}{\partial z}-\frac{\partial {\eta }_2}{\partial t}+{\eta }_1\frac{{\partial }^2{\mathrm{\Phi }}_1}{\partial z^2}-\frac{\partial {\eta }_1}{\partial x}\frac{\partial {\mathrm{\Phi }}_1}{\partial x}=0$$

(9)

$$\frac{\partial {\mathrm{\Phi }}_2}{\partial t}+g{\eta }_2+\frac{\partial }{\partial t}\left({\eta }_1\frac{\partial {\mathrm{\Phi }}_1}{\partial t}\right)+\frac{1}{2}\left[{\left(\frac{\partial {\mathrm{\Phi }}_1}{\partial x}\right)}^2+{\left(\frac{\partial {\mathrm{\Phi }}_1}{\partial z}\right)}^2\right]=0$$

(10)

The first-order PDEs represent the case of a linear wave. By substituting the velocity potential function Φ₁ and wave surface equation η₁ into the second-order PDEs, the second-order velocity potential function Φ₂ and wave surface equation η₂ that satisfy the Laplace equation and boundary conditions can be derived.

The velocity potential function for second-order Stokes waves is given by Equation (11):

$$\mathrm{\Phi }=\frac{\pi H}{\mathit{kT}}\cdot \frac{\mathit{ch}\left(k(z+d)\right)}{\mathit{sh}\left(\mathit{kd}\right)}\sin (\mathit{kx}-\omega t)+\frac{3\pi H}{8\mathit{kT}}\left(\frac{\pi H}{L}\right)\cdot \frac{\mathit{ch}\left(2k(z+d)\right)}{s{h}^4\left(\mathit{kd}\right)}\sin 2(\mathit{kx}-\omega t)$$

(11)

where $H$ is the wave height; $T$ is the wave period; $L$ is the wavelength; $k$ is the wave number ($k=2\pi /L$); $d$ is the water depth; $\omega$ is the wave frequency ($\omega =2\pi /T$); $\mathit{sh}$ is hyperbolic sine; and $\mathit{ch}$ is hyperbolic cosine.

The second-order wave surface Equation (12) is:

$$\eta =\frac{H}{2}\cos (\mathit{kx}-\omega t)+\frac{\pi H^2}{8L}\frac{\mathit{ch}\left(\mathit{kd}\right)}{{\mathit{sh}}^3\left(\mathit{kd}\right)}\cdot \left[2+\mathit{ch}\left(2\mathit{kd}\right)\right]\cos 2(\mathit{kx}-\omega t)$$

(12)

Combining the dispersion relation, the velocities of the horizontal and vertical mass points are given by Equations (13) and (14):

$$u=\frac{\pi H}{T}\cdot \frac{\mathit{ch}\left(k(z+d)\right)}{\mathit{sh}\left(\mathit{kd}\right)}\cos (\mathit{kx}-\omega t)+\frac{3\pi H}{4T}\left(\frac{\pi H}{L}\right)\cdot \frac{\mathit{ch}\left(2k(z+d)\right)}{{\mathit{sh}}^4\left(\mathit{kd}\right)}\cos 2(\mathit{kx}-\omega t)$$

(13)

$$v=\frac{\pi H}{T}\cdot \frac{\mathit{sh}\left(k(z+d)\right)}{\mathit{sh}\left(\mathit{kd}\right)}\sin (\mathit{kx}-\omega t)+\frac{3\pi H}{4T}\left(\frac{\pi H}{L}\right)\cdot \frac{\mathit{sh}\left(2k(z+d)\right)}{{\mathit{sh}}^4\left(\mathit{kd}\right)}\sin 2(\mathit{kx}-\omega t)$$

(14)

Taking the time derivative, the accelerations of the horizontal and vertical mass points are given by Equations (15) and (16):

$$a_x=\frac{2{\pi }^{2}H}{T^2}\cdot \frac{\mathit{ch}\left(k(z+d)\right)}{\mathit{sh}\left(\mathit{kd}\right)}\sin (\mathit{kx}-\omega t)+\frac{3{\pi }^3{H}^2}{T^{2}L}\cdot \frac{\mathit{ch}\left(2k(z+d)\right)}{{\mathit{sh}}^4\left(\mathit{kd}\right)}\sin 2(\mathit{kx}-\omega t)$$

(15)

$$a_{\mathrm{y}}=-\frac{2{\pi }^{2}H}{T^2}\cdot \frac{sh\left(k(z+d)\right)}{\mathit{sh}\left(\mathit{kd}\right)}\cos (\mathit{kx}-\omega t)-\frac{3{\pi }^3{H}^2}{T^{2}L}\cdot \frac{\mathit{sh}\left(2k(z+d)\right)}{{\mathit{sh}}^4\left(\mathit{kd}\right)}\cos 2(\mathit{kx}-\omega t)$$

(16)

Based on the actual water depth, wave height, and period, we can determine the wavelength and derive the equation for the wave profile, as well as the velocities and accelerations of the water mass points. Consequently, the velocity and acceleration at any point within the wave profile can be computed. Similar principles can be applied to higher-order Stokes formulas.

2.3. Wave Load Calculation Method

The formula for wave forces calculation, known as the Morison equation, was introduced in 1950 by Morrison, etc., from the University of California. It is commonly used to calculate wave forces on cylindrical components in fixed offshore platforms, particularly when viscosity plays a significant role. The drag force is derived by analyzing the forces exerted on the cylinder when a steady and uniform water flow passes around it, simulating a real viscous fluid. The inertia force, on the other hand, is determined by analyzing the unsteady potential flow of an ideal fluid [17].

When considering the relative motion between the structure and the water mass points, as well as the presence of currents, the original Morison equation needs to be further modified, as shown in Equation (17):

$$F=\frac{\pi }{4}{C}_M\rho D^2\left(\dot{u}\pm a_p\right)+\frac{1}{2}{C}_D\rho D\left|u\pm v_p\pm U\right|\left(u\pm v_p\pm U\right)$$

(17)

In Equation (17), C_M is the inertia coefficient; C_D is the drag coefficient; D is the diameter of the structure (m); u is wave velocity (m/s); U is flow velocity (m/s); ρ is fluid density (kg/m³); v_p is the velocity of the structural element (m/s), and a_p is the acceleration of the structural element (m/s²).

3. Establishment of the Finite Element Model of the Mariculture Ship

3.1. Structural Model

The numerical model was constructed referring to an actual mariculture ship, with some simplifications made [18]. The size specifications comply with relevant regulations set by the China Classification Society (CCS). Due to the fact that the structure is typical, the modeling approach is universal. The cage model mainly consists of three parts. The floating box model is established by a shell unit, and its basic parameters are shown in Figure 2. The grid model is established by beam units, and its basic parameters are shown in

Table 1. The basic parameters of the grid model.

Model	Component	Length (m)	Number	Density (g/cm3)	Elastic Modulus (Pa)
The grid model	Longitudinal main beam	91	12	7.85	2.1 × 1011
	Horizontal main beam	23	9	7.85	2.1 × 1011
		25	3	7.85	2.1 × 1011
		33	12	7.85	2.1 × 1011
	Upstand	9	116	7.85	2.1 × 1011
	Spur post	10	104	7.85	2.1 × 1011
	Lateral bracing	4.9	64	7.85	2.1 × 1011
		4.95	40	7.85	2.1 × 1011
		7	36	7.85	2.1 × 1011
	Short bracing	1.1	202	7.85	2.1 × 1011
		1.63	12	7.85	2.1 × 1011

. The basic parameters of the anchor chain model, established with a unidirectional line stiffness spring, are shown in

Table 2. Basic parameters of the grid model.

Model	Component	Length (m)	Offset Coordinates (m)	Elastic Modulus (Pa)
The anchor chain model	Anchor chain	85	X = −80	−2.1 × 1011
			Y = −30	0

. The schematic diagram of the established structural model is illustrated in Figure 3.

Considering that the density of the netting is approximately equal to that of seawater, the gravity and buoyancy of the netting need not be considered. The load applied on the floating tanks by the mooring chain is assigned to the deck load. In the simulation, the mooring chain is simplified as a spring, regardless of its weight.

3.2. Computational Loading Model

A three-dimensional model was established using the software Abaqus 2020, with a default convergence tolerance for residual forces under nonlinear conditions set at 0.5% of the average force over the entire time period. Dynamic loads were added using the Abaqus/Aqua module, and computations were performed using the implicit integration method based on the inversion of the integration operation matrices provided by Abaqus/Standard [19].

3.3. Wave–Current Load Model

The marine environmental parameters in the northern part of the South China Sea were provided by the SEAFINE JIP database and CCS-Metocean database. The extreme conditions refer to the data of super typhoon Mangkhut, which occurred in the South China Sea in 2018 [20]. This typhoon lasted for more than 10 h with an average wind speed of 48 m/s, gusts of wind speed of more than 63 m/s, and an irregular wave height of 3–4 m. A comprehensive approach was used, combining multiple methodologies and conducting comparative analyses to ascertain rational wave loads. This approach mitigated the potential inadequacy of safety margins resulting from reliance on a single method. Considering the actual marine geological conditions, we set certain parameters for the study. These included a water depth of less than or equal to 30 m, a current velocity of less than 2 m/s, wave heights of less than 6 m, and extreme conditions with wave periods ranging between 7 and 11 s [21].

3.4. Mooring Model

A ship with an irregular shape is generally fixed with a single-point mooring system. In this mooring mode, the structure can quickly adjust its position in response to wave forces from different directions, which is conducive to reducing the impact of wave forces on the structure and improving its ability to withstand typhoons. The model of the mooring chain uses a nonlinear spring, the property of which is that the tensile elastic modulus is similar to steel material when it is tautened, and the compressive elastic modulus is zero. The tensile elastic modulus of this mooring chain increases exponentially during the transition from a relaxed state to a tensioned state. After being tightened, the elastic modulus remains constant. The damping for the anchor chain is determined using actual calculations in the model.

A schematic representation of the structural model after preloading simulation is depicted in Figure 4.

4. Results and Discussion

4.1. Dynamic Response Analysis of Mooring Forces

Referring to the wave characteristics of the area near ‘Dehai-1’ when the typhoon “Mangkhut” passes through, the flow velocity is less than or equal to 1 m/s and the wave height is less than or equal to 4 m. Referring to the South China Sea wave scatter diagram, it can be observed that the significant wave height (Hs) is 4 m and more than 95% of the periods (Tp) are distributed within the range of 8 to 11 s [21]. Therefore, a flow velocity of 1 m/s, a wave height of 4 m, and a period of 8 s can be selected as the control group of the model in this paper, which is the basic working condition.

4.1.1. Analysis of Mooring Forces under Varying Flow Velocities

To analyze the effects of different flow velocities on a fully loaded mariculture ship, we extracted the time-dependent mooring forces in the X- and Y-directions of the anchor chains. The X-direction is along the ship, while the Y-direction is perpendicular to the ship. The variation in the mooring forces at the flow velocities of 1m/s, 1.5 m/s, and 2 m/s is depicted in Figure 5.

Under the action of periodic wave-induced forces with a fixed period, the structure shows periodic forced vibration. Increasing the flow velocity, the mooring forces exhibit a notable increase as the flow direction is opposite to the stretching direction of the mooring chain. The changing trend in mooring force and the distribution area at different flow velocities need to be observed by comparison. The peak and mean mooring forces under different flow velocities are presented in Figure 6.

At a certain flow velocity, the separation of the mooring force along the X-axis shows a larger range of fluctuations, while that along the Y-axis changes less. This indicates that the mooring force along the direction of the ship is more susceptible to flow velocity. On the same axis, there is no significant change in the range of mooring forces at different flow velocities. The magnitude of the mooring force increases approximately linearly as the flow velocity increases, with a more pronounced increase in the X-direction. This shows that the change in flow velocity does not have a significant effect on the relative value of the mooring force, but it does have a decisive effect on the absolute value of the mooring force.

4.1.2. Mooring Force Analysis under Wave Height Variations

A higher wave height corresponds to a shorter period. When the period decreases and the wave height increases to a certain extent, wave fragmentation occurs, which is not suitable for the current model. Therefore, wave heights ranging from 1 to 6 m were considered for the following analysis. The peak and mean values of mooring forces were comprehensively analyzed for different wave heights, and the variations in mooring forces on the structure were observed, as shown in Figure 7.

As the wave height increases, the mooring force increases in a quadratic function relationship. When the wave height is small, the change in mooring force is relatively moderate. Moreover, the higher the wave height, the larger the range of variation in mooring forces. This shows that non-wave factors are the main contributors to mooring force at small wave heights, but wave height is one of the main factors causing periodic and significant vibrations in mooring forces.

4.1.3. Analysis of Mooring Forces Variation with Period Changes

Referring to the basic working condition, the minimum wave period should not be less than 7 s, and the maximum wave period is unrestricted theoretically. However, considering practical circumstances, it should not exceed 17 s. The peak and average values of the mooring forces within the 7–17 s range were examined and analyzed, as shown in Figure 8.

The average values of the mooring force separation along the X- and Y-axes change slightly under different periodic conditions. When the period changes from 7 s to 17 s, the maximum and minimum values of the mooring force approximate the change pattern of a cubic curve. The behavior of the minimum value differs slightly from that of the maximum value, primarily due to some non-linearity when the mooring chain is not fully tightened.

When the waves are small, the forced vibration of the structural model floating in the ocean is dominated by the first-order vibration type. When the period increases, the structure will also be affected by other vibration patterns, presenting different vibration modes. When the fluctuation period of the wave exceeds 9 s, the fluctuation range of the mooring force of the structure increases continuously, and the vibration amplitude of the structure increases accordingly. As the period increases, the peak of the mooring force tends to reach a fixed value, and the vibration amplitude of the structure also stabilizes. However, the period does not increase indefinitely. Once the period extends to a certain limit, forced vibration of the structure will no longer occur.

4.2. Spatial Displacement Dynamic Response

As mentioned above, the horizontal vibrations of the structure lead to a periodic tightening or relaxation of the mooring chain, resulting in a periodic variation in the mooring force. There is a certain correspondence between the maximum mooring force and the spatial vibration of the structure. The fishery structure described in this paper is subject to a single-point mooring system, where the mooring chain is connected to the upper part of the first section of the structure. The fishery can rotate around the mooring point at the seabed. The spatial layout of the fishery structure is shown in Figure 9.

In order to describe the heave and pitch motion amplitude of the model under the influence of waves, four observation points were selected along the symmetry axis of the structure, denoted as A–D in Figure 9. By observing the overall spatial displacement at these points, the spatial displacement characteristics of the structure could be analyzed.

4.2.1. Spatial Displacement under Different Flow Velocities

Based on the basic working condition, the flow velocities were set at 1.0 m/s, 1.5 m/s, and 2.0 m/s. Oscillations at various points were simulated, where the wave height was 4 m and the period was 8 s. The time-dependent vertical displacements at different locations were captured at points A–D, as depicted in Figure 10.

The figure shows that the vertical heaving amplitude at points A and D, which are closer to the ends, is greater compared with points B and C in the middle. When the wave flows along the X-axis, the fishery undergoes longitudinal oscillations in the ZOX plane due to the mooring force. This results in periodic bow submergence and a certain degree of uplift at the tail end. This phenomenon becomes more pronounced as the flow velocity increases.

There is a certain degree of flexural deformation at different flow velocities. This bending always happens in the negative direction of the Z-axis. Due to the large span of the fishery structure, the gravity of the rigid frame structure is greater than its buoyancy. The deflection deformation of the structure is still dominated by its dead weight. It can be seen in the figure that deflection is inevitable, but in actual engineering, the deflection is minimal, and the influence on the structure can be ignored. This highlights the advantages of having a more integrated permeable rigid-frame structure and a larger moment of inertia on the weaker surface. Support points can be increased using multi-point suspension floating in the middle of the structure, reducing flexion deformation.

4.2.2. Spatial Displacement under Different Wave Heights

On the basis of the basic working condition, we increased the wave height by the same amplitude. We extracted the time-dependent displacement at points A to D along the Z-axis and observed the change in the oscillation amplitude of each point, as shown in Figure 11.

With the increase in wave height from 3m to 5m, the oscillation amplitude of each point increased significantly. This indicates that the wave height has a remarkable influence on the oscillation of the structure, making it more unstable. Points A, B, and C all shifted downward in their equilibrium positions, while only point D moved upward. The equilibrium position of point B had the largest downward offset, not that of point A. The subsidence amount at point C was roughly equivalent to that at point A.

Comprehensively, calculations were performed by varying wave heights from 1 to 6 m. Computing the absolute value of the heave difference in vertical displacement between the bow and the stern for each time step, the longitudinal oscillation differences under different flow velocities were obtained, as depicted in Figure 12.

As wave height increases, the heave difference between the bow and stern gradually increases. Though the bow will sink while the stern lifts at a certain flow velocity, the stability of the boat during these waves is mainly affected by its weight, and it remains stable when the waves are shorter. However, with the increase in wave height, the wave velocity potential and acceleration potential will increase, leading to a greater impact on the boat’s vibrations. This can result in a vertical deviation of the boat’s structure.

Similarly, the maximum deflection between the ends and the midsection could be calculated, where the deflection of downward bending is defined as positive. The median deflection values at different flow rates are shown in Figure 13.

Within the wave height range of 1 to 3 m, the mid-span deflection of the structure increases with the rise in the wave height, accompanied by a wider fluctuation amplitude. As the wave height increases from 4 m to 6 m, the mid-span deflection value stabilizes around 0.14–0.15 m, with a relatively minor fluctuation amplitude. Hence, the mid-span deflection and its fluctuation amplitude do not exhibit a continuous increase with the rise in wave height. When the wave height reaches 4–6 m, the spatial displacement feature of the structure is not only affected by heaving motion but also by pitch motion. This gradually starts to have a significant influence on the spatial deformation, resulting in the production of a higher order vibration pattern on the main body of the structure. Due to the interaction between the first order vibration caused by the dead weight and higher order vibration of the structure, the deflection value in the mid-span does not increase with the wave height.

4.2.3. Spatial Displacement under Different Periods

On the basis of the basic working condition, we increased the wave period from 8 s to 14 s. The oscillations of various points from A to D were observed. The time-dependent displacement along the Z-axis were extracted, as presented in Figure 14.

When the wave period increases, the heave of the structure at point A is not obvious, and the difference between the upper and lower peaks is within 0.3 m. This indicates that the front end of the fishery is less affected by the cycle change, which indicates that the mooring chain plays a certain role in restraining its vertical movement. The heaving range at the places from point B to point D increases rapidly, and the oscillation interval at point D increases from 0.05~0.15 m at 8 s to −0.7~0.9 m at 16 s, indicating that the heaving at the end of the fishery is greatly affected by the period.

A longer wave period corresponds to a longer wavelength. When the wavelength increases, the fishery can accommodate less waves in the length range. The lower order vibration is the dominant pattern, resulting in a larger swing in the tail of the structure. Similar analysis shows that the mid-span deflection does not increase significantly even though the stern heaves intensively because of higher wave height with longer wave period.

5. Conclusions

Numerical simulation is an important means for hydrodynamic analysis of mariculture ships. Structural models of a mariculture ship floating in seawater are established in this paper, using beam elements instead of shell elements. Thus, the modeling process is simplified by eliminating element division, and the computational efficiency is enhanced, by which the model establishment and analysis of complex full-scale structural models is ensured. By introducing the conception of dynamic equilibrium between gravity and buoyancy forces, as well as a non-linear uni-axial tension spring-damping mooring chains, the uncertainty in boundary conditions for floating structures is optimized.

Increasing the flow velocity leads to a linear increase in the mooring force. The change in the flow velocity has a greater influence on the mooring force in the X-axis direction. On the same axis, there is no significant change in the range of mooring forces at different flow velocities. The mooring force increases with the wave height in a quadratic relationship, and the vibration of the ship intensified remarkably. Increasing the wave period, the maximum and minimum values of the mooring force change in an approximately cubic curve. The change law of the minimum is slightly different from that of the maximum, which is mainly due to some non-linearity when the mooring chain is not fully tightened.

Under wave–current impact, the fishery generates negative bending moments within the ZOX plane because of the dragging effect of the mooring chain. The bow of the ship inevitably goes downward, while the stern end is uplifted. This phenomenon becomes more obvious with the increase in the flow velocity.

The wave height has a remarkable influence on the oscillation of the structure. As wave height increases, the heave difference between the bow and stern gradually increases. The mid-span deflection and its fluctuation amplitude do not exhibit a continuous increase with the rise in wave height.

Period variation predominantly impacts the vibration amplitude at the rear end of the fishery. The mid-span deflection and overall heaving of the fishery are both linked to the forced vibration order of the structure. Low order vibration is more likely to cause a large heaving in the tail end of the mariculture ship.