Hydrodynamic Study of a Fall Pipe Rock Dumping System

Abstract

The fall pipe rock dumping technique is extensively employed to create protection embankments around submarine cables, mitigating distortion and breakage resulting from bottom scouring. During the rock dumping operation, intricate interactions among the pipeline, rocks, and water currents can affect the stability and efficiency of the fall pipe system. This research proposed a method employing the fluid–structure interaction to analyze the interactions between the pipeline, rocks, and water currents. The paper begins with the design of an innovative fall pipe rock dumping system and presents a theoretical analysis of the applied model testing approach. The simulation parameters were determined according to the geometric, Froude, and Strouhal similarity criteria. A thorough numerical analysis was performed to investigate the hydrodynamic properties of the rockfall pipeline under fluid–structure interaction. The research examined the settling of rocks during rockfall, along with the forces and movements associated with the deposition process. The results show that the rockfall pipeline experienced vortex-induced vibrations (VIVs) caused by ocean currents during operation. The maximum settling velocity of the rocks throughout the rockfall process reached 2.2 m/s, with a final stable velocity of 1.5 m/s. These simulation results offer critical insights for improving the design and functionality of the rockfall pipeline, thereby enhancing the protection of underwater infrastructure.

1. Introduction

Submarine cables buried beneath the seabed frequently experience distortion, fractures, and leaks during deployment and operation due to external factors, including seabed scouring. In some cases, these cables remain exposed on the seabed, unburied and unprotected, owing to the existence of hard seabed rocks and unstable geological circumstances [1,2,3]. Moreover, undersea cables often cannot be buried due to obstacles like dense sediment, exposed rock formations, and unstable geological conditions, which exacerbate the risk of damage. Protective precautions for these cables are important. A feasible solution was to encase them in stones. In contrast to the technique of immediately hurling rocks from the water’s surface, a rock berm can be more accurately assembled over cables by dumping rocks on the seabed via a rockfall pipeline that extends from a vessel to the seabed. The newly developed fall pipe rock dumping system [4,5] was explicitly engineered for the insertion of submarine cable berms and comprised the following three principal components: a rock-throwing vessel, a rockfall pipeline, and a remotely operated vehicle (ROV). Thus far, developers have effectively built submerged rock berms at depths exceeding 100 m utilizing this method [4].

As the depth of water at which submarine cables are embedded grows, the length of the rockfall pipeline extending from the vessel likewise increases. The rockfall pipeline subjected to water currents produces vortex-induced vibrations (VIVs) [6,7]. This vibration disrupts the alignment of the rockfall pipeline outlet with the upper section of the cable to be buried. Moreover, the rocks employed to cover the submarine cables traverse a segment of ocean after departing from the rockfall pipeline before resting atop the cables. The fall pipe rock dumping system entails interactions among the rockfall pipeline, rocks, and the surrounding flow field. This interaction induces deformation and VIV in the rockfall pipeline due to water currents, while also redirecting the rocks ejected from the pipeline, resulting in their deposition at unintended sites. If the rockfall operation is executed without accounting for these considerations, it may result in inadequate burial efficacy or potential damage to the cables [8].

The considerable length and diameter of a real rockfall pipeline pose challenges to studying its hydrodynamic properties through model testing methods. Computational fluid dynamics (CFD) is more appropriate for numerical simulation. The bar theory CFD model was utilized to analyze the transverse VIV of a long flexible pipeline subjected to uniform water currents, demonstrating its multimodal vibration behavior under these flow conditions [9]. The wake oscillator model [10], grounded in bar theory, examined the deep-sea pipeline and its wakefield as a system, focusing on the characteristic response of a long cylinder to VIVs. Continuous deep-sea pipelines are predominantly utilized in offshore drilling platforms within real engineering applications. Most research on the dynamic response of deep-sea pipelines to water currents concentrated on continuous pipeline systems housed within watertight pipes [11]. The VIV characteristics of these systems are affected by operational parameters, including tension [12], top injection force, platform offset, and environmental factors such as water depth and wave characteristics [13]. Additionally, numerous methods for mitigating VIVs in cylinders have been established [14]. In contrast to the riser structure of offshore drilling platforms, which is restricted by both the platform and the seabed [15], the rockfall pipeline is constrained solely by the vessel platform. The existence of a free end complicates the forces acting on the rockfall pipeline in water currents, rendering the prediction of the pipeline outlet’s motion a significant challenge.

Rocks utilized for covering the submarine pipeline are initially released through the rockfall pipeline, subsequently flow out of the outlet, descend with the water currents, and ultimately settle on the seabed to encase the cables. Interactions take place between the rocks and the pipeline wall, among the rocks, and with water currents post-exit, as well as between the rocks and the cables intended for burial upon reaching the seabed [16]. Wang et al. [17] and Qi et al. [18] employed high-speed cameras to document the dynamics of objects descending into water. The Euler–Lagrange method can be utilized for numerical calculations of the interaction between rocks and water currents. Several computational methods for fluid–solid mixtures have been proposed and implemented, including the multiphase particle-in-cell method (MP-PIC) [19], the computational fluid dynamics discrete element method (CFD-DEM) [20], the arbitrary Lagrangian–Eulerian method (ALE) [21], and methods that couple the discrete element method (DEM) with smooth particle hydrodynamics (SPH) [22]. Ansys Fluent can be integrated with the Rocky module to model intricate particle–fluid interactions. Current research predominantly examined the relationship between falling rocks and submarine cables to prevent damage to the cables during the burial process [23]. After the rocks exit the pipeline, their interaction with water currents significantly affects the final burial of the cable. Analyzing the final rock settling process in the rockfall pipeline system is essential for preventing ineffective rock coverage that does not protect the submarine cable and for avoiding resource wastage.

This paper initially designs the essential structure of the rockfall pipe to address the aforementioned issues. The analysis employs CFD to examine the flow field, forces, and deformation surrounding the rockfall pipe during operation, grounded in the principle of model testing. The movement of rocks within the pipeline is subsequently analyzed utilizing DEM. The paper examines the interaction among the rockfall pipeline, rock, and water currents during the rock-throwing operation. Section 2 describes the numerical methods and models applied in this work. Section 3 provides an overview of the rockfall pipeline model and its basic components, explaining how to use similarity criteria to achieve fluid simulation of the physical model and verifying the convergence of the mesh. Section 4 examines the hydrodynamic properties of the joints and the complete rockfall pipeline in relation to water currents. Section 5 analyzes the movement of rocks upon their egress from the pipeline during laying operations. Section 6 summarizes the findings of the paper.

2. Numerical Methods

2.1. Volume of Fluid

The Volume of Fluid (VOF) model in CFD is a numerical method used for tracking and simulating free liquid surfaces or interfaces in multiphase flows [24]. This method employs Fractional Area/Volume Obstacle Representation (FAVOR) to introduce volume fractions or area fractions, which represent the distribution of different phases within the computational grid. Here, the volume fraction and area distribution are uniformly denoted as $\alpha$. When $\alpha$ = 1, the grid cell is completely occupied by the liquid phase. When $\alpha$ = 0, the grid cell is completely occupied by the gas phase. For 0 < $\alpha$ < 1, the grid cell contains both the liquid and gas phases, typically representing the free liquid surface.

The VOF model describes multiphase flow by using the conservation equation, and the transport equation for the volume fraction is given by the following:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \alpha }{\partial t}}+\nabla ·\left(\alpha \mathbf{u}\right)=0\end{array}$$

(1)

For an incompressible fluid, the density $\rho$ is constant, so the equation simplifies to the following:

$$\begin{array}{c}\hfill \nabla ·\mathbf{u}=0\end{array}$$

(2)

and its momentum conservation equation is as follows:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \mathbf{u}}{\partial t}}+(\mathbf{u}·\nabla )\mathbf{u}=-{\displaystyle \frac{1}{\rho }}\nabla p+\nu {\nabla }^2\mathbf{u}+\mathbf{g}\end{array}$$

(3)

At the free liquid level, the density of the unit fluid is $\rho =\alpha {\rho }_{\mathrm{water}}+(1-\alpha ){\rho }_{\mathrm{air}}$ and the viscosity is $\mu =\alpha {\mu }_{\mathrm{water}}+(1-\alpha ){\mu }_{\mathrm{air}}$. Thus, the governing equation for free liquid level flow can be expressed as follows:

$$\begin{array}{cc}\hfill \nabla ·\left(\rho \mathbf{u}\right)& =0\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \left(\rho \mathbf{u}\right)}{\partial t}}+\nabla ·\left(\rho \mathbf{u}\mathbf{u}\right)& =-\nabla p+\nabla ·\left[\mu \left(\nabla \mathbf{u}+{(\nabla \mathbf{u})}^T\right)\right]+\mathbf{F}\hfill \end{array}$$

(5)

Considering ${\rho }_{\mathrm{water}}\gg {\rho }_{\mathrm{air}},{\mu }_{\mathrm{water}}\gg {\mu }_{\mathrm{air}}$, the density and viscosity of the gas phase are negligible, and then, the continuity equation at the free liquid surface can be expressed in the following form for the three-dimensional case:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial x}}\left(u{A}_x\right)+{\displaystyle \frac{\partial }{\partial y}}\left(v{A}_y\right)+{\displaystyle \frac{\partial }{\partial z}}\left(w{A}_z\right)=0\end{array}$$

(6)

The Navier–Stokes (N-S) equations can be expressed in the following form for the three-dimensional case:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial u}{\partial t}}+{\displaystyle \frac{1}{V_F}}\left(u{A}_x{\displaystyle \frac{\partial u}{\partial x}}+v{A}_y{\displaystyle \frac{\partial u}{\partial y}}+w{A}_z{\displaystyle \frac{\partial u}{\partial z}}\right)=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial p}{\partial x}}+g_x+f_x\\ \hfill {\displaystyle \frac{\partial v}{\partial t}}+{\displaystyle \frac{1}{V_F}}\left(u{A}_x{\displaystyle \frac{\partial v}{\partial x}}+v{A}_y{\displaystyle \frac{\partial v}{\partial y}}+w{A}_z{\displaystyle \frac{\partial v}{\partial z}}\right)=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial p}{\partial y}}+g_y+f_y\\ \hfill {\displaystyle \frac{\partial w}{\partial t}}+{\displaystyle \frac{1}{V_F}}\left(u{A}_x{\displaystyle \frac{\partial w}{\partial x}}+v{A}_y{\displaystyle \frac{\partial w}{\partial y}}+w{A}_z{\displaystyle \frac{\partial w}{\partial z}}\right)=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial p}{\partial z}}+g_z+f_z\end{array}$$

(7)

In the above equations, $A_x$, $A_y$, and $A_z$ denote the area fraction of the fluid in the grid along the x, y, and z directions, respectively. The variables u, v, and w represent the components of the velocity in the x, y, and z directions, respectively. $V_F$ is the volume fraction, $\rho$ is the fluid density, and p is the mean fluid pressure. $g_x$, $g_y$, and $g_z$ are the components of the gravitational acceleration in the x, y, and z directions, respectively. $f_x$, $f_y$, and $f_z$ are the components of the viscous acceleration in the x, y, and z directions, respectively, which can be expressed as follows:

$$\begin{array}{c}\hfill f_x={\displaystyle \frac{{\Omega }_x}{\rho V_F}}\left[{\displaystyle \frac{\partial }{\partial x}}\left(A_x{\tau }_{xx}\right)+{\displaystyle \frac{\partial }{\partial y}}\left(A_y{\tau }_{xy}\right)+{\displaystyle \frac{\partial }{\partial z}}\left(A_z{\tau }_{xz}\right)\right]\\ \hfill f_y={\displaystyle \frac{{\Omega }_y}{\rho V_F}}\left[{\displaystyle \frac{\partial }{\partial x}}\left(A_x{\tau }_{xy}\right)+{\displaystyle \frac{\partial }{\partial y}}\left(A_y{\tau }_{yy}\right)+{\displaystyle \frac{\partial }{\partial z}}\left(A_z{\tau }_{yz}\right)\right]\\ \hfill f_z={\displaystyle \frac{{\Omega }_z}{\rho V_F}}\left[{\displaystyle \frac{\partial }{\partial x}}\left(A_x{\tau }_{xz}\right)+{\displaystyle \frac{\partial }{\partial y}}\left(A_y{\tau }_{yz}\right)+{\displaystyle \frac{\partial }{\partial z}}\left(A_z{\tau }_{zz}\right)\right]\end{array}$$

(8)

where ${\Omega }_x$, ${\Omega }_y$, and ${\Omega }_z$ represent the wall shear stress in the x, y, and z directions. The term ${\tau }_{ij}$ represents the shear stress in the i and j directions, where $i,j\in \left\{x,y,z\right\}$, and they are related to the dynamic viscosity $\mu$ and the velocity gradient, which can be expressed as follows:

$$\begin{array}{cc}\hfill {\tau }_{xx}& =-2\mu \left[{\displaystyle \frac{\partial u}{\partial x}}-{\displaystyle \frac{1}{3}}\left({\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{\partial v}{\partial y}}+{\displaystyle \frac{\partial w}{\partial z}}\right)\right]\hfill \\ \hfill {\tau }_{yy}& =-2\mu \left[{\displaystyle \frac{\partial v}{\partial y}}-{\displaystyle \frac{1}{3}}\left({\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{\partial v}{\partial y}}+{\displaystyle \frac{\partial w}{\partial z}}\right)\right]\hfill \\ \hfill {\tau }_{zz}& =-2\mu \left[{\displaystyle \frac{\partial w}{\partial z}}-{\displaystyle \frac{1}{3}}\left({\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{\partial v}{\partial y}}+{\displaystyle \frac{\partial w}{\partial z}}\right)\right]\hfill \\ \hfill {\tau }_{xy}& =-\mu \left({\displaystyle \frac{\partial v}{\partial x}}+{\displaystyle \frac{\partial u}{\partial y}}\right)\hfill \\ \hfill {\tau }_{xz}& =-\mu \left({\displaystyle \frac{\partial w}{\partial x}}+{\displaystyle \frac{\partial u}{\partial z}}\right)\hfill \\ \hfill {\tau }_{yz}& =-\mu \left({\displaystyle \frac{\partial w}{\partial y}}+{\displaystyle \frac{\partial v}{\partial z}}\right)\hfill \end{array}$$

(9)

2.2. Shear Stress Transport $k-\omega$ Model

Equations (7) and (8) are written in the Reynolds-Averaged (RANS) form and therefore require a turbulent viscosity ${\mu }_t$ to achieve closure. The effective viscosity in the momentum equations is ${\mu }_{\mathrm{eff}}=\mu +{\mu }_t$. In this study, ${\mu }_t$ is provided by the $k-\omega$ model as ${\mu }_t$ as follows:

$${\mu }_t=\rho a_1{\displaystyle \frac{k}{max(a_1\omega ,S{F}_2)}},$$

(10)

where S is the mean strain rate, $F_2$ is the blending function, and $a_1$ is a model constant. The transport equations for k and $\omega$ are given in Equations (11) and (12).

The Shear Stress Transport $k-\omega$ (SST $k-\omega$) model is a turbulence model proposed by Menter [25,26]. It employs the $k-\omega$ model in the near-wall region and gradually transforms to the $k-ϵ$ model in the free shear layer region. This approach combines the strengths of the standard $k-\omega$ model for near-wall regions with the advantages of the $k-ϵ$ model for free shear flows in the far field, resulting in more accurate turbulence predictions. The transport equation for the turbulent kinetic energy k in this model is as follows:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \rho k}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left[\rho u_{j}k-\left(\mu +{\sigma }_k{\mu }_t\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]=P_k-{\beta }^{*}\rho \omega k\end{array}$$

(11)

The transport equation for the specific dissipation rate $\omega$ is as follows:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \rho \omega }{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left[\rho u_j\omega -\left(\mu +{\sigma }_{\omega }{\mu }_t\right){\displaystyle \frac{\partial \omega }{\partial x_j}}\right]={\gamma }_{\omega }{\displaystyle \frac{P_k}{k}}-\beta \rho {\omega }^2+2(1-F_1){\sigma }_{\omega 2}\rho {\displaystyle \frac{1}{\omega }}{\displaystyle \frac{\partial k}{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}}\end{array}$$

(12)

where $u_j$ is the velocity component and $P_k$ is the production term of turbulent kinetic energy, defined as $P_k={\tau }_{tij}{S}_{ij}$. The turbulent viscous model of the Reynolds stress is given by the following:

$$\begin{array}{c}\hfill {\tau }_{tij}=2{\mu }_t\left(S_{ij}-S_m{\displaystyle \frac{{\delta }_{ij}}{3}}\right)-2\rho k{\displaystyle \frac{{\delta }_{ij}}{3}}\end{array}$$

(13)

In the above expression, ${\mu }_t$ represents the turbulent viscosity, $S_{ij}$ is the mean velocity strain rate tensor, and ${\Delta }_{ij}$ is the Kronecker delta. The coefficients ${\sigma }_k$, ${\sigma }_{\omega }$, and ${\sigma }_{{\omega }_2}$ are the turbulence model coefficients, with values of 0.85, 0.5, and 0.856, respectively. The constants $\gamma$, $\beta$, and ${\beta }^{*}$ are also part of the turbulence model, with values of 0.52, 0.075, and 0.09, respectively. $F_1$ is the blending function, which is used to smooth the transition between the near-wall region and the free shear layer.

2.3. Computational Fluid Dynamics-Discrete Element Method

The CFD-DEM [27] is a multiphase flow numerical simulation approach that integrates CFD with the DEM. This method is primarily used to simulate the complex interactions between particle systems and fluid flow. Due to the presence of particulate matter, the Navier–Stokes equations for incompressible fluids must include an additional term to account for solid–fluid interactions. Denoting the interaction force between the fluid and the particles as ${\mathbf{F}}_{fp}$, this term can be expressed as follows:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \left(\rho \mathbf{u}\right)}{\partial t}}+\nabla ·\left(\rho \mathbf{u}\mathbf{u}\right)=-\nabla p+\nabla ·\left[\mu \left(\nabla \mathbf{u}+{(\nabla \mathbf{u})}^T\right)\right]+\mathbf{f}+{\mathbf{F}}_{fp}\end{array}$$

(14)

The motion and interaction of particles, are primarily described by the translational and rotational equations of motion, respectively, as follows:

$$\begin{array}{cc}\hfill m_p{\displaystyle \frac{d{\mathbf{u}}_p}{dt}}& =\sum {\mathbf{F}}_p+{\mathbf{F}}_{fp}\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill I_p{\displaystyle \frac{d{\omega }_p}{dt}}& =\sum {\mathbf{T}}_p\hfill \end{array}$$

(16)

where $m_p$ is the particle mass; ${\mathbf{u}}_p$ is the particle velocity; $\sum {\mathbf{F}}_p$ represents the interaction forces between the particles, including contact force, friction force, etc.; ${\mathbf{F}}_{fp}$ is the fluid force acting on the particles; $I_p$ is the particle rotational inertia; ${\omega }_p$ is the particle angular velocity; and $\sum {\mathbf{T}}_p$ is the rotational torque between the particles. The fluid force ${\mathbf{F}}_{fp}$ acting on the particles primarily includes the drag force and buoyancy force, which can be expressed, respectively, as follows:

$$\begin{array}{cc}\hfill {\mathbf{F}}_d& ={\displaystyle \frac{1}{2}}{C}_d{\rho }_f{A}_p\left|{\mathbf{u}}_r\right|\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\mathbf{F}}_b& =-{\rho }_f{V}_{p}g\hfill \end{array}$$

(18)

where ${\mathbf{F}}_d$ is the drag force, ${\mathbf{F}}_b$ is the buoyancy force, $C_d$ is the coefficient of drag, ${\rho }_f$ is the fluid density, $A_p$ is the area of the particles facing the flow, ${\mathbf{u}}_r$ is the relative velocity defined as ${\mathbf{u}}_r=\mathbf{u}-{\mathbf{u}}_p$, $V_p$ is the volume of the particles, and g is the gravitational acceleration.

2.4. Fluid-Structure Interaction

Fluid-Structure Interaction (FSI) is a multi-physics phenomenon describing bidirectional coupling between deformable/movable structures and the surrounding fluid flow. In such systems, fluid forces (e.g., pressure and viscous stress) induce structural motion or deformation, while the structural response conversely modifies the fluid flow field through boundary displacement or velocity changes. This interdisciplinary methodology plays a pivotal role in applications ranging from biomedical devices to industrial processes involving multiphase flows [28].

In the FSI analysis of piping systems, the coupled simulation of CFD and structural mechanics is essential for understanding dynamic interactions. In this study, turbulent excitation in rockfall pipelines can induce pipe wall vibrations or deformations, while the structural response inversely affects the fluid flow field by altering velocity distributions and pressure gradients. First, a three-dimensional fluid model is established using CFD, employing VOF and SST $k-\omega$ models to solve the Navier–Stokes equations. This establishes a dynamic model for the wave flow and captures fluid velocity, pressure, and wall shear stress. Subsequently, fluid loads are imported into the structural mechanics module to analyze pipe stress, strain, and displacement fields. This requires a consideration of material nonlinearities and dynamic effects, such as resonance risks when fluid excitation frequencies approach the pipeline’s natural frequencies.

During the process of model solving, dynamic mesh technology is required to update fluid domain grids in real time, ensuring continuous interface velocity and stress equilibrium. Local mesh refinement is implemented in regions with significant deformation to balance accuracy and computational efficiency. In terms of solution strategies, weak coupling is suitable for small deformation problems, while strong coupling is employed for large deformations or high-frequency interactions, utilizing the Newton–Raphson method and adaptive time-stepping to control stability. Particularly in transient impact load analysis, damping terms must be introduced to suppress numerical oscillations. Taking submarine oil pipeline VIVs as an example, CFD simulations calculate vortex shedding frequency and lift coefficients of ocean currents around the pipeline, while structural analysis evaluates vibration amplitudes and fatigue life. Through coupled iterative corrections of vortex shedding patterns, this process forms a closed-loop fluid–structure interaction framework.

3. Model and Mesh

3.1. Designed Models

This paper details the design of a fall pipe rock dumping system, illustrated in Figure 1, which is capable of executing rock-throwing operations at water depths exceeding 100 m. The vessel features a stone storage silo designed to accommodate its capacity and the specific stone-throwing requirements of the project. A stone transport system is installed, segmented into multiple stages based on transmission length. This segmentation improves the efficiency of transporting stones from the storage silo to the rockfall pipeline operating platform. This system guarantees the delivery of rocks via the rockfall pipeline to the seabed located above the submarine cable. A ROV is utilized at the pipe’s terminus to modify the position of the rockfall pipeline, facilitating the accurate construction of a rock berm over the submarine cable.

This study examines the interaction between the submerged portion of the rockfall pipeline and seawater within the components of the fall pipe rock dumping system. The structure primarily comprises DN500 connecting pipes, pipe adapters, welding frames, and a ROV, as illustrated in Figure 2. The DN500 connecting pipe measures 4 m in length, featuring an outer diameter of 500 mm and an inner diameter of 480 mm. Pipe adapters are used to join the connecting pipes, as depicted in Figure 3. This configuration allows the fall pipe rock dumping system to operate at depths exceeding 100 m. A frame may be welded to the end of the pipe underwater to provide support for the ROV’s loading.

Figure 4 illustrates the assembly operation of the rockfall pipeline for lifting and dropping. The welding frame that supports the ROV is suspended at the water inlet of the lower deck by a crane from the upper deck. A connecting pipe is subsequently elevated and affixed to the upper section of the welding frame utilizing a pipe adapter. The assembled section is fastened at the water outlet using a connecting clamp. The crane continuously lifts subsequent connecting pipes, which are then joined using adapters. The steps are reiterated until the target length of the rockfall pipeline is achieved, modified in accordance with the operational water depth.

Upon installation of the rockfall pipeline, the rockfall operation may commence. The deck of the rock-throwing vessel features two stone storage silos, each linked to its corresponding conveyor belt, as illustrated in Figure 5. The conveyor belt facilitates the movement of rocks from the storage silos to the rockfall pipeline operating platform, enabling their delivery beneath the sea surface via the rockfall pipeline. Alongside the adjustable ROV positioned at the terminus of the rockfall pipeline, three fixed anchoring cables are affixed to the bow, stern, and hull of the vessel. These cables collectively support the forces exerted on the rockfall pipeline by the water current.

3.2. Similarity Criteria

As the operating depth of the fall pipe rock dumping system exceeds 100 m, it requires a rockfall pipeline that extends hundreds of meters. Therefore, direct numerical calculations on a 100 m scale are impractical. Consequently, a model-testing approach is necessary [29]. In conducting model tests for numerical calculations, the model of the fall pipe rock dumping system must exhibit geometric similarity to the actual vessel in appearance and fulfill the geometric similarity criteria for the performance-related parameters, including weight, center of gravity, and inertia. The marine environmental conditions outlined in the model test must adhere to both geometric and mechanical property similarity requirements.

The criterion for geometric similarity between the actual entity and the model is that the ratio of corresponding linear dimensions must be constant in both cases. Let $L_s$, $R_s$, and $d_s$ denote the length, diameter, and draft of the actual rockfall pipeline, respectively, while $L_m$, $R_m$, and $d_m$ represent the corresponding dimensions of the model, respectively. The relationship can be articulated as ${\displaystyle \frac{L_s}{L_m}}={\displaystyle \frac{B_s}{B_m}}={\displaystyle \frac{d_s}{d_m}}=\lambda$, where $\lambda$ denotes the scaling ratio. Alongside ensuring that the rockfall pipeline model and the actual entity adhere to geometric similarity conditions, it is essential that the environmental conditions during testing are proportionally scaled and simulated. The simulated water depth $h_m$, wave height $H_m$, and wavelength ${\lambda }_m$ of the model must align with the actual water depth $h_s$, wave heights $H_s$, and wavelengths ${\lambda }_s$ of the entity at sea, respectively, adhering to the geometric similarity conditions. The relationship can be represented as ${\displaystyle \frac{h_s}{h_m}}={\displaystyle \frac{H_s}{H_m}}={\displaystyle \frac{{\lambda }_s}{{\lambda }_m}}=\lambda$. In summary, when linear scale parameters are applied in a model test, it is essential to fulfill the geometric similarity condition. This ensures that the entities are scaled and simulated in accordance with the linear scaling ratios applied to the model.

The fall pipe rock dumping system is affected by the wind, wave, and water currents during operation. This paper focuses on simulating its motion and force under these effects. Gravity and inertia are the main factors determining its force. Therefore, the simulation calculation of the model should satisfy the Froude similarity criterion, which means that the Froude number (Fr) of the model and the entity should be equal to ensure the correct similarity relation of gravity and inertia forces between the model and the entity. In addition, the motion and force of the object on the wave carry the nature of periodic change. For example, in this paper’s study of the rockfall pipeline, when the water flow passes through it, its backside will periodically generate a vortex, thus causing the pipe to be subjected to periodic force, forming VIVs. The model, and the entity must also keep the Strouhal number (Sr) equal, i.e., ${\displaystyle \frac{V_m}{\sqrt{g{L}_m}}}={\displaystyle \frac{V_s}{\sqrt{g{L}_s}}},{\displaystyle \frac{V_m{T}_m}{L_m}}={\displaystyle \frac{V_s{T}_s}{L_s}}$, where $V_s$, $L_s$, $T_s$ represent the characteristic velocity, characteristic linear scale, and time period for the actual entity, respectively, and $V_m$, $L_m$, and $T_m$ represent these quantities for the model, respectively. Their relationship with the scaling factor $\lambda$ is as follows: ${\displaystyle \frac{V_s}{V_m}}=\sqrt{\lambda }$, ${\displaystyle \frac{T_s}{T_m}}=\sqrt{\lambda }$, and ${\displaystyle \frac{L_s}{L_m}}=\lambda$.

3.3. Grid Verification

This part of the study is a comparative validation of numerical computations under multiple grid sizes. It verifies the reliability of the adopted turbulence model and the irrelevance of the computational grids of the numerical model [30], and it analyzes the convergence and accuracy of the experimental cases under different grids. The flow field region and simulation objects set up for this validation experiment are shown in Figure 6, including a single connecting pipe (Figure 6a) at normal scale and a whole rockfall pipeline (Figure 6b) at a scaling ratio of 100, under the force conditions in the flow field. The dimensions of the flow field in Figure 6a are 15 m × 6 m × 5 m, with a flow rate magnitude of 0.6 m/s, and the dimensions of the flow field in Figure 6b are 1 m × 0.8 m × 0.2 m, with a flow rate magnitude of 0.06 m/s. The flow in the flow field is inflowed from the Inlet side and outflowed from the Outlet side, and the remaining four sides are wall surfaces through which the fluid cannot pass. In this study, a tetrahedral mesh was used for the flow field, a hexahedral mesh for the connecting pipe and rockfall pipeline, and mesh refinement was applied at the fluid–structure interface.

For the force analysis of a single connecting pipe in the flow field at a normal scale, we have chosen five experimental examples with cell grid sizes of 0.1 m, 0.2 m, 0.3 m, 0.4 m, and 0.5 m for comparative validation. The gridding structure of the five sizes is shown in Figure 7.

Figure 8 monitors the resistance of a single connecting pipe with five different grid sizes in a flow field with an incoming flow of 0.6 m/s at normal scales. It can be found from this figure that when the number of computational iteration steps exceeds 30, the results of the computation under the various grid sizes begin to converge to the convergence value. Specifically, when the grid sizes are 0.1 m, 0.2 m, 0.4 m, and 0.5 m, the change of the iterative results shows a very obvious tendency to converge. When the grid size is 0.5 m, the scaled residuals in the iterative process are the closest to 0. This indicates the strongest convergence under this condition. Considering the computational efficiency and accuracy, the grid size of 0.5 m is the best, and the number of computational iterations is set to 30 steps to ensure the convergence and accuracy of the model computation as well as the improvement of computational efficiency.

For the force analysis of the whole rockfall pipeline in the flow field under the scaling of 100, we chose four experimental cases with cell grid sizes of 0.025 m, 0.05 m, 0.075 m, and 0.1 m for comparative validation, and the mesh structure of the four sizes is shown in Figure 9.

Figure 10 monitors the resistance of the whole rockfall pipeline in a flow field with an incoming flow of 0.06 m/s for four different grid sizes at a scaling of 100. From this figure, it can be found that when the number of computational iteration steps exceeds 20, the grid sizes are 0.025 m, 0.05 m, and 0.075 m, the iterative results change in a very obvious tendency to converge. However, considering the subsequent need to calculate the deformation of the entire rockfall pipeline requires the use of dynamic mesh technology, the mesh size should not be too large to avoid the failure of the mesh update. Therefore, the mesh size of 0.05 m is optimal, and the number of iterations is set to 20 steps to ensure the convergence and accuracy of the model calculation, as well as to ensure the quality of the mesh update.

4. Numerical Simulation Analysis of Hydrodynamic of Rockfall Pipeline

The operation of the fall pipe rock dumping system is affected by several factors, notably the influence of the rockfall pipeline. Water flowing through the pipeline generates periodic vortices, which result in VIVs. The pipeline is influenced by the background flow generated by wave-induced water currents. The characteristics of this flow fluctuate according to different sea conditions, resulting in varying degrees of VIVs. This section employs computational fluid dynamics to analyze the two scenarios. The flow field distribution surrounding a single rockfall pipeline will be analyzed under various incoming flow conditions. Secondly, the analysis will focus on the forces and deformations affecting the entire pipeline within the framework of background flow fields exhibiting varying velocities. The objective is to analyze how these factors affect the rockfall process and mitigate the resistance experienced during this process.

4.1. Flow Field Around a Rockfall Pipe

The rockfall pipeline consists of several interconnected pipes. This section focuses on a single connecting pipe to analyze the characteristics of the surrounding flow field. The pipe is located within the flow field, as shown in Figure 11. The background flow speed is obtained from data gathered during a floating offshore platform project in the Gulf of Mexico, USA. The present operational sea state features a water current velocity of 0.62 m/s and a significant wave height of 3.05 m.

A comparative experiment will investigate the flow field distribution around a single rockfall pipe under four distinct incoming flow directions. Due to the symmetrical nature of the rockfall pipe, three scenarios, illustrated in Figure 12a–c, will be selected for demonstration. Condition 1, illustrated in Figure 12a, depicts the scenario in which the background currents flow into the connecting pipe from the front. Condition 2 (illustrated in Figure 12b) depicts the scenario in which the background currents flow into the rear of the connecting pipe. Condition 3, illustrated in Figure 12c, denotes the scenario in which the background currents ingress from the lateral side of the connecting pipe. The background flow in these tests traverses the rockfall pipeline from three distinct directions, each exhibiting a flow velocity of 0.06 m/s, to examine the flow field characteristics surrounding the rockfall pipeline.

Figure 13a–c illustrates the velocity distribution of the flow field within the vertical profile of the connecting pipe across three distinct conditions. The velocity within the pipe exhibits a significant decrease and remains largely independent of the background flow field. At both ends of the pipe, localized areas of increased flow velocity develop, resulting in a symmetrical configuration that preserves pressure equilibrium within the pipe. A boundary layer develops on the upstream side of the pipe as the background flow circulates around its outer wall, resulting in a decrease in flow velocity near the pipe. The thickness of the boundary layer increases with greater flow distance. The flow field on the downstream side of the pipe exhibits increased complexity. Conditions 1 and 2 exhibit the following similar characteristics: the flow velocity diminishes from the pipe ends toward the center, subsequently increasing and creating a significant low-velocity region at the midpoint, where the flow velocity decreases to 0.1 m/s. Conversely, the flow field under Condition 3 exhibits greater complexity, characterized by an initial decrease in velocity followed by an increase toward the midpoint, creating a significant high-velocity zone, where the local flow velocity can reach 0.75 m/s. The velocity gradient under Condition 3 is greater, enhancing the probability of vortex formation. The flow field velocities stabilize in all three conditions after exiting the pipe.

Figure 14a–c illustrates the velocity distribution of the flow field at the midpoint cross-section of the connecting pipe under three distinct situations. The velocity distribution in Conditions 1 and 2 is analogous. Upon contact with the pipe and subsequent escape, the flow field experiences a decrease in velocity to 0.1 m/s, followed by an increase to 0.6 m/s, resulting in a substantial low-velocity zone on the downstream side of the pipe that progressively dissipates. Moreover, a little high-velocity zone is evident in the lateral flow field perpendicular to the pipe’s background flow, exhibiting a more symmetrical high-speed flow on both sides of the pipe under Condition 2. This symmetry produces a more even force distribution on either side of the pipe. In Condition 3, the alteration in the flow field from contact with the pipe to its exit resembles the prior two conditions; however, the velocity distribution exhibits considerable asymmetry on both sides of the pipe. A substantial high-velocity zone with velocities approaching 1 m/s forms at the lower end of the pipe. This region is markedly larger than the corresponding high-velocity zone on the opposite side, creating a pressure differential between the two sides and inducing unstable forces during operation.

The velocity distribution around the rockfall pipe indicates that background flow characteristics require careful consideration in the design of the connecting pipe. Conditions 1 and 2 exhibit symmetrical flow patterns that facilitate pressure balance within the pipe, thereby ensuring stable performance. Condition 3 introduces complex flow dynamics characterized by a higher velocity gradient and an expanded high-velocity zone, which elevates the risk of vortex formation and potential instability. The findings underscore the necessity of meticulous management of flow conditions and pipe design to mitigate adverse effects, including vortex formation and pressure imbalances, that may jeopardize the system’s operational stability. The velocity distribution surrounding the pipe is significantly influenced by the direction and characteristics of the incoming flow. To validate these conclusions, the subsequent analysis will incorporate the vorticity distribution of the flow field for a more thorough examination.

Figure 15a–c illustrates the distribution of eddy viscosity within the flow field across the vertical profile for the three operational conditions. The eddy viscosity is elevated at the pipe’s opening, progressively decreasing as it extends inward before dissipating. No vortex is observed on the upstream side of the pipe, whereas vortex formation is significant on the downstream side. Under Conditions 1 and 2, the eddy viscosity initially increases, subsequently decreases, and then increases once more, resulting in the formation of a substantial vortex field at the midpoint of the pipe. Additionally, Eddy viscosity peaks at about 1.35 Pa·s right at the pipe mouth, falls to 0.90 Pa·s at mid-pipe, and decays below 0.20 Pa·s by one pipe diameter downstream, indicating a more uniform, stable wake. Conversely, in Condition 3, the inlet peak reaches 1.50 Pa·s, and even though it briefly falls to nearly 0 Pa·s at mid-pipe, the average eddy viscosity over the whole profile remains above 0.80 Pa·s, significantly higher than in the first two cases. This indicates that the turbulence intensity in Condition 3 is elevated [31], rendering it less appropriate for operation. In Condition 2, the eddy viscosity is generally lower, more uniformly distributed, and covers a reduced area, indicating that the flow field in Condition 2 exhibits greater stability and is more appropriate for rock-throwing operations.

When water flows through a single attached pipe, a high-speed region develops at the pipe’s opening, resulting in a boundary layer on the flow-facing side of the outer wall. A complex flow field develops on the downstream side, characterized by a significant velocity gradient that promotes vortex formation. Positioning the side with two tension bars downstream of the rockfall pipeline stabilizes the flow field around the pipe, minimizes vortex formation, and reduces turbulence intensity, thereby enhancing its suitability for rock-throwing operations.

4.2. Effect of Background Water Currents

This subsection examines the impact of background flow on the rockfall pipeline in the context of rock-throwing operations. The pipeline comprises multiple interconnected pipes, totaling 100 m in length, with an outer diameter of 500 mm and an inner diameter of 480 mm. The distance from the pipeline’s end to the seabed generally varies between 2 and 4 m, with 3 m employed in this study. The fall pipe rock dumping system typically functions at depths of approximately 100 m. Direct calculation of the flow field at this scale necessitates substantial computational resources and time, rendering it impractical. This study employs geometric similarity, Froude similarity, and Strouhal similarity criteria to analyze the rockfall pipeline comprehensively.

This section’s model test employs a scaling ratio $\lambda$ of 100. In accordance with the geometric similarity criterion, the rockfall pipeline model measures 1 m in length, with an outer diameter of 5 mm and an interior diameter of 4.8 mm. The water depth is established at 1 m, and the separation between the pipe’s end and the seabed is 0.03 m. The test wave height is established at 0.01 times the real wave height, specifically 0.03 m. According to the Froude and Strouhal similarity criteria, the background flow velocity in the model test is one-tenth of the real flow velocity (0.06 m/s). The specific parameters are delineated in

Table 1. Parameter conditions under a scaling ratio of 100.

Parameters	Realistic Parameters	Model Parameters
Length	100 m	1.0 m
Outer diameter	500 mm	5.0 mm
Inner diameter	480 mm	4.8 mm
Depth	100 m	1.0 m
Distance from the seabed	3 m	0.03 m
Wave height	3 m	0.03 m
water currents flow velocity	0.6 m/s	0.06 m/s

. To examine the influence of varying background flow velocities on the rockfall pipeline, two supplementary test sets are performed with background flow velocities of 0.05 m/s and 0.07 m/s in the model, which equate to 0.5 m/s and 0.7 m/s in real sea conditions. The computational domains of the model tests are illustrated in Figure 16.

Figure 17a–c depicts the velocity distribution of the vertical profile flow field at the terminus of the rockfall pipeline, affected by background flow at three distinct water current velocities. A substantial low-velocity zone develops on the downstream side of the rockfall pipeline, characterized by flow separations and vortices. With an increase in background flow velocity, the low-velocity zone downstream widens, resulting in a more chaotic flow field and diminished flow stability. This results in the creation of additional vortices, hence enhancing the probability of VIVs in the rockfall pipeline structure.

Figure 18a–c depicts the velocity distribution of the cross-sectional flow field near the terminus of the rockfall pipeline, influenced by three distinct background flows. A unique vortex region develops on the downstream side of the pipe, characterized by a non-uniform velocity distribution that includes both high- and low-velocity areas. With an increase in the background flow velocity, the vortex zone enlarges, resulting in a reduction of symmetry in the flow field on either side of the pipe. This indicates that vortex formation and flow separation significantly impact the downstream flow field, resulting in increased disturbances at the pipe’s terminus and a heightened probability of intense VIVs.

Figure 19 demonstrates the maximum deformation of the rockfall pipeline resulting from three distinct background flow velocities. As seen in

Table 2. Deformation of the rockfall pipeline at different water current flow velocities.

Flow Velocity	Maximum Deformation	Vibration Amplitude	Vibration Frequency
0.05 m/s	0.75 mm	0.3 mm	1.3 Hz
0.06 m/s	1.25 mm	0.3 mm	1.3 Hz
0.07 m/s	3.00 mm	1.0 mm	6.0 Hz

, when the background flow velocities are 0.05 m/s and 0.06 m/s, the maximum deformation fluctuates around 0.75 mm and 1.25 mm, respectively, with a vibration amplitude of around 0.3 mm and a vibration period of about 0.75 s. At a background flow velocity of 0.07 m/s, the maximum deformation progressively increases over the initial 6 s, followed by a rapid ascent, ultimately attaining an amplitude of about 3 mm and displaying high-frequency VIVs at around 6 Hz. The pipe deformation distribution graph beneath the primary figure illustrates that maximum distortion is observed at the outlet (Output), whereas least deformation is noted at the inlet (Input).

By jointly analyzing the above data with Figure 17 and Figure 18, several conclusions can be drawn.When the inflow velocity exceeds 0.07 m/s, flow separation on the upstream side of the pipeline intensifies and the shedding frequency aligns with the pipe’s natural frequency (Sr similarity). This resonance amplifies transient deformation and excites higher-order vibration modes, resulting in the irregular, large-amplitude oscillations observed in Figure 19. At lower speeds (0.05–0.06 m/s), vortex shedding remains weakly coupled to the pipe’s dynamics and produces only the fundamental VIV mode with lower, nearly sinusoidal amplitude.

Under normal operating conditions, with an entering velocity of 0.602 m/s, the deformation at the terminus of the 100 m-long rockfall pipeline during rock blasting activities approximates 125 mm, as per the Froude and Strouhal similarity criteria. Concurrently, VIVs with an amplitude of 30 mm and a period of 7.5 s arise from this distortion. When the inflow velocity surpasses 0.7 m/s, the deformation of the rockfall pipeline markedly intensifies, leading to high-frequency VIVs. These vibrations may induce metal fatigue and compromise the structural integrity of the rockfall pipeline.

According to the preceding research, at an operational depth of 100 m, the background flow, in conjunction with wave-induced currents, creates a substantial low-velocity zone on the backflow side of the rockfall pipeline, accompanied by flow separation and vortex generation. As the background flow velocity escalates, the low-speed zone enlarges, further perturbing the flow field and exacerbating vortex formation, which can easily induce VIVs. When the background flow velocity exceeds a critical threshold, the amplitude and frequency of these vibrations markedly increase, elevating the risk of metal fatigue and structural damage to the rockfall pipeline.

5. Simulation Analysis of Rockfall Process

The rock-throwing procedure at the termination of the rockfall pipeline is depicted in Figure 20. For submarine cables now in operation, the pipeline diameter is generally approximately 25 cm in shallow marine regions and 20 cm in deep-sea locations. This test utilized specifications for a 25 cm diameter pipeline representative of shallow sea conditions. The rocks descend from the terminus of the pipeline, traversing from 2 to 4 m of ocean before reaching the seabed to obscure the undersea cable. Throughout this process, the rocks are affected not only by the ambient water currents but also by an initial horizontal velocity $V_{vessel}$ imparted by the vessel’s motion when the rocks exit the rockfall pipeline, as seen in Figure 21. The distance between the rockfall site and the bottom influences the precision of rock coverage on the submarine cable. The background ocean currents and horizontal beginning velocity cause the rocks to deflect throughout their descent, leading to diminished accuracy in the coverage of the underwater cable. The greater the distance, the more significant the variance. This section will examine these factors.

The sizes of the gravel used in the rock-throwing operation are shown in Figure 22. They are divided into the following four main categories: 100 mm × 50 mm, 80 mm × 50 mm, 50 mm × 50 mm, and 50 mm × 20 mm.

5.1. Effect of Motion and Background Water Currents

The velocity of water currents near the seabed is generally about 0.1 m/s, whereas the operational speed of the rock-dropping vessel is normally 1 knot, roughly equivalent to 0.5 m/s. This section will analyze the impact of bottom water currents and vessel movement on the rock-throwing operation. As demonstrated in Figure 23, the initial velocity $v_0$ of the rocks upon entering the pipe is zero. As buoyancy F and gravity G interact, the rocks descend from the terminus of the rockfall pipeline into the seawater, with an initial vertical velocity $v_c$ at that instant. The overall length of the rockfall pipeline H is 100 m, and the vertical distance $\Delta h$ from the pipe’s outlet to the seabed is 2 m.

The process of rocks falling from within the rockfall pipeline is illustrated in Figure 24, which tracks the vertical velocity of the rocks at four different moments during the process of covering the submarine cable. The thicker cylinder at the top represents the rockfall pipeline, the thinner column below represents the submarine cable, and the gray areas depict the seabed. After entering the pipe from the inlet, the rocks take approximately 46 s to exit the pipe’s end. The batch of rocks with the best hydrodynamic performance falls first, reaching a bottom velocity of around 2.2 m/s. As time progresses, the bottom velocity of the subsequent rocks decreases gradually and eventually stabilizes at approximately 1.5 m/s.

Figure 25 similarly documents the lateral velocity of the rocks at four distinct intervals in relation to the pipeline. In this experiment, to more effectively assess the influence of the background flow on the rockfall operation, the orientation of the background flow was established perpendicular to the orientation of the submarine cable. The figure indicates that while the rocks remain within the pipeline, there exists a minor lateral velocity, fluctuating between −0.1 m/s and 0.1 m/s. It can be extrapolated in conjunction with the preceding section that the vortices within the pipeline are influencing the rocks. As the rocks escape the pipeline, their lateral velocity progressively rises due to the background flow, resulting in the majority of the rocks descending to one side of the underwater cable. As illustrated in Figure 26, nearly all the rocks accumulate on the downstream side of the undersea cable due to the background flow, with the majority concentrated within 0.4 m of the pipeline. A minor segment exceeds this range, with a maximum deviation of up to 0.8 m.

The movement of rocks falling from the rockfall pipeline while the vessel is moving at a speed of 1 knot is shown in Figure 27 and Figure 28. Figure 27 records the velocity of the rocks in the direction of the vessel’s movement, showing that the rocks velocity decreases sharply after leaving the pipeline due to water resistance, with their forward velocity dropping to zero before reaching the seabed. Figure 28 shows the coverage of the submarine cable by the rocks after it settles on the seabed, using the upper surface of the submarine cable as the reference plane. The distance from the seabed to this reference plane is 25 cm. In the figure, a smaller value (indicated by a color closer to blue) represents better coverage of the submarine cable. The results indicate that, at a vessel speed of 1 knot, a single rock-throwing operation using a 500 mm-diameter rockfall pipeline can only deposit two to three layers of rocks on both sides of the submarine cable, which is insufficient to fully cover the submarine cable. Multiple operations are necessary to achieve complete coverage and effective protection.

5.2. Effect of Distance Between Rockfall Pipeline Outlet and Seabed

To study the effect of the distance between the outlet of the rockfall pipeline and the seabed, three different heights are examined in this subsection as follows: 2 m, 3 m, and 4 m. In the preceding section, we noted that when the vessel travels at a speed of 1 knot, the rocks attain zero velocity before contacting the seabed due to water resistance; so, the impact of the vessel’s motion is not examined here. Figure 29 illustrates the distribution of rocks settling on the bottom influenced by background flow at three specified heights. The findings demonstrate that as height grows, the extent of rock deviation likewise escalates, with the maximum deviation distance attaining 0.8 to 1 m at heights ranging from 2 to 4 m. This deviation requires rectification by the use of a ROV to modify the outlet’s position during the rock-throwing operation. During this process, it is essential to account for both the pipeline’s VIV induced by the background flow and rectify the positional displacement of the rocks resulting from underwater currents. Furthermore, the vessel’s operational speed must not surpass 1 knot to avert the rockfall pipeline from producing significant turbulence and vortices downstream, which may result in severe VIVs.

Our simulations show that, under a vessel speed of 1 knot and a 2 m pipeline-to-seabed distance, most rocks land within ±0.4 m of the pipe centerline, while the maximum lateral offset reaches 0.8 m (Figure 26). When the height increases to 3 m and 4 m, the maximum offset grows to 0.9 m and 1.0 m, respectively (Figure 29). Given a nominal rock diameter of 0.5 m, any lateral shift exceeding this value creates uncovered gaps along the cable route. Such gaps compromise the intended protective layer, leaving sections of the submarine cable exposed to seabed currents and potential mechanical damage. To ensure complete protection, these deviations must be compensated, either by performing multiple rock-dropping passes or by actively adjusting the outlet position via ROV control. Incorporating these measures are expected eliminate uncovered spans and restore a continuous, minimum-thickness rock cover over the cable.

6. Conclusions

In this study, we numerically investigated how the rockfall pipeline, rocks and hydrodynamic loads jointly influence system performance and stability. These interactions affect the stability and efficiency of the fall pipe rock dumping system. A novel fall pipe rock dumping system was developed, and a fluid–structure interaction model was employed to examine the hydrodynamic properties and VIVs of the rockfall pipeline. The forces and movements of rocks during the settling and placement process were also analyzed. The findings of this investigation are summarized as follows:

When water flows through a single attached pipe, a high-speed region develops at the pipe’s opening. This forms a boundary layer and promotes downstream vortex formation. Two tension bars installed on the downstream side of the rockfall pipeline stabilize the flow, minimize vortex formation, and reduce turbulence. These improvements enhance the system’s suitability for rock-throwing operations. At a depth of 100 m, the combined effect of background flow and wave currents creates a low-speed zone behind the pipeline. This leads to flow separation and vortex formation, which may induce fluid–structure interaction and result in VIVs and pipeline deformations. Rocks entering the pipeline from the vessel take approximately 46 s to travel 100 m. However, their velocity decreases upon exiting the pipe, leading to inaccurate placement above the submarine cable due to the influence of water currents. To ensure accurate rock placement, a ROV should be used to adjust the outlet position, and the vessel speed should be maintained below 1 knot to prevent excessive turbulence and VIVs.

The construction of the fall pipe rock dumping system at the dock has been completed, and operational testing is expected to conclude between July and September 2025. This study provides a theoretical foundation and technical support for the optimized design of the rockfall pipeline system. Through numerical analysis of fluid–structure interactions among the pipeline, rocks, and ocean currents, the outcomes provide critical insights for refining the design and functionality of the system. This significantly enhances the protection performance and reliability of submarine cable protection dikes. The findings not only contribute to the high-quality development of submarine cable projects but also offer a valuable reference for the construction and maintenance of infrastructure in complex marine environments, promoting the advancement of marine engineering toward greater safety, efficiency, and intelligence.