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Article

Numerical Simulation Study of the Motion Characteristics of Autonomous Underwater Vehicles During Mooring Lurking Procedure

School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an 710072, China
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Author to whom correspondence should be addressed.
J. Mar. Sci. Eng. 2025, 13(2), 275; https://doi.org/10.3390/jmse13020275
Submission received: 5 January 2025 / Revised: 24 January 2025 / Accepted: 29 January 2025 / Published: 31 January 2025
(This article belongs to the Section Ocean Engineering)

Abstract

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A two-dimensional coupled dynamics model for a moored autonomous underwater vehicle (AUV) was developed using the lumped mass method for mooring cable dynamics and the Newton-Euler method for rigid body dynamics. This model enables the integrated simulation of AUV motion, flow field interactions, and mooring cable behavior. The study investigates the effects of varying ocean current velocities and mooring cable lengths on AUV motion responses. The results indicate that under the influence of mooring forces, the AUV stabilizes near its equilibrium position after release and undergoes periodic oscillatory motion. Specifically, when the X-direction oscillation completes two cycles and the Y-direction oscillation completes four cycles, the AUV demonstrates an 8-shaped trajectory, with maximum motion amplitudes observed. These findings provide insights into the dynamic behavior of moored AUVs in ocean environments, contributing to the design and operation of long-term underwater monitoring systems.

1. Introduction

Autonomous underwater vehicles (AUVs) are highly effective platforms for ocean exploration and underwater operations, extensively utilized in scientific research, commercial industries, and military applications [1]. Their versatility supports a wide range of tasks, including hydrological sampling [2] and near-seabed surveys [3]. However, conventional AUVs are constrained by limited onboard energy capacity, typically achieving an operational endurance of only 2–4 h [4]. Specifically, to ensure that the AUV maintains its spatial position and proper orientation, continuous control via thrusters is required, which results in significant energy consumption. This significant limitation poses challenges in meeting the demands of extended-duration missions, thereby restricting their potential for long-term and large-scale deployments.
The anchor-moored underwater vehicle remained stationary at a specified depth using anchors and cables, without the need for continuous adjustments to the vehicle’s orientation. In its moored state, the vehicle achieves static equilibrium, minimizing energy consumption. This allows for prolonged monitoring of the marine environment or extended seabed residence in anticipation of a secondary deployment. On the other hand, due to energy limitations, underwater vehicles typically need to dock through mooring systems and complete smooth docking with base stations [5]. Thus, the anchor-moored underwater vehicle system holds significant potential for a wide range of applications.
The anchor-moored strategy employs various underwater cables to connect the underwater vehicles to the anchor, transmitting the holding force generated by the anchor to the underwater vehicles. This enables the underwater vehicles to remain within a defined mooring radius near their designated position. However, under challenging marine environments, such as ocean currents, moored underwater vehicles often experience complex and irregular motion patterns [6], with significant vibrations and oscillations during their stay, resulting in reduced stability of the mooring. This lack of stability complicates tasks, such as attitude control and docking [7]. As a result, maintaining the stability of underwater vehicles during moored residence remains a significant challenge in this field of research. The stability issue of the mooring is related to the normal execution of detection tasks and the efficient development of energy harvesting processes. Therefore, it is crucial to develop an appropriate model to study the motion characteristics of anchor-moored underwater vehicles in flowing water.
Due to the presence of anchor chains in the mooring structure, there is strong coupling interference between the mooring system and the six degrees of freedom motion, and the motion response exhibits complex nonlinear characteristics. Currently, research on mooring stability mostly focuses on floating structures, but they remain quite limited. Mooring underwater vehicles are limited in their displacement and attitude variation range by the constraint force generated by anchor chains under the action of nonlinear loads in the ocean. Therefore, the key to studying the motion of underwater mooring systems lies in developing a coupled dynamic model of the mooring body and the anchor chain. Based on the single-point residence method, Wang [8] developed two-dimensional and three-dimensional single-point mooring systems for underwater vehicles and analyzed their motion and force characteristics. Hermawan and Furukawa [9] developed a coupled dynamic model for multicomponent mooring cables by extending the three-dimensional concentrated mass method. This approach allows for the motion of segment connection points. Since the lateral motion of the mooring line significantly affects the mooring line tension, the three-dimensional effects of the mooring line must be considered. Zheng [10] designed an environmentally adaptive, deep-sea, single-point mooring with no propellers and a hovering docking system, predicting and analyzing the dynamic motion states of the single-point mooring and hovering docking system under various conditions. Du and Zhang [11] established a kinematic model for the residence process of underwater gliders based on the concentrated mass method, integrating KANE and generalized force methods. The study indicates that the length of the anchor chain significantly affects the stability of the underwater mooring body. Zhu and Yoo [12] employed the lumped mass method to propose a dynamic equation for the anchor chain expressed in terms of unit direction vectors and fluid relative velocity vectors. This work enables a more accurate description of the hydrodynamic drag on the anchor chain. Tian [13] utilized both the Newton–Euler method and the concentrated mass method to model the rigid body of the underwater platform and the anchor chain, analyzing the effects of net buoyancy on the motion characteristics and stability of the underwater mooring platform. Vu [14,15] established the governing equations for underwater anchor chains based on the catenary method. Their simulations integrated the rigid body motion of the underwater vehicle with the flexible motion of the anchor chain. They employed the shooting method to solve the two-point boundary value problem of the catenary equation and conducted various motion simulations of the underwater vehicle, including ascent, turning, and oscillation. Jin [16] employed Newton’s laws, Hamilton’s variational principle, and the Morison equation to thoroughly account for fluid drag, buoyancy, and the added mass acting on each segment, deriving a dynamic model. Simulations of the proposed dynamic model were conducted, and data analysis revealed the nonlinear dynamic characteristics of the anchor chain. Zhou [17] proposed a catenary mooring length control method for motion suppression of semisubmersible floating wind turbines. Haider [18] developed a fully coupled mooring model for a semisubmersible floating offshore wind turbine using OpenFOAM, aimed at predicting the dynamic characteristics of the mooring system for the floating body under wind and wave conditions. Their study included the heave and pitch responses, as well as the tension loads on the mooring cables. Fakhruradzi and Mamat [19], based on a particle swarm optimization algorithm, studied the dynamic effects of waves on a two-dimensional floating structure in a moored state. As can be seen, current research methods mostly model anchor chains and platforms separately and then couple them through boundary conditions of force and motion. The research ideas of the above-mentioned scholars have important reference value in the modeling and analysis of underwater mooring suspension docking systems.
Through a comprehensive review of the literature, the study of fluid-structure interaction (FSI) and the dynamic behavior of moored autonomous underwater vehicles (AUVs) based on computational fluid dynamics (CFD) simulations of full-scale prototype suspension systems under real sea conditions has garnered increasing attention among researchers. In this study, a coupled numerical simulation framework was developed to investigate the dynamics of underwater moored vehicles. A mathematical model for anchor chain mooring dynamics was formulated using the concentrated mass method and the Newton-Euler approach, enabling iterative calculations of mooring tension in response to positional changes at mooring points. Subsequently, a rigid-body dynamics model of the underwater moored vehicle was established, incorporating four key parameters—hydrodynamic forces, anchor chain tension, velocity, and displacement of the AUV—as constraint conditions. This integration facilitated the coupled solution of the flow field and motion models. The primary focus of this study is to examine the motion state of moored underwater vehicles under realistic cable forces, as opposed to simplified spring oscillator systems, particularly at low ocean current speeds. Additionally, the study explores the effects of ocean current velocity and cable length on the vehicle’s motion dynamics.
This study proposes a novel coupled calculation method to evaluate the flow-induced vibration of cylindrical bodies, such as AUVs. Unlike traditional spring-mass models, this model more accurately reflects the effect of mooring forces under moored conditions. The results provide valuable insights for the design of mooring stability in underwater vehicles, offering a reference for the design of moored AUVs’ long-term stability. The remainder of the paper is organized as follows: Section 2 describes the physical and dynamic models. Section 3 introduces the grid of numerical simulation and the process of implementing a fluid-structure coupling algorithm. Section 4 discusses the results of numerical simulations, and finally, Section 5 summarizes the research content of this paper and presents conclusions.

2. Problem Formulation

2.1. Physical Model

The physical model of the anchor-moored underwater vehicle is depicted in Figure 1a. Assuming that the mass of the underwater vehicle is evenly distributed, it is simplified as a smooth cylinder. To simplify the calculation, one of the circular cross-sections is selected for two-dimensional calculation. Table 1 shows the physical parameters of AUV and cable.
This study is the double-point mooring system, with the research scenario set in the deep sea. It is assumed that the current speed is a uniform constant and that the area of the current is sufficiently large, with only horizontal flow and no vertical movement. Due to the fact that the circle in Fluent’s two-dimensional calculations is set as a cylinder with a height of 1 m, it is necessary to handle the anchor chain force acting on the calculation model. The forces of Cable 1 and Cable 2 are considered to act simultaneously at the midpoint of the cylinder, and the mass of the cylinder is 1/7 of the mass of all underwater vehicles. The final calculated anchor chain force for cable’ (as shown in Figure 1b) is (Fcable1 + Fcable2)/7 and the net buoyancy generated by the drainage volume of 1/7 underwater vehicles is 49 N.

2.2. Anchor Chain Dynamics Model

2.2.1. Dynamic Control Equation for Microsegment of Anchor Chain

To simplify the mathematical model, it is assumed that the anchor chain is a slender cylindrical body, and its bending and torsional resistance are neglected. The anchor chain is considered to be influenced only by gravity, buoyancy, fluid drag, and inertia. Based on Newton’s second law, the dynamic control equation for an infinitesimal segment of the anchor chain moving in the fluid is established as follows [13]:
T s + B + G + D = I
In this equation, T represents tension, B represents buoyancy, G represents gravity, D represents fluid drag, and I represents inertia force. I = M · X ¨ , M is the mass matrix, which includes both the inertial mass of the anchor chain and its added mass in water, X ¨ = x ¨ 0 c , y ¨ 0 c represents acceleration.

2.2.2. Additional Mass Force

When the motion state of the underwater cable unit changes in the fluid medium, it will inevitably drive a part of the surrounding fluid to produce the same motion state. This part of the fluid is completely consistent with the motion state of the underwater cable unit. Usually, anchor chains have a high aspect ratio, and when they undergo translational motion in a fluid, the tangential inertial hydrodynamic force of the anchor chain is negligible compared to the transverse force. Therefore, it can be considered that anchor chains only have additional mass in the transverse direction. In the local coordinate system of the anchor chain, the lateral additional mass per unit length of the anchor chain can be expressed as ρ k α σ . For anchor chains with circular cross-sections, their additional mass coefficient k α = 1 , ρ is the fluid density, σ is the cross-sectional area of the anchor chain.

2.2.3. Mathematical Model of Lumped Mass Method

Dynamic modeling uses discretization methods to describe the motion and forces of mooring cables, taking into account the inertia and damping effects of mooring cables. Common solving methods include the finite difference method, lumped mass method, finite element method, etc. This study uses the lumped mass method to analyze mooring cables [20], as shown in Figure 2. The anchor chain is discretized into N segments, corresponding to N + 1 nodes, with the upper mooring point as node i = 1, where the anchor chain length s = 0. The lower fixed point is defined as node i = N + 1, with the anchor chain length s = L.
The cable length corresponding to each node satisfies the following basic relationship:
0 = s 1 < s 2 < < s i < s i + 1 < < s N + 1 = L C
The stress-strain relationship between adjacent nodes is chosen as the continuity condition for the anchor chain: T = f ( ε ) . The dynamic equation of the anchor chain at the i node is:
M i X ¨ i = F i
In the formula, M i = m i I + M a i is the mass matrix, which includes the inertial mass of the anchor chain node m i = ( u i 1 / 2 l i 1 / 2 + u i + 1 / 2 l i + 1 / 2 ) / 2 and its additional mass in water M a i = ( M a ( 1 1 / 2 ) + M a ( i + 1 / 2 ) ) / 2 . Among them, l is the linear density of the anchor chain, subscript i + 1/2 represents the physical quantity between node i and i + 1, li+1/2 represents the length of the anchor chain between node i and i + 1, and Fi is all external forces acting on node i.

2.2.4. Force Analysis of Anchor Chain Nodes

The external forces acting on node i of the anchor chain include basic cable tension T i , buoyancy B i , gravity G i , and fluid drag D i , as follows:
F i = T i + B i + G i + D i
When the strain is less than zero, the stress-strain relationship becomes quite complex. In this study, T i is assumed to be 0 to simplify the problem.
The relationship between buoyancy and gravity can be expressed as:
B i + G i = 1 2 ρ g l i + 1 / 2 σ i + 1 / 2 + l i 1 / 2 σ i 1 / 2 m i g
As for the fluid drag on the anchor chain, its effect is only in the horizontal direction. In the local coordinate system of the anchor chain, the drag is divided into normal D n i and tangential components D t i , expressed mathematically as follows:
D n i = 1 2 ρ C n l i + 1 / 2 d i + 1 / 2 + l i 1 / 2 d i 1 / 2 v n i 2
D t i = 1 2 ρ C t l i + 1 / 2 d i + 1 / 2 + l i 1 / 2 d i 1 / 2 v t i 2
In the formula, d is the diameter of the anchor chain, Cn and Ct are the normal and tangential resistance coefficients of the anchor chain, and v n i and v t i are the normal and tangential velocities of the anchor chain node i relative to the water flow, respectively.

2.3. Dynamics Model for the Rigid Body

The vehicle has two degrees of freedom. Based on Newton’s law and Euler’s equation, assuming that the underwater vehicle is rigid and completely immersed in the fluid medium, ignoring the influence of the Earth’s rotation, the translational and rotational dynamic equations of the vehicle are expressed in a compact form [13].
M R B V ˙ + C R B ( V ) V = F
In the equation, the external force vector applied to the AUV is described as:
F = F μ + F B G + F C
where F μ is the hydrodynamic force, F B G is the gravity and buoyant force, and F C is the cable force.
MRB is the generalized mass matrix, expressed as:
M R B = m 0 0 0 0 m y c 0 m 0 0 0 m x c 0 0 0 m y c m x c 0 0 0 m y c J x x J x y 0 0 0 m x c J y x J y y 0 m y c m x c 0 0 0 0
CRB is the velocity matrix, expressed as:
C R B = 0 0 ω y 0 0 0 0 0 ω x 0 0 0 ω y ω x 0 0 0 0 0 0 v y 0 0 ω y 0 0 v x 0 0 ω x v y v x 0 ω y ω x 0
AUV itself has a certain mass and volume, and it is subject to the effects of gravity G and buoyancy B when moving underwater. Due to the fact that the volumetric force is directed in the vertical direction, gravity and buoyancy are generally combined into net buoyancy. The expression in a two-dimensional coordinate system is:
F B G = B G
F μ is directly solved through CFD software. The movement of the anchored AUV in the deployed state is constrained by the cable force F C , and its movement in turn affects the position and shape of the anchor chain.

3. Simulation and Computational Method

3.1. Grid Distribution

Figure 3a shows the rectangular fluid computational domain (38.74 D × 23 D) utilized for numerical simulations in this paper. According to existing studies [21], the effect of computational domain width is considered negligible when the blockage ratio (D/b, where b is the width of the domain) is less than 0.05. In this work, the blockage rate was calculated as D/b = 0.0431. To better capture the vibration in two degrees of freedom, the dynamic mesh of the model was treated using the nested grid technique, also known as overset mesh. This method involves overlapping background mesh and component mesh regions, with each region spatially overlapped yet independent, which necessitates preprocessing operations like hole-digging and interpolation point matching to establish connectivity.

3.2. Control Equation

The k-ω turbulence model used in this paper describes the intensity and scale of turbulence by introducing two additional variables, k (the turbulent kinetic energy) and ω (the turbulence frequency). The turbulent kinetic energy k describes the turbulent energy in the fluid and reflects the intensity of the turbulence, and its transport equation is [22]:
( ρ k ) t + · ( ρ U k ) = · ( μ t k σ k ) + P k ρ ε
where μ k is the turbulent viscosity, and σ k is the Prandtl number of turbulent kinetic energy (often taken to be σ k = 1.0). P k is the turbulence generation term and ε is the turbulent dissipation rate.
The turbulence frequency ω is related to the rate of dissipation of turbulence and describes the rate of conversion of turbulent energy with the transport equation [22]:
( ρ ω ) t + · ( ρ U ω ) = · ( μ t ω σ ω ) + β ω k P k α ω 2  
where σ ω is the Prandtl number of turbulence frequency (often taken as σ ω = 0.5). β is the turbulent productivity constant (often taken as β = 0.09). α is the turbulent dissipation rate constant (often taken as α = 0.5).

3.3. Fluid Structure Interaction (FSI) Calculation Method

This article proposes a calculation method for the rigid flexible coupling vortex-induced vibration of underwater mooring structures, which realizes the nonlinear coupling solution of anchor chains, rigid bodies, and flow fields. The specific solution process is shown as Figure 4.
A CFD model of the flow field around the underwater vehicle was established in FLUENT. A mooring dynamics model for the anchor chain was developed using MATLAB, compiled, and exported as a dynamic link library for use in the CFD software. Secondary development was carried out in the FLUENT UDF module by writing a C program to call the MATLAB dynamic link library to obtain the tension at the mooring points of the anchor chain. Using UDF macros and nonlinear fitting methods, the hydrodynamic forces and displacement information of the underwater vehicle were retrieved, along with all the necessary forces and motion constraint conditions for the calculations. In the UDF module, a system of rigid body dynamic differential equations for the underwater vehicle was established. Based on the force and motion constraint conditions, the velocity information for the next time step of the device was solved. The Fluent software utilizes the dynamic mesh motion of the rigid body to update the mesh of the flow field, allowing the simulation to proceed to the next time step.

3.4. Numerical Method Validation

The variation in grid density and time step size can affect the accuracy of numerical model predictions. Therefore, grid independence verification was first conducted to evaluate the impact of different grid densities on the accuracy of the calculation results. In this study, four grids with different densities were set up: coarse (20,000), refine 1 (30,000), refine 2 (40,000), and refine 3 (50,000). After calculating the stabilization, the vibration frequency in the Y-direction is used as a measure for comparison.
As shown in Table 2, when the number of grids is approximately 40,000, the deviation of the results after further densification of the grids is only 1.4%. At this point, it can be considered that the influence of the number of grids on the calculation results no longer exists. Taking into account the calculation accuracy and speed, this study selected 40,000 grids for calculation.
Subsequently, the independence of the time step was verified. In this study, four different time steps, T1 (0.1), T2 (0.07), T3 (0.05), and T4 (0.03), were set. As shown in Table 3, when the time step was 0.05, further reducing the step length resulted in an error of only 1.7%. At this point, it can be considered that the time step had no effect on the calculation results. Taking into account both calculation accuracy and speed, this study conducted calculations with a time step of 0.05 s.
To verify the accuracy of the numerical simulation method used in this article, the CFD calculation results were compared with existing data. Refer to the numerical simulation results of Zhang’s results [23] and compare the calculation results of this paper with them. The structural parameters of the underwater vehicle studied in this article are very similar to Zhang’s research. Their research operating parameters are as follows: ocean current velocity of 0.7716 m/s, cable length of 10 m, and the underwater vehicle is released at an initial position of 10 m.
In order to eliminate the influence of differences in operating conditions and make the results easier to compare, this study raised the release position coordinates of the AUV from (0, 0) to (0, 10) and then compared the vibration displacement in the Y-direction of the underwater vehicle moored with double anchor cables. In addition, based on the fact that the magnitude of the force is proportional to the square of the velocity, we compared the results of Zhang’s study with a 100× increase in the streamwise cable force used in this study. As shown in the Figure 5, the CFD results of this study show satisfactory consistency with the two sets of data from previous studies. Overall, the numerical methods used in this article are reliable and acceptable.

4. Results and Discussions

4.1. Influences of Ocean Current Velocity on the Motion Characteristics of the Moored System

In order to study the influence of ocean current velocity on the motion characteristics of the AUV, simulations for three different ocean current velocities v: 0.0486 m/s, 0.0648 m/s and 0.0809 m/s.

4.1.1. Force Analysis of Cable

The results of cable forces (including cable forces in the X-(Fcable_x) and Y-(Fcable_y) directions) changing with time at different ocean current velocities are shown in Figure 6. Both Fcable_x and Fcable_y generally exhibit regular periodic variations, but their stability is relatively low. Specifically, their peak values remain in a fluctuating state. The cable force Fcable_y in the Y-direction fluctuates approximately 49 N, which corresponds to the net buoyancy of the AUV cylinder. Overall, as the ocean velocity increases, the cable force required for the underwater vehicle to remain relatively stable also increases.

4.1.2. Force Analysis of Cable

Figure 7 shows the position of the AUV cylinder at different ocean current velocities. The ocean’s current velocity has a great influence on the movement of X and Y directions.
As shown in Figure 7a, in all the calculated cases with different current velocities, the X-movement curve first shows a slight increase followed by a rapid decrease, eventually starting to oscillate. This indicates that the moored AUV undergoes a physical underwater motion process where it is initially pushed away from the origin by the ocean current. During this motion, the cable gradually transitions from a relaxed state to a taut state, with the force on the cable continually increasing. Under the influence of the cable force, the AUV begins to move back toward the origin. Ultimately, under the combined effect of the cable force and the AUV’s hydrodynamic forces, the AUV oscillates cyclically in the X-direction.
Meanwhile, comparing the X-direction curves at different current velocities, it is observed that the lower the current velocity, the shorter the time it takes for the AUV to return. This is because the thrust is proportional to the square of the current velocity, as the current velocity decreases, the thrust becomes smaller. Consequently, the time required for the cable force and the hydrodynamic forces acting on the AUV to reach equilibrium is shorter, and the final equilibrium point is closer to the origin point. Besides, the oscillation of the AUV cylinder in the X-direction is mostly irregular, especially when the velocity is 0.0809 m/s and the time reaches 400 s, at which point the displacement curve begins to exhibit a stable periodic pattern and regularity.
Figure 7b shows the Y-direction vibration state of the AUV under the influence of ocean currents and cable tension. Unlike the oscillation in the X-direction, the vibration in the Y-direction is unstable, but its periodicity is very distinct. As the current velocity increases, the vibration in the Y-direction deviates more from y = 0, meaning the vibration amplitude becomes larger. When the v = 0.0809 m/s, the vibration amplitude reaches its maximum, with a peak amplitude of 1.112 m (Figure 8).
When v = 0.0809 m/s, the movement of the AUV cylinder tends to stabilize relatively during the period of 500–700 s, with the reciprocating movement in both the X and Y directions exhibiting complete periodicity. Therefore, we plotted the X-Y diagram to more intuitively observe the trajectory of the AUV cylinder. Additionally, data from other velocities during the same period were selected for plotting to facilitate comparison.
Figure 9 shows the motion trajectory of the AUV in the X-Y plane. As seen in the figure, when the ocean current velocities are v = 0.0486 m/s and v = 0.0648 m/s, the motion of the AUV’s cylindrical center of mass in the X-Y plane follows an irregular trajectory. It is worth noting that when v = 0.0809 m/s, the AUV’s motion trajectory exhibits an 8-shaped reciprocating motion, with a complete cycle of the trajectory lasting approximately 100 s. This phenomenon is closely related to the periodic oscillations of the AUV observed in the X-direction in Figure 7a. Specifically, when the AUV oscillates for 2 cycles in the X-direction and 4 cycles in the Y-direction, its motion trajectory in the X-Y plane takes on an 8-shaped pattern. Meanwhile, the amplitude in the Y-direction significantly increases, with the motion range exceeding twice the diameter of the AUV (2.1–2.3 D). Moreover, unlike the high-frequency characteristics of vortex-induced vibrations in spring systems, the 8-shaped motion observed in this study is low-frequency and slow.

4.1.3. Analysis of Flow Pattern

Figure 10 illustrates the cloud diagrams of velocity fields, streamlines, and vortex structures under different ocean current velocities, where the vortex structures are identified using the third-generation vortex identification method [24]. From the velocity field (Figure 10a–c), the underwater motion state of the moored AUV shows significant differences at various current speeds. When v = 0.0486 m/s, a high-velocity region forms behind the AUV cylinder. At v = 0.06479 m/s, the flow behind the AUV becomes stratified and uniform. However, at v = 0.0809 m/s, a low-velocity region appears behind the AUV, and the flow field distribution around the cylinder differs from the previous two cases. By introducing streamlines, as shown in Figure 10d–f, we observed that in the two cases with lower flow velocities, the points of significant streamline bending are typically located directly behind the cylinder, where a pair of vortices (one strong and one weak) forms. However, at v = 0.0809 m/s, the bending points of the streamlines shift further forward, and the vortex structures are concentrated along the cylinder’s surface.
The differences in velocity fields further lead to variations in vortex distribution and shedding patterns. As shown in Figure 10g–i, at the first two lower speeds, the vortex shedding pattern behind the AUV cylinder is characterized by the sequential formation of single vortices, referred to as the S (Single) mode. At this point, the detached vortices are very close to each other, and the vortex-induced pulsating forces acting on the cylinder are relatively small, resulting in a lower vibration amplitude. However, at v = 0.0809 m/s, the vortex shedding pattern transitions to the simultaneous formation of a pair of vortices (one with higher intensity and the other with lower intensity), referred to as the P (Pair) mode. When the flow is in this mode, the number of vortices shed per cycle gradually increases, indicating that the fluid forces cycle multiple times. As a result, the forces acting on the cylinder increase, leading to larger vibration amplitudes.

4.2. Influences of Cable Length on the Motion Characteristics of the Moored System

In order to study the influence of cable lengths on the motion characteristics of the AUV, simulations for three different cable lengths L: 12 m,14 m, and 16 m.

4.2.1. Force Analysis of Cable

The data of cable forces (including cable forces in the X and Y directions) changing with time at different cable lengths are shown in Figure 11. Both Fcable_x and Fcable_y exhibit regular periodic variations overall. Notably, as the cable length increases, the stability of the cable forces gradually improves. Specifically, their peak and trough values tend to converge and smooth out. Similar to previous observations, the cable force in the Y-direction ultimately fluctuates at approximately 49 N, corresponding to the net buoyancy of the AUV cylinder.

4.2.2. Movement and Trajectory

As shown in Figure 12a, in all calculated cases with varying cable lengths, the X-direction movement curves similarly exhibit a slight initial increase followed by a rapid decrease, eventually transitioning into oscillations. This indicates that the moored AUV is first pushed away from the origin by the ocean current and then reaches an equilibrium position under the influence of the cable force, where it continues to oscillate. Comparing the movement curves in the X-direction under different cable lengths, the shorter the cable, the shorter the return time of the AUV and the smaller the oscillation distance deviating from the origin point. Conversely, the longer the cable, the later the transition of the cable from slack to taut occurs, allowing the AUV to travel farther. Consequently, the equilibrium position is closer to the origin point.
Unlike the results under different ocean current velocities, an increase in cable length leads to relatively more stable changes in the centroid position of the AUV cylinder in the X-direction. Specifically, the amplitude is smaller, and the oscillation exhibits more regular patterns. Notably, when L = 14 m, the periodicity of the oscillation in the X-direction is the most pronounced (Figure 12a). The longer the cable, the greater its self-weight, resulting in a higher initial vertical cable tension. Since the net buoyancy of the AUV cylinder is constant, when the vertical cable tension exceeds the net buoyancy, the AUV cylinder will sink. During the sinking process, as the cable gradually slackens, the vertical cable tension continuously decreases until it is less than the net buoyancy. At this point, the sinking speed of the AUV cylinder gradually decreases to zero. Subsequently, under the influence of the upward buoyancy, the AUV cylinder begins to rise, eventually floating at the equilibrium position before starting to oscillate. Figure 12b reflects this change, showing that the longer the cable, the lower the vibration position in the Y-direction, which is caused by the sinking of the AUV. However, it is worth noting that, for all cable lengths, the period of the vertical vibration of the AUV changes very little and is nearly the same. When the L = 14 m, the vibration amplitude reaches its maximum, with a peak amplitude of 1.431 m, as shown in Figure 13.
Similar to the analytical approach used in the previous section, we plotted the X-Y diagram to more intuitively observe the trajectory of the AUV cylinder in different cable lengths. Figure 14 shows the motion trajectory of the AUV in the X-Y plane. As seen in the figure, when the cable length L = 12 m, the motion of the AUV’s cylindrical center of mass in the X-Y plane follows an irregular trajectory. When L = 14 m, the motion phenomenon changes; the AUV’s motion trajectory exhibits a distinct 8-shaped reciprocating motion. The displacement curve in Figure 12a reveals the underlying cause of this phenomenon: the AUV exhibits regular periodic oscillations in the X-direction, with two oscillation cycles in the X-direction accompanied by four vibration cycles in the Y-direction. When the cable length is further increased to L = 16 m, the regularity of the AUV’s oscillation in the X-direction begins to weaken. Simultaneously, its motion in the XY-plane no longer strictly adheres to the ‘two oscillation cycles in the X-direction corresponding to four vibration cycles in the Y-direction’ pattern. As a result, at a cable length of 16 m, the AUV’s figure-8 motion trajectory only appears intermittently.
Subsequently, the effects of varying cable lengths on the motion of the AUV cylinder were analyzed and discussed. For shorter cable lengths, abrupt changes in the Y-direction cable tension were observed. As illustrated in Figure 11b, these sudden variations in mooring cable tension exert significant influence on the motion of the moored body, causing the AUV cylinder to be rapidly pulled back in the Y-direction without a gradual transition in force. Consequently, this abrupt force interaction leads to irregular motion trajectories of the AUV cylinder in the XY plane. As the cable length increases, the initial gravitational force of the cable grows, which slows the upward motion of the AUV cylinder. Consequently, the variation in Y-direction cable tension becomes smoother, with no abrupt changes observed. This explains why 8-shaped trajectories appear for cable lengths of L = 14 m and L = 16 m. However, the case of L = 14 m is particularly notable, as the oscillation in the X-direction exhibits regular periodicity. This characteristic makes the 8-shaped reciprocating motion most pronounced at this specific cable length.

4.2.3. Analysis of Flow Pattern

Figure 15 shows the velocity field, streamlines, and vortex distribution around the AUV cylinder for different cable lengths. In terms of the velocity field (Figure 15a–c), the underwater motion state of the moored AUV exhibits significant differences at different cable lengths. When L = 12 m, the flow behind the AUV is evenly stratified. However, for L = 14 m and L = 16 m, a distinct low-velocity zone appears behind the AUV. Additionally, the flow field distribution around the cylinder also shows differences. From the streamlined diagram, as shown in Figure 15d–f, we observe that when L = 12 m, a pair of alternating strong and weak vortex structures forms behind the cylinder, with the vortices arranged in a front-to-back pattern. However, when L = 14 m and L = 16 m, the vortex structures are concentrated near the surface of the cylinder and are arranged side by side.
Figure 15g–i show that when L = 12 m, the vortex shedding pattern behind the AUV cylinder follows a single vortex forming sequentially, the S-mode. However, when L = 14 m and L = 16 m, the vortices are always generated in pairs, indicating the P-mode. Based on the results from Section 4.1, it can be inferred that when the vortex shedding pattern behind the moored AUV cylinder follows the P-mode, a more regular 8-shaped motion trajectory, or a motion state evolving from this pattern, will form.

5. Conclusions

This study develops a coupled computational fluid dynamics (CFD) numerical model for moored underwater vehicles, integrating fluid mechanics, rigid-body dynamics, and anchor chain dynamics. Using an anchored autonomous underwater vehicle (AUV) as the research subject, the investigation focuses on its underwater motion characteristics under varying ocean current velocities and cable lengths. The analysis encompasses displacement, motion trajectories, velocity fields, and vortex structures. The key findings are summarized as follows:
(1)
The ocean current velocity v has a significant impact on the underwater motion of the moored AUV. The higher the current velocity, the farther the AUV moves to reach its equilibrium position, the longer the time required, and the larger the vertical vibration amplitude. When v = 0.0809 m/s, the maximum amplitude is 1.112 m.
(2)
The cable length L primarily governs the change in the equilibrium position of the moored AUV. The longer the cable, the farther the AUV can move in both the flow direction and the vertical direction, and the equilibrium position shifts further from the initial release position. When L = 14 m, the vertical vibration amplitude of the moored AUV is the largest, with an amplitude of 1.431 m. Therefore, when designing the mooring system, attention should be paid to the compatibility between the anchor chain length and the AUV body.
(3)
The underwater movement of the moored AUV exhibits a relatively complex pattern. When the motion of the moored AUV in both the X and Y directions shows regular periodicity, with two oscillation cycles in the X-direction and four vibration cycles in the Y-direction—meaning the motion period in the X-direction is twice that of the Y-direction—the trajectory of the AUV’s centroid forms an 8-shaped low-frequency motion. It can be inferred that the mathematical relationship between the periods in the X and Y directions plays a significant role in the trajectory of the AUV’s underwater flow-induced motion. In practical mooring design, the ratio of the periods in the X and Y directions can be used as a reference for assessing the current motion state of the AUV.
(4)
In this study, the vortex shedding pattern of the moored AUV is classified into two types: S (single) and P (pair). The results show that when the vortex shedding pattern behind the AUV cylinder follows the P-mode, the trajectory of the AUV’s centroid approaches an ‘8’ shape. At this point, the number of vortices shed per cycle gradually increases, indicating that the fluid forces cycle multiple times, leading to larger vibration amplitudes.
The findings of this study provide valuable insights into the effectiveness of analyzing underwater motion characteristics of moored station-keeping AUVs. However, it is important to note that the coupled calculation method is constrained by the selection of model parameters. To improve the accuracy of the model, further investigations are required, particularly with respect to the rotational dynamics of the AUV rigid body, as well as the design of damping parameters in the displacement equations, which warrant additional research and experimental validation.

Author Contributions

Y.H.: Writing—original draft, Visualization, Data curation, Conceptualization. Z.M.: Writing—original draft, Visualization, Methodology, Conceptualization. B.C.: Resources, Investigation, Formal analysis, Conceptualization. B.L.: Validation, Software, Project administration, Formal analysis. W.T.: Supervision, Resources, Project administration, Funding acquisition. All authors have read and agreed to the published version of the manuscript.

Funding

This project was supported by the National Natural Science Foundation of China (Grant No. 52471346).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data are available from the corresponding author upon reasonable request.

Acknowledgments

The authors would like to express their gratitude to the editor and reviewers for their constructive comments and suggestions, which have contributed to the enhancement of the paper’s quality.

Conflicts of Interest

The authors declare no conflicts of interest.

Nomenclature

TCable tension
BBuoyancy
Ggravity
D C Fluid drag
DDiameter of AUV cylinder
IInitial force
LLength of cable
kTurbulent kinetic energy
ρ Fluid density
ω Turbulence frequency
ε Turbulent dissipation rate
α Turbulent turbulent dissipation rate constant
β Turbulent productivity constant
gGravitational acceleration
dDiameter of the anchor chain
vOcean current velocity
D n i Normal drag of anchor chain
D t i Tangential drag of anchor chain
C n i Normal resistance coefficients of anchor chain
C t i Tangential resistance coefficients of anchor chain
v n i Normal velocity of anchor chain
v t i Tangential velocity of anchor chain
μ k Turbulent viscosity
σ k Prandtl number of turbulence frequency
F μ Hydrodynamic force
F B G Gravity and buoyant force
F C Cable force
P k Turbulence generation term

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Figure 1. (a) Physical model of anchored underwater vehicle; (b) cylinder system and the corresponding computational domain.
Figure 1. (a) Physical model of anchored underwater vehicle; (b) cylinder system and the corresponding computational domain.
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Figure 2. Modelling of a dynamic mooring line.
Figure 2. Modelling of a dynamic mooring line.
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Figure 3. Grid distribution: (a) encrypted grid (I: static area; II: wake encrypted area; III: vibration encrypted area); (b) Grid of cylinder area (IV: square motion area).
Figure 3. Grid distribution: (a) encrypted grid (I: static area; II: wake encrypted area; III: vibration encrypted area); (b) Grid of cylinder area (IV: square motion area).
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Figure 4. Solving process of the numerical solution process of the FSI.
Figure 4. Solving process of the numerical solution process of the FSI.
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Figure 5. Numerical model validation: (a) comparison of vibration in y-direction; (b) comparison of cable tension in flow direction.
Figure 5. Numerical model validation: (a) comparison of vibration in y-direction; (b) comparison of cable tension in flow direction.
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Figure 6. The variation of cable force with time under different ocean current velocities. (a) Fcable_x; (b) Fcable_y.
Figure 6. The variation of cable force with time under different ocean current velocities. (a) Fcable_x; (b) Fcable_y.
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Figure 7. The variation of AUV position with time under different ocean current velocities. (a) X-position; (b) Y-position.
Figure 7. The variation of AUV position with time under different ocean current velocities. (a) X-position; (b) Y-position.
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Figure 8. Amplitude in the Y-direction at different ocean current velocities.
Figure 8. Amplitude in the Y-direction at different ocean current velocities.
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Figure 9. The trajectory of the center-of-mass motion for three ocean current velocities. (a) v = 0.0486 m/s; (b) v = 0.0648 m/s; (c) v = 0.0809 m/s.
Figure 9. The trajectory of the center-of-mass motion for three ocean current velocities. (a) v = 0.0486 m/s; (b) v = 0.0648 m/s; (c) v = 0.0809 m/s.
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Figure 10. Cloud diagrams of velocity field and vortex structure in different velocities.
Figure 10. Cloud diagrams of velocity field and vortex structure in different velocities.
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Figure 11. The variation of cable force with time under different cable lengths (a) Fcable_x; (b) Fcable_y.
Figure 11. The variation of cable force with time under different cable lengths (a) Fcable_x; (b) Fcable_y.
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Figure 12. The variation of AUV position with time under different cable lengths. (a) X-position; (b) Y-position.
Figure 12. The variation of AUV position with time under different cable lengths. (a) X-position; (b) Y-position.
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Figure 13. Amplitude in the Y-direction at different cable lengths.
Figure 13. Amplitude in the Y-direction at different cable lengths.
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Figure 14. The trajectory of the center-of-mass motion for three cable lengths. (a) L = 12 m; (b) L = 14 m; (c) L = 16 m.
Figure 14. The trajectory of the center-of-mass motion for three cable lengths. (a) L = 12 m; (b) L = 14 m; (c) L = 16 m.
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Figure 15. Cloud diagrams of velocity field and vortex structure in different cable lengths.
Figure 15. Cloud diagrams of velocity field and vortex structure in different cable lengths.
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Table 1. Main parameters of the AUV and the cable.
Table 1. Main parameters of the AUV and the cable.
ParametersValueParametersValue
Length of AUV7 mLength of cable12 m
Mass of AUV1438 kgLinear density of the cable1.56 kg/m
Diameter of AUV0.534 mDiameter of the cable0.02 m
Center of gravity(0, 0)
Table 2. Comparison of amplitude results for grid-independent validation.
Table 2. Comparison of amplitude results for grid-independent validation.
MeshElementsfy
Coarse20,0000.01992
Refine130,0000.01821(9.4%)
Refine240,0000.01703(6.9%)
Refine350,0000.01726(1.4%)
Table 3. Comparison of amplitude results for time step independence validation.
Table 3. Comparison of amplitude results for time step independence validation.
TimestepSteps(s)fy
T10.10.01548
T20.070.01633(5.2%)
T30.050.01712(4.6%)
T40.030.01741(1.7%)
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MDPI and ACS Style

Hu, Y.; Mao, Z.; Cheng, B.; Li, B.; Tian, W. Numerical Simulation Study of the Motion Characteristics of Autonomous Underwater Vehicles During Mooring Lurking Procedure. J. Mar. Sci. Eng. 2025, 13, 275. https://doi.org/10.3390/jmse13020275

AMA Style

Hu Y, Mao Z, Cheng B, Li B, Tian W. Numerical Simulation Study of the Motion Characteristics of Autonomous Underwater Vehicles During Mooring Lurking Procedure. Journal of Marine Science and Engineering. 2025; 13(2):275. https://doi.org/10.3390/jmse13020275

Chicago/Turabian Style

Hu, Yuyang, Zhaoyong Mao, Bo Cheng, Bo Li, and Wenlong Tian. 2025. "Numerical Simulation Study of the Motion Characteristics of Autonomous Underwater Vehicles During Mooring Lurking Procedure" Journal of Marine Science and Engineering 13, no. 2: 275. https://doi.org/10.3390/jmse13020275

APA Style

Hu, Y., Mao, Z., Cheng, B., Li, B., & Tian, W. (2025). Numerical Simulation Study of the Motion Characteristics of Autonomous Underwater Vehicles During Mooring Lurking Procedure. Journal of Marine Science and Engineering, 13(2), 275. https://doi.org/10.3390/jmse13020275

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