Numerical Simulation of Self-Propelled Dive Motion of a Virtual Mooring Buoy

Abstract

To verify the feasibility of the variable wing actuator for a virtual mooring buoy, this paper investigates the self-propelled dive motion of a virtual mooring buoy under hydrostatic variable density conditions using a computational fluid dynamics (CFD) approach. The virtual mooring buoy developed by our research group is used in this study, and the numerical simulation is performed using the Reynolds-averaged Navier–Stokes (RANS) equation and the SST K-Omega turbulence model to capture the turbulent flow. Grid convergence studies were conducted at three grid resolutions to ensure the accuracy of the numerical simulations. The effects of different wing angles on the self-propelled dive motion of the buoy are focused on and analyzed. The results show that the maximum velocity of the buoy in the horizontal direction can reach 0.31 m/s, with a wing angle of −8°, which is about 35% higher than that of 0°, effectively enhancing the buoy’s anti-disturbance capability against the horizontal currents. In addition, this study further analyzes the self-propelled dive motion of the buoy with variable wing angles. The results show that the velocity and attitude of the buoy at any moment are basically the same as those under the corresponding fixed wing angle. This shows that it is possible to change the motion of the buoy by varying the wing angle, verifying the feasibility of the variable wing actuator.

1. Introduction

The acquisition of parameters such as marine hydrological and meteorological elements is of great research significance for the three-dimensional observation of the marine environment. Autonomous profiling buoys are a transformative ocean observation technology capable of providing continuous, real-time subsurface observations of the global ocean, and the Argo buoys, which have been deployed in the global oceans since 1999, comprise the world’s first array for observing the subsurface ocean. As the only global subsurface ocean observing network, the Argo program was recognized as a major achievement of the Global Earth Observation System of Systems (GEOSS) in 2007 [1]. However, due to the vastness of the world’s oceans, it is difficult to cover all the oceans with sufficient density by increasing the number of Argo buoys, which are unable to achieve long-term observations of a given ocean area due to their wave-following operating characteristics [2,3]. In order to solve the above problems, observations of the designated sea area can be made by traditional observation methods such as artificial satellites, research vessels, or moored buoys. However, any critical scientific problem facing oceanography requires long sustained spatial observations, including measurements of the average state of the system as well as possible high-frequency variations in the system [4]. Artificial satellites are suited to collecting a wide range of data and are unable to monitor the underwater environment. Expedition vessels are unable to maintain monitoring at sea for extended periods of time. Moored buoy systems [5,6,7] allow for long-term monitoring at fixed points but are limited by long cables that are prone to tangling, difficulty in anchoring, and inconvenience in changing anchor points [3].

Fortunately, autonomous underwater vehicles (AUVs) have matured and are emerging as reliable tools for sustained ocean data collection, which fills a critical gap in ocean observation. After Henry Stommel [8] proposed the concept of underwater gliders, underwater gliders attracted attention and were widely used, such as Seaglider [9,10], Spary [11], and Slocum [12]. Subsequently, researchers have begun to explore the use of underwater gliders to observe designated areas through a technique called “virtual mooring”. The virtual mooring technique allows an underwater glider to move around a target location at sea level or to be stabilized at a certain depth underwater for fixed-point or fixed-depth mooring observations. This technique extends the applicability of underwater gliders while improving the scalability of existing mooring observations [4,13].

Referring to the design of underwater gliders and Argo buoys, this paper proposes a virtual mooring buoy with two modes of operation: current-resistant mode and vertical mode. In the current-resistant mode, the virtual mooring buoy is able to resist the effects of ocean currents and remain near the set mooring point, making them suitable for long-term monitoring, especially in the maintenance of maritime rights and interests and scientific research. In this way, the State can deploy clusters of buoys in disputed waters to form underwater fences, realize continuous monitoring of marine activities, provide real-time data support, and help safeguard maritime rights and interests. In scientific research, the virtual mooring buoy is capable of up and down movement within a set depth in vertical mode, which is suitable for monitoring complex phenomena such as medium- and microscale internal waves. This will provide scientists with the opportunity to study, in-depth, the dynamic changes of internal waves and analyze their impact on climate and ecosystem. In the exploration of marine resources, the virtual mooring buoy can be used as underwater platforms to realize communication between the seabed base and satellites, supporting the monitoring of changes in seabed resources, material migration, and the impact of environmental factors, thus providing strong support for the sustainable use of marine resources.

At present, most of the research on virtual mooring focuses on the field of control algorithms. As for the virtual mooring buoy driven by means of adjusting the buoyancy, it can only change its motion state by changing the hydrodynamic force it is subjected to when moving underwater. Therefore, the underwater maneuvering motion of virtual mooring buoys is one of the more concerned issues for designers.

With the continuous improvement of computer performance and the development of mesh technology, it has become possible to directly solve the multi-degree-of-freedom maneuvering motion of underwater vehicles by using computational fluid dynamics (CFD) methods, and the cost is much lower than that of the traditional self-propelled model test. CFD techniques have been widely used in the fields of submarines and ships and play an important role in simulating hydrodynamic performance and optimizing the maneuvering performance with high accuracy. Guo et al. [14] used the CFD method combined with the volume of fluid (VOF) method to realize the numerical simulation of near-surface submarine turning motion and captured the free-fluid surface dynamics effect during the simulation. This study enables more accurate simulation of near-free surface maneuvering performance of submarines by coupling the grid technique with the RANS (Reynolds-averaged Navier–Stokes Equation) model; in particular, the flow field characteristics and hydrodynamic parameters at different dive depths are effectively described. Similarly, Dubbioso et al. [15] applied the RANSE solver to deeply analyze the cross rudder and X rudder configurations in a submarine maneuverability study and found that there are significant differences in the maneuvering characteristics of the different rudder configurations; in particular, the X rudder configurations show better stability in the near-free surface environment. In ship design, Robin et al. [16] proposed a CFD-based dynamic velocity prediction program (VPP), which combines CFD hydrodynamics with real-time ship maneuvering control to optimize the ship’s speed under different wind conditions through fluid–structure interaction (FSI) technology. This method is able to simulate the real-time attitude changes and hydrodynamic characteristics of the ship in a single computation, thus achieving accurate speed optimization over the full range. Compared with traditional methods, this CFD-based VPP model not only improves the design efficiency but also reduces the error under complex sea state and dynamic load conditions. In the field of unmanned sailboats, Li et al. [17] successfully simulated the self-sailing performance prediction of an unmanned sailboat under multiple wind conditions by using the RANSE method combined with the overset grid technique. By coupling the interactions of hull, sail, and rudder, this method significantly improves the accuracy of speed prediction and effectively addresses the limitations of traditional experimental methods under multiphase flow conditions.

Based on these studies, this paper takes the virtual mooring buoy as the research object, controls the buoy body through the combination of the overset grid and dynamic fluid body interaction (DFBI) method, controls the wing angle through the sliding grid method, realizes the numerical simulation of the buoy’s self-propelled dive by using the STAR CCM+ CFD software (version 2306), verifies the feasibility of the buoy’s variable wing actuator, and provides the theory for the optimization of the future design.

2. Geometry Model of the Virtual Mooring Buoy

The virtual mooring buoy used in this study is designed with three compartments: buoyancy drive, center of gravity adjustment, and variable wing actuator. The virtual mooring buoy has a buoyancy adjustment system and an attitude adjustment system similar to those of traditional underwater gliders, with certain maneuvering performance, capable of directional, and steering gliding; it is also capable of realizing vertical profile movement, as in the case of traditional profiling buoys; coupled with a new type of variable wing, it improves the maneuverability of the buoy. The NACA0012 wing type is used for the variable wing, and the structural design is shown in Figure 1. The main body and wing size parameters of the virtual mooring buoy are shown in

Table 1. Detailed dimensional parameters of the virtual mooring buoys. (Units: m).

Main Body		Wing (NACA0012)
Length L0	2.4	Wing chord C	0.22
Maximum radius D	0.176	Wing span S	0.66

.

3. Numerical Methods and Grid Design

3.1. Numerical Methodology and Physical Model

In this study, the CFD software STAR CCM+ is used for the numerical simulation of the self-propelled and submerged motion of a virtual mooring buoy. The solver is an implicit non-constant separated flow solver with pressure–velocity coupling using the SIMPLE algorithm, and the time step is set to 0.1 s. Due to the fact that the buoy achieves self-submergence by regulating its own volume and sails at a low speed, the SST Menter K-Omega turbulence model is used. The SST Menter K-Omega turbulence model is suitable for surface or submerged vehicles and simulates the near-wall external flow field better than other fine turbulence models; it is computationally more economical for engineering problems compared to large eddy simulations, but it is more finely than other two-equation models. The hybrid wall treatment technique is used to achieve low y⁺ wall treatment for fine meshes and high y⁺ wall treatment for coarse meshes. The present calculations numerically simulate the autonomous dive of a buoy with a fixed wing angle and a variable wing angle under hydrostatic variable density conditions, respectively.

3.2. Coordinate and Computational Domain Configuration

This calculation reduces the 6-degrees-of-freedom motion of the buoy to a 3-degrees-of-freedom motion in a two-dimensional plane. In order to solve for the 3-degrees-of-freedom motion of the buoy, the flow field first needs to be calculated to determine the forces and moments acting on the buoy. Since this calculation is performed in a global (Earth-based) coordinate system, it is necessary to solve the equations of motion of the buoy by transforming the fluid forces and moments to the buoy body coordinate system through a transformation matrix constructed from Euler angles. In addition to the global coordinate system and the body coordinate system of the buoy, the rotational coordinate system of the wing also needs to be set up to achieve accurate control of the wing angle. Three different coordinate systems are used in this calculation, as shown in Figure 2. In fact, the global coordinate system (x, y, z) has been defined, and this coordinate system does not move with the buoy. The origin of the body coordinate system (u, v, w) of the buoy is located at the floating center of the buoy, the x-axis points to the bow of the buoy along the central axis of the body of the buoy, the y-axis is perpendicular to the x-axis pointing to the right side of the buoy, and the z-axis points to the base of the buoy perpendicularly to the plane formed with the x-axis and y-axis, with the same initial direction as the global coordinate system (x, y, z). The origin of the wing rotation coordinate system (p, q, r) is located at the position of 1/4 chord length of the wing, the x-axis points to the leading edge of the wing along the central axis, the y-axis is perpendicular to the x-axis pointing to the right side of the wing, and the z-axis is perpendicular to the plane formed with the x-axis and y-axis pointing to the bottom of the wing. The wing rotation coordinate system moves with the body coordinate system of the buoy.

The computational domain includes a background domain (a) and two buoy motion domains (b): a buoy body domain (c), and a wing rotation domain (d). The interface between the buoy motion domain and the background domain uses the linear interpolation method of the overset grid technique for data exchange; the interface between the wing rotation domain and the buoy body domain is a sliding intersection interface. The background domain is a rectangular body with dimensions of 12 m × 7.2 m × 12 m, as shown in Figure 3. The velocity at the entrance is 0 m/s, and the boundary conditions are shown in Figure 4.

3.3. Grid Generation

In this study, the computational grid utilized is a trimmed unstructured mesh arranged orthogonally. To capture the near-wall flow, a prismatic layer grid was used near the wall (buoy surface). With the dimensionless wall distance y⁺ taken as 30, in order to ensure that the boundary layer thickness of the fluid flow is equal to the prismatic layer grid thickness, the number of prismatic layers is 21, and the thickness of the first prismatic layer grid is 7.28 × 10⁻⁴ m. The exact distribution of y⁺ values can be found in Figure 5. In order to realize the 3-degrees-of-freedom motion of the buoy, overset grids are created in the buoy region. At the same time, the size of the overset grid needs to match the size of the background grid; therefore, grid refinement around the overlapping domain is necessary. During the grid refinement process, it is important to ensure that the buoy always stays within the refined grid region during motion, and the specific refinement dimensions are shown in

Table 2. Volume refinements: cell size given as basic size—(0.1 m) × N.

Domain	N [-]
	x	y	z
Background grid	100%	100%	100%
Refinement grid	50%	50%	50%
Buoy body domain	50%	50%	50%
Wing rotation domain	12.5%	12.5%	12.5%

. For the accuracy of the calculation, the surface grids of the buoy body and the wing are refined to achieve a finer grid resolution, and the specific refinement dimensions are shown in

Table 3. Isotropic surface refinements: cell size given as basic size—(0.1 m) × N.

Surface	N [-]
Buoy body surface	6%
Wing surface	3%

. The grid division around the buoy is shown in Figure 6.

3.4. Grid Dependence Analysis

The numerical uncertainty analysis adopted herein follows the recommendations by ITTC, and the grid counts used for convergence validation range from 8.7 × 10⁵ to 6.9 × 10⁶, with a grid scale refinement ratio $r_i$ of $\sqrt{2}$, defined as follows:

$$r_i={\displaystyle \frac{\Delta x_{i,2}}{\Delta x_{i,1}}}={\displaystyle \frac{\Delta x_{i,3}}{\Delta x_{i,2}}}={\displaystyle \frac{\Delta x_{i,m}}{\Delta x_{i,m-1}}}.$$

(1)

Under the hydrostatic constant density ($\rho =1025\mathrm{ k}\mathrm{g}/{\mathrm{m}}^{\mathrm{3}}$) condition, the buoy’s self-propelled motion at the three grid scales finally reaches convergence. In other words, the attitude and velocity of the buoy finally no longer change. Therefore, the resistance (based on the buoy coordinate system) when the buoy motion is stabilized is chosen as the judgment criterion for grid convergence in this study.

The verification of the resistance coefficient has been carried out with respect to grid convergence. The resistance coefficient $C_D$ is defined by the Equation (2):

$$C_D={\displaystyle \frac{F_D}{0.5\rho {L_0}^2{U}^2}},$$

(2)

where $F_D$ is the resistance on the buoy, $\rho$ is the density of seawater, $L_0$ is the characteristic length of the buoy, and $U$ is the velocity of the buoy. The calculation results of the three grid scales are shown in

Table 4. Resistance coefficients at different cell numbers.

Mesh	Symbol	Total Number	${\mathit{F}}_{\mathit{D}}$ (N)	${\mathit{C}}_{\mathit{D}}$
Coarse	${\phi }_{i,3}$	871,056	4.05	4.294 × 10−3
Medium	${\phi }_{i,2}$	2,383,718	3.76	4.180 × 10−3
Fine	${\phi }_{i,1}$	6,927,465	3.62	4.072 × 10−3

.

According to the ITTC (2021) Guideline for Computational Validation of CFD, the determination of convergence is based on the convergence parameter $R_i$, and the convergence study requires at least m = 3 solutions to evaluate the convergence of the input parameters. The criterion is defined as

$$R_i={\displaystyle \frac{{\epsilon }_{i,21}}{{\epsilon }_{i,32}}},$$

(3)

where ${\epsilon }_{i,21}={\phi }_{i,2}-{\phi }_{i,1}$ is the variation between the fine grid and the medium grid, and ${\epsilon }_{i,32}={\phi }_{i,3}-{\phi }_{i,2}$ is the variation between the medium grid and the coarse grid.

The coefficients were calculated to be monotonically convergent. When the numerical results satisfy monotonic convergence, the Richardson extrapolation (RE) [18] method is employed to evaluate the grid uncertainty. The discretization order $P_i$ is defined by Equation (4):

$$p_i={\displaystyle \frac{\ln (1/R_i)}{\ln (r_i)}}.$$

(4)

Finally, the convergence of the grid is verified by the Grid Convergence Index (GCI), where

$$GC{I}_i^{21}=F_S{\displaystyle \frac{{\epsilon }_{i,21}}{r_i^{p_i}-1}}.$$

(5)

In this context, $F_s$ represents a safety factor, which is set to 1.25 for the use of three or more grids. A smaller GCI value indicates a lower sensitivity of the numerical results to the grid. The calculated GCI value of 0.241% indicates that the effect of grid refinement on numerical predictions diminishes as the number of grids increases. The results of these studies are summarized in

Table 5. The computed convergence ratio, order of discretization and GCI.

Study	${\phi }_{i,3}-{\phi }_{i,2}$	${\phi }_{i,2}-{\phi }_{i,1}$	$R_i$	$p_i$	$GC{I}_i^{21}(\%)$
$C_D\times {10}^4$	1.14	1.08	0.947	0.157	0.241

. Consequently, to optimize computational resources, a medium-scale grid is chosen for subsequent calculations, with the safety factor $F_s$ ensuring a conservative approach to grid selection. The medium grid divisions are shown in Figure 7.

4. Numerical Simulation of Buoy Self-Propelled

The virtual mooring buoy is designed to change the buoyancy force on the buoy by adjusting the volume of the buoy’s external oil bladder, which, in turn, realizes the buoy’s dive and float, and to realize the change in the buoy’s attitude and speed by adjusting the size of the wing angle. At the same time, changes in seawater density have a non-negligible effect on the buoyancy force on the buoy, which, in turn, affects the movement of the buoy. Therefore, it is practical to use variable density medium in this study.

4.1. Simulation of Buoy Self-Propelled with Fixed Wing Angles

The basic CFD methodology and setup has been described in Section 3.1. This section will focus on the process of buoy self-sailing simulation. In the initial state, the buoy floats on the water surface (x-y plane), its center of gravity is located directly below the center of buoyancy, and the initial speed is 0. When the buoy dives, the external oil bladder returns 500 mL of oil to the inside of the buoy, the volume of the buoy decreases, the buoyant force decreases, the moving mass block is moved to the positive direction of the u-axis by 10 mm, and the buoy’s center of gravity changes. When the sum of resistance and buoyancy reaches equilibrium with gravity force, the buoy reaches the maximum speed; at the same time, due to the density of seawater gradually increases with depth, the buoyancy on the buoy will gradually increase, and finally, the buoy will be suspended at a certain depth. The specific parameters of the buoy are shown in

Table 6. Change in volume and center of gravity as the buoy begins to dive.

State	Parameter
	Volume (m3)	Center of Gravity (mm)
Initial	0.0592	(0.0, 0.0, 6.4)
Dive	0.0587	(8.4, 0.0, 6.4)

.

In this study, self-propelled simulations were performed under hydrostatic variable-density conditions for buoy wing angles from 0 to −12° (At −2° intervals). The density variation in the sea area where the buoy is applied is derived by fitting the Argo open-source data, defined by Equation (6) [19], and the variation of seawater density $\rho$ as a function of depth h was set by means of the user-defined field function in STAR CCM+.

$${\rho }_h=1027.458\times e^{4.508\times {10}^{-6}h}-2.254\times e^{-3.058\times {10}^{-3}h}, 0<h<1000$$

(6)

This study focuses on the motion of the buoy within a depth of 500 m. The release time is 1 s, the ramp time is 10 s, and the simulation time is 1200 s. The time step is 0.1 s, and the maximum number of iterations for each time step is 5. Under buoyancy drive, the buoy achieves autonomous dive in the working condition of 0~−12° (−2° interval) of the wing angle. The position and attitude of the buoy are shown in Figure 8, including the displacements in z-axis and x-axis directions as well as the angle of pitch. The sailing speed of the buoy is shown in Figure 9.

From Figure 8, it can be seen that as the wing angle increases, the sailing depth (z-direction displacement) of the buoy decreases gradually, and the angle of pitch of the buoy decreases gradually when it moves steadily; moreover, the sailing distance of the buoy in the horizontal direction is the largest when the wing angle is −8°, which gives the best performance of the buoy in gliding.

As can be seen in Figure 9, the buoy undergoes two phases: an acceleration phase and a deceleration phase. In the acceleration stage, the gravity on the buoy is greater than the sum of the buoyancy and resistance, and the buoy accelerates, but at the same time, due to the density of the seawater and the increase in the speed of the buoy, the resistance and buoyancy on the buoy are also increasing gradually; at about 50 s, the gravity on the buoy is equal to the sum of the buoyancy and the resistance and the buoy reaches equilibrium; at this time, the resistance and speed of the buoy reach their maximum; when the buoy continues diving, due to the increase in seawater density, the buoyancy on buoy gradually increases, and the gravity force on the buoy is less than the sum of the buoyancy force and the resistance force, and the buoy decelerates. It is worth noting that the horizontal velocity of the buoy with a wing angle of −8° increases by about 35% compared to 0°, which suggests that it is feasible to improve the buoy’s anti-disturbance capability to horizontal currents by adjusting the wing angle; the angular velocity of the buoy is close to zero about 50 s after the beginning of the dive, and the attitude of the buoy has stabilized.

4.2. Simulation of Buoy Self-Propelled with Variable Wing Angle

Relying on the above study, in this section, the motion of the buoy under variable wing angle conditions is further simulated in order to realize the fast dive of the virtual moored buoy and to increase the measurement data and observation range of the buoy in the target sea area (assumed to be 100~200 m and 300~400 m). The variation function of wing angle with sailing depth is set by the user-defined field function in STAR CCM+, and the wing angle of the buoy is 0° in the rapid dive stage of the buoy, and when it reaches the target depth, the wing angle is changed to −8° to increase the measurement data and observation range of the buoy in the target sea area. The motion track of the buoy is shown in Figure 10. The movement speed and attitude of the buoy are shown in Figure 11.

From Figure 10, it can be seen that through the change of wing angle, the buoy realizes rapid dive in 0~100 m and 200~300 m waters; at the same time, it enlarges the observation range in the horizontal direction in the target sea area, which verifies the feasibility of the variable wing actuator of the virtual mooring buoy.

Figure 11 indicates that when the wing angle changes from 0° to −8°, the velocity of the buoy in the vertical direction decreases, the velocity in the horizontal direction increases, and the angle of pitch decreases; the velocity and attitude of the buoy at any moment are basically the same as that under the corresponding fixed wing angle; and the velocity and attitude of the buoy tends to be stabilized in about 30 s after the change of the wing angle.

5. Discussion

In this study, the numerical simulation of the dive process of a virtual mooring buoy under hydrostatic variable density conditions is successfully implemented by STAR CCM+ software combined with overset grid and dynamic fluid body interaction (DFBI) methods, which can accurately predict the motion state of the buoy under different wing angles, including force, position, attitude, and velocity.

The simulation was performed on a 48-core, 96-thread processor with a time resolution of 0.1 s and a solution time of 1200 s; the computational time for each case took 48 h approximately. This highlights the computationally intensive nature of this type of analysis, which could be improved by employing more advanced computational equipment or optimizing the time step.

It is worth noting that in real ocean environments, changes in currents and water pressure to which a buoy is subjected can have a significant effect on its state of motion. Therefore, the motion process of the buoy in the ocean is a fluid–structure interaction (FSI) problem. With the increase in the depth of the buoy dive, the seawater density and pressure are also gradually increasing, and the volume of the buoy will decrease due to the increase in seawater pressure, which will affect the buoy’s buoyancy and the motion state; in addition, the change in the ocean current also affects the motion state of the buoy. In this study, the self-propelled simulation of the buoy is carried out under hydrostatic variable density conditions, and the effects of changes in ocean currents and water pressure on the kinematic state of the buoy are not considered for the time being. The focus of this study is to determine the optimal wing angle and optimal glide performance of the buoy via the CFD method, which is a method used for rapid qualitative analysis during the preliminary design and scheme selection stage.

Future studies can be conducted in more complex flow field environments to enhance the applicability of the simulation and provide more comprehensive technical support for the application of virtual moored buoys in real ocean observation.

6. Conclusions

In this paper, the autonomous submergence motion of a virtual mooring buoy is simulated under hydrostatic variable density conditions via the CFD method, and the autonomous submergence motion of the buoy is solved for both fixed wing angle and variable wing angle conditions.

During the buoy motion, the buoyancy on the buoy gradually increases due to the change in seawater density, and the speed of the buoy motion also decreases gradually. It is found that it takes about 50 s for the buoy to move from the beginning diving state to the stable state; with the increase in wing angle, the sailing depth of the buoy decreases gradually; when the wing angle is 0°, the buoy has the maximum speed in the vertical direction up to 0.55 m/s, and the sailing depth is 506.4 m; when the wing angle is −8°, the buoy has the maximum speed in the horizontal direction up to 0.31 m/s, which is about 35% higher than that of 0°, effectively enhancing the buoy’s anti-disturbance capability against the horizontal currents, and the sailing distance is 296.8 m, the best gliding performance of the buoy; when the wing angle continues to increase, the sailing distance of the buoy in the horizontal direction will not continue to increase. Through the variable wing simulation of the self-propelled buoy, the rapid dive of the buoy is realized, and the observation range in the target sea area is also increased; it takes about 30 s for the buoy motion to stabilize again each time the wing angle changes; the speed and attitude of the buoy at any moment are basically the same as those under the corresponding fixed wing angle, which also verifies the feasibility of the variable wing actuator of the buoy in a virtual mooring system.

Compared with traditional physical model experiments, the CFD method provides a more cost-effective and efficient means, especially in the early design stage of the buoy, which helps to qualitatively analyze the hydrodynamic performance of the buoy quickly and to optimize the design scheme.