Hydrodynamic Analysis and Drag-Reduction Design of an Unmanned Underwater Vehicle Based on Computational Fluid Dynamics

Abstract

In order to establish a proper geometry of an Unmanned Underwater Vehicle (UUV) for stable motion control and energy usage reduction, this paper analyzes the hydrodynamic performance of a complex shape underwater vehicle and develops a systematic Computational Fluid Dynamics (CFD) simulation method to solve the hydrodynamic parameters of the system. Based on the simulation method and their results, the streamlines and pressure distributions of the water flow around the underwater vehicle are analyzed, and the geometric model design is improved based on the drag characteristics. Also, a comparison scheme is designed to evaluate the vehicle model before and after the geometry improvement. Simulation result shows that the design schemes brings 18% drag reductions in surge direction and 32% in heave direction. Moreover, by establishing the physical and mathematical models of the UUV on a physical simulation platform, a complete model of the underwater vehicle is constructed, laying the foundation for further simulations and experiments.

1. Introduction

With the introduction of ocean energy policies and the development of related industries, marine-related industries have begun to focus on deep-sea mining and transportation of energy resources such as natural gas and oil. In addition, the installation and construction of submarine cables and other facilities that support communication between continents have also become a major hotspot [1]. However, due to the complex and variable environments of the ocean, it is difficult for divers to perform complex tasks through saturation diving, hence underwater vehicles has been widely applied to underwater exploration and operations. Among them, unmanned underwater vehicles (UUVs), which do not require divers to dive with them and can be remotely controlled from the surface, have gradually become the mainstream equipment for executing deep-sea operation projects [2]. The design and development of UUVs is a comprehensive and complex engineering task, with a high degree of technological intensity. UUVs can generally be divided into Remotely Operated Underwater Vehicles (ROVs) and Autonomous Underwater Vehicles (AUVs) [3]. ROVs can be further categorized into open-frame structures or enclosed structures. A frame with buoyancy block structure is widely adopted in most ROVs geometry designs, due to its various advantages. This structure enables the carrying and installation of large payloads, and exploration and operation equipment can be conveniently mounted on or removed from the frame-structure for further analysis and maintenance [4]. In the early stages of development, UUV system design placed greater emphasis on drag-reduction design to enhance navigation speed and energy efficiency. With the diversification of the UUVs’ application and functional requirements, loading capacity and motion stability are also be took into account. Thus, the hydrodynamic characteristics of the UUVs, especially drag-reduction performance, become an important research field [5].

For UUVs with complex geometries, such as ROVs, the structural integrity of the vehicles are important to withstand the influence from sea currents and vortices, and it is the cornerstone for ensuring the vehicle’s capability to effectively carry out tasks in deep-sea environments [6]. Accomplishing this objective entails a meticulous design of the UUVs’ geometry, along with the integration of well-engineered hydrodynamic components both on attachments and the main body, ensuring dynamic stability and drag reduction. Good geometry design not only entails maintaining robust structural stability, but also ensuring consistent hydrodynamic performance. Additionally, considerations must be given to endurance requirements, as well as the specific operational conditions and objectives [7]. Consequently, optimizing the geometry of underwater vehicles serves as the bedrock of the entire vehicle design process and represents a critical step.

To have a further knowledge of the hydrodynamic characteristic of the UUVs, the hydrodynamic model of the system should be established first. And it is fundamental to establish the dynamic mathematical models for UUVs, which play crucial roles in their maneuvering control and motion prediction. Traditional hydrodynamic model construction is based on a fundamental motion, typically uniform straight-line forward movement, to which a small perturbation is added. The multivariate function is then expanded using a Taylor series, with the fundamental motion serving as the reference point for the expansion. The coefficients obtained from the Taylor series expansion are known as hydrodynamic coefficients. Fossen, from the Norwegian University of Science and Technology, has proposed a simplified mathematical model for underwater robots [8]. This model considers the ROV to be symmetrical along three orthogonal planes: vertical, lateral, and fore/aft. In this model, the viscous hydrodynamic forces and moments acting on the underwater robot are also significantly simplified. It is assumed that the viscous hydrodynamic force (torque) in a particular direction is only related to the velocity (angular velocity) in that direction. This reduction in complexity significantly decreases the required hydrodynamic coefficients, thus easing the acquisition process. Chin et al. emphasized a systematic modeling of hydrodynamic damping using the CFD software ANSYS-CFX^TM on a complex-shaped ROV [9]. Also from NTNU, Eidsvik established hydrodynamic model and conducted experiments to identify hydrodynamic parameters of several ROVs [10]. Yang et al. [11] proposed a control-oriented modeling approach for a low-speed semi-AUV which has complex-shaped structures. They also used cost-efficient CFD softwares (ANSYS-CFX^TM, STAR-CCM+^TM to predict the two hydrodynamic key parameters.

The common methods for obtaining hydrodynamic coefficients currently in use are experimental methods, semi-empirical formula calculations, and numerical simulation methods based on CFD technology [12]. The experimental method is simple, intuitive, easy to implement, and has high reliability. Many researchers have predicted hydrodynamic coefficients using this method [13,14]. However, the high cost of experiments and the long cycle of the experimental process lead to difficulties using physical experimental method; the semi-empirical formula calculation method is suitable for calculating the hydrodynamic parameters of underwater vehicles with simple shapes and single geometric structures. However, the shapes of AUVs and ROVs are usually more complex, making it difficult for the semi-empirical formula calculation method to accurately obtain all of the added mass and added inertia moments and viscous hydrodynamic forces in the kinematic equations of underwater vehicles. In recent years, numerical simulation methods based on potential flow theory and finite element theory, which have developed from the principles of mass conservation, momentum conservation, and energy conservation, have advanced rapidly. Many researchers used CFD method to estimate hydrodynamic coefficients of AUVs with complex shapes, such as an axisymmetric body, a body with appendages, and a spherical body designed for amphibious tasks [15,16,17,18,19]. Also, the flow status of open-frame vehicles are studied numerically, which proves the possibility of analyzing the hydrodynamic characteristics of UUVs with complex geography accurately, using CFD method [20]. The co-researchers in Harbin Institute of Technology, Shenzhen, also conduct numerical studies on hydrodynamic characteristics for AUVs and ROVs, they proposed systemic methods and reviewed the CFD method of hydrodynamic analysis [21,22,23].

The primary strategy for enhancing the efficiency of UUVs involves refining the flow dynamics around the vehicle. Vortices and turbulent flow patterns can significantly diminish the ROV’s hydrodynamic capabilities. Altering the vehicle’s geometry while preserving its fundamental stability represents the most effective approach. The advancement of computer simulation technology has facilitated comprehensive optimization analyses of underwater vehicles. The iterative design paradigm, integrating ROV analysis with CFD simulation, is extensively utilized in underwater vehicle design processes [24]. For AUVs and other torpedo-shaped underwater robots, researchers focusing on reducing the drag in surge directions to lower the energy consumption [25,26,27,28,29]. And for ROVs and other work-class underwater robots, enhancing the suitability becomes the most important goal [6,30,31]. A comprehensive review on recent trends and applications of simulation-based design optimization (SBDO) of underwater vehicles [32], shows that the studies in SBDO still remains a predominant reliance on simpler, deterministic single-objective formulations. To model and navigate the uncertainties inherent in marine engineering, including factors like wave dynamics and ocean currents more accurately, advanced optimization approaches which are more sophisticated, multi-objective, and stochastic should be applied in to SBDO.

This paper initially establishes a six-degree-of-freedom (6-DOF) dynamics and kinematics equations for a ROV. Concurrently, it employs CFD simulation methods to solve for the most challenging hydrodynamic parameters within the dynamics. A more systematic CFD simulation method for complex-shape vehicles is proposed to calculate these hydrodynamic parameters, thereby establishing the vehicle’s dynamic model. By analyzing the motion state of the vehicle in the simulation, optimization schemes for the overall vehicle model based on hydrodynamic parameters are proposed, and the optimized model parameters are determined. Finally, comparative simulation experiments of the model’s motion before and after optimization are presented to demonstrate that the optimization scheme can effectively enhance the vehicle’s motion capabilities. To facilitate the simulation of underwater operation scenarios, an underwater simulation physical platform for the vehicle has been developed based on the Robot Operating System (ROS). This platform simulates the underwater environment and interference, and integrates various sensors for perception within the simulation environment, providing a channel for algorithm validation for different types of underwater vehicles.

2. Dynamic Model Establishment

To facilitate the determination of the motion state of underwater vehicles, two commonly used system coordinate frames are introduced, namely the Inertial Coordinate System (North East Down) and the Body Fixed Frame. For general types of underwater vehicles, the position and attitude of the vehicle can be described by six different variables, as shown in Figure 1.

For an underwater vehicle, six different motion components are conveniently defined as heave, sway, surge, roll, pitch, and yaw. The symbol definitions for motion parameters are defined by their direction and orientation, where clockwise rotation and movement along the positive direction of the coordinate system are considered positive, otherwise are negative. The descriptions and symbols for the relevant motions under each degree of freedom for underwater vehicles are shown in

Table 1. The motion definitions for the degrees of freedom of underwater vehicles.

DOF	Motion	Force	Velocity	Coordinate
1	Surge	X	u	x
2	Sway	Y	v	y
3	Heave	Z	w	z
4	Roll	K	p	$\varphi$
5	Pitch	M	q	$\theta$
6	Yaw	N	r	$\psi$

.

For underwater vehicles, the general motion of a six-degree-of-freedom (6-DOF) underwater vehicle with the origin O as the coordinate reference is defined as follows.

The velocity of the underwater vehicle can be decomposed into linear and angular velocities. In terms of linear velocity, the vehicle facilitates coordinate transformations through a rotation matrix, as shown in the following equation:

$$\begin{array}{c}\eta ={\left[{\eta }_{B/N}^N,{\Theta }_{B/N}^N\right]}^T={\left[x,y,z,\varphi ,\theta ,\psi \right]}^T\hfill \\ v={\left[v_{B/N}^B,{\omega }_{B/N}^B\right]}^T={\left[u,v,w,p,q,r\right]}^T\hfill \\ \tau ={\left[f^B,{\tau }^B\right]}^T={\left[X,Y,Z,K,M,N\right]}^T\hfill \end{array}$$

(1)

$${\eta }_{B/N}^N=R(\widehat{z},\psi )R(\widehat{y},\theta )R(\widehat{x},\varphi )v_{B/N}^B=J_{B/N}^V\left({\Theta }_{B/N}^N\right)v_{B/N}^B$$

(2)

In the given context, $R(\widehat{z},\psi )$ represents the coordinate transformation matrix for the rotation of the underwater vehicle about the Z-axis, $R(\widehat{y},\theta )$ is the coordinate transformation matrix for rotation about the Y-axis, and $R(\widehat{x},\varphi )$ denotes the coordinate transformation matrix for rotation about the X-axis. The aforementioned equation represents the linear velocity rotation matrix between the body-fixed coordinate system and the global (world) coordinate system, with the computation method as illustrated in the equation.

$$J_{B/N}^V\left({\Theta }_{B/N}^N\right)=\left(\begin{array}{ccc}c\psi c\theta & -s\psi c\varphi +c\psi s\theta s\varphi & s\psi s\varphi +c\psi s\theta c\varphi \\ s\psi c\theta & c\psi c\varphi +s\psi s\theta s\varphi & -c\psi s\varphi +s\psi s\theta c\varphi \\ -s\theta & c\theta s\varphi & c\theta c\varphi \end{array}\right)$$

(3)

The expressions for angular velocity and linear velocity are consistent, both derived from transformation matrices. Vectors are used to represent the kinematics, and the following equation illustrates the 6-DOF kinematic equations for the underwater vehicle:

$$\dot{\eta }=J_{\theta }\left(\eta \right)v$$

(4)

$$\left[\begin{array}{c}{\dot{\eta }}_{B/W}^W\\ {\dot{\Theta }}_{B/W}^W\end{array}\right]=\left[\begin{array}{cc}{J}_{B/W}^V& 0_{3\times 3}\\ 0_{3\times 3}& J_{B/W}^{\omega }\left({\theta }_{B/W}^W\right)\end{array}\right]\left[\begin{array}{c}{v}_{B/W}^B\\ {\omega }_{B/W}^B\end{array}\right]$$

(5)

Considering the analysis object of this paper is a general underwater vehicle model, the shape of the underwater vehicle does not change under normal circumstances, and its mass and center of gravity position are fixed. The body coordinate system is established based on the center of gravity of the underwater vehicle, and the three degrees of freedom inertial axes of the underwater vehicle are used as the coordinate axes of the body coordinate system. Additionally, due to the low-speed operational state of the underwater vehicle and the deep-sea working environment, the influence of surface waves does not need to be considered. The dynamic equations of the underwater vehicle can be described using relevant mechanics formulas, as shown in the following equation.

$$F=M_{RB}\dot{V}+C_{RB}\left(V\right)V=\left[\begin{array}{cc}m{I}_{3\times 3}& 0_{3\times 3}\\ 0_{3\times 3}& I_g\end{array}\right]\dot{V}+\left[\begin{array}{cc}mS\left(V_2\right)& 0_{3\times 3}\\ 0_{3\times 3}& -S\left(I_g{V}_2\right)\end{array}\right]V$$

(6)

In the equation, $M_{RB}$ represents the mass distribution of the UUV, while the Coriolis matrix, denoted as $C_{RB}$, represents the Coriolis and centrifugal forces acting on the UUV, and it satisfy the relationship $C_{RB}\left(V\right)=-C_{RB}^T\left(V\right)$; F represents the total force and moment acting on the underwater vehicle. In summary, by solving the mass matrix and the Coriolis and centrifugal force matrix, the general kinematic equations for the underwater vehicle can be obtained.

3. Hydrodynamic Coefficients Estimation for the Underwater Vehicle

To establish the dynamic model of the vehicle, this paper employs CFD simulations to solve for its hydrodynamic coefficients. CFD simulation requires meshing of the vehicle’s model, necessitating certain simplifications to the shape of the underwater vehicle before computation: (1) for minor structures, which have a negligible impact on drag, simplifications are made by filling in or removing them; (2) the propeller blades and external attachments, which have complex shapes that may cause issues with meshing, are simplified; and (3) for the minor recesses on the aluminum profile structure that have a minimal impact on hydrodynamics, they are optimized using solid square steel. For the load-bearing structures and other appendages that have a minor impact on drag, they are simplified by removal.

The basic numerical simulation process adopted in this paper is depicted in Figure 2. In the pre-processing stage, the underwater vehicle model is first simplified, followed by the meshing of the fluid domain. An unstructured meshing scheme is utilized to avoid meshing failures that may arise due to the irregular shape of the vehicle. In the numerical computation stage, the fluid domain of the vehicle is iteratively solved through CFD simulation. The fluid domain changes are studied frame by frame, and the simulation is iterated to the specified number of iterations to determine if the results have converged. If convergence is achieved, the process proceeds to the post-processing stage. Otherwise, the parameters in the pre-processing stage need to be readjusted. Finally, in the post-processing stage, the simulation software outputs parameters such as streamlines and pressure maps. These data are analyzed and solved using the aforementioned formulas to ultimately determine the hydrodynamic parameters and establish the model of the underwater vehicle.

3.1. Model Simplification and Fluid Domain Design

The basic model of the underwater vehicle obtained after applying the aforementioned simplification rules is shown in Figure 3. After the simplification, the model maintained the main structure of the original model, while it also improve the operability of the meshing process.

The CFD simulation defines the fluid domain as shown in Figure 4. Following the ITTC guidelines and baseline tests, by defining the length of the underwater vehicle in the direction of flow as L, the length from the entry surface to the body of the vehicle is designed as $4L$, and the length from the rear of the body to the exit surface is $8L$. The computational domain is characterized by a height and width of $8L$ each, and a length of $12L$.

The surrounding boundaries of the fluid domain are set as no-slip walls, the inlet face is set as a velocity control face for fluid inflow, the outlet face is set as a pressure control face for fluid outflow, and the surface of the UUV is set as a no-slip wall in order to specify the flow velocity and state of the fluid outside the UUV model. The definitions of each boundary are shown in Figure 5.

3.2. Mesh Generation

By obtaining the model of the underwater vehicle and the fluid domain, the mesh of the model is designed and generated using mesh generation software ANSYS Meshing, the mesh structure and the transition crossing the inflation layer are shown in Figure 6. The SST $k-\omega$ model is chosen as the turbulence model due to its greater accuracy by separately employing $k-\omega$ and $k-ϵ$ model in the near-wall inflation zone and the developing zone of the mesh grids. According to these features of SST $k-\omega$ model, the value of $y^{+}$ can be set to 1 to obtain a first layer of the inflation layer with lower height, which can ensure a smooth boundary layer transition with finer height changes. Additionally, controlling the surface mesh size of the ROV is essential to minimize boundary layer mesh flatness and enhance its quality. Drawing on the research of Lin et al. [33], and considering the scale of small structural components like the UUV framework, the first layer mesh size for this model is set at 0.025 m. This size ensures that the short-axis surface of the UUV framework is covered by at least five grids, maintaining calculation accuracy. It also allows for approximately 20 boundary layers, balancing computational cost and precision.

After simplifying the 3D model of the UUV, it is necessary to generate separate meshes for steady and unsteady motion simulations. The main parameters and settings for these two types of meshes are detailed in

Table 2. Mesh parameters and settings.

Parameters	Steady Motion	Unsteady Motion
Mesh type	Unstructured
Size Function	Curvature and Proximity
Element size	0.99 m
Max size	1 m
Inflation layer	With inflation
Face sizing	0.025 m
Body sizing	None	0.3 m
Geometry	Flow Zone	Sub Zone
Boundary	ROV Surface
Inflation option	First, Layer Thickness
Y plus	1

.

3.3. Simulation Settings and Parameters

After proper mesh generation, the hydrodynamic parameters of the vehicle can be calculated through numerical solvers in CFD simulation software ANSYS Fluent 2021. Once the model simplification and the computational fluid domain are established, mesh generation is performed using the built-in methods of the CFD software, and the momentum exchange approach for the simulation process is determined. The CFD simulation parameters and settings for the underwater vehicle are shown in

Table 3. Simulation parameters of CFD.

Items	Steady Motion	Unsteady Motion
Numerical solution method	FVM
Turbulence model	SST $k-\omega$
Solver algorithm	SIMPLEC
Discretization scheme	Second-order upwind scheme
Boundary conditions	Custom velocity inlet, no-slip wall
Grid type	Unstructured
Number of iterations	2000	100/T 1

¹T stands for the period of the sine motion in unsteady-state motion simulations.

. For steady-state simulations, the convergence residuals are set to ${10}^{-6}$, but the number of iterations is limited to 2000, which can satisfy our convergence criterion that the drag coefficient output in last 100 iterations stay stable (difference less than $0.5\%$), and decrease the computation load at the same time.

3.4. Mesh Independence Study

During mesh generation, variations in parameters such as face sizing, body sizing, and grid transition smoothness can affect grid refinement, resulting in differences in the quantity of grids for the same model. It is crucial to evaluate simulation accuracy across various grid scales and ensure that adjustments in grid density within an appropriate range maintain the precision of simulation outcomes.

To verify the grid’s independence within a specific range and its impact on hydrodynamic simulation results for ROV model with different mesh scales, experiments are conducted under five different face sizing settings. These experiments are conducted under the condition of Y-direction straight-line motion at 1.0 m/s. Drag in Y direction under different grid scale at $Re$ = 1,482,475.247 are shown in

Table 4. Drag in Y-directional with variation in mesh scale.

Face Sizing (m)	Mesh Scale	Drag (N)
0.022	5,062,880	2108.3
0.025	4,493,577	2105.168
0.03	3,511,002	2095.363
0.035	2,919,002	2046.842
0.05	1,985,662	1632.463
0.075	1,200,385	1738.659
0.1	abnormal grid	/

.

Analysis of the drag data reveals that increasing the face sizing of the ROV surface to 0.1 m results in the inability to generate the boundary layer of the framework properly. This occurs because the size of individual grids exceeds the width of some thin rods of the ROV frame, leading to the failure of the calculation. However, upon calculating the Y-directional drag under other mesh sizes, it is observed that the relative error of damping remains within 2.8 percent within the mesh quantity range of 2.9 to 5.0 million. This indicates that the influence of mesh quantity on the results begins to converge. The trend of Y-directional damping with the change in mesh number is depicted in Figure 7.

Therefore, balancing the accuracy and calculation load, 0.025 m was chosen as the face sizing value.

3.5. Validation of the Simulation Method

To validate the CFD method employed in the simulations in this work, A standard model called SUBOFF, which designed as a standard underwater vehicle model for testing from a past research conducted by DARPA, is simulated under same process and parameter decision method with this research. The original towing test data for the SUBOFF model can be sourced from a comprehensive series of experiments detailed in the experiment report [34]. By comparing the simulation results with these experimental data under identical operating conditions, we can ascertain the accuracy of the CFD method. Further details regarding the validation procedures can be found in our earlier publication [21]. Figure 8 shows the geometry of the SUBOFF model and also the simulation flow zone. And the errors between the simulations and physical experiments are shown in

Table 5. Comparison of the numerical results with the experimental data reported in [34].

Velocity (Kn)	Experimental Results of Drag-X (N)	Numerical Results of Drag-X (N)	Errors (%)
5.92	87.4	90.6	3.63
10.00	242.2	238.7	1.45
11.84	332.9	330.5	0.72
13.92	451.5	447.1	0.97
16.00	576.9	582.9	1.04
17.99	697.0	716.4	2.82

.

To further validate the simulation method with a open-framed model, we simplified the open-source BlueROV2 model as another validation model. The simplified model and the flow zone for the CFD simulation are shown in Figure 9. By conducting towing test simulations in surge direction under the same velocity gradient comparing with a experimental towing test for BlueROV2 from Li et al. [35], the drag force results and the errors between the simulations and the physical experiments are shown in

Table 6. Comparison of the numerical results with the experimental data reported in [35].

Velocity (m/s)	Experimental Results of Drag-X (N)	Numerical Results of Drag-X (N)	Errors (%)
0.2	1.169	1.209	3.45
0.4	6.259	6.621	5.78
0.6	16.167	16.366	1.23
0.8	29.162	31.142	6.79
1.0	44.701	45.248	2.67

.

3.6. Steady and Unsteady Motion Simulation

By employing the aforementioned CFD method, the hydrodynamic coefficients of the UUV can be obtained. These coefficients can be primarily divided into three parts, corresponding to the coefficients solved for straight-line motion in the X, Y, and Z directions, respectively. Using least squares method for the fitting, hydrodynamic coefficients can be calculated from the simulation results (force and moment). Detailed calculating process are presented in our previous work [21]. As examples, Z-direction straight-line motion and pure heave motion simulations settings are as follows.

3.6.1. Z-Direction Straight-Line Motion Simulation

In this simulation set, a total of 10 velocity conditions were simulated in the positive and negative directions for the Z-directional straight-line navigation. The experimental plan is shown in

Table 7. Experiment condition of Z-direction straight-line motion.

Motion Type	Motion Direction	Experiment Condition	Number of Simulation
Z-direction straight motion	Z-positive	relative velocity 0.1–1.0 m/s	10
	Z-negative	variation interval 0.1 m/s	10

.

3.6.2. Pure Heave Motion Simulation

In this simulation set, a total of five motion frequency conditions were simulated for the pure heave motion. The experimental plan is shown in

Table 8. Experiment condition of pure heave motion.

Motion Type	Experiment Condition	Number of Simulation
Pure heave motion	relative velocity: 0.5 m/s amplitude: 0.02 m frequency: 0.2–1.0 Hz variation interval: 0.2 Hz	5

.

3.6.3. Simulation Results

The final normalized hydrodynamic coefficients of the UUV model are presented in

Table 9. Normalized hydrodynamic coefficients.

Steady	Coefficients	Normalized Value	Unsteady	Coefficients	Normalized Value
X straight	$X_{uu}$	0.2326	heave	$Z_w$	−0.4045
	$X_{u\left|u\right|}$	0.2087		$M_w$	−0.0142
	$Z_{uu}$	0.0396		$Z_w$	−0.1223
	$Z_{u\left|u\right|}$	−0.00826		$M_w$	−0.1059
	$M_{uu}$	0.0260	sway	$Y_v$	−0.1461
	$M_{u\left|u\right|}$	0.00397		$K_v$	0.018
Y straight	$Y_{vv}$	0.4131		$N_v$	0.0077
Z straight	$Z_{ww}$	−0.4740		$Y_v$	−0.4097
	$Z_{w\left|w\right|}$	−0.0791		$K_v$	0.0109
XOY drift	$X_{uv}$	0.0381		$N_v$	0.1412
	$Y_{uv}$	−0.416	yaw	$Y_r$	−0.0044
	$Z_{uv}$	−0.0144		$K_r$	0.0009
	$K_{uv}$	0.0121		$N_r$	0.008
	$M_{uv}$	−0.00928		$Y_r$	−0.1208
	$N_{uv}$	0.00625		$K_r$	0.004
XOZ drift	$X_{uw}$	−0.00427		$N_r$	0.0269
	$X_{u\left|w\right|}$	−0.00041	pitch	$X_q$	−0.0056
	$Z_{uw}$	−0.501		$Z_q$	−0.0032
	$Z_{u\left|w\right|}$	0.0811		$M_q$	−0.0084
	$M_{uw}$	0.00625		$X_q$	−0.0338
	$M_{u\left|w\right|}$	−0.00607		$Z_q$	−0.0013
Surge	$X_u$	−0.1573		$M_q$	−0.0117

. Based on these obtained coefficients, the complete dynamic model of the UUV can be described and imported to the physical simulation platform in Section 7.

4. Baseline Parameter Selection of Underwater Vehicle Geometry Design

To identify the geometry of the underwater vehicle body with lower drag, a CFD simulation on the basic body model of the ROV is conducted to analyzed the flow pattern and the pressure distribution around the model, as Figure 10 shows. It can be observed from the velocity streamline and the pressure distribution contour that the highest pressure mainly distributed on the upstream surface of the buoyancy block. Thus, when designing the geometry of the vehicle, emphasis should be placed on optimizing the front part of the vehicle. The design should ensure that after the optimization of the parameters, the flow distribution around the vehicle is even, separations generation is minimized, and the pressure on the upstream surface of the buoyancy block is reduced.

The principle of the ROV’s geometry design is to optimize the front part of the buoyancy block while ensuring the mid and rear body of the buoyancy block remains in same volume to provide adequate buoyancy, as shown in Figure 11. By studying existing design schemes from some of the work-class ROV buoyancy block design in the industry, the shape of the front part of the buoyancy block is altered to streamlined forms in to reduce the drag in surge direction. Two geometry models, concave and convex model were selected, as shown in Figure 12.

By conducting simulations on the baseline and optimized models of the buoyancy block, as shown in Figure 13, it can be observed that the streamlines above and below the buoyancy block are relatively more sensitive to the cross-section shape at their same side. In light of this, main geometry parameters were designed to decide the cross-section shape of the upstream side in surge direction of the buoyancy block, as Figure 14 shows.

In both cross-section shape conditions, parameter 1 is used to optimize the flow state on the upper side independently. For the concave shape design, two exclusive parameters are used for cross-section shape constraint, while for convex shape design, one exclusive parameter is used. Each parameter is assigned with a range to obtain optimized drag-reduction results, as

Table 10. Ranges of the geometry parameters.

Parameters	Concave Shape	Convex Shape
Parameter 1	0°–30°
Parameter 2	100–200 mm	100–332 mm
Parameter 3	150 m–250 mm	/

shows.

These parameter ranges are designed reasonably considering avoiding excessive simulation conditions and be able to cover optimal result interval.

5. Underwater Vehicle Drag-Reduction Design Simulation

The general design strategy of the buoyancy block in this research is to reduce the drag from the upstream surface by introducing chamfer and curved surface that can guide the fluid flows smoother. Here, in this section, X-direction refers to the direction of surge motion, and Z-direction to the heave direction. And the complete process of the design strategy is shown in Figure 15.

Firstly, a CFD simulation is conducted for the design parameter 1 of the buoyancy block model of the underwater vehicle to observe the optimal results of parameter 1. With the underwater vehicle’s speed around 1 knot and 0.5 m/s, the results obtained from the simulation are presented in

Table 11. Simulation results of parameter 1 gradient under 0.5 m/s.

Parm1 (°)	Drag-X (N)	Drag-Z (N)
0 (Baseline)	249.4229	0.8633
5	235.1088	1.8906
10	220.8655	5.1278
15	203.1876	10.7220
20	179.0679	31.8028
25	165.0464	22.261
26	168.0316	15.3322
27	163.5313	7.929
28	163.5312	7.9326
29	166.7049	−8.6101
30	169.7156	−16.599

.

Using mathematical software tools, the results are fitted and analyzed. Figure 16 shows the fitting plot of the drag in the X direction. To aid in visualizing the optimization outcomes, polynomials are employed for fitting within the figure. As parameter 1 is set to 27.5°, the drag in the X direction of the buoyancy block achieves the lowest value. This phenomenon occurs due to the inclination angle of the buoyancy block’s windward surface, which induces the formation of water streamlines along the windward surface. Consequently, this reduces the impact force and stabilizes the streamlines in the upper region, thereby mitigating drag in the X direction.

The pressure distribution contour, velocity streamline, and comparison of the drag-reduced models are shown in Figure 17. When adjusting parameter 1, the angle of approach, the optimization results are best when the angle is between 20 and 25°. If the angle is too large, it will produce undesired forces in the Z direction, leading to counter-optimization results for the underwater vehicle in both directions. The conclusions are consistent with the aforementioned fitting results. In summary, it is concluded that when performing convex shape or concave shape design, the optimal result for the resistance of the buoyancy block’s windward surface is achieved when parameter 1 is around 27.5°, within the given parameter range.

After the determination of parameter 1, further simulations is conducted combining the concave design parameter 2, which is the approach length, and also parameter 3, which is the approach radius. With the premise that the optimal result of 27.5° has already been used for parameter 1, the simulation results for concave design are presented in

Table 12. Simulation results combining parameters 2 and 3.

Parm1 (°)	Parm2 (mm)	Parm3 (mm)	Drag_X (N)	Drag_Z (N)
27.5	0 (Baseline)	0 (Baseline)	163.5313	7.929
27.5	100	100	176.2649	118.1247
27.5	100	150	180.1904	120.8813
27.5	100	200	187.2354	126.9848
27.5	100	250	192.7954	125.1602
27.5	150	100	171.6083	111.8356
27.5	150	150	176.0831	114.4418
27.5	150	200	182.5689	116.6661
27.5	150	250	189.7901	117.7527

.

The combined fitting result for parameter 2 and 3 is shown in Figure 18. Polynomial fitting is used for demonstrating the variation trend, and the results were unexpected. The combined optimization of parameters 2 and 3 always showed an increasing trend in the drag in the X direction within the given range, which did not achieve the purpose of drag-reduction design.

The obtained pressure contour and velocity streamline are as Figure 19 shows. It can be observed that during the concave design process, the pressure at the bottom of the buoyancy block gradually increases with the optimization parameters. Additionally, due to the high-speed movement of the fluid along the surface of the vehicle, a larger separation is formed at the rear of the vehicle. Clearly, the concave design scheme has provided results in the opposite direction. The increasing separation acts on the rear of the buoyancy block, generating an oblique downward pull, which significantly increases the drag in both X and Z directions. This does not meet the purpose of this study.

In summary, under the condition of concave design, the shape optimization does not produce a positive effect. Due to the increased windward surface area, it leads to larger drag in X and Z directions. Thus, the concave design scheme should not be adopted.

For the convex design, with the premise that the optimal result of parameter 1 is 27.5°, the simulation results are as shown in

Table 13. Simulation results of convex design.

Parm1 (°)	Parm2 (mm)	Drag_X (N)	Drag_Z (N)
27.5	0	143.7221	−129.0891
27.5	100	133.6255	−108.5722
27.5	150	128.4418	−95.4861
27.5	200	122.7489	−81.0143
27.5	250	119.3638	−71.6148
27.5	260	117.0644	−67.5918
27.5	270	117.9888	−67.8878
27.5	280	116.7907	−64.4911
27.5	290	114.5748	−58.5837
27.5	300	116.7188	−61.3501
27.5	310	115.9669	−58.5325
27.5	320	116.6530	−56.7416
27.5	332	116.7736	−53.8360

.

The convex design pressure distribution contour and velocity streamline are shown in Figure 20. In this case, the pressure points on the model decrease, leading to a balanced and stable flow velocity distribution, thereby reducing the generation of separations. This results in a significant decrease in resistance for the vehicle in both the X and Z directions. However, when parameter 2 reaches a threshold value, the separations generated at the model’s tail gradually increase, leading to a reverse optimization outcome.

Figure 21 illustrates the fitting results of convex design simulations. When parameter 1 is set to 27.5°, an increase in parameter 2 leads to a quadratic distribution of resistance in the X direction, while resistance in the Z direction shows a linear trend. The growth in the Z direction is gradual, and the Z direction resistance gradually approaches zero within the parameter range. The irregular Z direction resistance generated due to water flow impact on the irregular exterior is mitigated through shape optimization. Moreover, the Z-direction resistance has minimal impact compared to the X direction. As depicted in Figure 12, convex optimization reaches its minimum value around 290 mm. Hence, for parameter 2, 290 mm is selected as the final value.

After obtaining our final simulation results, in order to verify the impact of different speeds on shape optimization and resistance, resistance experiments are conducted at different speeds for optimization parameter 1. The purpose was to confirm the independence of speed from the optimization results. The simulation results are presented in

Table 14. Simulation results under 1 m/s.

Parm1 (°)	Parm2 (mm)	Drag_X (N)	Drag_Z (N)
5	290	940.0161	1.6734
10	290	883.9527	19.0173
15	290	813.0202	41.2795
20	290	717.9258	128.3172
25	290	658.5213	92.1224
30	290	673.7852	−55.7875

.

As Figure 22 shows, the final data were compared with the condition where the velocity is 0.5 m/s, as shown in Figure 15. It was observed that there was not a significant difference in the resistance distribution at this velocity. The overall trend remained consistent. Consequently, the conclusion was drawn that at different velocities, there are no significant factors affecting the drag-reduction of the vehicle.

6. Result of Drag-Reduction Shape Design

Firstly, comparing the concave and convex design results by analyzing the velocity streamline of the these two strategy under their optimal parameter set, as Figure 23 shows. The concave strategy increase the flow velocity below the buoyancy block, but does not give much contribution to the reduction of the separations in the lower and rear area as the convex strategy.

Following the shape optimization of the buoyancy block at the vehicle’s front end, we have obtained optimized parameters that ensure the normal operation of the underwater vehicle in its current configuration. To assess the impact of locally optimizing the buoyancy block on the overall model, the partially designed model is re-integrated into the ROV model. The resulting comparison between the final model and the original model is shown in Figure 24.

Similarly, the optimized model of the ROV is simulated in the condition of velocity equals to 0.5 m/s. The obtained pressure contour and velocity streamline are as Figure 25 shows. A comparison between before and after optimization reveals that the optimized model has great characteristic to reduce pressure on the upstream side. The number of areas with high pressure decreases, leading to a more uniform distribution. Furthermore, the flow velocity field post-optimization indicates that the presence of streamlined surfaces results in the near elimination of separations at the rear of the vehicle. Consequently, with the absence of separations behind the vehicle, resistance in both the X and Z directions decreases.

And as

Table 15. Simulation resistance values before and after drag-reduction design.

	Drag_X (N)	Drag_Z (N)
Pre	338.0139	56.4946
Post	276.0298	38.3697
Percentage	18.338%	32.083%

shows, The simulation numerical results indicate that after drag-reduction design, the resistance in the X direction has decreased by approximately 18%, while the resistance in the Z direction has decreased by approximately 32%. The localized results have a significant overall impact, leading to a notable reduction in drag on the underwater vehicle.

After analyzing the simulation results and contours before and after the optimization design, what can be found is that the major factors that influencing the drag performance of the ROV are the streamlined cross-section shape of its main structure which has big volume proportion (e.g., buoyancy blocks), due to its open-frame design. By adopting convex shape design of the buoyancy block, the pressure at the front end are greatly reduced, and the volume of the separations at the rear end are also reduced, which decrease the drag from the tail.

7. Establishment and Validation of Underwater Physical Simulation Model

Since one of the objectives of the shape optimization of underwater vehicles is to enhance their mobility, it is essential to design a reasonable motion scheme to verify the improvement in the vehicle’s mobility before and after optimization. In this paper, a physical simulation platform is constructed for underwater vehicles, which is based on ROS and built using C++ and Python languages. It integrates Gazebo and Rviz for data visualization operations. In terms of the establishment of the vehicle model, the platform provides a sufficient number of interfaces and offers two different types of underwater vehicles for the current simulation.

The simulation platform uses Gazebo for construction. Although Gazebo itself provides very few underwater-specific modules, the development of Gazebo packages can rely on a continuously improving platform. In addition, integration with ROS, developed by OSRF, has been ensured through the Gazebo/ROS package. The simulation environment is mainly divided into the following parts: ocean environment, underwater vehicle body, thruster model, and sensors. The overall operation process design of the simulation platform is shown in Figure 26. After the underwater vehicle obtains waypoint data and the current state, the heading angle error required by the vehicle at this moment is sent to the controller through a path-tracking algorithm. At this time, the controller calculates the error to obtain the total thrust required by the vehicle. The underwater vehicle thruster model is responsible for decoupling the total thrust required by the underwater vehicle to the relevant thrusters to achieve the independent control of each thruster. The modeling of the underwater vehicle body is mainly divided into three parts: physical model establishment, physical parameter setting, and hydrodynamic parameter estimation. At this time, the thruster thrust can move the vehicle through the body model of the underwater vehicle. The sensors carried by the vehicle will record the movement state at this time and capture and send the current state to the path-tracking algorithm to complete the next cycle.

Figure 27 shows the laboratory underwater vehicle reproduced in the simulation platform, which is a fully actuated vehicle composed of seven thrusters. The control in the vertical plane is driven by three motors, and the control in the horizontal plane is driven by four motors. The simulation model is constructed based on the solved kinematic and dynamic models of the underwater vehicle.

To facilitate intuitive observation of the model establishment and drag-reduction design, the primary movement objective of the vehicle in the simulation platform is to follow a predetermined trajectory within a two-dimensional plane. The simulation mainly consists of four components: the preset trajectory, the controller, the vehicle model parameters, and the effect presentation. Five coordinate points are selected within the two-dimensional plane and connected. The vehicle starts from a given position and traverses the five coordinate points, ensuring that the control parameters remain unchanged between the original model and the final model, i.e., a consistent controller and control strategy are determined. The model parameters of the vehicle before and after the drag-reduction design have been solved, as mentioned earlier. The dynamic and kinematic models of the vehicle before and after optimization are written into the simulation, and the results are shown in Figure 28.

By examining the path tracking plot for the five given way-points, it can be observed that with all other variables remaining constant, the optimized model achieves better results with the specified control scheme. This allows the underwater vehicle to complete the set objectives more easily, with more accurate tracking of the established route. The vertical error of the vehicle from the preset trajectory is minimized at each navigation position, as shown in Figure 29. Since the vehicle needs to stably track the path, which involves converging its vertical error from the trajectory onto the vehicle’s path, the comparison of the vertical error from the optimized model before and after optimization with the preset trajectory indicates that the vehicle has acquired superior mobility capabilities after optimization. When tracking the trajectory, it can ensure that the vehicle better achieves the set objectives.

8. Conclusions

This work primarily focuses on modeling and drag-reduction design of a remotely operated underwater vehicle. By establishing the dynamic model of the vehicle, a comprehensive vehicle model that can represent the hydrodynamic characteristics of the vehicle is developed. CFD simulation methods are employed to determine the hydrodynamic parameters and analyze the fluid performance of underwater vehicle. Building upon this analysis, hypotheses concerning drag-reduction design are proposed, aiming to enhance the vehicle’s shape without compromising its overall operational effectiveness. The paper presents two distinct types of design strategy, which are evaluated against baseline parameters to derive the target shape for the underwater vehicle. Results demonstrate a reduction in resistance by 18% in the X direction and 32% in the Z direction during vehicle movement. To further validate the efficacy of the design scheme, a independence verification is conducted across different velocities. Based on the analysis of the simulation results and contours before and after the drag-reduction design, it is evident that the primary factors influencing the drag performance of the UUV are related to the streamlined cross-section shape of its main structure, particularly the buoyancy blocks, due to the open-frame design. By adopting a convex shape for the buoyancy blocks, the pressure at the front end is significantly reduced, and the volume of separations at the rear end is minimized, thereby decreasing the drag from the tail. Moreover, the paper establishes an underwater physical simulation platform to verify the mobility of the vehicle both before and after the drag-reduction design. The findings confirm that the proposed design scheme effectively enhances the mobility of underwater vehicles. Due to the limitations of the simulations, physical experiments should be conducted in the following studies to validate the effectiveness of the design strategy. Also, flow influence from waves and vortexes that applied on underwater vehicles can be tested in real-world to measure the positive impact on motion stabilization control of the vehicles.