Influence of Vehicle Wake on the Control of Towed Systems

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(1) This paper innovatively establishes a model of mechanical energy exchange, describes the characteristics of energy exchange between the cable and fluid dynamics, and divides the four regions of cable motion. (2) In the manipulation state, the dynamic energy exchange between the cable and the wake results in the transient vibration response of the cable.

Abstract

The hydrodynamic wake generated by the underwater vehicle’s motion has a considerable impact on the movement of the towed system underwater. This paper utilizes the lumped mass method to model the towed cable in order to improve the accuracy of predicting its position and attitude in the wake, and to determine the suitable cable-towed position. Velocity is transferred from the flow field to the cable dynamic model in an innovative way to imitate the motion of the cable. Several iterations are conducted to enhance the dynamic reactivity of the cable system. Numerical simulations are used to model both the straight towed and turning movements. The numerical calculation provides the characteristics of vorticity in the flow field formed by the energy exchange between the vorticity and the cable, as well as the gradually dissipating vorticity and momentum exchange characteristics at the trailing edge of the enclosure. The results indicate that the best location for the cable towed is where its motion does not cause any adhesion. On the other hand, the disadvantageous scenario for cable-towed systems occurs when the cable’s movement causes substantial adhesion. This paper innovatively establishes a model of mechanical energy exchange, describes the characteristics of energy exchange between the cable and fluid dynamics, and divides the four regions of cable motion. In the manipulation state, the dynamic energy exchange between the cable and the wake results in the transient vibration response of the cable. The fluid structure coupling method can accurately determine the separation region of the towed point of the vehicle based on its compatibility (non-adhesive) and incompatibility (adhesive). The boundary of the region is defined to distinguish a free tow point from a wall-attached tow point. A transition zone has the possibility to change the pattern from a free tow to a wall-attached tow. The wake can be divided into free tow region, transition zone, and adjacent wall tow region by this fluid structure interaction assessment method.

1. Introduction

In recent years, computational fluid dynamics (CFD) technology has rapidly become the main method for studying hydrodynamic wakes. The hydrodynamic wake generated during underwater navigation is a non-eliminable physical field information that can cause unpredictable effects on the motion of towed cables, making it crucial to find an appropriate towed position for underwater vehicles. Chase [1] performed self-propulsion model calculations on a SUBOFF model equipped with an E1619 propeller, employing the delayed separation eddy current simulation method of “CFD Ship-Iowa V4.5”. Funeno [2] analyzed the unsteady performance of a large skewed propeller using the viscous CFD method; Ablow and Schechter [3] pioneered the introduction of the finite difference method in cable dynamic analysis, obtaining the corresponding underwater attitude of the towed array; Yuan [4] conducted research on the ship–cable–body coupling model of ocean towed systems using the finite difference method; Guo [5] modeled and solved the attitude of towed cables influenced by eddy currents, achieving relatively accurate simulation results; Gonzalez [6] conducted research on simulating the deployment and posture of a towed system using the centralized mass method; Delmer [7] posited that the finite element method is only suitable for solving steady-state or quasi-steady-state problems, finding it challenging to apply to dynamic equations with nonlinear dynamic properties; Zhao [8] developed a fully coupled three-dimensional dynamic model for a towed cable system, employing the finite difference method to approximate the towed cable with axial elasticity, bending stiffness, and torsional stiffness; Sun [9] utilized the ANCF method to forecast the dynamic behavior of TCBS; Du [10] evaluated the fluid force acting on the cable section induced by the wake of the submarine’s propeller within the towed cable system, while Rega [11] provided an overview of the latest advancements in the study of elastic cable dynamics; Norve Eidsvik and Schjølberg [12] introduced a beam-theory-inspired umbilical cable model grounded in the finite element method; Ji [13] introduced a novel CFD simulation approach for modeling the dynamic behavior of quadcopter vehicles. Liu [14] focused on the underwater towed cable system, developing a lumped mass model for the cable and deriving its dynamic equations. Employing the fourth-order Runge–Kutta numerical integration algorithm, they conducted a nonlinear dynamic response analysis on the underwater cables; Zhou [15] presented a Hamiltonian global node position finite element method to investigate the dynamic response of submarine cables during bending deformations and prolonged large-scale global movements. Li Guo [16] considered the effect of marine towed cable on the towing cable position due to vortex shedding and vessel motion during towing movement. Cheng [17] considered under the condition of a fixed cable length, the position and attitude of the towed cable are sensitive to the current rate, liquid density, cable diameter, normal drag coefficient, and specific gravity. Yang [18] investigated real-time three-dimensional dynamic responses of underwater towed systems in ship propeller wakes. Wu [19] proposed a new hydrodynamic model to investigate the dynamic characteristics of an underwater towed system consisting of an unmanned surface vehicle (USV), a towing cable, and a towed vehicle under different operation modes. Yang [20] analyzed the dynamic characteristics of an entirely underwater towed system functioning under the undulating motion mode of the underwater towed vehicle. Jiang [21] established a dynamic model of an anchor chain based on the centralized mass method. The towing experiment of a submarine cable with an attached mass conducted by Suzuki [22] and Htun [23] further verifies the accuracy of NHMSS. Two video cameras were used to record the dynamic process of the submarine cable.

However, there are still some deficiencies in the dynamic response analysis of underwater towed systems. Most of the research on the control of vehicles concentrates on control algorithms, but there is virtually no research on the dynamic response, especially the three-dimensional hydrodynamic characteristics of towed systems in complex flow fields, especially wake and waves. The way in which energy is transferred from the cable to the fluid, the transfer rate, the smoothness of the transfer, and the mutual transfer are all key issues. Due to the effects of momentum exchange and wake non-uniformity, the pulling cable undergoes alternating changes of relaxation and tension, and the mechanisms behind the effects of cable vibration and cable shape instability need to be studied. Therefore, considering the interactions between the towed cable and the fluid, a new method is needed to address the three-dimensional dynamic response of underwater towed systems.

Building on the extensive work of previous researchers, this study proposes a new coupled three-dimensional hydrodynamic model for the dynamic analysis of underwater drag systems in complex flows. The hydrodynamic characteristics of the towed system are numerically simulated using the CFD method, discretizing the towed cable into a series of elastic segments using the lumped mass method. Velocity is transferred from the flow field to the cable dynamic model in an innovative way to imitate the motion of the cable. Several iterations are conducted to enhance the dynamic reactivity of the cable system.

The phenomenon of mutual influence between the cable and the wake around the spacecraft was discovered, and the energy method was constructed. The region division criterion was derived, and the region boundary was defined to distinguish between the free towed zone and the attached towed zone. The wake of an underwater vehicle is divided into a free towed zone, a transition zone, and an attached towed zone. The coupling mechanism of the unsteady non-uniformity of the wake caused by the manipulation of the underwater vehicle on cable vibration instability is studied.

2. Towed Cable Kinematic Model

The flow field generated by vehicles significantly influences the dynamic behavior of towed cables. Consequently, a motion model for the towed cable that considers the interference flow field generated by vehicles should be established. Then, an examination of the dynamic impact of vehicles on towed cables is conducted.

To establish a three-dimensional mathematical model of the coupled fluid mechanics acting on a towed cable in the complex flow field of a vehicle, appropriate simplifications and assumptions were made for modeling the towed cable and reduced computational complexity:

(1)

The forces acting on the towed cable are gravity, buoyancy, tension, and the hydrodynamic forces on the towed cable;

(2)

The towed cable is cylindrical and flexible, neglecting bending and torsional stiffness;

(3)

The cable is divided into a limited number of elastic segments connected by nodes.

2.1. Coordinate System

As shown in Figure 1, the lumped mass method is employed to discretize the towed cable into a series of elastic, frictionless springs interconnected by nodes.

To mathematically establish the motion model for the towed cable, three reference coordinate systems are defined: the ground inertial coordinate system ($E-XYZ$), the cable local coordinate system ($o-x_1{y}_1{z}_1$) ($m=1,\cdot \cdot \cdot ,N$), and the vehicle motion coordinate system ($S-xyz$). The spring is defined between Nm and Nm⁻¹. Damp between adjacent nodes is considered as a Rayleigh damp model.

From Figure 1, we define the position vector of the vehicle in $S-xyz$ as $P_S^E={[\begin{array}{ccc}{X}_S& Y_S& Z_S\end{array}]}^{\mathrm{T}}$. Additionally, to determine the posture of the vehicle in $S-xyz$, we refine the Euler angle into pitch angle $\theta$, yaw angle $\psi$, and roll angle $\phi$. These Euler angles represent the relationship between $S-xyz$ and $E-XYZ$:

$$\left[\begin{array}{ccc}x& y& z\end{array}\right]=\left[\begin{array}{ccc}X& Y& Z\end{array}\right]R_E^S(\theta ,\psi ,\phi )$$

(1)

where

$$R_E^S(\theta ,\psi ,\phi )=\left[\begin{array}{ccc}\cos \psi \cos \theta & \sin \psi \cos \phi +\cos \psi \sin \theta \sin \phi & -\sin \psi \sin \phi +\cos \psi \sin \theta \cos \phi \\ \sin \psi \cos \theta & \cos \psi \cos \phi +\sin \psi \sin \theta \sin \phi & -\cos \psi \sin \phi +\sin \psi \sin \theta \cos \phi \\ -\sin \theta & \cos \theta \sin \phi & \cos \theta \cos \phi \end{array}\right]$$

The towed cable is connected to the vehicle; therefore, the position of the link point between vehicle and the cable is defined as $P_L^S={[\begin{array}{ccc}{x}_L& y_L& z_L\end{array}]}^{\mathrm{T}}$. The position of the link point in the ground inertial coordinate system ($E-XYZ$) is represented as $P_L^E={(\begin{array}{ccc}{X}_L& Y_L& Z_L\end{array})}^{\mathrm{T}}$. The relationship between $P_L^S$ and $P_L^E$ can be determined by Equation (1) and $P_S^E$, which can be expressed as

$$P_L^E=P_S^E+R_S^E(\theta ,\psi ,\phi )P_L^S$$

(2)

The lumped mass method simulates the attitude and position of the towed cable approximately. The spatial location of cable profile is determined by a discrete system specification with consideration of buoyancy, gravity, hydrodynamics, and inertia force. The resolution of force is introduced in Section 2.3.

The position of the node m in the ground inertial coordinate system can be expressed as $P_m^E={(\begin{array}{ccc}{X}_m& Y_m& Z_m\end{array})}^{\mathrm{T}}$. The attitude and position of the elastic cable segment can be determined using Euler angles $\left({\xi }_m,{\eta }_m\right)$.

$${\xi }_m=\arcsin \left(Z_m-Z_{m-1}/l_m\right)$$

(3)

$${\eta }_m=\left\{\begin{array}{c} \arcsin \left(\frac{Y_m-Y_{m-1}}{l_m\cos {\xi }_m}\right) X_m-X_{m-1}\le 0 \\ \pi -\arcsin \left(\frac{Y_m-Y_{m-1}}{l_m\cos {\xi }_m}\right) X_m-X_{m-1}>0 \end{array} \right.$$

(4)

where $l_m$ is the length from node m to node m⁻¹, expressed as

$$l_m=\sqrt{{\left(X_m-X_{m-1}\right)}^2+{\left(Y_m-Y_{m-1}\right)}^2+{\left(Z_m-Z_{m-1}\right)}^2}$$

(5)

The relationship between $o-x_1{y}_1{z}_1$ and $E-XYZ$ can be expressed as

$$\left[\begin{array}{ccc}{x}_1& y_1& z_1\end{array}\right]=\left[\begin{array}{ccc}X& Y& Z\end{array}\right]R_E^m(\xi ,\eta )$$

(6)

where $R_E^m=\left(\begin{array}{ccc}\cos {\eta }_m\cos {\xi }_m& \sin {\eta }_m& \cos {\eta }_m\sin {\xi }_m\\ -\sin {\eta }_m\cos {\xi }_m& \cos {\eta }_m& -\sin {\eta }_m\sin {\xi }_m\\ -\sin {\xi }_m& 0& \cos {\xi }_m\end{array}\right)$.

2.2. Kinematic Model

The kinematic model describes the relationship between velocity and angular velocity in different reference frames. Therefore, to derive the kinematic model of a vehicle, we use $V_S={\left[\begin{array}{ccc}u& v& w\end{array}\right]}^{\mathrm{T}}$ and $W_S={\left[\begin{array}{ccc}p& q& r\end{array}\right]}^{\mathrm{T}}$ to represent the instantaneous velocity and angular velocity of a vehicle relative to the coordinate system. The velocity and angular velocity of a vehicle in the coordinate system can be represented as $V_S^E={\left[\begin{array}{ccc}{\dot{X}}_S& {\dot{Y}}_S& {\dot{Z}}_S\end{array}\right]}^{\mathrm{T}}$ and $W_S^E={\left[\begin{array}{ccc}\dot{\theta }& \dot{\psi }& \dot{\phi }\end{array}\right]}^{\mathrm{T}}$, respectively. The transformation relationship between the translational velocity and angular velocity vectors between coordinate systems $E-XYZ$ and $S-xyz$ can be written as follows

$$\left\{\begin{array}{c}{V}_S^E=R_E^S{V}_S\\ W_S^E=T_E^S{W}_S\end{array}\right.$$

(7)

where $T_E^S=\left(\begin{array}{ccc}0& \sin \phi & \cos \phi \\ \cos \phi /\tan \theta & 0& -\sin \phi /\tan \theta \\ 1& -\tan \theta \cos \phi & \tan \theta \sin \phi \end{array}\right)$.

Owing to the neglect of the bending and torsional stiffness of the towed cable, we posit that the node possesses only instantaneous translational velocity. The velocity of the node m relative to the cable’s local coordinate system can be denoted as $V_m={\left[\begin{array}{ccc}{\dot{x}}_m& {\dot{y}}_m& {\dot{z}}_m\end{array}\right]}^{\mathrm{T}}$. The velocity of node m relative to the coordinate system $E-XYZ$ can be denoted as $V_m^E={\left[\begin{array}{ccc}{\dot{X}}_m& {\dot{Y}}_m& {\dot{Z}}_m\end{array}\right]}^{\mathrm{T}}$. $\left({\dot{x}}_m,{\dot{y}}_m,{\dot{z}}_m\right)$ is the first-order derivative of the local coordinate system position $(x_m,y_m,z_m)$ of the cable with respect to time. Adding one point to the position coordinate mentioned in the article represents the first-order derivative of the position coordinate with respect to time, and adding two points represents the second-order derivative of the position coordinate with respect to time.

Hence, the relationship between the velocity of the coordinate system $E-XYZ$ and the cable’s local coordinate system can be expressed as

$$V_m^E=R_E^m{V}_m$$

(8)

By integrating the corresponding velocity and angular velocity, the position $\left(X_S,Y_S,Z_S\right)$ of vehicle and the position $\left(X_m,Y_m,Z_m\right)$ of the Euler angle $\left(\theta ,\psi ,\phi \right)$ relative to the ground inertial coordinate system can be obtained.

2.3. Force Analysis

To establish a three-dimensional mathematical model of the motion of a towed cable affected by the complex flow field of a vehicle, appropriate simplifications and assumptions were made for the modeling of the towed cable to reduce computational complexity. The force analysis mainly calculates the external forces on the towed cable and converts the forces on the towed cable into force analysis based on the lumped mass method for each node. Each node is subjected to at least five forces: gravity, buoyancy, tension, additional mass force, and fluid resistance. The force balance equation is obtained as follows.

$F_m$ is the external force acting on node m, including tension $\Delta T_m$, additional mass force $A_m$, buoyancy $B_m$, gravity $G_m$, and fluid resistance $D_m$.

$$F_m=G_m+B_m+\Delta T_m+A_m+D_m$$

(9)

The dynamic model describes the relationship between the generated force and acceleration. We have described the resultant force of the node in the ground inertial coordinate system above. Therefore, applying Newton’s second law yields the dynamic model for the node m, the external force acting on node m:

$$M_m\cdot {\dot{V}}_m^E=F_m=G_m+B_m+\Delta T_m+A_m+D_m\left(m=1,\cdot \cdot \cdot ,N\right)$$

(10)

where $M_m$ represents the mass matrix of the node m, expressed as $M_m=m_{m}I$. $I$ denotes a 3 × 3 identity matrix. ${\dot{V}}_m^E$ denotes the acceleration of the node m in the ground inertial coordinate system, which can be denoted as ${\dot{V}}_m^E={\left[\begin{array}{ccc}{\ddot{X}}_m& {\ddot{Y}}_m& {\ddot{Z}}_m\end{array}\right]}^{\mathrm{T}}$.

Given that the added mass force includes an acceleration term, we shift the added mass force to the left side of the equation, yielding the final differential equation for each node as follows

$$\left(M_m+M_{a,m}\right)\cdot {\dot{V}}_m^E=G_m+{{\rm B}}_m+T_m+D_m\left(m=1,\cdot \cdot \cdot ,N\right)$$

(11)

where $M_{a,m}$ represents the added mass matrix of the node m.

The initial point of the towed cable is attached to a vehicle; hence, it travels at the same speed as the vehicle:

$$\left(\begin{array}{c}u(0,t)\\ v(0,t)\\ w(0,t)\end{array}\right)=T_E^S{V}_m^E$$

(12)

The computational platform for modeling is a Fluent UDF development environment, as well as LPM and IMLS. The IMSL C Numerical Library (CNL) is a solver for solving quadratic systems of equations.

3. Construction and Validation of Coupled Methods

The vehicle’s flow field is solved to calculate the velocity generated on cable elements. This velocity is then interpolated to the respective cable element nodes using the Lagrange interpolation formula

$$V_m={\displaystyle {\sum }_{m=k}^{k+n}{V}_m}{\prod }_{j\ne m}^{\begin{array}{l}k+n\\ j=k\end{array}}\left[\left(z_m-z_j\right)/\left(z_m-z_j\right)\right]$$

(13)

where $V_m=\left(V_{xm},V_{ym},V_{zm}\right)$ is the interpolated value of water velocity at the towed cable node m. Figure 2 illustrates the construction of the coupling method.

The numerical method in this article is validated by comparing the numerical and experimental results. The comparative experimental results are the towed system experiment provided by Guan [24]. The underwater towing system experiment is carried out in the towing tank of Dalian University of Technology. The size of the towing tank is 170 m × 7 m × 3.5 m. Heavy object 1 is a cuboid with length of 7.5 cm, width of 4.8 cm, and height of 5.6 cm. The underwater photographic equipment and rulers are used for position measurements. The rulers are attached to the trailer and the high pole with photographic equipment, respectively. The horizontal position of the heavy object is measured by observing the ruler on the trailer. The depth position of the heavy object is measured by the ruler on the high pole with the underwater photographic equipment. The total cable length of the towed system is 3 m, the cable diameter of the towed cable is D = 0.01 m, the weight per unit length of the towed cable is 0.082 kg/m, the weight of the towed cable in water is 9.8 N, the towed cable is a nylon material cable, the initial set normal resistance coefficient is C_n = 1.2, the tangential resistance coefficient is C_t = 0.025, the towed cable is a flexible cable made of nylon material, the axial stiffness is 110 kN, the weight of the towed body is 1 kg, the underwater navigation resistance coefficient is C_dx = 1.05, C_dy = 0, C_dz = 0.1, the characteristic length of the towed body is 0.056 m, and the projected area of the towed body is 0.0036 m². The specific parameters are shown in Figure 3.

The calculation results of the position and attitude of the cable are shown in Figure 4, where the numerical simulation and experimental results at speeds of 0.3 m/s and 1.5 m/s are in good agreement. In summary, it can be determined that the numerical simulation method solved in this article ensures the accuracy of the calculation.

The error is as follows: the error at 0.3 m/s is 2%, and the error at 1.5 m/s is 3.06%. The error varies with the flow around the ellipse -> cylinder, and the prediction of the shear layer separation caused by the alternating flow around the cylinder due to the error is different from the surface flow separation position of the cable caused by the turbulence of the water flow in the experiment.

The cable motion involved in this article is AUV self-propulsion based on the SUBOFF model. In this case, the cable is a lightweight positively buoyant cable, with its head end attached to the vehicle’s hull (acting as a link point), and its free end connected to a lightweight buoy. The pertinent parameters are presented in

Table 1. Parameters of the towed system.

Physical Parameters of Towed Cable		Physical Parameters of Towed Body
Length	0.5 m/6 m	Mass	0.0001 kg
Diameter	0.01 m	Volume	0.00015 m3
Weight of per unit length	0.0628 kg/m	Drag coefficient	Cdx = 1.0, Cdy = 1.0, Cdz = 1.0
Bending/Torsional stiffness	0 kN·m2	Towed speeds	4 m/s
Axial stiffness	110 kN	Fluid density	1024 kg/m3
Poisson ratio	0.28	Kinematic viscosity	1.05 × 10−3 Pa·s
Cn/Ct	1.8/0.025

.

Xu [25] utilized adaptive mesh optimization techniques to investigate the maneuverability and hydrodynamics of the tethered vehicle. Cao [26] employed a multi-mesh block strategy to discretize the SUBOFF calculation domain of the bare hull, aiming to assess the capability of the viscous flow solver in predicting forces and moments around the hull and in the flow field. The grid resolution primarily encompasses the top-tier mesh resolution and the maximum resolution of grid subdivision [27]. Park [28] adopted the adaptive mesh method with local region refinement to model the flow field around a vehicle, with flow field boundary conditions and grid division depicted in Figure 5. The grid division employs block grid technology to generate high-quality adaptive grids for the external flow field, with grid refinement occurring in the region of spatial motion trajectory where the towed cable is located. The minimum grid resolution is set at 1/10 of the cable diameter, with spatial distribution range of the grid updated with time steps.

Referring to the calculation and verification of the flow field around SUBOFF [29,30], the velocity profile of the wake field, pressure coefficient distribution, and velocity vector distribution map behind the enclosure were compared with experimental data. The total number of 3D meshes used was 780,320. The region where SUBOFF’s tail wake emanates is densely discretized. The y+ is judiciously assigned values, resulting in a fine grid node division at the main hull bow, around the hull, and in the self-propelled wake drag cable path area, where local grid refinement is employed to satisfy computational accuracy requirements.

4. Mechanical Energy Transport Model of the Towed Cable

The impact of alternating changes in tension and relaxation of the cable due to the exchange of kinetic energy and non-uniformity of wake on the cable pulling area are examined. Based on the characteristics of energy exchange, four regions of cable pulling motion are divided. The energy transport equation is as follows:

$$\frac{\partial Q_0}{\partial t}+\frac{\partial Q_v}{\partial t}=\frac{\partial Q_c}{\partial t}+\frac{\partial Q_{fL}}{\partial t}+\frac{\partial Q_{fT}}{\partial t}$$

(14)

$$\frac{\partial Q_0}{\partial t}=q_0, \frac{\partial Q_c}{\partial t}=q_c(t), \frac{\partial Q_{fT}}{\partial t}=0, \frac{\partial Q_{fL}}{\partial t}=q_0-q_c(t)$$

(15)

$Q_0$ is the input energy from the towing point dependent on the velocity of the submarine. $Q_v$ is virtual disk input kinetic energy, $Q_c$ is the kinetic energy of the cable, $Q_{fL}$ is the kinetic energy of undisturbed laminar flow, $Q_{fT}$ is the kinetic energy of submarine wake. $q_0$ is a constant energy rate in a certain towing velocity, $q_c(t)$ is cable acquisition of kinetic energy from the towing point and virtual disk.

Cable pulling point kinetic energy is ½ mv²; during cable movement, a portion of energy is offset by fluid resistance. In various areas around the hull, there are the following forms of energy transport: how quickly the cable receives energy from the towing point and transfers it to the fluid. During energy exchange, the cable smoothly transfers the kinetic energy obtained inside the cable to the surrounding fluid through a very smooth channel. The fluid can quickly absorb the energy of the cable, causing it to fall down and align with the direction of the water flow. The unsteady absorption rate causes the cable to slowly sink and then slightly rise.

4.1. Difference between Towed with and without Propeller Suction Effect during Direct Towed

Initially, numerical simulation was conducted on the propeller at a speed of 4 m/s. The virtual disk model is based on the principle of representing propellers as actuator disks. The actuator disk is useful when providing information about the behavior of the propeller and requiring its influence on the surrounding flow field. The operation of the actuator disk in the flow field enters the momentum equation in the form of source terms distributed on the virtual disk.

Throughout the calculation process, it was assumed that the propeller rotation speed was n = 10 rps and an analysis of flow characteristics around the vessel was performed. As a result of suction effects, the propeller generated a jet region and wake influence region. However, it is important to note that the towed cable is positioned outside the core area of the jet formed by the propeller. Consequently, at this specific location, water velocity remains similar to that in an area without a propeller. The kinetic energy output from the virtual disk is mainly concentrated in the jet core area, where the kinetic energy is significantly greater than that without the virtual disk. However, due to the external formation of the jet core area by the cable virtual disk, the difference in kinetic energy obtained by the cable from the fluid is small. Therefore, it can be concluded that at this tow point, both cable position and attitude are minimally affected by the presence of the propeller as shown in Figure 6. Furthermore, in situations involving straight-line navigation, cables appear as nearly linear structures within an X–Z ground coordinate system. This observation suggests that when a vessel navigates with a constant speed along a straight path, the underwater cable approximates linearity.

4.2. Dynamic Response of Wake and Cable under Direct Navigation

The setup of the tow point is shown in Figure 7a, and the specific coordinates of the traction point position are shown in

Table 2. Parameters of the tow point.

Tow Point	Position	Tow Point	Position
1	(0.3, 0, 0.2)	7	(−0.5, 0, 0)
2	(0.2, 0, 0.2)	8	(−1, 0, 0)
3	(0.1, 0, 0.2)	9	(−1.5, 0, 0)
4	(0, 0, 0.2)	10	(−1.9, 0, 0)
5	(0, 0, 0.1)	11	(1.96, 0, 0)
6	(0, 0, 0)	12	(−2, 0, 0)

. The cable’s orientation along the vehicle body changes depending on the position of the tow cables, as shown in Figure 7a. When the tow cable is at the top of the coaming shell (positions 1–3), the angle between the cable and the vehicle body ranges from 0.3° to 0.37°. When the tow cable is at positions 4–6, the angle between the cable and the vehicle body ranges from 24° to 41°. The angle between the cable and the vehicle body is kept at 41° from positions 6 to 10 on the vehicle body surface. However, after position 10, this angle starts to decrease noticeably. The cable length continues to be lengthened, and the attaching motion does not occur at position 6–10, which is shown in Figure 7b.

4.3. Dynamic Response of Wake and Cable with Angle of Attack

The design speed V = 4 m/s was determined to evaluate the impact of hull attack angle on the form of the towed cable. Flow conditions behind the hull were examined at attack angles of 3°, 5°, and 8°. Figure 8 displays the positions and orientations of the towed system in relation to the wake at three distinct attack angles. When the submarine travels through a viscous fluid, objects like the casing and tail fins impede the flow, causing a reduction in flow velocity. Varying the attack angle has a significant impact on the final cable shapes, resulting in noticeable variances. Small variations in attack angles result in modest discrepancies in cable shapes. Moreover, placing cables at an 8° attack angle over various places, as depicted in Figure 9, resulted in changes that were similar to those observed under straight-line navigation conditions.

As shown in Figure 10, based on the position and attitude of the cable and the flow field distribution with direct navigation and angle of attack, the tow cable points of the vehicle are divided into regions according to suitable (unattached) and unsuitable (attached) tow cables. The tow cable area boundary is defined by the spatial distribution of wake to distinguish the free and attached tow cables. The boundary is positively correlated with the sailing angle of the vehicle. The transition zone has the possibility to change the pattern from a free tow to a wall-attached tow. According to the position and attitude of the cable, (1) (4) is defined as the attached tow region, (3) as the transition zone, and (2) as the free tow region.

During the cable movement, a portion of the energy is counteracted by fluid resistance, while a portion of the energy is transferred to fluid resistance through dragging the cable. The fluid dissipates energy in the form of vortices.

The fluid in regions (1) and (4) can quickly absorb the energy of the cable, causing it to fall down and align with the direction of the water flow. This region has high kinetic energy and low turbulence intensity. During energy exchange in region (2), the cable smoothly transfers the kinetic energy obtained inside the cable to the surrounding fluid through a very smooth channel. The part of the cable’s own kinetic energy allocated to water flow resistance is minimal, and the cable can be lifted up. The unstable absorption rate in region (3) causes the cable to sink and then slightly increase, resulting in an unstable energy absorption rate.

Figure 11 illustrates that the vorticity above the hull drops from 15 s⁻¹ to 35 s⁻¹ during direct navigation, with a gradual decline as the underwater flow field travels. The vorticity often shows an initial quick decrease followed by a stabilization. The vorticity dissipation rate is 80%, while the vorticity at the rear edge of the shell at an 8° angle of attack differs by a maximum of 1.18 s⁻¹ from the corresponding position in direct navigation, leading to a slight discrepancy in the cable angle at the rear position compared to direct navigation. The closer the cable is to the tail of the vehicle body, the stronger the adsorption owing to the action of the tail rudder coupled to the vortex hooves and flow separation.

4.4. Dynamic Response between Wake and Cable under Rotating Motion

To accurately capture the impact of the towed cable’s attitude and position on the complex flow of lateral separation in rotational motion, it is assumed that the vehicle and the fluid are rotating at the same angular velocity, and there is no relative motion between the fluid and the wall, only subject to the effect of the static pressure field generated by the fluid due to rotational motion. The specific parameters of the cable, the time step used in the calculation, and the basic assumptions are the same as those for straight navigation. By operating the rudder at angles of 2°, 5°, 8°, and 12°, respectively, we observed the relationship between the attitude and position of the cable and the wake.

As shown in Figure 12b, as the angle of drift increases, the lateral separation flow around the vehicle body gradually becomes stronger, and the wake motion generated by separation becomes more intense. Unlike the straight state, the spatial motion curvature of the cable under rotation motion is large and affected by the angle of drift, the wake shifts towards the y-axis, and the cable immersed in the wake also shifts in the y-direction. However, under different angles of drift, the wake changes slightly in the longitudinal direction, so the wake has a small difference in the longitudinal displacement of the cable under different angles of drift. However, it can be clearly observed that the larger the angle of drift, the greater the adsorption effect of the wake on the cable. As shown in Figure 12a, the larger the angle of drift, the closer the cable is to the wake.

Furthermore, under an 8° drift angle, multiple positions were selected for cable placement, as shown in Figure 13a,b. It was found that the attitude and position of the cable at each position was basically consistent with the trend of the cable under direct navigation, but the attitude and position of the cable under the effect of the drift angle had significant differences in both the longitudinal and transverse directions from direct navigation. In the longitudinal direction, it was closer to the vehicle body than direct navigation, and the ideal position of the cable angle with the vehicle body was 10°, which was approximately 75% less than the direct navigation case.

The lateral deviation angle of the tail end of the cable in the y-direction is greater than 0.4° but less than 0.9°, while this lateral deviation can be ignored in the case of direct navigation and angle of attack. As shown in Figure 13c, under the influence of an 8° drift angle, the vorticity at the trailing edge of the enclosure differs by up to 18 s⁻¹ from the corresponding position vorticity during direct navigation, with a relatively large difference.

5. Conclusions

The boundaries of a region were defined to distinguish a free tow point from a wall-attached tow point using a coupled system simulation method, which accurately captures the variations in position and attitude of the drag system cable resulting from its immersion in the wake. A transition zone has the possibility to change the pattern from a free tow to a wall-attached tow. Furthermore, the influence of wake on the towed cable is related to the maneuvering effect of submarines, resulting in a submergence effect and exhibiting significant differences compared to straight-line navigation.

(1) The interaction between the wake in the upper area of the hull and the cable is a strong coupled effect, with the wake causing the cable to completely adhere to the surface of the hull.

(2) During direct sailing, when the towed cable is not in the central area affected by the propeller’s operation (the strong jet flow area caused by the propeller), the suction effect of the propeller can be ignored.

(3) The low-speed wake area on the back of the hull and the hull absorbs energy, allowing the cable to adhere to the surface of the hull. During direct navigation and angle of attack movements, the attachment area absorbs energy, resulting in an angle of 1/105 between the cable shape and the direction of navigation in the free motion area. Under rotational motion, due to the more significant separation motion, the energy obtained in the free motion zone increases compared to direct navigation, and the angle between the cable shape in the attachment zone and the navigation direction is 1/25 of that in the free motion zone. The gap between the two has significantly narrowed.

(4) Under the action of the drift angle, the lateral separation flow around the vehicle gradually becomes stronger. Unlike the straight state, the spatial curvature of the cable motion under rotation is large, and due to the influence of the drift angle, the attitude and position of the cable is biased toward the y-axis, with the attitude and position of the cable being approximately 30° closer to the vessel body in the longitudinal direction compared to the straight-ahead cable.

(5) During the maneuvering motion, the distance between the attitude and position of the cable and the navigating body varies in a consistent trend at different towed positions. Based on the relationship between the wake and the cable, the distribution of the towed area for the submarine maneuvering action and straight motion can be divided into free tow region, transition zone, and adjacent wall tow region by this FSI assessment method.