## 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  
\[V_{m}=\sum_{m=k}^{k+n}V_{m}\prod_{j=m}^{ \begin{subarray}{c}k+n\\ j=k\end{subarray} }\ \ \left[\left(z_{m}-z_{j}\right)/\left(z_{m}-z_{j}\right)\right] \tag{13}\]  
where \(V_{m}=\left(V_{\chi m},V_{\eta m},V_{zm}\right)\) 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 [PERSON] [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 \(\times\) 7 m \(\times\) 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\({}_{\text{n}}\) = 1.2, the tangential resistance coefficient is C\({}_{\text{t}}\) = 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\({}_{\text{dx}}\) = 1.05, C\({}_{\text{dy}}\) = 0, C\({}_{\text{dz}}\) = 0.1, the characteristic length of the towed body is 0.056 m, and the projected area of the towed body is 0.0036 m\({}^{2}\). 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.  
Figure 2: Construction of the coupling method. Figure 4: Comparison of the position and attitude of the towed cable with [PERSON] [24]. Figure 3: Details of [PERSON] [24]. [PERSON] [25] utilized adaptive mesh optimization techniques to investigate the maneuverability and hydrodynamics of the tethered vehicle. [PERSON] [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].