## 1 Introduction
The hydrodynamic forces and moment of the hull and asymmetric rudders and stabilizers were predicted. In addition, CFD-based simulation of the BB2 submarine has been performed by [PERSON] et al. [7], in order to predict the hydrodynamic forces and moments acting on the hull and X-rudder plane configuration. Hydrodynamic forces, moments, and added masses of the submarine hull were obtained to evaluate the maneuvering characteristics of the submarine with six degrees of freedom; the thrust and moment of the propeller were estimated in open water conditions; and the turning ability of the submarine was verified according to the results obtained from free-running model tests. The normal force and yaw moment caused by rudders and stabilizers in the X-rudder plane configuration varied linearly over the deflection angle range from \(-15\) degrees to \(+15\) degrees. Additionally, a free-running test of an underwater vehicle with six degrees of freedom based on a CFD method has been implemented by [PERSON] et al. [10]. Straight line and steady turn maneuvering of the vehicle was also performed. The results showed that the maneuvering characteristics of the vehicle were in good agreement with the experimental results. An Autonomous Underwater Vehicle (AUV) with X-rudder plane configuration has been presented by [PERSON] et al. [11]. They predicted the hydrodynamic forces and moments of the X-rudder plane, and compared them with the results under the Cross-rudder plane configuration. An X-rudder configuration for an autonomous underwater vehicle has been tested by [PERSON] et al. [12], and the hydrodynamic characteristics of the body and rudders were then introduced into the equation of motion and control of the vehicle. The maneuvering behaviors of a model of the CRN-INSEAN 2475 submarine with X-rudder or C-rudder configurations were predicted by [PERSON] et al. [13]. The trajectory parameters of turning motion for the submarine were predicted under angle of attacks of 5, 10, and 15 degrees. It was shown that the turning characteristics under the flapped X-rudder configuration were better than the turning behavior of the submarine when using conventional X-plane rudder configurations. Captive model tests of the DARPA, SWE, and SMG submarines have been carried out by [PERSON] et al. [14], in order to predict hull and rudder forces for the prediction of maneuvering characteristics. The failure of stern steering of the submarine with X-rudder plane or Cross-rudder plane configurations was also investigated. Hydrodynamic forces and moments of an underwater vehicle with Cross-rudder plane configuration were obtained through CFD-based simulation of rotating arm test, drift test, rudder test, and combination tests (between rotating arm and rudder deflection or drift-rudder) [15]. The obtained hydrodynamic coefficients were then introduced into the equations of turning motion to evaluate the turning behavior of the vehicle. The propeller thrust and resistance of the submarine obtained from the virtual self-propulsion test were compared to determine the rotational speed of the propeller [16]. Furthermore, visualization of the unsteady fluid flow surrounding the submarine with the propeller and Cross-rudder plane configuration was investigated. The turning ability of a submarine with the X-rudder plane configuration and virtual propeller has been predicted through numerical simulation of turning circles with different deflection of the rudders in Star-CMM+ code in [17]. The complex fluid flow around the DARPA SUBOFF submarine with rudder deflection has been analyzed based on OpenFOAM [18], where the hydrodynamic forces and moments of the rudder with a wide range of deflection angles were calculated and compared with experimental data. The forward running of a submarine near the free water surface has been simulated using Ansys Fluent code to predict the resistance of the hull under the Cross-rudder plane configuration, as well as the thrust and torque of the propeller [19].