## 1 Introduction  
The maneuvering characteristics are some of the most important performance indicators of marine vehicles such as submarines. The rudder-plane configuration has a significant impact on controlling a ship's journey and, hence, the hydrodynamic forces and moments under different rudder plane configurations are needed for relevant analyses. Three types of rudder-plane configurations are often equipped on submarines; namely, Cross-plane, X-plane, and Y-plane rudders. Experimental studies on submarines appended with a control surface and/or marine propeller are frequently conducted, such as those detailed in [1; 2; 3; 4; 5]. For example, research on the prediction of hydrodynamic coefficients of rudders, stabilizers, and the submarine body has been presented by [PERSON] et al. [1]. The obtained results showed that the control forces of stabilizers and rudders have a linear variation in the range of deflection angle from \(-10\) to \(+10\) degrees. Experiments on both rudders and stabilizers were carried out in a towing tank with the deflection angle ranging from \(-30\) degrees to \(+30\) degrees. In addition, using an experimental method, [PERSON] et al. [2] predicted the lateral force and yaw moment under the Cross-skeg rudder and X-rudder configurations. The axial velocities of fluid flow induced by the rudder plane configurations were investigated. The stern of the SUBOFF-G submarine with a propeller and the Cross-rudder plane configuration was automatic maneuvered to carry out the forward running and diving motions. Moreover, the hydrodynamic force and moment of the cruciform stern configuration and the X-rudder plane configuration located at stern of an AUV have been predicted by [PERSON] et al. [3], using both an experimental method and a CFD-based approach. It was shown that the hydrodynamic coefficients of the cruciform stern configuration obtained from the numerical results are in good agreement with the experimental results; however, the differences between the CFD results and experimental results under the X-rudder plane configuration were 43.8% and 53.2% for the hydrodynamic coefficients of the normal force and torque, respectively. An experiment considering a submarine model in the Low-Speed Tunnel has been conducted to measure the cross-stream and in-plane velocities [4], and the experimental results were used for comparison with the CFD-based results for a submarine appended with rudders and stabilizers. On the other hand, measurement of the velocity in the stern wake of a SUBOFF submarine at a high angle of attack has been performed [5], and the axial velocity, tangential velocity, and vorticity were measured and presented in a rectangular domain to analyze the variation of fluid flow induced by the submarine body and the Cross-rudder plane configuration. Although the experimental methods are accurate for determining the hydrodynamic forces and moments of submarines with the stern rudder plane configuration, they also have disadvantages such as the limitations imposed by the water tank, measurement devices, and noise. Computational Fluid Dynamics (CFD) methods have been widely used for the prediction of hydrodynamics acting on marine vehicles, due to their various advantages. Each type of the rudder-plane configuration had been applied to predict the maneuvering characteristics of the submerged body as shown in [4; 6; 7; 8; 9; 10]. The straight ahead motion and side-slip motion at 10 degrees of a model of the DSTO generic submarine with full appendages have been investigated using the RANS (Reynolds-averaged Navier-stokes) and LES (Large Eddy Simulation) methods [4]. The fluid flow characteristics obtained from the numerical simulation, such as axial velocity profiles, displacement of vortex core, and turbulent intensity, were compared with the experimental results. [PERSON] [6] performed numerical analysis of a full-scale submarine with Cross-rudder plane configuration based on Ansys Fluent code.