### 1. Introduction

### 2. Case study

### 2.1 Objective

### 2.2 Test conditions

*α*,

*β*,

*δ*and

*ω*are called the angle of attack(AOA), drift angle, and deflection of control surfaces, and rotational velocity, respectively. Fig. 2(a) shows the definitions of straight running(

*α*,

*β*,

*δ*=0), oblique(

*α*or

*β*≠ 0,

*δ*= 0), and the deflection of control surfaces(

*α*,

*β*= 0,

*δ*≠ 0). The circular motion(

*α*,

*β*= 0) and combined circular motion(

*α*,

*β*≠ 0) are illustrated in Fig. 2(b). Following the International towing tank conference(ITTC) recommendations on Captive Model Tests Procedure(ITTC, 2014), the computing conditions of these cases are selected as shown in Table 1.

*u*,

*v*and

*w*are the velocity components along

*x*,

*y*and

*z*-axes, respectively.

*q*and

*r*are the angular velocities of

*y*and

*z*axes.

*X*,

*Y*and

*Z*are the hydrodynamic forces along

*x*,

*y*and

*z*-axes, respectively.

*M*and

*N*are hydrodynamic moments about

*y*and

*z*-axes.

### 2.3 Numerical modeling

*y*

^{+}value of the boundary layer. The value of

*y*

^{+}can be set about 3,000-5,000 for Re = 1.0E+9(Oh and Kang, 1992), and the boundary thickness of submarine in the trial(William, 1974) is in fair agreement with the empirical relation of Granville. Using these two references, the estimated first grid height is 9.0 mm corresponding to

*y*

^{+}of 2,500. Fig. 3 shows the mesh of submarine in the fluid domain.

*υ*stands for the fluid relative velocity,

_{r}*υ*denotes the fluid absolute velocity,

*u*is the velocity of the moving coordinate system including translational velocity

_{r}*υ*and rotational velocity

_{t}*ω*.

*R*is the radius of the motion.

### 3. Results and discussions

^{-5}, the hydrodynamic forces and moments acting on the ship will be obtained. It is then nondimensionalized and fitted with polynomial functions for determining the dimensionless derivatives. The velocity-dependent derivatives can be taken from straight and oblique motions while rotary derivatives are extracted from the circular motion. The cross-coupled derivatives are attained in the combined motion simulation. These derivatives are used to establish the mathematical model of the hydrodynamic force and moment acting on the hull(

*HD*) and control surfaces(

*C*). According to Feldman(1979), the hydrodynamic forces and moments are mathematically modeled as follows: