### Nomenclature

O – xyz: Earth-fixed coordinate

o – xbybzb: Body-fixed coordinate

U: Vehicle speed (m/s)

β: Drift angle (deg)

α: Angle of attack (deg)

u: Surge (axial) velocity (m/s)

v: Sway (lateral) velocity (m/s)

w: Heave (vertical) velocity (m/s)

p: Roll rate (deg/s)

q: Pitch rate (deg/s)

r: Yaw rate (deg/s)

δr: Rudder angle (deg)

δb: Bow plane angle (deg)

δs: Stern plane angle (deg)

Xu̇: Derivative of XHD with respect to u̇

Xu: Derivative of XHD with respect to u

Xu|u|: Derivative of XHD with respect to u|u|

Xδb|δb|: Derivative of XC with respect to δb|δb|

Xδs|δs|: Derivative of XC with respect to δs|δs|

Xδr|δr|: Derivative of XC with respect to δr|δr|

Yv̇: Derivative of YHD with respect to v̇

Yṗ: Derivative of YHD with respect to ṗ

Yṙ: Derivative of YHD with respect to ṙ

Yv: Derivative of YHD with respect to v

Yp: Derivative of YHD with respect to p

Yr: Derivative of YHD with respect to r

Yδr: Derivative of YC with respect to δr

Zẇ: Derivative of ZHD with respect to ẇ

Zq̇: Derivative of ZHD with respect to q̇

Zw: Derivative of ZHD with respect to w

Zq: Derivative of ZHD with respect to q

Zδb: Derivative of ZC with respect to δb

Zδs: Derivative of ZC with respect to δs

Kv̇: Derivative of KHD with respect to v̇

Kṗ: Derivative of KHD with respect to ṗ

Kṙ: Derivative of KHD with respect to ṙ

Kv: Derivative of KHD with respect to v

Kp: Derivative of KHD with respect to p

Kr: Derivative of KHD with respect to r

Kδr: Derivative of KC with respect to δr

Mẇ: Derivative of MHD with respect to ẇ

Mq̇: Derivative of MHD with respect to q̇

Mw: Derivative of MHD with respect to w

Mq: Derivative of MHD with respect to q

Mδb: Derivative of MC with respect to δb

Mδs: Derivative of MC with respect to δs

Nv̇: Derivative of NHD with respect to v̇

Nṗ: Derivative of NHD with respect to ṗ

Nṙ: Derivative of NHD with respect to ṙ

Nv: Derivative of NHD with respect to v

Np: Derivative of NHD with respect to p

Nr: Derivative of NHD with respect to r

Nδr: Derivative of NC with respect to δr

### 1. Introduction

### 2. Coordinate System and Test Conditions

### 2.1 Coordinate System

### 2.2 Subject Submerged Body

### 2.3 Test Conditions

*k*–

*ω*model is widely used to predict hydrodynamic forces applied on maneuvering ships (Quérard et al., 2008). The

*k*–

*ω*model for hydrodynamic derivatives is advantageous in terms of its CPU computation time. Because CFD calculations under several conditions are required in this study, the

*k*–

*ω*model was selected as the turbulence model. A 3D incompressible viscous flow was assumed, and a continuity equation and RANS equations were applied as governing equations (Jeon et al., 2016). The convection term was applied with the second-order upwind method, diffusion term was discretized using a second-order central differential method, and pressure and speed were linked by the semi-implicit method for pressure-linked equations (SIMPLE).

### 3. Analysis Results and Modeling

### 3.1 Static Test

*R*

^{2}was approximately 1. In addition, linear coefficients were identified within the drift angle range of ±4°. As previously mentioned, hydrodynamic forces were calculated by adjusting the angle of attack by up to ±90°, considering a large angle of attack; the results obtained from the static angle of attack test are presented in Fig. 6. A stall occurs at approximately ±50° and ±40° of the surge and pitch, respectively, and the tendency of forces should be modeled using the same model as the static drift test. Nonlinear coefficients constituting the nonlinear model must be analyzed using simple curve approximation results because physical implication is ambiguous. In contrast, linear coefficients comprising the linear model have a distinct physical implication, such that the effectiveness of numerical analysis results can be determined by examining the correlation among linear coefficients when there are no experimental results, as in this study. Fig. 7 presents the points of damping force applications when the sway

*v*and heave

*w*velocities are generated based on the linear coefficients of stability

*Y*,

_{v}*K*,

_{v}*N*,

_{v}*Z*, and

_{w}*M*, identified via the static drift and static angle of attack tests.

_{w}*N*/

_{v}*Y*, which is the application point of the damping force in the longitudinal direction due to the sway velocity, is positioned at approximately 0.066

_{v}*L*in front of the origin. Considering that the

*N*/

_{v}*Y*of a general slender-type vehicle such as a ship is approximately 0.25

_{v}*L*, the application point of the damping force moved backward considerably, toward the rear. Because the vertical shape is almost symmetrical, the application point

*K*/

_{v}*Y*of the damping force in the height direction is positioned slightly upward with respect to the geometrical origin; however, the size is negligible. In contrast,

_{v}*M*/

_{w}*Z*, which is the application point of the damping force in the longitudinal direction due to the heave velocity, is positioned approximately 0.277

_{w}*L*toward the rear, with respect to the origin. A large bow plane exists in the head part, which implies that it is disadvantageous in terms of vertical stability, owing to the predominant head part.

*r*and pitch velocity

*q*. The top area is more than two times greater than the side area of the hull, and the vertical hydrodynamic moment

*M*is substantially greater than the horizontal hydrodynamic moment

_{HD}*Y*. In contrast, hydrodynamic forces

_{HD}*Y*and

_{HD}*Z*are the forces generated from the difference in the shapes of the head and rear parts during the rotational motions in which the asymmetry of the shapes of the head and rear parts significantly influence the horizontal hydrodynamic force

_{HD}*Y*.

_{HD}*p*while a submerged body is advanced at speed

*U*. Generally, a submerged body with a cylindrical shape frequently experiences a large rolling motion if a controller is not applied because the roll damping coefficient applied on the hull is relatively small. The subject submerged body in this study has a relatively flat hull top surface and a large elevator area, thus exhibiting a large roll damping moment. As shown in Fig. 10, which illustrates the roll damping moment

*K*for the roll rate

_{HD}*p*, the order is greater than that of the same angular motion moment

*N*. A large roll damping moment indicates that a roll is not large during circular motions in which favorable dynamic characteristics can be obtained from the motion control perspective. A simulation must be conducted to verify whether such a phenomenon actually occurs.

_{HD}*w*and

*q*can be determined.

*u*,

*v*,

*w*,

*p*,

*q*, and

*r*were estimated via the above tests. Table 5 presents the coefficient of determination of the hydrodynamic force model and corresponding models generated in the static rudder and static elevator tests. Figs. 13–14 present the results of the static rudder and static elevator tests, respectively. In general, the center of pressure of the hydrodynamic forces applied on a rudder is positioned at approximately 1/4 the point of a rudder chord. Similar to the static drift test, hydrodynamic forces and the rudder angle have a linear relationship in a region with a small rudder angle; hence, a linear coefficient can be determined to estimate the center of pressure of the rudder force. Fig. 15 presents the center of pressure on a rudder, which is estimated based on a linear coefficient when the rudder was rotated by a small angle. The center of pressure in the longitudinal direction

*N*/

_{δr}*Y*when the rudder was rotated is positioned closer to 1/4 of a rudder chord. The center of pressure in the vertical direction

_{δr}*K*/

_{δr}*Y*is slightly more predominant in the top area from the side view of the hull; however, the size of

_{δr}*K*/

_{δr}*Y*is negligible because the vertical shape of the hull is almost symmetrical. The centers of pressure when the bow plane and stern elevator are turned,

_{δr}*M*/

_{δb}*Z*and

_{δb}*M*/

_{δs}*Z*, are also approximate to the 1/4 point of a rudder chord. Fig. 14 presents the results obtained from comparing the hydrodynamic forces when the bow plane was turned and when the stern elevator was turned. The area of a bow plane is approximately twice as large as the area of a stern elevator; hence, the linear control panel coefficient

_{δs}*Z*provided in Table 6 is approximately twice as large as

_{δb}*Z*. In contrast, moment coefficients

_{δs}*M*and

_{δb}*M*are associated with the distance to the pressure center of a control panel, and

_{δs}*M*is at least two times greater than

_{δb}*M*because the pressure center of a bow plane is far from the origin. Similarly, examining the physical relationship between linear coefficients appears to be an appropriate method for verifying the CFD analysis results when no experimental results are available.

_{δs}### 3.2 Dynamic Test

*v̇*and pure heave tests for estimating the added mass force with respect to heave acceleration

*ẇ*, respectively. The added mass moment of inertia coefficients

*K*and

_{v̇}*N*for

_{v̇}*v̇*are typically negligible, as well as the added mass moment of inertia coefficient

*M*for

_{ẇ}*ẇ*; therefore, the sizes of

*Y*and

_{v̇}*Z*need to be examined. The results obtained from comparing the sizes of

_{ẇ}*Y*and

_{v̇}*Z*with the mass of the submerged body are illustrated in Fig. 18.

_{ẇ}*Y*of a general ship has proportion of its mass. Furthermore,

_{v̇}*Y*and

_{v̇}*Z*are identical in a submerged body with symmetrical horizontal and vertical planes. However, the subject submerged body in this study is applied with a relatively greater added mass force owing to its plate shape. Considering that the top surface area is at least twice as large as the side area and the top surface shape is a plate shape, rather than a streamlined shape, it is feasible for the size of

_{ẇ}*Z*to be greater than that of

_{ẇ}*Y*and its own mass.

_{v̇}*ṙ*and pitch angular acceleration

*q̇*. Similar to the difference in the sizes of

*Y*and

_{v̇}*Z*, the size of

_{ẇ}*M*is greater than that of

_{q̇}*N*because the top surface shape is more predominant than that of the side shape. In general, the added mass moment of inertia is sufficiently small to be negligible compared to the mass moment of inertia (

_{ṙ}*I*) in a submerged body with a streamlined shape when a roll angular acceleration

_{xx}*ṗ*is generated. However, the subject submerged body has a relatively plate-like shape, and the slenderness ratio (

*L*/

*B*) is relatively small. Moreover, a large hydrodynamic moment is generated by a roll angular acceleration because the areas of bow and stern elevators are large. Consequently, the yaw induced mass moment of inertia

*K*is relatively larger than that of a general slender-type submerged body, as illustrated in Fig. 21.

_{ṗ}### 3.3 Dynamics Model

##### (2)

##### (3)

*x*

_{Ti},

*y*

_{Ti}, and

*z*

_{Ti}in Eq. (6) represent the distance from the origin of the

*i*-th thruster to the

*x*-,

*y*-,

*z*-axes, respectively. The thrust specified in the specifications of each thrust manufacturer was adopted as the thrust defined by

*T*

_{1–6}.

### 4. Dynamics Simulation

### 4.1 Stability Analysis

*δ*=

_{r}*δ*=

_{b}*δ*= 0). The sizes of the perturbation state variables

_{s}*v*,

*w*,

*q*, and

*r*were not significant when the control panel was fixed and the linearization of the equations of motion in Eq. (1) was possible; hence, the linear stability coefficients in Table 6 were adopted. Horizontal and vertical stability can be evaluated based on the stability margin, under the assumption that surge velocity

*u*has no influence on sway, yaw, heave, or pitch. Horizontal and vertical stability margins can be defined by adopting Eqs. (7) and (8), respectively, at infinite speed.

*m*and

*x*in Eqs. (7)–(8) are design variables, the stability margin results in Fig. 23 can be deduced for

_{g}*x*under the assumption that neutral buoyancy is applied. As mentioned in Section 3, the rear is more critical with regard to the side shape of the submerged body, where it was assumed to be more important than the vertical plane in terms of stability. The analysis results indicated that

_{g}*G*is greater than

_{h}*G*when the center of gravity is closer to zero.

_{v}### 4.2 Maneuverability Analysis

**°**, ±10

**°**, and ±15

**°**under an initial speed of 2.57 m/s, and the obtained results are illustrated in Figs. 24 and 25. The simulation results indicate that the tactical diameter is approximately 7.42 m when the rudder angle is 15

**°**. The hull length of the subject submerged body is 2 m, and the entire length is approximately 3.7 m, including a towing poll; hence, adequate turning performance is realized owing to the small side surface area and large rudder area. Through simulations, it was verified that the roll convergence was approximately 1

**°**during turning owing to a relatively large roll damping moment which was due to the large elevator and top surface areas of the hull, as observed in the pure roll tests mentioned in Section 3. The result of a meander test, which evaluates vertical stability in the maneuverability test of the submerged body, is illustrated in Fig. 26. The meander test investigates the stability of motions in the vertical plane direction over time, after the disturbance of a certain size is applied in the vertical plane direction (Jeon et al., 2020). After generating disturbance for a specific time in the vertical plane direction by turning the bow plane and stern elevator, the test examines whether the depth is maintained over time when the elevator is turned again at an angle of 0

**°**. Unlike horizontal plane motions, the vertical plane motions have a restoring moment, which is expressed as the sum of gravitational force and buoyancy, and thus converges to a specific value without diverging to a pitch angle. When the subject submerged body is assumed to be under neutral buoyancy, the pitch angle converges to zero, and the depth is maintained after the rudder angle is maintained at 0° after 30 s, as illustrated Fig. 26. Vertical stability can be examined again via a meander simulation, based on the results presented in Fig. 23. The motions in a large angle of attack were simulated by generating thrust and thrust moment in heave and pitch directions, while increasing the thrust of auxiliary thrusters

*T*

_{3},

*T*

_{4},

*T*

_{5}, and

*T*

_{6}attached on the hull presented in Fig. 22 to 49 N, 98 N, and 147 N; the obtained results are presented in Fig. 27.