### Nomenclature

O − xy: Earth-fixed coordinate

o− xbyb: Body-fixed coordinate

ψ: Heading angle

U: Ship speed

β: Drift angle

u: Surge velocity

v: Sway velocity

r: Yaw rate

u̇: Surge acceleration

v̇: Sway acceleration

ṙ: Angular acceleration

Izz: Yaw mass motion of inertia

xG: Longitudinal center of gravity

m: Ship mass

nP: Propeller revolution

t: Thrust deduction factor

xP: Longitudinal coordinate of propeller position

yP: Lateral coordinate of propeller position

wP: Wake coefficient in maneuvering motion

KT: Propeller thrust open water characteristic

JP: Propeller advanced ratio

tR: Steering resistance deduction factor

aH: Rudder increase factor

xH: Longitudinal coordinate of the rudder position

FN: Rudder normal force

UR: Resultant rudder inflow velocity

αR: Effective inflow angle to the rudder

uR: Longitudinal inflow velocity

vR: Lateral inflow velocity

fα: Rudder lift gradient coefficient

Λ: Rudder aspect ratio

ε: Ratio of a wake fraction at propeller and rudder positionsAn experimental constant for expressing uR

κ: Ratio of propeller diameter to rudder span

η: Flow straightening coefficient

γRβR: Effective inflow angle to the rudder in maneuvering motions

CP: Wake correction coefficient

βP: Geometrical inflow angle the to propeller in maneuvering motions

C: Course stability index

l′r: Yaw damping lever

l′v: Sway damping lever

### 1. Introduction

*h*/

*T*) of a ship as deep water:

*h*/

*T*> 3.0; medium-deep water: 1.5 <

*h*/

*T*< 3.0; shallow water: 1.2 <

*h*/

*T*< 1.5; and extremely shallow water:

*h*/

*T*< 1.2. In addition, the authors provided some aspects affected by shallow water as resistance, trim, checking, counter tuning ability, turning diameter, and rate of turn. Jachowski (2008), Yun et al. (2014), and Lee (2021) examined the ship squat, also known as sinkage and trim in shallow water. Delefortrie et al. (2016) conducted the captive model test based on the 6-DOF (degree of freedom) maneuvering model of KVLCC2 (KRISO Very large crude oil carrier 2) at various under keel clearances of 20%, 30%, and 80%. In the study, the ship was forced in the horizontal 3-DOF with free heave and pitch motion, while the roll was estimated from the roll decay test. In addition, some assumptions and numerical analyses were applied to assess the ship in vertical motion. Taimuri et al. (2020) studied the 6-DOF maneuvering model in deep and shallow water. It started from horizontal 3-DOF and non-linear unified seakeeping. The maneuvering time-domain using a numerical decay test was introduced as a rapid method for estimating the heave, roll, and pitch motion. Carrica et al. (2016) used CFD (Computational fluid dynamics) and an experimental study to develop a direct method for zigzag maneuvers in shallow water (

*h*/

*T*= 1.2) for KCS (KRISO container ship). A satisfactory relationship between CFD and the experimental study was observed for self-propulsion results and zigzag variables except for the yaw and yaw rate. Lee and Hong (2017) examined the course stability in shallow water for very large vessels like KVLCC2 and DTC using CFD. The study confirmed that the course stability was improved in very shallow water and was more significant in KVLCC2.

### 2. Maneuvering Simulation Model

### 2.1 Objective

### 2.2 Test Condition

*x*

_{1}and

*x*

_{2}, respectively. Furthermore, all motions were restrained throughout the experimental performance. Fig. 3 presents an experimental installation of the ship.

*β*and

*r′*dimensionless yaw rate, respectively.

*k*−

*ω*SST (Shear stress transport) turbulence model is applied extensively to predict the hydrodynamic forces and moments on a maneuvering ship because of several advantages in terms of its accuracy and time calculation (Quérard et al., 2008). The volume of fluid and open channel flow are the techniques to define the free surface and two flow phases of water and air. A SIMPLE (Semi-implicit method for pressure-linked equations) algorithm was used to solve the governing iteratively, adjusting the pressure to ensure that the resulting velocity field satisfied continuity. The least-squares cell-based method was used to evaluate the gradient of flow variables. The quantities at cell faces were calculated from the cell-centered values by the second-order upwind method. Unlike the static test, the circular motion test determines the rotating fluid zones using a multiple reference frame approach.

### 2.3 Mathematical Model

*Oxy*) and body-fixed (

*ox*) were set to determine the 3-DOF motion of the horizontal plane, as shown in Fig. 5. The earth-fixed coordinate defined the ship trajectory, orientation angle, and body-fixed (

_{b}y_{b}*ox*) defined the of motion and external force acting on the ship. The origin of the ship was located at the intersection of the midship, centerline, and draft.

_{b}y_{b}*H*,

*P*, and

*R*.

*X*,

_{vv}*Y*,

_{v}*Y*

_{v}_{|}

_{v}_{|},

*N*, and

_{v}*N*

_{v}_{|}

_{v}_{|}were estimated from the static drift test where the sway velocity was generated, the damping coefficients with respect to the yaw angular velocity

*X*,

_{rr}*Y*,

_{r}*Y*

_{r}_{|}

_{r}_{|},

*N*, and

_{r}*N*

_{r}_{|}

_{r}_{|}were obtained from the circular motion test, where the yaw angular velocity was given, and the coupling damping coefficients relative to sway velocity and yaw rate

*X*,

_{vr}*Y*,

_{vvr}*Y*,

_{vrr}*N*, and

_{vvr}*N*were taken from combined circular motion with drift, where both sway velocity and yaw angular velocity were generated. Furthermore, the added mass coefficients

_{vrr}*X*,

_{u̇}*Y*,

_{v̇}*Y*,

_{ṙ}*N*, and

_{v̇}*N*were determined from the pure surge, pure sway, and pure yaw test reported by Kim et al. (2021).

_{ṙ}*S*), so the formulae of the port side could be obtained by replacing them with subscript

*P*, and the formulae are the same except for the flow straightening coefficient (

*γ*) and wake correction coefficient (

_{R}*C*).

_{P}*γ*in the sway velocity (

_{R}*v*) formula of the rudder was different as

_{R}*β*). Considering the symmetry of twin-propeller and twin-rudder, the same value of

_{R}*β*≥ 0 in the starboard side and

_{R}*β*< 0 in the port side, and conversely. Similarly,

_{R}*C*in the surge velocity of the propeller was also considered symmetrical to the propeller position. According to the sign of the geometrical inflow angle to the propeller (

_{P}*β*), the same value of

_{P}*β*≥ 0 in the starboard side and

_{P}*β*< 0 in the port side, and vice versa.

_{P}### 3. Analysis results

### 3.1 Verification

### 3.2 CFD Simulation Results

*F′*=

*F*/(0.5

*ρU*

^{2}

*L*

^{2}) and

*M′*=

*M*/(0.5

*ρU*

^{2}

*L*

^{3}), respectively. This model was applied to the ship under deep and shallow water conditions. Fig. 7 presents the results of the static drift test at various water depths. The results were obtained by adjusting the drift angle by±18°. As the water depth became shallower, the hydrodynamic forces and moments were greater, particularly in extremely shallow water (

*h*/

*T*= 1.2). It increased two times compared to

*h*/

*T*= 1.5.

### 4. Dynamic Simulation

### 4.1 Course Stability

*l′*=

_{v}*N′*/

_{v}*Y′*), yaw damping lever (

_{v}*l′*=

_{r}*N′*/(

_{r}*Y′*−

_{r}*m′*)), and course stability index (

*C*=

*N′*/(

_{r}*Y ′*−

_{r}*m′*)−

*N′*/(

_{v}*Y′*)). The course was stable if the value of the course stability index was positive but was unstable if negative. Fig. 11 shows the results of the course stability based on the water depth. The sway damping lever decreased gradually while the yaw damping lever increased as the depth became shallower. In particular, a rapidly increasing yaw damping lever was observed in extremely shallow water (

_{v}*h*/

*T*= 1.2). The course stability index showed that it was unstable from deep water to

*h*/

*T*= 1.5 and improved in extremely shallow water. The yaw damping lever affected the course stability significantly.

### 4.2 Maneuverability Simulation

*h*/

*T*= 1.2. It was caused by the increasing hydrodynamic forces acting on the hull as the depth decreased. Therefore, the turning parameters in

*h*/

*T*= 1.2 did not satisfy the International Maritime Organization (IMO) (IMO, 2002). For the zigzag 10°/10° test, the first overshoot angle was more dominant at

*h*/

*T*= 1.5, 2.0, and smaller at

*h*/

*T*= 1.2 compared with those of deep water. By contrast, the second overshoot angle decreased gradually with increasing water depth to

*h*/

*T*= 1.2. Furthermore, the zigzag maneuver satisfied the IMO (2002) for deep and shallow waters.

*L*, as suggested by Fuji’s ellipse model based on the collision risk (Fuji and Tanaka., 1971). The rudder or propeller was then commanded to prevent the collision. In this case, the rudder of the own ship was turned 10° when it was a distance of 6

*L*from the target ship. Fig. 15 shows the collision avoidance simulation in deep and shallow water at different times. The collision avoidance ability of the ship was expressed well in both deep and shallow water because the target ship could reach a safe area when the own ship touched the trajectory of the target ship. On the other hand, the ability to avoid collisions in shallow water appeared to be superior.

### 5. Concluding Remarks

*h*/

*T*= 1.2. Furthermore, a simple collision avoidance of cross-right situation was executed to investigate the effect of shallow water on the collision avoidance ability. Shallow water looked better than deep water in the collision avoidance ability of the ship, even though the turning ability was better in deep water.