### Nomenclature

aH: Ratio of additional lateral force induced on ship hull by rudder action to the rudder force

CT: Coefficient of total resistance

DP: Propeller diameter

FCG: The roll and yaw arm of the fin

Fn: Froude number

FxR, FyR: Surge and sway forces acting on rudders

Ix: Moment of inertia of the ship about x-axis

Iz: Moment of inertia of the ship about z-axis

JP: Advance ratio

Jx: Added moment of inertia of ship with respect to x-axis

Jz: Added moment of inertia of ship with respect to z-axis

KH: Hull hydrodynamic moment in x direction at midship

Kfin: Hydrodynamic moment due to side fin acting on ship about x direction

Kv̇, Kṙ, Nv̇: Added moment of inertia

KR: Hydrodynamic moment acting about x-axis on ship due to twin rudders

KP: Hydrodynamic moment acting about x-axis on ship due to twin propellers

KT: Thrust coefficient

LPP: Ship length

m: Mass of the ship

NH: Hull hydrodynamic moment in z direction at midship

NR: Hydrodynamic moment due to twin rudders acting on ship About z direction

NH: Hull hydrodynamic moment in z direction at midship

NR: Hydrodynamic moment due to twin rudders acting on ship About z direction

NP: Hydrodynamic moment due to twin propellers acting on ship about z direction

Nfin: Hydrodynamic moment due to side fin acting on ship about z direction

nP: Propeller revolutions

p: Roll rate of ship about x-axis

R0: Resistance of ship in longitudinal direction

r: Yaw rate of ship about z-axis

T: Ship draft

tP: Thrust deduction factor

tR, xH: Rudder–hull interaction coefficients

u: Surge velocity of ship in x direction

v: Sway velocity of ship in y direction

vR: Sway inflow velocity twin rudders

wP: Propeller wake fraction

Xu̇ Yv̇: Added mass

XH: Hull hydrodynamic force in x direction at midship

XP: Hydrodynamic force due to twin propellers acting on ship in x direction

XR: Hydrodynamic force due to twin rudders acting on ship in x direction

Xfin: Hydrodynamic force due to side fin acting on ship in x direction

YH: Hull hydrodynamic force in y direction at midship

YR: Hydrodynamic force due to twin rudders acting on ship in y direction

yP: Offset distance of rudder stock from the ship center line

zH: Vertical distance between the acting point of sway hydrodynamic force on hull and the origin of the body-fixed frame

zR: Vertical distance between the acting point of lift force on rudder and the origin of the body-fixed frame

β: Ship drift angle

βR: Geometrical drift angle induced at the rudder position due to ship motions

γR: Rudder flow-straightening coefficients for drift angle

δ: Rudder angle

δR: Effective rudder angle where the rudder normal force becomes zero

∊: Ratio of effective wake fraction in way of propeller and rudder

κ: An experimental constant for expressing

ρ: Water density

ϕ: Roll angle of ship

### 1. Introduction

### 2. 4-DOF Maneuvering Motion Equation

### 2.1 Coordinate System

*O*

_{0}−

*X*

_{0}

*Y*

_{0}

*Z*

_{0}represents the Earth’s fixed coordinate system, and the horizontal plane is defined as an

*X*

_{0}−

*Y*

_{0}plane. The hull-fixed coordinate system is represented by

*O*−

*XYZ*,

*X*-axis is defined as the bow direction,

*Y*-axis is defined as the starboard direction, and positive direction of both

*Z*

_{0},

*Z*coordinate systems is defined as the direction of the bottom of the hull.

### 2.2 Mathematical Model of the Maneuvering Motion

*m*,

*m*,

_{x}*m*represent the ship’s mass and additional mass in the

_{y}*x*and

*y*directions, respectively.

*I*,

_{x}*I*and

_{z}*J*,

_{x}*J*are moments of inertia and additional inertia moments of the

_{z}*x*and

*z*axes, respectively.

*u*,

*v*,

*p*,

*r*are the velocities in each direction, and the dot on the velocity denotes acceleration component in each direction.

*X*,

*Y*,

*K*,

*N*on the right side represent the forces and moments in each direction, and the subscripts

*H*,

*R*,

*P*and

*F*represent the hull, rudder, propeller, and fins, respectively. In addition, the relationship between the angular velocity between the Earth-fixed and hull fixed coordinate systems is expressed as Eq. (2).

##### (3)

##### (5)

*D*and

_{fin}*L*expressed on the right side of Eq. (8) is described in Section 3.

_{fin}### 3. Numerical Analysis of the Side Fin Lift and Drag

### 3.1 Target Ship

### 3.2 Numerical Analysis Conditions

*k*− ϵ model was used for the stable and efficient numerical calculation of the turbulence model. As presented in Table 4, the boundary conditions of the numerical calculation are as follows: velocity inlet condition was designated to the inlet, top, bottom, and side boundaries, and a pressure outlet condition was designated to the outlet boundary, and infinite depth was configured. In addition, damping conditions were set at the inlet, outlet, and side boundaries to minimize disturbance due to the reflected waves.

### 3.3 Verification of the Numerical Analysis

*Fn =*0.25, according to the method suggested by the ITTC recommended procedures and guidelines, while the refinement ratio was set to

### 3.4 Simulation of the Side Fin Lift and Drag

### 4. 35° Turn Simulation in Consideration of the Side Fin Attack Angle

### 4.1 Conditions on the Turning Simulation

*δ*≤25°. In addition, the direction of the side fin is defined in two ways. The direction in which the side fin inclines the hull in the port side is defined as a positive direction, and the direction in which the hull is inclined to the starboard side is defined as a negative direction. Table 8 below presents the angle of attack and the direction of the generated moment of the fin in each direction based on the conditions.

_{fin}