### 1. Introduction

### 2. Mathematical Formulation

### 2.1 Coordinate System

*Oxy*and an earth-fixed coordinate system

*Ox*

_{0}

*y*

_{0}, as shown in Fig. 1. The body-fixed coordinate system advances with the ship’s forward speed

*U*and rotates with rotation speed

*r*. In addition,

*δ*,

*ψ*,

*μ*,

*β*, and

*ψ*are the rudder angle, heading angle, incident wave direction, drift angle, and incident wind direction, respectively. The ship’s position is expressed with respect to the earth-fixed coordinate system.

_{A}### 2.2 Equation of Motion

*u*and

*v*are the velocity components in the x-axis and y-axis direction, respectively.

*u̇*and

*v̇*are the surge acceleration and sway acceleration, respectively.

*ṙ*,

*I*,

_{zz}*m*, and

*x*are the angular acceleration, the moment of inertia about the

_{G}*z*axis, the ship’s mass, and the longitudinal position of the ship’s center of gravity, respectively.

*X*and

*Y*are the hydrodynamic forces, and

*N*is the moment around the z-axis. The subscripts

*H*,

*P*, and

*R*denote the hydrodynamic forces due to the hull, propeller, and rudder in calm water, respectively. The subscripts

*W*and

*A*denote the hydrodynamic forces induced by waves and wind, respectively.

### 2.3 Hull Force

*X*,

_{H}*Y*, and

_{H}*N*are modeled as functions of the non-dimensional sway velocity and non-dimensional yaw rate using Eq. (2).

_{H}*u*′ and

*v*′ are non-dimensional velocity components in the x-axis and y-axis directions, respectively.

*r*′ is the non-dimensional yaw rate, and

*X*

_{0}is the ship resistance.

*X*′

*,*

_{vv}*X*′

*,*

_{rr}*X*′

*,*

_{vr}*X*′

*,*

_{vvvv}*Y*′

*,*

_{v}*Y*′

*,*

_{r}*Y*′

*,*

_{vvv}*Y*′

*,*

_{vvr}*Y*′

*,*

_{vrr}*Y*′

*,*

_{rrr}*N*′

*,*

_{v}*N*′

*,*

_{r}*N*′

*,*

_{vvv}*N*′

*,*

_{vvr}*N*′

*, and*

_{vrr}*N*′

*are the hydrodynamic derivatives of the polynomials and are estimated based on empirical formulas for fishing vessels suggested by Yoshimura and Ma (2003).*

_{rrr}### 2.4 Rudder Force

*F*,

_{N}*a*, and

_{H}*t*are the rudder’s normal force, the rudder’s force increase factor, and the steering resistance deduction factor, respectively. The rudder’s normal force is expressed in Eq. (4).

_{R}*x*and

_{R}*x*are the longitudinal position of the rudder and the additional lateral force component, respectively.

_{H}*v*and

_{R}*u*are the lateral and longitudinal inflow velocities induced on the rudder by propeller rotation, respectively. Λ and

_{R}*A*are the rudder aspect ratio and the rudder area.

_{R}*f*

_{α}represents the rudder lift gradient coefficient, and

*U*represents the resultant inflow velocity to the rudder.

_{R}*η*denotes the ratio of the propeller diameter to the rudder span.

*∊*represents the ratio of the wake fraction at the rudder position to that of the propeller position.

*κ*represents the interaction between the propeller and rudder.

*α*denotes the effective inflow angle to the rudder.

_{R}*β*represents the effective inflow angle to the rudder, and

_{R}*γ*represents the flow straightening coefficient.

_{R}*l*′

*is an experimental constant that is used to express*

_{R}*v*accurately.

_{R}*w*represents the wave coefficient at propeller position. The interaction force coefficients

_{P}*t*,

_{R}*a*,

_{H}*l*′

*, and*

_{R}*∊*can be obtained from Yoshimura and Ma’s empirical formulas for fishing vessels.

### 2.5 Propeller Force

*X*can be estimated using Eq. (5).

_{P}*t*,

_{P}*ρ*, and

*D*are the thrust deduction factor, the water density, and the diameter of the propeller, respectively. For simplicity, the thrust deduction factor

_{P}*t*is assumed to be constant at any given propeller load.

_{P}*K*is the thrust coefficient, which can be expressed as the 3rd polynomial of the propeller advance ratio

_{T}*J*using Eq. (6).

_{P}*k*

_{0},

*k*

_{1},

*k*

_{2}, and

*k*

_{3}are coefficients representing

*K*. where

_{T}### 2.6 Wave Drift Force and Moment

*H*

_{1/3}and

*g*denote the significant wave height and the gravity acceleration, respectively.

*C̄*,

_{XW}*C̄*, and

_{YW}*C̄*denote the average value of the steady wave-induced force and moment coefficients in irregular waves estimated using Eq. (8).

_{NW}*C̄*,

_{XW}*C̄*and

_{YW}*C̄*are estimated based on the short-term prediction technique by Yasukawa et al. (2017). These coefficients are stored in files in databases and used when simulating the ship’s maneuverability in irregular waves.

_{NW}*S*

_{ζζ}(ω) and

*G*(θ) denote the wave spectrum and the wave direction distribution function, respectively. The ITTC spectrum is used as the wave spectrum

*S*

_{ζζ}(ω). The cosine-squared spreading function is used as the wave direction distribution function (

*θ*).

*C*(ω, χ),

_{XW}*C*(ω, χ), and

_{YW}*C*(ω, χ) denote the wave-induced steady force and moment coefficient in regular waves expressed as a function of the wave frequency

_{NW}*ω*and the wave direction

*χ*.

### 2.7 Wind Force and Moment

*L*is the overall length of the ship,

_{OA}*A*is the frontal projected area,

_{F}*A*is the lateral projected area, and

_{L}*C*is the distance from mid-ship to the centroid of

*A*.

_{L}*H*is the height of centroid of

_{C}*A*from the calm water level.

_{L}*C*,

_{XLI}*C*,

_{ALF}*C*, and

_{CF}*C*are coefficients of the multiple regression formulae. The hydrodynamic coefficients due to wind can be estimated using Eq. (10).

_{YLI}### 3. Numerical Method

### 3.1 Simulation Conditions

### 3.2 Hydrodynamic Coefficients

### 3.3 Hydrodynamic Force Due to Waves

### 3.4 Drifting Distance and Drifting Angle

*Drd*is the distance between lines that are perpendicular to the tangent to the turning trajectory, as shown in Fig. 4. For example,

*Drd*

_{90 − 450}represents the magnitude of the vector between two points corresponding to heading angles of 90 degrees and 450 degrees.

*Adr*is the angle between the tangential line of the turning trajectory and the ship’s approach direction in the

*x*direction, as shown in Fig. 5. For instance,

*Adr*

_{90 − 450}represents the angle between the ship’s approach direction in the

*x*direction and the tangential line of two points corresponding to heading angles of 90 degrees and 450 degrees. The relative drift angle

*Rdr*is the angle between the drifting angle and the wave propagation direction, which can be estimated using Eq. (11).

*δ*is the rudder angle, χ is the wave propagation direction, and

*Adr*is the drifting angle. The relative drifting distance

*Rdr*is suggested by Kim et al. (2019) to describe the ship’s turning trajectory in waves.