Abstract: Recently, the global warming has been one of important environmental problems. The reduction in the amount of the exhaust of carbon dioxide has to be also investigated in the field of the sea transportation. Then, the sail-assisted ship using natural energy has been concerned greatly. The novel merchant ship installed some large sails has been proposed from respect of the practical use at the early stage. This merchant sail-assisted ship cruises with a heel angle to obtain the thrust from wind and a small leeway angle to generate lateral force to cancel the component of sail force. The sailing attitude of ship has great influence on the sailing performances. We have thought to install the underwater fin in this ship in order to improve the sailing performance. The fin arrangements effect for the hydrodynamics of hull has been examined by towing tank test. We have constructed the simulation model of a sail-assisted ship equipped with the underwater fin by using results of these experiments. We have verified the sailing performance in steady cruising condition for various underwater fin arrangements in paper [1]. We have understood the effectiveness of underwater fin for sailing performance. In this paper, we have investigated the propulsive performance for actual wind conditions in North Pacific Seaways. We assume the averaged wind condition in North Pacific Seaways for one year. We have estimated the underwater fin arrangement effect for the propulsive performance in actual seaway.

　

m: Mass (kg)

Ixx, Izz: Inertia of x-axis and y-axis in body fixed coordinate system

XE, YE: The earth fixed coordinate system

β: Leeway angle (deg.)

φ: Heel angle (deg.)

ψw: Relative attack angle of the Sail (deg.)

ψB : Heading angle(deg.)

ψc: The direction of ship velocity in the earth fixed coordinate system (deg.)

U: Velocity at the position of Center of gravity

Uα: Relative Wind velocity (m/sec.)

L, B, d: Ship Length Breadth, draft (m)

Dp, n: Diameter and rotation velocity (rps) of propeller

X'S, Y'S, N'S, K'S: No-dimensional coefficients of force and moment on sail

X'US, Y'US, N'US, K'US: No-dimensional coefficients of force and moment on upper structure

w: The coefficient of wake

t: The coefficient of thrust reduction

As, AR: Area of sail and rudder (m²)

Afin: The area of the underwater fin (m²)

x'gf, zgf: The distance and height from the hydrodynamics force center of fin to the center of gravity

ρ: The density of water (N/m³)

ρα: The density of air (N/ m³)

g: Gravity acceleration (m/s²)

GM: The height of Meta-center

CL, CD: The lift and drag coefficient of the sail

CX, CY: The thrust and side force coefficient of the sail

KT, KQ: The coefficient of thrust and torque

Hac: Height from the aerodynamic center of sail to the center of gravity

d1, d2, d3, d4: The horizontal distance from the center of aero force on each 4 masts to Midship

DHP, DHP': The Delivery Horse Power and the integrated Average of the Delivery Effective Horse Power for navigation time

O, O+S, O+S+F: These suffix mean Original hull, Original hull + Sail, Original Hull + Sail + Fin

H, S, R: These subscript in the coefficient of motion equation mean Hull, Sail, Rudder

　

 The environmental problem comes to be concerned recently, the sail-assisted ship is paid attention. So the merchant ships should be equipped sailing devices practically and economically to carry many freights and passenger. The sailing performance needs to be improved in order to realize the sail-assisted merchant ship. But, as the sail area is larger to improve the sailing performance, leeway angle and heel angle are larger. We have thought that the underwater fin like sailing yachts seems to reduce leeway angle, heel angle and rudder angle. But the fin arrangements of sail- assisted ship (the novel ship type) have not been investigated until now. So we propose the 6 different patterns of fin configurations in paper [1]. We have estimated the sailing performance for steady conditions for fin arrangement. The hydrodynamics forces of fin configuration are examined by towing tank test in NMRI. We choose 3 typical patterns of fin configuration as simulation example in this study. We have constructed the simulation model of the sail-assisted ship has many components of motion than the novel ship in paper [1] . The motion of this ship becomes complex. More the hydrodynamics force has higher-order non-linear terms when this forces are expressed as polynomials. So the equations of motion become the Non-linear Simultaneous equations. The multi Newton-Raphson method is used to solve these simultaneous equations. The Equilibrium conditions of ship equipped with the fin are estimated by simulation model using the experiment results. We have applied the Delivery Horse Power (DHP) for estimation of the propulsive performance, since we don't know the characteristics of engine. We have compared the DHP of original hull with that of hull equipped with the underwater fin in order to estimate the sailing performance. We investigate the best fin arrangement for the sail-assisted ship from the viewpoint of the propulsive efficiency in actual seaways.

　

 The motion of the sail-assisted ship isn't only the horizontal movement. It is surge, sway and yaw motion. In addition, roll motion arise due to the inclination moment of sails. The coordinate systems of the sail-assisted ship show in Fig.1. The equations of motion in the equilibrium condition is written as

　

　

 if the steady motion is assumed, the equation of motion become as follows;

　

　

 The components of the external force and moment term are showed as

　

　

 The suffixes H, R, S, US mean Hull, Rudder, Sail and Upper structure

 More, the Hydrodynamics forces on hull are displayed in detail using the polynomial, if the motion is assumed to be small.

　

　

 The force and moment of rudder is predicted by Hirano's method [2]. The hydrodynamics force and moment of sail structure become the function of relative wind velocity and wind direction from the results of wind tunnel experiment. The force and moment on upper structure are predicted by Fujiwara's method and become functions of relative wind velocity and wind direction [3]. The thrust of propeller is described as

　

XP = (1 - t)KT(JP)・ρn²DP⁴　(5)

　

　

　

　

 The characteristic of propeller is estimated by experiment results. The characteristics of propeller is shown in Fig.2

　

　

 The equilibrium equation is obtained by substituting equation (4) into equation (2) expression.

　

X'0 + X'βββ² + X'βψβψ + X'ψψψ² + X'βββββ⁴ + X'δδδ² + X'S(Ψw)・(ρa / ρ) × (AS / Ld)(Ua / U)² + X'US(Ua, Ψw) = 0

　

{Y'β - Y'δγR / (1 - wR )}β + Y'ψψ + Y'ββββ³ + Y'ββψβ²ψ + Y'βψψβψ² + Y'ψψψψ³ + Y'δδ + Y'S(Ψw)・(ρα / ρ) × (AS / Ld)(Ua / U)² + Y'US(Ua, Ψw) = 0

　

{N'β - N'δγR / (1 - wR )}β + N'ψψ + N'ββββ³ + N'ββψβ²ψ + N'βψψβψ² + N'ψψψψ³ + N'δδ + N'S(Ψw)・(ρa / ρ) × (AS / Ld)(Ua / U)² + N'US(Ua, Ψw) = 0

　

{K'β - K'δγR / (1 - wR )}β + K'ψψ + K'ββββ³ + K'ββψβ²ψ + K'βψψβψ² + K'ψψψψ³ + K'δδ + K'S(Ψw)・(ρa / ρ) × (AS / Ld)(Ua / U)² - gCB(B/d)GM・sinψ / U² + KUS(Ua, Ψw) = 0　(6)

　

 These simultaneous equations are non-linear terms. These equations are difficult to be solved by analysis method. So these linear equations due to linear expansion of (5) are solved by the multi Newton-Raphson method. In this study, we apply the DHP as the evaluation index of propulsive performance. The DHP is used the rotation velocity when the thrust of propeller balances the resistance of ship. The DHP is defined by