A Mathematical Model of a Ship with Wings Propelled by Waves

This paper discusses mathematical modeling of a ship equipped with energy-saving wing devices. Therewith, the ship is mathematically represented by an elongated hull with high-aspect-ratio wings mounted near its bow and stern. Equations, describing ship motions in regular oncoming waves, are written in the spirit of strip theory with account of inertial and damping influence of energy-saving wing elements with the use of linear expansion of wing-related forces with respect to heave and pitch perturbations. This approach readily yields fast numerical solutions for the propulsion of a ship with wings in waves. The latter solutions are then used as an input for calculation of thrust on wing elements on the basis of classical unsteady foil theories corrected for finite aspect ratio. To evaluate speed of the ship in the modes which allow cruising exclusively by wave power, it is hypothetically assumed that in this case, the wave-generated thrust on the wings equals total drag of the ship-plus-wings system, the latter being defined as a sum of its viscous, wave-making, induced (for wing elements) and added-wave components. Excepting the added-wave term and wings’ contributions, the total drag is calculated herein by Holtrop method whereas added-wave resistance is evaluated with Beukelman-Gerritsma formula involving kinematic parameters of heaving and pitching motions of the ship calculated both without and with account of the wings. Also discussed in the paper is a decrease of added wave resistance for a ship with wings as compared to that of ship without wings. Finally, the energy efficiency design index (EEDI) introduced by the International Maritime Organization (IMO) is discussed for representative sea conditions as a measure of ship environmental friendliness.