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plied to the ship motion problem (e.g. King, et al., 1988). Lin and Yue (1994) extended this method to large amplitude ship motions. The exact hull surface boundary condition was satisfied on the instataneous wetted surface under the incident wave profile. The local incident wave elevations were used to transform the hull geometry and the free surface into a computational domain so that the transient Green function method could be applied. Another approach was adopted by de Kat and Paulling (1989) and by Magee (1994), in which the Froude-Krylov force and restoring force were calculated based on the instataneous wetted hull surface under the incident wave profile while linear radiated and diffracted wave forces were employed.

Typically, for the deck flow computation, the classical first order schemes have a strong dissipative effect on the numerical solution and the second order schemes produce numerical results with spurous oscillations near the discontinuity. In this paper, the governing equations of nonlinear shallow water flow on deck are derived in the flux vector form and are solved numerically by flux-Difference Splitting method. The Superbee flux limiter has been employed in the algorithm and the finite difference scheme is a second-order Total Variation Diminishing scheme which gives satisfactory results without non-physical spurious oscillations and is able to capture the hydraulic bore. The Fractional Step method is used so that solutions of the shallow water equation can be obtained by solving two sets of one-dimensional differential equations. The hydrodynamic forces caused by water flow on deck is considered in the time domain equation of ship motions. The time domain added mass, hydrodynamic damping and hydrodynamic restoring force coefficients are calculated using the impulse potential (King, et al., 1988). The equations of ship motion also include the nonlinear Froude-Krylov forces, nonlinear restoring forces and nonlinear viscous roll damping. However, the linear radiated and diffracted wave forces are used. Our approach can be summarized as follows:

  • the time domain added mass, hydrodynamic restoring force and damping force coefficients are computed using the impulse potential function (linear);

  • linear diffracted wave forces and retardation functions are computed based on the frequency domain diffracted wave forces and damping coefficients, respectively (linear);

  • F-K forces and hydrostatic restoring forces are computed at the instantaneous position (nonlinear);

  • forces due to water flow on deck, viscous damping, resistance, cross-flow drag, thrust, rudder and maneuvering forces are included (nonlinear); and

  • nonlinear equations of ship motion solved in the time domain.

The present work has been applied to fishing vessels with shallow draft. Computations of water flow on deck and ship motions are compared with model test results.

Ship Motion in the Time Domain

Three coordinate systems are employed for the the ship motion analysis as shown in Fig.1.

Fig. 1 Coordinate Systems for Ship Motions

OXYZ is the space-fixed coordinate system with the OXY plane on the calm water surface and the OZ axis be positive upwards. The second coordinate system omxmymzm is a moving system which moves with the same steady forward speed as the ship in OX direction. The omxmym plane always coincides with the OXY plane, the omxm axis is in the same direction as the OX axis and the omzm axis is positive upwards. The third coordinate system osxsyszs is fixed on the ship with the osxsys plane coincident with the OXY plane when the ship is at its static equilibrium position, and the oszs axis is positive upwards.

The oscillatory ship motion are described in the omxmymzm system. The ship motions are repre-

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