Ships often have to pass each other in proximity in harbor areas and waterways in dense shipping-traffic environment. Hydrodynamic interaction occurs when a ship is overtaking (or being overtaken) or encountering other ships. Such an interactive effect could be magnified in confined waterways, e.g., shallow and narrow rivers. Since Yeung published his initial work on ship interaction in shallow water, progress on unsteady interaction among multiple ships has been slow, though steady, over the following decades. With some exceptions, nearly all the published studies on ship-to-ship problem neglected free-surface effects, and a rigid-wall condition has often been applied on the water surface as the boundary condition. When the speed of the ships is low, this assumption is reasonably accurate as the hydrodynamic interaction is mainly induced by near-field disturbances. However, in many maneuvering operations, the encountering or overtaking speeds are actually moderately high (Froude number Fn > 0.2, where (equation) U is ship speed, g is the gravitational acceleration, and L is the ship length), especially when the lateral separation between ships is the order of ship length. Here, the far-field effects arising from ship waves can be important. The hydrodynamic interaction model must take into account the surface-wave effects. Classical potential-flow formulation is only able to deal with the boundary value problem when there is only one speed involved in the free-surface boundary condition. For multiple ships traveling with different speeds, it is not possible to express the free surface boundary condition by a single velocity potential. Instead, a superposition method can be applied to account for the velocity field induced by each vessel with its own and unique speed. The main objective of the present article is to propose a rational superposition method to handle the unsteady free-surface boundary condition containing two or more speed terms, and validate its feasibility in predicting the hydrodynamic behavior in ship encountering. The methodology used in the present article is a three-dimensional boundary-element method based on a Rankine-type (infinite-space) source function, initially introduced by Bai and Yeung. The numerical simulations are conducted by using an in-house-developed multibody hydrodynamic interaction program "MHydro." Waves generated and forces (or moments) are calculated when ships are encountering or passing each other. Published model-test results are used to validate our calculations, and very good agreement has been observed. The numerical results show that free-surface effects need to be taken into account for Fn > 0.2.

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