Abstract: Motion equations are derived for 2 ships maneuvering in close proximity within the potential theory under the assumption of rigid wall free-surface. By numerically solving the motion equations, the maneuvering motions of 2 ferries when one ship overtakes the other and passes away are simulated. Then, the added mass and the interaction forces are calculated by a 3D panel method as a function of the ship position in the time step for solving the motion equations. The collision caused by the hydrodynamic interactions between 2 ferries is realistically demonstrated in shallow water.

　

 When 2 ships move in close proximity, lateral force and yaw moment (interaction forces) act on the each hull due to unsymmetrical flow around the ship. In order to simulate the maneuvering motions of ships moving in close proximity, the interaction forces have to he rationally included in the simulation model.

 In existing models, the interaction forces are treated as a look-up table or a simple formula with 2 parameters of staggered position and lateral distance between 2 ships [1] [2]. However, it is presumed that the interaction forces change with not only position between 2 ships but also the heading angle difference and the ship moving condition such as the velocities and accelerations at the moment. Namely, effect of the actual maneuvering motions should he considered in the calculation of the interaction forces.

 Abkowitz et al. combined the interaction forces which were expressed by a line distribution of singularities with ordinary simulation model [3]. Landweber et al. studied the motions caused by the hydrodynamic interactions in the fluid for 2D bodies such as circular cylinders[4]. Their models are useful for better understanding of the hydrodynamic interactions, however, may be too simple to predict the ship maneuvering motions accurately.

  In this paper, first, motion equations are derived for ship maneuvering in close proximity within the potential theory under the assumption of rigid wall free-surface. By numerically solving the motion equations and calculating the interaction forces at the same time, we can simulate the maneuvering motions of 2 ships. The interaction forces are calculated by a 3D panel method as a function of the ship position in the time step for solving the motion equations. Such the simulation method will be described in this paper. Examples are given of the use of the simulation method for ships maneuvering when one ship overtakes the other and passes away. The ships behavior including a process of the collision caused by the hydrodynamic interactions is realistically demonstrated.

　

2.1 Coordinate Systems

　

 Let us consider 2 ships maneuvering in close proximity. For distinction of 2 ships, here, the ships are expressed as Ship i(i=1,2). The coordinate systems fixed in the space o xyz and fixed to Ship i o'i - x'iy'iz'i are employed. The x'i-axis is defined as direction from ship stern to the bow. y'i-axis to star- board and z-axis vertically downward. The x'i - y'i plane coincides with the still water surface. Sea bottom is assumed to be constant depth and expressed by z = h.

  Ship i is assumed to move with forward velocity u(i), lateral velocity v(i) and yaw angular velocity r(i). These velocity components are represented as a function of time t. The maneuvering motions with respect to surge. sway and yaw are denoted by X(i), Y(i) and ψ(i) respectively.

  For simplicity of the treatment, the following assumptions are employed:

　

 The neglect of free-surface effects and shed vortices removes all of the memory effects and allows the solution to be stepped through time as a series of independent hydrodynamic calculations.

　

2.2 Basic Equations

 The perturbation velocity potential due to ship motion is defined as φ(t,x,y,z). Then, φ has to fulfill the following boundary conditions;

　

　

 Eq.(1) is hull boundary condition of Ship i, eq.(2) the rigid wall free-surface condition and eq.(3) the boundary condition of sea bottom. SH⁽ⁱ⁾ means hull surface of Ship i. In eq.(1), n1⁽ⁱ⁾, n2⁽ⁱ⁾ and n3⁽ⁱ⁾ denote the outward normal vector components with respect to longitudinal, lateral and yaw directions of Ship i, respectively. The n3⁽ⁱ⁾ is defined by x'⁽ⁱ⁾n2⁽ⁱ⁾ - y'⁽ⁱ⁾n1⁽ⁱ⁾, where (x'⁽ⁱ⁾, y'⁽ⁱ⁾) is the coordinate of ship hull surface of Ship i.

 By integrating the hull pressure given from Bernoulli's equation, surge force 1⁽ⁱ⁾, lateral force 2⁽ⁱ⁾ and yaw moment 3⁽ⁱ⁾ acting on Ship i are obtained as follows:

　

　

 where ρ is water density. Replacing the term of φ/t appears in eqs.(4), (5) and (6), the following equations are obtained[5]:

　

　

　

2.3 Added Mass and Hydrodynamic Forces

 According to the form of eq.(1), φ(t,x,y,z) is expressed as:

　

　

 where U1 = u⁽¹⁾, U2 = υ⁽¹⁾,U3 = r⁽¹⁾, U4 = u⁽²⁾, U5 = v⁽²⁾ and U6 = r⁽²⁾. Then, the hull boundary conditions with respect to φi(i = 1 〜 6) are expressed as follows:

　

　

 The φi is expressed by intermediation of source strength σi as follows:

　

　

 where P = (x,y,z) is field point and Q = (ξ,η,ζ) the singular point. G(P; Q) is Green function to fulfill the conditions of eqs.(2) and (3), and expressed as infinite number of source taking mirror image with respect to z = ±2h[6]:

　

　

 where means non-dimensionalized value by water depth h and G0 is expressed as:

　

　

 Substituting eq.(13) into eqs.(11) and (12), the following equations are obtained for i = 1 〜 6: