Added mass appears in eq.(29) changes every moment when 2 ships approach. In this section, the added mass changes and the transient hydrodynamic forces using a 3D Rankine panel method are evaluated.

　

3.1 Added Mass Change of 2 Spheres

　

 Added mass is calculated by integrating the velocity potential on the ship hull surface as expressed in eq.(20). Then, the potential is evaluated by intermediation of the source strength which is obtained by solving eq. (16).

 First, we evaluate added mass change when 2 spheres move on the same line and approach each other. This situation is equivalent to the problem where 2 half-immersed spheres approach each other under the assumption of rigid wall free-surface. In this case, analytical solution of the added mass change has been derived by Lamb[7]:

　

　

 Here, radius of 2 spheres is denoted by a and b, and x means the distance between centers of 2 spheres.

 Fig.1 shows the comparison of the result calculated by a 3D panel method with analytical solution. Ratio of radius of 2 sphere b/a is assumed to be 0.8. The added mass is non-dimensionalized using 4pa³. The 1,024 constant panels per a half-immersed sphere were used for the numerical calculation. The calculated result captures well the tendency where the added mass considerably changes when 2 spheres are in close proximity. We see small discrepancy between the calculation and the analytical solution in the region of small x for △m'11 and △m'44. The use of the constant panel employed in the present method may cause the discrepancy since the constant panels do not accurately capture the spherical geometry. However, we judge no problems for practical purposes. The coupled terms of calculated added mass m'41 and m'14 are the same value and agree well with the analytical solutions.

　

3.2 Interaction Forces Between 2 Ships

　

 Next, we evaluate the transient hydrodynamic forces while the ships move parallel with constant speed, meet and pass away. As for the theoretical treatment of the interaction forces, the slender body theory which are based on 2D solution with respect to the hull section flow, has been presented by Tuck and Newman[9], Yeung[10], and Cohen and Beck[11]. In this paper, the fully 3D panel method which can capture the 3D hull form effect correctly is employed for the computations. The force computations using 3D panel method have been already carried out by Korsmeyer et al. [12] and Liu Zu-yuan et al. [13]. The present method is based on Hess and Smith method for non-lifting bodies as similar to Korsmeyer et al. [12].

　

　

 Here, no lateral and yawing motions are assumed in the present computations, namely U2 = 0, U3 = 0, U1 = 0, U5 = 0, U6 = 0 and U4 = 0. Then, the interaction forces Fj(j = 1, 2, 3) are expressed as:

　

　

FDj(j = 1, 2, 3) are expressed as:

　

　

　

 where

　

　

 FDj is composed of 3 terms with respect to square of velocity of Ship 1, product of 2 velocities of Ship 1 and 2 and square of velocity of Ship 2. Using these equations, the interaction forces are evaluated. The force and moment are non-dimensinalized using 1/2pLppdU1² and 1/2pLpp²dU1² respectively.

 A ferry with center skeg[8] was employed as an objective ship of the computation. The ship length Lpp breadth B and draft d are 156.0m, 22.0m and 6.0m respectively. The 2 ships are assumed to he the same hull form and the same size. Fig.2 shows the panel arrangement of 2 ships in meeting condition. Number of the hull panel is 1,152 per a ship. We assume that minimum lateral separation between the ship sides Sp, is 0.25B, and ratio of water depth and draft h/d is 1.2.

　

　

 Fig.3 shows calculations of the lateral force and yaw moment induced when the ships meet and pass away. We assume that Froude number based on ship length Fn, is 0.1 and the speed is the same for 2 ships. For comparison the result calculated by Yeung's method[10] is also plotted in the figure. It has been confirmed that Yeung s method can predict the interaction forces with sufficient accuracy for practical purposes[2]. The horizontal axis (x/L) means relative longitudinal distance of 2 midships, and x = 0 means that the ship stands side by side. F2 is the lateral force and F3 the yaw moment. Attraction lateral force near x/L = 0 and bow-in yaw moment near x/L = -0.7 and 0.2 act on the each ships. The lateral force and yaw moment calculated by 3D panel method are the same tendency and the same order of magnitude with the results calculated by Yeungs method.

 Fig.4 shows calculations of the lateral force and yaw moment induced when Ship I moves at starboard side of Ship 2 and overtakes Ship 2. . Ratio of 2 ship speeds U2/U1 is assumed to be 0.5. The result shows as follows: when bow part of overtaking ship (Ship 1) reaches to rear part of overtaken ship (Ship 2) around x/L = -0.5, bow-in moment is induced on Ship 1. Due to the bow-in moment, Ship 1 will approach to the stern of Ship 2. This is the dangerous situation where the ship collision between bow of Ship 1 and stern of Ship 2 may occur. Then, bow-out moment acts on Ship 2, so that both ships will shift to port side as the global behaviors. When Ship 1 gets ahead of Ship 2 around x/L = 0.1, the yaw moment acting on Ship 2 changes from bow-out to bow-in. However, this situation is not so serious in view of ship collision since Ship 1 advances faster than Ship 2. The lateral force and yaw moment calculated by the present method are the same tendency and the same order of magnitude with the results by Yeung's method again. The present 3D panel method can capture well the hydrodynamic characteristics of the transient interaction forces between ships.