The phenomenon is symmetric in z. Therefore, the source distribution (24) should involve only the Lame functions which are even in z, namely the classes K and L.

1.6 Added mass in the direction of motion

The Taylor added mass theorem[13] requires that the added mass M is given by

where the integration is performed over the area of the fundamental ellipse (4), in which x = μ ν/h3 and the differential area for λ=h2 will read

The source distribution has been already given in Eq.(24) and we note that only the even system of singularities is considered given that the problem is symmetric in z. Eq. (27) in its extended form will read

where (μ ,ν) = ( μ)ν). The integral in Eq.(29) is evaluated by orthogonality and is equal to / 2[3]. The final relation that provides the added mass is

and it is clearly seen that depends on only one coefficient. The unbounded case away from the wall(d →∞) can be recovered usingHence, the asymptotic value of M is

which can also be expressed in terms of elliptic integrals as given by Kochin et al.[14] and Lamb[15].Figures 1, 2 show some results for the attraction force and the added mass coefficients for the concerned mode of motion. It is clearly seen that the attraction force vanishes for large distances from the wall while the added-mass converges to the asymptotic value obtained by Eq. (31).

Fig. 1 (Color online) Attraction force acting on an ellipsoid 1 2 3 (a ,a ,a ) = (1,0.8,0.6) moving steadily close to a wall for increasing distance from it. The force has been normalized by 1 ρ a for unit velocity U

Fig. 2 (Color online) Added mass coefficients λ11=Mρ? for an ellipsoid ( a1 , a 2 )=(1,0.5) and several values of a3

2. Two ship encounter

Here we examine a common event that is always encountered during ship maneuvering operations,especially in ports, namely when two ships come across each other. In relevant situations, the forces exerted on ships, depending on the relative distances,the velocities and the direction of motions, may attract or repulse the ships. To analyze the phenomenon, we formulate the problem assuming significant simplifications and we approximate the ships by ellipsoidal geometries.

Fig. 3 Two ships encounter

We consider two ships maneuvering in a quiescent liquid in close proximity (Fig. 3) with two distinct steady velocities U 1 and U 2 with a relative course angle β. For simplicity we take the two ships to have the same geometry, represented by identical tri-axial ellipsoids, where the origin of the ship,expressed in an instantaneous system attached to the ship 1, is O 2( X, Y). The z coordinate in the direction of gravity (directed into the fluid) is the same for both ships. Thus, the relation between the two Cartesian coordinate systems is simply given by

For the analysis that follows, we need to interchange y and z in the equation of the ellipsoid (1)and the transformations between Cartesian and ellipsoidal coordinate systems (2), so that we will still have a 1 ＞ a 2 ＞a3. The small semi-axis now is the y-axis and all other parameters remain the same.

We assume that the draughts of the ships are larger than their beams. We also denote the induced source distribution on the fundamental “ellipse” of the ship 1 by σ 1 ( x 1 , 0,z1) and the velocity potential enforced around ship 1 by the rectilinear motion of ship 2, by σ 2 ( x1 , y1 , z1). Using further the steady Lagally theorem one can readily express the sway force and yawing moment exerted by ship 1 as

where S0 denotes the area of the fundamental ellipse(4). We further assume U 1 = U 2 =U. The leading order source distribution σ 1 (x 1 , 0,z1) as provided by Miloh[12] is

C1( K ) is literary λ) evaluated on λ=a1. The factor K denotes that the associated Lamé function is of class K. Eq. (36) enable us to express φ2 ( x1 , y1 , z1, X, Y ,β) by virtue of Eqs. (32) as

where σ 2(x 2 , 0,z 2)is again obtained by Eq. (35).Taking the derivative of the above with respect to y1 and evaluating on y1 =0 we obtain

Introducing Eq. (38) into Eqs. (33), (34), one may calculate the sought sway force and yaw moment.

Fig. 4 Normalized force exerted by ship 1 as the two ships move rectilinearly in opposite directions (β =180°) .The ships have been approximated by tri-axial ellipsoidswith geometrical particulars 1 2 3 (a ,a ,a ) = (2,1,0.2) . Negative force indicates attraction (opposite direction of 1 y shown in Fig. 3) and positive force indicates repulsion

Fig. 5 Norm alized momen t e xerted by ship 1 as the two ships moverectilinearlyinoppositedirections (β= 180°).The ships have been approximated by tri-axial ellipsoids with geometrical particulars ( a1 , a2 , a 3 )=(2,1,0.2)

Numerical examples of the attraction-repulsion phenomenon that occurs during two-ship encounter are depicted in Figs. 4 and 5. In particular, two ellipsoidal ships were considered traveling in opposite directions with steady velocities until the meet each other. The beam to draught ratio, here denoted by a3 /a2 was assumed to be sufficiently small. Also, the spacing between the ships was assumed to be sufficiently large to avoid time-dependent source distributions which must be considered within ship 1 in the case of small spacing. Clearly, when the ships approach each other a repulsion force is applied and accordingly when the bows of the ships are hypothetically attached in the (x, z) plane yielding a separation distance between centres 2 a1 (note that they were considered identical), repulsion changes to attraction the maximum of which occurs when the y-axes coincide. At that point the moment changes sign while when the y -axes coincide the moment is zero. The above discussion agrees with the remarks made in the study of Yeung[16].

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