Abstract: The hydrodynamic surge and sway forces and yaw moment on a moored ship induced by a another ship moving near and parallel to it are estimated. The results are compared with the values obtained by experiments carried out in the Towing tank for manoeuvres in shallow water (co-operation Flanders Hydraulics - Ghent University) in Antwerp. Subsequently, equations of motion of the moored ship are solved to determine the loads on the mooring ropes. The effect of underwater form of the ships on the forces is also discussed.

　

 A moored ship is subject to many external influences from the environment. The jetty and mooring system must be designed to keep the resultant displacement of the ship within the limits of operational requirements and to keep the mooring line loads at an acceptable level. The forces on the moored ship have impacts on cargo handling operations, mooring and fender systems and safety of people on board. In marine terminals and harbours, ships operate in close proximity resulting in imparting of forces and moments to both moored and moving ships. This hydrodynamic interaction effect is less on the moving ship and, moreover, this ship is under the helms control where a disturbance can be effectively counteracted.

　

 The hydrodynamic forces of prime considerations coming on the moored ship due to the proximity of a moving ship are the surge, sway and yaw components. Even though the sway force is much larger than the surge force, the effect of surge very often causes exceedingly high loading in mooring lines, due to much lower damping in surge mode. Hence it requires more attention in the design of mooring systems to keep the loads within permissible limits. The design and operation estimates of the mooring and fender systems, preferably in the early design stages, help to assess the operational limitations and also can lead to an optimal system design. A reliable mathematical model representing the motion of the moored body will certainly enable the designer to try for various mooring systems.

　

 The nature of the scenario in which the interaction takes place is very varied. The severity of consequences of the interaction depends on

　

* Speed of the passing ship

* Size and underwater form of vessels

* Separation distance between vessels

* Water depth of the region

* Mooring arrangement

* Material and type of construction of mooring ropes

　

 Increase in size and speed of new generation cargo and passenger vessels and the growth of traffic density envisage the importance of a clear understanding of hydrodynamic interaction effects, where a reliable tool to predict the consequent forces on ships and mooring/fender systems become essential. The transient sway force and yaw moment experienced by a ship when proceeding in the proximity of other ships in motion have been extensively studied by Varyani et al [4] [5]. Further extension of the work to moored ship case and its eventuality on the mooring ropes have been ascertained in the work carried out by Varyani and Krishnankutty [6]. Experimental studies of Remery [1] and Vantorre et al. [7] [8] reveals the effect of the size of the passing vessels and the separation distances on the interaction forces and moments on a moored vessel for different water depths.

　

 There are very few semi-empirical approaches, resulting in the estimation of the time histories of the forces and moments in the horizontal plane due to interaction with another ship as a function of geometry, speeds and environmental parameters. Comprehensive tests with ship models of both equal and different length in overtaking and encountering manoeuvre are described in [7] and [8]. Wang [14] estimated the hydrodynamic forces and moment on a moored ship resulting from the passing of another ship at various separation distances, water depths and passing ship sizes, using slender body theory. The hydrodynamic interaction problem between moored and passing ships was studied by Krishnankutty [10] [11] using the slender body theory with singularity distribution technique for the computation of forces in surge and sway modes and yaw moment acting on moored vessel. With an aim to consider the cross-coupling effects between different motions of interest, Dand [2] [3] used a numerical technique to solve the coupled equations of motion in surge, sway and yaw modes. The behaviour of the ship was deduced from a solution of the appropriate equations of motion with the exciting forces and moment for those motions deduced from measurements made on rigidly moored models.

 The approach stage of the initial transient is most critical for the purpose of navigational safety. Yeung [12] and Yeung and Tan [13] explained that this is because the ship is subject to an attraction force towards a fixed object (moored ship) and at the same time is subjected to a "bow-in" turning moment. In these circumstances an unconstrained ship with zero rudder angle would tend to head into the fixed object. The speed and separation distance limits of a passing ship can be regulated by knowing the movement limitations of the berthed ship, be due to the restriction imposed for cargo handling facilities. For berthed tankers, the cargo handling manifolds allow a maximum movement of only ±3.0m in surge and 3m in sway, OCIMF [9], and for berthed container ships the movements are to be more restricted due to the limited reach of the cranes.

　

 Studies have revealed that underwater form of the ships has substantial influence on the hydrodynamic interaction forces, predominantly in sway and yaw components [6]. In the present paper the loads acting on a moored container ship due to the passage of a bulk carrier at different lateral positions are computed based on a theoretical formulation, which uses slender body assumptions and employs singularity distribution technique. Idealised distribution of ship sectional area for both the moored and passing ships are generally used as a user-handy method, but the studies here also consider the actual ship form in place. Subsequently, the solution of the equations of motion give a good idea about the moored ship motion due to the hydrodynamic interaction induced by the passage of another ship in its proximity and the consequent mooring line forces. Comparison of the present numerical values with measured and estimated values shows that the present method is reliable for preliminary engineering estimate of the loads on a moored vessel induced by a passing ship and the loads on the mooring ropes.

　

 The hydrodynamic interaction problem between a moored ship and the passing ship are formulated here using a slender body theory with the following assumptions.

 The co-ordinate (xm, ym, zm) is fixed in the moored ship and (xp, p, zp) is fixed in the passing ship (Fig.1). Parameters with suffices m and p, respectively, refer to those related to the moored and passing ships.

　

　

 In addition to the governing (Laplace) equation applied to the fluid domain, the following boundary conditions are in order

　

　

 The velocity potential due to the passing ship with reference to a fixed co-ordinate system (x,y,z), Wang(1975), is

　

　

 where Sp and U are the mid-ship section area and speed of the passing ship, ξ and η are the stagger and lateral separation distances between the two ships.