Abstract: Throughout the world high speed is demanding in the development of passenger ferries. On the south coast of China there are dozens of waterjet propelled catamarans, which make the market of transportation booming. Meanwhile, the requirements to train operators of high speed vessels by simulator are getting more and more imminent. This paper investigates the manoeuvring simulations of a catamaran with waterjet propulsion based on the modular mathematical model of ship manoeuvring to be applied to a simulator. Due to the lack of knowledges about performance characteristics involved with the pump and the nozzle, the model is simplified by substituting twin rotatable propellers for the waterjet propulsions combined with the steering and reversing gear. The hydrodynamic characteristics of the catamaran are estimated for different speeds. Environmental influences, including wind, wave, current as well as shallow water effect, are taken into consideration. The simulation results are presented and compared with sea trial data, which demonstrates that the simplified model is applicable in practical use.

　

 Since 1980's many high speed crafts have been imported and built in China, with types varying from monohull, catamaran, hydrofoil to hovercraft, and new routes have been exploited. The south coast of China owns the superiority in geographic position, and its market of passenger transport by high speed vessels has been growing rapidly and the number of high speed crafts amounts to the largest part in the domestic passenger market. Among all these high speed crafts, waterjet propelled catamaran is the most typical ferry. Due to the potential risk of high speed ferry, it is essential to train pilots on simulator to enhance their ability of situation awareness. In order to train operators of high speed catamaran by simulator, it is necessary to have a deep insight into the simulation model of the catamaran manoeuvring.

　

 However, it is challenging to develop a precise model because waterjet is a special propulsion for ship and few theoretical and experimental studies were conducted in regard to the manoeuvrability of waterjet propelled catamaran. The hydrodynamic characteristics of the twin hulls and waterjets are complicated and the manoeuvrability may depend highly on speed, whereas it is nearly impossible to extrapolate to the manoeuvring characteristics at high speed from some results performed at slow speed.

　

 Reference [1] and [15] show the large dependence of the linear hydrodynamic derivatives on Fn for catamaran manoeuvring. The resistance of catamaran is quite different due to the ship form, appendage and demihull separation [3].

　

 Waterjet usually consists of the inlet, the pump, the nozzle and the bucket. The jet can be deflected to different direction with help of the bucket, which makes the craft highly maneuverable. Although it seems quite different between waterjet and propeller, the same fundamental principle can be applied to both of them, and in performance waterjet is very similar with propeller [6][10].

　

 For these reasons, the authors have investigated the mathematical model of the catamaran manoeuvring. In this model, twin rotatable propellers are substituted for the combination of steering and reversing gear in the waterjet propulsions, the hydrodynamic characteristics of the catamaran are estimated for different speed, and environmental influences including wind, wave, current as well as shallow water effects are taken into consideration.

　

 The proposed model has been used to simulate the manoeuvring motion of a waterjet propelled catamaran. Through the comparison of the simulated motions with the sea trial data, it is confirmed that the proposed model can be applied to prediction of manoeuvring motion of the catamaran for practical purpose.

　

 The principal particulars of the prototype catamaran [2][4] are listed in Table 1. The catamaran is shown in Fig.1.

　

　

　

3.1 Ship Motion Equation

　

 Two right-handed coordinate systems, the inertial system (earth-fixed) O-X0 Y0-Z0 and the non-inertial system (body-fixed) G-xyz are adopted to describe the manoeuvring motion, as shown in Fig.2. In the body-fixed system the ship motion equation can be written in the form of Eq. (1).

　

　

 where: m, mx, my, Iz, Jzz are mass of ship, added mass in x and y directions, moment of inertia and added moment of inertia around z-axis, respectively. u and v denote surge and sway velocity, and r yaw rate. X and y are the components of external force vector F acting on ship in the reference system of G-xyz and N is the moment around z-axis.

　

 In the present case the external forces consist of basic hydrodynamic forces due to hull and waterjets, the effects of shallow water and environmental forces such as wind, wave as well as current.

　

　

3.2 Waterjet Force

　

 As mentioned above, the forces produced by the waterjet and propeller can be explained by the same fundamental principle. Both are very similar in performance. However, the special characteristics of waterjet propelled catamaran have to be highlighted in the manoeuvring model [5]. Waterjet characterizes catamaran highly manoeuvrable in harbors or congested areas in the way of crabbing, pivoting, reversing move, dynamic positioning. It also has excellent braking capability especially at high speed.

　

3.2.1 Thrust

　

 With application of the simple momentum theory, the waterjet thrust takes the following form when approaching:

　

　

 where: Ta is the thrust derived under the ideal condition, Vj and V0 denote the velocity of flow at outlet and inlet, respectively; α is a coefficient representing the effect of boundary layer on the flow; ρ is the mass density of fluid, and Q is the volume rate of flow through the waterjet.

　

 Once the catamaran has yaw and/or sway motion, the lateral velocity v and yaw rate r will lead to the change of thrust. Validated by experimental data, Reference [12] expresses the thrust variation caused by v, r, and the bucket deflection δ as follows:

　

　

where: v, r denote the non-dimensional value of v, r

　

 Correspondingly, manoeuvring forces due to the bucket deflection 6 are simply expressed by:

　

　

 where. ljet denotes the x coordinate of the thrust center.

　

 Since theoretical evaluation of Ta in Eq.(2) is quite troublesome, to find equivalent propellers may well be a solution to the question. Consequently, Ta is expressed as follows:

　

　

 where: t is thrust deduction factor, n propeller revolution, Dp~ propeller diameter, J advance ratio, KT thrust coefficient for propeller open-water characteristics.

　

3.2.2 Equivalent propellers

　

 For a given operating condition, an ideal propeller (in inviscid flow) may be regarded as a pump with a variable nozzle area, and its stream tube surrounded by an imaginary thin duct. Such a duct would constitute the equivalent of a waterjet propulsion. KaMeWa once did an experiment on the comparison between waterjet and propeller propulsion, and found that propeller is similar in performance with waterjet. It was also testified that the characteristics of a Gawn propeller as propeller and as pump are very similar [6].

　

 Nowadays there are several series of propeller for ship propulsion, for example B series, MAU and so on, but many research works indicated that not all the conventional propellers match with high speed crafts. As non-cavitation propeller, the series of Gawn is suitable for high speed vessels [13][17].

　

 In present case, according to the ship's data and engine information, a Gawn propeller is exploited as the equivalent of the waterjet. The propeller's particulars are EAR = 0.95, D = 0.8288m, H/D = 1.4; and the characteristics of the propeller are shown in Fig 3.

　

　

3.3 Hydrodynamic Characteristics of Catamaran

　

 During recent decade, thorough studies have been carried out on the resistance characteristics of high speed catamaran using CFD or EFD, while few studies on the manoeuvring characteristics have been conducted. The available result of the hydrodynamic derivatives is quite scarce.

　

 The added mass of catamaran is quite deferent from that of monohull vessel. It takes the form [7]:

　

　

 where: mlx, mly, Jlzz denote the corresponding added mass and added moment of inertia of demihull , b is the width of demihull, d is the draft, and C is the separation distance between the hulls.