Abstract: Most formulations of mathematical modelling of ship manoeuvres in shallow water discussed in literature are based on expressions for the deep water case. Several usual and unusual phenomena occurring during manoeuvres at limited under keel clearance (10% to 50%) are not considered. A tabular model for the hull forces is proposed, taking the shallow water condition as starting point, with the intention to cover wide ranges of kinematical parameters so that a great variety of manoeuvres can be simulated. The implementation of the mathematical model is based on captive model tests with 4 m models of the tanker Esso Osaka and a fourth generation container carrier. The experimental program consists of well-known, classical PMM test types combined with alternative tests. Preliminary guidelines are formulated for the selection of test parameters, taking account of their influence on the hydrodynamic coefficients.

 The development of a mathematical model describing ship manoeuvres in varying conditions holds on many researchers since decades. Although some of them ask themselves whether "one can have a general, standardised manoeuvring model" [1], at this moment standardisation is still far away.

　

 The mathematical models published in literature can be divided into two groups: models based on a pure analytical approach (pure regression models) and models taking account of the underlying physical phenomena (modular simulation models). The latter provides many possibilities as the manoeuvring vessel is considered as a composition of interacting parts, hull, propeller and rudder, each represented by a separate module. On the other hand, the determination of the hydrodynamic forces acting on each part and, particularly, the interaction terms can become complex in case of manoeuvres in navigation areas characterised by limitations in both width and depth, such as harbours and approach channels.

　

 Although numerical methods based on CFD (Computational Fluid Dynamics) are becoming more and more popular to be used for the determination of hydrodynamic forces, the efforts have still not made model testing unnecessary, especially in shallow water. Calculations in very shallow water (h/T<1.2) and even shallow water (h/T<1.5) are scarce and the quantitative agreement in these conditions is rather poor [2].

　

 The paper intends:

2.1 Conditions

　

 The tabular model for hull forces in shallow and unrestricted water has to fulfil the following conditions :

　

　

 The model only describes the hydrodynamic forces acting on the ship hull during a combination of surge-sway-yaw motions. Roll motion is neglected due to the low ship velocities.

　

2.2 Hull forces and moment

　

The equations of motion of a manoeuvring ship are:

　

　

with H hull, p Propeller and R rudder.

　

Hull force components are mainly caused by accelerations and velocities.

　

　

 The velocity dependent forces in (2) are expressed as tabular models, the following angles varying over four quadrants from - 180° to 180°:

　

　

 The expressions f^(β)(u,v,0),f^(γ)(u,O,r)and f^(β,γ)(O,v,r) with f = X, Y or N are, respectively, the forces or moment measured during pure sway, pure yaw or the additional forces measured during a combination of sway and yaw. β is the drift angle, γ is the yaw rate angle and the angle indicated by Arctan(rl/v) has no specific name as it is based on the ratio of yaw rate to side velocity which are already used in the expressions (3) and (4) [3].

　

 Non-dimensional expressions for the velocity dependent forces and yawing moment are formulated as follows:

　

□ pure sway

　

　

□ pure yaw:

　

　

□ combination sway-yaw:

　

　

 Acceleration dependent coefficients are made non-dimensional based on the length between perpendiculars and the draught.

　

　

3.1 Ship models and test conditions

　

 In accordance to the full scale trials reported by Crane [4], the shallow water conditions during the captive tests with the Esso Osaka model correspond to under keel clearances of 20% and 50%. No model tests were carried out in deep water. The container carrier D is tested at under keel clearance values of 20% and 7% of the ship's draught. The captive model tests were executed with a fully automated PMM-carriage at the Towing Tank for Manoeuvres in Shallow Water (co-operation Flanders Hydraulics Research - Ghent University), Antwerp (Belgium). The main characteristics of the towing tank are 88 x 7 x 0.5 m³, with a useful length of 68 m.

　

 Hull characteristics are summarised in table 1 and body plans are shown in figure 1.

　

　

　

 The draught of the container carrier is slightly higher than the maximum draught of the largest existing containerships (d = 14.5 m).