3. MATHEMATICAL MODEL AND METHODS OF COMPARISON
3.1 Mathematical Model Overview
The mathematical model currently in use at BMT is based on the concept of the modular manoeuvring model (see ), whereby each element of the ship - hull, propellers, rudders, superstructure etc - is represented by a self-contained and discrete module.
Each module is constructed with reference to the detailed physical phenomena of the particular process and is combined with the various interaction effects and external forces (e.g. tugs, anchors) to provide the prediction of the ship's motion. In principle this means that data for one such module can be changed without having to alter other modules or the core mathematical model, as would be necessary with the formal Taylor Series based mathematical models.
The co-ordinate axes are fixed at amidships and the equations of motion in the horizontal plane are given by;
The right hand-side of these equations represents the hydrodynamic and external forces and moments that act on the ship. The left hand-side represents the inertial terms.
The construction of the hull hydrodynamic model is based upon that of Oltmann and Sharma and its principal structure is that of the Four Quadrant model where by the hull forces and broken down into;
・Hull lifting and ideal fluid effects
・Flow separation and vortex shedding effects
This is represented below in Figure 1.
Fig.1 Hull Forces with Drift Angle
At low drift angles, the forces and moments are dominated by ideal fluid and hull lifting effects. As the drift angle increases, flow separation and vortex shedding become significant and as the drift angle increases further, then the flow over the hull is dominated by the so-called Cross-Flow Drag. The effects of separation, vortex shedding and cross-flow are non-linear and are therefore represented by the higher-order derivatives.
A full-description of the method is given in ,  and .
The Four Quadrant model in use at BMT uses the linear and non-linear derivatives derived from regression formulae, combined with Cross-Flow Drag coefficients to calculate the hull forces and moments over the full range of ship motions.
Three hydrodynamic inflow angles, β,γ, γ*, are defined relating to pure drift, pure yaw and coupled drift - yaw respectively.
This allows the hull forces and moments to be split to represent those components due to pure drift, pure yaw and coupled drift-yaw respectively.
The total hydrodynamic forces and moments acting on the hull are therefore;
and contain both the linear and non-linear effects related to each component.
The equations that determine the linear and non-linear derivatives are derived from a series of model tests (, ) and are similar in form to those derived by Clarke, . The regression formulae are constructed from the following variables;
where αS is parameter describing the stern form of the vessel. The effect of trim is also included in the regression formulae.
This method of formulating the hydrodynamic forces and moments allows accurate representation of the forces acting on the hull at large drift angles and over the full range of motions. It is well established nowadays, the formal perturbation style models are not able to accurately represent the hull forces during 'close-quarters' manoeuvres, characterized by high drift angles and the frequent use of engines, rudders and thrusters to exert control of the vessel in all four quadrants of motion.
As is briefly described here, it is the linear and non-linear derivatives that are important in the definition of the hydrodynamic forces and moments in the BMT model and hence is it important that we are able to calculate them accurately.
3.1 Determination of Hydrodynamic Derivatives
Occasionally, we have access to model test data for the specific vessel to be modelled, but this is rare and hence for the most part we must rely on our empirical formulae for predicting the hydrodynamic derivatives for use in our mathematical model.
So far, this method has proven to be most successful and we have been able to validate our mathematical model against a large amount of fun-scale data. However, we feel that there is a need to improve some of the regression formulae to directly account for more modern hull-forms (for example ships with podded propulsion) and hence we instigated this study as the inception for a longer term plan at BMT.
To help achieve this we have developed a tool that allows the user to compare the hydrodynamic derivatives from a range of different regression analyses before constructing the four quadrant hull force and moment coefficients. In this manner, the user may make a reasoned assessment of the liner and non-linear derivatives to be used, including any scatter in the results. Thus, the user may work with the dataset that best suits the particular vessel to be modelled by making informed decisions as to the quality of the data used. Figure 2 below shows a 'screen-shot of the new software.'
Fig.2 BMT Ship Modelling Software
Having assessed the range of data available for the particular ship being modelled, the user may, in theory, use derivatives from one or more different sources providing they are all applicable to the vessel. When assessing this concept, we found that it was necessary to examine the effect of using multiple sources of hydrodynamic derivative data, when compared to using a single dataset.
3.2 Single Source or Multi-Source?
Clarke and Horn , suggested that data from a single source may bias the regression equations and it may therefore be preferable to use a multi-source approach in the determination of hydrodynamic derivatives. This therefore forms the key to our present study, namely;
"Will a combination of derivatives from different sources allow for a more accurate estimate of manoeuvring performance?"
It is well publicized (see , , ) that the direct comparison of derivatives from different sources is not easy and is affected by;
・Differences in experimental procedures and conditions.
・Model sizes; scale effects, wall effects.
・Data reduction techniques, which are rarely described in detail.
・Availability of raw data.
・The mathematical models used to breakdown the component forces and hence derive the derivatives.
However, the results presented in  suggest that such an approach may be taken (and indeed was) by re-analysing the different methods and unifying them so that they may be presented with a consistent form. In doing so, the eventual simulations for the ESSO OSAKA presented in  show a generally consistent comparison between the three datasets used with some difference occurring at larger drift angles and yaw rates. However, the results were sufficiently close for the ITTC ESSO OSAKA Committee to recommend the use of the mean derivative data as the future benchmark.
In this study we have set out to determine if this approach is feasible bearing in mind the following concepts:
・The single source approach may bias the regression equations due to the methods of one particular facility.
・A multi-source approach may introduce inaccuracies due to the aforementioned problems in comparing data from different establishments.
・If data from a relatively large number of sources is used, potential inaccuracies may be negated through the superposition of overestimations and underestimations.