# Simulation Tests of the Passing Distance of Ships on a Two-Way Fairway
## Materials and Methods
### Ship-to-Ship Interaction in Model and Simulation Studies
However, in the encounters of real vessels often of different types (shapes) and dimensions, the various phases may shift or even disappear. The number of factors necessary to consider and the complexity of the ship-to-ship interaction phenomenon for the passing case is relatively large. This dictates extreme caution in the interpretation and application (extrapolation) of existing results. Extensive systematic model studies of the ship-to-ship effect in terms of forces published in [8], among others, became the basis for our motion simulations. From the point of view of interaction forces acting on the own-ship, the transverse force and moment of the ship-ship effect were modelled in simulations as:  
Figure 1: Phases of ship-to-ship effect during passing. \[\left\{ \begin{array}{l}F_{ySS}\\ M_{zSS}\end{array} \right\}=0.5\rho LT\left(\frac{v_{O}^{2}+v_{T}^{2}}{2}\right) \cdot\left\{ \begin{array}{l}c_{fy\_SS}(\ldots)\\ c_{mz\_SS}(\ldots)\end{array} \right\} \tag{4}\]  
where:  
\(\rho\) - water density  
\(L\) - length (between perpendiculars) of the ship  
\(T\) - draft of the ship  
\(v_{o}\), \(v_{l}\) - longitudinal velocity of the own and target ship  
\(c_{fy\_SS},c_{mz\_SS}\) - dimensionless coefficients of transverse force and moment, which are functions of six arguments:  
* The mutual longitudinal position of the centers of the ships, usually represented as a dimensionless quantity \(\Delta x^{\prime}\) (in our case with reference to the length of the own ship, i.e., \(\Delta x^{\prime}\) [\(L\)]);
* The lateral distance between the centers of ships (\(y_{c\_B}\) [\(B\)] given in breadths of the own ship);
* The ratio of velocities (modules) \(\left|\,v_{o}\,\right|/\left|\,v_{l}\,\right|\);
* Type of encounter--passing or overtaking--which is represented by the sign of the ratio (negative in the case of passing);
* The relationship of the shape and size of own and target ships;
* The depth of water, represented by ratio \(h/T\) (where \(h\)--depth). The measurements of [8] for ships of similar size (length) were fitted into the above model of forces and their dimensionless coefficients--Figures 2 and 3. These were the tanker 'E' (according to the original designations, L = 287 m) as own-ship and the container ship 'D' (L = 290 m) as a target ship. The range of tested speeds in [8] for both vessels was 12, 10, 8, and 6 kn. Verification of ship-to-ship interaction was carried out for a tanker model L = 103.6 m on the desktop simulator (called SMART) of the Maritime University of Szczecin for the following conditions:  
* Side-to-side distance: \(<\)0.1\(B\), 0.25\(B\);
* \(\mathrm{v_{o}}\), \(\mathrm{v_{t}}\) = 8, 10 km and \(\mathrm{v_{o}}\) = 8, 10 km and \(\mathrm{v_{t}}\) = 0 km (stopped);
* \(h/T\) = 1.1, 1.2, 1.5. The hydrodynamic characteristics of hull, propeller and rudder of the tanker model were identified and calibrated (against full-scale trials and existing free-running model tests) in the authors' past research. Exemplary results of simulated passing at \(\mathrm{v_{t}}\) = 0 kt and \(\mathrm{v_{o}}\) = 10 km are shown in Figure 4, while Figure 5 shows the results of passing at \(\mathrm{v_{o}}\) = \(\mathrm{v_{t}}\) = 10 km. In these simulations, the initial bow-to-bow distance of own and target ships (both of equal size) was equal to 1 ship length for Figure 4, and 2 ship lengths for Figure 5. Ship contours are plotted every 5 s, Figure 4, or 15 s, Figure 5. The own-ship is without any active control in these scenarios, with her engine running ahead and rudder amidships, and the bank effect from either bank is switched off. The SMART software is a single own-ship simulator, at which the target ship in motion (moving at constant speed and heading/course) or in rest cannot have her dynamics like the own-ship. This is a certain drawback of the software, especially while testing the interaction at very close distance, e.g., corresponding to less than 0.1\(B\), mostly in both Figures 4 and 5.