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Figure 17. Longitudinal cuts for container ship, full draught The first iteration of RAPID (a Neumann-Kelvin approximation) was quite similar to the DAWSON results, which proved that the underprediction of diverging waves is caused by neglecting nonlinear effects and not by numerical dispersion or insufficient resolution in the DAWSON results. The same has been found for other cases. It appears that the short diverging waves have a stronger tendency to steepen than the long transverse waves, and thus display stronger nonlinear effects. For the same vessel in ballast condition the top of the bulbous bow is at 1.2 m above the still waterline (which is 15 % of the stagnation height). An important practical question is then, whether it still becomes completely immersed at the service speed; if not, a significant resistance penalty can be incurred. This cannot be studied using a linearized method. Inherent to the linearization is the transfer of the boundary condition towards the undisturbed free surface. The greater part of the bulbous bow is above that surface and plays no role in the calculation. Therefore the predicted bow wave form is unrealistic. The same is true if the top of the bulbous bow is extremely close to the undisturbed waterplane [16]. In RAPID however, the free surface boundary conditions are applied at the actual wave surface. If the bow becomes completely submerged due to the bow wave elevation, its effect is fully included in the mathematical model. The only precaution needed is to start with an increased draught to make the bow fully submerged, and to gradually reduce this draught in the course of the iteration until the equilibrium position is reached. RAPID here predicted that the bulbous bow would just emerge above the wave surface, by some 0.20 m. But as the free surface panel size was fairly large compared to the dimensions of the bulbous bow there was a risk of insufficient resolution of the large gradients that can occur above such bows. In fact, the towing test showed that an extremely thin sheet of water just wetted the upper side of the bulb. Immediately downstream a deep wave trough next to the hull was formed, and breaking phenomena affected the wave pattern. The less accurate prediction (Fig. 18) therefore comes as no surprise, although the major aspects of the wave pattern are captured. The calculations for this container ship thus show that also for fairly slender vessels at moderate speed, nonlinear effects can be surprisingly large in some respects, and for diverging waves in particular. The nonlinear code gives a much better agreement with the experimental data, and additionally it permits to make calculations for surface-piercing bows; but resolution of the rather violent flow features here requires care.
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