Analytical bow waves for fine ship bows with rake and flare

The bow wave generated by a steadily advancing ship is considered for a family of fine ruled ship bows with rake and flare. This family of ship bows is defined in terms of four parameters: the ship draft D, the entrance angles alpha and alpha' at the top and bottom waterlines, and the rake angle delta. The corresponding bow wave similarly depends on four parameters: the draft-based Froude number F and the three angles alpha, alpha', and delta. An extensive parametric study, based on thin-ship theory, is performed to explore the variations of the water height Z(0) at the ship stem X = 0, the location X(0) (measured from the ship stem) of the intersection of the bow-wave profile with the mean free-surface plane Z = 0, and the bow-wave profile, with respect to the four parameters F, alpha, alpha' and delta. This parametric study extends the previously reported similar study of the height Z(b) of the bow wave and the location X(b) of the bow-wave crest. These two complementary parametric studies yield simple analytical relations, which extend relations given previously for wedge-shaped ship bows without rake or flare. In spite of their remarkable simplicity, the analytical relations given here yield bow waves that are comparable to computational fluid dynamics (CFD) waves given by Euler-flow calculations. The analytical relations, which explicitly account for the influence of the four primary parameters F, alpha, alpha', and delta, can be used immediately-without hydrodynamic calculations-for ship design, notably at early design stages when the precise hull geometry is not yet known. The study also provides insight for ship bow design. Specifically, it suggests that a bow with positive rake and negative flare may be beneficial, and that a bulb located aft of the stem and integrated with the hull may be an advantageous alternative to a traditional bulb protruding ahead of the bow, in agreement with the results of a hull-form optimization analysis.