Istituto Nazionale per Studi ed Esperienze di Architettura Navale, Italy

  1. Do the number of panels of the submerged part affect the solution?

  2. What does “particular care in moving the free surface to conserve the mass” mean?


In our numerical calculation the number of elements on the wetted surface is kept constant, so the submergence will not affect the solution. Because we used lower order panel method, special care has been taken in order to satisfy the conservation of mass. That means that the rate of change with time of the water volume above z=0 should be equal to the rate of change with time of the body displacement below z=0. A detailed discussion about it can be found in Zhao and Faltinesen (1993).



University of Newcastle Upon Tyne, United Kingdom

In two-dimensional flow, if you consider planing action, then the momentum change in the jet of fluid thrown forward is proportional to the normal force on the planing surface. In this three-dimensional case, can this fact not be used to define the cut off of the jet, where it is considered as one-dimensional flow?


The two-dimensional planing problem is a steady state problem, which is different from the slamming problem. It can be shown that the momentum change in the jet can be neglected in the calculation of slamming force for asymptotically small deadrise angle (in using the conservation of momentum to predict slamming force).


Zhao, R. and Faltinsen, O. “Water entry of two-dimensional bodies,” Journal of Fluid Mechanics, Vol. 246, 1993.



Det Norske Veritas, Norway

Did you compare the 3D results with the results of your earlier published 2D method? I’m asking you this question because the 2D method can easily be applied to ship-forms by applying a ship-wise discretization. Probably it is not possible to do the same with this extreme 3D example, but it should be possible to compute a “2D” force by interpolating the 2D pressure distribution, to be calculated for the vertical symmetry plane, over the cone according to

where c is the contour of the symmetry plane.

It would be very interesting and relatively easy to quantify the 3D effect this way. Or is it maybe the same?


We have done the comparison of slamming pressure between a 2D wedge and a cone for different deadrise angles. The results of pressure distribution for the cones are presented in figure 7. The results for the wedges are given by Zhao and Faltinesen (1993). For small deadrise angle the maximum pressure is dependent on the wetted factor in square. Figure 10 shows the ratio of the wetted factor between the cones and wedges. The factor is dependent on the deadrise angles, which is about 0.8. That means that the maximum pressure for the cone is about 0.64 times the maximum pressure of the wedge with small deadrise angle. Increasing deadrise angle, this ratio will be changed. The ratio is about 0.72 for deadrise angle of 30 degrees. The average pressure for the cone is also lower than the average pressure of the wedge, which is also dependent on the deadrise angles. F2D=∝×F3D×∝.

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