A solution method for the nonlinear ship wave resistance problem
 Author Raven, H.C. Title A solution method for the nonlinear ship wave resistance problem Conference/Journal PhD-thesis, Delft University of Technology Month June Year 1996

Abstract
A ship in steady motion in a calm sea generates a wave pattern. Associated with this is a wave resistance force acting on the hull, which is a significant part of the total resistance. In order to be able to minimise this wave resistance by a proper hull form design we want to predict the wave pattern generated by a given hull form at a specified speed.

The mathematical problem to be solved is that of an incompressible potential flow, subject to a Neumann boundary condition on the hull, a Neumann boundary condition on the wave surface, and a given constant (atmospheric) pressure on the wave surface. The latter free surface boundary conditions are nonlinear, and the location and shape of the wave surface is still unknown.

Until recently this problem was always linearised in perturbations of an assumed base flow, and a boundary condition was imposed on the undisturbed water surface. In particular Dawsons's slow-ship linearisation is in widespread use, but it has several shortcomings. Chapter 4 studies the origin of these and compares various alternative linearisations. Specific attention is paid to the paradoxical occurrence of negative predicted wave resistances for slow ships, which is explained in terms of a spurious energy flux through the calculated wave surface. The general conclusion from this study is that a fully nonlinear method is required to remove the shortcomings of existing methods.

The nonlinear method proposed here uses distributions of Rankine source panels on the hull and free surface. An unconventional choice made is to put the free surface panels at a small distance above the wave surface, while keeping the corresponding collocation points on the wave surface itself. This has a number of practical advantages, and improves the smoothness of the velocity field. A theoretical analysis of the numerical dispersion, damping and stability of this approach reveals that moving the source panels off the wave surface almost eliminates the numerical dispersion errors caused by the source discretisation (which is a significant first order quantity in a conventional method). The only numerical dispersion remaining is that due to the difference scheme in the free surface condition, which is of 3rd order in the panel size. Since also the susceptibility to point-to-point oscillations is removed, the raised-panel method permits to achieve a much greater numerical accuracy than a conventional method.

To solve the nonlinear free surface problem an iterative procedure is used. Each iteration solves a problem in which a linearised boundary condition is imposed on the wave surface found in the previous iteration. Convergence is generally quick and robust, requiring 10 to 20 iterations. The total CPU time amounts to 2 to 20 minutes on a CRAY C98, using up to 6000 panels per symmetric half.

Special attention is paid to the flow off an immersed transom stern, a truncated ship afterbody with a sharp lower edge. A mathematical model is selected that takes into account the possibly weakly singular behaviour at the transom edge. Numerical and experimental validations confirm the usefulness of the model.

In a range of validations the wave pattern predictions have been found to be most accurate qualitatively, and often also quantitatively. The method responds correctly to changes in the hull geometry. The principal remaining deviations are due to viscous effects on stern waves. Improvement is still desired in the numerical evaluation of the resistance force, which now is too sensitive to numerical parameters; and in the stability with regard to transverse oscillations.

Nonlinear effects have been found to be far stronger than expected, in particular for slender vessels (containerships, ferries etc.) A study of the origin of these differences shows that the well-known underestimation of bow wave heights by linearised methods is entirely due to the fact that they impose the boundary condition on the still water surface rather than the true wave surface. In addition, refraction of waves by the variation of the velocity field near the ship is found to have a strong influence on the wave shape and amplitude, and this refraction is incompletely represented in linearised methods. This explains the large difference between linearised and nonlinear predictions of the amplitude of diverging bow waves.

The present nonlinear method has been implemented in the RAPID code, which now is in routine use at MARIN for practical ship hull form optimisation.

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