Seakeeping Calculations for Ships, taking into Account the Non-linear Steady Waves
A ship sailing at sea experiences a resistance due to its forward speed and incoming waves, which must be balanced by the propulsive power of the ship. To save fuel, this resistance is desired to be as small as possible, and a lot of expensive model tests in seakeeping basins are carried out to minimize the resistance. It is important to have a tool that can predict the forces on a ship sailing in waves and assist these model tests or partially replace them. Mathematics and computer simulations give us such a tool. The non-viscous part of the resistance of a ship has two major contributions. First, there is the resistance that a ship experiences when it sails at a constant speed in a calm sea, without incoming waves. This is called the wave resistance, and it can reasonably well be predicted by the method RAPID, developed at the MARIN, which calculates the nonlinear steady flow around a ship. Second, there is an ext ra resistance when a ship sails in incoming waves. Although harmonic incoming waves have a mean value which is zero, the forces due to these incoming waves can have a mean value which is non-zero. It is as if the waves want to push the ship ahead of them. This phenomenon is called drifting and is responsible for a resistance increase and can, in oblique waves, result in a change of a ship's course. The increase of the resistance is also known as the added resistance. This added resistance strongly depends on the forward speed of a ship. Prins [28) developed a method that can determine the drift forces for low speeds of a ship. In this thesis we presented a method that can determine the drift forces for moderate and high speeds of a ship as well. When the speed of a ship increases, the interaction between the steady waves on the one hand and the unsteady waves, the ship motions and the added resistance on the other hand becomes more and more important. Therefore, for moderate and high speeds of a ship, the non-linear steady flow has to be taken into account. We have modeled what happens to a ship when an incoming wave propagates over this non-linear steady wave, reaches the ship, and diffracts. We did this by assuming that the steepness of the incoming waves and the amplitude of the motions of the ship are small. This allowed us to linearize the condition that holds on the free surface, which describes the propagation of the waves, and the condition that holds on the hull of the ship, which relates the water flow and the motions of the ship. Due to the linearization, not only the steady fluid velocities are required , but also first and even second derivatives of these velocities. This sometimes gives problems due to inaccuracies in the steady velocities. This linear model was solved with a boundary-element method. The free surface and the hull of the ship were divided in small areas, called panels, and on each of these panels a pulsating source was placed with a constant strength. This way it is made sure that the Laplace equation inside the fluid is fulfilled. The strength of these sources was found by applying the boundary conditions in (or below) the cent re of these panels. Furthermore, the boundary condition on the free surface was discretized by using difference schemes for the time derivatives and the space derivatives. For the space derivatives, upwind difference schemes were used. With a stability analysis we showed that the use of the more accurate central difference schemes easily leads to instabilities. By using upwind difference schemes, the occurrence of instabilities can be avoided. We applied this model to a test ship, and investigated the grid dependence of the steady waves and the derivatives of the steady velocities, which are required to determine the unsteady waves. Then we investigated the dependence of the unsteady waves on the grid size and the time step and the dependence of the predicted forces on the panel density on the hull. In all cases a good convergence was obtained, except for the transfer term, a second derivative of steady velocities, which diverges near the ship, and which has to be obtained by means of extrapolation from the transfer term further away from the ship. To model incoming waves we implemented two methods: a method in which the incoming waves are generated by a wavemaker, and a method in which we separate the incoming waves from the diffracted waves and only calculate the latter. Although both methods gave satisfactory results, we prefer to separate the waves. After checking the numerical convergence of the model, we validated the model by comparing the predicted motions and added resistance of an LNG carrier with measurements from the MARIN. _A comparison was made for three moderate Froude numbers of the carrier, namely Fn = 0.14, Fn = 0.17 and Fn = 0.2. With the separation method, incoming head waves, bow-quartering waves and beam waves were simulated. For short waves, a very good agreement between the predicted and the measured motions and added resistance was found. For long waves there were some deviations, amongst others due to an insufficient size of the free surface. For all wavelengths, the roll motion could not be predicted because we neglect viscosity in our model. Finally, we compared our predictions with the predictions obtained by using the uniform flow and the double-body flow. It turned out that only by using the non-linear steady flow, an accurate prediction of the added resistance can be obtained.