The Motions of a Ship on a Sloped Seabed

In standard diffraction theory it is assumed that the water depth is constant and that the seabed is infinitely large. To account for a local varying bathymetry in shallow water (as it can occur for offshore LNG terminals) it is sometimes considered to introduce a second fixed body on the seabed representing this bathymetry in diffraction theory. Based on the results presented in this paper it can be concluded that this is (without special measures) not possible. The refraction and interference effects are too strong and affect the wave exciting forces on the LNG carrier in an incorrect way. A large size of the second body and smoother edges of this body do not improve the situation. However, a second body in diffraction theory, when chosen properly with respect to size and shape, can contribute to the correct calculation of the added mass and damping of vessels on sloped seabeds as this varies with the local water depth over the length of the vessel. This will clearly affect the motion response of the vessel. This can be seen for instance in the pitch-heave coupling. This will influence the motions of the ship in waves, as well as the resulting drift forces and related mooring loads. INTRODUCTION Recent experience with the development of offshore LNG terminals has shown that the issues related to shallow water hydrodynamics are at least of similar complexity as the ones in (ultra) deep water developments: In nearshore wave dynamics many different phenomena play a role, such as dispersion, diffraction, refraction, shoaling, reflection, nonlinear wave-wave interaction, wave-current interaction, wave breaking and bottom friction. The local bathymetry affects the waveloads on (and motions of) moored structures. As can be seen in Figures 1 and 2, both existing LNG jetties and new LNG mooring systems are in nearshore conditions where the local seabed can vary significantly. Low frequency wave effects such as set-down and shoaling can result in significant excitation. Streamlined LNG carrier hulls have a very low damping against low frequency motions. The combination of excitation and low damping can result in significant resonant motions and related mooring loads. Neglect of these important issues in shallow water motion and mooring prediction methods could result in problems for new offshore (LNG) terminals. The combined input from offshore hydrodynamics and coastal engineering is considered to be vital to solve these issues. Figure 1: Existing jetty-type moorings of LNG carriers in nearshore conditions Figure 2: New type moorings of LNG carriers are also considered in shallow waters 1 Copyright © 2006 by ASME The low frequency wave effects and resulting low frequency motions of moored LNG carriers in shallow water are discussed extensively by Naciri et al (2004), Van Dijk et al (2005) and Voogt et al (2005). Wave propagation over a nonconstant seabed has been a topic in coastal engineering for many years already, see for instance Mei et al (2005). In recent years very good results are achieved for complex bathymetries and coastal geometries with several methods using the Boussinesq equations, see for instance Borsboom et al (2000) and Madsen et al (2002). Bingham (2000) used Boussinesq-type wave modelling to calculate the motions of moored vessels in a de-coupled sense: Wave forcing: high-order Boussinesq theory Wave-body interaction: diffraction theory / panel method Ship response: time-domain equations of motion At the moment this type of method is being extended into a more coupled solution of the wave forcing and wavebody interaction. The present paper is part of this development and focuses on the local interaction between the varying seabed and the LNG carrier in diffraction theory: added mass, damping and wave forces. Even if the wave exciting forces are calculated with Boussinesq type models (Bingham, 2000), in the added mass and damping the local bathymetry effects (varying over the length of the vessel) should be taken into account. In standard diffraction theory it is assumed that the water depth is constant and that the seabed is infinitely large. To account for a local varying bathymetry, it is an option to introduce a second fixed body on the seabed representing this bathymetry, as shown schematically in Figure 3. Referring to Figures 1 and 2 this is a realistic situation for LNG terminals. Figure 3: Modelling a sloped seabed schematically as a second body in diffraction theory Teigen (2005) used the same methodology to investigate the motion of a spread moored barge close to an underwater ridge of limited size. He focussed on the general effect of water depth on added mass and damping and the effect of the underwater ridge on the wave field. The present paper focuses more on the situation where the vessel is actually above the varying seabottom, so that the added mass and damping are also directly affected by the bathymetry. Although the sloping seabed can be very large in reality (up to the coastline), it is not possible to model this completely in the diffraction theory, as this would result in extremely long computational times (and diffraction theory would not be able to simulate non-linear effects such as wave breaking on a beach or wave reflection on a breakwater). Consequently the paper discusses the following questions: Is it possible to model a sloping seabed as a second body in diffraction theory? What are the effects of the local bathymetry on the added mass, damping and wave forces on the vessel above it? How large should the second body (simulating the bathymetry) be and are there special requirements with respect to its shape? The paper focuses on the first order loads and motions, second order effects will be part of future investigations. METHODOLOGY As part of this study, both calculations and model tests were carried out. Model tests A 1:20 sloped bottom situation was tested in the MARIN Offshore Basin, using a wedge-type wooden slope of 550 by 550 m full scale. With this slope the water depth gradually decreased from 28 m at the bow to 15 m at the stern of the 274 m long LNG carrier (draft 11 m). The normal water depth was 35m. An overview of this set-up and the moored LNG carrier is shown in Figure 4. Figure 4: The modelling of a sloped seabed in the basin with a wooden wedge-shaped slope and the LNG carrier moored to an open jetty. 2 Copyright © 2006 by ASME Only head wave conditions were tested in this set-up. Diffraction calculations Two body diffraction analysis was carried out with the MARIN program DIFFRAC. As a start, the 550 x 550 m slope was calculated and checked against the model tests, as shown in Figure 5. Figure 5: The modelling of a sloped seabed in linear diffraction analysis as a second body with a narrow wedgeshaped slope of 550 by 550m. Originally it was believed that the dimensions of the 550m by 550m slope would be large enough to avoid end effects. However, it was discovered that the edge effects on the sides and end of the slope were much more important than originally expected, see the next section. As a result of that, the following further conditions were simulated: Much wider slope of 550 by 1650 m, see Figure 6. Figure 6: The situation with a much wider slope of 1650 by 550 m Sloped edges on the side and end of the slope, both for the 550 by 550 m and of 550 by 1650 m sloped, see Figure 7. Figure 7: Sloped edges for the slope of 550 by 550 m For reference, also calculations with 15m and 28m constant depth (without the slope) were carried out. RESULTS Based on existing literature in this field (Mei et al, 2005), the following effects were expected during the tests and simulations: Increased wave height on slope as the total wave energy remains constant while progressing into shallow water A shortening of the waves according to (with h as the water depth):