UNDA employs a collocation spectral method. The total velocity potential is constructed as a time-dependent linear superposition of basis potentials, each of which represents a steady flow solution that satisfies the boundary conditions imposed by bodies, tank walls and bottom, as well as representative boundary conditions at the top of the computational domain and at open boundaries. After the spatial problems of the basis potentials have been solved, the time evolution is integrated by imposing exact free-surface boundary conditions at the instantaneous water surface and flux boundary conditions at open boundaries. As an alternative, one also has the option of imposing linear surface boundary conditions.
The basis potentials are finite and continuous everywhere, including inside bodies and up to the highest elevation that the water can reach. They satisfy the Laplace equation in water and a higher-order equation in bodies. This gives enough freedom to satisfy continuity and Neumann boundary conditions at the interfaces. As far as satisfying the free-surface boundary conditions is concerned, one may forget about the difference between fluid and bodies.
Splines are used for interpolation between grid points in the basis potentials and for approximation of surface derivatives at the water surface.
Computation of Nonlinear Water Waves
Sample animations of a one-legged platform. A four-legged platform. Another four-legged platform.
UNDA is a fully nonlinear solver for potential flow around marine constructions. Particular attention has been made to multi-cylinder platforms with or without pontoons. Some typical situations are shown in these snapshots:
Long-crested waves hitting a four-legged platform
Click on the image above and to the right to see a video animation of the simulation.
The water depth is 345 m. The cylindrical legs have radius 15.5 m, with 80 m spacing between their centers. In the simulation, the legs extend from 77 m below to 23 m above the mean water level, while the animation only shows the legs extending from 23 m below the mean water level. The simulation was done in a domain extending from 1000 m in front to 1000 m behind the platform center, and extending 300 m to each side. The animation shows a limited region extending from 180 m in front to 125 m behind the platform center, and extending 89 m to each side. The vertical scale has been exaggerated 2.5 times in comparison with the horizontal scale.
The incoming wave has period 12 s and nominal amplitude 7 m. That gives a nominal steepness of ka = 0.2, which may be unrealistically high for waves in the ocean. The simulation starts from rest at time 0 s when waves begin to enter through one of the short sides of the simulation domain. The simulation stops at time 200 s in order to avoid contamination by waves reflected from the side walls.
Green water
Green water is the name used by seafarers for waves inundating a ship. The offshore oil and gas industry has recently been interested in green water due to significant damage suffered by ships and installations. However, the physical phenomenon can be observed at virtually any beach as waves overrun large rocks. We have investigated the special case of waves running over a truncated cylinder by using UNDA as a tool for computation of nonlinear potential flow.
UNDA was originally developed as a fully nonlinear solver for potential flow around surface-piercing columns or fully submerged bodies. The present project is an effort to show how the usage of UNDA can also be generalized to situations where an originally surface-piercing body is inundated by a large wave.
Potential flow cannot account for wave breaking or generation of vorticity, both of which will occur in the event of green water. Our limited goal is to describe the flow field at an early stage of inundation. Our results can thus be used as input to Navier-Stokes solvers or other appropriate tools for modeling of wave breaking etc. at later stages of a green water event.
Numerical example: A truncated circular cylinder.
The numerical wave tank has width 54 cm and depth 42 cm. The cylinder has diameter 24 cm and extends from 8 cm below to 1 cm above the quiescent water level. The incoming wave has period 1 s which corresponds to wavelength 1.48 m. The nominal wave amplitude is set to 2 cm. Therefore the largest wave crest is just high enough to inundate the top of the truncated cylinder.
In the series of six figures, we see how the initially dry top is inundated initially from the rim, and how the water then runs off to the sides. Notice that part of the dry top appears to remain dry.
It can be noticed from the simulation (though not from the snapshots) that the wave apparently slows down as it propagates on top of the cylinder. This it at least in qualitative agreement with the linear dispersion relation that predicts that the wave speed should be smaller for the small depth that is experienced on the top. Due to the small depth that is experienced on the top of the cylinder, the flow will be locally very nonlinear. This normally results in wave breaking, and thus a potential flow solver such as UNDA will eventually break down. UNDA can be used to predict the initial stage of a green water event. Impact velocities can be obtained and used as input to other computational tools appropriate for the local flow on top of the cylinder.
UNDA development has been funded by Statoil, Norsk Hydro, Norske Conoco, Saga Petroleum, SINTEF Applied Mathematics and Norske Shell from 1994 to 2002.
Nonlinear ocean wave modeling
The following simulations of bichromatic waves demonstrate how weakly nonlinear self-interaction can affect the evolution of a wave train on deep water. The figures show time series measured at four different locations along a wave tank. The bottom (blue) curves show linear evolution of the wave train. The middle (green) curves show the evolution predicted by the cubic nonlinear Schrödinger equation. The top (red) curves show the evolution predicted by the higher-order modified nonlinear Schrödinger equation.
According to linear theory there should be no amplitude or frequency modulation of the wave train. The cubic nonlinear Schrödinger equation predicts a symmetric amplitude modulation of the envelope, with the largest waves occurring in the middle of the group. At larger distances this modulation fades away, and the wave train may recover its initial shape. The higher-order nonlinear Schrödinger equation predicts that the initially symmetric group leans forward, with steep and large waves occurring near the front. At larger distances the group splits up in a large and steep group running away from a smaller group lagging behind.
Experiments of bichromatic waves reveal the behavior of the higher-order equation. We have done detailed validation against laboratory measurements to find the limiting distance for which these three wave models no longer faithfully predict the wave evolution (Trulsen & Stansberg 2001).
Using nonlinear wave models as those described above, we aim at describing various aspects of nonlinear ocean waves such as the evolution of the most energetic part of a wave spectrum, deterministic wave forecasting, and statistical analysis of extreme waves based on deterministic wave simulation.
This project is on a contract with the University of Bergen with funding from Norsk Hydro. It is part of the ocean wave activities that are coordinated by Professor Kristian B. Dysthe at the Department of Mathematics at the University of Bergen, with funding from both Norsk Hydro and Statoil. These activities are a supplement to the ongoing project on extreme waves funded by the Norwegian Research Council.
Wave loads on breakwaters in Hasvik harbour
References
- K. Trulsen and P. Teigen. Wave scattering around a vertical cylinder: Fully nonlinear potential flow calculations compared with low order perturbation results and experiment. In "Proceedings of OMAE'02", paper OMAE2002-28173, pp. 1-9, 2002.
- K. Trulsen, B. Spjelkavik and E. Mehlum. Green water computed with a spline-based collocation method for potential flow. International Journal of Applied Mechanics in Engineering, 7(1):107-123, 2002.
- P. Teigen and K. Trulsen. Numerical investigation of nonlinear wave effects around multiple cylinders. In "Proceedings of ISOPE 2001", Vol. 3, pp. 369-378, 2001.
- E. Mehlum. Splines and ocean wave modelling. In "Numerical Methods and Software Tools in Industrial Mathematics", Eds., M. Dæhlen and A. Tveito, pp. 235-253, Birkhäuser, 1997.
- X. Cai and E. Mehlum. Two fragments of a method for fully nonlinear simulations of water waves. In "Waves and Nonlinear Processes in Hydrodynamics". Eds., J. Grue, B. Gjevik and J. E. Weber, pp. 37-50, Kluwer Academic Publishers, 1996.
