Sand waves are self organizing patterns in non-cohesive sediment at the bottom of shallow seas. Their typical size and dynamics make them relevant for navigation safety and off-shore infrastructure. In the first chapters of this thesis, equations describing their behavior are derived. The hydrostatic incompressible Navier Stokes equations are used to describe the tidal flow and a generic bed-load transport model is used to estimate the resulting sediment transport. Combined this results in the self organizing behavior that creates sand wave patterns. To simulate the evolution of sand waves, we first use a Swift-Hohenberg equation. This model captures the main properties of the combined flow and sediment equations, but is much easier to analyze and solve. We analyze different solution techniques for this type of system and conclude that exponential time-stepping greatly improves stability and accuracy over traditional Euler methods. Using the Swift-Hohenberg model and the exponential time-stepping algorithm, we show that randomness in the initial condition from which sand waves start to grow, gives variations in the resulting field. Because of the small computing times the approach is convenient for infrastructure design computations. Although the Swift-Hohenberg equation captures the general behavior of sand waves, it does not capture the physical processes driving this behavior. To be able to investigate the influence of various physical conditions, a combined flow and sediment transport solver is constructed. The flow part uses a mimetic spatial and a spectral temporal discretization. The discrete flow equations are solved using a Newton iteration method. The resulting linear systems are solved using a combination of a block B-tree sparse storage LU decomposition and GMRES. For the sea floor update we use second order accurate finite differences in space and a semiimplicit time-stepping method. Due to computing time restrictions this approach is limited to 2D computations. Nevertheless we can show that linear stability is a good indication for non-linear wave length behavior and confirm that randomness in the initial condition results in variations in the resulting field. Through the use of a spectral discretization in the two horizontal spatial directions and a power series expansion instead of the Newton iteration method, full 3D simulations within reasonable computing times become feasible. Again the development of a pattern with variations from random initial conditions can be shown and the dominating wave length matches the one found in linear stability analysis. This last computing approach is most promising for further research, since it could be used to investigate the interactions between sand waves and sand banks.
|Award date||29 Jun 2007|
|Place of Publication||Netherlands|
|Publication status||Published - 29 Jun 2007|