· · · Institutional Repository

A boundary element method has been developed that can be used to calculate two-dimensional potential flow phenomena with a free surface. The unsteady Bernoulli equation is applied at the actual position of the free surface. A solution for corners arising in the geometry of the boundary has been found. Comparison of results of the present model with results obtained with other numerical methods shows that reliable results can be obtained with the present model as long as the gap between two adjacent normals at a nodal point is less than about 40 degrees. At this limit the calculations break down. Within this limit, a forward directed jet is well developed in the case of breaking waves. The calculations can proceed if two points at the tip of the jet are considered as corner points. With these special points, the calculations can be continued up to the moment the jet falls down the forward face of the wave.

The focus is on the numerical flow model TRIWAQ. It is developed as a hydrostatic free-surface flow model, which is currently being used by the KNMI and Rijkswaterstaat for predictions of water levels in the North Sea and Dutch estuaries. TRIWAQ has successfully been extended to the realm of non-hydrostatic modeling, TRIWAQ-NH, this allows the use of full Navier-Stokes equations. It has been validated to perform well for multiple processes such as dispersion and propagation. The goal is to assess the ability of TRIWAQ-NH for harbour problems. This has not been attempted before and poses a new challenge. Before an attempt is made to simulate a harbour, methods of imposing reflection are tested. For that matter, the thesis is split in two parts. The first part will investigate the ability of TRIWAQ-NH with respect to reflections. It will confine itself to four currently implemented open boundary conditions. It aims to provide an answer what condition is most suitable for future development into a partial reflecting boundary condition for harbour simulations. This is done by means of a literature survey, which inspects the background theory it stems from, its dependence on wave frequencies and limitations due to the angle of incidence. By means of 1 dimensional simulations the ability of each open boundary condition is tested when the non-hydrostatic method is used and will be referenced with a similar hydrostatic simulation. A monochromatic wave is used. The second part of the thesis focuses on validating 2 dimensional non-hydrostatic simulations with TRIWAQ-NH. This is done by modelling a simplified rectangular harbour basin of constant depth. A monochromatic wave is selected. From the 1 dimensional test cases, two conditions are selected and simulated, these are the Riemann and sponge layer condition. The last simulation has full reflective boundaries. Each is again referenced with a hydrostatic simulation. The current use of the model TRIWAQ-NH holds practical restrictions to the implementation of open boundary conditions. When such a definition spawns a length of more than grid cell, it needs to comply with the edges of the grid. For instance, a boundary under the angle of 77 degrees is only possible by creation of a multitude of smaller open boundaries, leading to human error. The thesis results in two recommendations. The first part covered reflections, from that segment of the thesis, it is recommended to further explore the option of a stronger and smaller grid sized sponge layer with either an open or closed boundary condition for future development as method for partial reflections. The second recommendation is of a more practical nature and does not emerge in the work. However, the current tools available for non-hydrostatic modelling are insufficient. The smaller wave lengths require smaller time frames then the current tools provide, which is in the order of hours, days and months.

Surface capturing is a technique for modelling the water surface in numerical computations of water flow: the computational grid is not deformed, a separate surface model gives the location of the water surface in the grid. Surface capturing is generally applicable and can handle complicated ship geometries. For steady flow problems, the major disadvantage is that most capturing methods do not allow the use of fast solution methods. This thesis shows that fast solution of a surface capturing model is possible. For this, a flow model is derived that consists of conservation laws only. As these equations allow coupled solution, they can be solved efficiently for steady flows. The flow equations are discretised with a finite-volume method. The convective part is discretised with linearised Riemann fluxes, which guarantee the stability of the discretisation and good performance of the relaxation methods. A RANS turbulence model is added to the system. A multigrid solver is combined with line Gauss-Seidel smoothing. The source term in the turbulence model can make the line smoothing unstable. Therefore, a local adaptive damping is added to the smoother. Also, the mixture surface model and the turbulence model cause large differences in the solutions on fine and coarse grids, so nonlinear multigrid is ineffective. Our multigrid method combines nonlinear smoothing on the finest grid with linear coarse grid corrections. The discretisation is made second-order accurate with a limited scheme. To keep the water surface sharp, a compressive limiter is used for the volume fraction, that indicates the surface. The second-order accurate equations are solved with defect correction. Results are presented for a 2D channel flow with a bottom bump. The capturing model gives good agreement with experiments and existing numerical models. The multigrid solution is up to 20 times faster than single-grid line smoothing. The thesis also contains two smaller topics: an unstructured grid refinement method for ship flow grids and a study of shock behaviour for a compressible two-fluid flow model.