Surface capturing and multigrid for steady free-surface water flows

Abstract

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.