This thesis is concerned with the application of adaptive local quasi-Newton coupled multigrid (ALQNMG) solvers to the numerical simulation of viscous incompressible fluids, using the multi-fluid model. The ALQNMG methodologyh as proven highly successful for single phase flows , leading to solution algorithms which are: (i) robust, (ii) efficient and (iii) accurate. Its extension to multiphase flows is very challenging because the governing equations are mathematically complex and their solutions are subject to constraints. The solver presented here has therefore required a considerable number of specific algorithmic developments. The outline of the thesis is as follows: firstly, the modelling and simulation of multiphase flows are reviewed, together with the different numerical techniques implemented in the solver. Finite volume discrete multiphase equations are then derived on structured, staggered grids. Next, having specified the solution algorithm, we consider the accuracy of the solver. Results from several test cases of varying complexity are compared with those of a widely used commercial CFD package and good agreementis obtained. The question of performance is then addressed in detail, both in terms of robustness and speed of convergence. Good accelerations are obtained using the multigrid method but the convergence rates are often not grid-independent. The most likely explanation is that the discrete operators are highly non-linear and therefore have different characteristics on different grids. Furthermore, the solution algorithm is shown to not handle certain multiphase diffusive terms very well. Convergence rates are much faster than those achieved by single grid solvers and commercial codes typically by one order of magnitude and often more, although the solver is not fully optimal. Finally, adaption is considered. Grids are generated automatically which facilitates the use of the code and allows error control. It is confirmed that multigrid methods offer a good framework for the implementation of adaption. Considerable gains in speed and memory usage, by one further order of magnitude, are achieved
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