The numerical treatment of curved boundary surfaces in an unfitted grid formulation for computational fluid dynamics

In this thesis a new Cartesian cut-cell scheme has been presented which solves the in- compressible Navier-Stokes equations using a finite volume approach and a staggered grid. The equations have been solved in conservation-law form upon a network of cell which were created using a Cartesian, cell-based grid generation procedure. This method used a recursive subdivision of fixed aspect ratio (Cartesian) cells creating arbitrarily shaped polygons when the Cartesian cell straddles a boundary, using a modified polygon clipping algorithm. The numerical scheme adopted is the finite volume formulation, adaptive multi- grid algorithm, and the Cartesian cut cell method. The scheme is modified to allow accurate calculation of flow variables at the intersection of curved body boundaries on non-aligned grids. A novel method to distinguish boundary cells is developed in this project. The outline of the thesis is as follows: firstly, the modelling and simulation of un- fitted Cartesian grids, structured grids, and unstructured grids are reviewed, together with the different numerical techniques implemented in the solver. Finite volume dis- crete equations are then derived on structured, staggered grid. Next, having specified the solution algorithm, we introduce the Cartesian cut cell method and consider the accuracy of the solver. Results from several test cases of varying complexity are com- pared with those of a widely used commercial CFD package and good agreement is obtained. The results of this method show that the Cartesian cut cell approach is second- order accurate for the two-dimensional flows. The question of performance of the algorithm is also addressed in detail, both in terms of robustness and speed of con- vergence by modified transformations. Good accelerations are obtained using the adaptive multigrid method but the convergence rates are often not grid-independent. Convergence rates are much faster than those achieved by single grid solvers and com- mercial codes, although the solver is not fully optimal. It is confirmed that multigrid methods offer a good framework for the implementation of grid adaptation, particu- larly in areas where the flow variable change abruptly/rapidly. Considerable gains in speed and memory usage, by one further order of magnitude, are achieved.