A Discontinuous Galerkin Finite Element Method with Turbulence Modelling for Incompressible Flows

This thesis explores the use of an innovative interior penalty Discontinuous Galerkin Finite Element Method (DGFEM) for the Reynolds averaged, incompressible Navier-Stokes equations, coupled with the k-w turbulence model. The simulation of incompressible flows is relatively inexpensive computationally, and, with appropriate assumptions, provides a good approximation to compressible flows. This makes them useful for large simulations, such as those required by the steam turbine industry. Current generation industrial CFD solvers require ad hoc user intervention with regards to solution refinement, in order to achieve numerical results with a sufficient degree of accuracy. Accurate simulations of curved blade geometries rely on a dense packing of straight edged elements in order to represent the geometry correctly. This results in extended simulation times and non-optimised numerical results.

Curved boundary elements allow highly curved geometries to be represented by fewer mesh elements, enabling effective mesh refinement perpendicular to the boundary, without increasing mesh density parallel to the boundary. To achieve this, we propose a novel approach using inverse estimates to derive a new discontinuity-penalisation function which stabilises the DGFEM for computations in both two and three dimensions, on meshes consisting of standard shaped elements with general polynomial faces. Automated solution refinement is achieved by considering the dual-weighted-residual approach, defining a suitable numerical approximation for the dual solution, along with a target functional to drive the refinement. A novel continuation and refinement algorithm, along with a prototype DGFEM solver is developed, producing a number of interesting numerical results for high Reynolds number flows. These ideas are extended to incorporate the recent results in the literature for DGFEMs on general computational meshes consisting of polygonal elements. For high Reynolds number turbulent flows, we show that polygonal elements can be used to significantly reduce mesh density and the computational resources required for fluid simulations through several numerical experiments.