Discontinuous spectral/hp element methods: development, analysis and applications to compressible flows

Abstract

This thesis is concerned with the development and analysis of discontinuous spectral/hp element methods and their applications to compressible aerodynamics with special focus on boundary-layer flows. In this thesis, we provide a detailed analysis on the connections between the discontinuous Galerkin method and the flux reconstruction approach for multidimensional nonlinear systems of conservation laws on irregular meshes (i.e. meshes with deformed and/or curved elements). The results help a better understanding of the broader class of discontinuous spectral/hp element methods and allow the direct applications to the flux reconstruction approach of the existing and more established techniques used in the discontinuous Galerkin community for tackling various issues of this class of schemes, including their aliasing problems. From this perspective, we present two dealiasing strategies based on the concept of consistent integration of the nonlinear terms (also referred to as over-integration of the linear terms). The first is a localised approach and it targets in each element the nonlinearities arising in the problem, while the second is a more global approach which involves a higher quadrature of the overall right-hand side of the discretised equation(s). The two dealiasing strategies have been observed to be effective in enhancing the numerical stability of both schemes, the flux reconstruction and the discontinuous Galerkin approaches. We finally present the direct numerical simulation of a high-speed subsonic flow past a roughness element, achieved by means of the discontinuous spectral/hp element methods developed. These results were successively compared to some data obtained from the asymptotic triple-deck theory. This work, besides demonstrating that the class of schemes analysed and developed is attractive for such aerodynamic problems, also addresses the lack of comparisons between theoretical models and numerical simulations.