Three-dimensional discontinuous spectral/hp element methods for compressible flows

Abstract

In this thesis we analyse and develop two high-order schemes which belong to the class of discontinuous spectral/hp element methods focusing on compressible aerodynamic studies and, more specifically, on boundary-layer flows. We investigate the discontinuous Galerkin method and the flux reconstruction approach providing a detailed analysis of the connections between these methods. The connections found enable a better understanding of the broader class of discontinuous spectral/hp element methods. From this perspective it was evident that some of the issues of the discontinuous Galerkin method are also encountered in the flux reconstruction approach, and in particular, the aliasing errors of the two schemes are identical. The techniques applied in the more famous discontinuous Galerkin method for tackling these errors can be also extended to the flux reconstruction approach. We present two dealiasing strategies based on the concept of consistent integration of the nonlinear terms. The first is a localised approach which 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). Both the strategies have been observed to be effective in enhancing the robustness of the schemes considered. We finally present the direct numerical simulation of a high-speed subsonic boundary-layer flow past a three-dimensional roughness element, achieved by means of the compressible aerodynamic solver developed. This type of analyses have been widely performed in the past with approximated theories. Only recently, has DNS been used due to the improvement of numerical techniques and an increase in computational resources for similar studies in low-speed subsonic, supersonic and hypersonic regimes. This thesis takes a first step to close the gap between the results for a high-speed subsonic regime and the results in supersonic and hypersonic regimes.