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

摘要

In this thesis we analyse and develop two high-order schemes which belong toudthe class of discontinuous spectral/hp element methods focusing on compressibleudaerodynamic studies and, more specifically, on boundary-layer flows. We investigateudthe discontinuous Galerkin method and the flux reconstruction approachudproviding a detailed analysis of the connections between these methods. The connectionsudfound enable a better understanding of the broader class of discontinuousudspectral/hp element methods.udFrom this perspective it was evident that some of the issues of the discontinuousudGalerkin method are also encountered in the flux reconstruction approach, andudin particular, the aliasing errors of the two schemes are identical. The techniquesudapplied in the more famous discontinuous Galerkin method for tackling these errorsudcan be also extended to the flux reconstruction approach. We present twouddealiasing strategies based on the concept of consistent integration of the nonlinearudterms. The first is a localised approach which targets in each element theudnonlinearities arising in the problem, while the second is a more global approachudwhich involves a higher quadrature of the overall right-hand side of the discretisedudequation(s). Both the strategies have been observed to be effective in enhancingudthe robustness of the schemes considered.udWe finally present the direct numerical simulation of a high-speed subsonicudboundary-layer flow past a three-dimensional roughness element, achieved by meansudof the compressible aerodynamic solver developed. This type of analyses have beenudwidely performed in the past with approximated theories. Only recently, has DNSudbeen used due to the improvement of numerical techniques and an increase inudcomputational resources for similar studies in low-speed subsonic, supersonic andudhypersonic regimes. This thesis takes a first step to close the gap between the resultsudfor a high-speed subsonic regime and the results in supersonic and hypersonicudregimes.