An automated reliable method for two-dimensional Reynolds-Averaged Navier-Stokes simulations

Abstract

The development of computational fluid dynamics algorithms and increased computational resources have led to the ability to perform complex aerodynamic simulations. Obstacles remain which prevent autonomous and reliable simulations at accuracy levels required for engineering. To consider the solution strategy autonomous and reliable, high quality solutions must be provided without user interaction or detailed previous knowledge about the flow to facilitate either adaptation or solver robustness. One such solution strategy is presented for two-dimensional Reynolds-averaged Navier-Stokes (RANS) flows and is based on: a higher-order discontinuous Galerkin finite element method which enables higher accuracy with fewer degrees of freedom than lower-order methods; an output-based error estimation and adaptation scheme which provides quantifiable measure of solution accuracy and autonomously drives toward an improved discretization; a non-linear solver technique based on pseudo-time continuation and line-search update limiting which improves the robustness for solutions to the RANS equations; and a simplex cut-cell mesh generation which autonomously provides higher-order meshes of complex geometries. The simplex cut-cell mesh generation method presented here extends methods previously developed to improve robustness with the goal of RANS simulations. In particular, analysis is performed to expose the impact of small volume ratios between arbitrarily cut elements on linear system conditioning and solution quality. Merging of the small cut element into its larger neighbor is identified as a solution to alleviate the consequences of small volume ratios. For arbitrarily cut elements randomness in the algorithm for generating integration rules is identified as a limiting factor for accuracy and recognition of canonical element shapes are introduced to remove the randomness. The cut-cell method is linked with line-search based update limiting for improved non-linear solver robustness and Riemannian metric based anisotropic adaptation to efficiently resolve anisotropic features with arbitrary orientations in RANS flows. A fixed-fraction marking strategy is employed to redistribute element areas and steps toward meshes which equidistribute elemental errors at a fixed degree of freedom. The benefit of the higher spatial accuracy and the solution efficiency (defined as accuracy per degree of freedom) is exhibited for a wide range of RANS applications including subsonic through supersonic flows. The higher-order discretizations provide more accurate solutions than second-order methods at the same degree of freedom. Furthermore, the cut-cell meshes demonstrate comparable solution efficiency to boundary-conforming meshes while significantly decreasing the burden of mesh generation for a CFD user.