Robust numerical formulations for compressible flows

Abstract

The current BEPE research project aims to investigate, develop and improve the formulation of high-order schemes based on the Flux Reconstruction / Correction Procedure via Reconstruction (FR/CPR) framework in order to accurately capture discontinuous solutions of the gas dynamics equations. High-order schemes, that is, third-order spatial accuracy or higher, can have numerous advantages when attempting to simulate natural phenomena that require a high spatial resolution. Moreover, they can be designed to have very well controlled amounts of artificial dissipation, which makes them particularly suited for handling problems in which different energy scales must be correctly represented. The use of high-order schemes for the solution of general fluid dynamic problems is gaining increasing popularity, given their potential to significantly reduce the number of discrete elements required to achieve a certain error threshold. However, when dealing with discontinuous solutions, such as shock waves, the formulation of this type of scheme requires special treatment in order to avoid problems of numerical nature. The solutions available in the literature today are far from ideal and are subjected to the introduction of further drawbacks to the overall scheme. Hence, the current work has three main objectives. The first one is to study and document the currently available formulations for dealing with this type of numerical problem. The second is to improve upon the existing formulations by proposing a new approach for the correct solution of shock wave structures when utilizing high-order schemes created under the FR/CPR framework. Lastly, the proposed formulation should be implemented in a computer code and tested against realistic aerospace engineering applications. Conclusions and main contributions arising from this research project will be documented and published in relevant scientific events and journals. (AU)