Abstract

The design of next-generation aircraft relies on computational fluid dynamics (CFD) to minimize testing requirements at reduced cost and risk. However, current industry reliance on Reynolds-averaged Navier-Stokes (RANS)-based CFD is limited in predicting transitional and turbulent flows. Large-eddy simulation (LES) offers accuracy where RANS methods fail, but can have prohibitive computational cost. To address this, we propose a high-order CFD framework to advance flux reconstruction (FR) methods toward industrial-scale simulations. FR is a family of high-order, unstructured schemes that provide accuracy at reduced cost per degree-of-freedom (DOF) compared to low-order methods, with proven potential for LES. We develop practical strategies to reduce the computational cost of FR methods for explicit and implicit formulations. Due to the low cost per time step, explicit time stepping is typically used in FR methods. However, stability constraints prohibitively limit time-step sizes in numerically stiff problems. Hence, implicit time stepping is preferred in these cases, but it requires solving large, nonlinear systems and can be computationally expensive.
This thesis introduces optimal Runge-Kutta methods to alleviate stability limits and reduce wall-clock times by approximately half in moderately low stiffness problems. For increased stiffness, we hybridize implicit FR methods using a trace variable, which allows a reduction of the implicit system via static condensation, decreasing implicit time stepping costs, especially at higher orders. Hybridization with both discontinuous (HFR) and continuous function spaces (EFR) is suitable for advection and advection-diffusion type problems within the FR method and enables significant speedup gains over standard FR. We incorporate polynomial adaptation to the hybridized framework, varying the solution polynomial’s degree locally within each element, which results in an overall reduction in DOF and significant speedup gains in a two-dimensional problem against standard polynomial-adaptive formulations. Finally, we combine implicit-explicit (IMEX) time stepping with hybridization to tackle geometry-induced numerical stiffness. The resulting method reduces computational cost at least fifteen times over explicit methods in a multi-element airfoil problem at Reynolds 1.7 million. Our proposed framework enables substantial reductions in both moderate and high stiffness problems, thus advancing high-order methods toward large industrial-scale problems.