The Development, Verification, and Validation of a Discontinuous Galerkin Method for the Navier-Stokes Equations

Abstract

The field of computational fluid dynamics is in many ways considered mature, in that CFD tools are regularly used in all fields of engineering analysis and design where they complement experiment and theory. As computing power has increased and algorithmic approaches have improved, the adoption of simulation tools have also increased. Traditional CFD methods have been applied to ever-broader problems at ever-greater resolution with generally good success, but with some notable weaknesses that are likely to be overcome only through further advances in efficient high-fidelity algorithms. While now-traditional second-order methods have been generally successful for simulating smooth attached flows, the simulation of vorticity- and turbulence-dominated flows still present many unmet challenges. As computing power increases with modern computer architectures composed of thousands of multicore processors and GPUs, there remains the need to develop further high-fidelity algorithms suitable for the simulation of high Reynolds number flows on the current generation of heterogeneous computing systems. This thesis describes the formulation, development, verification, and validation of a discontinuous Galerkin method for approximate solutions of the Navier-Stokes equations. The DG method holds promise, but its adoption in practice has been hindered by many issues such as lack of robustness and computational efficiency, as well as questionable evidence of superiority for purpose compared to second-order schemes. This work addresses some key elements that can enable the transition of high-order methods from basic research to a useful computational tool for the analysis of complex aerodynamic flows of importance. The subject ranges from the study of numerical properties of DG methods to their efficient implementation in software. The algorithmic portions of this work focus on the use of implicit time integration with black-box algebraic solvers. The verification study of this approach in subsequent portions suggests this is a viable method for the stable and efficient solution of high fidelity DG discretizations at large computational scale. The software portions of this work focus on the use of modern software development approaches for the reliable and efficient implementation of the underlying numerical methods. As CFD methodology becomes more complex, the challenges of software development grows. Verification of the approaches is performed using test cases ranging from simple scalar transport equations up to direct numerical simulation and large eddy simulations of the compressible Navier-Stokes equations on complex geometries. An implicit large eddy simulation over tandem spheres is performed at higher resolution than published elsewhere to date, as well as at Reynolds numbers not seen elsewhere. Following this is an implicit LES analysis over a high-lift aircraft configuration, a challenging problem meant to push the limits of existing high-fidelity CFD methods.