| dc.description.abstract | High-order discontinuous spectral element methods provide a potential direction for drastically reducing the computational cost of simulating complex unsteady fluid flows due to their high accuracy, geometric flexibility, and computationally efficient data structure. However, for certain application areas such as high Reynolds number flows, supersonic and hypersonic aeronautics, and rarefied gases, these methods can either become computationally intractable, suffer from numerical robustness problems, or are not well established as a numerical tool. This work presents various algorithmic developments for the use of discontinuous spectral element methods in the simulation of problems in continuum fluid mechanics and molecular gas dynamics.
First, as a mechanism for reducing the computational cost of simulating high Reynolds number flows, hybrid turbulence modeling within a high-order discontinuous spectral element framework was explored in the context of the Partially-averaged Navier–Stokes equations. It was observed that larger improvements were generally seen when increasing the discretization accuracy of the Partially-averaged Navier–Stokes method in comparison to methods without models. Furthermore, less sensitivity to the resolution-control parameter was observed with high-order discretizations.
Then, to allow for the use of discontinuous spectral element methods for flows in the supersonic and hypersonic regimes, a novel adaptive filtering approach was introduced as a shock capturing method by formulating convex invariants such as positivity of density and pressure and a local minimum entropy principle as constraints on the solution. The result of this approach is a provably robust, parameter-free method for resolving strong discontinuities that can be applied on general unstructured meshes with relatively low computational cost.
Finally, extensions to non-equilibrium and rarefied flow regimes were then performed through approximations of the polyatomic Boltzmann equation for molecular gas dynamics augmented with the Bhatnagar–Gross–Krook collision model. Through the combination of a positivity-preserving limiter and a discrete velocity model, the method guarantees discrete conservation and positivity of the macroscopic density and pressure. The approach was validated on experiments ranging from shock-dominated flows to direct numerical simulation of three-dimensional compressible turbulent flows, the latter of which is the first instance of such a flow computed by directly solving the Boltzmann equation. | |
