Robustness of High-Order Divergence-Free Finite Element Methods for Incompressible Computational Fluid Dynamics
von Philipp W. Schroeder
Datum der mündl. Prüfung:2019-03-01
Erschienen:2019-03-07
Betreuer:Prof. Dr. Gert Lube
Gutachter:Prof. Dr. Gert Lube
Gutachter:Prof. Dr. Andreas Dillmann
Gutachter:Prof. Dr. Leo G. Rebholz
Dateien
Name:Diss-PWS-2019-SUB.pdf
Size:35.1Mb
Format:PDF
Zusammenfassung
Englisch
In computational fluid dynamics, obtaining exactly divergence-free approximations to the incompressible Navier-Stokes equations, by means of finite element methods, has actually not been particularly popular in the last decade. This observation is in contrast to the fact that H(div)-conforming finite elements indeed facilitate the flexible construction of such methods in most diverse applications. In this context, from the methodical side, Discontinuous Galerkin (DG) methods play a key role and, from the computational point of view, the concept of hybridisation can and is exploited heavily. The present work demonstrates and explains why exactly divergence-free H(div) methods, especially in under-resolved simulations, show an excellent performance in several laminar and turbulent test scenarios. For convection-dominated problems, the use of upwinding, which is naturally incorporated into DG methods, is evaluated and assessed. Furthermore, a careful investigation of various numerical examples is provided; this includes, for example, a Kelvin-Helmholtz instability problem, 2D and 3D freely decaying turbulence and turbulent channel flows. Especially, it is shown that H(div) methods provide a framework for the robust simulation of turbulent flows for basically any Reynolds number. From a theoretical perspective, it is shown that exactly divergence-free methods allow the transfer of many crucial fluid dynamics properties directly to the discrete level. In fact, they allow for a comparably straightforward numerical error analysis as well, and it turns out that this success is strongly related to the concepts of pressure- and Reynolds-semi-robustness. One important consequence of pressure-robustness is that the accuracy of the resulting velocity approximation is completely independent of the quality of the pressure approximation. Finally, the role of high-order methods is investigated which shows that in the considered examples, using a moderate order promises to deliver a good compromise between accuracy and efficiency.
Keywords: computational fluid dynamics; incompressible Navier-Stokes equations; exactly divergence-free methods; H(div)-DG and HDG methods; structure preservation; Helmholtz decomposition; pressure- and Reynolds-semi-robustness; laminar and turbulent flows; Taylor-Green vortex; turbulent channel flow