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978-3-8439-2157-2, Reihe Ingenieurwissenschaften
High Order Discontinuous Galerkin Methods for the Simulation of Multiscale Problems
225 Seiten, Dissertation Universität Stuttgart (2015), Hardcover, A5
This work provides a contribution to the accurate, stable and efficient numerical simulation of non-linear multiscale problems with high order discretizations. Due to their wide range of spatial and temporal scale, these types of problems demand not only highly accurate and efficient numerical discretization schemes, but also careful code design with regards to supercomputing architectures. Still, as a rule, even for the most sophisticated algorithms and hardware, a full resolution of all scales remains infeasible.
Hydrodynamic turbulence is one example of these problems. In this work, a framework for the numerical solution of the compressible Navier-Stokes based on the Discontinuous Galerkin Spectral Element Method (DGSEM) is presented. This discretization scheme is highly efficient for the resolution of multiscale problems as it exhibits very low approximation errors over a wide range of scales. Since DGSEM supports an element-based discretization with a small communication footprint, it allows superior parallelization and the use of unstructured meshes. These features make it attractive for the full resolution of turbulence in a Direct Numerical Simulation (DNS) approach and highly competitive when compared to other discretization strategies.
These favorable discretization properties carry over into the under-resolved situation (Large Eddy Simulation, LES). However, depending on the discretization of the scale-producing mechanism, a self-feeding error can be introduced. The source and effects of these aliasing errors are investigated in this work. Strategies for countering or avoiding it are presented, and the code framework is extended accordingly. These strategies are compared and evaluated, showing that only the exact quadrature of the non-linear terms recovers the favorable approximation properties and thus the efficiency of the spectral approach. With this discretization strategy, it is shown that high order DGSEM can outperform established, lower-order LES formulations in terms of accuracy per invested degree of freedom for challenging test cases.
Through these investigations, a consistent strategy for stable and accurate DNS and LES of turbulent flows with high order DGSEM has been established. Further research strategies into LES modeling should take full advantage of the spectral character of DGSEM, and the associated scale range resolved within each element can be exploited in both an implicit as well as explicit closure approach.