06 Fakultät Luft- und Raumfahrttechnik und Geodäsie
Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/7
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Item Open Access High order discontinuous Galerkin methods for the simulation of multiscale problems(2015) Beck, Andrea; Munz, Claus-Dieter (Prof. Dr.)This work provides a contribution to the accurate, stable and efficient numerical simulation of hydrodynamic 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 occurring scales remains infeasible. Thus, an approximate solution with drastically reduced number of degrees of freedom is sought, which retains the most important characteristics of the full solution. This solution is obtained by solving a truncated multiscale problem, supplemented by a suitable modeling strategy for the omitted scales and their interaction with the truncated solution. This approach is only meaningful if the resolvable scales determine the mean solution features accurately, and if the non-resolved scales show some form of universality behavior, which allows the derivation of meaningful models. Hydrodynamic turbulence is one example of these types of problems. In this work, two frameworks for the numerical solution of the compressible Navier-Stokes equations are presented: A self-developed Fourier pseudo-spectral solver, and a co-developed framework based on the Discontinuous Galerkin Spectral Element Method (DGSEM). Both discretization schemes are highly efficient for the resolution of multiscale problems as they – due to their spectral character – exhibit very low approximation errors over a wide range of scales, and thus return a very high resolution capability per invested degree of freedom. Since DGSEM is based on the variational form of the governing equations, it allows an element-based discretization of the computational domain, which in turn leads to superior parallelization and the possibility for flexible, unstructured meshes. These features make it attractive for the full resolution of turbulence in a Direct Numerical Simulation (DNS) approach and – as demonstrated in this work – highly competitive when compared to other discretization strategies. These favorable discretization properties carry over into the under-resolved situation (Large Eddy Simulation, LES), where a lower-dimensional version of the problem is solved numerically. However, depending on the discretization of the scale-producing mechanism, its truncation can introduce a self-feeding error into the solution, that can lead to a global instability. The source and effects of these aliasing errors are investigate 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 at moderate Reynolds number turbulence. Extension to higher Reynolds numbers necessitates the introduction of some form of closure for the un-resolved scales, due to the increase in the truncation error. Aspects of two modeling approaches are discussed: An implicit modeling strategy for DGSEM can be based on the modification of the dissipation introduced by the inter-cell fluxes. The addition of an explicit modeling term which provides a subgrid dissipation mechanism raises the question whether de-aliasing remains essential in that situation. The de-aliasing strategy is revisited, and its interactions with an explicit closure model are examined. It is shown that only through a proper de-aliasing mechanism, the superior scale-resolving capabilities of the scheme can be recovered, and that a decoupling of explicit model and numerics is imperative. Through these investigations, a consistent strategy for stable and accurate DNS and LES of turbulent flows with high order DGSEM has been established. As an outlook, 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.