Lattice-Boltzmann-Methoden zur Simulation inkompressibler Wirbelströmungen

The present work develops novel methodological extensions to the lattice Boltzmann method (LBM). These extensions enable the method to efficiently simulate incompressible vortical flows. They cure two major drawbacks of the standard LBM: its instability in under-resolved turbulence and its restriction to regular computational grids. At first, a pseudo-entropic stabilizer (PES) is developed, which combines ideas from multiple-relaxation-time (MRT) models and entropic models. The new PES is local, explicit, and flexible. It modifies the collision step in a way that enables stable simulations and produces qualitatively matching results, even on strongly underresolved grids. To extend the LBM towards simulations on irregular grids, a recent discontinuous Galerkin lattice Boltzmann method is studied and enhanced by more stable time integrators. This study illustrates the severe shortcomings of existing off-lattice Boltzmann methods (OLBMs). Based on these findings, the present work succeeds in developing a semi-Lagrangian lattice Boltzmann method (SLLBM). This novel approach allows unstructured grids, large time steps, and high-order accurate representations of the solution to be used in a unique way. Applications to exemplary flows demonstrate how and why the new method outperforms other recent OLBMs in both efficiency and accuracy. Additionally, this work describes the development of a modular off-lattice Boltzmann code and shows that the method’s convergence order can be increased by implicit multistep methods.