Discrete Velocity Models for Mixtures and Non-Mixtures

In the present thesis we explore the application of discrete velocity models (DVM) to simulate gas mixtures. We provide a quick introduction to well-known theoretic results of both the continuous Boltzmann equation as well as DVM for mixtures. Moreover, we derive closure terms for the Navier-Stokes equation for mixtures from the continuous Boltzmann equation. We present efficient algorithms to set up DVM for mixtures with either two or three dimensional velocity spaces that are much faster than reasonable naive implementations. This is particularly useful for mixtures where we, depending on the mass ratio, often require a new DVM. We also explore how large the discrete velocity spaces should be made. For mixtures, we show that, besides supernormality, we must also achieve a certain balance for the momentum and energy transfer between the species. If the chosen velocity grids are too small, then there may be too few energy transferring collisions. We show in great detail that this can severely distort relaxation processes. Moreover, we investigate for which distributions a given velocity grid is suited and to what degree. For this, we develop a moment-based metric that enables us to automatically determine an optimal parameter range of both temperature and mean velocity for a given discrete velocity grid. Furthermore, we develop a generic method to automatically determine the collision weights that is suitable for DVM of all sizes. It allows sql-like grouping operations of collisions and, based on that, collision weight adjustments. In this thesis we use it to balance the intra- and interspecies collisions or mixtures or to balance the momentum and energy transfer between different species. Finally, we use such an approach to automatically remove discretization effects on the viscosity or Prandtl number.