# Derivation and numerical solution of hyperbolic moment equations for rarefied gas flows

- Herleitung und numerische Lösung hyperbolischer Momentengleichungen für Strömungen verdünnter Gase

Köllermeier, Julian; Torrilhon, Manuel (Thesis advisor); Li, Ruo (Thesis advisor)

*Aachen (2017)*

Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2017

Abstract

Hyperbolic moment equations for the simulation of rarefied gas flows are derived and solved using appropriate numerical schemes in this thesis.The numerical simulation of rarefied gases requires the solution of the Boltzmann equation as standard models like the Euler equations are invalid in the rarefied regime, which is characterized by moderate to large Knudsen numbers. An accurate direct discretization and solution of the Boltzmann equation needs many variables due to the high dimension of the phase space and is thus computationally expensive. It is the aim of this thesis to investigate efficient yet accurate moment models for the solution of the Boltzmann equation. The distribution function of the Boltzmann equation is expanded in velocity space resulting in a system of equations for the expansion coefficients, which is called moment system and can be seen as an extension of the Euler equations.However, standard approaches yield unstable solutions due to the lack of hyperbolicity, which has been a major drawback of moment models. The main contribution of this thesis is the derivation of new hyperbolic moment models and the explanation from different viewpoints that allow for a better understanding of the new models, called Quadrature-Based Moment Equations. The analytical properties of the new models are investigated, hyperbolicity is proven and explicit equations are derived.After the detailed one-dimensional derivation, an extension to the multi-dimensional case is performed in order to obtain equations for the simulation of more physically relevant test cases including a rotationally invariant version of the Quadrature-Based Moment Equations.Each hyperbolic moment model contains at least one equation that is only given in non-conservative form, which renders the whole system of equations a partially-conservative moment system. We study appropriate numerical schemes for the computation of the non-conservative products and develop an extension to a high resolution scheme on two-dimensional unstructured grids.The numerical schemes as well as the hyperbolic model equations are successfully used in various test cases that demonstrate a robust solution by the numerical schemes and accurate results of the new moment models. The new hyperbolic moment equations yield better accuracy than existing models and have a larger range of applicability due to their global hyperbolicity as shown by one-dimensional as well as two-dimensional numerical simulations.

### Identifier

- DOI: 10.18154/RWTH-2017-07475
- RWTH PUBLICATIONS: RWTH-2017-07475