A φ-divergence based finite element moment method for the polyatomic ES-BGK Boltzmann equation

Abstract

Bonafide numerical simulations of transport phenomena in rarefied gases are becoming increasingly important in science and engineering. To enhance the accuracy of the rarefied-gas model, it is important to account for the internal structure of the gas molecules and exceed the standard monatomic description. A suitable model for polyatomic rarefied-gas flows that balances between accuracy and complexity is the polyatomic ES-BGK Boltzmann equation. In this work we consider numerical approximation of the ES-BGK Boltzmann equation based on a Method of Moments (MoM) approximation in the velocity/internal-energy dependence, and DGFEM approximation in the spatial dependence. In the MoM setting, only two moments have to be considered in internal-energy dependence to capture the correct macroscopic gas properties, viz. the thermal conductivity, the adiabatic coefficient and the shear and dilational viscosity. In the presentation of the method, we focus specifically on the evaluation of the integrals in the velocity and intrinsic-variable dependence appearing in the MoM formulation and on linearization of the formulation. We present numerical results for the heat flux between two parallel heated plates, mass flow in a micro-channel, a poly-atomic lid-driven cavity problem, and rarefied-gas flow around a sphere. The computed results are compared with corresponding theoretical and experimental results. The results support the suitability of the FEM-MoM approximation of the ES-BGK Boltzmann equations as a model for predicting transport phenomena in rarefied-gas flows.