Comparison of the Shakhov and ellipsoidal models for the Boltzmann equation and DSMC for ab initio-based particle interactions

Abstract

In this paper, we consider the capabilities of the Boltzmann equation with the Shakhov and ellipsoidal models for the collision term to capture the characteristics of rarefied gas flows. The benchmark is performed by comparing the results obtained using these kinetic model equations with direct simulation Monte Carlo (DSMC) results for particles interacting via ab initio potentials. The analysis is restricted to channel flows between parallel plates and we consider three flow problems, namely: the heat transfer between stationary plates, the Couette flow and the heat transfer under shear. The simulations are performed in the non-linear regime for the 3He, 4He, and Ne gases. The reference temperature ranges between 1 K and 3000 K for 3He and 4He and between 20 K and 5000 K for Ne. While good agreement is seen up to the transition regime for the direct phenomena (shear stress, heat flux driven by temperature gradient), the relative errors in the cross phenomena (heat flux perpendicular to the temperature gradient) exceed 10% even in the slip-flow regime. The kinetic model equations are solved using the finite difference lattice Boltzmann algorithm based on half-range Gauss-Hermite quadratures with the third order upwind method used for the implementation of the advection.