Accuracy of higher-order lattice Boltzmann methods for microscale flows with finite Knudsen numbers

Abstract

Accuracy of the lattice Boltzmann (LB) method for microscale flows with finite Knudsen numbers is investigated. We employ up to the eleventh-order Gauss–Hermite quadrature for the lattice velocities and diffuse-scattering boundary condition for fluid–wall interactions. Detailed comparisons with the direct simulation Monte Carlo (DSMC) method and the linearized Boltzmann equation are made for planar Couette and Poiseuille flows. All higher-order LB methods considered here give improved results as compared with the standard LB method. The accuracy of the LB hierarchy, however, does not monotonically increase with the order of the Gauss–Hermite quadrature at moderate and large Knudsen numbers. The results also show the sensitivity to a quadrature chosen, even when the Gauss–Hermite quadratures have the same order of formal accuracy. Among the schemes investigated here, D2Q16 is the most efficient method and offers a quantitative prediction in the slip and transition regimes. The higher-order LB methods predict the Knudsen layer up to Kn = O ( 0.1 ) . The Knudsen layer, however, rapidly disappears when the Knudsen number approaches unity due to a finite number of the lattice velocities, while it is still present for Kn = O ( 1 ) in the Boltzmann equation. It is also found that the higher-order LB methods adopted here do not capture the asymptotic behavior of the Boltzmann equation at large Knudsen numbers.