Second-order convergence of the deviatoric stress tensor in the standard Bhatnagar-Gross-Krook lattice Boltzmann method

It is shown numerically that the deviatoric stress tensor is second-order accurate in the bulk Bhatnagar-Gross-Krook lattice Boltzmann (LB) method. In an earlier work [T. Krüger et al., Phys. Rev. E 79, 46704 (2009)], we have already predicted the second-order convergence. However, numerical simulations using a duct flow were not fully in line with this prediction. In particular, the convergence rate of the stress tensor was observed to depend on the LB boundary condition. In the present paper, we examine a pure bulk system, the decaying Taylor-Green vortex flow. Our prediction on the second-order accuracy of the stress tensor is unambiguously evidenced via these studies.

Figure 1

(a) The error of ${\sigma }_{xy}$ is shown for different inclination angles $\theta$. The errors of ${\sigma }_{xx}$ are nearly identical and not presented. (b) and (c) The errors of ${\sigma }_{xx}$ and ${\sigma }_{xy}$ are shown for different relaxation parameters $\tau$. The second-order slope is also provided as a guide for the eyes.