Analysis of the lattice Boltzmann Bhatnagar-Gross-Krook no-slip boundary condition: Ways to improve accuracy and stability

An analytical and numerical analysis of the no-slip boundary condition at walls at rest for the lattice Boltzmann Bhatnagar-Gross-Krook method is performed. The main result of this analysis is an alternative formulation for the no-slip boundary condition at walls at rest. Numerical experiments assess the accuracy and stability of this formulation for Poiseuille and Womersley flows, flow over a backward facing step, and unsteady flow around a square cylinder. This no-slip boundary condition is compared analytically and numerically to the boundary conditions of Inamuro [Phys. Fluids 7, 2928 (1995)] and Zou and He [Phys. Fluids 9, 1591 (1997)] and it is found that all three make use of the same mechanism for the off-diagonal element of the stress tensor. Mass conservation, however, is only assured by the present one. In addition, our analysis points out which mechanism lies behind the instabilities also observed by Laumltt [Phys. Rev. E 77, 056703 (2008)] for this kind of boundary conditions. We present a way to remove these instabilities, allowing one to reach relaxation frequencies considerably closer to 2.