A systematic study of hidden errors in the bounce-back scheme and their various effects in the lattice Boltzmann simulation of viscous flows

Abstract

Bounce-back schemes represent the most popular boundary treatments in the lattice Boltzmann method (LBM) when reproducing the no-slip condition at a solid boundary. While the lattice Boltzmann equation used in LBM for interior nodes is known to reproduce the Navier–Stokes (N–S) equations under the Chapman–Enskog (CE) approximation, the unknown distribution functions reconstructed from a bounce-back scheme at boundary nodes may not be consistent with the CE approximation. This problem could lead to undesirable effects such as nonphysical slip velocity, grid-scale velocity, pressure noises, the local inconsistency with the N–S equations, and sometimes even a reduction of the overall numerical-accuracy order of LBM. Here, we provide a systematic study of these undesirable effects. We first derive the explicit structure of the mesoscopic distribution function for interior nodes. Then, the bounce-back distribution function is examined to identify the hidden errors. It is shown that the relaxation parameters in the collision models play a key role in determining the magnitude of the hidden error terms, and there exists an optimal setting, which can suppress or eliminate most of these undesirable effects. While the existence of this optimal setting is derived previously for unidirectional flows, here, we show that this optimal setting can be extended to non-uniform flows under certain conditions. Finally, a systematic numerical benchmark study is carried out, including non-uniform and unsteady flows. It is shown that, in all these flows, our theoretical analyses of the hidden errors can guide us to significantly improve the quality of the simulation results.