Effects of the hidden errors in the bounce back scheme on the lattice Boltzmann simulation of the turbulent channel flow using the multiple-relaxation-time collision model

Abstract

In our recent paper [Dong et al., Phys. Fluids 34, 093608 (2022)], it is shown that hidden errors can be introduced by a bounce back scheme at the boundary nodes, due to the fact that it may not be entirely consistent with the Chapman–Enskog approximation of the lattice Boltzmann equation applied to the interior nodes. In this paper, we investigate the effects of these hidden errors on the lattice Boltzmann simulation of the turbulent channel flow with a multiple-relaxation-time (MRT) collision model, extending our previous study using the Bhatnagar–Gross–Krook and two-relaxation-time collision models applied to laminar non-uniform viscous flows. A theoretical framework for identifying the hidden errors in the MRT model is developed, and the hidden errors in two bounce back schemes, namely, the off-wall and on-wall bounce back schemes, are derived in terms of the hydrodynamic variables and relaxation rates. The results reveal several important differences in the expression of hidden errors between the two bounce back schemes. The analysis also points to a correction for the on-wall bounce back scheme when the external force is present. A set of six simulations of the turbulent channel flow, using the two bounce back schemes and three grid resolutions, are, then, performed to demonstrate that the magnitude of the hidden errors can significantly affect the simulated turbulence statistics, the local consistency with the Navier–Stokes equations, and the numerical stability.