Truncation errors and the rotational invariance of three-dimensional lattice models in the lattice Boltzmann method

Abstract

The application of the lattice Boltzmann method (LBM) in three-dimensional isothermal hydrodynamic problems often adopts one of the following models: D3Q15, D3Q19, or D3Q27. Although all of them retrieve consistent Navier–Stokes dynamics in the continuum limit, they are expected to behave differently at discrete level. The present work addresses this issue by performing a LBM truncation error analysis. As a conclusion, it is theoretically demonstrated that differences among the aforementioned cubic lattices lie in the structure of their non-linear truncation errors. While reduced lattice schemes, such as D3Q15 and D3Q19, introduce spurious angular dependencies through non-linear truncation errors, the complete three-dimensional cubic lattice D3Q27 is absent from such features. This result justifies the superiority of the D3Q27 lattice scheme to cope with the rotational invariance principle in three-dimensional isothermal hydrodynamic problems, particularly when convection is not negligible. Such a theoretical conclusion also finds support in numerical tests presented in this work: a Poiseuille duct flow and a weakly-rotating duct flow.