Analytic solutions of the variable force effect in lattice Boltzmann methods for Poiseuille flows

Various external forcing formulations of the lattice Boltzmann method (LBM) are analyzed by deriving the analytic solutions of the fully developed Poiseuille flows with and without the porous wall. For uniform driving force, all the forcing formulations recover the second-order accurate discretized Navier–Stokes equation. However, the analytic solutions show that extra force gradients arise due to variable force, and this form differs from the analysis using Chapman–Enskog expansion. It is possible to remove these extra terms of single relaxation time (SRT) LBM using specific relaxation time depending on the force formulation adopted. However, this limits the broader applicability of the SRT LBM. Moreover, the multiple-relaxation-time LBM may provide an option to remove the variable-force gradient term benefiting from separating relaxation parameters for each moment.