First- and second-order forcing expansions in a lattice Boltzmann method reproducing isothermal hydrodynamics in artificial compressibility form

The isothermal Navier–Stokes equations are determined by the leading three velocity moments of the lattice Boltzmann method (LBM). Necessary conditions establishing the hydrodynamic consistency of these moments are provided by multiscale asymptotic techniques, such as the second-order Chapman–Enskog expansion. However, for simulating incompressible hydrodynamics the structure of the forcing term in the LBM is still a discordant issue as far as its correct velocity expansion order is concerned. This work uses the traditional second-order Chapman–Enskog expansion analysis to demonstrate that the truncation order of the forcing term may depend on the time regime in this case. This is due to the fact that LBM does not reproduce exactly the incompressibility condition. It rather approximates it through a weakly compressible or an artificial compressible system. The present study shows that for the artificial compressible setup, as the incompressibility flow condition is singularly perturbed by the time variable, such a connection will also affect the LBM forcing formulation. As a result, for time-independent incompressible flows the LBM forcing must be truncated to first order whereas for a time-dependent case it is convenient to include the second-order term. The theoretical findings are confirmed by numerical tests carried out in several distinct benchmark flows driven by space- and/or time-varying body forces and possessing known analytical solutions. These results are verified for the single relaxation time, the multiple relaxation time and the regularized collision models.