A comparative study of the axisymmetric lattice Boltzmann models under the incompressible limit

A comparative study on four axisymmetric lattice Boltzmann (LB) models, namely, the kinetic theory based model by Guo et al. (2009), the consistent model by Li et al. (2010), the centered scheme model by Zhou (2011), and our model (based on applying the centered scheme to the Guo et al. (2009) model), is conducted both theoretically and numerically. The finite difference interpretation of the LB method by Junk (2001) is applied to evaluate the accuracy of the models under the incompressible limit. Particularly, the finite difference stencils adopted for the spatial gradient terms in the macroscopic axisymmetric Navier-Stokes (N-S) equations are compared. Besides, the numerical performance (i.e., the numerical accuracy, stability and the convergence efficiency) of the models is compared by two benchmark tests, namely, the unsteady-state Womersley flow and the cylindrical cavity flow. The numerical results accord well with the theoretical analysis. Additionally, it is also found that the numerical stability of the axisymmetric LB models is effectively improved by removing the effects from the non-hydrodynamic variables.