Three-dimensional discrete-velocity BGK model for the incompressible Navier–Stokes equations

The lattice Boltzmann method (LBM) has been widely used for the simulations of the incompressible Navier–Stokes (NS) equations. The finite difference Boltzmann method (FDBM) in which the discrete-velocity Boltzmann equation is solved instead of the lattice Boltzmann equation has also been applied as an alternative method for simulating the incompressible flows. The particle velocities of the FDBM can be selected independently from the lattice configuration. In this paper, taking account of this advantage, we present the discrete velocity Boltzmann equation that has a minimum set of the particle velocities with the lattice Bharnagar–Gross–Krook (BGK) model for the three-dimensional incompressible NS equations. To recover incompressible NS equations, tensors of the particle velocities have to be isotropic up to the fifth rank. Thus, we propose to apply the icosahedral vectors that have 13 degrees of freedom to the particle velocity distributions. Validity of the proposed model (D3Q13BGK) is confirmed by numerical simulations of the shear-wave decay problem and the Taylor–Green vortex problem. With respect to numerical accuracy, computational efficiency and numerical stability, we compare the proposed model with the conventional lattice BGK models (D3Q15, D3Q19 and D3Q27) and the multiple-relaxation-time (MRT) model (D3Q13MRT) that has the same degrees of freedom as our proposal. The comparisons show that the compressibility error of the proposed model is approximately double that of the conventional lattice BGK models, but the computational efficiency of the proposed model is superior to that of the others. The linear stability of the proposed model is also superior to that of the lattice BGK models. However, in non-linear simulations, the proposed model tends to be less stable than the others.