Lattice kinetic scheme for the Navier-Stokes equations coupled with convection-diffusion equations

In this paper, we target two basic issues residing in some modified lattice kinetic schemes for the Navier-Stokes (NS) equations coupled with convection-diffusion equations (CDEs). First, a lattice Boltzmann (LB) model motivated by the lattice kinetic scheme (LKS) is presented for the NS equations coupled with CDEs. Due to the nonequilibrium schemes for the gradient terms contained in the equilibria as well as the discrete source term, the collision process of the present model can be implemented locally in both time and space. The Chapman-Enskog analysis shows that the macroscopic equations can be correctly recovered from the present model without additional assumptions. Second, we prove that the present modified LKS model, though written in the Bhatnagar-Gross-Krook (BGK) form, has two relaxation rates essentially. Based on this theoretical result, the modified lattice kinetic schemes in the literature should not be grouped as the BGK model, and the better numerical stability is intrinsically attributed to the adjustment of their two relaxation rates. Several benchmark thermal flow problems are simulated to validate the present model and the local nonequilibrium schemes for the shear rate and temperature gradient. The accuracy of the present model as well as its better numerical stability compared with the BGK model are verified, which supports our theoretical results. In addition, we also demonstrate that the regularized LB (RLB) model has two relaxation times as the present LKS model.

Figure 1

Schematic of the two-dimensional thermal Couette flow.

Figure 2

Comparisons of numerical results [(a) velocity; (b) temperature; (c) velocity gradient; and (d) temperature gradient] and analytical solutions of the thermal Couette flow for different Ec at $\text{Pr}=0.71$. Solid lines: analytical solutions; symbols: numerical results.

Figure 3

Comparisons of numerical results [(a) velocity; (b) temperature; (c) velocity gradient and (d) temperature gradient] and analytical solutions of the thermal Poiseuille flow for different Ec at $\text{Pr}=0.71$ and $\text{Re}=20$. Solid lines: analytical solutions; symbols: numerical results.

Figure 4

The relative differences between errors of present model and (a) the previous model [32]; (b) the previous model [33].

Figure 5

Schematic of natural convection in a fluid-saturated porous cavity.

Figure 6

Isotherms of the natural convection in a cavity. The results are obtained by (a) the standard BGK model at $t=200{\delta }_t$, and (b) the present model at $t=2000{\delta }_t$.