Lattice Boltzmann model for the correct convection-diffusion equation with divergence-free velocity field

A lattice Boltzmann (LB) model for the convection-diffusion equation (CDE) with divergence-free velocity field is proposed, and the Chapman-Enskog analysis shows that the CDE can be recovered correctly. In the present model, the convection term is treated as a source term in the lattice Boltzmann equation (LBE) rather than being directly recovered by LBE; thus the CDE is intrinsically solved as a pure diffusion equation with a corresponding source term. To avoid the adoption of a nonlocal finite-difference scheme for computing the convection term, a local scheme is developed based on the Chapman-Enskog analysis. Most importantly, by properly specifying the discrete source term in the moment space, the local scheme can reach the same order $\left({\varepsilon }^2\right)$ at which the CDE is recovered by a LB model. Numerical tests, including a one-dimensional periodic problem, diffusion of a Gaussian hill, diffusion of a rectangular pulse, and natural convection in a square cavity, are carried out to verify the present model. Numerical results are satisfactorily consistent with analytical solutions or previous numerical results, and show higher accuracy due to the correct recovery of CDE.

Figure 1

The profiles of the scalar variable $\varphi$ along the $x$ coordinate when $t=T$ for different dimensionless relaxation times $\tau$. The symbols and solid lines represent the numerical results and analytical solutions, respectively.

Figure 2

Comparisons of the relative error $E_{\varphi }$ during $0\le t/T\le 4$ for dimensionless relaxation time (a) $\tau =0.6$, (b) $\tau =0.8$, (c) $\tau =1.2$, and (d) $\tau =2.0$.

Figure 3

The time-averaged values of relative error ${\overline{E}}_{\varphi }$ versus the lattice sizes N for $\tau =0.8$. The additional line inserted in the figure is used to show the second-order convergence rate of the LB models.

Figure 4

Distribution of the scalar variable $\varphi$ along the $x$ coordinate and across the Gaussian hill peak at time $t_m={\sigma }_0^2/\left(2D\right)$ for $N=16$, $\mathrm{Pe}=16$, and $\tau =1.1$.

Figure 5

The relative errors of scalar variable $E_{\varphi }$ versus the lattice sizes N at time $t_m={\sigma }_0^2/\left(2D\right)$ for $\mathrm{Pe}=16$ and $\tau =0.8$.

Figure 6

The relative errors of scalar variable $E_{\varphi }$ versus ${\beta }_1$ at time $t_m={\sigma }_0^2/\left(2D\right)$ for $s_e$ being $10/13$, $1/\tau$, and $50/49$ and the corresponding ${\beta }_{1,\mathrm{opt}}$ [calculated by Eq. (26)] being −0.5, 0, and 0.5, respectively, when $N=16$, $\mathrm{Pe}=16$, and $\tau =1.1$.

Figure 7

The profiles of the scalar variable $\varphi$ along the $x$ coordinate at time $t=T$ for different dimensionless relaxation times $\tau$. The colorful dotted and black solid lines represent the numerical results and analytical solutions, respectively.

Figure 8

Variation of the relative error $E_{\varphi }$ with time during $0\le t/T\le 4$ for different dimensionless relaxation times $\tau$.

Figure 9

The profiles of the scalar variable $\varphi$ along the $x$ coordinate at the beginning of the convection-diffusion process for dimensionless relaxation time (a) $\tau =0.6$, (b) $\tau =0.8$, (c) $\tau =1.2$, and (d) $\tau =2.0$.

Figure 10

Schematic diagram of natural convection in a square cavity with coordinates and boundary conditions shown.

Figure 11

Streamlines (left) and isothermals (right) of natural convection in a square cavity at (a) $\mathrm{Ra}={10}^3$, (b) $\mathrm{Ra}={10}^4$, (c) $\mathrm{Ra}={10}^5$, and (d) $\mathrm{Ra}={10}^6$.