Generalized modification in the lattice Bhatnagar-Gross-Krook model for incompressible Navier-Stokes equations and convection-diffusion equations

In this paper, two modified lattice Boltzmann Bhatnagar-Gross-Krook (LBGK) models for incompressible Navier-Stokes equations and convection-diffusion equations are proposed via the addition of correction terms in the evolution equations. Utilizing this modification, the value of the dimensionless relaxation time in the LBGK model can be kept in a proper range, and thus the stability of the LBGK model can be improved. Although some gradient operators are included in the correction terms, they can be computed efficiently using local computational schemes such that the present LBGK models still retain the intrinsic parallelism characteristic of the lattice Boltzmann method. Numerical studies of the steady Poiseuille flow and unsteady Womersley flow show that the modified LBGK model has a second-order convergence rate in space, and the compressibility effect in the common LBGK model can be eliminated. In addition, to test the stability of the present models, we also performed some simulations of the natural convection in a square cavity, and we found that the results agree well with those reported in the previous work, even at a very high Rayleigh number (Ra = ${10}^{12})$.

Figure 1

Comparisons between numerical results and analytical solutions of velocities at different Re numbers: (a) Re = 10, $A$ = 0.02; (b) Re = 15, $A$ = 0.04; (c) Re = 20, $A$ = 0.06; (d) Re = 25, $A$ = 0.08. Solid line: analytical solutions; circle: numerical solutions.

Figure 2

GRE at different lattice spacings and Re numbers: (a) nonmodified LBGK model, (b) modified LBGK model.

Figure 3

GRE at different values of $A$.

Figure 4

Comparisons between numerical results and analytical solutions at different time and Womersley numbers: (a) $\alpha =2.99$, (b) $\alpha =4.22$, (c) $\alpha =7.49$, (d) $\alpha =11.85$. Solid line: analytical solutions; symbols: numerical solutions.

Figure 5

GRE obtained by the nonmodified LBGK model at different lattice spacings and Womersley numbers: (a) $t=T$, (b) $t=T/4$.

Figure 6

GRE obtained by the modified LBGK model at different lattice spacings and Womersley numbers: (a) $t=T$, (b) $t=T/4$.

Figure 7

Schematic diagram of the natural convection.

Figure 8

Streamlines of the natural convection: (a) Ra = ${10}^3$, (b) Ra = ${10}^4$, (c) Ra = ${10}^5$, (d) Ra = ${10}^6$.

Figure 9

Isotherms of the natural convection: (a) Ra = ${10}^3$, (b) Ra = ${10}^4$, (c) Ra = ${10}^5$, (d) Ra = ${10}^6$.

Figure 10

Temperature profiles at the midheight of the cavity from Ra = ${10}^3$ to ${10}^6$.

Figure 11

The instantaneous streamlines of the natural convection: (a) Ra = ${10}^7$, (b) Ra = ${10}^8$, (c) Ra = ${10}^9$, (d) Ra = ${10}^{10}$, (e) Ra = ${10}^{11}$, (f) Ra = ${10}^{12}$.

Figure 12

The instantaneous isotherms of the natural convection: (a) Ra = ${10}^7$, (b) Ra = ${10}^8$, (c) Ra = ${10}^9$, (d) Ra = ${10}^{10}$, (e) Ra = ${10}^{11}$, (f) Ra = ${10}^{12}$.

Figure 13

Temperature profiles (time-averaged) at the midheight of the cavity from Ra = ${10}^7$ to ${10}^{12}$: (a) the entire cavity, (b) enlarged view near the left wall.

Figure 14

A comparison between present and previous works on temperature profiles (time-averaged) at the midheight of the cavity: (a) Ra = ${10}^7$–${10}^8$, (b) Ra = ${10}^9$–${10}^{10}$.

Figure 15

Time-averaged Nu along the hot wall: (a) Ra = ${10}^7$–${10}^9$, (b) Ra = ${10}^{10}$–${10}^{12}$.

Figure 16

The variation of Nu${}_{\text{av}}$ as a function of time: (a) Ra = ${10}^{11}$, (b) Ra = ${10}^{12}$.

Figure 17

(a) The variation of Nu${}_{\text{av}}$ as a function of time at different Ra numbers; (b) the variation of $u_{max}$ as a function of time at different Ra numbers.

Figure 18

The variation of Nu${}_{\text{av}}$ as a function of Ra.