Temperature-scaled collision process for the high-order lattice Boltzmann model

We postulate that the relaxations of the distribution function in the lattice Boltzmann model should be self-similar under temperature scaling. Based on this postulation, a multiple-relaxation-time collision model in the relative, temperature-scaled reference frame is devised with Hermite expansion. Resorting to the relation between the Hermite basis with the temperature-scaled relative velocity and the Hermite basis with the raw velocity, the relaxations in the temperature-scaled reference frame can be converted to those in the raw reference frame with some correction terms to eliminate the cross-talk effects among the relaxations of different orders. The highest-order nonequilibrium relative central moment is filtered due to the insufficient discrerization in the velocity space. The highest-order collision term can be recursively obtained from the lower-order collision terms. The improved performance is validated by the double shear layer flow, shock tube flow, and the Taylor-Green vortex flow.

Figure 1

The diagrammatic sketch for the evaluations of the highest-order nonequilibrium moment for different models.

Figure 2

The vorticity contours at $t=T$ for the double shear layer. The LBGK model is applied in the simulation with resolution $4096\times 4096$. The Mach number is 0.35 and the Reynolds number is $30000$.

Figure 3

Double shear layer at Mach number is 0.35 and Reynolds number is $30000$ for the D2V37 lattice structure. From top to bottom: (a) mean kinetic energy, (b) standard deviation of the kinetic energy, (c) mean enstrophy, and (d) standard deviation of the enstrophy in dimensionless form. The reference results are obtained from the LBGK model resolved by $4096\times 4096$ resolution.

Figure 4

Stability boundaries of the present and PR3rd models in thermal double-shear-layer simulations. The resolution is $128\times 128$, $\mathrm{Pr}=1,$ and 2/3.

Figure 5

Shock wave: Pressure ratio is 4, temperature ratio is 1, the resolution is $L=2000$, the viscosity relaxation time is ${\tau }_2=0.6$. The results of the present model and the existing models, including PR4th, PR3th, and Mattila-2017, are compared. The relaxation rates are set to be identical to ${\tau }_2$. The results are extracted at $t=200$ steps.

Figure 6

Shock wave: Pressure ratio is 10, temperature ratio is 6, half length of the domain is $L=800$, and the viscosity relaxation time is $\tau =0.76$. The Prandtl number is set to $2/3$. The results of the present collision model are compared with the PR3rd model and the model proposed by K. Mattila et al. The results are extracted at $t=0.1t_c$; $t_c=L/\sqrt{\gamma T}$, $\gamma =2$, and $T$ is the temperature in the lower region.

Figure 7

Shock wave: Pressure ratio is 10, temperature ratio is 6, half length of the domain is $L=800$, and the viscosity relaxation time is $\tau =0.76$. The Prandtl number is set to $2/3$. The results of the present collision model are compared with the model proposed by Shan recently with different ${\omega }_4$. The results are extracted at $t=0.1t_c$; $t_c=L/\sqrt{\gamma T}$, $\gamma =2$, and $T$ is the temperature in the lower region.

Figure 8

Convergence of the kinetic energy decay for different collision models with the D3V39 lattice structure.

Figure 9

Convergence of the kinetic energy decay for different collision models with the D3V103 lattice structure.