A multi-domain decomposition strategy for the lattice Boltzmann method for steady-state flows

Figures & data

Figure 1. A two-domain decomposition strategy, Block1 and Block2.

Figure 2. The multi-domain decomposition strategy for (a) two blocks; (b) four blocks.

Table 1. The iteration steps for each block (two-domain decomposition strategy) for a two-dimensional lid-driven cavity flow.

Re		Block1	Block2	Block12	Steps
400	0	–	–	–	58,216
	2	0	15,333	47,675	63,008
1000	0	–	–	–	205,075
	2	0	20,500	114,915	135,415
5000	0	–	–	–	>1,000,000
	2	581	36,013	261,633	298,227
10,000	0	–	–	–	>1,000,000
	2	287	85,986	850,871	937,144

Note: The column headings Block1, Block2, and Block12 represent the iteration steps of Block1, Block2 and the whole flow field (Block1 + Block2), respectively. Steps is the total iteration number, and when , this indicates that the traditional lattice BGK model is adopted.

Figure 3. Streamlines in the two-dimensional cavity flow at for: (a) Case (I); (b) Case (II); (c) Case (III).

Figure 4. Streamlines in the two-dimensional cavity flow at for: (a) Case (I); (b) Case (II); (c) Case (III).

Figure 5. Streamlines in the two-dimensional cavity flow at for: (a) Case (I); (b) Case (II); (c) Case (III).

Figure 6. Streamlines in the two-dimensional cavity flow at for: (a) Case (I); (b) Case (III).

Table 2. The central locations of vortices for the lid-driven cavity flow for , and .

		Primary Vortex		Lower-left Vortex		Lower-right Vortex
Re		x	y	x	y	x	y
400	a	0.5547	0.6055	0.0508	0.0469	0.8906	0.1250
	b	0.5668	0.6069	0.0471	0.0482	0.8857	0.1231
	c	0.5573	0.6074	0.0482	0.0462	0.8854	0.1228
	d	0.5580	0.6084	0.0476	0.0453	0.8850	0.1237
1000	a	0.5313	0.5625	0.0859	0.0781	0.8594	0.1094
	b	0.5336	0.5675	0.0820	0.0731	0.8644	0.1128
	c	0.5335	0.5674	0.0815	0.0753	0.8648	0.1129
	d	0.5320	0.5654	0.0817	0.0741	0.8650	0.1131
5000	a	0.5117	0.5352	0.0703	0.1367	0.8086	0.0742
	b	0.5158	0.5458	0.0742	0.1337	0.80554	0.0745
	c	0.5161	0.5346	0.0744	0.1333	0.8091	0.0747
	d	0.5170	0.5372	0.0738	0.1365	0.8120	0.0733
10000	a	0.5117	0.5333	0.0586	0.1641	0.7656	0.0586
	b	–	–	–	–	–	–
	c	0.5123	0.5304	0.0603	0.1585	0.7783	0.0614
	d	0.5123	0.5289	0.0587	0.1628	0.7862	0.0611
Re		Upper-left Vortex		Second-lower-right Vortex
Re		x	y	x	y
5000	a	0.0625	0.9102	0.9805	0.0195
	b	0.0638	0.9084	0.9836	0.0160
	c	0.0630	0.9082	0.9842	0.0155
	d	0.0616	0.9088	0.9822	0.0163
10000	a	0.0703	0.9141	0.9336	0.0625
	b	–	–	–	–
	c	0.0698	0.9117	0.9420	0.0557
	d	0.0695	0.9115	0.9412	0.0604

Note: a = Ghia et al. (1982) (with FDM NS solver); b = LBGK model (He et al., 2009); c = the multi-domain decomposition strategy for the LBM (257×257); d = the multi-domain decomposition strategy for the LBM ().

Figure 7. Comparison of the profiles of the u-velocity through the geometric center in the present results and those of Ghia et al. (1982) at along: (a) the vertical line; (b) the horizontal line.

Figure 8. Comparison of the profiles of the u-velocity through the geometric center in the present results and those of Ghia et al. (1982) at Re = 1000 along: (a) the vertical line; (b) the horizontal line.

Figure 9. Comparison of the profiles of the u-velocity through the geometric center in the present results and those of Ghia et al. (1982) at Re = 5000 along: (a) the vertical line; (b) the horizontal line.

Figure 10. Comparison of the profiles of the u-velocity through the geometric center in the present results and those of Ghia et al. (1982) at Re = 10,000 along: (a) the vertical line; (b) the horizontal line.

Table 3. The iteration steps of each block (four-domain decomposition strategy) in a two-dimensional lid-driven cavity flow.

Re	Block1	Block2	Block3	Block4	Block12	Block34	Block1234	Steps
5000	0	0	470	198	519	27,577	232,024	260,788
10,000	2262	8870	3549	13,032	5613	62,043	712,016	807,385

Table 4. The central locations of vortices for the lid-driven cavity flow for and 10,000.

		Primary Vortex		Lower-left Vortex		Lower-right Vortex
Re		x	y	x	y	x	y
5000	a	0.5117	0.5352	0.0703	0.1367	0.8086	0.0742
	b	0.5169	0.5354	0.0734	0.1352	0.8107	0.0746
10000	a	0.5117	0.5333	0.0586	0.1641	0.7656	0.0586
	b	0.5122	0.5301	0.0603	0.1591	0.7809	0.0616
Re		Upper-left Vortex		Second-lower-right Vortex
		x	y	x	y
5000	a	0.0625	0.9102	0.9805	0.0195
	b	0.0624	0.9083	0.9850	0.0147
10000	a	0.0703	0.9141	0.9336	0.0625
	b	0.0698	0.9117	0.9427	0.0548

Note: a = Ghia et al. (1982) (with FDM NS solver); b = the multi-domain decomposition strategy.

Figure 11. The multi-domain decomposition strategy in the flow around a circular cylinder (three blocks).

Figure 12. Streamlines of the steady-state flow around a circular cylinder at .

Figure 13. Streamlines of the steady-state flow around a circular cylinder at .

Figure 14. Streamlines of the steady-state flow around a circular cylinder at .

Table 5. The iteration steps for each block (three-domain decomposition strategy) in the flow around a circular cylinder.

Re		Block1	Block2	Block3	Block23	Block123	Steps
10	0	–	–	–	–	–	6455
	2	327	4	94	430	5445	6300
20	0	–	–	–	–	–	11,535
	2.1	432	152	76	296	7764	8720
40	0	–	–	–	–	–	19,900
	2.1	471	1211	2757	839	13,284	18,562

Figure 15. The computational distribution of each block at . Note: Blue represents the steps during which Block1 is computed, and green represents the steps during which Block2 is computed, while all flow fields are being computed in the remaining steps.

Figure 16. A Von Karman Vortex Street and streamlines of the flow around a circular cylinder at .

Table 6. The iterations of each block (two-domain decomposition strategy) in the flow around a circular cylinder.

Re		Block1	Block2	Block12	Steps
100	2	66	447	19,487	20,000

Figure 17. The two-domain decomposition strategy for a three-dimensional lid-driven cavity flow.

Table 7. The iterations of each block for the three-dimensional lid-driven cavity flow.

Re		Block1	Block2	Block12	Steps
1000	0	–	–	–	30,100
	2	0	6423	21,545	27,968

Figure 18. (a) The streamlines and (b) the velocity for the three-dimensional lid-driven cavity flow at .

Figure 19. The two-domain decomposition strategy in the flow around a sphere.

Table 8. The iterations of each block in the flow around a sphere.

Re		Block1	Block2	Block12	Step
100	0	–	–	–	11,746
	2	1249	906	9477	11,631

Figure 20. The streamlines of the flow around a sphere .

Ghia U., Ghia K. N., & Shin C. T. (1982). High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. Journal of Computational Physics, 48, 387–411.

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He Y. L., Wang Y., & Li Q. (2009). Lattice Boltzmann method: Theory and applications. Beijing: Science Press.

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