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In order to improve the computational efficiency of multiple-relaxation-time lattice Boltzmann model (MRT), a 12-velocity multiple-relaxation-time lattice Boltzmann model (iD3Q12 MRT model) for three-dimensional incompressible flows is proposed in this work by using an inversion method. This model has higher computational efficiency than the commonly used D3Q13 MRT model in principle. In numerical simulations, the accuracy and stability of iD3Q12 MRT model are validated by simulating different flows, including steady Poiseuille flow driven by pressure, unsteady pulsatile flow driven by periodic pressure and lid-driven cavity flow. We also compare the iD3Q12 MRT model with the 13-velocity multiple-relaxation-time lattice Boltzmann model(He-Luo D3Q13 MRT model). For the Poiseuille flow and pulsatile flow, the numerical solutions of the iD3Q12 MRT model agree well with the analytical solutions. In terms of accuracy, the iD3Q12 MRT model and He-Luo D3Q13 MRT model are used to simulate Poiseuille flow with different parameters. The global relative errors of the two models are identical. Similarly, we also simulate the pulsatile flow to calculate the global relative errors of flow fields at different times and different lattice spacing. It is found that the global relative errors of the iD3Q12 MRT model are smaller than those of the He-Luo D3Q13 MRT model, and both models have the second-order spatial accuracy. Furthermore, we also simulate the pulsatile flow by changing the lattice spacing or relaxation time when the maximal pressure drop of the channel is increased, and it is found that the global relative errors calculated by the iD3Q12 MRT model are smaller than those by the He-Luo D3Q13 MRT model in most cases, but the iD3Q12 MRT model diverges when the maximal pressure drop of the channel is large. This indicates that the iD3Q12 MRT model is more accurate than the He-Luo D3Q13 MRT model in simulating unsteady pulsatile flow, but less stable. For the lid-driven cavity flow, the results show that the numerical results of the iD3Q12 MRT model agree well with those given by Ku et al [Ku H C, Hirsh R S, Taylor T D 1987 J. Comput. Phys. 70439]. In terms of stability, the iD3Q12 MRT model is quantitatively less stable than He-Luo D3Q13 MRT model. -
Keywords:
- 12-velocity in three dimensions/
- multiple-relaxation-time/
- lattice Boltzmann model/
- incompressible flows
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${\rm GRE}_u$ Lattice spacing $\text{δ} x$ Model 1/8 1/16 1/32 1/64 ${\lambda}_{\nu}=0.8,$ ${\lambda}'_{\nu}=1.143$ $3.090\times10^{-2}$ $7.700\times10^{-3}$ $1.900\times10^{-3}$ $4.623\times10^{-4}$ iD3Q12 MRT $3.090\times10^{-2}$ $7.700\times10^{-3}$ $1.900\times10^{-3}$ $4.623\times10^{-4}$ D3Q13 MRT ${\lambda}_{\nu}=1.0,$ ${\lambda}'_{\nu}=1.333$ $5.990\times10^{-2}$ $1.660\times10^{-2}$ $4.400\times10^{-3}$ $1.100\times10^{-3}$ iD3Q12 MRT $5.990\times10^{-2}$ $1.660\times10^{-2}$ $4.400\times10^{-3}$ $1.100\times10^{-3}$ D3Q13 MRT ${\lambda}_{\nu}=1.3,$ ${\lambda}'_{\nu}=1.576$ $8.720\times10^{-2}$ $2.500\times10^{-2}$ $6.700\times10^{-3}$ $1.700\times10^{-3}$ iD3Q12 MRT $8.720\times10^{-2}$ $2.500\times10^{-2}$ $6.700\times10^{-3}$ $1.700\times10^{-3}$ D3Q13 MRT 
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Lattice spacing ${\rm GRE}_u$ Model $T/4$ $T/2$ $3 T/4$ T ${\rm{\text{δ}} } x= {1}/{20}$ $1.483\times10^{-2}$ $4.214\times10^{-2}$ $1.805\times10^{-2}$ $4.028\times10^{-2}$ iD3Q12 MRT $1.662\times10^{-2}$ $4.733\times10^{-2}$ $2.118\times10^{-2}$ $4.299\times10^{-2}$ D3Q13 MRT ${\rm{\text{δ}} } x= {1}/{40}$ $3.803\times10^{-3}$ $1.199\times10^{-2}$ $4.651\times10^{-3}$ $1.153\times10^{-2}$ iD3Q12 MRT $4.172\times10^{-3}$ $1.324\times10^{-2}$ $5.398\times10^{-3}$ $1.217\times10^{-2}$ D3Q13 MRT ${\rm{\text{δ}} } x= {1}/{60}$ $1.702\times10^{-3}$ $5.569\times10^{-3}$ $2.085\times10^{-3}$ $5.369\times10^{-3}$ iD3Q12 MRT $1.855\times10^{-3}$ $6.116\times10^{-3}$ $2.412\times10^{-3}$ $5.648\times10^{-3}$ D3Q13 MRT ${\rm{\text{δ}} } x= {1}/{80}$ $9.605\times10^{-4}$ $3.204\times10^{-3}$ $1.177\times10^{-3}$ $3.092\times10^{-3}$ iD3Q12 MRT $1.043\times10^{-3}$ $3.509\times10^{-3}$ $1.360\times10^{-3}$ $3.247\times10^{-3}$ D3Q13 MRT DownLoad:
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Adjacent spacing Order Model $T/4$ $T/2$ $3 T/4$ T Average 1.978 1.875 1.974 1.869 iD3Q12 MRT 1.998 1.891 1.984 1.879 D3Q13 MRT ${1}/{20} \to {1}/{40}$ 1.963 1.813 1.956 1.805 iD3Q12 MRT 1.994 1.838 1.972 1.821 D3Q13 MRT ${1}/{40}\to {1}/{60}$ 1.983 1.891 1.979 1.885 iD3Q12 MRT 1.999 1.905 1.987 1.893 D3Q13 MRT ${1}/{60} \to {1}/{80}$ 1.989 1.922 1.988 1.918 iD3Q12 MRT 2.001 1.931 1.992 1.924 D3Q13 MRT DownLoad:
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$ \Delta p $ Lattice spacing ${\rm{\text{δ}} } x$ Model 1/20 1/40 1/60 1/80 $0.005 $ $9.919\times10^{-2}$ $3.030\times10^{-2}$ $1.442\times10^{-2}$ $8.402\times10^{-3}$ iD3Q12 MRT $1.121\times10^{-1}$ $3.326\times10^{-2}$ $1.568\times10^{-2}$ $9.084\times10^{-3}$ D3Q13 MRT $0.010$ $1.172\times10^{-1}$ $3.445\times10^{-2}$ $1.618\times10^{-2}$ $9.362\times10^{-3}$ iD3Q12 MRT $1.679\times10^{-1}$ $4.763\times10^{-2}$ $2.199\times10^{-2}$ $1.260\times10^{-2}$ D3Q13 MRT $0.020$ $1.777\times10^{-1}$ $5.110\times10^{-2}$ $2.365\times10^{-2}$ $1.355\times10^{-2}$ iD3Q12 MRT $2.940\times10^{-1}$ $8.630\times10^{-2}$ $3.987\times10^{-2}$ $2.279\times10^{-2}$ D3Q13 MRT $0.050$ $1.243\times10^{-1}$ $5.848\times10^{-2}$ $3.386\times10^{-2}$ iD3Q12 MRT $2.025\times10^{-1}$ $9.868\times10^{-2}$ $5.757\times10^{-2}$ D3Q13 MRT $0.080$ $6.073\times10^{-2}$ iD3Q12 MRT $1.575\times10^{-2}$ $9.405\times10^{-2}$ D3Q13 MRT $0.100$ iD3Q12 MRT $1.192\times10^{-1}$ D3Q13 MRT $0.120$ iD3Q12 MRT $1.454\times10^{-1}$ D3Q13 MRT DownLoad:
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$\Delta p$ τ Model 0.55 0.60 0.70 0.90 $0.005 $ $1.302\times10^{-1}$ $6.311\times10^{-2}$ $2.955\times10^{-2}$ $1.744\times10^{-2}$ iD3Q12 MRT $1.556\times10^{-1}$ $6.560\times10^{-3}$ $3.023\times10^{-2}$ $1.993\times10^{-2}$ D3Q13 MRT $0.010$ $1.612\times10^{-1}$ $6.830\times10^{-2}$ $2.711\times10^{-2}$ $1.736\times10^{-2}$ iD3Q12 MRT $2.435\times10^{-1}$ $8.735\times10^{-2}$ $2.661\times10^{-2}$ $2.058\times10^{-2}$ D3Q13 MRT $0.020$ $2.475\times10^{-1}$ $9.926\times10^{-2}$ $2.624\times10^{-2}$ $1.656\times10^{-2}$ iD3Q12 MRT $4.182\times10^{-1}$ $1.542\times10^{-1}$ $2.757\times10^{-2}$ $2.195\times10^{-2}$ D3Q13 MRT $0.030$ $1.430\times10^{-1}$ $3.421\times10^{-2}$ $1.509\times10^{-2}$ iD3Q12 MRT $5.482\times10^{-1}$ $2.193\times10^{-1}$ $3.616\times10^{-2}$ $2.343\times10^{-2}$ D3Q13 MRT $0.040$ $5.001\times10^{-2}$ $1.349\times10^{-2}$ iD3Q12 MRT $4.693\times10^{-2}$ $2.502\times10^{-2}$ D3Q13 MRT $0.050$ $1.291\times10^{-2}$ iD3Q12 MRT $2.674\times10^{-2}$ D3Q13 MRT DownLoad:
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Re Model iD3Q12 MRT He-Luo D3Q13 MRT 100 $\checkmark$ $\checkmark$ 400 $\checkmark$ $\checkmark$ 1000 $\checkmark$ $\checkmark$ 1500 $\checkmark$ $\checkmark$ 1600 $\checkmark$ $\checkmark$ 1700 divergent $\checkmark$ 1800 divergent divergent DownLoad:
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