Acta Physica Sinica, Volume. 68, Issue 23, 234701-1(2019)
Fig. 1. The schematic of three-dimensional Poiseuille flow.三维泊肃叶流示意图
Fig. 2. Comparison between numerical and analytical solutions of Poiseuille flow: (a) The variation of
with
Fig. 3. The variation of of velocity field with the lattice spacing at different for Poiseuille flow. Symbols represent numerical solutions, lines represent fitting line. 不同的 下, 模拟泊肃叶流得到的速度场的全局相对误差 随空间步长 的变化, 符号代表数值解, 连线表示拟合直线
Fig. 4. The variation of horizontal velocity
Fig. 5. The variation of with the lattice spacing at four different times in a period for pulsatile flow. 同一周期四个不同时刻下变量 随空间步长 的变化
Fig. 6. The schematic of three-dimensional lid-driven cavity flow三维顶盖驱动的方腔流示意图
Fig. 7. The velocity distribution in the vertical and horizontal center lines at section
for cavity flows at different
: (a)
; (b)
; (c)
.
不同的雷诺数下模拟方腔流, 在截面
处竖直和水平中心线的速度分布 (a)
Table 1. The of velocity field for Poiseuille flow computed by iD3Q12 MRT and D3Q13 MRT models under different relaxation parameters and different lattice spacings. iD3Q12 MRT和D3Q13 MRT模型在不同松弛因子 和不同空间步长下计算得到的泊肃叶流的速度场的全局相对误差
View tableView in ArticleTable 1. The of velocity field for Poiseuille flow computed by iD3Q12 MRT and D3Q13 MRT models under different relaxation parameters and different lattice spacings. iD3Q12 MRT和D3Q13 MRT模型在不同松弛因子 和不同空间步长下计算得到的泊肃叶流的速度场的全局相对误差
${\rm GRE}_u$ 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
Table 2. The global relative errors of the velocity field at different times for pulsatile flow simulated by iD3Q12 MRT and D3Q13 MRT models at different lattice spacings, . 在 时, 不同空间步长下用iD3Q12 MRT模型和D3Q13 MRT模型模拟脉动流所得的不同时刻下的速度场的全局相对误差
View tableView in ArticleTable 2. The global relative errors of the velocity field at different times for pulsatile flow simulated by iD3Q12 MRT and D3Q13 MRT models at different lattice spacings, . 在 时, 不同空间步长下用iD3Q12 MRT模型和D3Q13 MRT模型模拟脉动流所得的不同时刻下的速度场的全局相对误差
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
Table 3.
The orders of the spatial accuracy of iD3Q12 MRT and D3Q13 MRT models under adjacent spacings.
相邻空间步长下的iD3Q12 MRT和D3Q13 MRT模型的空间精度的阶
View tableView in ArticleTable 3.
The orders of the spatial accuracy of iD3Q12 MRT and D3Q13 MRT models under adjacent spacings.
相邻空间步长下的iD3Q12 MRT和D3Q13 MRT模型的空间精度的阶
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
Table 4. The global relative error calculated by the velocity field at time T of pulsatile flow simulated by the iD3Q12 MRT and D3Q13 MRT models under different lattice spacings. The maximal pressure drop of the channel increases, , are fixed. The blank indicates that the computation is divergent. 在 , , 最大压差 增大时不同的空间步长下由iD3Q12 MRT和D3Q13 MRT模型模拟的脉动流在时刻T下的速度场所计算的全局相对误差 , 空白处表示计算发散
View tableView in ArticleTable 4. The global relative error calculated by the velocity field at time T of pulsatile flow simulated by the iD3Q12 MRT and D3Q13 MRT models under different lattice spacings. The maximal pressure drop of the channel increases, , are fixed. The blank indicates that the computation is divergent. 在 , , 最大压差 增大时不同的空间步长下由iD3Q12 MRT和D3Q13 MRT模型模拟的脉动流在时刻T下的速度场所计算的全局相对误差 , 空白处表示计算发散
$ \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
Table 5. The global relative error of the velocity field at time T of the pulsatile flow simulated by the iD3Q12 MRT and D3Q13 MRT models under different relaxation time τ. The maximal pressure drop of the channel is increased and is fixed. The blank indicates that the computation is divergent. 在 时, 最大压差 增大时不同的松弛时间τ下由iD3Q12 MRT和D3Q13 MRT模型模拟的脉动流由T时刻的速度场计算得出的全局相对误差 , 空白处表示计算发散
View tableView in ArticleTable 5. The global relative error of the velocity field at time T of the pulsatile flow simulated by the iD3Q12 MRT and D3Q13 MRT models under different relaxation time τ. The maximal pressure drop of the channel is increased and is fixed. The blank indicates that the computation is divergent. 在 时, 最大压差 增大时不同的松弛时间τ下由iD3Q12 MRT和D3Q13 MRT模型模拟的脉动流由T时刻的速度场计算得出的全局相对误差 , 空白处表示计算发散
$\Delta p$ 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
Table 6. Comparing the stability of iD3Q12 MRT and He-Luo D3Q13 MRT models for three-dimensional cavity flows when the Reynolds number is continuously increased. The tick represents convergence, the convergence criterion is formula (
39 ). 不断增大雷诺数比较iD3Q12 MRT和He-Luo D3Q13 MRT模型在模拟方腔流时的稳定性. 代表收敛, 收敛准则是(39 )式View tableView in ArticleTable 6. Comparing the stability of iD3Q12 MRT and He-Luo D3Q13 MRT models for three-dimensional cavity flows when the Reynolds number is continuously increased. The tick represents convergence, the convergence criterion is formula (
39 ). 不断增大雷诺数比较iD3Q12 MRT和He-Luo D3Q13 MRT模型在模拟方腔流时的稳定性. 代表收敛, 收敛准则是(39 )式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
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Jia-Yi Hu, Wen-Huan Zhang, Zhen-Hua Chai, Bao-Chang Shi, Yi-Hang Wang.
Received: Jun. 26, 2019
Accepted: --
Published Online: Sep. 17, 2020
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