图  1  D2Q9 离散速度模型示意图

Figure  1.  Schematic diagram of D2Q9 discrete velocity model

图  2  粗细网格排布示意图. 灰色区域为粗细网格的重叠区域

Figure  2.  Schematic diagram of the coarse and fine grid arrangement. The grey area is the overlap between the coarse and fine grids

图  3  强迫泰勒−格林涡各宏观量的分布云图($Q=0.5,\;tU_0/L=1$, 黑色矩形框内的区域为加密区域)

Figure  3.  The contours of each macroscopic quantity of the forced Taylor-Green vortex ($Q=0.5,\;tU_0/L=1$, the area inside the black rectangular box is the refinement region)

图  4  强迫泰勒−格林涡中在$ (x/L,y/L)=(0.4,0.25) $点处各宏观量随时间的变化. (a) ~ (c)分别为BGK模型和MRT模型下的相对压力 $ P-P_0 $、水平方向速度 $ u_x $和黏性应力 $ \sigma_{xx} $与理论结果的对比

Figure  4.  The variation of each macroscopic quantity with time at the point $ (x/L,y/L)=(0.4,0.25) $ in the forced Taylor-Green vortex. (a) ~ (c) are comparisons between the relative pressure $ P-P_0 $, velocity in the $ x $ direction $ u_x $ and viscous stress $ \sigma_{xx} $ under BGK model and MRT model, respectively, and the theoretical results

图  5  强迫泰勒−格林涡中各宏观量的剖线图($tU_0/L=1,\;y/L=0.25$, 两条黑色虚线间的区域为加密区域). (a) ~ (c)分别为BGK模型和MRT模型下相对压力 $ P-P_0 $、水平方向速度 $ u_x $和黏性应力 $ \sigma_{xx} $与理论结果的对比

Figure  5.  Profile diagram of each macroscopic quantity of forced Taylor-Green vortex ($ tU_0/L=1,\;y/L=0.25 $, the area between the two black dashed lines is refined). (a) ~ (c) are comparisons between the relative pressure $ P-P_0 $, velocity in the $ x $ direction $ u_x $ and viscous stress $ \sigma_{xx} $ under BGK model and MRT model, respectively, and the theoretical results

图  6  平板泊肃叶流中对流−扩散问题示意图

Figure  6.  Schematic diagram of diffusion problem in a planar Poiseuille flow

图  7  平板泊肃叶流中对流−扩散问题(黑色虚线为加密界面). (a) ~ (d)分别为BGK模型下水平方向速度 $ u_x $、标量 $ \phi $、通量 $ J_x $和$ J_y $与理论结果的对比

Figure  7.  Diffusion problem in a planar Poiseuille flow (the black dashed lines are the refinement interface). (a) ~ (d) represent the velocity in x direction $ u_x $, scalar $ \phi $, flux $ J_y $ and $ J_x $ compared with theoretical results under the BGK model

图  8  顶盖驱动方腔流4个角点示意图, 其中黑色粗实线代表壁面

Figure  8.  Diagram of four corner points of the square cavity flow driven by the top lid, where the thick black line represents the wall surface

图  9  $ Re=1000 $时顶盖驱动方腔流的压力分布云图对比, (a) ~ (d)分别为UCG, UFG, EQ和UCG-L对应的压力分布云图 ((d)中的黑色直线为加密界面, 界面以上为加密区域)

Figure  9.  When $ Re=1000 $, the contours of pressure of the lid-driven cavity flow are compared. (a) ~ (d) are the contours of pressure corresponding to UCG, UFG, EQ and UCG-L, respectively (the black line in (d) is the grid refinement interface, and the area above the black line is the refinement area)

图  10  $ Re=1000 $时顶盖驱动方腔流中无量纲速度的剖线图, UCG, UFG, EQ和UCG-L与参考数据的对比. (a) 沿垂直直线穿过几何中心的$ u_x/U_w $, (b)为其局部放大, (c) 沿水平直线穿过几何中心的$ u_y/U_w $, (d)为其局部放大

Figure  10.  Profile diagram of dimensionless velocity in a lid-driven cavity flow when $ Re=1000 $, comparison of UCG, UFG, EQ and UCG-L with reference data. (a) $ u_x/U_w $ in a vertical straight line through the center of the geometry, (b) local amplification of (a), (c) $ u_y/U_w $ in a horizontal straight line through the center of the geometry, (d) local amplification of (c)

图  11  一维剪切波问题示意图

Figure  11.  Schematic diagram of one-dimensional shear wave problem

图  12  一维剪切波问题在不同加密区域下均匀网格和非均匀网格对应的数值黏性和物理黏性之比的分布情况(黑色虚线为加密界面). (a) ~ (d)分别为不同加密区域下UCG-L和UCG对应的数值黏性和物理黏性之比随$ y/H $变化的剖线图 (续)

Figure  12.  For one-dimensional shear wave problem, the distribution of the ratio of numerical viscosity and physical viscosity corresponding to uniform and non-uniform grids in different refinement regions (the black dashed line is the refinement interface). (a) ~ (d) are the cross-section plots of the ratio of numerical viscosity and physical viscosity corresponding to UCG-L and UCG with $ y/H $ in different refinement regions, respectively (continued)