AN AXISYMMETRIC IMMERSED BOUNDARY METHOD BASED ON 2D MESH
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摘要: 浸没边界法是处理颗粒两相流中运动边界问题的一种常用数值模拟方法. 当研究的物理问题的无量纲参数满足一定要求时, 该流场结构呈现轴对称状态. 为此本文提出了一种基于2D笛卡尔网格和柱坐标系的轴对称浸没边界法. 该算法采用有限体积法(FVM)对动量方程进行空间离散, 并通过阶梯状锐利界面替代真实的固体浸没边界来封闭控制方程. 为了提高计算效率, 本文采用自适应网格加密技术提高浸没边界附近网格分辨率. 由于柱坐标系的使用, 使得动量方程中的黏性项产生多余的源项, 我们对其作隐式处理. 此外, 在对小球匀速近壁运动进行直接数值模拟时, 由于球壁间隙很小, 间隙内的压力变化比较剧烈. 因此想要精确地解析流场需要很高的网格分辨率. 此时, 需要在一个时间步内多次实施投影步来保证计算的稳定性. 而在小球自由碰壁运动中, 我们通过引入一个润滑力模型使得低网格分辨率下也能模拟小球近壁处的运动. 最后通过小球和圆盘绕流、Stokes流小球近壁运动以及小球自由下落碰壁弹跳算例验证本算法对于轴对称流的静边界和动边界问题均是适用和准确的.Abstract: Immersed boundary methods is a common numerical simulation method to deal with moving boundary problems in particle two-phase flows. When the dimensionless parameters of studied physical problems meet certain requirements, its corresponding flow structure becomes axisymmetric. Hence, an axisymmetric immersed boundary method based on a 2D mesh and cylindrical coordinates is proposed in the present paper. A finite volume method is used as the spatial discretization in the present algorithm. And the governing equation is closed by a sharp stepped interface, which is used to replace the real solid immersed boundary. In order to improve the efficiency of computation, an adaptive mesh refinement technology is used to improve the mesh resolution near the immersed boundary. The using of cylindrical coordinates will produce a redundant source item from the viscous term in the momentum equation. The additional source term will be handled by using an implicit scheme. Moreover, in the direct numerical simulation of a sphere approaching a wall with a constant velocity, the pressure of the fluid in the gap dramatically changes because of the small gap. So, in order to accurately analyze the flow field, the required grid resolution is very high in the gap. Multiple projection-step calculations are carried out in one time step for maintaining the stability of the simulation. In the movement of a sphere free impacting a wall, a lubrication force model will be introduced to simulate the movement of the sphere near the wall even with a low grid resolution. Finally, simulation results on the flow past a fixed sphere, the flow past a circular disk, the stokes flow by a sphere approaching a wall and the flow caused by the sphere-wall collision prove that the present axisymmetric IBM algorithm is applicable and accurate for dealing with stationary and moving boundary problems in an axisymmetric flow.
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Key words:
- axisymmetric /
- 2D mesh /
- immersed boundary method /
- sphere-wall collision
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图 1 浸没边界和网格单元划分示意图
Figure 1. Schematic diagram of immersed boundary and dividing cells type
图 7 小球回流区长度和阻力系数对比
Figure 7. Comparisons of the length of recirculation zone
$ {L}_{re} $ and drag coefficient$ {C}_{d} $ of sphere
图 10 圆盘回流区长度和阻力系数对比
Figure 10. Comparisons of the length of recirculation zone
$ {L}_{re} $ and drag coefficient$ {C}_{d} $ of circular disk
图 11 小球匀速靠近壁面几何参数
Figure 11. Configuration for a sphere with a constant velocity approaching a wall
图 12 压力云图间隙为(a)
$ \varepsilon = 1.1 $ 和(b)$ \varepsilon = 0.1 $ Figure 12. Contour of pressure around sphere with (a)
$\varepsilon =1.1$ and (b)$\varepsilon =0.1$ 表 1 雷诺数Re = 200网格无关性验证
Table 1. Grid independence test at Re = 200
Case $D/\Delta x$ $ {C_d} $ $ {L_{re}} $ coarse mesh 160 0.768 1.435 fine mesh 320 0.770 1.430 
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表 2 雷诺数Re = 100网格无关性验证
Table 2. Grid independence test at Re = 100
Case $D/\Delta x$ $ {C_d} $ $ {L_{re}} $ coarse mesh 160 1.327 1.960 fine mesh 320 1.329 1.965
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表 3 物理计算参数
Table 3. Physical and computational parameters
Case A B $ S{t_c} $ 27 152 $ {Re} $ 30 165 $ {U_{in}} $/(m·s−1) 0.518 0.585 $ D $/m 0.006 0.003 $ {\rho _p} $/(kg·m−3) 7800 7800 $ {\rho _f} $/(kg·m−3) 965 935 $ \nu $/(s·m−2) $1.036\;3 \times {10^{ - 4} }$ $1.069\;5 \times {10^{ - 5} }$ $ D/\Delta x $ 32 32 $ {\varepsilon _{al}} $ 0.5 0.5 $ {\varepsilon _w} $ 0.001 0.001
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