Numerical Study of the Interaction between High-Speed Gas and Elliptical Column Cloud
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摘要: 高速颗粒流在天文、自然灾害、工业安全、医疗工业和国防等领域有着重要应用。采用基于分层流模型的直接数值模拟方法，对平面激波与椭圆柱云的相互作用进行数值研究，重点关注椭圆柱横截面的不同长短轴之比和椭圆柱横截面长轴与来流方向所成角度对流场的影响，从气体来流方向上的速度、x轴和y轴方向上的均方根速度、动能、内能和湍动能的分布上进行分析，对能量在计算域的上游区域、椭圆柱云区域和下游区域进行定量分析。同时针对椭圆柱改进了一维体积平均模型，利用该模型拟合了由直接数值模拟得到的反射激波和透射激波位置，获得了最适配的一维体积平均模型中的人工有效阻力系数，并探讨此系数的分布规律。Abstract: High-speed particle-laden flow has important applications in astronomy, natural disasters, industrial safety, medical industry, and national defense. In this work, a direct numerical simulation method based on the stratified flow model is used to study the interaction between a planar shock wave and an elliptical column cloud. The influence of the aspect ratio and the tilt angle, the distributions of the flow velocity, RMS velocities along x axis, kinetic energy, internal energy, and turbulent kinetic energy are analyzed; energy values in the upstream region, the elliptical column cloud region and the downstream region of the computational domain are quantitatively analyzed. The 1-D volume-average model is improved for elliptical columns. Based on this model, the appropriate artificial effective drag coefficients are decided by fitting the positions of the reflected shock and the transmitted shock from the direct numerical simulation results, and the distribution of the optimal artificial effective drag coefficients is also discussed.
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Key words:
- shock wave /
- elliptical column /
- direct numerical simulation /
- 1-D volume-averaged model
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图 1 椭圆柱横截面几何示意图
Figure 1. Illustration of the geometry for the cross-section of the elliptical cylinder
图 2 分层流模型示意图[25]（i–1、i和i + 1表示网格索引号，界面
$\overline {ab} $ 和$\overline {ef} $ 为气-气界面，$\overline {bc} $ 和$\overline {fg} $ 为气-固界面，$\overline {cd} $ 和$\overline {gh} $ 为固-固界面）Figure 2. Illustration of the stratified flow model[25] (Where i –1, i and i + 1 are the indexes of the cell.
$\overline {ab} $ and$\overline {ef} $ are the interfaces between the gas phases,$\overline {bc} $ and$\overline {fg} $ are the interfaces between the gas phase and solid phase,$\overline {cd} $ and$\overline {gh} $ are the interfaces between the solid phases.)
图 4 x-y平面计算区域设置示意图（右图为初始椭圆柱云分布图，蓝色表示低压区域，红色表示高压区域）
Figure 4. Illustration of the computational domain setting in the x-y plane (The right plot shows the initial distribution of the elliptical cylinder cloud. The red and blue regions represent the high-pressure and low-pressure regions, respectively.)
图 5 当λ = 2、θ = 0°时，不同无量纲时间下流场的无量纲压强分布
Figure 5. Distributions of the dimensionless pressure at different dimensionless time when λ = 2 and θ = 0°
图 6 t = 3.5时不同λ下θ分别为0°、45°、90°、135°时的流场速度、流场内能和流场动能分布（灰色矩形区域表示椭圆柱云，RS、TS、UFC、DFC分别表示反射激波、透射激波、椭圆柱云上游边界、椭圆柱云下游边界）
Figure 6. Distributions of the fluid velocity, fluid internal energy and fluid kinetic energy with different λ when θ equals to 0°, 45°, 90°, 135° at dimensionless time t = 3.5 (The gray rectangular regions stand for the elliptical cylinder cloud. Hereafter, RS, TS,UFC and DFC mean reflected shock, transmitted shock, the upstream front of elliptical column cloud, and the downstream front of elliptical column cloud, respectively.)
图 7 t = 3.5时不同θ下λ分别为2、3、4时，流场速度、流场内能和流场动能的分布（灰色矩形区域表示椭圆柱云）
Figure 7. Distributions of the fluid velocity, fluid internal energy and fluid kinetic energy with different θ, when λ equals to 2, 3, 4, at dimensionless time t = 3.5, where the gray rectangular regions stand for the elliptical cylinder cloud
图 8 t = 3.5，θ = 0°, 45°, 90°, 135°时流场RMS速度
${u''}$ 、${v''}$ 以及湍动能k在不同λ下沿x方向的分布Figure 8. Distributions of the fluid RMS velocity
${u''}$ ,${v''}$ and turbulent kinetic energy k in the x-direction at different λ, when θ is equal to 0°, 45°, 90°, 135° at dimensionless time t = 3.5
图 9 t = 3.5时不同θ和λ下流场内能、流场动能和流场湍动能在计算域上游区域（
$x \in \left[ { - 3.0, - 0.5} \right]$ ）、椭圆柱云区域（$x \in \left[ { - 0.5,0.5} \right]$ ）和计算域下游区域（$x \in \left[ {0.5, 4.0} \right]$ ）分布Figure 9. Distributions of the fluid internal energy, fluid kinetic energy and fluid turbulent kinetic energy at different θ and λ in three different regions, that is the upstream area of the domain
$x \in \left[ { - 3.0, - 0.5} \right]$ , elliptical column cloud area$x \in \left[ { - 0.5,0.5} \right]$ , the downstream area of the domain$x \in \left[ {0.5, 4.0} \right]$ at dimensionless time t = 3.5
图 10 不同角度θ和不同λ下流场内能、流场动能和流场湍动能随无量纲时间t的变化
Figure 10. Variations of the fluid internal energy, fluid kinetic energy and fluid turbulent kinetic energy with dimensionless time t at different θ and λ
图 11 t = 3.5时不同的λ和θ下一维体积平均模型与DNS拟合结果
Figure 11. Fitting results of the 1-D volume-averaged model and DNS at different λ and θ when the dimensionless time is equal to 3.5
图 12 人工有效阻力系数Cd的最优取值分布
Figure 12. Distribution for the optimal value of artificial effective drag coefficient Cd
表 1 网格收敛性分析实验中使用的4种网格
Table 1. Four meshes used in the convergence analysis experiment
Mesh nb Nx Ny 1 8 112 64 2 16 224 128 3 32 448 256 4 64 896 512 
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表 2 平面激波与椭圆柱云相互作用数值模拟使用的网格设置
Table 2. Mesh settings in numerical simulation of the interaction between plane shock and elliptical column cloud
λ Np a b Δx/10–4 Nx Ny N/106 2 440 0.029 44 0.014 72 4.59 3 894 2 176 8.5 3 440 0.036 04 0.012 02 3.75 4 763 2 666 12.7 4 440 0.041 64 0.010 41 3.25 5 500 3 078 16.9
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表 3 人工有效阻力系数Cd的最优取值
Table 3. Optimal values of artificial effective drag coefficient Cd
λ Cd θ = 0° θ = 15° θ = 30° θ = 45° θ = 60° θ = 90° θ = 135° 2 1.19 1.06 1.06 1.42 2.22 6.00 1.68 3 0.86 0.67 0.69 1.48 2.80 14.00 1.29 4 0.57 0.58 0.47 1.30 3.00 36.00 1.15
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