A Chebyshev Spectral Method for the Unsteady Maxwell Oblique Stationary Point Flow on an Axially Cosine Oscillating Cylinder
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摘要: 研究了非稳态Maxwell流体斜撞击轴向余弦振荡圆柱的斜驻点流动. 首先，基于斜驻点流动特性，在柱面坐标系下求得关于压力的二阶常微分方程，对压强进行修正，建立了非稳态Maxwell流体在振荡圆柱上斜驻点流动的边界层模型. 接着，合理的相似变换将模型转化，使用Chebyshev谱方法求得模型的数值解. 结果表明，在贴近圆柱表面的流体随着圆柱体做周期性运动；圆柱的曲率越大越会使在同一时刻同一位置处的流体质点的速度越大；相反，非稳态参数及流体的记忆特性也会在更靠近圆柱壁面处阻碍流体流动.
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关键词:
- 非稳态斜驻点流动 /
- Maxwell流体 /
- 振荡圆柱 /
- 修正压强场 /
- Chebyshev谱方法
Abstract: The oblique stationary point flow of the Maxwell fluid impacting an axially cosine oscillating cylinder was studied. Firstly, based on the oblique stationary point flow characteristics, the pressure was corrected with the 2nd-order ordinary differential equation of pressure obtained in the cylindrical coordinate system. Later, the boundary layer model for the unsteady Maxwell fluid on an oscillating cylinder was established. The model was converted through the reasonable similarity transform, and the numerical solutions were obtained with the Chebyshev spectral method. The results show that, the fluid near the surface of the cylinder moves periodically with the cylinder. The larger the curvature of the cylinder is, the higher the velocity of the fluid particle will be in the same position at the same time. In contrast, the unsteady state parameter and the memory properties of the fluid hinder the flow closer to the cylinder wall. -
图 2 不同振荡参数ω下的速度三维图
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 2. 3D plots of velocities u for different values of oscillation parameter ω
图 4 不同圆柱曲率参数β下的速度u
Figure 4. Velocities u for different values of curvature parameter β
图 5 不同速度比参数s1下的速度u
Figure 5. Velocities u for different values of velocity ratio parameter s1
图 6 不同Deborah拉数De下的速度u
Figure 6. Velocities u for different values of Deborah number De
表 1 现有f″(0)的数据与文献[33-34]中f″(0)的数据对比结果
Table 1. Comparison results between the present data and the data in ref. [33-34]
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表 2 选取不同长度L时f″(0)的数据对比结果
Table 2. Data comparison results of f″(0) with different values of length L
s1 β=0，De=0，N=120 L=6 L=7 L=8 L=9 0.15(a/c=0.3) -0.849 420 808 3 -0.849 420 047 2 -0.849 420 014 1 -0.849 420 022 6 0.25(a/c=0.5) -0.667 263 677 5 -0.667 263 660 4 -0.667 263 666 5 -0.667 263 681 3 0.4(a/c=0.8) -0.299 388 804 7 -0.299 388 802 1 -0.299 388 810 2 -0.299 388 830 4 1(a/c=2) 2.017 502 833 5 2.017 502 837 70 2.017 502 821 3 2.017 502 787 82 1.5(a/c=3) 4.729 282 401 84 4.729 282 403 29 4.729 282 384 3 4.729 282 346 45 2(a/c=4) 8.000 429 507 31 8.000 429 504 18 8.000 429 481 5 8.000 429 443 28
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