期刊菜单

Study on Interaction between Moving Fluid and Flexible Rod with Variable Cross-Section
DOI: 10.12677/IJM.2021.101003, PDF, HTML, XML, 下载: 258  浏览: 356  国家自然科学基金支持

Abstract: In order to study the interaction between moving fluid and flexible rod with variable cross-section, a model of flexible rod with variable cross-section clamped at both ends under the action of axial wind is established. Based on the ideal fluid assumption and the Euler Bernoulli beam assumption, it establishes the basic equation of the interaction between flexible rod with variable cross-section and the wind field. Then through the linear stability analysis, the expression of the temporal growth rate of the perturbation and the characteristic function of the flexible rod is obtained. From this, the long-wave instability and instability mechanism of the flexible rod are studied, and the change law of the growth rate of the frequency and the temporal growth rate of the perturbation is discussed. The results show that the section variation coefficient and position function of flexible rod have a significant influence on the value of the frequency and the temporal growth rate of the perturbation; the modulus of elasticity has a greater influence on the temporal growth rate of the perturbation, and the influence on the frequency is negligible. It derives the expressions of the wave number and frequency when the coupled motion is the most unstable, and the frequency at the most unstable time is linearly related to the wind velocity.

1. 引言

2. 控制方程

(a) 变截面柔性杆(Flexible rod with variable cross-section)(b) 运动模型(Motion model)

Figure 1. Model of interaction between airflow and flexible rod with variable cross-section

$\frac{{\partial }^{2}}{\partial x}\left[EI\left(x\right)\frac{{\partial }^{2}\xi }{\partial x}\right]+{\rho }^{\prime }A\left(x\right)\frac{{\partial }^{2}\xi }{\partial t}+\delta p\cdot D\left(x\right)=0$ (1)

$\nabla \cdot u=0$ (2)

$\frac{\partial u}{\partial t}+\left(u\cdot \nabla \right)u=-\frac{1}{\rho }\nabla p$ (3)

$u\left(y=\xi \right)\cdot n=\frac{\text{d}\xi }{\text{d}t}$ (4)

3. 线性稳定性分析

$\xi =A{\text{e}}^{ikx-i\Omega t}$ (5)

${u}_{x}=V+UVkA{\text{e}}^{-i\Omega t+ikx-\kappa y}$ (6)

${u}_{y}=WVkA{\text{e}}^{-i\Omega t+ikx-\kappa y}$ (7)

$p={p}_{0}+{P}^{\prime }\rho {V}^{2}kA{\text{e}}^{-i\Omega t+ikx-\kappa y}$ (8)

${k}^{2}={\kappa }^{2}$ (9)

$U=±\frac{kV-\Omega }{kV}$ (10)

$W=i\frac{kV-\Omega }{kV}$ (11)

${P}^{\prime }=\mp \frac{{\left(kV-\Omega \right)}^{2}}{{\left(kV\right)}^{2}}$ (12)

$\delta p=p\left(y\to {0}^{+}\right)-p\left(y\to {0}^{-}\right)=-2\rho \frac{{\left(kV-\Omega \right)}^{2}}{k}\xi$ (13)

$\frac{\pi E}{64}\left[-{k}^{2}\frac{{\partial }^{2}D{\left(x\right)}^{4}}{\partial {x}^{2}}-2i{k}^{3}\frac{\partial D{\left(x\right)}^{4}}{\partial x}+{k}^{4}D{\left(x\right)}^{4}\right]-{\Omega }^{2}{\rho }^{\prime }D{\left(x\right)}^{2}-\frac{2\rho {\left(kV-\Omega \right)}^{2}}{k}D\left(x\right)=0$ (14)

4. 色散关系

$\stackrel{¯}{\Omega }=\frac{{d}_{0}}{V}\Omega$, $\stackrel{¯}{k}={d}_{0}k$, $\stackrel{¯}{x}=\frac{x}{{d}_{0}}$, $\stackrel{¯}{E}=\frac{E}{\rho {V}^{2}}$, $\stackrel{¯}{\rho }=\frac{{\rho }^{\prime }}{\rho }$.

$\frac{\pi \stackrel{¯}{E}}{4}\left[{\stackrel{¯}{k}}^{2}{\alpha }^{2}\left(1+2\alpha \stackrel{¯}{x}\right)+i{\stackrel{¯}{k}}^{3}\alpha {\left(1+2\alpha \stackrel{¯}{x}\right)}^{2}-\frac{1}{16}{\stackrel{¯}{k}}^{4}{\left(1+2\alpha \stackrel{¯}{x}\right)}^{3}\right]+\frac{\pi }{4}\stackrel{¯}{\rho }\left(1+\alpha \stackrel{¯}{x}\right){\stackrel{¯}{\Omega }}^{2}+\frac{2}{\stackrel{¯}{k}}{\left(\stackrel{¯}{k}-\stackrel{¯}{\Omega }\right)}^{2}=0$ (15)

$-\frac{\pi \stackrel{¯}{E}}{64}{\stackrel{¯}{k}}^{4}{\left(1+2\alpha \stackrel{¯}{x}\right)}^{3}+\frac{\pi }{4}\stackrel{¯}{\rho }\text{ }{\stackrel{¯}{\Omega }}^{2}\left(1+2\alpha \stackrel{¯}{x}\right){\stackrel{¯}{\Omega }}^{2}+\frac{2}{\stackrel{¯}{k}}{\left(\stackrel{¯}{k}-\stackrel{¯}{\Omega }\right)}^{2}=0$ (16)

$\stackrel{¯}{\Omega }=\frac{8\stackrel{¯}{k}}{8+\pi \stackrel{¯}{\rho }\stackrel{¯}{k}\left(1+2\alpha \stackrel{¯}{x}\right)}±\frac{\stackrel{¯}{k}\sqrt{\frac{\pi \stackrel{¯}{E}}{4}{\stackrel{¯}{k}}^{4}{\left(1+2\alpha \stackrel{¯}{x}\right)}^{3}\left[\frac{2}{\stackrel{¯}{k}}+\frac{\pi }{4}\stackrel{¯}{\rho }\left(1+2\alpha \stackrel{¯}{x}\right)\right]-8\pi \stackrel{¯}{\rho }\stackrel{¯}{k}\left(1+2\alpha \stackrel{¯}{x}\right)}}{8+\pi \stackrel{¯}{\rho }\stackrel{¯}{k}\left(1+2\alpha \stackrel{¯}{x}\right)}$ (17)

$\frac{\pi \stackrel{¯}{E}}{4}{\stackrel{¯}{k}}^{4}{\left(1+2\alpha \stackrel{¯}{x}\right)}^{3}\left[\frac{2}{\stackrel{¯}{k}}+\frac{\pi }{4}\stackrel{¯}{\rho }\left(1+2\alpha \stackrel{¯}{x}\right)\right]-8\pi \stackrel{¯}{\rho }\stackrel{¯}{k}\left(1+2\alpha \stackrel{¯}{x}\right)=0$ (18)

$\stackrel{¯}{\omega }=\frac{8\stackrel{¯}{k}}{8+\pi \stackrel{¯}{\rho }\stackrel{¯}{k}\left(1+2\alpha \stackrel{¯}{x}\right)}$ (19)

$\stackrel{¯}{\sigma }=±\frac{\stackrel{¯}{k}\sqrt{8\pi \stackrel{¯}{\rho }\stackrel{¯}{k}\left(1+2\alpha \stackrel{¯}{x}\right)-\frac{\pi \stackrel{¯}{E}}{4}{\stackrel{¯}{k}}^{4}{\left(1+2\alpha \stackrel{¯}{x}\right)}^{3}\left[\frac{2}{\stackrel{¯}{k}}+\frac{\pi }{4}\stackrel{¯}{\rho }\left(1+2\alpha \stackrel{¯}{x}\right)\right]}}{8+\pi \stackrel{¯}{\rho }\stackrel{¯}{k}\left(1+2\alpha \stackrel{¯}{x}\right)}$ (20)

(a) ( $\alpha =0.001$ ) (b) ( $\alpha =0.003$ )(c) ( $\alpha =0.005$ ) (d) ( $\stackrel{¯}{x}=1000$ )

Figure 2. The effect of dimensionless wave number on dimensionless angular frequency with respectively different the cross-section change coefficient and the transverse position

(a) ( $\alpha =0.001$ ) (b) ( $\stackrel{¯}{E}={10}^{3}$ )

Figure 3. The effect of dimensionless wave number on dimensionless the temporal growth rate on $\stackrel{¯}{x}=0$

5. 结果和讨论

5.1. 最不稳定波数

$\begin{array}{l}\left[8+\pi {k}_{m}\stackrel{¯}{\rho }\left(1+2\alpha \stackrel{¯}{x}\right)\right]\left\{24\pi {k}_{m}^{2}\stackrel{¯}{\rho }\left(1+2\alpha \stackrel{¯}{x}\right)-\frac{3}{2}\pi \stackrel{¯}{E}{k}_{m}^{5}{\left(1+2\alpha \stackrel{¯}{x}\right)}^{3}\left[\frac{2}{{k}_{m}}+\frac{1}{4}\pi \stackrel{¯}{\rho }\left(1+2\alpha \stackrel{¯}{x}\right)\right]+\frac{1}{2}\pi \stackrel{¯}{E}{k}_{m}^{4}{\left(1+2\alpha \stackrel{¯}{x}\right)}^{3}\right\}\\ -2\pi \stackrel{¯}{\rho }\left(1+2\alpha \stackrel{¯}{x}\right)\left\{8\pi {k}_{m}^{3}\stackrel{¯}{\rho }\left(1+2\alpha \stackrel{¯}{x}\right)-\frac{1}{4}\pi \stackrel{¯}{E}{k}_{m}^{6}\stackrel{¯}{\rho }{\left(1+2\alpha \stackrel{¯}{x}\right)}^{3}\left[\frac{2}{{k}_{m}}+\frac{1}{4}\pi \stackrel{¯}{\rho }\left(1+2\alpha \stackrel{¯}{x}\right)\right]\right\}=0\end{array}$ (21)

(a) ( $\alpha =0.001$ ) (b) ( $\alpha =0.003$ )(c) ( $\alpha =0.005$ ) (d) ( $\stackrel{¯}{x}=1000,\stackrel{¯}{E}={10}^{3}$ )

Figure 4. The effect of dimensionless wave number on dimensionless the temporal growth rate of the perturbation with respectively different the cross-section change coefficient and the elastic modulus on $\stackrel{¯}{x}=1000$

${k}_{m}\approx {\left(\frac{32}{\pi \stackrel{¯}{E}}\right)}^{\frac{1}{3}}{\left(1+2\alpha \stackrel{¯}{x}\right)}^{-1}$ (22)

${\stackrel{¯}{\omega }}_{m}\approx \frac{8}{\pi \stackrel{¯}{\rho }\left(1+2\alpha \stackrel{¯}{x}\right)}$ (23)

${\stackrel{¯}{\sigma }}_{m}\approx {\left(\frac{6912}{\stackrel{¯}{E}}\right)}^{\frac{1}{6}}{\left(\pi \right)}^{-\frac{2}{3}}{\left(\stackrel{¯}{\rho }\right)}^{-\frac{1}{2}}{\left(1+2\alpha \stackrel{¯}{x}\right)}^{-1}$ (24)

5.2. 角频率和波数与风速的关系

${\omega }_{m}\approx \frac{8\rho }{\pi {\rho }^{\prime }\left({d}_{0}+2\alpha x\right)}V$ (25)

${k}_{m}\approx {\left(\frac{32\rho }{\pi E{\left({d}_{0}+2\alpha x\right)}^{3}}\right)}^{\frac{1}{3}}{V}^{\frac{2}{3}}$ (26)

6. 结论

1) 变截面柔性杆角频率的值随着波数的增长逐渐趋于一个定值；在相同横向位置处其值随着截面变化系数的增大而逐渐减小；在截面变化系数相同时，其值随着横向位置的增大而减小。

2) 变截面柔性杆扰动时间增长率的值随着弹性模量值的增大而减小；在相同横向位置处，其值随着截面变化系数的增大而减小。

 [1] Liao, J.C., Beal, D.N., Lauder, G.V., et al. (2003) Fish Exploiting Vortices Decrease Muscle Activity. Science, 302, 1566-1569. https://doi.org/10.1126/science.1088295 [2] Muller, U.K. (2003) Physiology. Fish ‘n Flag. Science, 302, 1511-1512. https://doi.org/10.1126/science.1092367 [3] Lighthill, M.J. (2006) Note on the Swimming of Slender Fish. Journal of Fluid Mechanics, 9, 305-317. https://doi.org/10.1017/S0022112060001110 [4] 陆夕云, 杨基明, 尹协振, 等. 飞行和游动的生物运动力学和仿生技术研究[J]. 中国科学技术大学学报, 2007, 37(10): 1159-1163. [5] 董帝渤. 仿生柔性结构与流固耦合系统的数值方法及运动机理研究[D]: [博士学位论文]. 哈尔滨: 哈尔滨工业大学, 2016. [6] Huang, L. (1995) Flut-ter of Cantilevered Plates in Axial Flow. Journal of Fluids and Structures, 9, 127-147. https://doi.org/10.1006/jfls.1995.1007 [7] Balint, T.S. and Lucey, A.D. (2005) Instability of a Cantilevered Flexible Plate in Viscous Channel Flow. Journal of Fluids and Structures, 20, 893-912. https://doi.org/10.1016/j.jfluidstructs.2005.05.005 [8] Lim, W.L., Chew, Y.T., Low, H.T., et al. (2003) Cavitation Phenomena in Mechanical Heart Valves: The Role of Squeeze Flow Velocity and Contact Area on Cavitation Initiation between Two Impinging Rods. Journal of Biomechanics, 36, 1269-1280. https://doi.org/10.1016/S0021-9290(03)00161-1 [9] Gerbeau, J.F., Vidrascu, M. and Frey, P. (2005) Flu-id-Structure Interaction in Blood Flows on Geometries Based on Medical Imaging. Computers & Structures, 83, 155-165. https://doi.org/10.1016/j.compstruc.2004.03.083 [10] Watanabe, Y., Isogai, K., Suzuki, S., et al. (2002) A Theoretical Study of Paper Flutter. Journal of Fluids and Structures, 16, 543-560. https://doi.org/10.1006/jfls.2001.0436 [11] 张成瑶. 单个或多个柔性体自主推进的流固耦合数值研究[D]: [博士学位论文]. 合肥: 中国科学技术大学, 2020. [12] Jia, P., Andreotti, B. and Claudin, P. (2015) Paper Waves in the Wind. Physics of Fluids, 27, Article ID: 104101. https://doi.org/10.1063/1.4931777 [13] Taneda, S. (1968) Waving Motions of Flags. Journal of the Physical So-ciety of Japan, 24, 392-401. https://doi.org/10.1143/JPSJ.24.392 [14] Datta, S.K. and Gottenberg, W.G. (1975) Instability of an Elastic Strip Hanging in an Airstream. Journal of Applied Mechanics, 42, 195-198. https://doi.org/10.1115/1.3423515 [15] Lemaitre, C., Hemon, P. and de Langre, E. (2005) Instability of a Long Ribbon Hanging in Axial Air Flow. Journal of Fluids and Structures, 20, 913-925. https://doi.org/10.1016/j.jfluidstructs.2005.04.009 [16] Yamaguchi, N., Sekiguchi, T., Yokota, K., et al. (2000) Flutter Limits and Behavior of a Flexible Thin Sheet in High-Speed Flow II: Experimental Results and Predicted Be-haviors for Low Mass Ratios. Journal of Fluids Engineering, 122, 74-83. https://doi.org/10.1115/1.483228 [17] Argentina, M. and Mahadevan, L. (2005) Fluid-Flow-Induced Flutter of a Flag. Proceedings of the National Academy of Sciences of the United States of America, 102, 1829-1834. https://doi.org/10.1073/pnas.0408383102 [18] Eloy, C., Kofman, N. and Schouveiler, L. (2011) The Origin of Hysteresis in the Flag Instability. Journal of Fluid Mechanics, 691, 583-593. https://doi.org/10.1017/jfm.2011.494 [19] 孙传宝, 贾来兵, 李发尧, 等. 单个柔性旗帜在均匀流中摆动的测力实验[J]. 实验力学, 2011, 25(2): 1-6. [20] 王思莹, 孙传宝, 尹协振. 均匀来流条件下并行排列旗帜耦合运动模式的实验[J]. 实验力学, 2010, 25(4): 401-407. [21] Kim, D., Cossé, J., Huertas, C.C., et al. (2013) Flapping Dynam-ics of an Inverted Flag. Journal of Fluid Mechanics, 736, R1-R7. https://doi.org/10.1017/jfm.2013.555 [22] Cisonni, J., Lucey, A.D. and Elliott, N.S.J. (2019) Flutter of Structurally in Homogenous Cantilevers in Laminar Channel Flow. Journal of Fluids and Structures, 90, 177-189. https://doi.org/10.1016/j.jfluidstructs.2019.06.006 [23] 郭日修. 弹性力学与张量分析[M]. 北京: 高等教育出版社, 2003: 177-179. [24] 原渭兰. 气体动力学[M]. 北京: 科学出版社, 2013: 55-56. [25] Yang, T. and Mason, M.S. (2019) Aerodynamic Characteristics of Rectangular Cylinders in Steady and Accelerating Wind Flow. Journal of Fluids and Structures, 90, 246-262. https://doi.org/10.1016/j.jfluidstructs.2019.07.004 [26] 高云, 邹丽, 宗智. 两端铰接的细长柔性体圆柱体涡激振动响应特征数值研究[J]. 力学学报, 2018, 50(1): 9-20.