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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。

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