ANALYSIS OF THE SURFACE WAVE INSTABILITY OF A SEMI-SPHERICAL DROPLET UNDER VERTICAL EXCITATION
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摘要: 外部激励作用下液滴的表面失稳特性一直都是流体力学领域十分关注的问题. 在不同的振动激励参数下, 表面会产生不同形态的波形或破碎生成二次液滴. 本文对于液滴表面纬向波和经向波发展的动态特性和产生机理开展了研究. 首先建立激励振幅和频率可控的液滴振荡实验系统. 实验结果表明, 激励振幅的改变会影响液滴表面波形, 振幅较大时经向波才会产生, 演化频率为驱动频率的一半, 而纬向波一直存在, 其频率等于驱动频率. 驱动频率改变会引起失稳模式的转变, 驱动频率增加, 表面波模态数增加、波长减小. 驱动频率超过一定阈值, 波形会从只存在纬向波模式向纬向波叠加经向波模式转变. 同时, 基于三维数值模拟, 通过研究液滴的速度场与压力场, 结合液滴顶点位移与惯性力的相位关系, 阐明液滴形成纬向波的机理: 在惯性力和表面张力的共同作用下, 液滴表面波完成周期性的能量转化和传递. 通过对比分析竖直方向与沿液滴径向加速度下Faraday不稳定性主导的表面波特性, 发现液滴的几何特征使得接触线处产生法向的径向力, 当竖直惯性力增加使得径向力达到一定阈值, 液滴发生经向失稳, 相应经向波频率为驱动频率的一半.
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关键词:
- 竖直振动 /
- 界面失稳 /
- 实验研究 /
- 仿真分析 /
- Faraday 不稳定性
Abstract: Surface destabilization of droplets under external excitation has always been a matter of interest in the field of fluid dynamics. Waveforms with different morphologies or secondary droplets would appear in the surface under different excitation conditions. In this paper, the analysis of the dynamic characteristics and generation mechanism of latitudinal and longitudinal waves was conducted. Firstly, an experimental system of droplet oscillation with controllable excitation amplitude and frequency was established. The experimental results show that different forcing amplitudes lead to different droplet interface instability modes. The longitudinal waves are generated only when the amplitude is large enough, and its evolution frequency is half of the driving frequency, while the latitudinal waves are always present whose frequency equals to the driving frequency. A change in driving frequency causes a shift in destabilization modes, and an increase in driving frequency increases the number of surface wave modes and decreases the wavelength of the surface waves. When the driving frequency exceeds a certain threshold, the waveform will shift from a latitudinal wave mode only to a latitudinal wave superimposed on a longitudinal wave mode. Meanwhile, three-dimensional numerical simulations were conducted. By studying the velocity and pressure fields of droplets, combined with the phase relationship between droplet vertex displacement and inertial force, the mechanism of droplet formation of latitudinal waves is elucidated: under the combined action of inertial force and surface tension, the droplet surface wave completes periodic energy conversion and transition. The surface wave characteristics dominated by the Faraday instability are analyzed comparatively for vertical versus radial acceleration direction. It is found that the geometrical characteristics of the droplet generate radial forces normal to the contact line, and when the vertical inertial force increases so that the radial force reaches a certain threshold, the droplet undergoes longitudinal instability, and the corresponding longitudinal wave frequency is half of the driving frequency. -
图 2 不同频率下的激励振幅与激励电压的关系
Figure 2. The relationship between forcing amplitude and excitation voltage of the plate under different frequencies
图 3 垂直惯性力和径向惯性力作用物理模型
Figure 3. Physical model with vertical inertia force and radial inertia force
图 4 液滴顶端位移的仿真和实验结果对比
Figure 4. Comparison between computational results of the droplet vertex with the experiment measurement
图 5 实验(工况E-A)和仿真(工况S-A)表面波形态的比较
Figure 5. Comparison of surface wave morphology of experiment (case E-A) and simulation (case S-A)
图 6 不同激励振幅下表面波形态的时间演化
Figure 6. Time evolution of surface wave morphology under different forcing amplitudes
图 7 不同驱动频率下表面波形态的时间演化
Figure 7. Time evolution of surface wave morphology under different forcing frequencies
图 8 液滴纬向表面波平均波长随频率的变化
Figure 8. Variation of the average wavelength of droplet latitudinal surface waves with frequency
图 9 液滴纬向表面波演化仿真结果(工况S-A)
Figure 9. Simulation results of droplet latitudinal surface wave generation process (case S-A)
图 10 稳定循环周期液滴顶端位移和惯性力随时间的演变(工况S-A)
Figure 10. Evolution of droplet vertex displacement and inertial force with time during the stabilization cycle (case S-A)
图 11 工况S-A在xoy截面上不同时刻的液滴压力场和速度矢量场
Figure 11. Droplet pressure field and velocity vector field at different moments for case S-A in the xoy cross section
图 12 液滴在x方向压力分布(工况S-A, $ t = 0 $)
Figure 12. Pressure distribution from top to bottom surface of droplet(case S-A, $ t = 0 $)
图 13 顶端表面波和次级表面波作用下压力随时间的变化(工况S-A)
Figure 13. The variation of pressure with time under the top surface wave and secondary surface wave (case S-A)
图 15 经向波波形和表面压力分布(工况S-B, $ t = 23.35 T $)
Figure 15. Longitudinal wave form and surface pressure distribution(case S-B,$ t = 23.35 T $)
图 16 液滴底部的压力场和速度矢量场的轮廓(工况S-B, y = 0)
Figure 16. Contour of the pressure field and velocity vector field at the bottom of the droplet (case S-B, y = 0)
图 17 径向加速度下Faraday不稳定性引起的液滴界面不稳定性(工况S-B)
Figure 17. Droplet interface instability dominated by Faraday instability under radial acceleration (case S-B)
表 1 水滴在室温、标准大气压下的物理特性
Table 1. Physical properties of water at atmospheric temperature and pressure
Diameter/mm Density/ (kg·m−3) Dynamic viscosity/ (mPa∙s) Surface tension coefficient/(mN·m−1) 7.5 ± 0.02 998 1.01 72.7 
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表 2 实验工况: 激励振幅的影响 (300 Hz)
Table 2. Experimental condition: influence of forcing amplitude (300 Hz)
Cases E-A E-B E-C Amplitude/μm 40 160 200
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表 3 实验工况: 激励频率的影响
Table 3. Experimental condition: influence of forcing frequency
Variables Values frequency/Hz 200, 250, 300, 350, 400, 450, 500, 800, 1000, 4000 voltage/V 5 ~ 30
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表 4 仿真工况设置
Table 4. Simulation operating condition parameters
Cases $ Re $ $ We $ ${\varDelta }_{D}$ S-A 16875 520 0.211 S-B 16875 520 0.421
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表 5 不同加速度形式下经向波的物理特征值
Table 5. Physical characteristic values of longitudinal surface waves in different acceleration forms
Wavelength diameter
ratio ($\varLambda /D)$Amplitude diameter
ratio ($ a/D $)radial acceleration 0.398 0.080 vertical acceleration 0.402 0.064 linear analysis 0.418 —
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