1.School of Precision Instruments and Optoelectronics Engineering, Tianjin University, Tianjin 300072, China 2.College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China 3.China Aerodynamics Research and Development Center, Mianyang 621000, China 4.Beijing Institute of Spacecraft Environment Engineering, Beijing 100094, China
Fund Project:Project supported by the Major Research Plan of the National Natural Science Foundation of China (Grant No. 91741107)
Received Date:14 April 2020
Accepted Date:11 May 2020
Available Online:07 June 2020
Published Online:20 September 2020
Abstract:Acoustically-excited bubble dynamics is the foundation of pipeline bubble detection based on acoustic technology. Due to the existence of multiple bubbles in pipeline flow, the Bjerknes forces among arbitrary bubbles under acoustic excitation may enforce bubble-bubble interaction and then change the features of bubble dynamics. Based on traditional free bubble’s Rayleigh-Plesset (R-P) model, this paper tries to establish bubble-bubble interaction model in consideration of the second Bjerknes force and bubble distribution in the pipeline axial direction. Meanwhile, the influence of finite wave speed in compressible fluid is considered. The proposed model is numerically calculated by the fourth-order Runge-Kutta method. Firstly, the differences in bubble feature between the free bubble’s R-P model and bubble-bubble interaction model are compared under excitation with different frequencies and amplitudes. Results show that the differences in bubble dynamics are minor when the bubble’s distance is large enough. When the bubble’s distance is fixed, the differences are significant on condition that the frequency of acoustic excitation is nearly the resonant frequency of bubbles. Secondly, through establishing compressible model and incompressible fluid model, we compare the differences between the two models. Numerical calculations show that the second Bjerknes force under the compressible assumption acts as an external force and forces the bubble to vibrate. On the other hand, the second Bjerknes force under the incompressible assumption changes the dynamics of bubble-bubble interaction as well as the resonant features. Finally, we study the effect of bubble-bubble distance and bubble’s axial position on bubble vibration characteristics. The bubble-bubble distance affects the second Bjerknes force and may lead the bubbles to vibrate nonlinearly. The bubble’s axial position changes the phase of external acoustic force and leads to the difference in initial vibration feature. When this difference is coupled with the second Bjerknes force, the bubble-bubble interaction may be changed even into nonlinear vibration, leading the bubble’s oscillation spectrum to differ from linear vibrations significantly. These results demonstrate that the resonant state of a small bubble may be converted into nonlinear vibration state if the second Bjerknes force is present. On the other hand, the resonant state of a large bubble can keep linear vibration when the second Bjerknes force is not obvious. Keywords:bubble dynamics/ second Bjerknes force/ compressible fluid/ bubble detection
本小节分析不同声波频率与幅度激励下的气泡动力学特征, 并对比双气泡与自由气泡的扰动动力学特征. 为降低气泡二次辐射力的影响, 在双气泡动力学计算过程中, 两气泡之间的距离设置为$h = 8$ mm, 对应气泡间二次辐射力延时时间约为$4.73 \times {10^{ - 6}}$ s. 数值计算中时间迭代步长为10–7 s, 气泡的初始状态设置为${R_{1, 2}} = {R_{10, 20}}$以及${\dot R_{1, 2}} = 0$. 为保证激励信号对双气泡的影响一致, 气泡球心连接线与轴向坐标$x$轴的夹角设置为${{\text{π}} / 2}$. 图2给出了相应的气泡半径的演化过程. 图 2 三种不同频率(2, 15和50 kHz)与两类幅度(1 Pa与1 kPa)声波激励下的自由气泡与双气泡设置下半径振动特征 (a) 1 Pa声波激励下气泡1在三种激励频率工况下自由气泡与双气泡约束对应的气泡半径振动特征; (b) 1 Pa声波激励下气泡2在三种激励频率工况下自由气泡与双气泡约束对应的气泡半径振动特征; (c) 1 kPa声波激励下气泡1在三种激励频率工况下自由气泡与双气泡约束对应的气泡半径振动特征; (d) 1 kPa声波激励下气泡2在三种激励频率工况下自由气泡与双气泡约束对应的气泡半径振动特征; 图例在图(d)中给出 Figure2. Effects of acoustic excitations with different frequencies (2, 15 and 50 kHz) and amplitudes (1 Pa and 1 kPa) on the bubble dynamics under single free and regulated two-bubble vibrations: (a) Relative radius of bubble 1 between two configurations (single free and regulated two-bubble vibrations) with the amplitude of acoustic excitation being 1 Pa; (b) relative radius of bubble 2 between two configurations (single free and regulated two-bubble vibrations) with the amplitude of acoustic excitation being 1 Pa; (c) relative radius of bubble 1 between two configurations (single free and regulated two-bubble vibrations) with the amplitude of acoustic excitation being 1 kPa; (d) relative radius of bubble 2 between two configurations (single free and regulated two-bubble vibrations) with the amplitude of acoustic excitation being 1 kPa. The figure legend is given in panel (d).
本小节分析气泡距离对双气泡动力学的影响. 具体而言, 设置两个工况, 即气泡间距离$h$分别为2与8 mm, 对应平衡状态下时间延时分别为$6.76 \;\times {10^{ - 7}}$与$4.73 \times {10^{ - 6}}$ s. 数值计算中时间迭代步长为10–7 s, 气泡的初始状态设置为${R_{1, 2}} = {R_{10, 20}}$以及${\dot R_{1, 2}} = 0$. 外界声波的激励频率和幅度与3.1小节保持一致. 图3给出两个气泡的相对半径动力学特征, 图4给出了两个气泡受到的相应次Bjerknes辐射力. 图 3 三种不同频率(2, 15和50 kHz)与两类幅度(1 Pa与1 kPa)声波激励下的不同气泡距离(2与8 mm)对气泡振动动力学特征影响 (a) 1 Pa声波激励下气泡1在三种激励频率工况下不同气泡距离对应的气泡半径振动特征; (b) 1 Pa声波激励下气泡2在三种激励频率工况下不同气泡距离对应的气泡半径振动特征; (c) 1 kPa声波激励下气泡1在三种激励频率工况下不同气泡距离对应的气泡半径振动特征; (d) 1 kPa声波激励下气泡2在三种激励频率工况下不同气泡距离对应的气泡半径振动特征; 图例在图(a)和(c)中给出 Figure3. Effects of acoustic excitations with different frequencies (2, 15 and 50 kHz) and amplitudes (1 Pa and 1 kPa) on the bubble dynamics under different distances, with 2 mm and 8 mm, between the two bubbles: (a) Relative radius of bubble 1 under two different bubble’s distance with the amplitude of acoustic excitation being 1 Pa; (b) relative radius of bubble 2 under two different bubble’s distance with the amplitude of acoustic excitation being 1 Pa; (c) relative radius of bubble 1 under two different bubble’s distance with the amplitude of acoustic excitation being 1 kPa; (d) relative radius of bubble 2 under two different bubble’s distance with the amplitude of acoustic excitation being 1 kPa. The figure legend is given in panel (a) and (c).
图 4 三种不同频率(2, 15和50 kHz)与两类幅度(1 Pa与1 kPa)声波激励下的不同气泡距离(2与8 mm)形成的次Bjerknes辐射力 (a) 1 Pa声波激励下气泡1在三种激励频率工况下不同气泡距离对应的次Bjerknes辐射力; (b) 1 Pa声波激励下气泡2在三种激励频率工况下不同气泡距离对应的次Bjerknes辐射力; (c) 1 kPa声波激励下气泡1在三种激励频率工况下不同气泡距离对应的次Bjerknes辐射力; (d) 1 kPa声波激励下气泡2在三种激励频率工况下不同气泡距离对应的次Bjerknes辐射力 Figure4. Effects of acoustic excitations with different frequencies (2, 15 and 50 kHz) and amplitudes (1 Pa and 1 kPa) on the second Bjerknes force under different distances, with 2 mm and 8 mm, between the two bubbles: (a) Relative radius of bubble 1 under two different bubble’s distance with the amplitude of acoustic excitation being 1 Pa; (b) relative radius of bubble 2 under two different bubble’s distance with the amplitude of acoustic excitation being 1 Pa; (c) relative radius of bubble 1 under two different bubble’s distance with the amplitude of acoustic excitation being 1 kPa; (d) relative radius of bubble 2 under two different bubble’s distance with the amplitude of acoustic excitation being 1 kPa.
本小节对气泡轴向位置的影响进行分析, 通过改变两气泡球心连线与轴向坐标$x$轴夹角$\theta $的大小实现轴向位置的改变. 在数值计算过程中, $\theta $分别取值0, ${{\text{π}} / 2}$, ${\text{π}}$, 对应三种不同位置. 此外, 气泡球心距离设置为2 mm, 激励声波幅度为1 kPa. 数值计算中时间迭代步长为10–7 s, 气泡的初始状态设置为${R_{1, 2}} = {R_{10, 20}}$以及${\dot R_{1, 2}} = 0$. 图5—图7给出了气泡半径的动态演化过程图. 图 5 2 kHz声波激励下不同气泡位置对气泡振动过程的影响 (a)三种不同相对位置工况下气泡1相对半径动态变化图; (b)三种不同相对位置工况下气泡2相对半径动态变化图 Figure5. Effects of different bubble locations on the bubble’s dynamics with the frequency of acoustic excitation being 2 kHz: (a) The radius dynamics of bubble 1 under three different locations; (b) the radius dynamics of bubble 2 under three different locations
图 7 50 kHz声波激励下不同气泡位置对气泡振动过程的影响 (a)三种不同相对位置工况下气泡1相对半径动态变化图; (b)三种不同相对位置工况下气泡2相对半径动态变化图 Figure7. Effects of different bubble locations on the bubble’s dynamics with the frequency of acoustic excitation being 50 kHz: (a) The radius dynamics of bubble 1 under three different locations; (b) the radius dynamics of bubble 2 under three different locations.
物理上而言, 当$\theta = {{\text{π}} / 2}$时, 两个气泡的轴向位置一致, 激励信号对两个气泡的相位和幅度保持一致. 当$\theta = 0$时, 两个气泡的轴向位置不同, 且满足${x_2} - {x_1} = 2 $ mm. 当$\theta = {\text{π}}$时, 两气泡的轴向位置满足${x_2} - {x_1} = - 2$ mm. 针对$\theta = 0$以及$\theta = {\text{π}}$的工况, 外界激励声波对两气泡存在相位的差别. 当声波激励频率为2 kHz时, 三种位置工况($\theta = 0$, $\theta = {{\text{π}} / 2}$与$\theta = {\text{π}}$)对应的相位延时分别为$0.017{\text{π}}$, 0以及$ - 0.017{\text{π}}$. 由图5可以看出, 在上述延迟工况下, 气泡1与气泡2的相对半径变化趋势在三种工况下基本保持一致. 当声波激励频率为15 kHz时, 三种位置工况对应的相位延时分别为$0.1274{\text{π}}$, 0以及$ - 0.1274{\text{π}}$. 由图6可得, 气泡1与气泡2的相对半径趋势在初始状态存在微小区别. 随着时间的增加, 由于次Bjerknes辐射力的影响, 气泡1与气泡2的相对半径变化在三种工况下的演化趋势差别明显. 相位的不一致带来次Bjerknes辐射力的变化, 次Bjerknes辐射力的改变不仅影响相位的变化, 而且明显改变气泡半径的变化幅度. 当声波激励频率为50 kHz时, 三种位置工况对应的相位延时分别为$0.4245{\text{π}}$, 0以及$ - 0.4245{\text{π}}$. 图 6 15 kHz声波激励下不同气泡位置对气泡振动过程的影响 (a)三种不同相对位置工况下气泡1相对半径动态变化图; (b)三种不同相对位置工况下气泡2相对半径动态变化图 Figure6. Effects of different bubble locations on the bubble’s dynamics with the frequency of acoustic excitation being 15 kHz: (a) The radius dynamics of bubble 1 under three different locations; (b) the radius dynamics of bubble 2 under three different locations.