Search

Article

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

Coupled resonance of bubbles in spherical cavitation clouds

Zhang Peng-Li Lin Shu-Yu Zhu Hua-Ze Zhang Tao

Coupled resonance of bubbles in spherical cavitation clouds

Zhang Peng-Li, Lin Shu-Yu, Zhu Hua-Ze, Zhang Tao
PDF
HTML
Get Citation
  • The interaction between bubbles in bubble group mainly acts on the other bubble through radiation sound pressure between the bubbles. In this paper, based on the bubble dynamics equation in bubble clouds, the equation of bubble wall motion is linearly reduced, the expression of bubble resonance frequency in spherical bubble group is obtained and the correction coefficient of bubble resonance frequency and single bubble are given. Furthermore, the effects of the initial radius, the number of bubbles and the distance between bubbles on the resonance frequency are discussed. The results show that the phase of bubbles is taken into account. Considering the interaction between bubbles, the resonance frequency of bubbles in spherical bubble group is obviously less than that of single bubble. With the decrease of the number of bubbles in bubble group, the distance between bubbles increases, the interaction between bubbles in bubble group decreases, and the resonance frequency of bubbles returns to the resonance frequency of Minnaert single bubble. At the same time, the resonance frequency of bubbles in bubble group changes gradient with the increase of the distance between bubbles and the number of bubbles. However, when the number of bubbles increases a certain value, the resonant frequency of the bubble is almost constant. When the bubble group has a certain radius, the more the number of bubbles, the smaller the resonance frequency of the bubble is, but there exists a critical value. It is also found that a smaller correction coefficient is held by the bubble group with larger initial radius, which indicates the same number of bubble groups. Under the same bubble spacing, the interaction of small bubbles with smaller bubbles is more significant, and the resonance frequency of the bubble is obviously affected. Because the frequency and amplitude of driving sound pressure can only be given values in ultrasonic cavitation, the resonant frequency of cavitation bubbles will be reduced by properly injecting air bubbles into liquid, which makes most of cavitation bubbles undergo intense non-linear oscillating steady-state cavitation. Therefore, the occurrence of cavitation can be effectively suppressed.
      Corresponding author: Lin Shu-Yu, 253383739@qq.com
    [1]

    陈伟中 2014 声空化物理 (北京, 科学出版社)第2—5页

    Chen W Z 2014 Sound Cavitation Physics (Beijing: Science Press) pp2–5 (in Chinese)

    [2]

    Rayleigh J W 1917 Philosophical Magazine 34 94

    [3]

    Cole R H 1948 Underwater Eplosion (Princeton: Princeton U.P)pp60–65

    [4]

    Npltingk B E 2002 Proc. Phys. Soc. 63 674

    [5]

    Plesset M S, Chapmam R B 1977 J. Fluid Mech. 9 145

    [6]

    Mason T J, J P Lorimer 1988 Lorimer Application and Use of Ultrasound in Chemisty (USA: Ellis Horworrd Limited) pp130—135

    [7]

    Madrazo A, Garcia N, Nieto-Vesperinas M 1998 Phys. Rev. Lett. 80 4590

    [8]

    Shim A 1971 J. Basic Engin. 93 426

    [9]

    An Y 2011 Phys. Rev. E 83 66313−1

    [10]

    王成会, 林书玉 2010 力学学报 42 1050

    Wang C H, Lin S Y 2010 Acta Mech. Sin. 42 1050

    [11]

    胡静, 林书玉, 王成会, 李锦 2013 物理学报 62 114334

    Hu J, Lin S Y, Wang C H, Li J 2013 Acta Phys. Sin. 62 114334

    [12]

    王成会, 莫润阳, 胡静 2016 物理学报 65 144301

    Wang C H, Mo R Y, Hu J 2016 Acta Phys. Sin. 65 144301

    [13]

    王成会, 程建春 2014 物理学报 63 134301

    Wang C H, Cheng J C 2014 Acta Phys. Sin. 63 134301

    [14]

    Yasui K, Iida Y, Tuziuti T, et al. 2008 Phys. Rev. E 77 66313−1

    [15]

    Barber B P, Hiller R A, Ritva L, et al. 1997 Phys.Reports 281 65

    [16]

    Kwak H Y, Na J H 1996 Phys. Rev. Lett. 77 4454

    [17]

    Wang Q X 2004 Phys. Fluids 165 1610

    [18]

    Hsiao C T, Choi J K, Singh S, Chahine G L, et al 2013 J. Fluid Mech. 716 137

    [19]

    Keith W, Seth J P 1996 Phys. Rev. E 54 R2205

    [20]

    Cui P, Wang Q X, Wang S P, Zhang A M 2016 Phys. Fluids 28 94

    [21]

    苗博雅, 安宇 2015 物理学报 64 204301−1

    Miao B Y, An Y 2015 Acta Phys. Sin. 64 204301−1

    [22]

    蔡军, 淮秀兰等 2015 科学通报 56 947

    Cai J, Huai X L, et al. 2015 Chinese Sci Bull. 56 947

    [23]

    Wang C, Khoo B C, Yeo K S 2003 Comput. Fluids 32 1195

  • 图 1  球形空化云中气泡的运动 (a)气泡初始半径5 μm; (b)泡群中气泡数量为100; (c), (d)气泡初始半径5 μm, 泡群中气泡数量N = 100

    Figure 1.  Movement of bubbles in spherical cavitation clouds: (a) Bubble initial radius 5 μm; (b) the number 100; (c), (d) bubble initial radius 5 μm, N = 100.

    图 2  泡群中气泡的谐振频率 (a)气泡谐振频率与泡群中气泡之间距离关系; (b)气泡谐振频率与泡群中气泡数量之间关系, 气泡的初始半径均为20 μm

    Figure 2.  Resonance frequency of bubbles in bubble group: (a) The relationship between bubble resonance frequency and distance in the bubble group; (b) the relationship between bubble resonance frequency and the number of bubbles in the bubble group, the initial radius of the bubbles is 20 μm.

    图 3  修正系数M与泡群中气泡距离关系

    Figure 3.  Relationship between correction coefficient and bubble distance.

    图 4  气泡初始半径与泡群中气泡谐振频率的关系

    Figure 4.  Relationship between the initial radius of bubbles and the resonant frequency of bubbles.

    图 5  单气泡体积变化图

    Figure 5.  Volume change of single bubble

    图 7  球状气泡云体积变化图(N = 5)

    Figure 7.  Volume change of spherical cavitation cloud N = 5.

    图 6  球状空化云气泡体积变化图(N = 9)

    Figure 6.  Volume change of spherical cavitation cloud N = 9.

  • [1]

    陈伟中 2014 声空化物理 (北京, 科学出版社)第2—5页

    Chen W Z 2014 Sound Cavitation Physics (Beijing: Science Press) pp2–5 (in Chinese)

    [2]

    Rayleigh J W 1917 Philosophical Magazine 34 94

    [3]

    Cole R H 1948 Underwater Eplosion (Princeton: Princeton U.P)pp60–65

    [4]

    Npltingk B E 2002 Proc. Phys. Soc. 63 674

    [5]

    Plesset M S, Chapmam R B 1977 J. Fluid Mech. 9 145

    [6]

    Mason T J, J P Lorimer 1988 Lorimer Application and Use of Ultrasound in Chemisty (USA: Ellis Horworrd Limited) pp130—135

    [7]

    Madrazo A, Garcia N, Nieto-Vesperinas M 1998 Phys. Rev. Lett. 80 4590

    [8]

    Shim A 1971 J. Basic Engin. 93 426

    [9]

    An Y 2011 Phys. Rev. E 83 66313−1

    [10]

    王成会, 林书玉 2010 力学学报 42 1050

    Wang C H, Lin S Y 2010 Acta Mech. Sin. 42 1050

    [11]

    胡静, 林书玉, 王成会, 李锦 2013 物理学报 62 114334

    Hu J, Lin S Y, Wang C H, Li J 2013 Acta Phys. Sin. 62 114334

    [12]

    王成会, 莫润阳, 胡静 2016 物理学报 65 144301

    Wang C H, Mo R Y, Hu J 2016 Acta Phys. Sin. 65 144301

    [13]

    王成会, 程建春 2014 物理学报 63 134301

    Wang C H, Cheng J C 2014 Acta Phys. Sin. 63 134301

    [14]

    Yasui K, Iida Y, Tuziuti T, et al. 2008 Phys. Rev. E 77 66313−1

    [15]

    Barber B P, Hiller R A, Ritva L, et al. 1997 Phys.Reports 281 65

    [16]

    Kwak H Y, Na J H 1996 Phys. Rev. Lett. 77 4454

    [17]

    Wang Q X 2004 Phys. Fluids 165 1610

    [18]

    Hsiao C T, Choi J K, Singh S, Chahine G L, et al 2013 J. Fluid Mech. 716 137

    [19]

    Keith W, Seth J P 1996 Phys. Rev. E 54 R2205

    [20]

    Cui P, Wang Q X, Wang S P, Zhang A M 2016 Phys. Fluids 28 94

    [21]

    苗博雅, 安宇 2015 物理学报 64 204301−1

    Miao B Y, An Y 2015 Acta Phys. Sin. 64 204301−1

    [22]

    蔡军, 淮秀兰等 2015 科学通报 56 947

    Cai J, Huai X L, et al. 2015 Chinese Sci Bull. 56 947

    [23]

    Wang C, Khoo B C, Yeo K S 2003 Comput. Fluids 32 1195

  • [1] Zhang A-Man, Yao Xiong-Liang. The law of the bubble motion near the wall. Acta Physica Sinica, 2008, 57(3): 1662-1671. doi: 10.7498/aps.57.1662
    [2] Zhang A-Man, Yao Xiong-Liang. The law of the underwater explosion bubble motion near free surface. Acta Physica Sinica, 2008, 57(1): 339-353. doi: 10.7498/aps.57.339
    [3] Zhang Hua-Wei, Li Yan-Xiang. Study on bubble nucleation in liquid metal. Acta Physica Sinica, 2007, 56(8): 4864-4871. doi: 10.7498/aps.56.4864
    [4] Wang Shi-Ping, Zhang A-Man, Liu Yun-Long, Yao Xiong-Liang. Numerical simulation of bubbles coupled with an elastic membrane. Acta Physica Sinica, 2011, 60(5): 054702. doi: 10.7498/aps.60.054702
    [5] Zhang A-Man, Wang Chao, Wang Shi-Ping, Cheng Xiao-Da. Experimental study of interaction between bubble and free surface. Acta Physica Sinica, 2012, 61(8): 084701. doi: 10.7498/aps.61.084701
    [6] Wu Wei, Sun Dong-Ke, Dai Ting, Zhu Ming-Fang. Modeling of dendritic growth and bubble formation. Acta Physica Sinica, 2012, 61(15): 150501. doi: 10.7498/aps.61.150501
    [7] Wang Shi-Ping, Zhang A-Man, Liu Yun-Long, Wu Chao. Experimental research on bubble dynamics near circular hole of plate. Acta Physica Sinica, 2013, 62(6): 064703. doi: 10.7498/aps.62.064703
    [8] Zhang A-Man, Xiao Wei, Wang Shi-Ping, Cheng Xiao-Ou. Experimental study of the interactions between a pulsating bubble and sand particles with different diameters. Acta Physica Sinica, 2013, 62(1): 014703. doi: 10.7498/aps.62.014703
    [9] Li Shuai, Zhang A-Man, Wang Shi-Ping. Experimental and numerical studies on “crown” spike generated by a bubble near free-surface. Acta Physica Sinica, 2013, 62(19): 194703. doi: 10.7498/aps.62.194703
    [10] Li Shuai, Zhang A-Man. Study on a rising bubble bouncing near a rigid boundary. Acta Physica Sinica, 2014, 63(5): 054705. doi: 10.7498/aps.63.054705
    [11] Jiang Dan, Li Song-Jing, Bao Gang. Parameter identification of gas bubble model in pressure pulsations using genetic algorithms. Acta Physica Sinica, 2008, 57(8): 5072-5080. doi: 10.7498/aps.57.5072
    [12] Liu Yun-Long, Zhang A-Man, Wang Shi-Ping, Tian Zhao-Li. Research on interaction between bubble and surface waves based on BEM. Acta Physica Sinica, 2012, 61(22): 224702. doi: 10.7498/aps.61.224702
    [13] Ni Bao-Yu, Li Shuai, Zhang A-Man. Jet splitting after bubble breakup at the free surface. Acta Physica Sinica, 2013, 62(12): 124704. doi: 10.7498/aps.62.124704
    [14] Liang Shan-Yong, Wang Jiang-An, Zong Si-Guang, Wu Rong-Hua, Ma Zhi-Guo, Wang Xiao-Yu, Wang Le-Dong. Laser detection method of ship wake bubbles based on multiple scattering intensity and polarization characteristics. Acta Physica Sinica, 2013, 62(6): 060704. doi: 10.7498/aps.62.060704
    [15] Liu Yun-Long, Zhang A-Man, Wang Shi-Ping, Tian Zhao-Li. Study on bubble dynamics near plate with hole based on boundary element method. Acta Physica Sinica, 2013, 62(14): 144703. doi: 10.7498/aps.62.144703
    [16] Shi Dong-Yan, Wang Zhi-Kai, Zhang A-Man. Study on coupling characteristics between bubble and complex walls at the same scale. Acta Physica Sinica, 2014, 63(17): 174701. doi: 10.7498/aps.63.174701
    [17] Wang Shu-Shan, Li Mei, Ma Feng. Dynamics of the interaction between explosion bubble and free surface. Acta Physica Sinica, 2014, 63(19): 194703. doi: 10.7498/aps.63.194703
    [18] Li Hong-Xing, Tao Chun-Hui, Liu Fu-Lin, Zhou Jian-Ping. Effect of gas bubble on acoustic characteristic of sediment: taking sediment from East China Sea for example. Acta Physica Sinica, 2015, 64(10): 109101. doi: 10.7498/aps.64.109101
    [19] Zheng Jian, Zhang Duo, Jiang Bang-Hai, Lu Fang-Yun. Formation mechanism of water jets induced by the interaction between bubble and free surface. Acta Physica Sinica, 2017, 66(4): 044702. doi: 10.7498/aps.66.044702
    [20] Ai Xu-Peng, Ni Bao-Yu. Influence of viscosity and surface tension of fluid on the motion of bubbles. Acta Physica Sinica, 2017, 66(23): 234702. doi: 10.7498/aps.66.234702
  • Citation:
Metrics
  • Abstract views:  427
  • PDF Downloads:  14
  • Cited By: 0
Publishing process
  • Received Date:  14 March 2019
  • Accepted Date:  28 April 2019
  • Available Online:  06 June 2019
  • Published Online:  05 July 2019

Coupled resonance of bubbles in spherical cavitation clouds

    Corresponding author: Lin Shu-Yu, 253383739@qq.com
  • 1. Institute of Applied Acoustics Shaanxi Normal University, Xi’an 710062, China
  • 2. College of Science, Xi’an University of Science and Technology, Xi’an 710054, China

Abstract: The interaction between bubbles in bubble group mainly acts on the other bubble through radiation sound pressure between the bubbles. In this paper, based on the bubble dynamics equation in bubble clouds, the equation of bubble wall motion is linearly reduced, the expression of bubble resonance frequency in spherical bubble group is obtained and the correction coefficient of bubble resonance frequency and single bubble are given. Furthermore, the effects of the initial radius, the number of bubbles and the distance between bubbles on the resonance frequency are discussed. The results show that the phase of bubbles is taken into account. Considering the interaction between bubbles, the resonance frequency of bubbles in spherical bubble group is obviously less than that of single bubble. With the decrease of the number of bubbles in bubble group, the distance between bubbles increases, the interaction between bubbles in bubble group decreases, and the resonance frequency of bubbles returns to the resonance frequency of Minnaert single bubble. At the same time, the resonance frequency of bubbles in bubble group changes gradient with the increase of the distance between bubbles and the number of bubbles. However, when the number of bubbles increases a certain value, the resonant frequency of the bubble is almost constant. When the bubble group has a certain radius, the more the number of bubbles, the smaller the resonance frequency of the bubble is, but there exists a critical value. It is also found that a smaller correction coefficient is held by the bubble group with larger initial radius, which indicates the same number of bubble groups. Under the same bubble spacing, the interaction of small bubbles with smaller bubbles is more significant, and the resonance frequency of the bubble is obviously affected. Because the frequency and amplitude of driving sound pressure can only be given values in ultrasonic cavitation, the resonant frequency of cavitation bubbles will be reduced by properly injecting air bubbles into liquid, which makes most of cavitation bubbles undergo intense non-linear oscillating steady-state cavitation. Therefore, the occurrence of cavitation can be effectively suppressed.

    • 超声空化现象是发生在液体中的强声现象, 是由于液体中压力的变化引发气泡的生长、溃灭现象, 液体中几乎所有的强声技术都伴随着超声空化现象的发生[13]. 声空化是一种经典的物理现象, 但空化泡的微米尺度和高速脉动阻碍了人们对它内部微观过程的理解, 探索空化泡内部极端物理条件、声致发光微观机理等仍是当前声学和物理学界的热门课题[48].

      在实际空化过程中, 空化现象是以空化云形式存在的, 由于气泡的振动会在液体中形成次级声辐射, 气泡之间的相互作用对每个气泡的振动特性影响将都不能忽略. 为了更切合实际, 人们开始将注意力从单一空化泡转向双空化泡和空化泡群. 1971年, Shim[8]对双泡问题进行了讨论. An[9]给出了气泡呈链式泡群和球状泡群内气泡动力学方程, 并分析了泡群内气泡的运动行为和多泡声致发光特征. 文献[1013]研究了超声波作用下泡群的非线性响应、共振响应、耦合振动等; Yasui等[14]给出了两束空化云相互作用的气泡动力学方程. 泡群动力学的研究有利于发展控制超声空化效应强弱的理论和方法. 气泡在泡群内由于受到其他气泡的相互作用, 其自身振动会受到抑制. Barber等[15]研究发现在液体中注入大气泡能够抑制空化的发生和发展, 而其研究主要是讨论气泡之间的相互作用对泡壁运动的影响. 文献[16,17]模拟了自有液面下两个水平排列的气泡的耦合作用. 还有研究表明, 当气泡初始距离较近时, 气泡之间还会发生融合现象[18]. 然而气泡在声场中的振动是与外界驱动声压频率和声压幅值有着密切的关系, 能否发生瞬态空化效应, 主要取决于外界驱动声压频率和气泡自身谐振频率之间的关系. 本文则通过对球状泡群内气泡的运动方程进行线性约化, 得到气泡的谐振频率, 并研究了泡群中气泡的谐振频率与单泡的谐振频率的关系, 以及影响气泡谐振频率的主要参数. 希望对多气泡环境中气泡的受迫振动及多气泡空化理论提供依据.

    2.   理论模型
    • 由于气泡间距和气泡尺寸相对于液体中声波波长来说非常小, 故可以认为气泡处于相同的声场中. 设气泡间距为${r_{ij}}$, 球状泡群半径为r, 声场中空化泡的运动方程的得到基于以下几个假设[1923]:

      1)由于表面张力的作用, 空化泡膨胀和收缩期间始终为球形, 在声波作用下气泡仍能保持完好的球对称性, 做几乎纯径向脉动, 且始终在同一位置振动, 没有发生平动;

      2)不考虑空化泡在谐振过程中的热交换、水蒸汽的相变、气体质量交换及泡内的化学反应;

      3)考虑液体黏滞性, 表面张力及液体的蒸汽压.

      此时球状泡群内空化泡的动力学方程可以表示为[9,12,20]:

      式中最后一项即体现出空化泡之间相互作用项. 式中${R_i}$为任意时刻气泡半径, ${R_{i0}}$为其初始半径, ρ为气泡外液体密度, ${p_\infty }$为气泡泡壁外侧液体的压力, u为液体黏滞系数, σ为表面张力系数, ${p_{i{{\rm{g}}_0}}}$为气泡内部气体压强. 由于空化过程极短, 只有短短的几十微秒, 在空化过程中体积又非常小, 可近似认为空化过程为绝热等熵过程.

      图1给出了不同气泡数量、初始半径、驱动声频率、驱动声强下气泡的半径随时间的变化关系. 图1(a)图1(b)为不同气泡数量、不同初始半径下泡壁半径、速度随时间的变化关系, 其中外界驱动频率20 kHz, 强度${p_a} = 1.2 \times {10^5}\;{\rm{Pa}}$. 从图1(a)中不难看出空化云中气泡数量影响气泡的膨胀比. 当空化云气泡数量增多时, 气泡受其他气泡之间的抑制作用越强烈, 会使得气泡的膨胀比显著减小. 同时气泡生长到最大和溃灭时间也会延迟, 这就说明气泡数量越多, 气泡之间抑制作用越强, 气泡越难生长起来. 当空化云中气泡数量由10增大到500时, 气泡的膨胀比减小, 由最大28减小到15. 由图1(b)可以看出空化泡的初始半径对其膨胀比影响比较明显, 当其由5 μm增大到20 μm时, 气泡的膨胀比由25减小到2.8. 综上可得空化云中气泡数量越多, 气泡的初始半径越大气泡越不容易生长起来, 同样的气泡数量, 小气泡则更容易生长起来. 曾有学者指出, 为了控制空化强度, 可以在水中注入不同数量的大气泡, 通过气泡之间的相互作用来调节空化强弱[15]. 图1(c)图1(d)给出了不同驱动声参数时气泡半径随时间的变化关系, 选取气泡的初始半径为5 μm, 气泡数量为100. 由图1(c)图1(d)不难看出驱动声压频率小、声压幅值小时, 气泡的膨胀比反倒大一些.

      Figure 1.  Movement of bubbles in spherical cavitation clouds: (a) Bubble initial radius 5 μm; (b) the number 100; (c), (d) bubble initial radius 5 μm, N = 100.

      实际上气泡在声场作用下是一种受迫振动, 而影响受迫振动的振幅或气泡膨胀比的主要因素则是气泡自身谐振频率、外界驱动声压幅值、频率等参数, 为了研究空化泡谐振频率与气泡数量、气泡初始半径、气泡之间距离的关系, 对(1)式进行线性约化.

    3.   泡壁运动方程的线性约化
    • 假设球形空化云中有N个气泡, 一气泡处于球心位置, 其余气泡均匀分布在距离为d的球面上. 为了简化处理, 认为气泡具有相同初始半径, 被研究的气泡处于球心位置. 此时(1)式化简后变为:

      在气泡做小幅振动情况下, 考虑液体黏滞系数的影响时, 令

      同时对(2)式做以下线性化处理[1]:

      将(3)和(4)式代入(2)式, 同时考虑到x是一微小量, 忽略二阶无穷小量${\dot x^2}$, 化简可得到

      注意到[1]${\left( {1 + x} \right)^2} = \left( {1 + 2x + {x^2}} \right) \approx 1 + x$, $\displaystyle\frac{1}{{1 + x}} \approx$$ 1 - x$. 代入(5)式, 化简并整理后得到

      (6)式左边第二项对应气泡泡壁受迫振动的阻尼项, 第三项对应气泡的本征频率. 由(6)式可以看出超声波对气泡的驱动, 是一种强迫驱动下的阻尼振动. 除了强迫振动外, 还包含参数驱动, 即使在小幅驱动下, 气泡的脉动也包含基波、谐波、次谐波等成分. 式中当$d \to \infty $, 或者N = 1时, 上式回归到单气泡的小幅振动动力学方程. 上式中${\omega _r}$为气泡谐振的角频率, 且有

      或者写成

      式中$ {\omega _r} = 2{\text{π}}{f_r}$.

      (8)式即为泡群中气泡的谐振频率, 因泡群中气泡运动过程中受其他气泡的相互作用力, 其谐振频率明显不同于单泡的谐振频率. 从式中不难看出气泡的谐振频率除了与气泡的初始半径有关外, 还与球状空化云中气泡数量、气泡之间距离等参数有关.

    4.   数值模拟
    • 本文将水作为液体介质, 计算参数为[19]: ${\rho} = $$ 1000\;{\rm{kg}}/{{\rm{m}}^{\rm{3}}}$, $\delta = 0.072\;{\rm{N}}/{\rm{m}}$. (8)式中气泡的谐振频率除了与气泡的初始半径有关外, 还与球状空化云中气泡数量、气泡之间距离有密切关系.

      图2中选取空化泡的初始半径为${R_0}$ = 20 μm, 得到空化泡之间距离、球状空化云中气泡数量等参数与气泡谐振频率之间关系. 图2(a)为气泡之间距离与气泡谐振频率之间关系, 对于初始半径为${R_0}$ = 20 μm 的气泡群, 当气泡之间距离为1到8个气泡初始半径时, 气泡的谐振频率受气泡之间相互作用影响较大. 再增大气泡之间距离时, 气泡之间相互作用几乎可以不计, 此时气泡谐振频率趋于一恒定值. 不同数量的气泡其谐振频率也不同, 在气泡之间距离相等时, 泡群中数量少的气泡受其他气泡之间的相互作用小, 气泡谐振频率较大. 同时泡群中气泡数量少的其谐振频率在气泡之间距离较近时受其他气泡的影响较为显著, 变化梯度要明显大于气泡数量多的泡群. 图2(b)为空化云中气泡数量与气泡谐振频率之间关系, 可以看出气泡谐振频率随气泡数量的增大而减小, 当气泡数量从1增加到150时, 气泡的谐振频率随数量变化显著, 急剧减小, 当气泡数量增加一定值后, 气泡的谐振频率几乎不变. 也就是说当泡群半径一定时, 不是气泡数量越多气泡的谐振频率越小, 而是有一个临界值.

      Figure 2.  Resonance frequency of bubbles in bubble group: (a) The relationship between bubble resonance frequency and distance in the bubble group; (b) the relationship between bubble resonance frequency and the number of bubbles in the bubble group, the initial radius of the bubbles is 20 μm.

    • 由上面分析可知气泡的谐振频率与气泡之间距离、气泡数量等参数有关, 当气泡之间距离较大时, 气泡之间相互作用可以忽略, 气泡的谐振频率趋于一定值. 为了研究泡群中气泡谐振频率与单泡谐振频率的关系, 先给出对于在密度为ρ的液体中初始半径为${R_0}$的空化泡自然谐振频率${f_0}$, Minnaert给出的表达式为[1]

      为了描述泡群中气泡谐振频率与单泡的Minnaert频率之间关系, 对单泡的Minnaert频率给一修正系数M, 令${f_r} = M{f_0}$, 或者

      将(8)和(9)式代入(10)式有

      (11)式即为球状泡群中气泡的谐振频率和单泡Minnaert频率的修正关系. 不难看出只要N ≠ 1时, 便有M < 1, 即只要泡群中气泡数量多与1个, 气泡的谐振频率就会小于单泡的谐振频率. 这就说明泡群中的气泡因受到其他气泡的抑制作用, 降低了自身谐振频率. 而空化现象能否发生与气泡谐振频率有关, 那么在液体中注入气泡后气泡之间相互作用增强会降低泡群中气泡的谐振频率, 使得能发生瞬态空化的气泡数相应减小, 空化强度就会减弱. (11)式中当N = 1时M = 1, 泡群的谐振频率回归到单泡的Minnaert频率.

      考虑到气泡之间相互作用后, 泡群中气泡的谐振频率与单气泡的频率比值M与气泡的初始半径、泡群中气泡数量、气泡之间距离有很大关系. 图3(a)为修正系数M与气泡之间距离的关系, 取气泡的初始半径为20 μm, 泡群中气泡数量分别为10, 20, 50, 100, 150, 200. 由图可以看出当气泡之间距离增大时M的值逐渐增大, 最后均趋于一定值. 图3(b)中泡群中气泡数量取10, 气泡的初始半径分别为2, 6, 10, 20, 40 μm. 对比发现初始半径较大的气泡群具有较小的M, 说明同样数量的泡群在同样的气泡间距下, 大气泡较小气泡的相互作用要显著一些, 对气泡的谐振频率影响明显.

      图4为泡群中气泡谐振频率与气泡初始半径之间关系, 由图可知气泡初始半径越大时, 气泡的修正系数越小. 这也说明气泡初始半径越大, 气泡之间的相互影响越明显, 气泡的谐振频率越小, 越难发生空化. 即相同参数下, 大气泡相比小气泡更难发生空化现象. 观察图4, 当气泡半径均为10 μm时, 数量较少的泡群, 或是间距较大的泡群内气泡将越稀疏, 此时气泡受到的抑制作用越小, 修正系数则越接近1, 气泡的谐振频率趋近于单泡谐振频率.

      Figure 3.  Relationship between correction coefficient and bubble distance.

      Figure 4.  Relationship between the initial radius of bubbles and the resonant frequency of bubbles.

    • 选定气泡初始半径为50 μm, 超声波频率为20 kHz、幅值为1.2个大气压, 采用FLUENT流体分析软件对有限流体域内超声波作用下单个气泡、球状气泡云气泡生长及溃灭过程进行对比分析计算. 球状泡群中选取9个、5个相互作用气泡进行研究, 计算结果如图5图7所示.

      Figure 5.  Volume change of single bubble

      Figure 7.  Volume change of spherical cavitation cloud N = 5.

      图5, 图6图7可看出, 无论是单气泡也好, 气泡云也好, 在外声压的作用下, 气泡均会随着时间的推移先缓慢膨胀后快速塌陷、溃灭. 尽管所处的压力环境一样, 外界驱动声参也数完全相同, 但处于球心位置和球面位置的气泡振动形态却完全不同. 在气泡膨胀初期, 所有气泡几乎能同步膨胀, 但外围球面处的气泡体积会率先达到最大值. 在溃灭时, 气泡从球形到椭球形, 再塌陷直至完全溃灭. 到了溃灭后期, 由于其内侧界面与中心气泡的相互制约, 导致中心气泡形状能较好地保持为球面. 而外侧界由于压力梯度变化急剧, 使得球面处气泡呈现内凹形状, 在溃灭瞬间, 会产生指向中心的射流. 这一过程中心气泡因受其他气泡之间的相互制约, 仍能保持为球形以达到自身最小体积衡量. 对比单气泡可以看出, 因受到气泡之间相互作用, 使得球心位置处的空化泡溃灭时间则相对延迟.

      Figure 6.  Volume change of spherical cavitation cloud N = 9.

    5.   结 论
    • 空化泡溃灭时会产生很大的瞬时压强, 会造成流体机械装置的空蚀破坏并产生噪声和剧烈振动, 有时候需要抑制空化的产生. 而空化现象实际上是一种受迫振荡所产生的结果, 是气泡在声场作用下生长、振荡、溃灭的一系列过程. 空化现象能否发生与外界驱动声参数密切关系. 本文从气泡动力学方程出发, 得到了气泡群中气泡的谐振频率, 给出了球状气泡群中气泡谐振频率与单泡Minnaert频率的修正关系. 研究结果表明: 泡群中气泡的谐振频率受气泡的初始半径、泡群中气泡数量、气泡之间距离等多种因素的影响. 当泡群中气泡数量越多、气泡的初始半径越大时, 气泡的谐振频率越小. 超声空化时由于驱动声压的频率和幅值只能是某一给定值, 那么在液体中适当注入大气泡就会使得空化泡的谐振频率减小, 使得大多数空化泡在做剧烈的非线性振荡稳态空化, 不发生激烈的溃灭过程, 从而有效地抑制空化现象的发生.

Reference (23)

Catalog

    /

    返回文章
    返回