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A Numerical Study of the Formation of a Conical Cavitation Bubble Structure at Low Ultrasonic Frequency
Available online at www.sciencedirect.com
ScienceDirect Physics Procedia 70 (2015) 1070 – 1073
2015 International Congress on Ultrasonics, 2015 ICU Metz
A numerical study of the formation of a conical cavitation bubble structure at low ultrasonic frequency C. Vanhillea , C. Campos-Pozuelob , C. Grangerc and B. Dubusc,* a
Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain Consejo Superior de Investigaciones Científicas, Serrano 144, 28006 Madrid, Spain c IEMN - département ISEN, UMR CNRS 8520, 41 Boulevard Vauban, 59800 Lille, France b
Abstract This paper presents a study of the formation of a conical bubble structure due to cavitation at 20 kHz in water. This analysis is performed by using the numerical code SNOW-BL, which solves the interaction of finite amplitude pressure waves and a population of oscillating bubbles. Axisymmetric simulations show that a thin bubbly layer with inhomogeneous bubble density at the horn surface induces the strong focusing of the transmitted ultrasonic field and three-dimensional resonances of the layer. Waves at high amplitudes also exhibit nonlinear distortion due to the bubbles. Similarities are found with the self-stabilized conical bubble structure observed experimentally at the surface of a horn driven at 20 kHz in water. © by Elsevier B.V. This is an open access article under the CC BY-NC-ND license © 2015 2015Published The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of ICU 2015
Peer-review under responsibility of the Scientific Committee of 2015 ICU Metz. Keywords: ultrasonic cavitation; bubble structure
1. Introduction Distribution of cavitation bubbles plays an important role in ultrasonic devices used in sonochemistry and other cavitation-assisted processes [Lauterborn and Kurz (2010); Lauterborn and Mettin (2014)]. The cylindrical horntype transducer is a technology widely used for generating high intensity cavitation at low frequency. It produces a bubble structure of conical shape localized at the vicinity of the horn [Moussatov et al (2003); Skokov et al (2005); Madroyan et al (2009), Bai et al (2014)]. Very specific acoustic features are associated with this conical bubble
* Corresponding author. E-mail address: [email protected]
1875-3892 © 2015 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of ICU 2015 doi:10.1016/j.phpro.2015.08.228
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C. Vanhille et al. / Physics Procedia 70 (2015) 1070 – 1073
structure such as shock wave generation and focusing at short distance from horn surface [Campos-Pozuelo et al (2005); Dubus et al (2010)]. In this paper, recent numerical and experimental results on conical bubble structures are displayed. The 3D axisymmetric finite difference time domain model coupling finite amplitude acoustic waves and population of bubbles used here is presented and numerical simulations of acoustic wave transmission through bubbly layers are analyzed in Section 2. They are compared to high speed images of conical bubble structure in Section 3. 2. Numerical model and simulation results The three-dimensional semi-infinite space is simplified by considering axial symmetry, cylindrical coordinates r and z, and excited by a prescribed pressure at its boundary. The following coupled differential system based on the Rayleigh-Plesset approximation and the wave equation is solved to track bubble volume variation v (from its initial volume vb ) and acoustic pressure p in time t [Vanhille and Campos-Pozuelo (2013)]: prr + pr r + pzz − ptt cl2 = − ρl N b vtt , 0 < z < Lz , 0 < r < Lr , 0 < t < T ° 2 2 2 °vtt + δωb vt + ωb v = av + b ( 2vvtt + vt ) − η p , 0 ≤ z ≤ Lz , 0 ≤ r ≤ Lr , 0 < t < T ° °v = vt = 0, t = 0, 0 ≤ z ≤ Lz , 0 ≤ r ≤ Lr °° p = p = 0, t = 0, 0 < z ≤ L , 0 ≤ r ≤ L t z r ® ° p = p0 sin (ω0 t ) , z = 0, 0 ≤ r ≤ Lr , 0 < t ≤ T °p = −1 c p , z = L , 0 ≤ r ≤ L , 0 < t ≤ T l) t z r ° z ( ° pr = 0, r = 0, 0 ≤ z ≤ Lz , 0 < t ≤ T ° °¯ pr = ( −1 cl ) pt , r = Lr , 0 ≤ z ≤ Lz , 0 < t ≤ T ,
(1)
in which Lz , Lr , T define the computational limits in space and time, ρl and cl are the equilibrium density and the small-amplitude sound speed of liquid, Nb ( z, r ) is the bubble density in the liquid, δ is the viscous damping coefficient of the bubbly liquid, and ωb is the bubble resonance frequency. Coefficients η , a , and b depend on the characteristics of gas bubbles and liquid. The time-dependent pressure function at the source is of amplitude p0 and circular frequency ω0 . More details about this model can be found in [Vanhille and Campos-Pozuelo (2013)]. Based on this model, nonlinear simulations of acoustic wave transmission through an axisymetrical inhomogeneous bubbly layer are solved using SNOW-BL code previously used to model infinite bubbly media and bubbly layers in an homogeneous infinite liquid [Vanhille and Campos-Pozuelo (2009), Vanhille and CamposPozuelo (2013),]. Water is considered for the liquid and air for the gas. Volumetric bubble fraction V f = N b vb is assumed uniform along layer thickness and varies as a sine curve along radius. Ultrasonic frequency is f 0 = ω0 2π = 20 kHz . Figures 1 and 2 display the peak of negative pressure calculated during the last period of the simulations at each point of the space domain for several incident pressure amplitudes p0 and void fractions V f . Focusing at short distance is observed with a slight decrease of focal distance from 1.40d to 1.3d (where d is layer diameter) when acoustic pressure varies between 1 Pa and 15 kPa and a decrease of focal distance from 1.91d to 1.40d when void fraction varies between 3.8 10-8 and 9.5 10-5. The focal distance is thus shorter as the nonlinearity of the medium is raised. Standing waves also appear in the bubble layer. They involve wavenumbers along layer thickness and diameter and could be associated to three-dimensional resonances of the layer. 3. Experimental results Ultrasonic vibration is generated by a sandwich piezoelectric transducer coupled to a mechanical amplifier and a 70 mm diameter horn. The horn is partially immersed (~1 cm deep) in a tank filled with tap water. Images of cavitation bubble structures are obtained with a digital camera or a high speed camera at 7000 frames per second.
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Pictures of bubble structure are displayed in Fig. 3 for various values of the acoustic intensity defined as the active electrical power provided by the amplifier divided by horn surface area. In concordance with simulations, displacement of focal point is small when acoustic intensity increases. The ratio of the focal point distance to the horn diameter is around 0.73, a value smaller than those obtained in the simulations. This difference could be due to experimental acoustic pressures which are much larger than the simulated ones, and then create denser bubble layers (higher void fractions) by cavitation than the simulated ones. When driving the transducer with continuously varying acoustic intensities, snapshots shows changes of bubble structure geometry parallel to horn surface (Fig. 4) which could be associated to three-dimensional resonances of the bubble layer observed in the numerical simulations.
Fig. 1. Peak of negative pressure vs. p0 at V f = 0.0095 % .
Fig. 2. Peak of negative pressure vs. V f at p0 = 15 kPa .
C. Vanhille et al. / Physics Procedia 70 (2015) 1070 – 1073
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Fig. 3. Conical bubble structure at various acoustic intensity: (a) 1.8 W.cm-2; (b) 3.5 W.cm-2; (c) 5.3 W.cm-2; (d) 8.2 W.cm-2.
Fig. 4. Snapshots of conical bubble structure for continuously varying acoustic intensities.
Acknowledgements This work is funded by the research project DPI2012-34613 (Spain). References Bai, L., Xu, W., Deng, J., Li, C., Xu, D. Gao, Y., 2014. Generation and control of acoustic cavitation structure. Ultrasonics Sonochemistry 21, 1696–1706. Campos-Pozuelo, C., Granger, C., Vanhille, C., Dubus, B., 2005. Experimental and theoretical investigation of the mean acoustic pressure in the cavitation field. Ultrasonics Sonochemistry 12, 79–84. Dubus, B., Vanhille, C., Campos-Pozuelo, C., Granger, 2010. On the physical origin of conical bubble structure under ultrasonic horn. Ultrasonics Sonochemistry 12, 79–84. Lauterborn, W., Kurz, T., 2010. Physics of bubble oscillations. Report on Progress in Physics 73, 106501. Lauterborn, W., Mettin, R., 2014. Acoustic cavitation: bubble dynamics in high power ultrasonic fields, in “Power Ultrasonics”. In: GallegoJuarez, J.A., Graff K.F. (Eds.). Woodhead Publishing, Cambridge, pp. 377. Moussatov, A., Granger, C., Dubus, B., 2003. Cone-like bubble formation in ultrasonic cavitation field. Ultrasonics Sonochemistry 10, 191–195. Mandroyan, A., Viennet, R., Bailly, Y., Doche, M.-L., Hihn, J.-Y., 2009. Modification of the ultrasound-induced activity by the presence of an electrode in a sonoreactor working at two low frequencies (20 and 40 kHz) – Part I: Active zone visualization by laser tomography. Ultrasonics Sonochemistry 16, 88–96. Skokov, V., Koverda, V.P., Reshetnikov, A.V., Vinogradov, A.V., 2005. 1/f fluctuations under acoustic cavitation of liquids. Physica A 364, 63– 69. Vanhille, C., Campos-Pozuelo, C., 2009. Nonlinear ultrasonic waves in bubbly liquids with nonhomogeneous bubble distribution: numerical experiments, Ultrasonics Sonochemistry 16, 669-685. Vanhille, C., Campos-Pozuelo, C., 2013. Numerical simulations of three-dimensional nonlinear acoustic waves in bubbly liquids, Ultrasonics Sonochemistry 20, 963-969.
ScienceDirect Physics Procedia 70 (2015) 1070 – 1073
2015 International Congress on Ultrasonics, 2015 ICU Metz
A numerical study of the formation of a conical cavitation bubble structure at low ultrasonic frequency C. Vanhillea , C. Campos-Pozuelob , C. Grangerc and B. Dubusc,* a
Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain Consejo Superior de Investigaciones Científicas, Serrano 144, 28006 Madrid, Spain c IEMN - département ISEN, UMR CNRS 8520, 41 Boulevard Vauban, 59800 Lille, France b
Abstract This paper presents a study of the formation of a conical bubble structure due to cavitation at 20 kHz in water. This analysis is performed by using the numerical code SNOW-BL, which solves the interaction of finite amplitude pressure waves and a population of oscillating bubbles. Axisymmetric simulations show that a thin bubbly layer with inhomogeneous bubble density at the horn surface induces the strong focusing of the transmitted ultrasonic field and three-dimensional resonances of the layer. Waves at high amplitudes also exhibit nonlinear distortion due to the bubbles. Similarities are found with the self-stabilized conical bubble structure observed experimentally at the surface of a horn driven at 20 kHz in water. © by Elsevier B.V. This is an open access article under the CC BY-NC-ND license © 2015 2015Published The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of ICU 2015
Peer-review under responsibility of the Scientific Committee of 2015 ICU Metz. Keywords: ultrasonic cavitation; bubble structure
1. Introduction Distribution of cavitation bubbles plays an important role in ultrasonic devices used in sonochemistry and other cavitation-assisted processes [Lauterborn and Kurz (2010); Lauterborn and Mettin (2014)]. The cylindrical horntype transducer is a technology widely used for generating high intensity cavitation at low frequency. It produces a bubble structure of conical shape localized at the vicinity of the horn [Moussatov et al (2003); Skokov et al (2005); Madroyan et al (2009), Bai et al (2014)]. Very specific acoustic features are associated with this conical bubble
* Corresponding author. E-mail address: [email protected]
1875-3892 © 2015 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of ICU 2015 doi:10.1016/j.phpro.2015.08.228
1071
C. Vanhille et al. / Physics Procedia 70 (2015) 1070 – 1073
structure such as shock wave generation and focusing at short distance from horn surface [Campos-Pozuelo et al (2005); Dubus et al (2010)]. In this paper, recent numerical and experimental results on conical bubble structures are displayed. The 3D axisymmetric finite difference time domain model coupling finite amplitude acoustic waves and population of bubbles used here is presented and numerical simulations of acoustic wave transmission through bubbly layers are analyzed in Section 2. They are compared to high speed images of conical bubble structure in Section 3. 2. Numerical model and simulation results The three-dimensional semi-infinite space is simplified by considering axial symmetry, cylindrical coordinates r and z, and excited by a prescribed pressure at its boundary. The following coupled differential system based on the Rayleigh-Plesset approximation and the wave equation is solved to track bubble volume variation v (from its initial volume vb ) and acoustic pressure p in time t [Vanhille and Campos-Pozuelo (2013)]: prr + pr r + pzz − ptt cl2 = − ρl N b vtt , 0 < z < Lz , 0 < r < Lr , 0 < t < T ° 2 2 2 °vtt + δωb vt + ωb v = av + b ( 2vvtt + vt ) − η p , 0 ≤ z ≤ Lz , 0 ≤ r ≤ Lr , 0 < t < T ° °v = vt = 0, t = 0, 0 ≤ z ≤ Lz , 0 ≤ r ≤ Lr °° p = p = 0, t = 0, 0 < z ≤ L , 0 ≤ r ≤ L t z r ® ° p = p0 sin (ω0 t ) , z = 0, 0 ≤ r ≤ Lr , 0 < t ≤ T °p = −1 c p , z = L , 0 ≤ r ≤ L , 0 < t ≤ T l) t z r ° z ( ° pr = 0, r = 0, 0 ≤ z ≤ Lz , 0 < t ≤ T ° °¯ pr = ( −1 cl ) pt , r = Lr , 0 ≤ z ≤ Lz , 0 < t ≤ T ,
(1)
in which Lz , Lr , T define the computational limits in space and time, ρl and cl are the equilibrium density and the small-amplitude sound speed of liquid, Nb ( z, r ) is the bubble density in the liquid, δ is the viscous damping coefficient of the bubbly liquid, and ωb is the bubble resonance frequency. Coefficients η , a , and b depend on the characteristics of gas bubbles and liquid. The time-dependent pressure function at the source is of amplitude p0 and circular frequency ω0 . More details about this model can be found in [Vanhille and Campos-Pozuelo (2013)]. Based on this model, nonlinear simulations of acoustic wave transmission through an axisymetrical inhomogeneous bubbly layer are solved using SNOW-BL code previously used to model infinite bubbly media and bubbly layers in an homogeneous infinite liquid [Vanhille and Campos-Pozuelo (2009), Vanhille and CamposPozuelo (2013),]. Water is considered for the liquid and air for the gas. Volumetric bubble fraction V f = N b vb is assumed uniform along layer thickness and varies as a sine curve along radius. Ultrasonic frequency is f 0 = ω0 2π = 20 kHz . Figures 1 and 2 display the peak of negative pressure calculated during the last period of the simulations at each point of the space domain for several incident pressure amplitudes p0 and void fractions V f . Focusing at short distance is observed with a slight decrease of focal distance from 1.40d to 1.3d (where d is layer diameter) when acoustic pressure varies between 1 Pa and 15 kPa and a decrease of focal distance from 1.91d to 1.40d when void fraction varies between 3.8 10-8 and 9.5 10-5. The focal distance is thus shorter as the nonlinearity of the medium is raised. Standing waves also appear in the bubble layer. They involve wavenumbers along layer thickness and diameter and could be associated to three-dimensional resonances of the layer. 3. Experimental results Ultrasonic vibration is generated by a sandwich piezoelectric transducer coupled to a mechanical amplifier and a 70 mm diameter horn. The horn is partially immersed (~1 cm deep) in a tank filled with tap water. Images of cavitation bubble structures are obtained with a digital camera or a high speed camera at 7000 frames per second.
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Pictures of bubble structure are displayed in Fig. 3 for various values of the acoustic intensity defined as the active electrical power provided by the amplifier divided by horn surface area. In concordance with simulations, displacement of focal point is small when acoustic intensity increases. The ratio of the focal point distance to the horn diameter is around 0.73, a value smaller than those obtained in the simulations. This difference could be due to experimental acoustic pressures which are much larger than the simulated ones, and then create denser bubble layers (higher void fractions) by cavitation than the simulated ones. When driving the transducer with continuously varying acoustic intensities, snapshots shows changes of bubble structure geometry parallel to horn surface (Fig. 4) which could be associated to three-dimensional resonances of the bubble layer observed in the numerical simulations.
Fig. 1. Peak of negative pressure vs. p0 at V f = 0.0095 % .
Fig. 2. Peak of negative pressure vs. V f at p0 = 15 kPa .
C. Vanhille et al. / Physics Procedia 70 (2015) 1070 – 1073
(a)
(b)
(c)
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(d)
Fig. 3. Conical bubble structure at various acoustic intensity: (a) 1.8 W.cm-2; (b) 3.5 W.cm-2; (c) 5.3 W.cm-2; (d) 8.2 W.cm-2.
Fig. 4. Snapshots of conical bubble structure for continuously varying acoustic intensities.
Acknowledgements This work is funded by the research project DPI2012-34613 (Spain). References Bai, L., Xu, W., Deng, J., Li, C., Xu, D. Gao, Y., 2014. Generation and control of acoustic cavitation structure. Ultrasonics Sonochemistry 21, 1696–1706. Campos-Pozuelo, C., Granger, C., Vanhille, C., Dubus, B., 2005. Experimental and theoretical investigation of the mean acoustic pressure in the cavitation field. Ultrasonics Sonochemistry 12, 79–84. Dubus, B., Vanhille, C., Campos-Pozuelo, C., Granger, 2010. On the physical origin of conical bubble structure under ultrasonic horn. Ultrasonics Sonochemistry 12, 79–84. Lauterborn, W., Kurz, T., 2010. Physics of bubble oscillations. Report on Progress in Physics 73, 106501. Lauterborn, W., Mettin, R., 2014. Acoustic cavitation: bubble dynamics in high power ultrasonic fields, in “Power Ultrasonics”. In: GallegoJuarez, J.A., Graff K.F. (Eds.). Woodhead Publishing, Cambridge, pp. 377. Moussatov, A., Granger, C., Dubus, B., 2003. Cone-like bubble formation in ultrasonic cavitation field. Ultrasonics Sonochemistry 10, 191–195. Mandroyan, A., Viennet, R., Bailly, Y., Doche, M.-L., Hihn, J.-Y., 2009. Modification of the ultrasound-induced activity by the presence of an electrode in a sonoreactor working at two low frequencies (20 and 40 kHz) – Part I: Active zone visualization by laser tomography. Ultrasonics Sonochemistry 16, 88–96. Skokov, V., Koverda, V.P., Reshetnikov, A.V., Vinogradov, A.V., 2005. 1/f fluctuations under acoustic cavitation of liquids. Physica A 364, 63– 69. Vanhille, C., Campos-Pozuelo, C., 2009. Nonlinear ultrasonic waves in bubbly liquids with nonhomogeneous bubble distribution: numerical experiments, Ultrasonics Sonochemistry 16, 669-685. Vanhille, C., Campos-Pozuelo, C., 2013. Numerical simulations of three-dimensional nonlinear acoustic waves in bubbly liquids, Ultrasonics Sonochemistry 20, 963-969.