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Linearized dynamics of spherical bubble clouds

Published online by Cambridge University Press:  26 April 2006

Luca D'Agostino  and
Christopher E. Brennen
Luca D'Agostino
Affiliation:
California Institute of Technology, Pasadena, CA 91125, USA
Christopher E. Brennen
Affiliation:
California Institute of Technology, Pasadena, CA 91125, USA
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Abstract

The present work investigates the dynamics of the one-dimensional, unsteady flow of a spherical bubble cloud subject to harmonic far-field pressure excitation. Bubble dynamics effects and energy dissipation due to viscosity, heat transfer, liquid compressibility and relative motion of the two phases are included. The equations of motion for the average flow and the bubble radius are linearized and a closed-form solution is obtained. The results are then generalized by means of Fourier synthesis to the case of arbitrary far-field pressure excitation. The flow displays various regimes (sub-resonant, trans-resonant and super-resonant) with different properties depending on the value of the relevant flow parameters. Examples are discussed in order to show the effects of the inclusion of the various energy dissipation mechanisms. Finally the results for the case of Gaussian-shaped far-field pressure change are presented and the most important limitations of the theory are briefly discussed. The simple linearized dynamical analysis developed so far clearly demonstrates the importance of the complex phenomena connected to the interaction of the dynamics of the bubbles with the flow and provides an introduction to the more realistic study of the same flows with nonlinear bubble dynamics.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 199 , February 1989 , pp. 155 - 176
Copyright
© 1989 Cambridge University Press

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References

Arakeri, V. H. & Shanmuganathan, V., 1985 On the evidence for the effect of bubble interference on cavitation noise. J. Fluid Mech. 159, 131–150.Google Scholar
Biesheuvel, A. & van Wijngaarden, L. 1984 Two phase flow equations for a dilute dispersion of gas bubbles in liquid. J. Fluid Mech. 148, 301–318.Google Scholar
Carstensen, E. L. & Foldy, L. L., 1947 Propagation of sound through a liquid containing bubbles. J. Acoust. Soc. Am. 19, 481–501.Google Scholar
Chahine, G. L.: 1982a Pressure field generated by the collective collapse of cavitation bubbles. IAHR Symp. on Operating Problems of Pump Stations and Power Plants, Amsterdam, Netherlands, Vol. 1, Paper 2.Google Scholar
Chahine, G. L.: 1982b Cloud cavitation theory. 14th Symp. on Naval Hydrodynamics, August 1982, Session I, p. 51.Google Scholar
d'Agostino, L. & Brennen, C. E. 1983 On the acoustical dynamics of bubble clouds. ASME Cavitation and Multiphase Flow Forum, pp. 72–75.Google Scholar
d'Agostino, L., Brennen, C. E. & Acosta, A. J., 1984 On the linearized dynamics of two-dimensional bubbly flows over wave-shaped surfaces. ASME Cavitation and Multiphase Flow Forum, pp. 8–13.Google Scholar
d'Agostino, L., Brennen, C. E. & Acosta, A. J., 1988 Linearized dynamics of two-dimensional bubbly and cavitating flows over slender surfaces. J. Fluid Mech. 192, 485–509.Google Scholar
Foldy, L. L.: 1945 The multiple scattering of waves. Phys. Rev. 67, 107–119.Google Scholar
Fox, S. E., Curley, S. R. & Larson, G. S., 1955 Phase velocity and absorption measurements in water containing air bubbles. J. Acoust. Soc. Am. 27, 534–539.Google Scholar
Ishii, M.: 1975 Thermo-Fluid Dynamic Theory of Two-Phase Flow. Eyrolles.
Knapp, R. T., Daily, J. W. & Hammit, F. G., 1970 Cavitation. McGraw Hill.
Macpherson, J. D.: 1957 The effect of gas bubbles on sound propagation in water. Proc. Phys. Soc. Lond. B B70, 85–92.Google Scholar
Marboe, R. C., Billet, M. L. & Thompson, D. E., 1986 Some aspects of traveling bubble cavitation and noise. Intl Symp. on Cavitation and Multiphase Flow Noise, Anaheim, California.Google Scholar
Mørch, K. A.: 1980 On the collapse of cavity cluster in flow cavitation. Proc. 1st Intl Conf. on Cavitation and Inhomogeneities in Underwater Acoustics. Springer Series in Electrophysics, Vol 4, pp. 95–100. Springer.
Mørch, K. A.: 1981 Cavity cluster dynamics and cavitation erosion. ASME Cavitation and Polyphase Flow Forum, pp. 1–10.Google Scholar
Mørch, K. A.: 1982 Energy considerations on the collapse of cavity cluster. Appl. Sci. Res. 38, 313.Google Scholar
Muir, T. F. & Eichhorn, R., 1963 Compressible flow of an air–water mixture through a vertical two-dimensional converging–diverging nozzle. Proc. Heat Trans., Fluid Mech. Inst., Stanford. Stanford University Press.
Noordzij, L.: 1973 Shock waves in mixtures of liquids and air bubbles. Doctoral thesis, Technische Hogeschool, Twente, Netherlands.
Noordzij, L. & van Wijngaarden, L.: 1974 Relaxation effects, caused by relative motion, on shock waves in gas-bubble/liquid mixtures. J. Fluid Mech. 66, 115–143.Google Scholar
Omta, R.: 1987 Oscillations of a cloud of bubbles of small and not so small amplitude. J. Acoust. Soc. Am. 82, 1018–1033.Google Scholar
Plesset, M. S. & Prosperetti, A., 1977 Bubble dynamics and cavitation. Ann. Rev. Fluid. Mech. 9, 145–185.Google Scholar
Prosperetti, A.: 1984 Bubble phenomena in sound fields: part one. Ultrasonics, March, 69–78.Google Scholar
Pylkkänen, J. V.: 1986 Characteristics of spherical cloud cavity. Advancements in Aerodynamics, Fluid Mechanics and Hydraulics (ed. R. Arndt et al.), ASCE Conference, Minneapolis, Minnesota, pp. 96–103.
Silberman, E.: 1957 Sound velocity and attenuation in bubbly mixtures measured in standing wave tubes. J. Acoust. Soc. Am. 18, 925–933.Google Scholar
Stewart, H. B. & Wendroff, B., 1984 Two-phase flows: models and methods. J. Comput. Phys. 56, 363–409.Google Scholar
Tangren, R. F., Dodge, C. H. & Seifert, H. S., 1949 Compressibility effects in two-phase flows. J. Appl. Phys. 20, 637–645.Google Scholar
Voinov, O. V.: 1973 Force acting on a sphere in an inhomogeneous flow of an ideal incompressible fluid. Plenum (transl. from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 182–184, July–August 1973).
van Wijngaarden, L.: 1964 On the collective collapse of a large number of gas bubbles in water. Proc. 11th Intl Cong. Appl. Mech., pp. 854–861. Springer.
van Wijngaarden, L.: 1968 On the equations of motion of mixtures of liquid and gas bubbles. J. Fluid Mech. 33, 465–474.Google Scholar
van Wijngaarden, L.: 1972 One-dimensional flow of liquids containing small gas bubbles. Ann. Rev. Fluid Mech. 4, 369–396.Google Scholar
van Wijngaarden, L.: 1980 Sound and shock waves in bubbly liquids. In Cavitation and Inhomogeneities in Underwater Acoustics (ed. W. Lauterborn), pp. 127–140. Springler.