Mathematical Theory and Numerical Simulation of Bubbly Cavitating Nozzle Flows

  • Can F. Delale
  • Şenay Pasinlioğlu
  • Zafer Başkaya


Unsteady quasi-one-dimensional and two-dimensional bubbly cavitating nozzle flows are considered using a homogeneous bubbly flow model. For quasi-one-dimensional nozzle flows, the system of model equations is reduced to two evolution equations for the flow speed and bubble radius and the initial and boundary value problems for the evolution equations are formulated. Results obtained for quasi-one-dimensional nozzle flows capture the measured pressure losses due to cavitation, but they turn out to be insufficient in describing the two-dimensional structures. For this reason, model equations for unsteady two-dimensional bubbly cavitating nozzle flows are considered and, by suitable decoupling, they are reduced to evolution equations for the bubble radius and for the velocity field, the latter being determined by an integro-partial differential system for the unsteady acceleration. This integro-partial differential system constitutes the fundamental equations for the evolution of the dilation and vorticity in two-dimensional cavitating nozzle flows. The initial and boundary value problem of the evolution equations are then discussed and a method to integrate the equations is introduced.


Void Fraction Flow Speed Bubble Radius Cavitation Number Cavitating Flow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This paper is dedicated to Professor Yu.N. Savchenko on the occasion of his 70th birthday.


  1. 1.
    Tangren RF, Dodge CH, Seifert HS. Compressibility effects in two-phase flow. J Appl Phys. 1949;20:637–45.zbMATHCrossRefGoogle Scholar
  2. 2.
    Ishii R, Umeda Y, Murata S, Shishido N. Bubbly flows through a converging-diverging nozzle. Phys Fluids A. 1993;5:1630–43.zbMATHCrossRefGoogle Scholar
  3. 3.
    van Wijngaarden L. On the equations of motion for mixtures of liquid and gas bubbles. J Fluid Mech. 1968;33:465–74.zbMATHCrossRefGoogle Scholar
  4. 4.
    van Wijngaarden L. One-dimensional flow of liquids containing small gas bubbles. Ann Rev Fluid Mech. 1972;4:369–96.CrossRefGoogle Scholar
  5. 5.
    Noordzij L, van Wijngaarden L. Relaxation effects, caused by the relative motion, on shock waves in gas-bubble/liquid mixtures. J Fluid Mech. 1974;66:115–43.zbMATHCrossRefGoogle Scholar
  6. 6.
    Wang YC, Brennen CE. One dimensional bubbly cavitating flows through a converging-diverging nozzle. ASME J Fluids Eng. 1998;120:166–70.CrossRefGoogle Scholar
  7. 7.
    Delale CF, Schnerr GH, Sauer J. Quasi-one-dimensional steady-state cavitating nozzle flows. J Fluid Mech. 2001;427:167–204.zbMATHCrossRefGoogle Scholar
  8. 8.
    Pasinlioğlu Ş, Delale CF, Schnerr GH. On the temporal stability of quasi-one-dimensional steady-state bubbly cavitating nozzle flow solutions. IMA J Appl Math. 2009;74:230–49.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Preston AT, Colonius T, Brennen CE. A numerical investigation of unsteady bubbly cavitating nozzle flows. Phys Fluids. 2002;14:300–11.CrossRefGoogle Scholar
  10. 10.
    Saffman PG. Vortex dynamics. Cambridge: Cambridge University Press; 1992.Google Scholar
  11. 11.
    Brennen CE. Cavitation and bubble dynamics. Oxford: Oxford University Press; 1995.Google Scholar
  12. 12.
    Wang YC, Chen E. Effect of phase relative motion on critical bubbly flows through a converging-diverging nozzle. Phys Fluids. 2002;14:3215–23.CrossRefGoogle Scholar
  13. 13.
    Mørch KA. Cavitation nuclei and bubble formation: a dynamic liquid-solid interface problem. ASME J Fluids Eng. 2000;122:494–8.CrossRefGoogle Scholar
  14. 14.
    Delale CF, Hruby J, Marsik F. Homogeneous bubble nucleation in liquids: the classical theory revisited. J Chem Phys. 2003;118:792–806.CrossRefGoogle Scholar
  15. 15.
    Delale CF, Okita K, Matsumoto Y. Steady state cavitating nozzle flows with nucleation. ASME J Fluids Eng. 2005;127:770–7.CrossRefGoogle Scholar
  16. 16.
    Brennen CE. Fission of collapsing cavitation bubbles. J Fluid Mech. 2002;472:153–66.zbMATHCrossRefGoogle Scholar
  17. 17.
    Delale CF, Tunç M. A bubble fission model for collapsing cavitation bubbles. Phys Fluids. 2004;16:4200–3.CrossRefGoogle Scholar
  18. 18.
    Blake JR, Gibson DC. Cavitation bubbles near boundaries. Ann Rev Fluid Mech. 1987;19:99–123.CrossRefGoogle Scholar
  19. 19.
    Kubota A, Kato H, Yamaguchi H. A numerical study of unsteady cavitation on a hydraulic section. J Fluid Mech. 1992;240:59–96.CrossRefGoogle Scholar
  20. 20.
    Nigmatulin RI, Khabeev NS, Nagiev FB. Dynamics, heat and mass transfer of vapor-gas bubbles in a liquid. Int J Heat Mass Tran. 1981;24:1033–44.zbMATHCrossRefGoogle Scholar
  21. 21.
    Prosperetti A, Crum LA, Commander KW. Nonlinear bubble dynamics. J Acoust Soc Am. 1988;83:502–14.CrossRefGoogle Scholar
  22. 22.
    Prosperetti A. The thermal behavior of oscillating gas bubbles. J Fluid Mech. 1991;222:587–616.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Delale CF. Thermal damping in cavitating nozzle flows. ASME J Fluids Eng. 2002;124:969–76.CrossRefGoogle Scholar
  24. 24.
    Franc JP, Michel JM. Fundamentals of cavitation. Dordrecht: Kluwer; 2004.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Can F. Delale
    • 1
  • Şenay Pasinlioğlu
  • Zafer Başkaya
  1. 1.Department of Mechanical EngineeringIşιk University, ŞileIstanbulTurkey

Personalised recommendations