Shock Waves

, Volume 19, Issue 1, pp 49–58 | Cite as

Shock and wave dynamics in cavitating compressible liquid flows in injection nozzles

  • I. H. SezalEmail author
  • S. J. Schmidt
  • G. H. Schnerr
  • M. Thalhamer
  • M. Förster
Original Article


Due to the exceptional high inlet pressures up to 2,000 bar flow dynamics and efficiency of modern injection systems are controlled by high frequency wave dynamics of the compressible liquid flow. Corresponding to alternating shock and expansion waves the liquid fluid evaporates and recondenses instantaneously. Here we present CFD simulations of the time accurate evolution of cavitating flows in 2-D plane and in six-hole injection nozzles with focus on the wave dynamics just after initialisation of the flow and within the time scale Δt ≤ 10−4 s of pilot and multi-point injection. Due to shock reflections at the bottom of the sack hole the instantaneous maximum pressure increases more than three times higher as compared with the prescribed pressure at the nozzle inlet. For instance, in case of an inlet pressure of 600 bar the maximum pressure in the sack and therefore ahead of the nozzle bore holes reaches about 2,100 bar. It is quite reasonable that this amplification of the pressure affects the evolution of the convective flow and therefore the mass flow through the nozzle bore holes.


Compressible liquid flow Multiphase flow Cavitation Injection nozzle 


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  1. 1.
    Alajbegovic A., Meister G., Greif D., Basara B.: Three phase cavitating flows in high-pressure swirl injectors. Exp. Thermal Fluid Sci. 26, 667–681 (2002)CrossRefGoogle Scholar
  2. 2.
    Brennen C.E.: Fission of collapsing cavitating bubbles. J. Fluid Mech. 472, 153–166 (2002)zbMATHCrossRefGoogle Scholar
  3. 3.
    Delale C.F.: Thermal damping in cavitating nozzle flows. ASME J. Fluids Eng. 24, 969–976 (2002)CrossRefGoogle Scholar
  4. 4.
    Preston A.T., Colonius T., Brennen C.E.: A reduced-order model of diffusive effects on the dynamics of bubbles. Phys. Fluids 19, 123302 (2007)CrossRefGoogle Scholar
  5. 5.
    Delale C.F., Schnerr G.H., Sauer J.: Quasi-one-dimensional steady-state cavitating nozzle flows. J. Fluid Mech. 427, 167–204 (2001)zbMATHCrossRefGoogle Scholar
  6. 6.
    Schmidt, S.J., Sezal, I.H., Schnerr, G.H.: Compressible simulation of high-speed hydrodynamics with phase change. In: Wesseling, P., Onate, E., Periaux, J. (eds.) Proceedings ECCOMAS CFD 2006—European Conference on Computational Fluid Dynamics, CD-ROM Publication, The Netherlands, September 5–8 2006Google Scholar
  7. 7.
    Berg A., Iben U., Meister A., Schmidt J.: Modeling and simulation of cavitation in hydraulic pipelines based on the thermodynamic properties of liquid and steam. Shock Waves 14, 111–121 (2005)CrossRefGoogle Scholar
  8. 8.
    Hoeijmakers, H.W.M., Janssens, M.E., Kwan, W.: Numerical simulation of sheet cavitation. In: Michel J.M., Kato, H. (eds.) Proceedings 3rd International Symposium on Cavitation, Grenoble, April 7–10, pp. 257–262 (1998)Google Scholar
  9. 9.
    Schmidt, D.P., Rutland, C.J., Corradini, M.L., Roosen, P., Genge, O.: Cavitation in two-dimensional asymmetric nozzles. SAE pap. 1999-01-0518 (1999)Google Scholar
  10. 10.
    Iben U., Wrona F., Munz C.-D., Beck M.: Cavitation in hydraulic tools based on thermodynamic properties of liquid and gas. J. Fluids Eng. 124, 1011–1017 (2002)CrossRefGoogle Scholar
  11. 11.
    Schnerr G.H., Sezal I.H., Schmidt S.J.: Numerical investigation of 3-D cloud cavitation with special emphasis on collapse induced shock dynamics. Phys. Fluids 20, 040703 (2008)CrossRefGoogle Scholar
  12. 12.
    Franc J.-P., Michel J.-M.: Fundamentals of Cavitation. Kluwer Academic Publishers, Dordrecht (2004)zbMATHGoogle Scholar
  13. 13.
    Oldenbourg R.: Properties of Water and Steam in SI Units. Springer, Berlin (1999)Google Scholar
  14. 14.
    Saurel R., Cocchi C.P., Butler P.B.: Numerical study of cavitation in the wake of a hypervelocity profile. J. Propul. Power. 15, 513–522 (1999)CrossRefGoogle Scholar
  15. 15.
    IAPWS. Inernational association for the properties of water and steamGoogle Scholar
  16. 16.
    Toro E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, Berlin (1999)zbMATHGoogle Scholar
  17. 17.
    Guillard H., Viozat C.: On the behavior of upwind schemes in the low Mach number limit. Comput. Fluids 28, 63–86 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Munz C.-D., Roller S., Klein R., Geratz K.J.: The extension of incompressible flow solvers to the weakly compressible regime. Comput. Fluids 32, 173–196 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Meister A.: Asymptotic single and multiple scale expansions in the low Mach number limit. SIAM J. Appl. Math. 60, 256–271 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Schmidt, S.J., Sezal, I.H., Schnerr, G.H., Thalhamer, M.: Riemann techniques for the simulation of compressible liquid flows with phase-transition at all Mach numbers—shock and wave dynamics in cavitating 3-D micro and macro systems. In: 46th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada. AIAA paper 2008-1238 (2008)Google Scholar
  21. 21.
    Shu, C.W.: Essentailly non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. ICASE Report no. 97–65 (1997)Google Scholar
  22. 22.
    LeVeque R.J.: Finite volume methods for hyperbolic problems. Cambridge University Press, London (2002)zbMATHGoogle Scholar
  23. 23.
    Rudy D.H., Strickwerda J.D.: A non-reflecting outflow boundary condition for subsonic Navier–Stokes calculations. J. Comput. Phys. 36, 55–70 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Schnerr, G.H., Schmidt, S.J., Sezal, I.H., Thalhamer, M.: Shock and wave dynamics of compressible liquid flows with special emphasis on unsteady load on hydrofoils and on cavitation in injection nozzles. Invited Lecture. In: Proceedings CAV2006—Sixth International Symposium on Cavitation, CD-ROM publication, The Netherlands, September 11–15 2006Google Scholar
  25. 25.
    Yuan W., Schnerr G.H.: Numerical simulation of two-phase flow in injection nozzles: interaction of cavitation and external jet formation. J. Fluids Eng. 125, 963–969 (2003)CrossRefGoogle Scholar
  26. 26.
    Zierep J.: Theoretische Gasdynamik. G. Braun, Karlsruhe (1991)Google Scholar
  27. 27.
    Busch, R.: Untersuchung von Kavitationsphänomenen in Dieseleinspritzdüsen. PhD Thesis, Universität Hannover, Hannover (2001)Google Scholar
  28. 28.
    Chaves, H., Knapp, M., Kubitzek, A., Obermeier, F., Schneider, T.: Experimental study of cavitation in the nozzle hole of diesel injectors using transparent nozzles. SAE Paper 950290, pp. 645–657 (1995)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • I. H. Sezal
    • 1
    Email author
  • S. J. Schmidt
    • 1
  • G. H. Schnerr
    • 1
  • M. Thalhamer
    • 1
  • M. Förster
    • 1
  1. 1.Technische Universität München, Lehrstuhl für FluidmechanikGarchingGermany

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