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Numerical simulation and analysis of condensation shocks in cavitating flow

Published online by Cambridge University Press:  22 January 2018

Bernd Budich*
Technical University of Munich, Department of Mechanical Engineering, Chair of Aerodynamics and Fluid Mechanics, Boltzmannstr. 15, D-85748 Garching bei München, Germany
S. J. Schmidt
Technical University of Munich, Department of Mechanical Engineering, Chair of Aerodynamics and Fluid Mechanics, Boltzmannstr. 15, D-85748 Garching bei München, Germany
N. A. Adams
Technical University of Munich, Department of Mechanical Engineering, Chair of Aerodynamics and Fluid Mechanics, Boltzmannstr. 15, D-85748 Garching bei München, Germany
Email address for correspondence:


We analyse unsteady cavity dynamics, cavitation patterns and instability mechanisms governing partial cavitation in the flow past a sharp convergent–divergent wedge. Reproducing a recent reference experiment by numerical simulation, the investigated flow regime is characterised by large-scale cloud cavitation. In agreement with the experiments, we find that cloud shedding is dominated by the periodic occurrence of condensation shocks, propagating through the two-phase medium. The physical model is based on the homogeneous mixture approach, the assumption of thermodynamic equilibrium, and a closed-form barotropic equation of state. Compressibility of water and water vapour is taken into account. We deliberately suppress effects of molecular viscosity, in order to demonstrate that inertial effects dominate the flow evolution. We qualify the flow predictions, and validate the numerical approach by comparison with experiments. In agreement with the experiments, the vapour volume fraction within the partial cavity reaches values ${>}80\,\%$ for its spanwise average. Very good agreement is further obtained for the shedding Strouhal number, the cavity growth and collapse velocities, and for typical coherent flow structures. In accordance with the experiments, the simulations reproduce a condensation shock forming at the trailing part of the partial cavity. It is demonstrated that it satisfies locally Rankine–Hugoniot jump relations. Estimation of the shock propagation Mach number shows that the flow is supersonic. With a magnitude of only a few kPa, the pressure rise across the shock is much lower than for typical cavity collapse events. It is thus far too weak to cause cavitation erosion directly. However, by affecting the dynamics of the cavity, the flow aggressiveness can be significantly altered. Our results indicate that, in addition to classically observed re-entrant jets, condensation shocks feed an intrinsic instability mechanism of partial cavitation.

JFM Papers
© 2018 Cambridge University Press 

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Arndt, R. E. A., Song, C. C. S., Kjeldsen, M. & Keller, A. 2001 Instability of partial cavitation: a numerical/experimental approach. In Proceedings of the 23rd Symposium on Naval Hydrodynamics, pp. 599615. National Academies Press.Google Scholar
Brennen, C. E. 1995 Cavitation and Bubble Dynamics. Oxford University Press.Google Scholar
Callenaere, M., Franc, J.-P., Michel, J.-M. & Riondet, M. 2001 The cavitation instability induced by the development of a re-entrant jet. J. Fluid Mech. 444, 223256.CrossRefGoogle Scholar
Campbell, I. J. & Pitcher, A. S. 1958 Shock waves in a liquid containing gas bubbles. Proc. R Soc. Lond. A 243 (1235), 534545.CrossRefGoogle Scholar
Crespo, A. 1969 Sound and shock waves in liquids containing bubbles. Phys. Fluids 12 (11), 2274.CrossRefGoogle Scholar
Egerer, C. P., Hickel, S., Schmidt, S. J. & Adams, N. A. 2014 Large-eddy simulation of turbulent cavitating flow in a micro channel. Phys. Fluids 26 (8), 085102.CrossRefGoogle Scholar
Egerer, C. P., Schmidt, S. J., Hickel, S. & Adams, N. A. 2016 Efficient implicit LES method for the simulation of turbulent cavitating flows. J. Comput. Phys. 316, 453469.CrossRefGoogle Scholar
Eskilsson, C. & Bensow, R. E. 2012 A compressible model for cavitating flow: comparison between Euler, RANS and LES simulations. In Proceedings of the 29th Symposium on Naval Hydrodynamics, pp. 113. IOS Press.Google Scholar
Foeth, E.-J., van Terwisga, T. J. C. & van Doorne, C. 2008 On the collapse structure of an attached cavity on a three-dimensional hydrofoil. Trans. ASME J. Fluids Engng 130 (7), 071303.CrossRefGoogle Scholar
Franc, J.-P. & Michel, J.-M. 2005 Fundamentals of Cavitation. Springer Science & Business Media.Google Scholar
Furness, R. A. & Hutton, S. P. 1975 Experimental and theoretical studies of two-dimensional fixed-type cavities. Trans. ASME J. Fluids Engng 97 (4), 515521.CrossRefGoogle Scholar
Ganesh, H., Mäkiharju, S. A. & Ceccio, S. L. 2015 Interaction of a compressible bubbly flow with an obstacle placed within a shedding partial cavity. J. Phys.: Conf. Ser. 656, 012151.Google Scholar
Ganesh, H., Mäkiharju, S. A. & Ceccio, S. L. 2016 Bubbly shock propagation as a mechanism for sheet-to-cloud transition of partial cavities. J. Fluid Mech. 802, 3778.CrossRefGoogle Scholar
Gnanaskandan, A. & Mahesh, K. 2016 Large eddy simulation of the transition from sheet to cloud cavitation over a wedge. Intl J. Multiphase Flow 83, 134.CrossRefGoogle Scholar
Gopalan, S. & Katz, J. 2000 Flow structure and modeling issues in the closure region of attached cavitation. Phys. Fluids 12 (4), 895911.CrossRefGoogle Scholar
Jakobsen, J. K. 1964 On the mechanism of head breakdown in cavitating inducers. Trans. ASME J. Basic Engng 86 (2), 291305.CrossRefGoogle Scholar
Karypis, G. & Kumar, V. 1998 A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20 (1), 359392.CrossRefGoogle Scholar
Kato, H., Yamaguchi, H., Maeda, M., Kawanami, Y. & Nakasumi, S. 1999 Laser holographic observation of cavitation cloud on a foil section. J. Vis. 2 (1), 3750.CrossRefGoogle Scholar
Kawanami, Y., Kato, H., Yamaguchi, H., Tanimura, M. & Tagaya, Y. 1997 Mechanism and control of cloud cavitation. Trans. ASME J. Fluids Engng 119 (4), 788794.CrossRefGoogle Scholar
Knapp, R. T. 1955 Recent investigations of the mechanics of cavitation and cavitation damage. Trans. ASME 77 (5), 10451054.Google Scholar
Koren, B. 1993 A robust upwind discretisation method for advection, diffusion and source terms. In Numerical Methods for Advection–Diffusion Problems, pp. 117138. Vieweg.Google Scholar
Laberteaux, K. R. & Ceccio, S. L. 2001 Partial cavity flows. Part 1. Cavities forming on models without spanwise variation. J. Fluid Mech. 431, 141.CrossRefGoogle Scholar
Le, Q., Franc, J.-P. & Michel, J.-M. 1993 Partial cavities: pressure pulse distribution around cavity closure. Trans. ASME J. Fluids Engng 115, 249254.CrossRefGoogle Scholar
Leroux, J.-B., Astolfi, J. A. & Billard, J. Y. 2004 An experimental study of unsteady partial cavitation. Trans. ASME J. Fluids Engng 126 (1), 94101.CrossRefGoogle Scholar
Lush, P. A. & Skipp, S. R. 1986 High speed cine observations of cavitating flow in a duct. Intl J. Heat Fluid Flow 7 (4), 283290.CrossRefGoogle Scholar
Mallock, A. 1910 The damping of sound by frothy liquids. Proc. R Soc. Lond. A 84 (572), 391395.CrossRefGoogle Scholar
Mihatsch, M. S., Schmidt, S. J. & Adams, N. A. 2015 Cavitation erosion prediction based on analysis of flow dynamics and impact load spectra. Phys. Fluids 27 (10), 103302103321.CrossRefGoogle Scholar
Noordzij, L. & van Wijngaarden, L. 1974 Relaxation effects, caused by relative motion, on shock waves in gas-bubble/liquid mixtures. J. Fluid Mech. 66 (1), 115143.CrossRefGoogle Scholar
Pham, T. M., Larrarte, F. & Fruman, D. H. 1999 Investigation of unsteady sheet cavitation and cloud cavitation mechanisms. Trans. ASME J. Fluids Engng 121 (2), 289296.CrossRefGoogle Scholar
Philipp, A. & Lauterborn, W. 1998 Cavitation erosion by single laser-produced bubbles. J. Fluid Mech. 361, 75116.CrossRefGoogle Scholar
Reisman, G. E., Wang, Y. C. & Brennen, C. E. 1998 Observations of shock waves in cloud cavitation. J. Fluid Mech. 355, 255283.CrossRefGoogle Scholar
Saurel, R., Cocchi, P. & Butler, P. B. 1999 Numerical study of cavitation in the wake of a hypervelocity underwater projectile. J. Propul. Power 15 (4), 513522.CrossRefGoogle Scholar
Schmidt, E. & Grigull, U. 1989 Properties of Water and Steam in SI-units: 0–800 ° C 0–1000 bar = Zustandsgrößen von Wasser und Wasserdampf in SI-Einheiten, 4th edn. Springer.Google Scholar
Schmidt, S. J.2015 A low Mach number consistent compressible approach for simulation of cavitating flows. PhD thesis, Technische Universität München, Germany.Google Scholar
Schmidt, S. J., Mihatsch, M. S., Thalhamer, M. & Adams, N. A. 2014 Assessment of erosion sensitive areas via compressible simulation of unsteady cavitating flows. In Advanced Experimental and Numerical Techniques for Cavitation Erosion Prediction (ed. Kim, K.-H., Chahine, G. L., Franc, J.-P. & Karimi, A.), pp. 329344. Springer.CrossRefGoogle Scholar
Schmidt, S. J., Thalhamer, M. & Schnerr, G. H. 2009 Inertia controlled instability and small scale structures of sheet and cloud cavitation. In Proceedings of the 7th International Symposium on Cavitation, CAV 2009, pp. 6275. University of Michigan.Google Scholar
Schnerr, G. H., Sezal, I. H. & Schmidt, S. J. 2008 Numerical investigation of three-dimensional cloud cavitation with special emphasis on collapse induced shock dynamics. Phys. Fluids 20 (4), 040703.CrossRefGoogle Scholar
Wade, R. B. & Acosta, A. J. 1966 Experimental observations on the flow past a plano-convex hydrofoil. Trans. ASME J. Basic Engng 88 (1), 273282.CrossRefGoogle Scholar
Wagner, W. & Pruß, A. 2002 The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use. J. Phys. Chem. Ref. Data 31 (2), 387535.CrossRefGoogle Scholar