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Numerical study of a planar shock interacting with a cylindrical water column embedded with an air cavity

Published online by Cambridge University Press:  24 July 2017

Gaoming Xiang Open the ORCID record for Gaoming Xiang [Opens in a new window]  and
Bing Wang Open the ORCID record for Bing Wang [Opens in a new window]
Gaoming Xiang
Affiliation:
School of Aerospace Engineering, Tsinghua University, Beijing, 100084, China
Bing Wang*
Affiliation:
School of Aerospace Engineering, Tsinghua University, Beijing, 100084, China
*
†Email address for correspondence: [email protected]
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Abstract

This paper performs a numerical study on the interaction of a planar shock wave with a water column embedded with/without a cavity of different sizes at high Weber numbers. The conservative-type Euler and non-conservative scalar two-equations representing the transportation of two-phase properties consist of the diffusion interface capture models. The numerical fluxes are computed by the Godunov-type Harten-Lax–van Leer contact Riemann solver coupled with an incremental fifth-order weighted essentially non-oscillatory (WENO) scheme. A third-order total variation diminishing (TVD) Runge–Kutta scheme is used to advance the solution in time. The morphology and dynamical characteristics are analysed qualitatively and quantitatively to demonstrate the breakup mechanism of the water column and formation of transverse jets under different incident shock intensities and embedded-cavity sizes. The jet tip velocities are extracted by analysing the interface evolution. The liquid column is prone to aerodynamic breakup with the formation of micro-mist at later stages instead of liquid evaporation because of the weakly heating effects of the surrounding air. It is numerically confirmed that the liquid-phase pressure will drop below the saturated vapour pressure, and the low pressure can be sustained for a certain time because of the focusing of the expansion wave, which accounts for the cavitation inside the liquid water column. The geometrical parameters of the deformed water column are identified, showing that the centreline width decreases but the transverse height increases nonlinearly with time. The deformation rates are nonlinearly correlated under different Mach numbers. The first transverse jet is found for a water column with an embedded cavity, whereas the water hammer shock and second jet do not occur under the impact of low intensity incident shock waves. The $x$-velocity component recorded at the rear stagnation point can remain unchanged for a comparable time after a declined evolution, which indicates that the downstream wall of the shocked water ring somehow moves uniformly. It can be explained that the acceleration of the downstream wall is balanced by the trailing shedding vortex, and this effect is more evident under higher Mach numbers. The increased enstrophy, mainly generated at the interface, demonstrates the competition of the baroclinic effects of the shock wave impact over dilatation.

JFM classification

Drops and Bubbles: Cavitation Drops and Bubbles: Drops Compressible Flows: Shock waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 825 , 25 August 2017 , pp. 825 - 852
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Copyright
© 2017 Cambridge University Press 

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References

Abgrall, R. & Karni, S. 2001 Computations of compressible multifluids. J. Comput. Phys. 169 (2), 594–623.CrossRefGoogle Scholar
Allaire, G., Clerc, S. & Kokh, S. 2002 A five-equation model for the simulation of interfaces between compressible fluids. J. Comput. Phys. 181 (2), 577–616.CrossRefGoogle Scholar
Apazidis, N. 2016 Numerical investigation of shock induced bubble collapse in water. Phys. Fluids 28 (4), 046101.CrossRefGoogle Scholar
Ball, G. J., Howell, B. P., Leighton, T. G. & Schofield, M. J. 2000 Shock-induced collapse of a cylindrical air cavity in water: a free-lagrange simulation. Shock Waves 10 (4), 265–276.CrossRefGoogle Scholar
Barber, B. P., Hiller, R. A., Löfstedt, R., Putterman, S. J. & Weninger, K. R. 1997 Defining the unknowns of sonoluminescence. Phys. Rep. 281 (2), 65–143.CrossRefGoogle Scholar
Betney, M. R, Tully, B., Hawker, N. A. & Ventikos, Y. 2015 Computational modelling of the interaction of shock waves with multiple gas-filled bubbles in a liquid. Phys. Fluids 27 (3), 036101.CrossRefGoogle Scholar
Bhattacharya, S. 2016 Interfacial wave dynamics of a drop with an embedded bubble. Phys. Rev. E 93 (2), 023119.Google ScholarPubMed
Bourne, N. K. & Field, J. E. 1999 Shock–induced collapse and luminescence by cavities. Phil. Trans. R. Soc. Lond. A 357 (1751), 295–311.CrossRefGoogle Scholar
Brennen, C. E. 2013 Cavitation and Bubble Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Brenner, M. P., Hilgenfeldt, S. & Lohse, D. 2002 Single-bubble sonoluminescence. Rev. Mod. Phys. 74 (2), 425–484.CrossRefGoogle Scholar
Brujan, E. A., Keen, G. S., Vogel, A. & Blake, J. R. 2002 The final stage of the collapse of a cavitation bubble close to a rigid boundary. Phys. Fluids 14 (1), 85–92.CrossRefGoogle Scholar
Coralic, V. & Colonius, T. 2014 Finite-volume weno scheme for viscous compressible multicomponent flows. J. Comput. Phys. 274, 95–121.CrossRefGoogle ScholarPubMed
Field, J. E., Camus, J. J., Tinguely, M., Obreschkow, D. & Farhat, M. 2012 Cavitation in impacted drops and jets and the effect on erosion damage thresholds. Wear 290, 154–160.CrossRefGoogle Scholar
Field, J. E., Dear, J. P. & Ogren, J. E. 1989 The effects of target compliance on liquid drop impact. J. Appl. Phys. 65 (2), 533–540.CrossRefGoogle Scholar
Hanson, A. R., Domich, E. G. & Adams, H. S. 1963 Shock tube investigation of the breakup of drops by air blasts. Phys. Fluids 6 (8), 1070–1080.CrossRefGoogle Scholar
Hawker, N. A. & Ventikos, Y. 2012 Interaction of a strong shockwave with a gas bubble in a liquid medium: a numerical study. J. Fluid Mech. 701, 59–97.CrossRefGoogle Scholar
Hsiang, L. & Faeth, M. 1992 Secondary drop breakup in the deformation regime. Ann. Arbor 1001, 110.Google Scholar
Hsiang, L. P. & Faeth, G. M.1993 Deformation and secondary breakup of drops, AIAA Paper 93-0814.CrossRefGoogle Scholar
Hsiang, L. P. & Faeth, G. M. 1995a Drop deformation and breakup due to shock wave and steady disturbance. Prev. Heat Mass Transfer 5 (21), 415.Google Scholar
Hsiang, L. P. & Faeth, G. M. 1995b Drop deformation and breakup due to shock wave and steady disturbances. Intl J. Multiphase Flow 21 (4), 545–560.CrossRefGoogle Scholar
Igra, D. & Takayama, K. 2001a Investigation of aerodynamic breakup of a cylindrical water droplet. Atomiz. Sprays 11 (2), 167–185.Google Scholar
Igra, D. & Takayama, K. 2001b Numerical simulation of shock wave interaction with a water column. Shock Waves 11 (3), 219–228.CrossRefGoogle Scholar
Johnsen, E. & Colonius, T. 2006 Implementation of weno schemes in compressible multicomponent flow problems. J. Comput. Phys. 219 (2), 715–732.CrossRefGoogle Scholar
Johnsen, E. & Colonius, T. 2009 Numerical simulations of non-spherical bubble collapse. J. Fluid Mech. 629, 231–262.CrossRefGoogle ScholarPubMed
Joseph, D. D., Belanger, J. & Beavers, G. S. 1999 Breakup of a liquid drop suddenly exposed to a high-speed airstream. Intl J. Multiphase Flow 25 (6), 1263–1303.CrossRefGoogle Scholar
Klein, A. L., Bouwhuis, W., Visser, C. W., Lhuissier, H., Sun, C., Snoeijer, J. H., Villermaux, E., Lohse, D. & Gelderblom, H. 2015 Drop shaping by laser-pulse impact. Phys. Rev. Appl. 3 (4), 044018.CrossRefGoogle Scholar
Kondo, T. & Ando, K. 2016 One-way-coupling simulation of cavitation accompanied by high-speed droplet impact. Phys. Fluids 28 (3), 033303.CrossRefGoogle Scholar
Koukouvinis, P., Gavaises, M., Supponen, O. & Farhat, M. 2016 Numerical simulation of a collapsing bubble subject to gravity. Phys. Fluids 28 (3), 032110.CrossRefGoogle Scholar
Kush, E. A. & Schetz, J. A. 1973 Liquid jet injection into a supersonic flow. AIAA J. 11 (9), 1223–1224.CrossRefGoogle Scholar
Li, S. C. 2015 Tiny bubbles challenge giant turbines: three gorges puzzle. Interface Focus 5 (5), 20150020.CrossRefGoogle ScholarPubMed
Meng, J. C. & Colonius, T. 2015 Numerical simulations of the early stages of high-speed droplet breakup. Shock Waves 25 (4), 399–414.CrossRefGoogle Scholar
Nagy, J., Horvath, A., Jordan, C. & Harasek, M. 2012 Turbulent phenomena in the aerobreakup of liquid droplets. CFD Lett. 4 (3), 112–126.Google Scholar
Niederhaus, J. H. J., Greenough, J. A., Oakley, J. G., Ranjan, D., Anderson, M. H. & Bonazza, R. 2008 A computational parameter study for the three-dimensional shock–bubble interaction. J. Fluid Mech. 594, 85–124.CrossRefGoogle Scholar
Obreschkow, D., Dorsaz, N., Kobel, P., de Bosset, A., Tinguely, M., Field, J. & Farhat, M. 2011 Confined shocks inside isolated liquid volumes: a new path of erosion? Phys. Fluids 23 (10), 101702.CrossRefGoogle Scholar
Obreschkow, D., Kobel, P., Dorsaz, N., De Bosset, A., Nicollier, C. & Farhat, M. 2006 Cavitation bubble dynamics inside liquid drops in microgravity. Phys. Rev. Lett. 97 (9), 094502.CrossRefGoogle ScholarPubMed
Ohl, C. D. & Ikink, R. 2003 Shock-wave-induced jetting of micron-size bubbles. Phys. Rev. Lett. 90 (21), 214502.CrossRefGoogle ScholarPubMed
Pilch, M. & Erdman, C. A. 1987 Use of breakup time data and velocity history data to predict the maximum size of stable fragments for acceleration-induced breakup of a liquid drop. Intl J. Multiphase Flow 13 (6), 741–757.CrossRefGoogle Scholar
Ranjan, D., Oakley, J. & Bonazza, R. 2011 Shock-bubble interactions. Annu. Rev. Fluid Mech. 43, 117–140.CrossRefGoogle Scholar
Sankin, G. N., Simmons, W. N., Zhu, S. L. & Zhong, P. 2005 Shock wave interaction with laser-generated single bubbles. Phys. Rev. Lett. 95 (3), 034501.CrossRefGoogle ScholarPubMed
Sembian, S., Liverts, M., Tillmark, N. & Apazidis, N. 2016 Plane shock wave interaction with a cylindrical water column. Phys. Fluids 28 (5), 056102.CrossRefGoogle Scholar
Shaw, S. J. & Spelt, P. D. M. 2010 Shock emission from collapsing gas bubbles. J. Fluid Mech. 646, 363–373.CrossRefGoogle Scholar
Shpak, O., Verweij, M., de Jong, N. & Versluis, M. 2016 Droplets, bubbles and ultrasound interactions. In Therapeutic Ultrasound, pp. 157–174. Springer.CrossRefGoogle Scholar
Shpak, O., Verweij, M., Vos, H. J., de Jong, N., Lohse, D. & Versluis, M. 2014 Acoustic droplet vaporization is initiated by superharmonic focusing. Proc. Natl Acad. Sci. 111 (5), 1697–1702.CrossRefGoogle ScholarPubMed
Supponen, O., Kobel, P., Obreschkow, D. & Farhat, M. 2015 The inner world of a collapsing bubble. Phys. Fluids 27, 091113.CrossRefGoogle Scholar
Supponen, O., Obreschkow, D., Tinguely, M., Kobel, P., Dorsaz, N. & Farhat, M. 2016 Scaling laws for jets of single cavitation bubbles. J. Fluid Mech. 802, 263–293.CrossRefGoogle Scholar
Theofanous, T. G. & Li, G. J. 2008 On the physics of aerobreakup. Phys. Fluids 20 (5), 052103.CrossRefGoogle Scholar
Theofanous, T. G., Li, G. J. & Dinh, T. N. 2004 Aerobreakup in rarefied supersonic gas flows. Trans. ASME J. Fluids Engng 126 (4), 516–527.CrossRefGoogle Scholar
Tully, B., Hawker, N. & Ventikos, Y. 2016 Modeling asymmetric cavity collapse with plasma equations of state. Phys. Rev. E 93 (5), 053105.Google ScholarPubMed
Wang, B., Xiang, G. M., Zhang, W. B. & Hu, X. Y.2016 An incremental-stencil weno reconstruction for simulation of compressible two-phase flows. Preprint, submitted to Elsevier.Google Scholar
Wang, Q. X., Liu, W. K., Zhang, A. M. & Sui, Y. 2015 Bubble dynamics in a compressible liquid in contact with a rigid boundary. Interface Focus 5 (5), 20150048.CrossRefGoogle Scholar
Wierzba, A. & Takayama, K. 1988 Experimental investigation of the aerodynamic breakup of liquid drops. AIAA J. 26 (11), 1329–1335.CrossRefGoogle Scholar