Skip to main content
SpringerLink
Log in

Numerical modelling of shock-bubble interactions using a pressure-based algorithm without Riemann solvers

  • Research Article
  • Published:
Experimental and Computational Multiphase Flow Aims and scope Submit manuscript

A Correction to this article was published on 05 February 2022

This article has been updated

Abstract

The interaction of a shock wave with a bubble features in many engineering and emerging technological applications, and has been used widely to test new numerical methods for compressible interfacial flows. Recently, density-based algorithms with pressure-correction methods as well as fully-coupled pressure-based algorithms have been established as promising alternatives to classical density-based algorithms based on Riemann solvers. The current paper investigates the predictive accuracy of fully-coupled pressure-based algorithms without Riemann solvers in modelling the interaction of shock waves with one-dimensional and two-dimensional bubbles in gas-gas and liquid-gas flows. For a gas bubble suspended in another gas, the mesh resolution and the applied advection schemes are found to only have a minor influence on the bubble shape and position, as well as the behaviour of the dominant shock waves and rarefaction fans. For a gas bubble suspended in a liquid, however, the mesh resolution has a critical influence on the shape, the position and the post-shock evolution of the bubble, as well as the pressure and temperature distribution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Change history

References

  • Abgrall, R., Kami, S. 2001. Computations of compressible multifluids. J Comput Phys, 169: 594–623.

    Article  MathSciNet  MATH  Google Scholar 

  • Abgrall, R., Saurel, R. 2003. Discrete equations for physical and numerical compressible multiphase mixtures. J Comput Phys, 186: 361–396.

    Article  MathSciNet  MATH  Google Scholar 

  • Allaire, G., Clerc, S., Kokh, S. 2002. A five-equation model for the simulation of interfaces between compressible fluids. J Comput Phys, 181: 577–616.

    Article  MathSciNet  MATH  Google Scholar 

  • Anderson, J. D. 2003. Modern Compressible Flow: With a Historical Perspective. McGraw-Hill New York.

    Google Scholar 

  • Ando, K., Liu, A.-Q., Ohl, C.-D. 2012. Homogeneous nucleation in water in microuidic channels. Phys Rev Lett, 109: 044501.

    Article  Google Scholar 

  • Baer, M. R., Nunziato, J. W. 1986. A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials. Int J Multiphase Flow, 12: 861–889.

    Article  MATH  Google Scholar 

  • Bagabir, A., Drikakis, D. 2001. Mach number effects on shock-bubble interaction. Shock Waves, 11: 209–218.

    Article  MATH  Google Scholar 

  • Balay, S., Abhyankar, S., Adams, M. F., Brown, J., Brune, P., Buschelman, K., Dalcin, L. D., Eijkhout, V., Gropp, W., Kaushik, D., Knepley, M., May, D., McInnes, L. C., Munson, T., Rupp, K., Sanan, P., Smith, B., Zampini, S., Zhang, H., Zhang, H. 2017. PETSc users manual revision 3.8. Technical Report. ANL-95/11 — Revision 3.8. Argonne National Laboratory.

  • Bartholomew, P., Denner, F., Abdol-Azis, M. H., Marquis, A., van Wachem, B. G. M. 2018. Unified formulation of the momentum-weighted interpolation for collocated variable arrangements. J Comput Phys, 375: 177–208.

    Article  MathSciNet  MATH  Google Scholar 

  • Bo, W., Grove, J. W. 2014. A volume of fluid method based ghost fluid method for compressible multi-fluid flows. Comput Fluid, 90: 113–122.

    Article  MathSciNet  MATH  Google Scholar 

  • Brouillette, M. 2002. The Richtmyer-Meshkov instability. Ann Rev Fluid Mech, 34: 445–468.

    Article  MathSciNet  MATH  Google Scholar 

  • Chang, C.-H., Liou, M.-S. 2007. A robust and accurate approach to computing compressible multiphase flow: Stratified flow model and AUSM+-up scheme. J Comput Phys, 225: 840–873.

    Article  MathSciNet  MATH  Google Scholar 

  • Chorin, A. J. 1967. A numerical method for solving incompressible viscous flow problems. J Comput Phys, 2: 12–26.

    Article  MathSciNet  MATH  Google Scholar 

  • Chorin, A. J., Marsden, J. E. 1993. A Mathematical Introduction to Fluid Mechanics. Springer Verlag.

  • Coralic, V., Colonius, T. 2014. Finite-volume WENO scheme for viscous compressible multicomponent flows. J Comput Phys, 274: 95–121.

    Article  MathSciNet  MATH  Google Scholar 

  • Cordier, F., Degond, P., Kumbaro, A. 2012. An asymptotic-preserving all-speed scheme for the Euler and Navier-Stokes equations. J Comput Phys, 231: 5685–5704.

    Article  MathSciNet  MATH  Google Scholar 

  • Delale, C. F. 2013. Bubble Dynamics and Shock Waves. Springer Berlin Heidelberg.

    Book  MATH  Google Scholar 

  • Demirdžić, I., Lilek, Ž., Perić, M. 1993. A collocated finite volume method for predicting flows at all speeds. Int J Numer Meth Fluids, 16: 1029–1050.

    Article  MATH  Google Scholar 

  • Denner, F. 2018. Fully-coupled pressure-based algorithm for compressible flows: Linearisation and iterative solution strategies. Comput Fluid, 175: 53–65.

    Article  MathSciNet  MATH  Google Scholar 

  • Denner, F., van Wachem, B. 2015. TVD differencing on three-dimensional unstructured meshes with monotonicity-preserving correction of mesh skewness. J Comput Phys, 298: 466–479.

    Article  MathSciNet  MATH  Google Scholar 

  • Denner, F., van Wachem, B. G. M. 2014. Compressive VOF method with skewness correction to capture sharp interfaces on arbitrary meshes. J Comput Phys, 279: 127–144.

    Article  MathSciNet  MATH  Google Scholar 

  • Denner, F., Xiao, C.-N., van Wachem, B. G. M. 2018. Pressure-based algorithm for compressible interfacial flows with acoustically-conservative interface discretisation. J Comput Phys, 367: 192–234.

    Article  MathSciNet  MATH  Google Scholar 

  • Fedkiw, R. P., Aslam, T., Merriman, B., Osher, S. 1999a. A non-oscillatory eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J Comput Phys, 152: 457–492.

    Article  MathSciNet  MATH  Google Scholar 

  • Fedkiw, R. P., Aslam, T., Xu, S. J. 1999b. The ghost fluid method for deflagration and detonation discontinuities. J Comput Phys, 154: 393–427.

    Article  MathSciNet  MATH  Google Scholar 

  • Fuster, D. 2018. A review of models for bubble clusters in cavitating flows. Flow Turbulence Combust, 102: 497–536.

    Article  Google Scholar 

  • Fuster, D., Popinet, S. 2018. An all-Mach method for the simulation of bubble dynamics problems in the presence of surface tension. J Comput Phys, 374: 752–768

    Article  MathSciNet  MATH  Google Scholar 

  • Goncalves, E., Hoarau, Y., Zeidan, D. 2019. Simulation of shock-induced bubble collapse using a four-equation model. Shock Waves, 29: 221–234.

    Article  Google Scholar 

  • Haas, J.-F., Sturtevant, B. 1987. Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities. J Fluid Mech, 181: 41.

    Article  Google Scholar 

  • Haimovich, O., Frankel, S. H. 2017. Numerical simulations of compressible multicomponent and multiphase flow using a high-order targeted ENO (TENO) finite-volume method. Comput Fluid, 146: 105–116.

    Article  MathSciNet  MATH  Google Scholar 

  • Harlow, F. H., Amsden, A. A. 1971a. A numerical fluid dynamics calculation method for all flow speeds. J Comput Phys, 8: 197–213.

    Article  MATH  Google Scholar 

  • Harlow, F., Amsden, A. 1971b. Fluid Dynamics, Monograph LA-4700. Los Alamos National Laboratory.

  • Hauke, G., Hughes, T. J. R. 1998. A comparative study of different sets of variables for solving compressible and incompressible flows. Comput Method Appl M, 153: 1–44.

    Article  MathSciNet  MATH  Google Scholar 

  • Hejazialhosseini, B., Rossinelli, D., Koumoutsakos, P. 2013. Vortex dynamics in 3D shock-bubble interaction. Phys Fluid, 25: 110816.

    Article  Google Scholar 

  • Hirt, C. W., Nichols, B. D. 1981. Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys, 39: 201–225.

    Article  MATH  Google Scholar 

  • Hou, T. Y., Floch, P. G. L. 1994. Why nonconservative schemes converge to wrong solutions: Error analysis. Math Comput, 62: 497–530.

    Article  MathSciNet  MATH  Google Scholar 

  • Hu, X. Y., Khoo, B. C. 2004. An interface interaction method for compressible multifluids. J Comput Phys, 198: 35–64.

    Article  MATH  Google Scholar 

  • Johnsen, E. 2007. Numerical simulations of non-spherical bubble collapse: With applications to shockwave lithotripsy. Ph.D. Thesis. California Institute of Technology, USA.

    Google Scholar 

  • Johnsen, E. R. I. C., Colonius, T. I. M. 2009. Numerical simulations of non-spherical bubble collapse. J Fluid Mech, 629: 231–262.

    Article  MathSciNet  MATH  Google Scholar 

  • Johnsen, E., Colonius, T. 2006. Implementation of WENO schemes in compressible multicomponent flow problems. J Comput Phys, 219: 715–732.

    Article  MathSciNet  MATH  Google Scholar 

  • Karimian, S. M. H., Schneider, G. E. 1994. Pressure-based computational method for compressible and incompressible flows. J Thermophys Heat Tr, 8: 267–274.

    Article  Google Scholar 

  • Kokh, S., Lagoutière, F. 2010. An anti-diffusive numerical scheme for the simulation of interfaces between compressible fluids by means of a five-equation model. J Comput Phys, 229: 2773–2809.

    Article  MathSciNet  MATH  Google Scholar 

  • Kunz, R. F., Cope, W. K., Venkateswaran, S. 1999. Development of an implicit method for multi-fluid flow simulations. J Comput Phys, 152: 78–101.

    Article  MathSciNet  MATH  Google Scholar 

  • Layes, G., Jourdan, G., Houas, L. 2003. Distortion of a spherical gaseous interface accelerated by a plane shock wave. Phys Rev Lett, 91: 174502.

    Article  Google Scholar 

  • Layes, G., Jourdan, G., Houas, L. 2005. Experimental investigation of the shock wave interaction with a spherical gas inhomogeneity. Phys Fluid, 17: 028103.

    Article  MATH  Google Scholar 

  • Liu, C., Hu, C. H. 2017. Adaptive THINC-GFM for compressible multi-medium flows. J Comput Phys, 342: 43–65.

    Article  MathSciNet  MATH  Google Scholar 

  • Liu, T. G., Khoo, B. C., Yeo, K. S. 2003. Ghost fluid method for strong shock impacting on material interface. J Comput Phys, 190: 651–681.

    Article  MATH  Google Scholar 

  • Michael, L., Nikiforakis, N. 2019. The evolution of the temperature field during cavity collapse in liquid nitromethane. Part I: Inert case. Shock Waves, 29: 153–172.

    Article  Google Scholar 

  • Moguen, Y., Bruel, P., Dick, E. 2015. Solving low Mach number Riemann problems by a momentum interpolation method. J Comput Phys, 298: 741–746.

    Article  MathSciNet  MATH  Google Scholar 

  • Moguen, Y., Bruel, P., Dick, E. 2019. A combined momentum-interpolation and advection upstream splitting pressure-correction algorithm for simulation of convective and acoustic transport at all levels of Mach number. J Comput Phys, 384: 16–41.

    Article  MathSciNet  MATH  Google Scholar 

  • Moguen, Y., Kousksou, T., Bruel, P., Vierendeels, J., Dick, E. 2012. Pressure-velocity coupling allowing acoustic calculation in low Mach number flow. J Comput Phys, 231: 5522–5541.

    Article  MathSciNet  MATH  Google Scholar 

  • Moukalled, F., Mangani, L., Darwish, M. 2016. The Finite Volume Method in Computational Fluid Dynamics: An Advanced Introduction with OpenFOAM and Matlab. Springer.

  • Murrone, A., Guillard, H. 2005. A five equation reduced model for compressible two phase flow problems. J Comput Phys, 202: 664–698.

    Article  MathSciNet  MATH  Google Scholar 

  • Niederhaus, J. H. J., Greenough, J. A., Oakley, J. G., Bonazza, R. 2008a. Vorticity evolution in two- and three-dimensional simulations for shock-bubble interactions. Phys Scripta, T132: 014019.

    Article  Google Scholar 

  • Niederhaus, J. H. J., Greenough, J. A., Oakley, J. G., Ranjan, D., Anderson, M. H., Bonazza, R. 2008b. A computational parameter study for the three-dimensional shock-bubble interaction. J Fluid Mech, 594: 85–124.

    Article  MATH  Google Scholar 

  • Nourgaliev, R. R., Dinh, T. N., Theofanous, T. G. 2006. Adaptive characteristics-based matching for compressible multifluid dynamics. J Comput Phys, 213: 500–529.

    Article  MATH  Google Scholar 

  • Ohl, S.-W., Ohl, C.-D. 2016. Acoustic cavitation in a microchannel. In: Handbook of Ultrasonics and Sonochemistry. Springer Singapore, 99–135.

    Chapter  Google Scholar 

  • Pan, S., Adami, S., Hu, X., Adams, N. A. 2018. Phenomenology of bubble-collapse-driven penetration of biomaterial-surrogate liquid-liquid interfaces. Phys Rev Fluids, 3: 114005.

    Article  Google Scholar 

  • Park, J. H., Munz, C.-D. 2005. Multiple pressure variables methods for fluid flow at all Mach numbers. Int J Numer Meth Fluids, 49: 905–931.

    Article  MathSciNet  MATH  Google Scholar 

  • Quirk, J. J., Karni, S. 1996. On the dynamics of a shock-bubble interaction. J Fluid Mech, 318: 129.

    Article  MATH  Google Scholar 

  • Ranjan, D., Niederhaus, J., Motl, B., Anderson, M., Oakley, J., Bonazza, R. 2007. Experimental investigation of primary and secondary features in high-Mach-number shock-bubble interaction. Phys Rev Lett, 98: 024502.

    Article  Google Scholar 

  • Ranjan, D., Oakley, J., Bonazza, R. 2011. Shock-bubble interactions. Annu Rev Fluid Mech, 43: 117–140.

    Article  MathSciNet  MATH  Google Scholar 

  • Roe, P. 1986. Characteristic-based schemes for the Euler equations. Ann Rev Fluid Mech, 18: 337–365.

    Article  MathSciNet  MATH  Google Scholar 

  • Saurel, R., Abgrall, R. 1999. A simple method for compressible multifluid flows. SIAM J Sci Comput, 21: 1115–1145.

    Article  MathSciNet  MATH  Google Scholar 

  • Saurel, R., Le Métayer, O., Massoni, J., Gavrilyuk, S. 2007. Shock jump relations for multiphase mixtures with stiff mechanical relaxation. Shock Waves, 16: 209–232.

    Article  MATH  Google Scholar 

  • Saurel, R., Pantano, C. 2018. Diffuse-interface capturing methods for compressible two-phase flows. Ann Rev Fluid Mech, 50: 105–130.

    Article  MathSciNet  MATH  Google Scholar 

  • Shukla, R. K. 2014. Nonlinear preconditioning for efficient and accurate interface capturing in simulation of multicomponent compressible flows. J Comput Phys, 276: 508–540.

    Article  MathSciNet  MATH  Google Scholar 

  • Shukla, R. K., Pantano, C., Freund, J. B. 2010. An interface capturing method for the simulation of multi-phase compressible flows. J Comput Phys, 229: 7411–7439.

    Article  MathSciNet  MATH  Google Scholar 

  • Shyue, K.-M. 2006. A volume-fraction based algorithm for hybrid barotropic and non-barotropic two-fluid flow problems. Shock Waves, 15: 407–423.

    Article  MATH  Google Scholar 

  • Terashima, H., Tryggvason, G. 2009. A front-tracking/ghost-fluid method for fluid interfaces in compressible flows. J Comput Phys, 228: 4012–4037.

    Article  MATH  Google Scholar 

  • Tian, B. L., Toro, E. F., Castro, C. E. 2011. A path-conservative method for a five-equation model of two-phase flow with an HLLC-type Riemann solver. Comput Fluid, 46: 122–132.

    Article  MathSciNet  MATH  Google Scholar 

  • Tokareva, S. A., Toro, E. F. 2010. HLLC-type Riemann solver for the Baer-Nunziato equations of compressible two-phase flow. J Comput Phys, 229: 3573–3604.

    Article  MathSciNet  MATH  Google Scholar 

  • Toro, E. F., Spruce, M., Speares, W. 1994. Restoration of the contact surface in the HLL-Riemann solver. Shock Waves, 4: 25–34.

    Article  MATH  Google Scholar 

  • Turkel, E. 2006. Numerical methods and nature. J Sci Comput, 28: 549–570.

    Article  MathSciNet  MATH  Google Scholar 

  • Turkel, E., Fiterman, A., van Leer, B. 1993. Preconditioning and the limit to the incompressible flow equations. Technical Report. NASA CR-191500. Institute for Computer Applications in Science and Engineering Hampton VA, USA.

    Google Scholar 

  • Ubbink, O., Issa, R. I. 1999. A method for capturing sharp fluid interfaces on arbitrary meshes. J Comput Phys, 153: 26–50.

    Article  MathSciNet  MATH  Google Scholar 

  • Van der Heul, D. R., Vuik, C., Wesseling, P. 2003. A conservative pressure-correction method for flow at all speeds. Comput Fluid, 32: 1113–1132.

    Article  MathSciNet  MATH  Google Scholar 

  • Van Doormaal, J. P., Raithby, G. D., McDonald, B. H. 1987. The segregated approach to predicting viscous compressible fluid flows. J Turbomach, 109: 268–277.

    Article  Google Scholar 

  • Wang, C. W., Liu, T. G., Khoo, B. C. 2006. A real ghost fluid method for the simulation of multimedium compressible flow. SIAM J Sci Comput, 28: 278–302.

    Article  MathSciNet  MATH  Google Scholar 

  • Wesseling, P. 2001. Principles of Computational Fluid Dynamics. Springer.

  • Wong, M. L., Lele, S. K. 2017. High-order localized dissipation weighted compact nonlinear scheme for shock- and interface-capturing in compressible flows. J Comput Phys, 339: 179–209.

    Article  MathSciNet  MATH  Google Scholar 

  • Xiang, G., Wang, B. 2017. Numerical study of a planar shock interacting with a cylindrical water column embedded with an air cavity. J Fluid Mech, 825: 825–852.

    Article  MathSciNet  Google Scholar 

  • Xiao, C.-N., Denner, F., van Wachem, B. G. M. 2017. Fully-coupled pressure-based finite-volume framework for the simulation of fluid flows at all speeds in complex geometries. J Comput Phys, 346: 91–130.

    Article  MathSciNet  MATH  Google Scholar 

  • Xiao, F. 2004. Unified formulation for compressible and incompressible flows by using multi-integrated moments I: One-dimensional inviscid compressible flow. J Comput Phys, 195: 629–654.

    Article  MathSciNet  MATH  Google Scholar 

  • Yoo, Y.-L., Sung, H.-G. 2018. Numerical investigation of an interaction between shock waves and bubble in a compressible multiphase flow using a diffuse interface method. Int J Heat Mass Tran, 127: 210–221.

    Article  Google Scholar 

  • Zhai, Z., Si, T., Luo, X., Yang, J. 2011. On the evolution of spherical gas interfaces accelerated by a planar shock wave. Phys Fluid, 23: 084104.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Chair of Mechanical Process Engineering, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Fabian Denner & Berend G. M. van Wachem

Authors
  1. Fabian Denner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Berend G. M. van Wachem
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fabian Denner.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Denner, F., van Wachem, B.G.M. Numerical modelling of shock-bubble interactions using a pressure-based algorithm without Riemann solvers. Exp. Comput. Multiph. Flow 1, 271–285 (2019). https://doi.org/10.1007/s42757-019-0021-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s42757-019-0021-2

Keywords

Search

Navigation