Advertisement

Experimental and Computational Multiphase Flow

, Volume 1, Issue 4, pp 271–285 | Cite as

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

  • Fabian DennerEmail author
  • Berend G. M. van Wachem
Research Article

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.

Keywords

shock-bubble interaction shock capturing interfacial flows finite-volume methods volume-of-fluid methods 

References

  1. Abgrall, R., Kami, S. 2001. Computations of compressible multifluids. J Comput Phys, 169: 594–623.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Abgrall, R., Saurel, R. 2003. Discrete equations for physical and numerical compressible multiphase mixtures. J Comput Phys, 186: 361–396.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  4. Anderson, J. D. 2003. Modern Compressible Flow: With a Historical Perspective. McGraw-Hill New York.Google Scholar
  5. Ando, K., Liu, A.-Q., Ohl, C.-D. 2012. Homogeneous nucleation in water in microuidic channels. Phys Rev Lett, 109: 044501.CrossRefGoogle Scholar
  6. 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.CrossRefzbMATHGoogle Scholar
  7. Bagabir, A., Drikakis, D. 2001. Mach number effects on shock-bubble interaction. Shock Waves, 11: 209–218.CrossRefzbMATHGoogle Scholar
  8. 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.Google Scholar
  9. 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.MathSciNetCrossRefGoogle Scholar
  10. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  11. Brouillette, M. 2002. The Richtmyer-Meshkov instability. Ann Rev Fluid Mech, 34: 445–468.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Chorin, A. J. 1967. A numerical method for solving incompressible viscous flow problems. J Comput Phys, 2: 12–26.CrossRefzbMATHGoogle Scholar
  14. Chorin, A. J., Marsden, J. E. 1993. A Mathematical Introduction to Fluid Mechanics. Springer Verlag.Google Scholar
  15. Coralic, V., Colonius, T. 2014. Finite-volume WENO scheme for viscous compressible multicomponent flows. J Comput Phys, 274: 95–121.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  17. Delale, C. F. 2013. Bubble Dynamics and Shock Waves. Springer Berlin Heidelberg.CrossRefzbMATHGoogle Scholar
  18. 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.CrossRefzbMATHGoogle Scholar
  19. Denner, F. 2018. Fully-coupled pressure-based algorithm for compressible flows: Linearisation and iterative solution strategies. Comput Fluid, 175: 53–65.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  24. Fedkiw, R. P., Aslam, T., Xu, S. J. 1999b. The ghost fluid method for deflagration and detonation discontinuities. J Comput Phys, 154: 393–427.MathSciNetCrossRefzbMATHGoogle Scholar
  25. Fuster, D. 2018. A review of models for bubble clusters in cavitating flows. Flow Turbulence Combust, 102: 497–536.CrossRefGoogle Scholar
  26. 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: 752–768MathSciNetCrossRefzbMATHGoogle Scholar
  27. Goncalves, E., Hoarau, Y., Zeidan, D. 2019. Simulation of shock-induced bubble collapse using a four-equation model. Shock Waves, 29: 221–234.CrossRefGoogle Scholar
  28. Haas, J.-F., Sturtevant, B. 1987. Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities. J Fluid Mech, 181: 41.CrossRefGoogle Scholar
  29. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  30. Harlow, F. H., Amsden, A. A. 1971a. A numerical fluid dynamics calculation method for all flow speeds. J Comput Phys, 8: 197–213.CrossRefzbMATHGoogle Scholar
  31. Harlow, F., Amsden, A. 1971b. Fluid Dynamics, Monograph LA-4700. Los Alamos National Laboratory.Google Scholar
  32. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  33. Hejazialhosseini, B., Rossinelli, D., Koumoutsakos, P. 2013. Vortex dynamics in 3D shock-bubble interaction. Phys Fluid, 25: 110816.CrossRefGoogle Scholar
  34. Hirt, C. W., Nichols, B. D. 1981. Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys, 39: 201–225.CrossRefzbMATHGoogle Scholar
  35. Hou, T. Y., Floch, P. G. L. 1994. Why nonconservative schemes converge to wrong solutions: Error analysis. Math Comput, 62: 497–530.MathSciNetCrossRefzbMATHGoogle Scholar
  36. Hu, X. Y., Khoo, B. C. 2004. An interface interaction method for compressible multifluids. J Comput Phys, 198: 35–64.CrossRefzbMATHGoogle Scholar
  37. 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
  38. Johnsen, E. R. I. C., Colonius, T. I. M. 2009. Numerical simulations of non-spherical bubble collapse. J Fluid Mech, 629: 231–262.MathSciNetCrossRefzbMATHGoogle Scholar
  39. Johnsen, E., Colonius, T. 2006. Implementation of WENO schemes in compressible multicomponent flow problems. J Comput Phys, 219: 715–732.MathSciNetCrossRefzbMATHGoogle Scholar
  40. Karimian, S. M. H., Schneider, G. E. 1994. Pressure-based computational method for compressible and incompressible flows. J Thermophys Heat Tr, 8: 267–274.CrossRefGoogle Scholar
  41. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  42. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  43. Layes, G., Jourdan, G., Houas, L. 2003. Distortion of a spherical gaseous interface accelerated by a plane shock wave. Phys Rev Lett, 91: 174502.CrossRefGoogle Scholar
  44. Layes, G., Jourdan, G., Houas, L. 2005. Experimental investigation of the shock wave interaction with a spherical gas inhomogeneity. Phys Fluid, 17: 028103.CrossRefzbMATHGoogle Scholar
  45. Liu, C., Hu, C. H. 2017. Adaptive THINC-GFM for compressible multi-medium flows. J Comput Phys, 342: 43–65.MathSciNetCrossRefzbMATHGoogle Scholar
  46. 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.CrossRefzbMATHGoogle Scholar
  47. 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.CrossRefGoogle Scholar
  48. Moguen, Y., Bruel, P., Dick, E. 2015. Solving low Mach number Riemann problems by a momentum interpolation method. J Comput Phys, 298: 741–746.MathSciNetCrossRefzbMATHGoogle Scholar
  49. 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.MathSciNetCrossRefGoogle Scholar
  50. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  51. Moukalled, F., Mangani, L., Darwish, M. 2016. The Finite Volume Method in Computational Fluid Dynamics: An Advanced Introduction with OpenFOAM and Matlab. Springer.Google Scholar
  52. Murrone, A., Guillard, H. 2005. A five equation reduced model for compressible two phase flow problems. J Comput Phys, 202: 664–698.MathSciNetCrossRefzbMATHGoogle Scholar
  53. 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.CrossRefGoogle Scholar
  54. 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.CrossRefzbMATHGoogle Scholar
  55. Nourgaliev, R. R., Dinh, T. N., Theofanous, T. G. 2006. Adaptive characteristics-based matching for compressible multifluid dynamics. J Comput Phys, 213: 500–529.CrossRefzbMATHGoogle Scholar
  56. Ohl, S.-W., Ohl, C.-D. 2016. Acoustic cavitation in a microchannel. In: Handbook of Ultrasonics and Sonochemistry. Springer Singapore, 99–135.CrossRefGoogle Scholar
  57. 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.CrossRefGoogle Scholar
  58. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  59. Quirk, J. J., Karni, S. 1996. On the dynamics of a shock-bubble interaction. J Fluid Mech, 318: 129.CrossRefzbMATHGoogle Scholar
  60. 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.CrossRefGoogle Scholar
  61. Ranjan, D., Oakley, J., Bonazza, R. 2011. Shock-bubble interactions. Annu Rev Fluid Mech, 43: 117–140.MathSciNetCrossRefzbMATHGoogle Scholar
  62. Roe, P. 1986. Characteristic-based schemes for the Euler equations. Ann Rev Fluid Mech, 18: 337–365.MathSciNetCrossRefzbMATHGoogle Scholar
  63. Saurel, R., Abgrall, R. 1999. A simple method for compressible multifluid flows. SIAM J Sci Comput, 21: 1115–1145.MathSciNetCrossRefzbMATHGoogle Scholar
  64. 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.CrossRefzbMATHGoogle Scholar
  65. Saurel, R., Pantano, C. 2018. Diffuse-interface capturing methods for compressible two-phase flows. Ann Rev Fluid Mech, 50: 105–130.MathSciNetCrossRefzbMATHGoogle Scholar
  66. Shukla, R. K. 2014. Nonlinear preconditioning for efficient and accurate interface capturing in simulation of multicomponent compressible flows. J Comput Phys, 276: 508–540.MathSciNetCrossRefzbMATHGoogle Scholar
  67. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  68. Shyue, K.-M. 2006. A volume-fraction based algorithm for hybrid barotropic and non-barotropic two-fluid flow problems. Shock Waves, 15: 407–423.CrossRefzbMATHGoogle Scholar
  69. Terashima, H., Tryggvason, G. 2009. A front-tracking/ghost-fluid method for fluid interfaces in compressible flows. J Comput Phys, 228: 4012–4037.CrossRefzbMATHGoogle Scholar
  70. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  71. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  72. Toro, E. F., Spruce, M., Speares, W. 1994. Restoration of the contact surface in the HLL-Riemann solver. Shock Waves, 4: 25–34.CrossRefzbMATHGoogle Scholar
  73. Turkel, E. 2006. Numerical methods and nature. J Sci Comput, 28: 549–570.MathSciNetCrossRefzbMATHGoogle Scholar
  74. 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
  75. Ubbink, O., Issa, R. I. 1999. A method for capturing sharp fluid interfaces on arbitrary meshes. J Comput Phys, 153: 26–50.MathSciNetCrossRefzbMATHGoogle Scholar
  76. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  77. 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.CrossRefGoogle Scholar
  78. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  79. Wesseling, P. 2001. Principles of Computational Fluid Dynamics. Springer.Google Scholar
  80. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  81. 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.MathSciNetCrossRefGoogle Scholar
  82. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  83. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  84. 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.CrossRefGoogle Scholar
  85. 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.CrossRefGoogle Scholar

Copyright information

© Tsinghua University Press 2019

Authors and Affiliations

  1. 1.Chair of Mechanical Process EngineeringOtto-von-Guericke-Universität MagdeburgMagdeburgGermany

Personalised recommendations