1932

Abstract

Simulation of compressible flows became a routine activity with the appearance of shock-/contact-capturing methods. These methods can determine all waves, particularly discontinuous ones. However, additional difficulties may appear in two-phase and multimaterial flows due to the abrupt variation of thermodynamic properties across the interfacial region, with discontinuous thermodynamical representations at the interfaces. To overcome this difficulty, researchers have developed augmented systems of governing equations to extend the capturing strategy. These extended systems, reviewed here, are termed diffuse-interface models, because they are designed to compute flow variables correctly in numerically diffused zones surrounding interfaces. In particular, they facilitate coupling the dynamics on both sides of the (diffuse) interfaces and tend to the proper pure fluid–governing equations far from the interfaces. This strategy has become efficient for contact interfaces separating fluids that are governed by different equations of state, in the presence or absence of capillary effects, and with phase change. More sophisticated materials than fluids (e.g., elastic–plastic materials) have been considered as well.

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2018-01-05
2024-06-13
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Literature Cited

  1. Abgrall R. 1996. How to prevent pressure oscillations in multicomponent flow calculations: a quasi conservative approach. J. Comput. Phys. 125:150–60 [Google Scholar]
  2. Abgrall R, Saurel R. 2003. Discrete equations for physical and numerical compressible multiphase mixtures. J. Comput. Phys. 186:361–96 [Google Scholar]
  3. Abramzon B, Sirignano W. 1989. Droplet vaporization model for spray combustion calculations. Int. J. Heat Mass Transf. 32:1605–18 [Google Scholar]
  4. 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 [Google Scholar]
  5. Anderson DM, McFadden GB, Wheeler AA. 1998. Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30:139–65 [Google Scholar]
  6. Aslam T, Bdzil J, Stewart D. 1996. Level set methods applied to modeling detonation shock dynamics. J. Comput. Phys. 126:390–409 [Google Scholar]
  7. Baer M, Nunziato J. 1986. A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiphase Flow 12:861–89 [Google Scholar]
  8. Bdzil J, Menikoff R, Son S, Kapila A, Stewart D. 1999. Two-phase modeling of deflagration-to-detonation transition in granular materials: a critical examination of modeling issues. Phys. Fluids 11:378–402 [Google Scholar]
  9. Brackbill J, Kothe D, Zemach C. 1992. A continuum method for modeling surface tension. J. Comput. Phys. 100:335–54 [Google Scholar]
  10. Cahn J, Hilliard J. 1958. Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28:258–67 [Google Scholar]
  11. Chiapolino A, Boivin P, Saurel R. 2017a. A simple and fast phase transition relaxation solver for compressible multicomponent two-phase flows. Comput. Fluids 150:31–45 [Google Scholar]
  12. Chiapolino A, Boivin P, Saurel R. 2017b. A simple phase transition relaxation solver for liquid–vapor flows. Int. J. Numer. Methods Fluids 83:583–605 [Google Scholar]
  13. Chiapolino A, Saurel R, Nkonga B. 2017c. Sharpening diffuse interfaces with compressible fluids on unstructured meshes. J. Comput. Phys. 340:389–417 [Google Scholar]
  14. Cockburn B, Shu CW. 1989. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comput. 52:411–35 [Google Scholar]
  15. Dal Maso G, LeFloch P, Murat F. 1995. Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74:483–548 [Google Scholar]
  16. Dervieux A, Thomasset F. 1980. A finite element method for the simulation of a Rayleigh-Taylor instability. Approx. Methods Navier-Stokes Prob.: Proc. Symp. Int. Union Theor. Appl. Mech., Paderborn, Ger., 9–15 Sept. R Rautmann 145–58 Berlin: Springer-Verlag [Google Scholar]
  17. Dumbser M, Hidalgo A, Castro M, Parés C, Toro E. 2010. FORCE schemes on unstructured meshes II: non-conservative hyperbolic systems. Comput. Methods Appl. Mech. Eng. 199:625–47 [Google Scholar]
  18. Favrie N, Gavrilyuk S, Saurel R. 2009. Solid–fluid diffuse interface model in cases of extreme deformations. J. Comput. Phys. 228:6037–77 [Google Scholar]
  19. Fedkiw R, Aslam T, Merriman B, Osher S. 1999. A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152:457–92 [Google Scholar]
  20. Franquet E, Perrier V. 2012. Runge–Kutta discontinous Galerkin method for the approximation of Baer and Nunziato type multiphase models. J. Comput. Phys. 231:4096–141 [Google Scholar]
  21. Furfaro D, Saurel R. 2015. A simple HLLC-type Riemann solver for compressible non-equilibrium two-phase flows. Comput. Fluids 111:159–78 [Google Scholar]
  22. Furfaro D, Saurel R. 2016. Modeling droplet phase change in the presence of a multi-component gas mixture. Appl. Math. Comput. 272:518–41 [Google Scholar]
  23. Glimm J, Grove J, Li X, Shyue K, Zeng Y, Zhang Q. 1998. Three-dimensional front tracking. SIAM J. Sci. Comput. 19:703–27 [Google Scholar]
  24. Glowinski R, Osher S, Yin W. 2010. Splitting Methods in Communication and Imaging, Science, and Engineering Cham, Switz.: Springer Int. [Google Scholar]
  25. Godunov SK. 1959. A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. 89:271–306 [Google Scholar]
  26. Harten A, Engquist B, Osher S, Chakravarthy S. 1987. Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71:231–303 [Google Scholar]
  27. Harten A, Lax P, van Leer B. 1983. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25:35–61 [Google Scholar]
  28. Hejazialhosseini B, Rossinelli D, Koumoutsakos P. 2013. Vortex dynamics in 3D shock-bubble interaction. Phys. Fluids 25:110816 [Google Scholar]
  29. Hirt C, Amsden A, Cook J. 1974. Arbitrary Lagrangian-Eulerian computing method for all flow speeds. J. Comput. Phys. 14:227–53 [Google Scholar]
  30. Hirt C, Nichols B. 1981. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39:201–25 [Google Scholar]
  31. Johnsen E, Colonius T. 2009. Numerical simulations of non-spherical bubble collapse. J. Fluid Mech. 629:231–62 [Google Scholar]
  32. Kapila A, Menikoff R, Bdzil J, Son S, Stewart D. 2001. Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations. Phys. Fluids 13:3002–24 [Google Scholar]
  33. Lallemand M-H, Chinnayya A, Le Metayer O. 2005. Pressure relaxation procedures for multiphase compressible flows. Int. J. Numer. Methods Fluids 49:1–56 [Google Scholar]
  34. Lallemand M-H, Saurel R. 2000. Pressure relaxation procedures for multiphase compressible flows Tech. Rep. RR-4038, Inst. Natl. Rech. Inform. Autom. (INRIA), Rocquencourt, Fr. [Google Scholar]
  35. Layes G, Le Metayer O. 2007. Quantitative numerical and experimental studies of the shock accelerated heterogeneous bubbles motion. Phys. Fluids 19:042105 [Google Scholar]
  36. Le Martelot S, Nkonga B, Saurel R. 2013. Liquid and liquid–gas flows at all speeds. J. Comput. Phys. 255:53–82 [Google Scholar]
  37. Le Martelot S, Saurel R, Nkonga B. 2014. Towards the direct numerical simulation of nucleate boiling flows. Int. J. Multiphase Flow 66:62–78 [Google Scholar]
  38. Le Metayer O, Saurel R. 2016. The Noble-Abel stiffened-gas equation of state. Phys. Fluids 28:046102 [Google Scholar]
  39. Leveque RJ. 1997. Wave propagation algorithms for multidimensional hyperbolic systems. J. Comput. Phys. 131:327–53 [Google Scholar]
  40. Leveque RJ. 2002. Finite Volume Methods for Hyperbolic Problems Cambridge, U.K.: Cambridge Univ. Press [Google Scholar]
  41. Loubere R, Dumbser M, Diot S. 2014. A new family of high order unstructured MOOD and ADER finite volume schemes for multidimensional systems of hyperbolic conservation laws. Commun. Comput. Phys. 16:718–63 [Google Scholar]
  42. Lund H. 2012. A hierarchy of relaxation models for two-phase flow. SIAM J. Appl. Math. 72:1713–41 [Google Scholar]
  43. Massoni J, Saurel R, Nkonga B, Abgrall R. 2002. Some models and Eulerian methods for interface problems between compressible fluids with heat transfer. Int. J. Heat Mass Transf. 45:1287–307 [Google Scholar]
  44. Meng J. 2016. Numerical simulation of droplet aerobreakup PhD Thesis, Calif. Inst. Technol. [Google Scholar]
  45. Murrone A, Guillard H. 2005. A five equation reduced model for compressible two phase flow problems. J. Comput. Phys. 202:664–98 [Google Scholar]
  46. Ndanou S, Favrie N, Gavrilyuk S. 2015. Multi-solid and multi-fluid diffuse interface model: applications to dynamic fracture and fragmentation. J. Comput. Phys. 295:523–55 [Google Scholar]
  47. Olsson E, Kreiss G, Zahedi S. 2007. A conservative level set method for two phase flow II. J. Comput. Phys. 225:785–807 [Google Scholar]
  48. Osher S, Solomon F. 1982. Upwind difference schemes for hyperbolic systems of conservation laws. Math. Comput. 38:339–74 [Google Scholar]
  49. Perigaud G, Saurel R. 2005. A compressible flow model with capillary effects. J. Comput. Phys. 209:139–78 [Google Scholar]
  50. Petitpas F, Franquet E, Saurel R, Metayer OL. 2007. A relaxation-projection method for compressible flows. Part II: artificial heat exchanges for multiphase shocks. J. Comput. Phys. 225:2214–48 [Google Scholar]
  51. Petitpas F, Saurel R, Franquet E, Chinnayya A. 2009. Modelling detonation waves in condensed energetic materials: multiphase CJ conditions and multidimensional computations. Shock Waves 19:377–401 [Google Scholar]
  52. Roe P. 1981. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43:357–72 [Google Scholar]
  53. Rossinelli D, Hejazialhosseini B, Hadjidoukas P, Bekas C, Curioni A. 2013. 11 PFLOP/s simulation of cloud cavitation collapse. Proc. Int. Conf. High Perform. Comput., Netw., Storage Anal., Denver, Colo., 17–21 Nov. New York: Assoc. Comput. Mach. [Google Scholar]
  54. Saurel R, Abgrall R. 1999a. A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150:425–67 [Google Scholar]
  55. Saurel R, Abgrall R. 1999b. A simple method for compressible multifluid flows. SIAM J. Sci. Comput. 21:1115–45 [Google Scholar]
  56. Saurel R, Boivin P, Le Metayer O. 2016. A general formulation for cavitating, boiling and evaporating flows. Comput. Fluids 128:53–64 [Google Scholar]
  57. Saurel R, Chinnayya A, Carmouze Q. 2017. Modelling compressible dense and dilute two-phase flows. Phys. Fluids 29:063301 [Google Scholar]
  58. Saurel R, Franquet E, Daniel E, Le Metayer O. 2007a. A relaxation-projection method for compressible flows. Part I: the numerical equation of state for the Euler equations. J. Comput. Phys. 223:822–45 [Google Scholar]
  59. Saurel R, Gavrilyuk S, Renaud F. 2003. A multiphase model with internal degrees of freedom: application to shock–bubble interaction. J. Fluid Mech. 495:283–321 [Google Scholar]
  60. Saurel R, Le Martelot S, Tosello R, Lapébie E. 2014. Symmetric model of compressible granular mixtures with permeable interfaces. Phys. Fluids 26:123304 [Google Scholar]
  61. Saurel R, Le Metayer O, Massoni J, Gavrilyuk S. 2007b. Shock jump relations for multiphase mixtures with stiff mechanical relaxation. Shock Waves 16:209–32 [Google Scholar]
  62. Saurel R, Petitpas F, Abgrall R. 2008. Modelling phase transition in metastable liquids: application to cavitating and flashing flows. J. Fluid Mech. 607:313–50 [Google Scholar]
  63. Saurel R, Petitpas F, Berry R. 2009. Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures. J. Comput. Phys. 228:1678–712 [Google Scholar]
  64. Scardovelli R, Zaleski S. 1999. Direct numerical simulation of free-surface and interfacial flow. Annu. Rev. Fluid Mech. 31:567–603 [Google Scholar]
  65. Schoch S, Nikiforakis N, Lee B, Saurel R. 2013. Multi-phase simulation of ammonium nitrate emulsion detonations. Combust. Flame 160:1883–99 [Google Scholar]
  66. Sethian JA, Smereka P. 2003. Level set methods for fluid interfaces. Annu. Rev. Fluid Mech. 35:341–72 [Google Scholar]
  67. Shu CW, Osher S. 1988. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77:439–71 [Google Scholar]
  68. Shukla R. 2014. Nonlinear preconditioning for efficient and accurate interface capturing in simulation of multicomponent compressible flows. J. Comput. Phys. 276:508–40 [Google Scholar]
  69. Shukla RK, Pantano C, Freund JB. 2010. An interface capturing method for the simulation of multi-phase compressible flows. J. Comput. Phys. 229:7411–39 [Google Scholar]
  70. Shyue KM. 1998. An efficient shock-capturing algorithm for compressible multicomponent problems. J. Comput. Phys. 142:208–42 [Google Scholar]
  71. Shyue KM, Xiao F. 2014. An Eulerian interface sharpening algorithm for compressible two-phase flow: the algebraic THINC approach. J. Comput. Phys. 268:326–54 [Google Scholar]
  72. Titarev V, Toro E. 2002. ADER: arbitrary high order Godunov approach. J. Sci. Comput. 17:609–18 [Google Scholar]
  73. Tiwari A, Freund J, Pantano C. 2013. A diffuse interface model with immiscibility preservation. J. Comput. Phys. 252:290–309 [Google Scholar]
  74. Tiwari A, Pantano C, Freund J. 2015. Growth-and-collapse dynamics of small bubble clusters near a wall. J. Fluid Mech. 775:1–23 [Google Scholar]
  75. Tokareva S, Toro E. 2010. HLLC-type Riemann solver for the Baer-Nunziato equations of compressible two-phase flow. J. Comput. Phys. 229:3573–604 [Google Scholar]
  76. Toro E. 2009. Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction Berlin: Springer-Verlag [Google Scholar]
  77. Toro E, Spruce M, Spears W. 1994. Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4:25–34 [Google Scholar]
  78. van Leer B. 1979. Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method. J. Comput. Phys. 32:101–36 [Google Scholar]
  79. von Neumann J, Richtmyer R. 1950. A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys. 21:232–37 [Google Scholar]
  80. Wood AB. 1930. A Textbook of Sound New York: Macmillan [Google Scholar]
  81. Zein A, Hantke M, Warnecke G. 2010. Modeling phase transition for compressible two-phase flows applied to metastable liquids. J. Comput. Phys. 229:2964–98 [Google Scholar]
  82. Zhang X, Shu CW. 2012. Positivity-preserving high order finite difference WENO schemes for compressible Euler equations. J. Comput. Phys. 231:2245–58 [Google Scholar]
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