Shock Waves

, Volume 29, Issue 1, pp 235–261 | Cite as

Methods for compressible multiphase flows and their applications

  • H. Kim
  • Y. Choe
  • H. Kim
  • D. Min
  • C. KimEmail author
Original Article


This paper presents an efficient and robust numerical framework to deal with multiphase real-fluid flows and their broad spectrum of engineering applications. A homogeneous mixture model incorporated with a real-fluid equation of state and a phase change model is considered to calculate complex multiphase problems. As robust and accurate numerical methods to handle multiphase shocks and phase interfaces over a wide range of flow speeds, the AUSMPW+_N and RoeM_N schemes with a system preconditioning method are presented. These methods are assessed by extensive validation problems with various types of equation of state and phase change models. Representative realistic multiphase phenomena, including the flow inside a thermal vapor compressor, pressurization in a cryogenic tank, and unsteady cavitating flow around a wedge, are then investigated as application problems. With appropriate physical modeling followed by robust and accurate numerical treatments, compressible multiphase flow physics such as phase changes, shock discontinuities, and their interactions are well captured, confirming the suitability of the proposed numerical framework to wide engineering applications.


Homogeneous mixture model Low-Mach-number preconditioning Multiphase shock capturing Cryogenic flows Phase change 



This research is supported by the program of National Research Foundation of Korea (NRF-2014M1A3A3A02034856), by Advanced Research Center Program (NRF-2013R1A5A1073861) through the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) contracted through Advanced Space Propulsion Research Center at Seoul National University, and by the Civil-Military Technology Cooperation Program. This work is also supported by the KISTI Supercomputing Center (KSC-2016-C3-0067, KSC-2017-G2-0004). The authors appreciate the valuable experimental data provided by Doosan Heavy Industries and Chungnam National University Cavitation Tunnel (CNU-CT). Finally, the comments and suggestions of the reviewers on the original manuscript are highly appreciated.


  1. 1.
    Baer, M.R., Nunziato, J.W.: A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiph. Flows 12, 861–889 (1986). zbMATHGoogle Scholar
  2. 2.
    Romenski, E., Resnyansky, A.D., Toro, E.F.: Conservative hyperbolic formulation for compressible two-phase flow with different phase pressures and temperatures. Q. Appl. Math. 65(2), 259–279 (2007). MathSciNetzbMATHGoogle Scholar
  3. 3.
    Zeidan, D.: Assessment of mixture two-phase flow equations for volcanic flows using Godunov-type methods. Appl. Math. Comput. 272, 707–719 (2016). MathSciNetGoogle Scholar
  4. 4.
    Bruce Stewart, H., Wendroff, B.: Two-phase flow: Models and methods. J. Comput. Phys. 56(3), 363–409 (1984). MathSciNetzbMATHGoogle Scholar
  5. 5.
    Liou, M.S., Chang, C.H., Nguyen, L., Theofanous, T.G.: How to solve compressible multifluid equations: a simple, robust, and accurate method. AIAA J. 46(9), 2345–2356 (2008). Google Scholar
  6. 6.
    Saurel, R., Le Metayer, O., Massoni, J., Gavrilyuk, S.: Shock jump relations for multiphase mixtures with stiff mechanical relaxation. Shock Waves 16(3), 209–232 (2007). zbMATHGoogle Scholar
  7. 7.
    Hosangadi, A., Ahuja, V.: Numerical study of cavitation in cryogenic fluids. J. Fluids Eng. 127(2), 267–281 (2005). Google Scholar
  8. 8.
    Kunz, R.F., Boger, D.A., Stinebring, D.R., Chyczewski, T.S., Lindau, J.W., Gibeling, H.J., Venkateswaran, S., Govindan, T.: A preconditioned Navier–Stokes method for two-phase flows with application to cavitation prediction. Comput. Fluids 29(8), 849–875 (2000). zbMATHGoogle Scholar
  9. 9.
    Saurel, R., Boivin, P., Le Métayer, O.: A general formulation for cavitating, boiling and evaporating flows. Comput. Fluids 128, 53–64 (2016). MathSciNetzbMATHGoogle Scholar
  10. 10.
    Utturkar, Y., Wu, J., Wang, G., Shyy, W.: Recent progress in modeling of cryogenic cavitation for liquid rocket propulsion. Prog. Aerosp. Sci. 41(7), 558–608 (2005). Google Scholar
  11. 11.
    Kim, H., Kim, H., Kim, C.: Computations for homogeneous multi-phase real fluid flows at all speeds. AIAA J. (2018).
  12. 12.
    Flåtten, T., Lund, H.: Relaxation two-phase flow models and the subcharacteristic condition. Math. Models Methods Appl. Sci. 21(12), 2379–2407 (2011). MathSciNetzbMATHGoogle Scholar
  13. 13.
    Harlow, F.H., Amsden, A.A.: Fluid Dynamics: A LASL Monograph. Technical Report LA-4700, Los Alamos, New Mexico (1971)Google Scholar
  14. 14.
    Wagner, W., Kretzschmar, H-J.: International steam tables: properties of water and steam based on the industrial formulation IAPWS-IF97. In: IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam, pp. 7–150. Springer, Berlin, Heidelberg (2008).
  15. 15.
    NIST: NIST reference fluid thermodynamic and transport properties database (REFPROP): version 8.0. (2010)
  16. 16.
    Kunick, M., Kretzschmar, H.-J.: Guideline on the fast calculation of steam and water properties with the spline-based table look-up method (SBTL). Technical Report, The International Association for the Properties of Water and Steam (2015)Google Scholar
  17. 17.
    Merkle, C.L., Feng, J.Z., Buelow, P.E.O.: Computational modeling of the dynamics of sheet cavitation. In: 3rd International Symposium on Cavitation. Grenoble, France (1998)Google Scholar
  18. 18.
    Sauer, J., Schnerr, G.H.: Unsteady cavitating flow—a new cavitation model based on modified front capturing method and bubble dynamics. In: Summer Meeting, American Society of Mechanical Engineers; Fluids Engineering Division. American Society of Mechanical Enginners, Boston, Massachusetts (2000)Google Scholar
  19. 19.
    Hertz, H.: Ueber die Verdunstung der Flüssigkeiten, insbesondere des Quecksilbers, im luftleeren Raume. Ann. Phys. 253, 177–193 (1882). Google Scholar
  20. 20.
    Hill, P.G.: Condensation of water vapour during supersonic expansion in nozzles. J. Fluid Mech. 25(03), 593–620 (1966). Google Scholar
  21. 21.
    Menter, F.R., Kuntz, M., Langtry, R.: Ten years of industrial experience with the SST turbulence model. Turbul. Heat Mass Transf. 4, 625–632 (2003)Google Scholar
  22. 22.
    Weiss, J.M., Smith, W.A.: Preconditioning applied to variable and constant density flows. AIAA J. 33(11), 2050–2057 (1995). zbMATHGoogle Scholar
  23. 23.
    Venkateswaran, S., Merkle, C.L.: Dual time-stepping and preconditioning for unsteady computations. In: 33rd Aerospace Sciences Meeting and Exhibit, Aerospace Sciences Meetings. American Institute of Aeronautics and Astronautics, Reno, Nevada, AIAA Paper 1995-78 (1995).
  24. 24.
    Kim, K.H., Kim, C., Rho, O.H.: Methods for the accurate computations of hypersonic flows: I. AUSMPW+ Scheme. J. Comput. Phys. 174(1), 38–80 (2001). MathSciNetzbMATHGoogle Scholar
  25. 25.
    Kim, S., Kim, C., Rho, O.H., Hong, S.K.: Cures for the shock instability: Development of a shock-stable Roe scheme. J. Comput. Phys. 185(2), 342–374 (2003). MathSciNetzbMATHGoogle Scholar
  26. 26.
    Liou, M.S.: A sequel to AUSM, Part II: AUSM\(^+\)-up for all speeds. J. Comput. Phys. 214(1), 137–170 (2006). MathSciNetzbMATHGoogle Scholar
  27. 27.
    Ihm, S.W., Kim, C.: Computations of homogeneous-equilibrium two-phase flows with accurate and efficient shock-stable schemes. AIAA J. 46(12), 3012–3037 (2008). Google Scholar
  28. 28.
    Roe, P.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43(2), 357–372 (1981). MathSciNetzbMATHGoogle Scholar
  29. 29.
    Gottlieb, S., Shu, C.W.: Total variation diminishing Runge–Kutta schemes. Math. Comput. 67(221), 73–85 (1998). MathSciNetzbMATHGoogle Scholar
  30. 30.
    Yoon, S., Jameson, A.: Lower–upper symmetric-Gauss–Seidel method for the Euler and Navier–Stokes equations. AIAA J. 26(9), 1025–1026 (1988). Google Scholar
  31. 31.
    Yoon, S.H., Kim, C., Kim, K.H.: Multi-dimensional limiting process for three-dimensional flow physics analyses. J. Comput. Phys. 227(12), 6001–6043 (2008). MathSciNetzbMATHGoogle Scholar
  32. 32.
    Quirk, J.J.: A contribution to the great Riemann solver debate. Int. J. Numer. Meth. Fluids 18, 555–574 (1994). MathSciNetzbMATHGoogle Scholar
  33. 33.
    Kitamura, K., Liou, M.S., Chang, C.H.: Extension and comparative study of AUSM-family schemes for compressible multiphase flow simulations. Commun. Comput. Phys. 16(3), 632–674 (2014). MathSciNetzbMATHGoogle Scholar
  34. 34.
    Müller, B.: Low-Mach-number asymptotics of the Navier–Stokes equations. J. Eng. Math. 34(1), 97–109 (1998). MathSciNetzbMATHGoogle Scholar
  35. 35.
    Pelanti, M.: Low Mach number preconditioning techniques for Roe-type and HLLC-type methods for a two-phase compressible flow model. Appl. Math. Comput. 310, 112–133 (2017). MathSciNetGoogle Scholar
  36. 36.
    Meng, H., Yang, V.: A unified treatment of general fluid thermodynamics and its application to a preconditioning scheme. J. Comput. Phys. 189(7), 277–304 (2003). zbMATHGoogle Scholar
  37. 37.
    Abgrall, R.: How to prevent oscillations in multicomponent flow calculations: A quasi conservative approach. J. Comput. Phys. 125(1), 150–160 (1996). MathSciNetzbMATHGoogle Scholar
  38. 38.
    Lee, B.J., Toro, E.F., Castro, C.E., Nikiforakis, N.: Adaptive Osher-type scheme for the Euler equations with highly nonlinear equations of state. J. Comput. Phys. 246, 165–183 (2013). MathSciNetzbMATHGoogle Scholar
  39. 39.
    Shyue, K.M.: A fluid-mixture type algorithm for compressible multicomponent flow with van der Waals equation of state. J. Comput. Phys. 156(1), 43–88 (1999). MathSciNetzbMATHGoogle Scholar
  40. 40.
    Hord, J.: Cavitation in Liquid Cryogens II: Hydrofoil. Technical Report CR-2156, Cleveland, Ohio (1973)Google Scholar
  41. 41.
    Moore, M.J., Walters, P.T., Crane, R.I., Davidson, B.J.: Predicting the fog drop size in wet steam turbines. Wet Steam 4, 101–109 (1973)Google Scholar
  42. 42.
    Kermani, M.J., Gerber, A.G.: A general formula for the evaluation of thermodynamic and aerodynamic losses in nucleating steam flow. Int. J. Heat Mass Transf. 46(17), 3265–3278 (2003). zbMATHGoogle Scholar
  43. 43.
    Ahuja, V., Hosangadi, A., Mattick, S., Lee, C.P., Field, R.E., Ryan, H.: Computational analyses of pressurization in cryogenic tanks. In: 44th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit. Hartford, CT, AIAA Paper 2008-4752 (2008).
  44. 44.
    Haselmaier, L.H., Field, R.E., Ryan, H.M., Dickey, J.C.: Overview of propellant delivery systems at the NASA John C. Stennis Space Center. In: 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, AIAA Paper 2006-4757 (2006).

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringSeoul National UniversitySeoulRepublic of Korea
  2. 2.Hyundai Maritime Research InstituteHyundai Heavy IndustriesSeoulRepublic of Korea
  3. 3.Institute of Advanced Aerospace TechnologySeoul National UniversitySeoulRepublic of Korea

Personalised recommendations