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Shock Waves

pp 1–19 | Cite as

Extension of AUSM-type fluxes: from single-phase gas dynamics to multi-phase cryogenic flows at all speeds

  • H. Kim
  • C. KimEmail author
Original Article
  • 27 Downloads

Abstract

Despite their simple formulations, advection upstream splitting method (AUSM)-type flux schemes treat linear and nonlinear waves of complex flow fields in a robust and accurate manner. This paper presents the progress of AUSM-type fluxes augmented with pressure-based weight functions introduced by the authors and their colleagues. Starting from a flux designed for single-phase gas dynamics (AUSMPW+), its extensions to capture multi-phase flow physics with phase transition (AUSMPW+_N) have been carried out. The accuracy of the computed results by the AUSMPW+_N scheme for multi-phase flows is then further improved by introducing a simple phase interface-sharpening procedure, which scales the volume fraction in a mass-conserving manner. Various all-speed compressible tests ranging from interactions between a shock and phase interfaces, two- and three-dimensional interface-only problems, to a cryogenic three-component flow with phase change are computed to demonstrate the effectiveness of the proposed method.

Keywords

AUSMPW+_N Interface sharpening Multi-phase shock-capturing Multi-phase flow 

Notes

Acknowledgements

This research was supported by the programs of the National Research Foundation of Korea (NRF-2014M1A3A3A02034856, NRF-2013R1A5A1073861).

References

  1. 1.
    Tryggvason, G., Bunner, B., Esmaeeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han, J., Nas, S., Jan, Y.-J.: A front-tracking method for the computations of multiphase flow. J. Comput. Phys. 169(2), 708–759 (2001).  https://doi.org/10.1006/jcph.2001.6726 MathSciNetzbMATHGoogle Scholar
  2. 2.
    Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988).  https://doi.org/10.1016/0021-9991(88)90002-2 MathSciNetzbMATHGoogle Scholar
  3. 3.
    Fedkiw, R.P., Aslam, T., Merriman, B., Osher, S.: A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152(2), 457–492 (1999).  https://doi.org/10.1006/jcph.1999.6236 MathSciNetzbMATHGoogle Scholar
  4. 4.
    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. Flow 12(6), 861–889 (1986).  https://doi.org/10.1016/0301-9322(86)90033-9 zbMATHGoogle Scholar
  5. 5.
    Kapila, A.K., Menikoff, R., Bdzil, J.B., Son, S.F., Stewart, D.S.: Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations. Phys. Fluids 13(10), 3002–3024 (2001).  https://doi.org/10.1063/1.1398042 zbMATHGoogle Scholar
  6. 6.
    Saurel, R., Abgrall, R.: A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150(2), 425–467 (1999).  https://doi.org/10.1006/jcph.1999.6187 MathSciNetzbMATHGoogle Scholar
  7. 7.
    Liou, M.-S., Steffen, C.J.: A new flux splitting scheme. J. Comput. Phys. 107(1), 23–39 (1993).  https://doi.org/10.1006/jcph.1993.1122 MathSciNetzbMATHGoogle Scholar
  8. 8.
    Liou, M.-S.: A sequel to AUSM: AUSM\(^+\). J. Comput. Phys. 129(2), 364–382 (1996).  https://doi.org/10.1006/jcph.1996.0256 MathSciNetzbMATHGoogle Scholar
  9. 9.
    Liou, M.-S.: A sequel to AUSM, part II: AUSM\(^+\)-up for all speeds. J. Comput. Phys. 214(1), 137–170 (2006).  https://doi.org/10.1016/j.jcp.2005.09.020 MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chang, C.-H., Liou, M.-S.: A robust and accurate approach to computing compressible multiphase flow: Stratified flow model and AUSM\(^+\)-up scheme. J. Comput. Phys. 225(1), 840–873 (2007).  https://doi.org/10.1016/j.jcp.2007.01.007 MathSciNetzbMATHGoogle Scholar
  11. 11.
    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).  https://doi.org/10.1006/jcph.2001.6873 MathSciNetzbMATHGoogle Scholar
  12. 12.
    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).  https://doi.org/10.2514/1.35097 Google Scholar
  13. 13.
    Kim, H., Kim, H., Kim, C.: Computations of homogeneous multiphase real fluid flows at all speeds. AIAA J. 56(7), 2623–2634 (2018).  https://doi.org/10.2514/1.J056497 Google Scholar
  14. 14.
    Shukla, R.K., Pantano, C., Freund, J.B.: An interface capturing method for the simulation of multi-phase compressible flows. J. Comput. Phys. 229(19), 7411–7439 (2010).  https://doi.org/10.1016/j.jcp.2010.06.025 MathSciNetzbMATHGoogle Scholar
  15. 15.
    So, K.K., Hu, X.Y., Adams, N.A.: Anti-diffusion interface sharpening technique for two-phase compressible flow simulations. J. Comput. Phys. 231(11), 4304–4323 (2012).  https://doi.org/10.1016/j.jcp.2012.02.013 MathSciNetzbMATHGoogle Scholar
  16. 16.
    Shyue, K.-M., Xiao, F.: An Eulerian interface sharpening algorithm for compressible two-phase flow: The algebraic THINC approach. J. Comput. Phys. 268, 326–354 (2014).  https://doi.org/10.1016/j.jcp.2014.03.010 MathSciNetzbMATHGoogle Scholar
  17. 17.
    Chiapolino, A., Saurel, R., Nkonga, B.: Sharpening diffuse interfaces with compressible fluids on unstructured meshes. J. Comput. Phys. 340, 389–417 (2017).  https://doi.org/10.1016/j.jcp.2017.03.042 MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kinzel, M.P., Lindau, J.W., Kunz, R.F.: A multiphase level-set approach for all-Mach numbers. Comput. Fluids 167, 1–16 (2018).  https://doi.org/10.1016/j.compfluid.2018.02.026 MathSciNetzbMATHGoogle Scholar
  19. 19.
    Brackbill, J.U., Kothe, D.B., Zemach, C.: A continuum method for modeling surface tension. J. Comput. Phys. 100(2), 335–354 (1992).  https://doi.org/10.1016/0021-9991(92)90240-Y MathSciNetzbMATHGoogle Scholar
  20. 20.
    Merkle, C.L., Feng, J.Z., Buelow, P.E.O.: Computational modeling of the dynamics of sheet cavitation. 3rd International Symposium on Cavitation, Grenoble, France (1998)Google Scholar
  21. 21.
    Abgrall, R.: How to prevent pressure oscillations in multicomponent flow calculations: A quasi conservative approach. J. Comput. Phys. 125, 150–160 (1996).  https://doi.org/10.1006/jcph.1996.0085 MathSciNetzbMATHGoogle Scholar
  22. 22.
    Lemmon, E.W., Huber, M.L., McLinden, M.O.: NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, Version 8.0 [Online]. http://www.nist.gov/srd/nist23.cfm. National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg (2007)
  23. 23.
    International Association for the Properties of Water and Steam. Guideline on the Fast Calculation of Steam and Water Properties with the Spline-Based Table Look-Up Method (SBTL). Technical Report (2015)Google Scholar
  24. 24.
    Weiss, J.M., Smith, W.A.: Preconditioning applied to variable and constant density flows. AIAA J. 33(11), 2050–2057 (1995).  https://doi.org/10.2514/3.12946 zbMATHGoogle Scholar
  25. 25.
    Venkateswaran, S., Merkle, C.L.: Dual time-stepping and preconditioning for unsteady computations. 33rd Aerospace Sciences Meeting and Exhibit, AIAA Paper 95-0078 (1995).  https://doi.org/10.2514/6.1995-78
  26. 26.
    Kim, H., Choe, Y., Kim, H., Min, D., Kim, C.: Methods for compressible multiphase flows and their applications. Shock Waves 29(1), 235–261 (2019).  https://doi.org/10.1007/s00193-018-0829-x Google Scholar
  27. 27.
    Wada, Y., Liou, M.-S.: A flux splitting scheme with high-resolution and robustness for discontinuities. 32nd Aerospace Sciences Meeting and Exhibit, AIAA Paper 94-0083 (1994).  https://doi.org/10.2514/6.1994-83
  28. 28.
    Kim, K.H., Lee, J.H., Rho, O.H.: An improvement of AUSM schemes by introducing the pressure based weight functions. Comput. Fluids 27(3), 311–346 (1998).  https://doi.org/10.1016/S0045-7930(97)00069-8 MathSciNetzbMATHGoogle Scholar
  29. 29.
    Kim, K.H., Kim, C.: Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows. Part I: Spatial discretization. J. Comput. Phys. 208(2), 527–569 (2005).  https://doi.org/10.1016/j.jcp.2005.02.021 MathSciNetzbMATHGoogle Scholar
  30. 30.
    Edwards, J.R., Liou, M.-S.: Low-diffusion flux-splitting methods for flows at all speeds. AIAA J. 36(9), 1610–1617 (1998).  https://doi.org/10.2514/2.587 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, 6001–6043 (2008).  https://doi.org/10.1016/j.jcp.2008.02.012 MathSciNetzbMATHGoogle Scholar
  32. 32.
    Gottlieb, S., Shu, C.-W.: Total variation diminishing Runge–Kutta schemes. Math. Comput. 67(221), 73–85 (1998).  https://doi.org/10.1090/S0025-5718-98-00913-2 MathSciNetzbMATHGoogle Scholar
  33. 33.
    Nonomura, T., Kitamura, K., Fujii, K.: A simple interface sharpening technique with a hyperbolic tangent function applied to compressible two-fluid modeling. J. Comput. Phys. 258, 95–117 (2014).  https://doi.org/10.1016/j.jcp.2013.10.021 MathSciNetzbMATHGoogle Scholar
  34. 34.
    Yoon, S., Jameson, A.: Lower-upper symmetric-Gauss–Seidel method for the Euler and Navier–Stokes equations. AIAA J. 26(9), 1025–1026 (1988).  https://doi.org/10.2514/3.10007 Google Scholar
  35. 35.
    Grace, J.R.: Shapes and velocities of bubbles rising in infinite liquids. Trans. Inst. Chem. Eng. 51, 116–120 (1973)Google Scholar
  36. 36.
    van Sint Annaland, M., Deen, N.G., Kuipers, J.A.M.: Numerical simulation of gas bubbles behaviour using a three-dimensional volume of fluid method. Chem. Eng. Sci. 60(11), 2999–3011 (2005).  https://doi.org/10.1016/j.ces.2005.01.031 Google Scholar
  37. 37.
    Clift, R., Grace, J.R., Weber, M.E., Weber, M.F.: Bubbles, Drops, and Particles. Academic Press, London (1978)Google Scholar
  38. 38.
    Pan, S., Han, L., Hu, X., Adams, N.A.: A conservative interface-interaction method for compressible multi-material flows. J. Comput. Phys. 371, 870–895 (2018).  https://doi.org/10.1016/j.jcp.2018.02.007 MathSciNetGoogle Scholar
  39. 39.
    Ahuja, V., Hosangadi, A., Mattick, S., Lee, C.P., Field, R.E., Ryan, H.: Computational analyses of pressurization in cryogenic tanks. 44th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, AIAA Paper 2008-4752 (2008).  https://doi.org/10.2514/6.2008-4752
  40. 40.
    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

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringSeoul National UniversitySeoulRepublic of Korea

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