Skip to main content
SpringerLink
Log in

An enhanced AUSM\(^{+}\)-up scheme for high-speed compressible two-phase flows on hybrid grids

  • Original Article
  • Published:
Shock Waves Aims and scope Submit manuscript

Abstract

An enhanced \(\hbox {AUSM}^+\)-up scheme is presented for high-speed compressible two-phase flows using a six-equation two-fluid single-pressure model. Based on the observation that the \(\hbox {AUSM}^+\)-up flux function does not take into account relative velocity between the two phases and thus is not stable and robust for computation of two-phase flows involving interaction of strong shock waves and material interfaces, the enhancement is in the form of a volume fraction coupling term and a modification of the velocity diffusion term, both proportional to the relative velocity between the two phases. These modifications in the flux function obviate the need to employ the exact Riemann solver, leading to a significantly less expensive yet robust flux scheme. Furthermore, the Tangent of Hyperbola for INterface Capturing (THINC) scheme is used in order to provide a sharp resolution for material interfaces. A number of benchmark test cases are presented to assess the performance and robustness of the enhanced \(\hbox {AUSM}^+\)-up scheme for compressible two-phase flows on hybrid unstructured grids. The numerical experiments demonstrate that the enhanced \(\hbox {AUSM}^+\)-up scheme along with THINC scheme can efficiently compute high-speed two-fluid flows such as shock–bubble interactions, while accurately capturing material interfaces.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20

Similar content being viewed by others

References

  1. Ishii, M., Hibiki, T.: Thermo-Fluid Dynamics of Two-Phase Flow. Springer, Berlin (2010). https://doi.org/10.1007/978-1-4419-7985-8

    Book  MATH  Google Scholar 

  2. Wallis, G.B.: One-Dimensional Two-Phase Flow. McGraw-Hill, London (1969)

    Google Scholar 

  3. Toumi, I., Kumbaro, A.: An approximate linearized Riemann solver for a two-fluid model. J. Comput. Phys. 124(2), 286–300 (1996). https://doi.org/10.1006/jcph.1996.0060

    Article  MathSciNet  MATH  Google Scholar 

  4. 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

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.2514/1.34793

    Article  Google Scholar 

  6. 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(03), 632–674 (2014). https://doi.org/10.4208/cicp.020813.190214a

    Article  MathSciNet  MATH  Google Scholar 

  7. Lhuillier, D., Chang, C.-H., Theofanous, T.G.: On the quest for a hyperbolic effective-field model of disperse flows. J. Fluid Mech. 731, 184–194 (2013). https://doi.org/10.1017/jfm.2013.380

    Article  MathSciNet  MATH  Google Scholar 

  8. Niu, Y.-Y.: Computations of two-fluid models based on a simple and robust hybrid primitive variable Riemann solver with AUSMD. J. Comput. Phys. 308, 389–410 (2016). https://doi.org/10.1016/j.jcp.2015.12.045

    Article  MathSciNet  MATH  Google Scholar 

  9. Niu, Y.-Y., Wang, H.-W.: Simulations of the shock waves and cavitation bubbles during a three-dimensional high-speed droplet impingement based on a two-fluid model. Comput. Fluids 134, 196–214 (2016). https://doi.org/10.1016/j.compfluid.2016.05.018

    Article  MathSciNet  MATH  Google Scholar 

  10. Dinh, T., Nourgaliev, R., Theofanous, T.: Understanding the ill-posed two-fluid model. In: Proceedings of the 10th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH03) (2003)

  11. Nourgaliev, R., Dinh, T., Theofanous, T.: A characteristics-based approach to the numerical solution of the two-fluid model. In: ASME/JSME 2003 4th Joint Fluids Summer Engineering Conference, pp. 1729–1748. American Society of Mechanical Engineers (2003). https://doi.org/10.1115/FEDSM2003-45551

  12. Vazquez-Gonzalez, T., Llor, A., Fochesato, C.: Ransom test results from various two-fluid schemes: Is enforcing hyperbolicity a thermodynamically consistent option? Int. J. Multiph. Flow 81, 104–112 (2016). https://doi.org/10.1016/j.ijmultiphaseflow.2015.12.007

    Article  MathSciNet  Google Scholar 

  13. Baer, M., Nunziato, J.: 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

    Article  MATH  Google Scholar 

  14. Saurel, R., Abgrall, R.: A simple method for compressible multifluid flows. SIAM J. Sci. Comput. 21(3), 1115–1145 (1999). https://doi.org/10.1137/S1064827597323749

    Article  MathSciNet  MATH  Google Scholar 

  15. 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

    Article  MathSciNet  MATH  Google Scholar 

  16. Saurel, R., Lemetayer, O.: A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation. J. Fluid Mech. 431, 239–271 (2001). https://doi.org/10.1017/S0022112000003098

    Article  MATH  Google Scholar 

  17. Abgrall, R., Saurel, R.: Discrete equations for physical and numerical compressible multiphase mixtures. J. Comput. Phys. 186(2), 361–396 (2003). https://doi.org/10.1016/S0021-9991(03)00011-1

    Article  MathSciNet  MATH  Google Scholar 

  18. Kitamura, K., Nonomura, T.: Simple and robust HLLC extensions of two-fluid AUSM for multiphase flow computations. Comput. Fluids 100, 321–335 (2014). https://doi.org/10.1016/j.compfluid.2014.05.019

    Article  MathSciNet  MATH  Google Scholar 

  19. 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

    Article  MathSciNet  MATH  Google Scholar 

  20. Houim, R., Oran, E.: A multiphase model for compressible granular-gaseous flows: formulation and initial tests. J. Fluid Mech. 789, 166–220 (2016). https://doi.org/10.1017/jfm.2015.728

    Article  MathSciNet  MATH  Google Scholar 

  21. Pandare, A.K., Luo, H.: A finite-volume method for compressible viscous multiphase flows. 2018 AIAA Aerospace Sciences Meeting, Kissimmee, FL, AIAA Paper 2018-1814 (2018). https://doi.org/10.2514/6.2018-1814

  22. Pandare, A.K., Luo, H.: A robust and efficient finite volume method for compressible inviscid and viscous two-phase flows. J. Comput. Phys. 371, 67–91 (2018). https://doi.org/10.1016/j.jcp.2018.05.018

    Article  MathSciNet  MATH  Google Scholar 

  23. Abgrall, R.: How to prevent pressure oscillations in multicomponent flow calculations: a quasi conservative approach. J. Comput. Phys. 125(1), 150–160 (1996). https://doi.org/10.1006/jcph.1996.0085

    Article  MathSciNet  MATH  Google Scholar 

  24. Parés, C.: Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J. Numer. Anal. 44(1), 300–321 (2006). https://doi.org/10.1137/050628052

    Article  MathSciNet  MATH  Google Scholar 

  25. Xiao, F., Honma, Y., Kono, T.: A simple algebraic interface capturing scheme using hyperbolic tangent function. Int. J. Numer. Methods Fluids 48(9), 1023–1040 (2005). https://doi.org/10.1002/fld.975

    Article  MATH  Google Scholar 

  26. Xie, B., Ii, S., Xiao, F.: An efficient and accurate algebraic interface capturing method for unstructured grids in 2 and 3 dimensions: The THINC method with quadratic surface representation. Int. J. Numer. Methods Fluids 76(12), 1025–1042 (2014). https://doi.org/10.1002/fld.3968

    Article  MathSciNet  Google Scholar 

  27. 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

    Article  MathSciNet  MATH  Google Scholar 

  28. Städtke, H.: Gasdynamic Aspects of Two-Phase Flow: Hyperbolicity, Wave Propagation Phenomena, and Related Numerical Methods. Wiley, London (2006). https://doi.org/10.1002/9783527610242

    Book  Google Scholar 

  29. Stuhmiller, J.: The influence of interfacial pressure forces on the character of two-phase flow model equations. Int. J. Multiph. Flow 3(6), 551–560 (1977). https://doi.org/10.1016/0301-9322(77)90029-5

    Article  MATH  Google Scholar 

  30. Chang, C.-H., Sushchikh, S., Nguyen, L., Liou, M.-S., Theofanous, T.: Hyperbolicity, discontinuities, and numerics of the two-fluid model. In: 5th ASME/JSME Fluids Engineering Summer Conference, 10th International Symposium on Gas–Liquid Two-phase Flows, San Diego (2007). https://doi.org/10.1115/FEDSM2007-37338

  31. Kuzmin, D.: A vertex-based hierarchical slope limiter for \(p\)-adaptive discontinuous Galerkin methods. J. Comput. Appl. Math. 233(12), 3077–3085 (2010). https://doi.org/10.1016/j.cam.2009.05.028

    Article  MathSciNet  MATH  Google Scholar 

  32. 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

    Article  MathSciNet  MATH  Google Scholar 

  33. Ii, S., Xie, B., Xiao, F.: An interface capturing method with a continuous function: The THINC method on unstructured triangular and tetrahedral meshes. J. Comput. Phys. 259, 260–269 (2014). https://doi.org/10.1016/j.jcp.2013.11.034

    Article  MathSciNet  MATH  Google Scholar 

  34. Hérard, J.-M., Hurisse, O.: A simple method to compute standard two-fluid models. Int. J. Comput. Fluid Dyn. 19(7), 475–482 (2005). https://doi.org/10.1080/10618560600566885

    Article  MathSciNet  MATH  Google Scholar 

  35. Gallouët, T., Helluy, P., Hérard, J.-M., Nussbaum, J.: Hyperbolic relaxation models for granular flows. ESAIM Math. Model. Numer. Anal. 44(2), 371–400 (2010). https://doi.org/10.1051/m2an/2010006

    Article  MathSciNet  MATH  Google Scholar 

  36. Liou, M.-S., Edwards, J.R.: Numerical speed of sound and its application to schemes for all speeds. 14th Computational Fluid Dynamics Conference, Norfolk, VA, AIAA Paper 1999-3268 (1999). https://doi.org/10.2514/6.1999-3268

  37. Edwards, J.R.: Towards unified CFD simulations of real fluid flows. 15th AIAA Computational Fluid Dynamics Conference, Anaheim, CA, AIAA Paper 2001-2524 (2001). https://doi.org/10.2514/6.2001-2524

  38. Niu, Y.-Y., Lin, Y.-C., Chang, C.-H.: A further work on multi-phase two-fluid approach for compressible multi-phase flows. Int. J. Numer. Methods Fluids 58(8), 879–896 (2008). https://doi.org/10.1002/fld.1773

    Article  MathSciNet  MATH  Google Scholar 

  39. Castro, M., Fernández-Nieto, E., Ferreiro, A., García-Rodríguez, J., Parés, C.: High order extensions of Roe schemes for two-dimensional nonconservative hyperbolic systems. J. Sci. Comput. 39(1), 67–114 (2009). https://doi.org/10.1007/s10915-008-9250-4

    Article  MathSciNet  MATH  Google Scholar 

  40. Dumbser, M., Toro, E.: A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems. J. Sci. Comput. 48(1), 70–88 (2011). https://doi.org/10.1007/s10915-010-9400-3

    Article  MathSciNet  MATH  Google Scholar 

  41. Munkejord, S., Evje, S., FlÅtten, T.: A musta scheme for a nonconservative two-fluid model. SIAM J. Sci. Comput. 31(4), 2587–2622 (2009). https://doi.org/10.1137/080719273

    Article  MathSciNet  MATH  Google Scholar 

  42. 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

    Article  MathSciNet  MATH  Google Scholar 

  43. Paillere, H., Corre, C., Cascales, J.G.: On the extension of the AUSM\(^+\) scheme to compressible two-fluid models. Comput. Fluids 32(6), 891–916 (2003). https://doi.org/10.1016/S0045-7930(02)00021-X

    Article  MathSciNet  MATH  Google Scholar 

  44. Haimovich, O., Frankel, S.H.: Numerical simulations of compressible multicomponent and multiphase flow using a high-order targeted ENO (TENO) finite-volume method. Comput. Fluids 146, 105–116 (2017). https://doi.org/10.1016/j.compfluid.2017.01.012

    Article  MathSciNet  MATH  Google Scholar 

  45. Nourgaliev, R., Dinh, T.-N., Theofanous, T.-G.: Adaptive characteristics-based matching for compressible multifluid dynamics. J. Comput. Phys. 213(2), 500–529 (2006). https://doi.org/10.1016/j.jcp.2005.08.028

    Article  MATH  Google Scholar 

  46. Lauer, E., Hu, X., Hickel, S., Adams, N.: Numerical investigation of collapsing cavity arrays. Phys. Fluids 24(5), 052104 (2012). https://doi.org/10.1063/1.4719142

    Article  Google Scholar 

  47. Haas, J.-F., Sturtevant, B.: Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities. J. Fluid Mech. 181, 41–76 (1987). https://doi.org/10.1017/S0022112087002003

    Article  Google Scholar 

  48. Burton, D.E., Morgan, N.R., Carney, T.C., Kenamond, M.A.: Reduction of dissipation in lagrange cell-centered hydrodynamics (CCH) through corner gradient reconstruction (CGR). J. Comput. Phys. 299, 229–280 (2015). https://doi.org/10.1016/j.jcp.2015.06.041

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research was partially supported by the Consortium for Advanced Simulation of Light Water Reactors, an Energy Innovation Hub for Modeling and Simulation of Nuclear Reactors under U.S. Department of Energy Contract No. DE-AC05-00OR22725. The authors would like to thank C.-H. Chang, N. Dinh, J. R. Edwards, and R. Nourgaliev for fruitful discussions.

Author information

Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC, 27695, USA

    A. K. Pandare & H. Luo

  2. Computational Physics Group (CCS-2), Computer, Computational and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    J. Bakosi

Authors
  1. A. K. Pandare
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. Bakosi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Luo.

Additional information

Communicated by H.D. Ng.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pandare, A.K., Luo, H. & Bakosi, J. An enhanced AUSM\(^{+}\)-up scheme for high-speed compressible two-phase flows on hybrid grids. Shock Waves 29, 629–649 (2019). https://doi.org/10.1007/s00193-018-0861-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00193-018-0861-x

Keywords

Search

Navigation