DOI: https://doi.org/10.26089/NumMet.v24r214

Numerical algortihms for solving two-phase flows based on relaxation Baer-Nunziato model

Authors

  • Mikhail V. Alekseev

Keywords:

Baer-Nunziato model
discontinuous Galerkin method
shock-bubble interaction tests

Abstract

The paper considers the issues of numerical simulation of two-phase flows in the Eulerian model of the Baer-Nunziato model. The description of the mathematic model, the numerical algorithm for solving the problem based on discontinuous Galerkin method are formulated. A description of the developed software package is provided with demonstration the possibility of its application for solving problems in actual grid sizes.

Downloads

Published

2023-05-01

Issue

Vol. 24 (2023): Issue 2.

Section

Methods and algorithms of computational mathematics and their applications

Author Biography

Mikhail V. Alekseev

Keldysh Institute of Applied Mathematics of RAS
• Junior Researcher

References

  1. M. R. Baer and J. W. Nunziato, “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).
    doi 10.1016/0301-9322(86)90033-9.
  2. D. A. Drew and S. L. Passman, Theory of Multicomponent Fluids (Springer, New York, 1999). doi10.1007/b97678.
  3. N. Favrie, S. L. Gavrilyuk, and R. Saurel, “Solid-Fluid Diffuse Interface Model in Cases of Extreme Deformations,” J. Comput. Phys. 228 (16), 6037-6077 (2009).
    doi 10.1016/j.jcp.2009.05.015.
  4. A. K. Kapila, S. F. Son, J. B. Bdzil, and R. Menikoff, “Two-Phase Modeling of DDT: Structure of the Velocity-Relaxation Zone,” Phys. Fluids 9 (12), 3885-3897 (1997).
    doi 10.1063/1.869488.
  5. A. K. Kapila, R. Menikoff, J. B. Bdzil, et al., “Two-Phase Modeling of Deflagration-to-Detonation Transition in Granular Materials: Reduced Equations,” Phys. Fluids 13 (10), 3002-3024 (2001).
    doi 10.1063/1.1398042.
  6. A. Murrone and H. Guillard, “A Five-Equation Reduced Model for Compressible Two Phase Flow Problems,” J. Comput. Phys. 202 (2), 664-698 (2005).
    doi 10.1016/j.jcp.2004.07.019.
  7. S. A. Tokareva and E. F. Toro, “HLLC-Type Riemann Solver for the Baer-Nunziato Equations of Compressible Two-Phase Flow,” J. Comput. Phys. 229 (10), 3573-3604 (2010).
    doi 10.1016/j.jcp.2010.01.016.
  8. M. Dumbser and E. F. Toro, “A Simple Extension of the Osher Riemann Solver to Non-conservative Hyperbolic Systems,” J. Sci. Comput. 48 (1-3), 70-88 (2011).
    doi 10.1007/s10915-010-9400-3.
  9. E. Franquet and V. Perrier, “Runge-Kutta Discontinuous Galerkin Method for the Approximation of Baer and Nunziato Type Multiphase Models,” J. Comput. Phys. 231 (11), 4096-4141 (2012).
    doi 10.1016/j.jcp.2012.02.002.
  10. M. T. H. de Frahan, S. Varadan, and E. Johnsen, “A New Limiting Procedure for Discontinuous Galerkin Methods Applied to Compressible Multiphase Flows with Shocks and Interfaces,” J. Comput. Phys. 280, 489-509 (2015).
    doi 10.1016/j.jcp.2014.09.030.
  11. B. Cockburn and C.-W. Shu, “The Runge-Kutta Local Projection P^1-Discontinuous-Galerkin Finite Element Method for Scalar Conservation Laws,” ESAIM Math. Model. Numer. Anal. 25 (3), 337-361 (1991).
  12. X. Zhong and C.-W. Shu, “A Simple Weighted Essentially Nonoscillatory Limiter for Runge-Kutta Discontinuous Galerkin Methods,” J. Comput. Phys. 232 (1), 397-415 (2013).
    doi 10.1016/j.jcp.2012.08.028.
  13. M. V. Alekseev and E. B. Savenkov, Two-Phase Hyperelastic Model., “Scalar” Case , Preprint No. 40 (Keldysh Institute of Applied Mathematics, Moscow, 2022).
    doi 10.20948/prepr-2022-40.
  14. R. R. Polekhina, M. V. Alekseev, and E. B. Savenkov, “Validation of a Computational Algorithm Based on the Discontinuous Galerkin Method for the Baer-Nunziato Relaxation Model,” Differ. Uravn. 58 (7), 977-994 (2022) [Differ. Equ. 58 (7), 966-984 (2022)].
    doi 10.1134/S0012266122070096.
  15. N. Andrianov and G. Warnecke, “The Riemann Problem for the Baer-Nunziato Two-Phase Flow Model,” J. Comput. Phys. 195 (2), 434-464 (2004).
    doi 10.1016/j.jcp.2003.10.006.
  16. F. Daude, R. A. Berry, and P. Galon, “A Finite-Volume Method for Compressible Non-Equilibrium Two-Phase Flows in Networks of Elastic Pipelines Using the Baer-Nunziato Model,” Comput. Methods Appl. Mech. Eng. 354, 820-849 (2019).
    doi 10.1016/j.cma.2019.06.010.
  17. R. Saurel and R. Abgrall, “A Simple Method for Compressible Multifluid Flows,” SIAM J. Sci. Comput. 21 (3), 1115-1145 (1999).
    doi 10.1137/S1064827597323749.
  18. R. Nigmatulin, Dynamics of Multiphase Media (Nauka, Moscow, 1987; Hemisphere, New York, 1990).
  19. G. Dal Maso, P. Le Floch, and F. Murat, “Definition and Weak Stability of Nonconservative Products,” J. Math. Pures Appl. 74 (6), 483-548 (1995).
  20. C. Parés, “Numerical Methods for Nonconservative Hyperbolic Systems: A Theoretical Framework,” SIAM J. Numer. Anal. 44 (1), 300-321 (2006).
    doi 10.1137/050628052.
  21. C. Michoski, C. Dawson, E. J. Kubatko, et al., “A Comparison of Artificial Viscosity, Limiters, and Filters, for High Order Discontinuous Galerkin Solutions in Nonlinear Settings,” J. Sci. Comput. 66 (1), 406-434 (2016).
    doi 10.1007/s10915-015-0027-2.
  22. P.-O. Persson and J. Peraire, “Sub-Cell Shock Capturing for Discontinuous Galerkin Methods,” in Proc. 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, USA, January 9-12, 2006.
    doi 10.2514/6.2006-112.
  23. R. C. Moura, R. C. Affonso, A. F. de Castro da Silva, and M. A. Ortega, “Diffusion-Based Limiters for Discontinuous Galerkin Methods – Part I: One-Dimensional Equations,” in Proc. 22nd Int. Congress of Mechanical Engineering, Ribeirão Preto, Brazil, November 3-7, 2013.
    https://www.researchgate.net/publication/270273162_Diffusion-Based_Limiters_for_Discontinuous_Galerkin_Methods_-_Part_I_One-Dimensional_Equations/link/54a40bc10cf267bdb9066b22/download . Cited April 9, 2023.
  24. L. Krivodonova, “Limiters for High-Order Discontinuous Galerkin Methods,” J. Comput. Phys. 226 (1), 879-896 (2007).
    doi 10.1016/j.jcp.2007.05.011.
  25. V. A. Balashov, E. B. Savenkov, and B. N. Chetverushkin, “DIMP-HYDRO Solver for Direct Numerical Simulation of Fluid Microflows within Pore Space of Core Sample,” Mat. Model. 31 (7), 21-44 (2019) [Math. Models Comput. Simul. 12 (2), 110-124 (2020)].
    doi 10.1134/S2070048220020027.

License

Copyright (c) 2023 М. В. Алексеев

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.