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.
Section
Methods and algorithms of computational mathematics and their applications
References
- 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.
- D. A. Drew and S. L. Passman, Theory of Multicomponent Fluids (Springer, New York, 1999). doi10.1007/b97678.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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).
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- R. Nigmatulin, Dynamics of Multiphase Media (Nauka, Moscow, 1987; Hemisphere, New York, 1990).
- 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).
- C. Parés, “Numerical Methods for Nonconservative Hyperbolic Systems: A Theoretical Framework,” SIAM J. Numer. Anal. 44 (1), 300-321 (2006).
doi 10.1137/050628052.
- 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.
- 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.
- 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.
- 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.
- 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.