Abstract
This paper is devoted to the numerical study of a finite-volume scheme with an HLLEM flux for solving equations from the family of Baer-Nunziato models. Three versions of the model are considered, differing in the degree of “nonequilibrium.” A brief description of the models and their differences are provided. To approximate the equations of nonequilibrium models with stiff right-hand sides, describing the process of mechanical and thermodynamic relaxation, the method of splitting into physical processes is used. Spatial approximations are constructed using the 1st and 2nd (TVD) order finite volume method. The HLLEM flux is used as a numerical flux, for which a simple algorithm for determining the parameter of the method that guarantees the physicality of the solution is proposed. A feature of the study is that all three considered models are applied to analyze numerically the same physical setting.
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REFERENCES
M. Dumbser and D. S. Balsara, “A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems,” J. Comput. Phys. 304, 275–319 (2016).
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. Multiphase Flow 12 (6), 861–889 (1986).
D. A. Drew and S. L. Passman, Theory of Multicomponent Fluids (Springer Science & Business Media, New York, 2006).
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).
A. K. Kapila, S. F. Son, J. B. Bdzil, R. Menikoff, and D. S. Stewart, “Two-phase modeling of DDT: Structure of the velocity-relaxation zone,” Phys. Fluids 9 (12), 3885–3897 (1997).
A. K. Kapila, R. Menikoff, J. B. Bdzil, S. F. Son, and D. S. Stewart, “Two-phase modeling of deflagration-to-detonation transition in granular materials: Reduced equations,” Phys. Fluids 13 (10), 3002–3024 (2001).
L. van de Leur, “Assessment of the Baer-Nunziato seven-equation model applied to steam-water transients: Calibration of the stiffened gas equation of state based on steam-water tables,” Master Thesis (Eindhoven University of Technology Eindhoven, 2015).
A. Murrone and H. Guillard, “A five equation reduced model for compressible two phase flow problems,” J. Comput. Phys. 202 (2), 664–698 (2005).
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).
F. Fraysse, C. Redondo, G. Rubio, and E. Valero, “Upwind methods for the Baer–Nunziato equations and higher-order reconstruction using artificial viscosity,” J. Comput. Phys. 326, 805–827 (2016).
I. Menshov and A. Serezhkin, “A generalized Rusanov method for the Baer–Nunziato equations with application to DDT processes in condensed porous explosives,” Int. J. Numer. Methods Fluids 86 (5), 346–364 (2018).
R. Saurel and R. Abgrall, “A simple method for compressible multifluid flows,” SIAM J. Sci. Comput. 21 (3), 1115–1145 (1999).
D. Gidaspow, “Modeling of two phase flow,” in Proc. International Heat Transfer Conference 5 (Tokyo, Japan, 3–7 September, 1974), pp. 163–168, International Heat Transfer Conference Digital Library (Begel House, New York, 1974).
L. van Wijngaarden, “Some problems in the formulation of the equations for gas/liquid flows,” in Theoretical and Applied Mechanics, Proc. 14th Int. Congress (Delft, Netherlands, August 30–September 4, 1976), Ed. by W. T. Koiter (North-Holland, Amsterdam, 1977), pp. 249–260.
R. W. Lyczkowski, D. Gidaspow, C. W. Solbrig, and E. D. Hughes, “Characteristics and stability analyses of transient one-dimensional two-phase flow equations and their finite difference approximations,” Nucl. Sci. Eng. 66 (3), 378–396 (1978).
G. Allaire, S. Clerc, and S. Kokh, “A five-equation model for the simulation of interfaces between compressible fluids,” J. Comput. Phys. 181 (2), 577–616 (2002).
R. Saurel, O. Le Métayer, J. Massoni, and S. Gavrilyuk, “Shock jump relations for multiphase mixtures with stiff mechanical relaxation,” Shock Waves 16 (3), 209–232 (2007).
A. V. Rodionov, “Methods of increasing the accuracy in Godunov’s scheme,” USSR Comput. Math. Math. Phys. 27 (6), 164–169 (1987).
P. Le Floch and T.-P. Liu, “Existence theory for nonlinear hyperbolic systems in nonconservative form,” Forum Math. 5 (3), 261–280 (1993).
G. Dal Maso, P. G. LeFloch, and F. Murat, “Definition and weak stability of nonconservative products,” J. Math. Pures Appl. 74 (6), 483–548 (1995).
M. J. Castro, P. G. LeFloch, M. L. Muñoz-Ruiz, and C. Parés, “Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes,” J. Comput. Phys. 227 (17), 8107–8129 (2008).
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This study was supported by the Russian Science Foundation, project 17-71-30014.
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Korneev, B.A., Tukhvatullina, R.R. & Savenkov, E.B. Numerical Study of Two-Phase Hyperbolic Models. Math Models Comput Simul 13, 1002–1013 (2021). https://doi.org/10.1134/S2070048221060090
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DOI: https://doi.org/10.1134/S2070048221060090