Abstract
We present an implicit–explicit finite volume scheme for isentropic two phase flow in all Mach number regimes. The underlying model belongs to the class of symmetric hyperbolic thermodynamically compatible models. The key element of the scheme consists of a linearisation of pressure and enthalpy terms at a reference state. The resulting stiff linear parts are integrated implicitly, whereas the non-linear higher order and transport terms are treated explicitly. Due to the flux splitting, the scheme is stable under a CFL condition which is determined by the resolution of the slow material waves and allows large time steps even in the presence of fast acoustic waves. Further the singular Mach number limits of the model are studied and the asymptotic preserving property of the scheme is proven. In numerical simulations the consistency with single phase flow, accuracy and the approximation of material waves in different Mach number regimes are assessed.
Funding statement: A.T. and M.L. have been partially supported by the Gutenberg Research College, JGU Mainz. Further, M.L. research was partially supported by the German Science Foundation under the SFB/TRR 146 Multiscale Simulation Methods for Soft Matter Systems and by the Mainz Institute of Multiscale Modelling. G.P. is a member of GNCS and acknowledges the support of PRIN2017 and Sapienza, Progetto di Ateneo [RM120172B41DBF3A].
References
[1] A. Ambroso, C. Chalons, and P.-A. Raviart, A Godunov-type method for the seven-equation model of compressible two-phase flow, Comput. & Fluids 54 (2012), 67–9110.1016/j.compfluid.2011.10.004Search in Google Scholar
[2] N. Andrianov, R. Saurel, and G.Warnecke, A simple method for compressible multiphase mixtures and interfaces, Int. J. Numer. Methods Fluids 41 (2003), No. 2, 109–13110.1002/fld.424Search in Google Scholar
[3] N. Andrianov and G.Warnecke, The Riemann problem for the Baer–Nunziato two-phase flow model, J. Comput. Phys 195 (2004), No. 2, 434–46410.1016/j.jcp.2003.10.006Search in Google Scholar
[4] 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 (1986), No. 6, 861–88910.1016/0301-9322(86)90033-9Search in Google Scholar
[5] G. Bispen, M. Lukáčová-Medvid’ová, and L. Yelash, Asymptotic preserving IMEX finite volume schemes for low Mach number Euler equations with gravitation, J. Comput. Phys 335 (2017), 222–24810.1016/j.jcp.2017.01.020Search in Google Scholar
[6] S. Busto, L. Río-Martín, M. E. Vázquez-Cendón, and M. Dumbser, A semi-implicit hybrid finite volume/finite element scheme for all Mach number flows on staggered unstructured meshes, Appl. Math. Comput 402 (2021), 12611710.1016/j.amc.2021.126117Search in Google Scholar
[7] S. Chiocchetti and C. Müller, A solver for stiff finite-rate relaxation in Baer–Nunziato two-phase flow models, Fluid Mech. Appl 121 (2020), 31–4410.1007/978-3-030-33338-6_3Search in Google Scholar
[8] F. Cordier, P. Degond, and A. Kumbaro, An asymptotic-preserving all-speed scheme for the Euler and Navier–Stokes equations, J. Comput. Phys 231 (2012), No. 17, 5685–570410.1016/j.jcp.2012.04.025Search in Google Scholar
[9] V. Daru, P. Le Quéré, M.-C. Duluc, and O. Le Maitre, A numerical method for the simulation of low Mach number liquid–gas flows, J. Comput. Phys 229 (2010), No. 23, 8844–886710.1016/j.jcp.2010.08.013Search in Google Scholar
[10] P. Degond and M. Tang, All speed scheme for the low Mach number limit of the isentropic Euler equations, Commun. Comput. Phys 10 (2011), No. 1, 1–3110.4208/cicp.210709.210610aSearch in Google Scholar
[11] S. Dellacherie, Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number, J. Comput. Phys 229 (2010), No. 4, 978–101610.1016/j.jcp.2009.09.044Search in Google Scholar
[12] A. D. Demou, N. Scapin, M. Pelanti, and L. Brandt, A pressure-based diffuse interface method for low-Mach multiphase flows with mass transfer, J. Comput. Phys 448 (2022), 11073010.1016/j.jcp.2021.110730Search in Google Scholar
[13] G. Dimarco, R. Loubère, V. Michel-Dansac, and M.-H. Vignal, Second-order implicit–explicit total variation diminishing schemes for the Euler system in the low Mach regime, J. Comput. Phys 372 (2018), 178–20110.1016/j.jcp.2018.06.022Search in Google Scholar
[14] M. Dumbser, I. Peshkov, E. Romenski, and O. Zanotti, High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: Viscous heat-conducting fluids and elastic solids, J. Comput. Phys 314 (2016), 824–86210.1016/j.jcp.2016.02.015Search in Google Scholar
[15] M. Feistauer, V. Dolejší, and V. Kučera, On the discontinuous Galerkin method for the simulation of compressible flow with wide range of Mach numbers, Comput. Vis. Sci 10 (2007), No. 1, 17–2710.1007/s00791-006-0051-8Search in Google Scholar
[16] M. Feistauer and V. Kučcera, A new technique for the numerical solution of the compressible Euler equations with arbitrary Mach numbers. In: Hyperbolic Problems: Theory, Numerics, Applications Springer, 2008, pp. 523–53110.1007/978-3-540-75712-2_50Search in Google Scholar
[17] S. K. Godunov, An interesting class of quasilinear systems, Dokl. Akad. Nauk SSSR 139 (1961), No. 3, 521–523Search in Google Scholar
[18] S. K. Godunov and E. I. Romenskii, Elements of Continuum Mechanics and Conservation Laws Kluwer Academic/Plenum Publishers, 200310.1007/978-1-4757-5117-8Search in Google Scholar
[19] S. K. Godunov and E. I. Romensky, Thermodynamics, conservation laws and symmetric forms of differential equations in mechanics of continuous media, Computational Fluid Dynamics Review 95 (1995), 19–31Search in Google Scholar
[20] H. Gouin and S. Gavrilyuk, Hamilton’s principle and Rankine–Hugoniot conditions for general motions of mixtures, Meccanica 34 (1999), No. 1, 39–4710.1023/A:1004370127958Search in Google Scholar
[21] H. Guillard and C. Viozat, On the behaviour of upwind schemes in the low Mach number limit, Comput. & Fluids 28 (1999), No. 1, 63–8610.1016/S0045-7930(98)00017-6Search in Google Scholar
[22] S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput 21 (1999), No. 2, 441–45410.1137/S1064827598334599Search in Google Scholar
[23] K. Kaiser, J. Schütz, R. Schöbel, and S. Noelle, A new stable splitting for the isentropic Euler equations, J. Sci. Comput 70 (2017), No. 3, 1390–140710.1007/s10915-016-0286-6Search in Google Scholar
[24] 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 (2001), No. 10, 3002–302410.1063/1.1398042Search in Google Scholar
[25] S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Commun. Pur. Appl. Math 34 (1981), No. 4, 481–52410.1002/cpa.3160340405Search in Google Scholar
[26] R. Klein, Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics I: One-dimensional flow, J. Comput. Phys 121 (1995), No. 2, 213 – 23710.1016/S0021-9991(95)90034-9Search in Google Scholar
[27] J. J. Kreeft and B. Koren, A new formulation of Kapila’s five-equation model for compressible two-fluid flow, and its numerical treatment, J. Comput. Phys 229 (2010), No. 18, 6220–624210.1016/j.jcp.2010.04.025Search in Google Scholar
[28] V. Kučera, M. Lukáčová-Medvid’ová, S. Noelle, and J. Schütz, Asymptotic properties of a class of linearly implicit schemes for weakly compressible Euler equations, Numer. Math 150 (2022), No. 1, 79–10310.1007/s00211-021-01240-5Search in Google Scholar
[29] S. Müller, M. Hantke, and P. Richter, Closure conditions for non-equilibrium multi-component models, Contin. Mech. Thermodyn 28 (2016), No. 4, 1157–118910.1007/s00161-015-0468-8Search in Google Scholar
[30] J. H. Park and C.-D. Munz, Multiple pressure variables methods for fluid flow at all Mach numbers, Int. J. Numer. Methods Fluids 49 (2005), No. 8, 905–93110.1002/fld.1032Search in Google Scholar
[31] M. Pelanti, Low Mach number preconditioning techniques for Roe-type and HLLC-type methods for a two-phase compressible flow model, Appl. Math. Comput 310 (2017), 112–13310.1016/j.amc.2017.04.014Search in Google Scholar
[32] I. Peshkov, M. Dumbser, W. Boscheri, E. Romenski, S. Chiocchetti, and M. Ioriatti, Simulation of non-Newtonian viscoplastic flows with a unified first order hyperbolic model and a structure-preserving semi-implicit scheme, Comput. & Fluids 224 (2021), 10496310.1016/j.compfluid.2021.104963Search in Google Scholar
[33] I. Peshkov, M. Pavelka, E. Romenski, and M. Grmela, Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations, Contin. Mech. Thermodyn 30 (2018), No. 6, 1343–137810.1007/s00161-018-0621-2Search in Google Scholar
[34] B. Re and R. Abgrall, A pressure-based method for weakly compressible two-phase flows under a Baer–Nunziato type model with generic equations of state and pressure and velocity disequilibrium, Int. J. Numer. Methods Fluids 94 (2022), No. 8, 1183–123210.1002/fld.5087Search in Google Scholar
[35] E. Romenski, A. A. Belozerov, and I. M. Peshkov, Conservative formulation for compressible multiphase flows, Quart. Appl. Math 74 (2016), No. 1, 113–13610.1090/qam/1409Search in Google Scholar
[36] E. Romenski, D. Drikakis, and E. Toro, Conservative models and numerical methods for compressible two-phase flow, J. Sci. Comput 42 (2010), No. 1, 6810.1007/s10915-009-9316-ySearch in Google Scholar
[37] E. Romenski, G. Reshetova, I. Peshkov, and M. Dumbser, Two-phase computational model for small-amplitude wave propagation in a saturated porous medium. In: Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy Springer International Publishing, Cham, 2020, pp. 313–32010.1007/978-3-030-38870-6_41Search in Google Scholar
[38] E. Romenski and E. F. Toro, Compressible two-phase flows: Two-pressure models and numerical methods, Comput. Fluid Dyn. J 13 (2004), 403–41610.1007/s00193-004-0229-2Search in Google Scholar
[39] E. I. Romensky, Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics, Mathematical and Computer Modelling 28 (1998), No. 10, 115–13010.1016/S0895-7177(98)00159-9Search in Google Scholar
[40] R. Saurel and R. Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys 150 (1999), No. 2, 425–46710.1006/jcph.1999.6187Search in Google Scholar
[41] F. Thein, E. Romenski, and M. Dumbser, Exact and numerical solutions of the Riemann problem for a conservative model of compressible two-phase flows, J. Sci. Comput 93 (2022), No. 3, 8310.1007/s10915-022-02028-xSearch in Google Scholar
[42] J. Zeifang, J. Schütz, K. Kaiser, A. Beck, M. Lukáčová-Medvid’ová, and S. Noelle, A novel full-Euler low Mach number IMEX splitting, Commun. Comput. Phys 27 (2020), 292–32010.4208/cicp.OA-2018-0270Search in Google Scholar
[43] Z. Zou, E. Audit, N. Grenier, and C. Tenaud, An accurate sharp interface method for two-phase compressible flows at low-Mach regime, Flow, Turbulence and Combustion 105 (2020), No. 4, 1413–144410.1007/s10494-020-00125-1Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
