×

Derivation and closure of Baer and Nunziato type multiphase models by averaging a simple stochastic model. (English) Zbl 1457.76176

Summary: In this article, we show how to derive a multiphase model of Baer and Nunziato type with a simple stochastic model. Baer and Nunziato models are known to be unclosed, namely, they depend on modeling parameters, as interfacial velocity and pressure, and relaxation terms, whose exact expression is still an open question. We prove that with a simple stochastic model, interfacial and relaxation terms are equivalent to the evaluation of an integral, which cannot be explicitly computed in general. However, in different particular cases matching with a large range of applications (topology of the bubbles/droplets, or special flow regime conditions), the interfacial and relaxation parameters can be explicitly computed, leading to different models that are either nonlinear versions or slight modifications of previously proposed models. The validity domains of previously proposed models are clarified, and some modeling parameters of the averaged system are linked with the local topology of the flow. Last, we prove that usual properties like entropy dissipation are ensured with the new closures found.

MSC:

76T10 Liquid-gas two-phase flows, bubbly flows
76M50 Homogenization applied to problems in fluid mechanics
35L65 Hyperbolic conservation laws
35L60 First-order nonlinear hyperbolic equations

Software:

SageMath
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] R. Abgrall and S. Karni, A comment on the computation of non-conservative products, J. Comput. Phys., 229 (2010), pp. 2759-2763. · Zbl 1188.65134
[2] R. Abgrall and R. Saurel, Discrete equations for physical and numerical compressible multiphase mixtures, J. Comput. Phys., 186 (2003), pp. 361-396, https://doi.org/10.1016/S0021-9991(03)00011-1. · Zbl 1072.76594
[3] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, 1972. · Zbl 0543.33001
[4] R. J. Adler and J. E. Taylor, Random Fields and Geometry, Springer, New York, 2009. · Zbl 1149.60003
[5] 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), pp. 861-889. · Zbl 0609.76114
[6] D. Bresch and M. Hillairet, A compressible multifluid system with new physical relaxation terms, Ann. Sci. Éc. Norm. Supér. (4), 52 (2019), pp. 255-295. · Zbl 1421.35241
[7] Y. Chen, J. Glimm, D. H. Sharp, and Q. Zhang, A two-phase flow model of the Rayleigh-Taylor mixing zone, Phys. Fluids, 8 (1996), p. 816. · Zbl 1025.76535
[8] A. Chinnayya, E. Daniel, and R. Saurel, Modelling detonation waves in heterogeneous energetic materials, J. Comput. Phys., 196 (2004), pp. 490-538. · Zbl 1109.76335
[9] F. Coquel, T. Gallouët, P. Helluy, J.-M. Hérard, O. Hurisse, and N. Seguin, Modelling compressible multiphase flows, ESAIM Proc., 40 (2013), pp. 34-50. · Zbl 1329.76349
[10] F. Coquel, T. Gallouët, J.-M. Hérard, and N. Seguin, Closure laws for a two-fluid two-pressure model, C. R. Math., 334 (2002), pp. 927-932. · Zbl 0999.35057
[11] F. Coquel, J.-M. Hérard, K. Saleh, and N. Seguin, A class of two-fluid two-phase flow models, in Proceedings of the 42nd AIAA Fluid Dynamics Conference and Exhibit, 2012, p. 3356.
[12] P. Cordesse and M. Massot, Entropy supplementary conservation law for non-linear systems of PDEs with non-conservative terms: Application to the modelling and analysis of complex fluid flows using computer algebra, Commun. Math. Sci., 18 (2020), pp. 515-534, https://doi.org/10.4310/CMS.2020.v18.n2.a10. · Zbl 1460.35225
[13] 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. Engrg., 354 (2019), pp. 820-849. · Zbl 1441.76076
[14] G. B. Deane and M. D. Stokes, Scale dependence of bubble creation mechanisms in breaking waves, Nature, 418 (2002), pp. 839-844.
[15] J. M. Delhaye and J. A. Bouré, General equations and two-phase flow modeling, in Handbook of Multiphase Systems, Vol. 1, G. Hestroni, ed., Hemisphere, Washington, DC, 1982.
[16] D. A. Drew and S. L. Passman, Theory of Multicomponent Fluids, Appl. Math. Sci. 135, Springer, New York, 1999. · Zbl 0919.76003
[17] E. Franquet and V. Perrier, Runge-Kutta discontinuous Galerkin method for the approximation of Baer and Nunziato type multiphase models, J. Comput. Phys., 231 (2012), pp. 4096-4141, https://doi.org/10.1016/j.jcp.2012.02.002. · Zbl 1426.76530
[18] T. Gallouët, J.-M. Hérard, and N. Seguin, Numerical modeling of two-phase flows using the two-fluid two-pressure approach, Math. Models Methods Appl. Sci., 14 (2004), pp. 663-700. · Zbl 1177.76428
[19] C. Garrett, M. Li, and D. Farmer, The connection between bubble size spectra and energy dissipation rates in the upper ocean, J. Phys. Oceanography, 30 (2000), pp. 2163-2171.
[20] S. Gavrilyuk and R. Saurel, Mathematical and numerical modeling of two-phase compressible flows with micro-inertia, J. Comput. Phys., 175 (2002), pp. 326-360. · Zbl 1039.76067
[21] J. Glimm, D. Saltz, and D. H. Sharp, Renormalization group solution of two-phase flow equations for Rayleigh-Taylor mixing, Phys. Lett. A, 222 (1996), pp. 171-176. · Zbl 0972.76512
[22] E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Appl. Math. Sci. 118, Springer, New York, 2013. · Zbl 1063.65080
[23] V. Guillemaud, Modélisation et simulation numérique des écoulements diphasiques par une approche bifluide à deux pressions, Ph.D. thesis, Université de Provence-Aix-Marseille I, 2007; also available online from https://tel.archives-ouvertes.fr/tel-00169178.
[24] E. Han, M. Hantke, and S. Müller, Modeling of Multi-Component Flows with Phase Transition and Application to Collapsing Bubbles, Technical report, Institut für Geometrie une Praktische Mathematik, 2014; also available online from https://www.igpm.rwth-aachen.de/Download/reports/mueller/sieema-report.pdf.
[25] E. Han, M. Hantke, and S. Müller, Efficient and robust relaxation procedures for multi-component mixtures including phase transition, J. Comput. Phys., 338 (2017), pp. 217-239. · Zbl 1415.76468
[26] J.-M. Hérard, A three-phase flow model, Math. Comput. Model., 45 (2007), pp. 732-755. · Zbl 1165.76382
[27] J. O. Hinze, Fundamentals of the hydrodynamic mechanism of splitting in dispersion processes, AIChE J., 1 (1955), pp. 289-295.
[28] T. Y. Hou and P. G. LeFloch, Why nonconservative schemes converge to wrong solutions: Error analysis, Math. Comp., 62 (1994), pp. 497-530. · Zbl 0809.65102
[29] H. Jin, J. Glimm, and D. H. Sharp, Entropy of averaging for compressible two-pressure two-phase flow models, Phys. Lett. A, 360 (2006), pp. 114-121. · Zbl 1236.76081
[30] 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 (1997), pp. 3885-3897.
[31] D. Lhuillier, A mean-field description of two-phase flows with phase changes, Int. J. Multiph. Flow, 29 (2003), pp. 511-525. · Zbl 1136.76561
[32] G. Linga and T. Fl\ratten, A hierarchy of non-equilibrium two-phase flow models, ESAIM Proc. Surveys, 66 (2019), pp. 109-143. · Zbl 1443.76227
[33] H. Lochon, F. Daude, P. Galon, and J.-M. Hérard, Comparison of two-fluid models on steam-water transients, ESAIM Math. Model. Numer. Anal., 50 (2016), pp. 1631-1657. · Zbl 1353.76089
[34] H. Lochon, F. Daude, P. Galon, and J.-M. Hérard, HLLC-type Riemann solver with approximated two-phase contact for the computation of the Baer-Nunziato two-fluid model, J. Comput. Phys., 326 (2016). · Zbl 1373.76136
[35] R. Menikoff and B. J. Plohr, The Riemann problem for fluid flow of real materials, Rev. Modern Phys., 61 (1989), p. 75. · Zbl 1129.35439
[36] S. Müller, M. Hantke, and P. Richter, Closure conditions for non-equilibrium multi-component models, Contin. Mech. Thermodyn., 28 (2016), pp. 1157-1189. · Zbl 1355.76069
[37] A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, McGraw-Hill Higher Education, New York, 2002.
[38] V. H. Ransom and D. L. Hicks, Hyperbolic two-pressure models for two-phase flow, J. Comput. Phys., 53 (1984), pp. 124-151. · Zbl 0537.76070
[39] C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning, Adapt. Comput. Mach. Learn., MIT Press, Cambridge, MA, 2006. · Zbl 1177.68165
[40] S. O. Rice, Mathematical analysis of random noise, Bell Syst. Tech. J., 24 (1945), pp. 46-156. · Zbl 0063.06487
[41] K. Saleh and N. Seguin, Some mathematical properties of a barotropic multiphase flow model, to appear ESAIM Proc. Surveys 2020; also available online from https://hal.archives-ouvertes.fr/hal-01921027v2/file/Note_Triph.pdf.
[42] R. Saurel and R. Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys., 150 (1999), pp. 425-467, https://doi.org/10.1006/jcph.1999.6187. · Zbl 0937.76053
[43] R. Saurel, S. Gavrilyuk, and F. Renaud, A multiphase model with internal degrees of freedom: Application to shock-bubble interaction, J. Fluid Mech., 495 (2003), pp. 283-321, https://doi.org/10.1017/S002211200300630X. · Zbl 1080.76062
[44] R. Saurel and C. Pantano, Diffuse-interface capturing methods for compressible two-phase flows, Annu. Rev. Fluid Mech., 50 (2018), pp. 105-130. · Zbl 1384.76054
[45] R. Saurel, F. Petitpas, and R. A. Berry, Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures, J. Comput. Phys., 228 (2009), pp. 1678-1712. · Zbl 1409.76105
[46] The Sage Developers, SageMath, the Sage Mathematics Software System (Version 7.5.1), 2017, https://www.sagemath.org.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.