A novel ES-BGK model for non-polytropic gases with internal state density independent of the temperature. (English) Zbl 07922463
Summary: A novel ES-BGK-based model of non-polytropic rarefied gases in the framework of kinetic theory is presented. Key features of this model are: an internal state density function depending only on the microscopic energy of internal modes (avoiding the dependence on temperature seen in previous reference studies); full compliance with the H-theorem; feasibility of the closure of the system of moment equations based on the maximum entropy principle, following the well-established procedure of rational extended thermodynamics. The structure of planar shock waves in carbon dioxide \((\text{CO}_2)\) obtained with the present model is in general good agreement with that of previous results, except for the computed internal temperature profile, which is qualitatively different with respect to the results obtained in previous studies, showing here a consistently monotonic behavior across the shock structure, rather than the non monotonic behavior previously found.
MSC:
| 76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
| 76L05 | Shock waves and blast waves in fluid mechanics |
Keywords:
Boltzmann equation; ES-BGK model; polyatomic gas; non-polytropic gas; maximum entropy principle; shock wave structureReferences:
| [1] | McCourt, FRW; Beenakker, JJM; Köhler, WE; Kuščer, I., Nonequilibrium Phenomena in Polyatomic Gases, Vol. 1: Dilute Gases, 1990, Oxford: Clarendon Press, Oxford |
| [2] | Nagnibeda, E.; Kustova, E., Non-equilibrium Reacting Gas Flows: Kinetic Theory of Transport and Relaxation Processes, 2009, Berlin: Springer, Berlin · Zbl 1186.82003 |
| [3] | Boyd, ID; Schwartzentruber, TE, Nonequilibrium Gas Dynamics and Molecular Simulation, 2017, Cambridge: Cambridge University Press, Cambridge · Zbl 1366.82001 |
| [4] | Borsoni, T.; Bisi, M.; Groppi, M., A general framework for the kinetic modelling of polyatomic gases, Commun. Math. Phys., 393, 1, 215-266, 2022 · Zbl 07545281 |
| [5] | Li, Z-H; Zhang, H-X, Gas-kinetic numerical studies of three-dimensional complex flows on spacecraft re-entry, J. Comput. Phys., 228, 4, 1116-1138, 2009 · Zbl 1330.76094 |
| [6] | Mathiaud, J., Models and Methods for Complex Flows: Application to Atmospheric Reentry and Particle/Fluid Interactions (Habilitation à Diriger des Recherches), 2018, Bordeaux: University of Bordeaux, Bordeaux |
| [7] | Borgnakke, C.; Larsen, PS, Statistical collision model for Monte Carlo simulation of polyatomic gas mixture, J. Comput. Phys., 18, 4, 405-420, 1975 |
| [8] | Bourgat, J-F; Desvillettes, L.; Le Tallec, P.; Perthame, B., Microreversible collisions for polyatomic gases and Boltzmann’s theorem, Eur. J. Mech. B, 13, 2, 237-254, 1994 · Zbl 0807.76067 |
| [9] | Kosuge, S.; Kuo, H-W; Aoki, K., A kinetic model for a polyatomic gas with temperature-dependent specific heats and its application to shock-wave structure, J. Stat. Phys., 177, 2, 209-251, 2019 · Zbl 1448.76136 |
| [10] | Ruggeri, T.; Sugiyama, M., Classical and Relativistic Rational Extended Thermodynamics of Gases, 2021, Cham: Springer, Cham · Zbl 1461.82002 |
| [11] | Pavić, M.; Ruggeri, T.; Simić, S., Maximum entropy principle for rarefied polyatomic gases, Physica A, 392, 6, 1302-1317, 2013 |
| [12] | Ruggeri, T., Maximum entropy principle closure for 14-moment system for a non-polytropic gas, Ricerche mat., 70, 1, 207-222, 2021 · Zbl 1468.76060 |
| [13] | Kogan, MN, Rarefied Gas Dynamics, 359-368, 1967, New York: Academic Press, New York |
| [14] | Dreyer, W., Maximisation of the entropy in non-equilibrium, J. Phys. A, 20, 18, 6505-6517, 1987 · Zbl 0633.76081 |
| [15] | Müller, I.; Ruggeri, T., Extended Thermodynamics, 1993, New York: Springer, New York · Zbl 0801.35141 |
| [16] | Arima, T.; Taniguchi, S.; Ruggeri, T.; Sugiyama, M., Extended thermodynamics of dense gases, Continuum Mech. Thermodyn., 24, 4-6, 271-292, 2012 · Zbl 1318.80001 |
| [17] | Arima, T.; Taniguchi, S.; Ruggeri, T.; Sugiyama, M., Recent results on non linear extended thermodynamics of real gas with six fields. Part I: general theory, Ricerche Mat., 65, 1, 263-277, 2016 · Zbl 1352.82014 |
| [18] | Bisi, M.; Ruggeri, T.; Spiga, G., Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and extended thermodynamics, Kinet. Rel. Models, 11, 2, 1-25, 2018 |
| [19] | Arima, T.; Carrisi, MC; Pennisi, S.; Ruggeri, T., Which moments are appropriate to describe gases with internal structure in rational extended thermodynamics?, Int. J. Non Linear Mech., 137, 2021 |
| [20] | Baranger, C.; Dauvois, Y.; Marois, G.; Mathé, J.; Mathiaud, J.; Mieussens, L., A BGK model for high temperature rarefied gas flows, Eur. J. Mech. B, 80, 1-12, 2020 · Zbl 1472.76078 |
| [21] | Bisi, M.; Spiga, G., On kinetic models for polyatomic gases and their hydrodynamic limits, Ricerche mat., 66, 1, 113-124, 2017 · Zbl 1373.76292 |
| [22] | Rahimi, B.; Struchtrup, H., Capturing non-equilibrium phenomena in rarefied polyatomic gases: a high-order macroscopic model, Phys. Fluids, 26, 5, 2014 |
| [23] | Struchtrup, H., The BGK model for an ideal gas with an internal degree of freedom, Transp. Theor. Stat. Phys., 28, 4, 369-385, 1999 · Zbl 0949.76078 |
| [24] | Holway, LHJ, New statistical models for kinetic theory: methods of construction, Phys. Fluids, 9, 9, 1658-1673, 1966 |
| [25] | Andries, P.; Le Tallec, P.; Perlat, J-P; Perthame, B., The Gaussian-BGK model of Boltzmann equation with small Prandtl number, Eur. J. Mech. B, 19, 6, 813-830, 2000 · Zbl 0967.76082 |
| [26] | Brull, S.; Schneider, J., On the ellipsoidal statistical model for polyatomic gases, Continuum Mech. Thermodyn., 20, 8, 489-508, 2009 · Zbl 1234.76046 |
| [27] | Brau, CA, Kinetic theory of polyatomic gases: models for the collision processes, Phys. Fluids, 10, 1, 48-55, 1967 |
| [28] | Gorji, MH; Jenny, P., A Fokker-Planck based kinetic model for diatomic rarefied gas flows, Phys. Fluids, 25, 6, 2013 |
| [29] | Mathiaud, J.; Mieussens, L., A Fokker-Planck model of the Boltzmann equation with correct Prandtl number for polyatomic gases, J. Stat. Phys., 168, 5, 1031-1055, 2017 · Zbl 1373.82059 |
| [30] | Arima, T.; Ruggeri, T.; Sugiyama, M., Rational extended thermodynamics of a rarefied polyatomic gas with molecular relaxation processes, Phys. Rev. E, 96, 4, 2017 |
| [31] | Arima, T.; Ruggeri, T.; Sugiyama, M., Extended thermodynamics of rarefied polyatomic gases: 15-field theory incorporating relaxation processes of molecular rotation and vibration, Entropy, 20, 4, 301, 2018 |
| [32] | Dauvois, Y.; Mathiaud, J.; Mieussens, L., An ES-BGK model for polyatomic gases in rotational and vibrational nonequilibrium, Eur. J. Mech. B, 88, 1-16, 2021 · Zbl 1494.76074 |
| [33] | Mathiaud, J.; Mieussens, L.; Pfeiffer, M., An ES-BGK model for diatomic gases with correct relaxation rates for internal energies, Eur. J. Mech. B, 96, 65-77, 2022 · Zbl 1500.76075 |
| [34] | Müller, I.; Ruggeri, T., Rational Extended Thermodynamics, 1998, New York: Springer, New York · Zbl 0895.00005 |
| [35] | Boillat, G.; Ruggeri, T., Moment equations in the kinetic theory of gases and wave velocities, Continuum Mech. Thermodyn., 9, 4, 205-212, 1997 · Zbl 0892.76075 |
| [36] | Ruggeri, T., Galilean invariance and entropy principle for systems of balance laws, Continuum Mech. Thermodyn., 1, 1, 3-20, 1989 · Zbl 0759.35039 |
| [37] | Hamburger, HL, Hermitian transformations of deficiency-index (1, 1), Jacobi matrices and undetermined moment problems, Am. J. Math., 66, 4, 489-522, 1944 · Zbl 0061.25808 |
| [38] | Arima, T.; Taniguchi, S.; Ruggeri, T.; Sugiyama, M., Extended thermodynamics of real gases with dynamic pressure: an extension of Meixner’s theory, Phys. Lett. A, 376, 44, 2799-2803, 2012 · Zbl 1396.80001 |
| [39] | Arima, T.; Taniguchi, S.; Ruggeri, T.; Sugiyama, M., Dispersion relation for sound in rarefied polyatomic gases based on extended thermodynamics, Continuum Mech. Thermodyn., 25, 6, 727-737, 2013 · Zbl 1341.80004 |
| [40] | Taniguchi, S.; Arima, T.; Ruggeri, T.; Sugiyama, M., Thermodynamic theory of the shock wave structure in a rarefied polyatomic gas: beyond the Bethe-Teller theory, Phys. Rev. E, 89, 1, 2014 |
| [41] | Gilbarg, G.; Paolucci, D., The structure of shock waves in the continuum theory of fluids, J. Rational Mech., 2, 4, 617-642, 1953 · Zbl 0051.18102 |
| [42] | Kosuge, S.; Aoki, K., Shock-wave structure for a polyatomic gas with large bulk viscosity, Phys. Rev. Fluids, 3, 2, 2018 · Zbl 1475.82020 |
| [43] | Taniguchi, S.; Arima, T.; Ruggeri, T.; Sugiyama, M., Overshoot of the non-equilibrium temperature in the shock wave structure of a rarefied polyatomic gas subject to the dynamic pressure, Int. J. Non Linear Mech., 79, 66-75, 2016 |
| [44] | Jaynes, ET, Information theory and statistical mechanics, Phys. Rev., 106, 4, 620-630, 1957 · Zbl 0084.43701 |
| [45] | Jaynes, ET, Information theory and statistical mechanics II, Phys. Rev., 108, 2, 171-190, 1957 · Zbl 0084.43701 |
| [46] | Levermore, CD, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83, 5-6, 1021-1065, 1996 · Zbl 1081.82619 |
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