Macroscopic Gas-Dynamic Approximations of the Local Nonequilibrium Molecular Velocity Distribution Function

   Aim. The investigation into the applicability of classical macroscopic approximations to obtain the nonequilibrium local distribution function inside the structure of a strong shock wave.

   Methodology. This paper examines the capability of various macroscopic models (the Navier – Stokes – Fourier equations, the Burnett equations, and the original and regularized 13-moment Grad equations) to approximate a nonequilibrium molecular velocity distribution function.

   Results. The locally reconstructed distribution functions obtained from the flow macro-parameters for the considered models are compared with each other and with a benchmark solution at different locations within the structure of a planar shock wave. The benchmark solution is provided by the Direct Simulation Monte Carlo (DSMC) method, which supplies the flow macro-parameters required for the reconstruction of the distribution function.

   Research implications. All considered models predict the distribution function rather poorly in the supersonic part of the shock-wave structure, where strong oscillations and nonphysical negative values are observed.

1. Cercignani, C. (1988). The Boltzmann Equation and Its Applications. Berlin: Springer.

2. Aristov, V. V. & Cheremisin, F. G. (1980). The conservative splitting method for solving Boltzmann's equation. In: USSR Computational Mathematics and Mathematical Physics, 20 (1), 191–207 (in Russ.).

3. Ohwada, T. (1993). Structure of normal shock waves: Direct numerical analysis of the Boltzmann equation for hard‐sphere molecules. In: Physics of Fluids A: Fluid Dynamics, 5 (1), 217–234. DOI: 10.1063/1.858777.

4. Bird, G. A. (1994). Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford: Clarendon Press.

5. Ivanov, M. S. & Rogazinskii, S. V. (1988). Comparative analysis of direct simulation algorithms in rarefied gas dynamics. In: USSR Computational Mathematics and Mathematical Physics, 28 (7), 1058–1070 (in Russ.).

6. Ivanov, M. S. & Gimelshein, S. F. (1998). Computational Hypersonic Raredied Flows. In: Annual Review of Fluid Mechanics, 30, 469–505. DOI: 10.1146/annurev.fluid.30.1.469.

7. Struchtrup, H. (2005). Macroscopic Transport Equations for Rarefied Gas Flows. Berlin: Springer.

8. Torrilhon, M. (2016). Modeling Nonequilibrium Gas Flow Based on Moment Equations. In: Annual Review of Fluid Mechanics, 48 (1), 429–458. DOI: 10.1146/annurev-fluid-122414-034259.

9. Kogan, M. N. (1967). Dynamics of rarefied gas: kinetic theory. Moscow: Nauka (in Russ.).

10. Chapman, S. & Cowling, T. G. (1991). The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases. Cambridge: Cambridge University Press.

11. Grad, H. (1949). On the kinetic theory of rarefied gases. In: Communications on Pure and Applied Mathematics, 2 (4), 331–407. DOI: 10.1002/cpa.3160020403.

12. Pham-Van-Diep, G. C., Erwin, D. A. & Muntz, E. P. (1991). Testing continuum descriptions of low-Mach-number shock structures. In: Journal of Fluid Mechanics, 232, 403–413. DOI: 10.1017/S0022112091003749.

13. Erofeev, A. I. & Friedlander, O. G. (2007). Macroscopic Models for Non-equilibrium Flows of Monatomic Gas and Normal Solutions. In: Rarefied Gas Dynamics: Proceedings of 25^th International Symposium. Novosibirsk: Siberian Branch of the Russian Academy of Sciences, pp. 117–124.

14. Ivanov, I. E., Timokhin, M. Y., Kryukov, I. A., Bondar, Y. A., Kokhanchik, A. A. & Ivanov, M. S. (2012). Study of the Shock Wave Structure by Regularized Grad's Set of Equations. In: AIP Conference Proceedings, 1501 (1), 215–222. DOI: 10.1063/1.4769507.

15. Timokhin, M. Yu., Ivanov, I. E. & Kryukov, I. A. (2014). Moment equations and gas-kinetic scheme application to numerical simulation of gas flows in micro scale devices. In: AIP Conference Proceedings, 1628 (1), 748–755. DOI: 10.1063/1.4902668.

16. Westerkamp, A. & Torrilhon, M. (2019). Finite element methods for the linear regularized 13-moment equations describing slow rarefied gas flows. In: Journal of Computational Physics, 389, 1–21. DOI: 10.1016/j.jcp.2019.03.022.

17. Zheng, Y. & Struchtrup, H. (2004). Burnett equations for the ellipsoidal statistical BGK model. In: Continuum Mechanics and Thermodynamics, 16, 97–108. DOI: 10.1007/s00161-003-0143-3.

18. Timokhin, M. Yu., Struchtrup, H., Kokhanchik, A. A. & Bondar, Ye. A. (2016). The analysis of different variants of R13 equations applied to the shock-wave structure. In: AIP Conference Proceedings, 1786 (1), 140006. DOI: 10.1063/1.4967637.

19. Gu, X. J. & Emerson, D. R. (2009). A high-order moment approach for capturing non-equilibrium phenomena in the transition regime. In: Journal of Fluid Mechanics, 636, 177–216. DOI: 10.1017/S002211200900768X.

20. Timokhin, M. Yu., Ivanov, I. E. & Kryukov, I. A. (2012). 2D Numerical Simulation of Gas Flow Interaction with Solid Wall by Regularized Grad’s Set of Equations. In: AIP Conference Proceedings, 1501 (1), P. 843–848. DOI: 10.1063/1.4769630.

21. Kashkovsky, A. V., Vashchenkov, P. V., Shevyrin, A. A., Molchanova, A. N. & Litvintsev, A. S. et al. (2025). SMILE family of software systems for the direct Monte Carlo simulations of rarefied gas flows. In: Journal of engineering physics and thermophysics, 98, 1872–1897. DOI: 10.1007/s10891-026-03266-8.

22. Ivanov, M. S. Kashkovsky, A. V., Vashchenkov, P. V. & Bondar, Ye. A. (2011). Parallel Object-Oriented Software System for DSMC Modeling of High-Altitude Aerothermodynamic Problems. In: AIP Conference Proceedings, 1333 (1), 211–218. DOI: 10.1063/1.3562650.

23. Becker, R. (1922). Stoßwelle und Detonation. In: Zeitschrift für Physik, 8 (1), 321–362. DOI: 10.1007/BF01329605.

24. Mott-Smith, H. M. (1951). The Solution of the Boltzmann Equation for a Shock Wave. In: Physical Review, 82, 885–892. DOI: 10.1103/PhysRev.82.885.

25. Yen, S. M. (1966). Temperature Overshoot in Shock Waves. In: Physics of Fluids, 9 (1), 1417–1418. DOI: 10.1063/1.1761862.

26. Xu, K. & Huang, J.-C. (2010). A unified gas-kinetic scheme for continuum and rarefied flows. In: Journal of Computational Physics, 229 (20), 7747–7764. DOI: 10.1016/j.jcp.2010.06.032.

27. Hansen, K. & Hornig, D. F. (1960). Thickness of Shock Fronts in Argon. In: Journal of Chemical Physics, 33 (3), 913–916. DOI: 10.1063/1.1731288.

28. Alsmeyer, H. (1976). Density Profiles in Argon and Nitrogen Shock Waves Measured by the Absorption of an Electron Beam. In: Journal of Fluid Mechanics, 74 (3), 497–513. DOI: 10.1017/S0022112076001912.

29. Pham-Van-Diep, G., Erwin, D. & Muntz, E. P. (1989). Nonequilibrium Molecular Motion in a Hypersonic Shock Wave. In: Science, 245 (4918), 624–626. DOI: 10.1126/science.245.4918.624.

30. Timokhin, M. Yu., Tikhonov, M., Mursenkova, I. V. & Znamenskaya, I. A. (2020). Shock-wave Thickness Influence to the Light Diffraction on a Plane Shock Wave. In: Physics of Fluids, 32 (11), 116103. DOI: 10.1063/5.0029612.

31. Rankine, W. J. M. (1870). On the thermodynamic theory of waves of finite longitudinal disturbance. In: Philosophical Transactions of the Royal Society of London, 160, 277–288. DOI: 10.1098/rstl.1870.0015.

32. Lockerby, D. A., Reese, J. M. & Struchtrup, H. (2009). Switching criteria for hybrid rarefied gas flow solvers. In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 465 (2105), 1581–1598. DOI: 10.1098/rspa.2008.0497.

33. Timokhin, M. & Rukhmakov, D. (2021). Local non-equilibrium phase density reconstruction with Grad and Chapman-Enskog methods. In: Journal of Physics: Conference Series, 1959 (1), 012049. DOI: 10.1088/1742-6596/1959/1/012049.

34. Cai, Z. & Torrilhon, M. (2019). On the Holway-Weiss debate: Convergence of the Grad-moment-expansion in kinetic gas theory. In: Physics of Fluids, 31 (12), 126105. DOI: 10.1063/1.5127114.