Mathematical Models and Computer Simulations

, Volume 10, Issue 2, pp 237–248 | Cite as

On the Relaxation of Molecules’ Rotational Energy in the Direct Simulation Monte Carlo Method

  • A. L. Kusov


Rotational-translational energy exchange simulation in the direct simulation Monte Carlo method is considered for the problem of the entry of hypersonic space vehicles in the atmosphere of the Earth, Mars, Venus, Titan, and gas giants. The diatomic and polyatomic molecules’ discrete quantum energy levels are systematized. The energy exchange Larsen-Borgnakke algorithm is described for the molecules with discrete rotational energy levels. The parameters of the model of hard spheres of variable diameters (VHS) are derived for molecules present in the atmospheres from the experimental data on the viscosity of gases and calculation of cross sections of clashing molecules elastic. Analogously, the parameters in Parker’s formula, describing the rotational-translational relaxation, are chosen from the experimental data on the times of relaxation of the rotational energy of the molecules. Close agreement between The calculated and experimental data are in close agreement for moving away and the width of the shock wave before the sphere in nitrogen gas. This means that the used models and their parameters are adequate.


direct simulation Monte Carlo method rarefied gases nonequilibrium flow rotational energy relaxation entry of planets into the atmosphere 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. N. Zalogin, “Rotational relaxation of nitrogen in a viscous shock layer at low Reynolds numbers,” Fluid Dyn. 12, 629–632 (1977).CrossRefzbMATHGoogle Scholar
  2. 2.
    V. V. Ryabov, “Numerical investigation of the flow of nitrogen past a sphere with allowance for rotational relaxation,” Fluid Dyn. 15, 320–324 (1980).CrossRefGoogle Scholar
  3. 3.
    V. I. Vlasov and A. B. Gorshkov, “Comparison of the calculated results for hypersonic flow past blunt bodies with the OREX flight test data,” Fluid Dyn. 36, 812–819 (2001).CrossRefzbMATHGoogle Scholar
  4. 4.
    A. L. Kusov, “Comparison of the calculation heat flux with OREX flight data,” Fiz.-Khim. Kinet. Gaz. Din. 17 (1), 4 (2016).Google Scholar
  5. 5.
    M. J. Wright, J. Olejniczak, L. Walpot, E. Raynaud, T. Magin, L. Caillaut, and B. R. Hollis, “A code calibration study for huygens entry aeroheating,” in Proceedings of the 44th AIAA Aerospace Sciences Meeting and Exhibit, Jan. 9–12, 2006, Reno, NV, AIAA Paper No. 2006-382, pp. 1–16.Google Scholar
  6. 6.
    C. Park, “Viscous shock layer calculation of stagnation-region heating environment in neptune aerocapture,” in Proceedings of the 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Jan. 4–7, 2011, Orlando, FL, AIAA Paper No. 2011-248, pp. 1–7.7.Google Scholar
  7. 7.
    L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 3: Quantum Mechanics: Non-Relativistic Theory (Nauka, Moscow, 1989, 4th ed.; Pergamon, New York, 1977, 3rd ed.).Google Scholar
  8. 8.
    L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 5: Statistical Physics (Nauka, Moscow, 1995; Pergamon, Oxford, 1980).Google Scholar
  9. 9.
    A. S. Davydov, Quantum Mechanics (Pergamon, Oxford, 1965; Nauka, Moscow, 1973).Google Scholar
  10. 10.
    S. F. Gimelshein, I. D. Boyd, and M. S. Ivanov, “Modeling of internal energy transfer in plume flows of polyatomic molecules by the DSMC method,” in Proceedings of the 37th Aerospace Sciences Meeting and Exhibit, Jan. 11–14, 1999, Reno, NV, AIAA Paper No. 99-0738, pp. 1–9.Google Scholar
  11. 11.
    Thermodynamical Properties of Individual Substances, The Handbook, Ed. by V. P. Glushko (Nauka, Moscow, 1978), Vol. 1, pt. 1 [in Russian].Google Scholar
  12. 12.
    G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows (Clarendon, Oxford, 1994).Google Scholar
  13. 13.
    A. L. Kusov, “Numerical simulation of the flow around cylinder with sphere nose using direct simulation Monte-Carlo method,” Mat. Model. 27 (12), 33–47 (2015).MathSciNetzbMATHGoogle Scholar
  14. 14.
    A. L. Kusov and V. V. Lunev, “Use of Monte-Carlo direct static modeling method for solving problem of nonstationary rarefied gas dispersion when evaporating from overheated material surface in vacuum,” Kosmonavt. Raketostroen., No. 1 (58), 36–45 (2010).Google Scholar
  15. 15.
    J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1954).zbMATHGoogle Scholar
  16. 16.
    Physicochemical Processes in Gas Dynamics. Computer Handbook, Vol. 1: Dynamics of Physicochemical Processes in Gas and Plasma, Ed. by G. G. Chernyi and S. A. Losev (Mosk. Gos. Univ., Moscow, 1995) [in Russian].Google Scholar
  17. 17.
    M. J. Wright, D. Bose, G. E. Palmer, and E. Levin, “Recommended collision integrals for transport property computations, Part 1: Air species,” AIAA J. 43, 2558–2564 (2005).CrossRefGoogle Scholar
  18. 18.
    M. J. Wright, H. H. Hwang, and D. W. Schwenke, “Recommended collision integrals for transport property computations, Part 2: Mars and Venus entries,” AIAA J. 45, 281–288 (2007).CrossRefGoogle Scholar
  19. 19.
    Physical Values, The Handbook, Ed. by I. S. Grigorev and E. Z. Meilikhov (Energoatomizdat, Moscow, 1991) [in Russian].Google Scholar
  20. 20.
    A. L. Kusov, “On the possibility of the oxygen dissociation modeling in the shock wave using classical models of the direct simulation Monte Carlo method,” Fiz.-Khim. Kinet. Gaz. Din. 17 (1), 5 (2016).Google Scholar
  21. 21.
    J. G. Parker, “Rotational and vibrational relaxation in diatomic gases,” Phys. Fluids 2, 449–462 (1959).MathSciNetCrossRefGoogle Scholar
  22. 22.
    I. D. Boyd, “Rotational and vibrational nonequilibrium effects in rarefied hypersonic flow,” J. Thermophys. 4, 478–484 (1990).CrossRefGoogle Scholar
  23. 23.
    I. D. Boyd, “Rotational-translational energy transfer in rarefied nonequilibrium flows,” Phys. Fluids A 2, 447–452 (1990).CrossRefGoogle Scholar
  24. 24.
    Y. V. Stupochenko, S. A. Losev, and A. I. Osipov, Relaxation in Shock Waves (Springer, Berlin, Heidelberg, 1967; Nauka, Moscow, 1965).Google Scholar
  25. 25.
    A. V. Bogdanov, G. V. Dubrovskii, A. I. Osipov, and V. M. Strelchenia, Rotational Relaxation in Gases and Plasma (Energoatomizdat, Moscow, 1991) [in Russian].Google Scholar
  26. 26.
    Ya. B. Zel’dovich and Yu. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena (Academic, New York, 1968; Nauka, Moscow, 1973).Google Scholar
  27. 27.
    I. M. Sobol’, A Primer for the Monte Carlo Method (CRC, Boca Raton, FL, 1994; Nauka, Moscow, 1973).zbMATHGoogle Scholar
  28. 28.
    D. A. Russell, “Density disturbance ahead of a sphere in rarefied supersonic flow,” Phys. Fluids 11, 1679 (1968).CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Central Research Institute of Machine BuildingKorolev, Moscow oblastRussia

Personalised recommendations