Navier-Stokes-Fourier model options for supersonic and hypersonic flows

DOI: 10.34759/trd-2020-112-3


Budanova S. Y.*, Krasavin E. E.**, Nikitchenko Y. A.***

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia



The flows of the high-degree non-equilibrium are of considerable interest at present stage of the engineering development. Such flows are being realized, for example, while the hypersonic flying vehicles flow-around, reentry spacecraft, in vacuum installations and other technical devices.

The basic physico-mathematical model of a gas medium flow is the Navier-Stokes-Fourier (NSF) model. This model is theory-based for weakly non-equilibrium flows, but it can be applied for the flows of a high-degree non-equilibrium. In such flows, the NSF model coarsens the solution. For example, when computing shock waves, the disturbance area is narrowed. The model is short of viscosity. The opposite sign effects are being observed while the gas intensive expansion.

The presented work analyses the processes of the non-equilibrium stresses and heat flows forming at the shock wave front, employing the model kinetic equation (MKE). The hypersonic flow at the Mach number of M= 5 is under consideration.

The article shows that the heat flows gradients, absent in the NSE, contribute mainly to the non-equilibrium stresses forming process. The basic factor of the heat flows development are the fourth-order gradients, which are missed in the model as well.

Several options for improving the viscous properties of the NSF model are under consideration. It is known that in the case of multi-atomic gases a significant effect can be achieved by accounting for the voluminous viscosity in the equations of non-equilibrium stresses. Besides, the voluminous viscosity coefficient allows computing the temperature of the translational degrees of freedom molecules in the first approximation. Defining the shear viscosity coefficient by this temperature, will improve the viscous properties of the model.

The Stokes’s friction law can be obtained by using the moment stress equation as its strict first approximation. If one accounts for the terms of the second order of vanishing in the equation terms, containing gradients and divergence of the flow velocity, , then the shear viscosity coefficient will take a tensor form. The NSF model with this shear viscosity coefficient describes better the processes of viscosity and heat conductivity.

The article demonstrates that the improved option of the NSF model allows obtaining rather wide area of disturbances on the example density, velocity and temperature profiles in the flat shock wave. The profiles shapes differ slightly from real ones. This, probably, is a consequence of the artificial approach to the model improvement.


Navier-Stokes-Fourier model, viscosity factor, hypersonic flow, numerical tests


  1. Nikitchenko Yu.A. Izvestiya Rossiiskoi akademii nauk. Mekhanika zhidkosti i gaza, 2018, no. 2, pp. 128 – 138.

  2. Zhdanov V.M., Alievskii M.Ya. Protsessy perenosa i relaksatsii v molekulyarnykh gazakh (Transfer and relaxation processes in molecular gases), Moscow, Nauka, 1989, 336 p.

  3. Kuznetsov M.M., Kuleshova Yu.D., Reshetnikova Yu.G., Smotrova L.V. Trudy MAI, 2017, no. 95, available at:

  4. Khatuntseva O.N. Trudy MAI, 2019, no. 104, available at:

  5. Gred G. Mekhanika, 1952, no. 5, pp. 62 – 96.

  6. Nikitchenko Yu.A. Modeli neravnovesnykh techenii (Non-equilibrium flow models), Moscow, Izd-vo MAI, 2013, 160 p.

  7. Buzykin O.G., Galkin V.S. Izvestiya Rossiiskoi akademii nauk. Mekhanika zhidkosti i gaza, 2001, no. 3, pp. 185 – 199.

  8. Kogan M.N. Dinamika razrezhennogo gaza (Rarefied gas dynamics), Moscow, Nauka, 1967, 440 p.

  9. Gusev V.N., Egorov I.V., Erofeev A.I., Izvestiya Rossiiskoi akademii nauk. Mekhanika zhidkosti i gaza, 1999, no. 2, pp. 128 – 137.

  10. Brykina I.G. Izvestiya Rossiiskoi akademii nauk. Mekhanika zhidkosti i gaza, 2017, no. 4, pp. 125 – 139.

  11. Brykina I.G., Rogov B.V., Tirskii G.A., Titarev V.A., Utyuzhnikov S.V. Prikladnaya matematika i mekhanika, 2013, no. 77(1), pp. 15 – 26.

  12. Nikitchenko Yu.A. Zhurnal vychislitel’noi matematiki i matematicheskoi fiziki, 2017, vol. 57, no. 11, pp. 117 – 129.

  13. Nikitchenko Yu.A. Uchenye zapiski TsAGI, 2015, vol. XLVI, no. 1, 72 – 84.

  14. Shakhov E.M. Metod issledovaniya dvizhenii razrezhennogo gaza (Method of rarefied gas motions studying), Moscow, VTs AN SSSR, 1975, 207 p.

  15. Holtz T., Muntz E.P. Molecular velocity distribution functions in an argon normal shock wave at Mach number 7, Physics of Fluids, September 1983, no. 26 (9), pp. 2425 – 2436.

  16. Alsmeyer H. Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam, Journal of Fluid Mechanics, 1976, vol. 74, no. 3, pp. 497 – 513. DOI:10.1017/S0022112076001912

  17. Robben F., Talbot L. Experimental study of the rotational distribution function of nitrogen in a shock wave, Physics of Fluids, 1966, vol. 9, no. 4, pp. 653 – 662. DOI: 10.1063/1.1761730

  18. Bird G.A. Molecular gas dynamics and the direct simulation of gas flows, Oxford, Clarendon Press, 1994, 458 p.

  19. Degond P., Jin S., Mieussens L. A smooth transition model between kinetic and hydrodynamic equations, Journal of Computational Physics, 2005, no. 209 (2), pp 665 – 694. DOI: 10.1016/

  20. Rovenskaya O.I., Croce G. Numerical simulation of gas flow in rough micro channels: hybrid kinetic–continuum approach versus Navier–Stokes, Microfluidics and Nanofluidics, 2016, no. 20 (5). DOI: 10.1007/s10404-016-1746-x

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