Joint of kinetic and hydrodynamic models by the example of the Couette flow

Fluid, gas and plasma mechanics


Berezko M. E.*, Nikitchenko Y. A.**, Tikhonovets A. V.***

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



The problem of setting the boundary conditions on a solid surface for the viscous heat-conducting gas equations becomes very relevant in the case of hypersonic and moderately rarefied flows. The statement of the boundary conditions at the level of gas-dynamic variables results in significant errors for relatively large Mach values (M) and Knudsen (Kn) numbers. Physically adequate formulation of the boundary conditions on active (absorbing or releasing the gas) surfaces is impossible without considering molecular processes.

The kinetic models describing the gas flow at the molecular level make it possible to set the physically adequate conditions on the surfaces with different properties for any flow regime. However, the kinetic calculation of the flow field of complex geometry by  is rather inefficient.

The purpose of this work is the development of a physical-mathematical flow model (KIN_NSF) containing the Navier-Stokes-Fourier model (NSF), «joined» to the kinetic model equation of polyatomic gases. The kinetic model is used in the near-wall Knudsen layer. The remaining flow range is described by the NSF model. In the range of the model, joining the approximating velocity distribution function of molecules is recovered by the parameters determined by the NSF model. It represents the expansion of the local-balanced distribution function in terms of the thermal velocity. The expansion coefficients (unbalanced stresses and heat fluxes) are presented in the Navier-Stokes approximation.

The developed KIN_NSF model is efficient enough for the practical applications and at the same time allows setting the boundary conditions at the kinetic level of the gas with the surface interaction processes description.

A series of test calculations has been performed on the example of a flat Couette flow in the intervals . The kinetic model equation for polyatomic gases, NSF and KIN_NSF models were tested.

The results of the calculations revealed that the KIN_NSF model is not much inferior in accuracy to the kinetic model, and it substantially exceeds it in the efficiency. With  the KIN_NSF model required a few dozen times less CPU time than the kinetic model.

When describing flows of the dense gases, the economy of the KIN-NSF model does not depend on the Kn number and depends weakly on the M number, which is typical for the NSF model.


boundary conditions, Navier-Stokes-Fourier model, kinetic equation, joining the models, Couette flow


  1. Gred G. Mekhanika, 1952, no. 4, pp. 71-97.

  2. Kogan M.N. Dinamika razrezhennogo gaza (Rarefied Gas Dynamics), Moscow, Nauka, 1967, 440 p.

  3. Barantsev R.G. Vzaimodeistvie razrezhennykh gazov s obtekaemymi poverkhnostyami (Rarefied Gas and Streamlined Surfaces Interaction), Moscow, Nauka, 1975, 343 p.

  4. Sakabekov A.S. Matematicheskii sbornik, 1992, Vol. 183, no. 9, pp. 67-88.

  5. Latyshev A.V., Yushkanov A.A. Izvestiya RAN. Mekhanika zhidkosti i gaza, 2004, no. 2, pp. 193-208.

  6. Nikitchenko Yu.A. Vestnik Moskovskogo aviatsionnogo instituta, 2012, Vol. 19, no. 3, pp. 5-14.

  7. Becker M., Boyland D.E. Flow field and surface pressure measurements in the fully merged and transition flow regimes on a cooled sharp flat plate. Rarefied Gas Dynamics, Suppl. 4, 1967, vol. 2, pp. 993-1014.

  8. Tannehill J.C. ,Mohling R.A., Rakich J.V. Numerical computation of the hypersonic rarefied flow near the sharp leading edge of a flat plate. AIAA Paper, 1973, №73-200, pp. 1-13.

  9. Nikitchenko Yu.A. Trudy MAI, 2014, no. 77, available at

  10. Glinkina V.S., Nikitchenko Yu.A., Popov S.A., Ryzhov Yu.A. Drag Coefficient of an Absorbing Plate Set Transverse to a Flow. Fluid Dynamics, 2016, vol. 51, no. 6, pp. 791-798.

  11. Nikitchenko Yu.A. Modeli neravnovesnykh techenii (Models of the Unbalanced Flows), Moscow, Izd-vo MAI, 2013, 160 p.

  12. Koshmarov Yu.A., Ryzhov Yu.A. Prikladnaya dinamika razrezhennogo gaza (Applied Dynamics of the Rarefied Gas), Moscow, Mashinostroenie, 1977, 184 p.

Download — informational site MAI

Copyright © 2000-2019 by MAI