First approximation models for nonequilibrium flows of polyatomic gases

Aviation technics and technology


Nikitchenko Y. A.

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



Physical-chemical processes, which take place in the gas and on the surface that is streamlined by this gas, depend on the energy of thermal motion of molecules. In nonequilibrium flows, the energy of thermal motion is distributed unevenly between translational and internal degrees of freedom of molecules. Under these conditions it is necessary to consider the energy of the translational degrees of freedom (translational temperature Tt) separately from the energy of the internal degrees of freedom (internal temperature TΩ).

The physical-mathematical models of the first approximation are most common in practical applications. These models can be obtained from the system of moment equations of polyatomic gases [1]. A method for constructing models of the first approximation is described in [2].

Two models of the first approximation are considered. The first model is written down by using the thermodynamic variables. Its system of equations contains 5 scalar equations (5-moment model). This model is the Navier-Stokes-Fourier model. The coefficient of bulk viscosity is adduced in an explicit form. The system of equations does not contain Tt and TΩ temperatures. These parameters are defined by a special dependency through the coefficient of bulk viscosity.

The second model contains 6 scalar equations. The Tt and TΩ temperatures are defined by independent moment equations (two-temperature model). The heat flow is represented by two components. The first component is the energy flow of the translational degrees of freedom. The second component is the energy flow of the internal degrees of freedom. The coefficient of bulk viscosity is not used. The exchange of energy between translational and internal degrees of freedom is described explicitly.

Numerical testing of the models was conducted on the sample problem of the shock wave shape. The tests have shown that 5-moment model produces qualitatively incorrect values of Tt and TΩ temperatures. The second law of thermodynamics was violated in some profile areas. The two-temperature models only produce errors of quantitative nature. These errors can be reduced by the adjustment of free parameters of the model.

The two-temperature model has significant advantages over the 5-moment model during the modeling of the nonequilibrium flows. These advantages are essentially important for the description of physical-chemical processes.


moment equations, first approximation, bulk viscosity, two-temperature model, shock wave shape


  1. 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.
  2. Nikitchenko Yu. A. Modeli neravnovesnykh techenii (Models of nonequilibrium flows), Moscow, Publishing house of the Moscow Aviation Institute, 2013, 160 p.
  3. Nikitchenko Yu. A. Polet. 2010. no. 11. pp. 43-51.
  4. Nikitchenko Yu. A. Vestnik Moskovskogo Aviatsionnogo Instituta, 2012, vol.19, no. 2, pp. 11-17.
  5. Bird G.A. Molecular gas dynamics and the direct simulation of gas flows. Oxford: Clarendon Press, 1994, 340 p.
  6. Alsmeyer H. Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam. J. Fluid Mech., 1976, vol. 74, pt. 3, pp. 497-513.

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