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A Version of Closing the System of Moment Equations of an Arbitrary Order

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Abstract

A method for closing the system of moment equations of order greater than two that does not use the approximation function of speed distribution of molecules is proposed. The moments closing this system are built as combinations of moments of lower orders. Systems of moment equations up to the sixth order, inclusive, are constructed. Using the shock wave profile problem as an example, it is shown that an increase in the order of the system of moment equations does not result in improving the solution. This is explained by the fact that the terms of the closing moment equations that cannot be expressed in terms of lower moments introduce an error comparable in magnitude with the magnitude of the closing moment. The solutions are verified using the model kinetic equation of polyatomic gases.

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REFERENCES

  1. M. N. Kogan, Dynamics of Rarefied Gas (Nauka, Moscow, 1967) [in Russian].

    Google Scholar 

  2. Yu. A. Nikitchenko, “On the reasonability of taking the volume viscosity coefficient into account in gas dynamic problems,” Fluid Dyn. 53, 305–314 (2018).

    Article  MathSciNet  Google Scholar 

  3. H. Grad, “On the kinetic theory of rarefied gases,” Commun. Pure Appl. Math. 2, 331–407 (1949).

    Article  MathSciNet  Google Scholar 

  4. H. Grad, Principles of Kinetic Theory of Gases. Handbuch der Physik, Ed. by S. Flügge (Springer, Berlin, 1958), vol. 12.

    Google Scholar 

  5. H. Grad, “The profile of a steady plane shock wave,” Commun. Pure Appl. Math. 5 (3), 257–300 (1952).

    Article  MathSciNet  Google Scholar 

  6. H. Struchtrup and M. Torrilhon, “Regularization of Grad’s 13 moment equations: Derivation and linear analysis,” Phys. Fluids 15, 2668–2680 (2003).

    Article  MathSciNet  Google Scholar 

  7. M. Yu. Timokhin, H. Struchtrup, A. A. Kokhanchik, and Ye. A. Bondar, “Different variants of R13 moment equations applied to the shock-wave structure,” Phys. Fluids 29, 049901 (2017).

  8. Yu. A. Nikitchenko, “Decreasing the short-wave instability of a system of moment equations by its expansion,” Uch. Zap. TsAGI 46 (1), 72–84 (2015).

    Google Scholar 

  9. J. A. Lordi and R. E. Mates, “Rotational relaxation in nonpolar diatomic gases,” Phys. Fluids 13, 291–308 (1970).

    Article  Google Scholar 

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

    Google Scholar 

  11. Yu. A. Nikitchenko, “Model kinetic equation for polyatomic gases,” Comput. Math, Mat. Phys. 57, 1843–1855 (2017).

    MATH  Google Scholar 

  12. H. Holtz and E. P. Muntz, “Molecular velocity distribution functions in an argon normal shock wave at Mach number 7,” Phys. Fluids 26, 2425–2436 (1983).

    Article  Google Scholar 

  13. H. Alsmeyer, “Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam,” J. Fluid Mech. 74, 497–513 (1976).

    Article  Google Scholar 

  14. F. Robben and L. Talbot, “Experimental study of the rotational distribution function of nitrogen in a shock wave,” Phys. Fluids 9, 653–662 (1966).

    Article  Google Scholar 

  15. T. G. Elizarova, I. A. Shirokov, and S. Montero, “Numerical simulation of shock-wave structure for argon and helium,” Phys. Fluids 17, 068101 (2005).

  16. T. G. Elizarova, A. A. Khokhlov, and S. Montero, “Numerical simulation of shock wave structure in nitrogen,” Phys. Fluids 19, 068102 (2007).

  17. Yu. A. Nikitchenko, S. A. Popov, and A. V. Tikhonovets, “Combined kinetic-hydrodynamic model of polyatomic gas flow,” Math. Models Comput. Simul. 11, 740–749 (2019).

    Article  MathSciNet  Google Scholar 

  18. Yu. A. Nikitchenko and A. V. Tikhonovets, “Testing the kinetic-hydrodynamic model by calculating the flow above an absorbing surface,” Math. Models Comput. Simul. 13, 426–436 (2021).

    Article  MathSciNet  Google Scholar 

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ACKNOWLEDGMENTS

I am grateful to A.V. Tikhonovets for her help in writing this paper.

Funding

This work was supported by the Ministry for Science and Education of Russian Federation, project no. FSFF-2020-0013.

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Authors and Affiliations

  1. Moscow Aviation Institute (National Research University), 125993, Moscow, Russia

    Yu. A. Nikitchenko

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  1. Yu. A. Nikitchenko
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Correspondence to Yu. A. Nikitchenko.

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Translated by A. Klimontovich

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Nikitchenko, Y.A. A Version of Closing the System of Moment Equations of an Arbitrary Order. Comput. Math. and Math. Phys. 62, 487–507 (2022). https://doi.org/10.1134/S0965542522030125

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  • DOI: https://doi.org/10.1134/S0965542522030125

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