Semi-Lagrangian Approximation of Conservation Laws of Gas Flow in a Channel with Backward Step

  • Vladimir V. ShaydurovEmail author
  • Maksim V. Yakubovich
Conference paper
Part of the Smart Innovation, Systems and Technologies book series (SIST, volume 133)


In the chapter, the numerical modeling of a supersonic flow of a viscous heat-conducting gas in a flat channel with the backward step is considered. A numerical algorithm is proposed for the initial boundary-value problem for the Navier-Stokes equations. These equations are modified and supplemented by new boundary conditions to provide the conservation law for the full energy: kinetic and inner. Then the combination of the Lagrangian approximation for the transfer operators and the conforming finite element method for other terms provides an efficient algorithm. Particular attention has been given to the approximation providing the conservation laws for mass and full energy at discrete level. Test calculations have been performed for a wide range of Mach and Reynolds numbers.


Viscous heat-conducting gas Navier-Stokes equations Lagrangian approximations of transfer operator Conservation laws Finite element method Gas flow in a flat channel 



The work was supported by Russian Foundation for Basic Research to project No. 17-01-000270 and by Russian Foundation for Basic Research, Government of Krasnoyarsk Territory, Krasnoyarsk Region Science and Technology Support Fund to the research project No. 18-41-243006.


  1. 1.
    Shaydurov, V., Shchepanovskaya, G., Yakubovich, M.: A semi-Lagrangian approximation in the Navier-Stokes equations for the gas flow around a wedge. AIP Conf. Proc. 1684, 090011 (2015)CrossRefGoogle Scholar
  2. 2.
    Johnson, R.W. (ed.): The handbook of fluid dynamics. CRC Press LLC & Springer, New York (1998)Google Scholar
  3. 3.
    Lodato, G., Domingo, P., Vervisch, L.: Three-dimensional boundary conditions for direct and large-eddy simulation of compressible viscous flows. J. Comput. Phys. 227(1), 5105–5143 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Farbar, E., Boyd, I.D.: Subsonic flow boundary conditions for the direct simulation Monte Carlo method. Comput. Fluids 102, 99–110 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bosnyakov, S., Kursakov, I., Lysenkov, A., Matyash, S., Mikhailov, S., Vlasenko, V., Quest, J.: Computational tools for supporting the testing of civil aircraft configurations in wind tunnels. Prog. Aerosp. Sci. 44(2), 67–120 (2008)CrossRefGoogle Scholar
  6. 6.
    Shaydurov, V., Shchepanovskaya, G., Yakubovich, M.: A semi-Lagrangian approach in the finite element method for the Navier-Stokes equations of viscous heat-conducting gas. AIP Conf. Proc. 1629, 19 (2014)CrossRefGoogle Scholar
  7. 7.
    Timoshenko, V.I.: Supersonic flows of viscous gas. Naukova Dumka, Kiev (1987). (in Russian)Google Scholar
  8. 8.
    Anderson, D., Tannehill, J., Pletcher, R.: Computational fluid mechanics and heat transfer. Hemisphere Publishing Corporation, New York (1984)zbMATHGoogle Scholar
  9. 9.
    Rannacher, R.: Incompressible Viscous Flow. In: Stein, E., De Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of computational mechanics. Wiley, New York (2011)Google Scholar
  10. 10.
    Rannacher, R.: Methods for numerical flow simulation. University of Heidelberg, Germany (2007)zbMATHGoogle Scholar
  11. 11.
    Shaydurov, V., Shchepanovskaya, G., Yakubovich, M.: Numerical simulation of supersonic flows in a channel. Russian J. Numer. Anal. Math. Model. 27(6), 585–601 (2012)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Pironneau, O.: On the transport-diffusion algorithm and its applications to the Navier-Stokes equations. Numer. Math. 38(3), 309–332 (1982)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Shaydurov, V., Vyatkin, A., Kuchunova, E.: Semi-Lagrangian difference approximations with different stability requirements. Russ. J. Numer. Anal. Math. Model. 33(2), 123–135 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Marchuk, G.I., Agoshkov, V.I.: Introduction in the projection-grid methods. Nauka, Moscow (1981). (in Russian)zbMATHGoogle Scholar
  15. 15.
    Fletcher, K.: Numerical methods on the base of Galerkin method. Mir, Moscow (1988). (in Russian)Google Scholar
  16. 16.
    Voevodin, V.V., Kuznetsov, Yu.A: Matrices and computations. Nauka, Moscow (1984). (in Russian)zbMATHGoogle Scholar
  17. 17.
    Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. Academic Press, New York, London (1970)zbMATHGoogle Scholar

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Authors and Affiliations

  • Vladimir V. Shaydurov
    • 1
    • 2
    Email author
  • Maksim V. Yakubovich
    • 1
  1. 1.Institute of Computational Modeling, SB of the RASAkademgorodok, KrasnoyarskRussian Federation
  2. 2.Tianjin University of Finance and EconomicsHexi District, TianjinChina

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