Computational Mathematics and Modeling

, Volume 28, Issue 1, pp 37–59 | Cite as

Application of Quasi-Gas Dynamic Equations to Numerical Simulation of Near-Wall Turbulent Flows

  • I. A. ShirokovEmail author
  • T. G. Elizarova

We describe the possibilities of a numerical method based on quasi-gas dynamic (QGD) equations for the numerical simulation of a turbulent boundary layer. A subsonic Couette flow in nitrogen is used as an example, with dynamic Reynolds numbers of 153 and 198. The QGD system differs from the system of Navier–Stokes equations by additional nonlinear dissipative terms with a small parameter as a coefficient. In turbulent flow simulation, these terms describe small-scale effects that are not resolved on the grid. Comparison of our results (velocity profiles and mean-square velocity pulsations) with direct numerical simulation (DNS) results and benchmark experiments show that the QGD algorithm adequately describes the viscous and the logarithmic layers near the wall. Compared with high-accuracy DNS methods, the QGD algorithm permits using a relatively large spatial grid increment in the interior part of the viscous sublayer. Thus, the total number of grid points in the turbulent boundary layer may be relatively small. Unlike various versions of the large-eddy simulation (LES) method, the QGD algorithm does not require introduction of near-wall functions, because the additional terms vanish near the wall. For small Reynolds numbers, the QGD algorithm describes laminar Couette flow.


turbulent boundary layer quasi-gas dynamic equations subsonic Couette flow subgrid dissipation 


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  1. 1.
    H. Schlichting, Boundary-Layer Theory, McGraw-Hill, New York (1968).zbMATHGoogle Scholar
  2. 2.
    M. Lesieur, Turbulence in Fluids, Springer (2008).Google Scholar
  3. 3.
    P. Sagaut, “Theoretical background: large-eddy simulation,” in: C. Wagner, T. Huttl, and P. Sagaut, editors, Large-Eddy Simulation for Acoustics, Cambridge University Press, Cambridge (2007), pp. 89–127.CrossRefGoogle Scholar
  4. 4.
    S. Pirozzoli, M. Bernardini, and P. Orlandi, “Turbulence statistics in Couette flow at high Reynolds number,” J. Fluid Mech., 758, 327–343 (2014).CrossRefGoogle Scholar
  5. 5.
    K. N. Volkov, “Formulation of wall boundary conditions in turbulent flow computations on unstructured meshes,” Comput. Math. Math. Phys., 54, No. 2, 353–367 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    J. M. Robertson, “On turbulent plane Couette flow,” Proc. 6th Midwestern Conf. Fluid Mech., Univ. Texas, Austin (1959), pp. 169–182.Google Scholar
  7. 7.
    M. M. M. El Telbany and A. J. Reynolds, “Velocity distributions in plane turbulent channel flows,” J. Fluid Mech., 100 (part 1), 1–29 (1980).Google Scholar
  8. 8.
    J. A. Clark, “A study of incompressible turbulent boundary layers in channel flow,” J. Basic Engineering, 90, 455 (1968).CrossRefGoogle Scholar
  9. 9.
    A. K. M. F. Hussain and W. C. Reynolds, “Measurements in fully developed turbulent channel flow,” J. Fluids Engineering, 97, 568–578 (1975).CrossRefGoogle Scholar
  10. 10.
    M. M. M. El Telbany and A. J. Reynolds, “The structure of turbulent plane Couette flow,” J. Fluids Engineering, 104, 367–372 (1982).CrossRefGoogle Scholar
  11. 11.
    B. L. Rozhdestvenskii, I. N. Simakin, and M. I. Stoinov, “Modeling turbulent Couette flow in a plane channel,” J. Appl. Mech. Tech. Phys., 30, No. 2, 223–229 (1989).CrossRefGoogle Scholar
  12. 12.
    K. Bech, N. Tillmark, P. Alfredsson, and H. Andersson, “An investigation of turbulent plane Couette flow at low Reynolds numbers,” J. Fluid Mech., 286, 291–325 (1995).CrossRefGoogle Scholar
  13. 13.
    O. Kitoh, K. Nakabyashi, and F. Nishimura, “Experimental study on mean velocity and turbulence characteristics of plane Couette flow: low-Reynolds-number effects and large longitudinal vortical structure,” J. Fluid Mech., 539, 199–227 (2005).CrossRefzbMATHGoogle Scholar
  14. 14.
    T. Tsukahara, H. Kawamura, and K. Shingai, “DNS of turbulent Couette flow with emphasis on the large-scale structure in the core region,” J. Turbulence, 7, No. 19 (2006).Google Scholar
  15. 15.
    P. A. Skovorodko, “Slip effects in compressible turbulent channel flow,” 28th International Symposium on Rarefied Gas Dynamics, 2012, AIP Conf. Proc. 1501 (2012), pp. 457–464.Google Scholar
  16. 16.
    V. Avsarkisov, S. Hoyas, M. Oberlack, and J. P. Garcia-Galache, “Turbulent plane Couette flow at moderately high Reynolds number,” J. Fluid Mech., 751, No. R1, (2014).Google Scholar
  17. 17.
    B. N. Chetverushkin, Kinetic Schemes and Quasi-Gas Dynamic System of Equations, CIMNE, Barcelona (2008).zbMATHGoogle Scholar
  18. 18.
    Yu. V. Sheretov, Continuum Dynamics under Spatiotemporal Averaging. SPC Regular and Chaotic Dynamics [in Russian], Moscow-Izhevsk (2009).Google Scholar
  19. 19.
    T. G. Elizarova, Quasi-Gas Dynamic Equations, Springer, Dordrecht (2009).CrossRefzbMATHGoogle Scholar
  20. 20.
    M. V. Popov and T. G. Elizarova, “Smoothed MHD equations for numerical simulations of ideal quasi-neutral gas dynamic flows,” Computer Physics Communications, 196, No. 348–361 (2015).Google Scholar
  21. 21.
    I. A. Shirokov and T. G. Elizarova, “Simulation of laminar–turbulent transition in compressible Taylor–Green flow basing on quasigas dynamic equations,” J. Turbulence, 15, No. 10, 707–730 (2014).MathSciNetCrossRefGoogle Scholar
  22. 22.
    V. G. Priimak, “Direct numerical simulation of spatially localized structures and wave motions in turbulent shear flows: Numerical requirements,” Mat. Model., 20, No. 12, 27–43 (2008).MathSciNetGoogle Scholar
  23. 23.
    G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford (1998).Google Scholar
  24. 24.
    H. W. Liepmann and A. Roshko, Elements of Gasdynamics, California Institute of Technology, John Willey and Sons, New York, 1957.zbMATHGoogle Scholar
  25. 25.
    T. G. Elizarova and I. A. Shirokov, “Numerical simulation of a nonstationary flow in the vicinity of a hypersonic vehicle,” Math. Model. Comput. Simul., 4, No. 4, 410–418 (2012).CrossRefzbMATHGoogle Scholar
  26. 26.
    T. G. Elizarova and I. A. Shirokov, “Direct simulation of laminar-turbulent transition in a viscous compressible gas layer,” Comput. Math. Model., 25, No. 1, 27–48 (2012).MathSciNetGoogle Scholar
  27. 27.
    K-100 System, Keldysh Institute of Applied Mathematics RAS, Moscow; Available at
  28. 28.
    J. Kim, P. Moin, and R. Moser, “Turbulence statistics in fully developed channel flow at low Reynolds number,” J. Fluid Mech., 177, 133–166 (1987).CrossRefzbMATHGoogle Scholar
  29. 29.
    J. Jeong and F. Hussain, “On the identification of a vortex,” J. Fluid Mech., 285, 69–94 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    J. Fang, Y. Yao, Z. Li, and L. Lu, “Investigation of low-dissipation monotonicity-preserving scheme for direct numerical simulation of compressible turbulent flows,” Computers & Fluids, 104, 55–72 (2014).MathSciNetCrossRefGoogle Scholar
  31. 31.
    T. G. Elizarova and P. N. Nikolskii, “Numerical simulation of the laminar–turbulent transition in the flow over a backward-facing step,” Vestn. MGU, Ser. 3: Fiz. Astron., No. 4, 14–17 (2007).Google Scholar
  32. 32.
    T. G. Elizarova, P. N. Nikolskii, and J. C. Lengrand, “A new variant of subgrid dissipation for LES method and simulation of laminar-turbulent transition in subsonic gas flows,” in: Shia-Hui Peng and Werner Haase, eds., Advances in Hybrid RANS-LES Modeling, Springer-Verlag, Berlin, (2008), pp. 289–298.CrossRefGoogle Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Faculty of Computational Mathematics and CyberneticsMoscow State UniversityMoscowRussia
  2. 2.Keldysh Institute of Applied MathematicsRussian Academy of SciencesMoscowRussia

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