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

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.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    H. Schlichting, Boundary-Layer Theory, McGraw-Hill, New York (1968).

    Google Scholar 

  2. 2.

    M. Lesieur, Turbulence in Fluids, Springer (2008).

  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.

    Google 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).

    Article  Google 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).

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    J. M. Robertson, “On turbulent plane Couette flow,” Proc. 6th Midwestern Conf. Fluid Mech., Univ. Texas, Austin (1959), pp. 169–182.

  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).

  8. 8.

    J. A. Clark, “A study of incompressible turbulent boundary layers in channel flow,” J. Basic Engineering, 90, 455 (1968).

    Article  Google 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).

    Article  Google 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).

    Article  Google 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).

    Article  Google 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).

    Article  Google 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).

    Article  MATH  Google 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).

  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.

  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).

  17. 17.

    B. N. Chetverushkin, Kinetic Schemes and Quasi-Gas Dynamic System of Equations, CIMNE, Barcelona (2008).

    Google Scholar 

  18. 18.

    Yu. V. Sheretov, Continuum Dynamics under Spatiotemporal Averaging. SPC Regular and Chaotic Dynamics [in Russian], Moscow-Izhevsk (2009).

  19. 19.

    T. G. Elizarova, Quasi-Gas Dynamic Equations, Springer, Dordrecht (2009).

    Google 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).

  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).

    MathSciNet  Article  Google 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).

    MathSciNet  Google 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.

    Google 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).

    Article  MATH  Google 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).

    MathSciNet  Google Scholar 

  27. 27.

    K-100 System, Keldysh Institute of Applied Mathematics RAS, Moscow; Available at http://www.kiam.ru/MVS/resourses/k100.htm.

  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).

    Article  MATH  Google Scholar 

  29. 29.

    J. Jeong and F. Hussain, “On the identification of a vortex,” J. Fluid Mech., 285, 69–94 (1995).

    MathSciNet  Article  MATH  Google 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).

    MathSciNet  Article  Google 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).

  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.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to I. A. Shirokov.

Additional information

Translated from Prikladnaya Matematika i Informatika, No. 51, 2016, pp. 52–80.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Shirokov, I.A., Elizarova, T.G. Application of Quasi-Gas Dynamic Equations to Numerical Simulation of Near-Wall Turbulent Flows. Comput Math Model 28, 37–59 (2017). https://doi.org/10.1007/s10598-016-9344-z

Download citation

Keywords

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