Applied Mathematics and Mechanics

, Volume 31, Issue 1, pp 97–108 | Cite as

Direct numerical simulation of flow in channel with time-dependent wall geometry

  • Ming-wei Ge (葛銘纬)
  • Chun-xiao Xu (许春晓)Email author
  • Gui-xiang Cui (崔桂香)


A numerical scheme is developed to extend the scope of the spectral method without solving the covariant and contravariant forms of the Navier-Stokes equations in the curvilinear coordinates. The primitive variables are represented by the Fourier series and the Chebyshev polynomials in the computational space. The time advancement is accomplished by a high-order time-splitting method, and a corresponding high-order pressure condition at the wall is introduced to reduce the splitting error. Compared with the previous pseudo-spectral scheme, in which the Navier-Stokes equations are solved in the covariant and contravariant forms, the present scheme reduces the computational cost and, at the same time, keeps the spectral accuracy. The scheme is tested in the simulations of the turbulent flow in a channel with a static streamwise wavy wall and the turbulent flow over a flexible wall undergoing the streamwise traveling wave motion. The turbulent flow over an oscillating dimple is studied with the present numerical scheme, and the periodic generation of the vortical structures is analyzed.

Key words

spectral method time-dependent wall geometry turbulent flow 

Chinese Library Classification


2000 Mathematics Subject Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Moin, P. and Mahesh, K. Direct numerical simulation: a tool in turbulence research. Annual Review of Fluid Mechanics 30, 539–578 (1998)CrossRefMathSciNetGoogle Scholar
  2. [2]
    Mani, R., Lagoudas, D. C., and Rediniotis, O. K. Active skin for turbulent drag reduction. Smart Materials & Structures 17(3), 035004 (2008)CrossRefGoogle Scholar
  3. [3]
    Gad-el-Hak, M. Compliant coatings for drag reduction. Progress in Aerospace Sciences 38(1), 77–99 (2002)CrossRefGoogle Scholar
  4. [4]
    Orszag, S. A. and Patterson, G. S. Numerical simulation of three-dimensional homogeneous isotropic turbulence. Physical Review Letters 28(2), 76–79 (1972)CrossRefGoogle Scholar
  5. [5]
    Kim, J., Moin, P., and Moser, R. Turbulence statistics in fully developed channel flow at low Reynolds number. Journal of Fluid Mechanics 177, 133–166 (1987)zbMATHCrossRefGoogle Scholar
  6. [6]
    Carlson, H. A., Berkooz, G., and Lumley, J. L. Direct numerical simulation of flow in a channel with complex time-dependent wall geometries: a pseudospectral method. Journal of Computational Physics 121(1), 155–175 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Luo, H. and Bewley, T. R. On the contravariant form of the Navier-Stokes equations in timedependent curvilinear coordinate systems. Journal of Computational Physics 199(1), 355–375 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Kang, S. and Choi, H. Active wall motions for skin-friction drag reduction. Physics of Fluids 12(12), 3301–3304 (2000)CrossRefGoogle Scholar
  9. [9]
    Shen, L., Zhang, X., Yue, D. K. P., and Triantafyllou, M. S. Turbulence flow over a flexible wall undergoing a streamwise traveling wave motion. Journal of Fluid Mechanics 484, 197–221 (2003)zbMATHCrossRefGoogle Scholar
  10. [10]
    Karniadakis, G. E., Isreali, M., and Orszag, S. A. High-order splitting methods for the incompressible Navier-Stokes equations. Journal of Computational Physics 97(2), 414–443 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Xu, C., Zhang, Z., den Toonder, J. M. J., and Nieuwstadt, F. T. M. Origin of high kurtosis levels in the viscous sublayer. Direct numerical simulation and experiment. Physics of Fluids 8(7), 1938–1944 (1996)CrossRefGoogle Scholar
  12. [12]
    Angelis, V. D., Lombardi, P., and Banerjee, S. Direct numerical simulation of turbulent flow over a wavy wall. Physics of Fluids 9(8), 2429–2442 (1997)CrossRefGoogle Scholar
  13. [13]
    Dukowicz, J. K. and Dvinsky, A. S. Approximate factorization as a high order splitting for the implicit incompressible flow equations. Journal of Computational Physics 102(2), 336–347 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Gresho, P. M. On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 1: theory. International Journal for Numerical Methods in Fluids 11(5), 587–620 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Marcus, P. S. Simulation of the Taylor-Couette flow. Part 1: numerical methods and comparison with experiments. Journal of Fluid Mechanics 146, 45–46 (1984)zbMATHCrossRefGoogle Scholar
  16. [16]
    Kleizer, L. and Schumman, U. Treatment of incompressibility and boundary conditions in 3-D numerical spectral simulations of plane channel flows. Proceedings of the Third GAMM-Conference on Numericsl Methods in Fluid Mechanics (ed. Hirschel, E. H.), Vieweg-Verlag, Braunschweig, 165–173 (1980)Google Scholar
  17. [17]
    Hudson, J. D., Dykhno, L., and Hanratty, T. J. Turbulence production in flow over a wavy wall. Experiments in Fluids 20(4), 257–265 (1996)CrossRefGoogle Scholar
  18. [18]
    Cherukat, P., Na, Y., Hanratty, T. J., and McLaughlin, J. B. Direct numerical simulation of a fully developed turbulent flow over a wavy wall. Theoretical and Computational Fluid Dynamics 11(2), 109–134 (1998)zbMATHCrossRefGoogle Scholar
  19. [19]
    Hunt, J. C. F., Wray, A. A., and Moin, P. Eddies, streams, and convergence zones in turbulent flows. Center for Turbulence Research Report, CTR-S88, 193–208 (1988)Google Scholar

Copyright information

© Shanghai University and Springer Berlin Heidelberg 2010

Authors and Affiliations

  • Ming-wei Ge (葛銘纬)
    • 1
  • Chun-xiao Xu (许春晓)
    • 1
    Email author
  • Gui-xiang Cui (崔桂香)
    • 1
  1. 1.School of AerospaceTsinghua UniversityBeijingP. R. China

Personalised recommendations