Applied Mathematics and Mechanics

, Volume 36, Issue 8, pp 1057–1072 | Cite as

Optimal transient growth in turbulent pipe flow

  • Yang Song
  • Chunxiao Xu
  • Weixi HuangEmail author
  • Guixiang Cui


The optimal transient growth process of perturbations driven by the pressure gradient is studied in a turbulent pipe flow. A new computational method is proposed, based on the projection operators which project the governing equations onto the subspace spanned by the radial vorticity and radial velocity. The method is validated by comparing with the previous studies. Two peaks of the maximum transient growth amplification curve are found at different Reynolds numbers ranging from 20 000 to 250 000. The optimal flow structures are obtained and compared with the experiments and DNS results. The location of the outer peak is at the azimuthal wave number n = 1, while the location of the inner peak is varying with the Reynolds number. It is observed that the velocity streaks in the buffer layer with a spacing of 100δ v are the most amplified flow structures. Finally, we consider the optimal transient growth time and its dependence on the azimuthal wave length. It shows a self-similar behavior for perturbations of different scales in the optimal transient growth process.


turbulent pipe flow optimal transient growth coherent structure 

Chinese Library Classification


2010 Mathematics Subject Classification

76F02 76E05 76M22 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Kline, S. J., Reynolds, W. C., Schraub, F. A., and Runstadler, P. W. The structure of turbulent boundary layers. Journal of Fluid Mechanics, 30(4), 741–773 (1967)CrossRefGoogle Scholar
  2. [2]
    Jimenez, J. and Moser, R. What are we learning from simulating wall turbulence. Philosophical Transactions of the Royal Society A, 365, 715–732 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Nagata, M. Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. Journal of Fluid Mechanics, 217, 519–527 (1990)MathSciNetCrossRefGoogle Scholar
  4. [4]
    Waleffe, F. Homotopy of exact coherent structures in plane shear flows. Physics of Fluids, 15(6), 1517–1534 (2003)MathSciNetCrossRefGoogle Scholar
  5. [5]
    Waleffe, F. Exact coherent structures in channel flow. Journal of Fluid Mechanics, 435, 93–102 (2001)CrossRefzbMATHGoogle Scholar
  6. [6]
    Hamilton, J. M., Kim, J., and Waleffe, F. Regeneration mechanisms of near-wall turbulence structures. Journal of Fluid Mechanics, 287, 317–348 (1995)CrossRefzbMATHGoogle Scholar
  7. [7]
    Moffatt, H. K. The interaction of turbulence with strong wind shear. Proceedings of the URSIIUGG Colloquium on Atoms, Turbulence and Radio Wave Propation, Nauka, Moscow (1967)Google Scholar
  8. [8]
    Farrell, B. F. and Ioannou, P. J. Dynamics of streamwise rolls and streaks in turbulent wallbounded shear flow. Jounal of Fluid Mechanics, 708, 149–196 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Jimenez, J. and Hoyas, S. Turbulent fluctuations above the buffer layer of wall-bounded flow, Journal of Fluid Mechanics, 611, 215–236 (2008)CrossRefzbMATHGoogle Scholar
  10. [10]
    Marusic, I., McKeon, B. J., and Monkewitz, P. A. Wall-bounded turbulent flows at high Reynolds numbers: recent advances and key issues. Physics of Fluids, 22(6), 065103 (2010)CrossRefGoogle Scholar
  11. [11]
    Mathis, R., Hutchins, N., and Marusic, I. Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers. Jounal of Fluid Mechanics, 628, 311–337 (2009)CrossRefzbMATHGoogle Scholar
  12. [12]
    O’Sullivan, P. L. and Breuer, K. S. Transient growth in circular pipe flow, linear disturbances. Physics of Fluids, 6(11), 3643–3651 (1994)CrossRefzbMATHGoogle Scholar
  13. [13]
    Schmid, P. J. and Henningson, D. S. Stability and Transition in Shear Flows, Springer, New York (2001)CrossRefzbMATHGoogle Scholar
  14. [14]
    Butler, K. M. and Farrell, B. F. Optimal perturbations and streak spacing in wall-bounded turbulent shear flow. Physics of Fluids, 5(3), 774–777 (1993)CrossRefGoogle Scholar
  15. [15]
    Reynolds, W. C. and Tiederman, W. G. Stability of turbulent channel flow, with application to Malkuss theory. Jounal of Fluid Mechanics, 27(2), 253–272 (1967)CrossRefGoogle Scholar
  16. [16]
    del Alamo, J. C. and Jimenez, J. Linear energy amplification in turbulent channels. Jounal of Fluid Mechanics, 559, 205–213 (2006)CrossRefzbMATHGoogle Scholar
  17. [17]
    Pujals, G., Villalba, M. G., Cossu, C., and Depardon, S. A note on optimal transient growth in turbulent channel flows. Physics of Fluids, 21(1), 015109 (2009)Google Scholar
  18. [18]
    Cess, R. D. A Survey of the Literature on Heat Transfer in Turbulent Tube Flow, Westinghouse Research, Resarch Report No. 8-0529-R24 (1958)Google Scholar
  19. [19]
    Willis, P. A., Hwang, Y. Y., and Cossu, C. Optimally amplified large-scale streaks and drag reduction in turbulent pipe flow. Physical Review E, 8(23), 336–354 (2010)Google Scholar
  20. [20]
    Burridge, D. M. and Drazin, P. G. Comments on ‘Stability of Pipe Poiseuille Flow’. Physics of Fluids, 12, 264–265 (1969)CrossRefGoogle Scholar
  21. [21]
    Trefethen, L. Spectral Methods in Matlab, Tsinghua University Press, Beijing (2011)Google Scholar
  22. [22]
    Schmid, P. J. and Henningson, D. S. Optimal energy density growth in Hagen-Poiseuille flow. Journal of Fluid Mechanics, 277, 197–225 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Meseguer, A. and Mellibovsky, F. On a solenoidal Fourier-Chebyshev spectral method for stability analysis of the Hagen-Poiseuille flow. Applied Numerical Mathematics, 57, 920–938 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Zagarola, M. V. and Smits, A. J. Mean-flow scaling of turbulent pipe flow. Journal of Fluid Mechanics, 373, 33–79 (1998)CrossRefzbMATHGoogle Scholar
  25. [25]
    Deng, B. Q., Xu, C. X., Huang, W. X., and Cui, G. X., Effect of active control on optimal structures in wall turbulence. Science China Physics, Mechanics and Astronomy, 56, 290–297 (2013)CrossRefGoogle Scholar
  26. [26]
    Bailey, S. C. C. and Smits, A. J. Experimental investigation of the structure of large- and verylarge-scale motions in turbulent pipe flow. Journal of Fluid Mechanics, 651, 339–356 (2010)CrossRefzbMATHGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Yang Song
    • 1
  • Chunxiao Xu
    • 1
  • Weixi Huang
    • 1
    Email author
  • Guixiang Cui
    • 1
  1. 1.Key Laboratory of Applied Mechanics, Department of Engineering MechanicsTsinghua UniversityBeijingChina

Personalised recommendations