Nonlinear coherent structures in a square duct

  • Håkan WedinEmail author
  • Alessandro Bottaro
  • Masato Nagata
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 132)


The transition to turbulence in a square duct is an intriguing problem of hydrodynamics and has been studied since the work by Nikuradse [5]. The mean secondary flow of the turbulent state is made up by 8 vortices in the cross-sectional plane with 2 vortices in each corner, symmetric about the diagonals [2, 3, 5, 6, 7, 8]. The underlying mechanism causing this flow has been related to anisotropic turbulent fluctuations. Recently it has been shown through numerical simulations that the flow at transitional Reynolds numbers (\(Re_{b} = \widehat{U}_{b}\widehat{b}/\widehat{v}\), where \(\widehat{U}_{b}\) is the bulk speed, \(\widehat{v}\) the kinematic viscosity and \(\widehat{b}\) the half duct height) can feature instantaneous 4-vortex states (Biau & Bottaro [1] and Uhlmann et al [7]). The lower limit for transition in Re b is between 865 and 1077 [1, 7].


Direct Numerical Simulation Coherent Structure Travel Wave Solution Nonlinear Solution Constant Pressure Gradient 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    D. Biau and Alessandro Bottaro, “An optimal path to transition in a duct,” Phil. Trans. Roy. Soc. A, 367, 529, (2009).zbMATHCrossRefMathSciNetADSGoogle Scholar
  2. 2.
    A. Huser and S. Biringen, “Direct numerical simulation of turbulent flow in a square duct,” J. Fluid Mech. 257, 65 (1993).zbMATHCrossRefADSGoogle Scholar
  3. 3.
    F.B. Gessner, “The origin of secondary flow in turbulent flow along a corner,” J. Fluid Mech. 58, 1 (1973).CrossRefADSGoogle Scholar
  4. 4.
    T. Tatsumi and T. Yoshimura, “Stability of the laminar flow in a rectangular duct,” J. Fluid Mech. 212, 437 (1990).zbMATHCrossRefADSGoogle Scholar
  5. 5.
    J. Nikuradse, “Untersuchungen uber die Geschwindigkeitsverteilung in turbulenten Stromungen,” PhD Thesis, Göttingen, 1926.Google Scholar
  6. 6.
    S. Gavrilakis, “Numerical simulation of low-Reynolds-number turbulent flow through a straight square duct,” J. Fluid Mech., 244, 101 (1992).CrossRefADSGoogle Scholar
  7. 7.
    M. Uhlmann, A. Pinelli, G. Kawahara and A. Sekimoto, “Marginally turbulent flow in a square duct,” J. Fluid Mech., 588, 153 (2007).zbMATHCrossRefADSGoogle Scholar
  8. 8.
    D. Biau, H. Soueid and A. Bottaro, “Transition to turbulence in duct flow,” J. Fluid Mech., 596, 133 (2008).zbMATHCrossRefADSGoogle Scholar
  9. 9.
    B. Hof, C.W.H. van Doorne, J. Westerweel, F.T.M. Nieuwstadt, H. Faisst, B. Eckhardt, H. Wedin, R.R. Kerswell and F. Waleffe, “Experimental observation of nonlinear travelling waves in turbulent pipe flow,” Science, 305, 1594 (2004).CrossRefADSGoogle Scholar
  10. 10.
    H. Wedin, D. Biau A. Bottaro and M. Nagata, “Coherent Flow States in a Square Duct,” Phys. Fluids., 20, 094105 (2008).CrossRefADSGoogle Scholar
  11. 11.
    H. Wedin and R.R. Kerswell, “Exact coherent structures in pipe flow: travelling wave solutions,” J. Fluid Mech., 508, 333 (2004).zbMATHCrossRefMathSciNetADSGoogle Scholar
  12. 12.
    F. Waleffe, “Homotopy of exact coherent structures in plane shear flows,” Phys. Fluids., 15, 1517 (2003).CrossRefMathSciNetADSGoogle Scholar
  13. 13.
    H. Faisst and B. Eckhardt, “Travelling waves in pipe flow,” Phys. Rev. Lett., 91, 224502 (2003).CrossRefADSGoogle Scholar
  14. 14.
    M. Nagata, “Three-dimensional finite amplitude solutions in plane Couette flow: bifurcation from infinity,” J. Fluid Mech., 217, 519 (1990).CrossRefMathSciNetADSGoogle Scholar
  15. 15.
    Y. Duguet, A.P. Willis and R.R. Kerswell, “Transition in pipe flow: the saddle structure on the boundary of turbulence,” J. Fluid Mech., 613, 255 (2008).zbMATHCrossRefMathSciNetADSGoogle Scholar
  16. 16.
    C.C.T. Pringle, Y. Duguet and R.R. Kerswell, “Highly symmetric travelling waves in pipe flow,” Phil. Trans. Roy. Soc. A, 367, 457 (2009).zbMATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    C. Pringle and R.R. Kerswell, “Asymmetric, helical and mirror-symmetric travelling waves in pipe flow,” Phys. Rev. Lett., 99, 074502 (2007).CrossRefADSGoogle Scholar
  18. 18.
    Y. Duguet, C.C.T. Pringle and R.R. Kerswell, “Relative periodic orbits in transitional pipe flow,” Phys. Fluids, 20, 114102 (2008).CrossRefADSGoogle Scholar
  19. 19.
    C.C.T. Pringle, Y. Duguet and R.R. Kerswell, “Highly symmetric travelling waves in pipe flow,” Phil. Trans. Roy. Soc. A, 367, 457 (2009).zbMATHCrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Håkan Wedin
    • 1
    Email author
  • Alessandro Bottaro
    • 1
  • Masato Nagata
    • 2
  1. 1.DICATUniversity of GenovaGenovaItaly
  2. 2.Graduate School of EngineeringKyoto UniversitySakyo-kuJapan

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