Advertisement

Exact Coherent Structures in Turbulent Shear Flows

  • Fabian Waleffe
Part of the Notes on Numerical Fluid Mechanics and Multidisciplinary Design book series (NNFM, volume 105)

Abstract

Exact coherent structures are three-dimensional, nonlinear traveling wave solutions of the Navier-Stokes equations. These solutions are typically unstable from onset, yet they capture the basic statistical and structural features of low Reynolds number turbulent shear flows remarkably well. These exact coherent structures have now been found in all canonical shear flows: plane Couette, Poiseuille and pipe flow. They are generic for shear flows and exist for both no-slip and stress boundary conditions. Their discovery opens up new avenues for turbulence research and forces a fundamental rethinking of the true nature of turbulence.

Keywords

Travel Wave Solution Unstable Manifold Stable Manifold Lower Branch Branch Solution 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Acarlar, M.S., Smith, C.R.: A study of hairpin vortices in a laminar boundary layer. J. Fluid Mech. 175, 1–41 and 45–83 (1987)Google Scholar
  2. 2.
    Benney, D.J.: The evolution of disturbances in shear flows at high Reynolds numbers. Stud. Appl. Math. 70, 1–19 (1984)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Busse, F.H.: The optimum theory of turbulence. In: Advances in Applied Mechanics (A79-47538 21-34), New York, vol. 18, pp. 77–121. Academic Press, London (1978)Google Scholar
  4. 4.
    Clever, R.M., Busse, F.H.: Three-dimensional convection in a horizontal fluid layer subjected to constant shear. J. Fluid Mech. 234, 511–527 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Clever, R.M., Busse, F.H.: Tertiary and quaternary solutions for plane Couette flow. J. Fluid Mech. 344, 137–153 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cvitanović, P., Artuso, R., Mainieri, R., Tanner, G., Vattay, G.: Chaos: Classical and Quantum. Niels Bohr Institute, Copenhagen (2005), ChaosBook.org Google Scholar
  7. 7.
    Drazin, P., Reid, W.H.: Hydrodynamic Stability. Cambridge University Press, Cambridge (1981)zbMATHGoogle Scholar
  8. 8.
    Faisst, H., Eckhardt, B.: Traveling waves in pipe flow. Phys. Rev. Lett. 91, 224502 (2003)CrossRefGoogle Scholar
  9. 9.
    Hamilton, J., Kim, J., Waleffe, F.: Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348 (1995)zbMATHCrossRefGoogle Scholar
  10. 10.
    Hof, B., Juel, A., Mullin, T.: Scaling of the turbulence transition threshold in a pipe. Phys. Rev. Lett. 91, 244502 (2003)CrossRefGoogle Scholar
  11. 11.
    Hof, B., van Doorne, C.W.H., Westerweel, J., Nieuwstadt, F.T.M., Faisst, H., Eckhardt, B., Wedin, H., Kerswell, R.R., Waleffe, F.: Experimental observation of nonlinear traveling waves in turbulent pipe flow. Science 305(5690), 1594–1598 (2004)CrossRefGoogle Scholar
  12. 12.
    Itano, T., Toh, S.: The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70, 703–716 (2001)CrossRefGoogle Scholar
  13. 13.
    Jimenez, J., Kawahara, G., Simens, M.P., Nagata, M.: Characterization of near-wall turbulence in terms of equilibrium and ‘bursting’ solutions. Phys. Fluids 17(015105), 16 p. (2005)Google Scholar
  14. 14.
    Kawahara, G., Kida, S.: Periodic motion embedded in Plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291–300 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kerswell, R.R.: Elliptical instability. Annual Review of Fluid Mechanics 34(1), 83–113 (2002)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Kline, S.J., Reynolds, W.C., Schraub, F.A., Rundstadler, P.W.: The structure of turbulent boundary layers. J. Fluid Mech. 30, 741–773 (1967)CrossRefGoogle Scholar
  17. 17.
    Manneville, P.: Spots and turbulent domains in a model of transitional plane Couette flow. Theoretical and Computational Fluid Dynamics 18, 169–181 (2004)zbMATHCrossRefGoogle Scholar
  18. 18.
    Moehlis, J., Faisst, H., Eckhardt, B.: A low-dimensional model for turbulent shear flows. New Journal of Physics 6, 56+17 (2004)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Moehlis, J., Faisst, H., Eckhardt, B.: Periodic orbits and chaotic sets in a low-dimensional model for shear flows. SIAM J. Applied Dynam. Systems 4, 352–376 (2004)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Nagata, M.: Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519–527 (1990)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Panton, R.L. (ed.): Self-Sustaining Mechanisms of Wall Bounded Turbulence. Computational Mechanics Publications, Southampton (1997)Google Scholar
  22. 22.
    Piomelli, U., Balaras, E.: Wall-layer models for large-eddy simulations. Annual Review of Fluid Mechanics 34(1), 349–374 (2002)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  24. 24.
    Robinson, S.K.: Coherent motions in the turbulent boundary layer. Annual Review of Fluid Mechanics 23(1), 601–639 (1991)CrossRefGoogle Scholar
  25. 25.
    Viswanath, D.: Recurrent motions within plane Couette turbulence. Journal of Fluid Mechanics 580, 339–358 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Waleffe, F.: Hydrodynamic stability and turbulence: Beyond transients to a self-sustaining process. Stud. Applied Math. 95, 319–343 (1995)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Waleffe, F.: Transition in shear flows: Nonlinear normality versus non-normal linearity. Phys. Fluids 7, 3060–3066 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Waleffe, F.: On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900 (1997)CrossRefGoogle Scholar
  29. 29.
    Waleffe, F.: Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81, 4140–4148 (1998)CrossRefGoogle Scholar
  30. 30.
    Waleffe, F.: Exact coherent structures in channel flow. J. Fluid Mech. 435, 93–102 (2001)zbMATHCrossRefGoogle Scholar
  31. 31.
    Waleffe, F.: Exact coherent structures and their instabilities: Toward a dynamical-system theory of shear turbulence. In: Kida, S. (ed.) Proceedings of the International Symposium on Dynamics and Statistics of Coherent Structures in Turbulence: Roles of Elementary Vortices, pp. 115–128. National Center of Sciences, Tokyo (2002)Google Scholar
  32. 32.
    Waleffe, F.: Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 1517–1543 (2003)CrossRefMathSciNetGoogle Scholar
  33. 33.
    Waleffe, F., Kim, J., Hamilton, J.: On the origin of streaks in turbulent shear flows. In: Durst, F., Friedrich, R., Launder, B.E., Schmidt, F.W., Schumann, U., Whitelaw, J.H. (eds.) Turbulent Shear Flows 8: selected papers from the Eighth International Symposium on Turbulent Shear Flows, Munich, Germany, September 9-11, 1993, pp. 37–49. Springer, Berlin (1993)Google Scholar
  34. 34.
    Waleffe, F., Wang, J.: Transition threshold and the self-sustaining process. In: Mullin, T., Kerswell, R.R. (eds.) Non-uniqueness of Solutions to the Navier-Stokes Equations and their Connection with Laminar-Turbulent Transition, pp. 85–106. Kluwer, Dordrecht (2005)Google Scholar
  35. 35.
    Wedin, H., Kerswell, R.R.: Exact coherent structures in pipe flow. J. Fluid Mech. 508, 333–371 (2004)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Fabian Waleffe
    • 1
  1. 1.Departments of Mathematics and Engineering PhysicsUniversity of WisconsinMadisonUSA

Personalised recommendations