The Stagnation Point Structure of Wall-Turbulence and the Law of the Wall in Turbulent Channel Flow

Conference paper
Part of the ERCOFTAC Series book series (ERCO, volume 14)


DNS of turbulent channel flows propose the following picture. (a) The Taylor microscale λ(y) is proportional to s (y), the average distance between stagnation points of the fluctuating velocity field, i.e. λ(y)=B 1 s (y) where B 1 is constant, for δ νy<δ. (b) The number density of stagnation points n s varies with height as \(n_{s}=C_{s}y_{+}^{-1}/\delta_{\nu}^{3}\) with C s constant in the range δ νy<δ. (c) In that same range, the kinetic energy dissipation rate per unit mass, \(\epsilon=\frac{2}{3}E_{+}u_{\tau}^{3}/(\kappa_{s}y)\) where \(E_{+}=E/u_{\tau}^{2}\) is the normalised total kinetic energy per unit mass and \(\kappa_{s}=B_{1}^{2}/C_{s}\) is the stagnation point von Kármán coefficient. (d) For Re τ ≫1, large enough for the production to balance dissipation locally and for \(-\langle{uv}\rangle \sim u_{\tau}^{2}\) in the range δ νyδ, \(d\langle{u}\rangle /dy\simeq\frac{2}{3}E_{+}u_{\tau}/(\kappa_{s}y)\) in that same range. (e) The von Kármán coefficient κ is a meaningful and well-defined coefficient and the log-law holds only if E + is independent of y + and Re τ in that range, in which case κκ s . The universality of \(\kappa_{s}=B_{1}^{2}/C_{s}\) depends on the universality of the stagnation point structure of the turbulence via B 1 and C s , which are conceivably not universal.



We are grateful to Dr. Sylvain Laizet for providing the Navier–Stokes solver and to Halliburton for the financial support.


  1. 1.
    Barenblatt, G.I., Chorin, A.J., Prostokishin, V.M.: Proc. Natl. Acad. Sci. 94, 773–776 (1997) zbMATHCrossRefGoogle Scholar
  2. 2.
    Brouwers, J.J.H.: Phys. Fluids 19, 101702 (2007) CrossRefGoogle Scholar
  3. 3.
    George, W.K.: Phil. Trans. R. Soc. A 365, 789–806 (2007) zbMATHCrossRefGoogle Scholar
  4. 4.
    Goto, S., Vassilicos, J.C.: Phys. Fluids 21, 035104 (2009) CrossRefGoogle Scholar
  5. 5.
    Laizet, S., Lamballais, E.: J. Comp. Phys. 228(16), 5989–6015 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Mazellier, N., Vassilicos, J.C.: Phys. Fluids 20, 015101 (2008) CrossRefGoogle Scholar
  7. 7.
    McKeon, B. (ed.): Theme Issue: Scaling and Structure in High Reynolds Number Wall-Bounded Flows. Phil. Trans. R. Soc. A 365 (2007) Google Scholar
  8. 8.
    Moser, R.D., Kim, J., Mansour, N.N.: Phys. Fluids 11, 943 (1999) zbMATHCrossRefGoogle Scholar
  9. 9.
    Nagib, H.M., Chauhan, K.A.: Phys. Fluids 20, 101518 (2008) CrossRefGoogle Scholar
  10. 10.
    Min, T., Kang, S.M., Speyer, J.L., Kim, J.: J. Fluid Mech. 558, 309–318 (2006) zbMATHCrossRefGoogle Scholar
  11. 11.
    Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge (2000) zbMATHCrossRefGoogle Scholar
  12. 12.
    Salazar, J.P.L.C., Collins, L.R.: Annu. Rev. Fluid Mech. 41, 405–432 (2009) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Townsend, A.A.: The Structure of Turbulent Shear Flow. Cambridge University Press, Cambridge (1976) zbMATHGoogle Scholar
  14. 14.
    Xu, J., Dong, S., Maxey, M.R., Karniadakis, G.E.: J. Fluid Mech. 582, 79–101 (2007) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Institute for Mathematical Sciences & Department of AeronauticsImperial College LondonLondonUK

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