On the influence of wavy riblets on the slip behaviour of viscous fluids

  • Matthieu Bonnivard
  • Francisco J. Suárez-Grau
  • Giordano Tierra


In this work, we use the homogenization theory to investigate the capability of wavy riblet patterns to influence the behaviour of a viscous flow near a ribbed boundary. Starting from perfect slip conditions on the wall, we show that periodic oscillations of wavy riblets in the lateral direction may induce a friction effect in the direction of the flow, contrary to what happens with straight riblets. Finally, we illustrate this effect numerically by simulating riblet profiles that are widely used in experimental studies: the V-shape, U-shape, and blade riblets.

Mathematics Subject Classification

35B27 35Q35 74A55 


Viscous fluids Slip condition Rough boundary Straight riblets Wavy riblets Homogenization 


  1. 1.
    Allaire G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23, 1482–1518 (1992)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Amirat Y., Simon J.: Influence de la rugosit en hydrodynamique laminaire. C. R. Acad. Sci. Paris Sér. I 323, 313–318 (1996)MathSciNetGoogle Scholar
  3. 3.
    Arbogast T., Douglas J., Hornung U.: Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J. Math. Anal. 21, 823–836 (1990)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Brezzi F., Fortin M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)CrossRefMATHGoogle Scholar
  5. 5.
    Březina J.: Asymptotic properties of solutions to the equations of incompressible fluid mechanics. J. Math. Fluid Mech. 12(4), 536–553 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bucur D., Feireisl E., Nec̆asová N.: On the asymptotic limit of flows past a ribbed boundary. J. Math. Fluid Mech. 10(4), 554–568 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bucur D., Feireisl E., Nec̆asová N.: Boundary behaviour of viscous fluids: influence of wall roughness and friction-driven boundary conditions. Arch. Ration. Mech. Anal. 197, 117–138 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bucur D., Feireisl E., Nečasová S., Wolf J.: On the asymptotic limit of the Navier–Stokes system on domains with rough boundaries. J. Differ. Equ. 244, 2890–2908 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Casado-Díaz J.: Two-scale convergence for nonlinear Dirichlet problems in perforated domains. Proc. R. Soc. Edinb. 130(A), 249–276 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Casado-Díaz J., Fernández-Cara E., Simon J.: Why viscous fluids adhere to rugose walls: a mathematical explanation. J. Differ. Equ. 189, 526–537 (2003)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Casado-Díaz J., Luna-Laynez M., Martín-Gómez J.D.: An adaptation of the multi-scale methods for the analysis of very thin reticulated structures. C. R. Acad. Sci. Paris Sér. I. 332, 223–228 (2001)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Casado-Díaz J., Luna-Laynez M., Suárez-Grau F.J.: Asymptotic behaviour of a viscous fluid with slip boundary conditions on a slightly rough wall. Math. Models Methods Appl. Sci. 20, 121–156 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Cioranescu D., Damlamian A., Griso G.: Periodic unfolding and homogenization. C.R. Acad. Sci. Paris Sér. I. 335, 99–104 (2002)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Dean B., Bhushan B.: Shark-skin surfaces for fluid-drag reduction in turbulent flow: a review. Philos. Trans. R. Soc. A 368, 4775–4806 (2010)CrossRefGoogle Scholar
  15. 15.
    Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986). MR 851383 (88b:65129)Google Scholar
  16. 16.
    Hecht F.: New development in FreeFem++. J. Numer. Math. 20, 251–265 (2012)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ito, K., Kunisch, K.: Lagrange Multiplier Approach to Variational Problems and Applications. Advances in Design and Control, vol. 15. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008). MR 2441683 (2009g:49001)Google Scholar
  18. 18.
    Jaeger W., Mikelić A.: On the roughness-induced effective boundary conditions for an incompressible viscous flow. J. Differ. Equ. 170, 96–122 (2001)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kramer, F., Grueneberger, R., Wassen, E., Hage, W., Meyer, R.: Wavy riblets for turbulent drag reduction. In: AIAA 5th flow control conference, AIAA-2010-4583 (2010)Google Scholar
  20. 20.
    Mohammadi B., Pironneau O., Valentin F.: Rough boundaries and wall laws. Int. J. Numer. Methods Fluids 27(1–4), 169–177 (1998)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Nguetseng G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20, 608–623 (1989)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Viotti C., Quadrio M., Luchini P.: Streamwise oscillation of spanwise velocity at the wall of a channel for turbulent drag reduction. Phys. Fluids. 21, 115109 (2009)CrossRefMATHGoogle Scholar
  23. 23.
    Wassen, E., Grueneberger, R., Kramer, F., Hage, W., Meyer, R., Thiele, F.: Turbulent drag reduction by oscillating riblets. In: AIAA 4th Flow Control Conference. AIAA-2008-3771 (2008)Google Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Matthieu Bonnivard
    • 1
  • Francisco J. Suárez-Grau
    • 1
    • 2
  • Giordano Tierra
    • 3
  1. 1.Univ. Paris Diderot, Sorbonne Paris Cité, Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRSParisFrance
  2. 2.Dpto. Ecuaciones Diferenciales y Análisis NuméricoUniversidad de SevillaSevilleSpain
  3. 3.Mathematical Institute, Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic

Personalised recommendations