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Boundary Layer for the Navier-Stokes-alpha Model of Fluid Turbulence

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Abstract

We study boundary-layer turbulence using the Navier-Stokes-alpha model obtaining an extension of the Prandtl equations for the averaged flow in a turbulent boundary layer. In the case of a zero pressure gradient flow along a flat plate, we derive a nonlinear fifth-order ordinary differential equation, which is an extension of the Blasius equation. We study it analytically and prove the existence of a two-parameter family of solutions satisfying physical boundary conditions. Matching these parameters with the skin-friction coefficient and the Reynolds number based on momentum thickness, we get an agreement of the solutions with experimental data in the laminar and transitional boundary layers, as well as in the turbulent boundary layer for moderately large Reynolds numbers.

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References

  1. Barenblatt, G. I., Chorin, A. J., Prostokishin, V. M.: Self-similar intermediate structures in turbulent boundary layers at large Reynolds numbers. J. Fluid Mech. 410, 263–283 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Belhachmi, Z., Brighi, B., Taous, K.: On the concave solutions of the Blasius equation. Acta Math. Univ. Comenian. (N.S.) 69, 199–214 (2000)

    MathSciNet  MATH  Google Scholar 

  3. Blasius, H.: Grenzschichten in Flüssigkeiten mit Kleiner Reibung. Z. Math. u. Phys. 56, 1–37 (1908)

    Google Scholar 

  4. Cebeci, T., Cousteix, J.: Modeling and Computation of Boundary-Layer Flows. Horizons Publishing Inc., Long Beach, CA, 1999

  5. Chen, S., Foias, C., Holm, D. D., Olson, E., Titi, E. S., Wynne, S.: The Camassa-Holm equations as a closure model for turbulent channel and pipe flow. Phys. Rev. Lett. 81, 5338–5341 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, S., Foias, C., Holm, D. D., Olson, E., Titi, E. S., Wynne, S.: A connection between the Camassa-Holm equations and turbulent flows in pipes and channels. Phys. Fluids, 11, 2343–2353 (1999)

    Google Scholar 

  7. Chen, S., Foias, C., Holm, D. D., Olson, E., Titi, E. S., Wynne, S.: The Camassa-Holm equations and turbulence. Physica D 133, 49–65 (1999)

    Article  MathSciNet  Google Scholar 

  8. Cheskidov, A.: Turbulent boundary layer equations. C. R. Acad. Sci. Paris, Ser. I 334, 423–427 (2002)

    Google Scholar 

  9. Cheskidov, A., Holm, D. D., Olson, E., Titi, E. S.: On a Leray-α Model of Turbulence. Preprint

  10. Constantin, P.: Filtered viscous fluid equations. Computer and Mathematics with Applications 46, 537–546 (2003)

    Google Scholar 

  11. Coppel, W. A.: On a differential equation of boundary layer theory. Phil. Trans. Roy. Soc. London Ser. A 253, 101–136 (1960)

    MATH  Google Scholar 

  12. Domaradzki, J. A., Holm, D. D.: Navier Stokes-alpha model: LES equations with nonlinear dispersion, Special LES volume of ERCOFTAC Bulletin, Modern Simulations Strategies for turbulent flow. B. J. Geurts, editor, Edwards Publishing

  13. Foias, C., Holm, D. D., Titi, E. S.: The Navier-Stokes-alpha model of fluid turbulence. Physica D 152, 505–519 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Foias, C., Holm, D. D., Titi, E. S.: The three dimensional viscous Camassa–Holm equations, and their relation to the Navier–Stokes equations and turbulence theory. Journal of Dynamics and Differential Equations 14, 1–35 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Furuya, S.: Note on a boundary value problem. Comment. Math. Univ. St Paul 1, 81–83 (1953)

    MathSciNet  MATH  Google Scholar 

  16. Hartman, P.: Ordinary Differential Equations. John Wiley & Sons, Inc., New York, London, Sidney, 1964

  17. Holm, D. D., Marsden, J. E., Ratiu, T. S.: Euler-Poincaré Models of Ideal Fluids with Nonlinear Dispersion. Phys. Rev. Lett. 80, 4173–4176 (1998)

    Article  Google Scholar 

  18. Holm, D. D., Putkaradze, V., Weidman, P. D., Wingate, B. A.: Boundary effects on exact solutions of the Lagrangian-averaged Navier-Stokes-α equations. J. Stat Phys. 113, 841–854 (2003)

    Article  Google Scholar 

  19. Iglisch, R.: Elementarer Existenzbeweis für die Strömung in der laminaren Grenzschicht zur Potentialströmung U =u 1 x m mit m > 0 bei Absaugen und Ausblasen. Z. Angew. Math. Mech. 33, 143–147 (1953)

    MathSciNet  MATH  Google Scholar 

  20. Ishimura, N., Matsui, S.: On blowing-up solutions of the Blasius equation. Discrete Contin. Dyn. Syst. 9, 985–992 (2003)

    MathSciNet  MATH  Google Scholar 

  21. von Kármán, Th.: Mechanische Ähnlichkeit und Turbulenz. Nachr. Ges. Wiss. Göttingen, Math. Phys. Klasse, 58, (1930)

  22. Landau, L. D., Lifshitz, E. M.: Fluid Mechanics. Pergamon, New York, 1959

  23. Majda, A. J., Kramer, P. R.: Simplified models for turbulent diffusion: theory, numerical modeling, and physical phenomena. Physics Reports 314, 237–574 (1999)

    Article  MathSciNet  Google Scholar 

  24. Marsden, J. E., Shkoller, S.: Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-α) equations on bounded domains. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 359, 1449–1468 (2001)

    Google Scholar 

  25. Marsden, J. E., Shkoller, S.: The anisotropic Lagrangian averaged Euler and Navier-Stokes equations. Arch. Ration. Mech. Anal. 166, 27–46 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mohseni, K., Kosović, B., Shkoller, S., Marsden, J. E.: Numerical simulations of the Lagrangian averaged Navier-Stokes equations for homogeneous isotropic turbulence. Phys. Fluids. 15 524–544 (2003)

    Google Scholar 

  27. Prandtl, L.: Über die ausgebildete Turbulenz. ZAMM 5, 136–139 (1925)

    MATH  Google Scholar 

  28. Prandtl, L.: Über Flüssigkeitsbewegung bei sehr kleiner Reibung. Proc. III. Intern. Math. Congr., Heidelberg (1904). Reprinted in Vier Abhdlg. zur Hudro- u. Aerodynamik, Göttingen (1927)

  29. Putkaradze, V., Weidman, P.: Wakes and boundary layer flows in Navier-Stokes α equations. Physical Review E, 67, 036304 (2003)

    Google Scholar 

  30. Roach, P. E., Brierlay, D. H.: The influence of a turbulent free-stream on zero pressure gradient transitional boundary layer development including the condition test cases T3A and T3B. In Numerical Simulation of Unsteady Flows and Transition to Turbulence (eds O. Pironneau et al.), Cambridge University Press, Cambridge, 1989

  31. Ščerbina, G. V.: On a boundary problem for an equation of Blasius. Sov. Math., Dokl. 2, 1219–1222 (1961)

    Google Scholar 

  32. Schlichting, H.: Boundary Layer Theory. 7th ed., Mc Graw-Hill, 2000

  33. Tam, K. K.: An elementary proof for the Blasius equation. Z. Angew. Math. Mech. 51, 318–319 (1971)

    MATH  Google Scholar 

  34. Weyl, H.: On the differential equations of the simplest boundary-layer problems. Ann. Math. 43, 381–407 (1942)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Indiana University, Bloomington, IN 47405

    A. Cheskidov

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  1. A. Cheskidov
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Correspondence to A. Cheskidov.

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Communicated by P. J. Holmes

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Cheskidov, A. Boundary Layer for the Navier-Stokes-alpha Model of Fluid Turbulence. Arch. Rational Mech. Anal. 172, 333–362 (2004). https://doi.org/10.1007/s00205-004-0305-x

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