Skip to main content
SpringerLink
Log in

Further solutions for laminar boundary layers with cross flows driven by boundary motion

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

Laminar boundary layers with cross flows driven by transverse plate motions are considered. In each case, the cross flows are generated either by uniform plate motion or by transverse uniform shear. The three streamwise laminar boundary layers studied are the two-dimensional wall jet, the Blasius boundary layer flow with streamwise uniform plate motion, and uniform shear flow over a plate with streamwise stretching plate motion. In each case, a coordinate decomposition reduces the governing equations to a primary ordinary differential equation for the streamwise flow and a secondary linear equation coupled to the primary solution describing the fully developed cross flow. The one-parameter family of Blasius and uniform shear flow solutions, dependent on the streamwise wall motion parameter \(\lambda \), have dual solutions, and previous studies have shown that the upper branches of these dual solutions are stable, while the lower branches are unstable. This limits the range of admissible solutions for the new cross flows studied here.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Prandtl, L.: Über Flüssigkeitsbewegung bei sehr kleiner Reibung, Verh. III. Int. Math. Kongr. Heidelberg, pp. 484–491 (Teubner, Leipzig, 1946)

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

    MATH  Google Scholar 

  3. Prandtl, L.: On boundary layers in three-dimensional flow. Rep. Aero. Res. Coun. Lond. No. 9829 (1946)

  4. Cooke, J.C., Hall, M.G.: Boundary layers in three-dimensional. Prog. Aeron. Sci. 2, 221–282 (1962)

    Article  Google Scholar 

  5. Eichelbrenner, E.A.: Three-dimensional Boudary Layers, Ann. Rev. Fluid Mech. 5, eds. M. Van Dyke, W. G. Vincinti and J. V. Wehausen (Annual Reviews Inc., Palo Alto, 1973)

  6. Jones, R.T.: Effects of sweepback on boundary-layer separation. Rep. Nat. Adv. Comm. Aer., Washington, No. 884 (1947)

  7. Klemp, J.B., Acrivos, A.A.: A method for integrating the boundary-layer equations through a region of reverse flow. J. Fluid Mech. 53, 177–199 (1972)

    Article  MATH  Google Scholar 

  8. Klemp, J.B., Acrivos, A.A.: A moving-wall boundary layer with reverse flow. J. Fluid Mech. 76, 363–381 (1976)

    Article  MATH  Google Scholar 

  9. Hussaini, M.Y., Lakin, W.D., Nachman, A.: On similarity solutions of a boundary layer problem with an upstream moving wall. SIAM J. Appl. Math. 47, 699–709 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  10. Weidman, P.D.: New solutions for laminar boundary layers with cross flow. Z. Math. Phys. 48, 341–356 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Glauert, M.B.: The wall jet. J. Fluid Mech. 1, 625–643 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  12. Weidman, P.D., Kubitschek, D.G., Davis, A.M.J.: The effect of transpiration on self-similar boundary layer flow over moving surfaces. Int. J. Eng. Sci. 44, 730–737 (2006)

    Article  MATH  Google Scholar 

  13. Weidman, P.D., Davis, A.M.J., Kubitschek, D.G.: Crocco variable formulation for uniform shear flow over a stretching surface with transpiration: multiple solutions and stability. Z. Math. Phys. 58, 313–332 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Weidman, P.D., Turner, M.R.: Stagnation-point flow with stretching surfaces: a unified formulation and new results. Eur. J. Mech. B/Fluids 61, 144–153 (2017)

    Article  MathSciNet  Google Scholar 

  15. Hiemenz, K.: Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder. Dinglers Poly. J. 326, 321–410 (1911)

    Google Scholar 

  16. Homann, F.Der: Einfluss grosser Zähigkeit bei der Strömung um den Zylinder und um die Kugel. Z. Angew. Math. Mech. 16, 153–164 (1936)

    Article  MATH  Google Scholar 

  17. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  18. Tetervin, N.: Laminar flow of a slightly viscous incompressible fluid that issues from a slip and passes over a flat plate. NACA TN 1644, Washington DC, p. 40 (1948)

  19. Akatnov, N.I.: Development of 2D laminar jet along a solid surface. Leningrad Politek. Inst. Trudy 5, 4–31 (1953)

    Google Scholar 

  20. Fang, T., Lee, C.F.: Three-dimensional wall bounded laminar boundary layer with span-wise cross free stream and moving boundary. Acta Mech. 204, 235–248 (2009)

    Article  MATH  Google Scholar 

  21. Merkin, J.H.: On dual solutions occurring in mixed convection in a porous medium. J. Eng. Math. 20, 171–179 (1985)

    Article  MATH  Google Scholar 

  22. Rosenhead, L. (ed.): Laminar Boundary Layers. Oxford University Press, Oxford (1963)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of Colorado, Boulder, CO, 80309-0427, USA

    Patrick Weidman

Authors
  1. Patrick Weidman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Patrick Weidman.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Weidman, P. Further solutions for laminar boundary layers with cross flows driven by boundary motion. Acta Mech 228, 1979–1991 (2017). https://doi.org/10.1007/s00707-017-1810-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-017-1810-y

Search

Navigation