Skip to main content
SpringerLink
Log in

Three-dimensional instability analysis of boundary layers perturbed by streamwise vortices

  • Original Article
  • Published:
Theoretical and Computational Fluid Dynamics Aims and scope Submit manuscript

Abstract

A parametric study is presented for the incompressible, zero-pressure-gradient flat-plate boundary layer perturbed by streamwise vortices. The vortices are placed near the leading edge and model the vortices induced by miniature vortex generators (MVGs), which consist in a spanwise-periodic array of small winglet pairs. The introduction of MVGs has been experimentally proved to be a successful passive flow control strategy for delaying laminar-turbulent transition caused by Tollmien–Schlichting (TS) waves. The counter-rotating vortex pairs induce non-modal, transient growth that leads to a streaky boundary layer flow. The initial intensity of the vortices and their wall-normal distances to the plate wall are varied with the aim of finding the most effective location for streak generation and the effect on the instability characteristics of the perturbed flow. The study includes the solution of the three-dimensional, stationary, streaky boundary layer flows by using the boundary region equations, and the three-dimensional instability analysis of the resulting basic flows by using the plane-marching parabolized stability equations. Depending on the initial circulation and positioning of the vortices, planar TS waves are stabilized by the presence of the streaks, resulting in a reduction in the region of instability and shrink of the neutral stability curve. For a fixed maximum streak amplitude below the threshold for secondary instability (SI), the most effective wall-normal distance for the formation of the streaks is found to also offer the most stabilization of TS waves. By setting a maximum streak amplitude above the threshold for SI, sinuous shear layer modes become unstable, as well as another instability mode that is amplified in a narrow region near the vortex inlet position.

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.

Similar content being viewed by others

References

  1. Andersson, P., Berggren, M., Henningson, D.S.: Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11, 134–150 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Andersson, P., Brandt, L., Bottaro, A., Henningson, D.S.: On the breakdown of boundary layer streaks. J. Fluid Mech. 428, 29–60 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bagheri, S., Hanifi, A.: The stabilizing effect of streaks on Tollmien-Schlichting and oblique waves: A parametric study. Phys. Fluids 19(7), 078103 (2007)

  4. Bridges, T.J., Morris, P.J.: Differential eigenvalue problems in which the parameter appears nonlinearly. J. Comput. Phys. 55, 437–460 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen, C.S.: Numerical method for predicting three-dimensional steady compressible flow in long microchannels. J. Micromech. Microeng. 14, 1091–1100 (2004)

    Article  Google Scholar 

  6. Chen, C.S.: Reduced Navier–Stokes simulation of incompressible microchannel flows. Numer. Heat Transf. Part A 53, 71–87 (2008)

    Article  Google Scholar 

  7. Cossu, C., Brandt, L.: Stabilization of Tollmien–Schlichting waves by finite amplitude optimal streaks in the blasius boundary layer. Phys. Fluids 14(8), L57–L60 (2002)

    Article  MATH  Google Scholar 

  8. Cossu, C., Brandt, L.: On Tollmien–Schlichting-like waves in streaky boundary layers. Eur. J. Mech. B. Fluids 23, 815–833 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. De Tullio, N., Paredes, P., Sandham, N.D., Theofilis, V.: Roughness-induced instability and breakdown to turbulence in a supersonic boundary-layer. J. Fluid Mech. 735, 613–646 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Denissen, N.A., White, E.B.: Secondary instability of roughness-induced transient growth. Phys. Fluids 25(11), 114108 (2013)

    Article  Google Scholar 

  11. Fletcher, C.A.J.: Computational Techniques for Fluid Dynamics, vol. II. Springer Verlag, Berlin (1988)

    Book  MATH  Google Scholar 

  12. Fransson, J.H.M., Talamelli, A.: On the generation of steady streamwise streaks in flat-plate boundary layers. J. Fluid Mech. 698, 211–234 (2012)

    Article  MATH  Google Scholar 

  13. Fransson, J.H.M., Brandt, L., Talamelli, A., Cossu, C.: Experimental and theoretical investigation of the nonmodal growth of steady streaks in a flat plate boundary layer. Phys. Fluids 16(10), 3627–3638 (2004)

    Article  MATH  Google Scholar 

  14. Fransson, J.H.M., Brandt, L., Talamelli, A., Cossu, C.: Experimental study of the stabilization of Tollmien–Schlichting waves by finite amplitude streaks. Phys. Fluids 17(5), 054110 (2005)

    Article  MATH  Google Scholar 

  15. Fransson, J.H.M., Talamelli, A., Brandt, L., Cossu, C.: Delaying transition to turbulence by a passive mechanism. Phys. Rev. Lett. 96, 064501–064501 (2006)

    Article  Google Scholar 

  16. Goldstein, M.E., Sescu, A.: Boundary-layer transition at high free-stream disturbance levels-beyond Klebanoff modes. J. Fluid Mech. 613, 95–124 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Goldstein, M.E., Wundrow, D.W.: On the environmental realizability of algebraically growing disturabances and their relation to Klebanoff modes. Theor. Comput. Fluid Dyn. 10, 171–186 (1998)

    Article  MATH  Google Scholar 

  18. Goldstein, M.E., Sescu, A., Duck, P.W., Choudhari, M.: The long range persistence of wakes behind a row of roughness elements. J. Fluid Mech. 644, 123–163 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  19. Goldstein, M.E., Sescu, A., Duck, P.W., Choudhari, M.: Algebraic/transcendental disturbance growth behind a row of roughness elements. J. Fluid Mech. 668, 236–266 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hall, P.: The nonlinear development of Görtler vortices in growing boundary-layers. J. Fluid Mech. 193, 243–266 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hall, P., Smith, F.T.: On strongly nonlinear vortex wave interactions in boundary-layer-transition. J. Fluid Mech. 227, 641–666 (1991)

    Article  MATH  Google Scholar 

  22. Herbert, T.: Parabolized stability equations. Annu. Rev. Fluid Mech. 29, 245–283 (1997)

    Article  MathSciNet  Google Scholar 

  23. Kemp, N.: The laminar three-dimensional boundary layer and a study of the flow past a side edge. M.Ae.S. Thesis, Cornell University (1951)

  24. Klebanoff, P.S.: Effect of free-stream turbulence on a laminar boundary layer. Bull. Am. Phys. Soc. 10(11), 1323 (1971)

    Google Scholar 

  25. Leib, S.J., Wundrow, D.W., Goldstein, M.E.: Effect of free-stream turbulence and other vortical disturbances on a laminar boundary layer. J. Fluid Mech. 380, 169–203 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  26. Martin, J.A., Martel, C.: Nonlinear streak computation using boundary region equations. Fluid Dyn. Res. 44, 045503 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  27. Paredes, P.: Advances in global instability computations: from incompressible to hypersonic flow. Phd thesis, Technical University of Madrid (2014)

  28. Paredes, P., Hermanns, M., Le Clainche, S., Theofilis, V.: Order \(10^4\) speedup in global linear instability analysis using matrix formation. Comput. Meth. Appl. Mech. Eng. 253, 287–304 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  29. Paredes, P., Hanifi, A., Theofilis, V., Henningson, D.: The nonlinear PSE-3D concept for transition prediction in flows with a single slowly-varying spatial direction. Procedia IUTAM 14C, 35–44 (2015)

    Google Scholar 

  30. Ricco, P.: The pre-transitional Klebanoff modes and other boundary-layer disturbances induced by small-wavelength free stream vorticity. J. Fluid Mech. 638, 267–303 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  31. Ricco, P., Luo, J., Wu, X.: Evolution and instability of unsteady nonlinear streaks generated by free-stream vortical disturbances. J. Fluid Mech. 677, 1–38 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  32. Rubin, S.G., Tannehill, J.C.: Parabolized reduced Navier–Stokes computational techniques. Annu. Rev. Fluid Mech. 24, 117–144 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  33. Sattarzadeh, S.S., Fransson, J.H.M., Talamelli, A., Fallenius, B.E.G.: Consecutive turbulence transition delay with reinforced passive control. Phys. Rev. E 89(6), 061001 (2014)

  34. Schlatter, P., Brandt, L., de Lange, H.C., Henningson, D.S.: On streak breakdown in bypass transition. Phys. Fluids 20, 101505 (2008)

    Article  MATH  Google Scholar 

  35. Shahinfar, S., Sattarzadeh, S.S., Fransson, J.H.M., Talamelli, A.: Revival of classical vortex generators now for transition delay. Phys. Rev. Lett. 109(7), 074501 (2012)

  36. Shahinfar, S., Fransson, J.H.M., Sattarzadeh, S.S., Talamelli, A.: Scaling of streamwise boundary layer streaks and their ability to reduce skin-friction drag. J. Fluid Mech 733, 1–32 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  37. Shahinfar, S., Sattarzadeh, S.S., Fransson, J.H.M.: Passive boundary layer control of oblique disturbances by finite-amplitude streaks. J. Fluid Mech. 749, 1–36 (2014)

    Article  MATH  Google Scholar 

  38. Siconolfi, L., Camarri, S., Fransson, J.H.M.: Boundary layer stabilization using free-stream vortices. J. Fluid Mech. 764, R2 (2015a)

  39. Siconolfi, L., Camarri, S., Fransson, J.H.M.: Stability analysis of boundary layers controlled by miniature vortex generators. J. Fluid Mech. 784, 596–618 (2015b)

    Article  Google Scholar 

  40. Theofilis, V.: Spatial stability of incompressible attachment line flow. Theor. Comput. Fluid Dyn. 7, 159–171 (1995)

    Article  MATH  Google Scholar 

  41. Theofilis, V.: Advances in global linear instability of nonparallel and three-dimensional flows. Prog. Aero. Sci. 39(4), 249–315 (2003)

    Article  Google Scholar 

  42. Theofilis, V.: Global linear instability. Annu. Rev. Fluid Mech. 43, 319–352 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  43. Wu, X., Choudhari, M.: Linear and nonlinear instabilities of a blasius boundary layer perturbed by streamwise vortices. part 2. intermitent instability induced by long-wavelength klebanoff modes. J. Fluid Mech. 483, 249–286 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  44. Wu, X., Luo, J.: Linear and nonlinear instabilities of a blasius boundary layer perturbed by streamwise vortices. part 1. steady streaks. J. Fluid Mech. 483, 225–248 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  45. Wu, X., Zhao, D., Luo, J.: Excitation of steady and unsteady Görtler vortices by free-stream vortical disturbances. J. Fluid Mech. 682, 66–100 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  46. Wundrow, D.W., Goldstein, M.E.: Effect on a laminar boundary layer of small-amplitude streamwise vorticity in the upstream flow. J. Fluid Mech. 426, 229–262 (2001)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Aeronautics, Universidad Politécnica de Madrid, 28040, Madrid, Spain

    Juan A. Martín & Pedro Paredes

Authors
  1. Juan A. Martín
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pedro Paredes
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juan A. Martín.

Additional information

Communicated by Dr. Vassilios Theofilis.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Martín, J.A., Paredes, P. Three-dimensional instability analysis of boundary layers perturbed by streamwise vortices. Theor. Comput. Fluid Dyn. 31, 505–517 (2017). https://doi.org/10.1007/s00162-016-0403-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00162-016-0403-z

Keywords

Search

Navigation