Skip to main content
SpringerLink
Log in

Excitation of unsteady Görtler vortices by localized surface nonuniformities

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

Abstract

A combined theoretical and numerical analysis of an experiment devoted to the excitation of Görtler vortices by localized stationary or vibrating surface nonuniformities in a boundary layer over a concave surface is performed. A numerical model of generation of small-amplitude disturbances and their downstream propagation based on parabolic equations is developed. In the framework of this model, the optimal and the modal parts of excited disturbance are defined as solutions of initial-value problems with initial values being, respectively, the optimal disturbance and the leading local mode at the location of the source. It is shown that a representation of excited disturbance as a sum of the optimal part and a remainder makes it possible to describe its generation and downstream propagation, as well as to predict satisfactorily the corresponding receptivity coefficient. In contrast, the representation based on the modal part provides only coarse information about excitation and propagation of disturbance in the range of parameters under investigation. However, it is found that the receptivity coefficients estimated using the modal parts can be reinterpreted to preserve their practical significance. A corresponding procedure was developed. The theoretical and experimental receptivity coefficients are estimated and compared. It is found that the receptivity magnitudes grow significantly with the disturbance frequency. Variation of the span-wise scale of the nonuniformities affects weakly the receptivity characteristics at zero frequency. However, at high frequencies, the efficiency of excitation of Görtler vortices depends substantially on the span-wise scale.

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. Aihara, Y., Tomita, Y., Ito, A.: Generation, development and distortion of longitudinal vortices in boundary layers along concave and flat plates. In: Kozlov, V.V. (ed.) Laminar-Turbulent Transition, IUTAM Symposium, Novosibirsk, USSR 9–13 July 1984, pp. 447–454. Springer, Berlin (1985)

  2. Bake, S., Ivanov, A.V., Fernholz, H.H., Neemann, K., Kachanov, Y.S.: Receptivity of boundary layers to three-dimensional disturbances. Eur. J. Mech. B/Fluids 21, 29–48 (2002)

    Article  MATH  Google Scholar 

  3. Bassom, A.P., Hall, P.: Concerning the interaction of non-stationary crossflow vortices in a three-dimensional boundary layer. Q. J. Mech. Appl. Math. 44, 147–172 (1991)

    Article  MATH  Google Scholar 

  4. Benjamin, T.B.: Shearing flow over a wavy boundary. J. Fluid Mech. 6, 161–205 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bertolotti, F.P.: Vortex generation and wave-vortex interactionover a concave plate with roughness and suction. Report 93–101, ICASE (1993)

  6. Bippes, H., Görtler, H.: Dreidimensionale Störungen in der Grenzschicht an einer konkaven Wand. Acta Mech. 14, 251–267 (1972)

    Article  Google Scholar 

  7. Boiko, A.V., Dovgal, A.V., Grek, G.R., Kozlov, V.V.: Physics of Transitional Shear Flows. Springer, Berlin (2011)

    Google Scholar 

  8. Boiko, A.V., Ivanov, A.V., Kachanov, Y.S., Mischenko, D.A.: Quasi-steady and unsteady Goertler vortices on concave wall: experiment and theory. In: Palma, J.M.L.M., Lopes, A.S. (eds.) Advances in Turbulence XI: Proceedings of the 11th EUROMECH European Turbulence Conference, 25–28 June 2007 Porto, Portugal, pp. 173–175. Springer, Berlin (2007)

  9. Boiko, A.V., Ivanov, A.V., Kachanov, Y.S., Mischenko, D.A.: Steady and unsteady Görtler boundary-layer instability on concave wall. Eur. J. Mech. B/Fluids 29(2), 61–83 (2010)

    Article  MATH  Google Scholar 

  10. Boiko, A.V., Nechepurenko, Y.M.: Numerical spectral analysis of temporal stability of laminar duct flows with constant cross-sections. Comput. Math. Math. Phys. 48(10), 1699–1714 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bottaro, A., Luchini, P.: Görtler vortices: are they amenable to local eigenvalue analysis? Eur. J. Mech. B/Fluids 18(1), 47–65 (1999)

    Article  MATH  Google Scholar 

  12. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)

    MATH  Google Scholar 

  13. Choudhari, M., Duck, P.W.: Nonlinear excitation of inviscid stationary vortex instabilities in a boundary-layer flow. In: Duck, P.W., Hall, P. (eds.) IUTAM Symposium on Nonlinear Instability and Transition in Three-Dimensional Boundary Layers: Proceedings of the IUTAM Symposium held in Manchester, 17–20 July 1995, pp. 409–422. Springer (1996)

  14. Clauser, M., Clauser, F.: The effect of curvature on the transition from laminar to turbulent boundary layer. TN 613, NACA (1937)

  15. Crouch, J.D., Spalart, P.R.: A study of non-parallel and nonlinear effects on the localized receptivity of boundary layers. J. Fluid Mech. 290, 29–37 (1995)

    Article  MATH  Google Scholar 

  16. Denier, J.P., Hall, P., Seddougui, S.O.: On the receptivity problem for Görtler vortices: vortex motions induced by roughness. Philos. Trans. R. Soc. Lond. A 335, 51–85 (1991)

    Article  MATH  Google Scholar 

  17. Faris, W., Asrar, W., Omerbegovic, A.: Micropump modeling: current status and challenge. Aust. J. Basic Appl. Sci. 6(1), 134–142 (2012)

    Google Scholar 

  18. Finnis, M.V., Brown, A.: The linear growth of görtler vortices. Int. J. Heat Fluid Flow 18(04), 389–399 (1997)

    Article  Google Scholar 

  19. Floryan, J.M.: On the Görtler instability of boundary layers. J. Aerosp. Sci. 28, 235–271 (1991)

    MATH  Google Scholar 

  20. Floryan, J.M., Saric, W.S.: Stability of Görtler vortices in boundary layers. AIAA J. 20(3), 316–324 (1982)

    Article  MATH  Google Scholar 

  21. Gaponenko, V.R., Ivanov, A.V., Kachanov, Y.S.: Experimental study of 3D boundary-layer receptivity to surface vibrations. In: Duck, P.W., Hall, P. (eds.) IUTAM Symposium on Nonlinear Instability and Transition in Three-Dimensional Boundary Layers: Proceedings of the IUTAM Symposium held in Manchester, 17–20 July 1995, pp. 389–398. Kluwer, Dordrecht (1996)

  22. Gaponenko, V.R., Ivanov, A.V., Kachanov, Y.S., Crouch, J.D.: Swept-wing boundary-layer receptivity to surface nonuniformities. J. Fluid Mech. 461, 93–126 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gaurav, Shankar, V.: Stability of pressure-driven flow in a deformable neo-hookean channel. J. Fluid Mech. 659, 318–350 (2010)

  24. Görtler, H.: Inistabilität laminaren Grenzschichten an konkaven Wänden gegenüber gewissen dreidimensionalen Störungen. Z. Angew. Math. Mech. 21, 250–252 (1941)

    Article  MATH  Google Scholar 

  25. Gregory, N., Walker, W.S.: The effect on transition of isolated surface excrescences in the boundary-layer. R&M 2779, ARC (1956)

  26. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd edn. Springer, Berlin (2008)

    MATH  Google Scholar 

  27. Hall, P.: Taylor-Görtler vortices in fully developed or boundary-layer flows: linear theory. J. Fluid Mech. 124, 475–494 (1982)

    Article  MATH  Google Scholar 

  28. Hall, P.: The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41–58 (1983)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  30. Hall, P.: Görtler vortices in growing boundary layers: the leading edge receptivity problem, linear growth and the nonlinear breakdown stage. Mathematika 37, 151–189 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hämmerlin, G.: Über das Eigenwertproblem der dredimensionalen Instabilität laminarer Grenzschichten an konkaven Wänden. J. Ration. Mech. Anal. 4(2), 279–321 (1955)

    MATH  Google Scholar 

  32. Hämmerlin, G.: Zur Theorie der dreidimensionalen Instabilität laminarer Grenzschichten. Z. Angew. Math. Phys. 7, 156–164 (1956)

    Article  MATH  Google Scholar 

  33. Herron, I.H., Clark, A.D.: Instabilities in the Görtler model for wall bounded flows. Appl. Math. Lett. 13, 105–110 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. Hœpffner, J., Bottaro, A., Favier, J.: Mechanisms of non-modal energy amplification in channel flow between compliant walls. J. Fluid Mech. 642, 489–507 (2010)

    Article  MATH  Google Scholar 

  35. Ito, A.: Visualization of boundary layer transition along a concave wall. In: Proceedings of the 4th International Symposium on Flow Visualization (Paris), pp. 339–344. Hemisphere, Washington (1987)

  36. Ivanov, A.V., Kachanov, Y.S., Mischenko, D.A.: Excitation of unsteady Görtler vortices by surface non-uniformities. In: Euromech Fluid Mechanics Conference-8 (EFMC-8), 13–16 September. Abstracts, pp. S4–12. Bad-Reichenhall, Germany (2010)

  37. Ivanov, A.V., Kachanov, Y.S., Mischenko, D.A.: Boundary-layer receptivity to surface non-uniformities leading to generation of Görtler vortices. J. Phys. Conf. Ser. 318, 1–10 (2011)

    Google Scholar 

  38. Ivanov, A.V., Kachanov, Y.S., Mischenko, D.A.: Generation of nonstationary Görtler vortices by localized surface nonuniformities. Receptivity coefficients. Thermophys. Aeromech. 19(4), 523–539 (2012)

    Article  Google Scholar 

  39. Ivanov, A.V., Kachanov, Y.S., Obolentseva, T.G.: Experimental study of the blasius boundary layer receptivity to localized surface vibrations. Thermophys. Aeromech. 6(2), 179–192 (1999)

    Google Scholar 

  40. Ivanov, A.V., Kachanov, Y.S., Obolentseva, T.G., Michalke, A.: Receptivity of the Blasius boundary layer to surface vibrations. Comparison of theory and experiment. In: Kharitonov A.M. (ed.) Proceedings of Ninth International Conference on Methods of Aerophysical Research, vol. 1, pp. 93–98. RAS. Sib. Branch Inst. Theoret. Appl. Mech., Novosibirsk (1998)

  41. Kachanov, Y.S.: Three-dimensional receptivity of boundary layers. Eur. J. Mech. B/Fluids 19(5), 723–744 (2000)

    Article  MATH  Google Scholar 

  42. Kierzenka, J., Shampine, L.F.: A BVP solver based on residual control and the Matlab PSE. ACM TOMS 27(3), 299–316 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  43. Liepmann, H.W.: Investigations on laminar boundary layer stability and transition on curved boundaries. Wartime Report W-107, NACA (1943)

  44. Luchini, P.: Linearized no-slip boundary conditions at a rough surface. J. Fluid Mech. 737, 349–367 (2013)

    Article  MATH  Google Scholar 

  45. Luchini, P., Bottaro, A.: Görtler vortices: a backward in time approach to the receptivity problem. J. Fluid Mech. 363, 1–23 (1998)

    Article  MATH  Google Scholar 

  46. Murphy, J.S.: Extensions of the Falkner-Skan similar solutions to flows with surface curvature. AIAA J. 3(11), 2043–2049 (1965)

    Article  Google Scholar 

  47. Nechepurenko, Y.M.: On the dimension reduction of linear differential-algebraic control systems. Doklady Math. 86(1), 457–459 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  48. Saric, W.S.: Görtler vortices. Annu. Rev. Fluid Mech. 26, 379–409 (1994)

    Article  MATH  Google Scholar 

  49. Saric, W.S., Reed, H.L., Kerschen, E.J.: Boundary-layer receptivity to freestream disturbances. Annu. Rev. Fluid Mech. 34, 291–319 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  50. Saric, W.S., Reed, H.L., White, E.B.: Stability and transition of three-dimensional boundary layers. Annu. Rev. Fluid Mech. 35, 413–440 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  51. Schlichting, H., Gersten, K.: Boundary Layer Theory, 8th edn. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  52. Schmid, P.J., Henningson, D.S.: Stability and Transition in Shear Flows. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  53. Schultz, M.P., Volino, R.J.: Effects of concave curvature on boundary layer transition under high free-stream turbulence conditions. ASME J. Fluids Eng. 125, 18–27 (2003)

    Article  Google Scholar 

  54. Shampine, L.F., Gladwell, I., Thompson, S.: Solving ODEs with MATLAB. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  55. Smith, A.M.O.: On the growth of Taylor–Görtler vortices along highly concave walls. Q. Appl. Math. 13, 223 (1955)

    Article  MATH  Google Scholar 

  56. Spall, R.E., Malik, M.R.: Goertler vortices in supersonic boundary layers. Paper 88-3678, AIAA (1988)

  57. Tani, I.: Production of longitudinal vortices in the boundary-layer along a curved wall. J. Geophys. Res. 67, 3075–3080 (1962)

    Article  Google Scholar 

  58. Tannehill, J.C., Anderson, D.A., Pletcher, R.H.: Computational Fluid Mechanics and Heat Transfer, 3rd edn. Series in Computational and Physical Processes in Mechanics and Thermal Sciences. CRC Press, Boca Raton (2012)

  59. Taylor, G.I.: Stability of a viscous liquid contained between two rotating cylinders. Proc. R. Soc. Lond. A 223, 289–343 (1923)

    MATH  Google Scholar 

  60. Tempelmann, D., Schrader, L.U., Hanifi, A., Brandt, L., Henningson, D.S.: Swept wing boundary-layer receptivity to localized surface roughness. J. Fluid Mech. 711, 516–544 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  61. Timoshenko, S.P., Woinowsky-Krieger, S.: Theory of Plates and Shells, Engineering Societies Monographs, 2nd edn. McGraw-Hill, New York (1959)

    MATH  Google Scholar 

  62. Tomita, Y., Funatsu, K., Tsuzuki, S., Miyazaki, K.: Neutral stability of laminar boundary layers along the concave wall. Bull. J. Jpn. Soc. Mech. Eng. 28, 2288–2293 (1985)

    Article  Google Scholar 

  63. Tumin, A., Reshotko, E.: Spatial theory of optimal disturbances in boundary layers. Phys. Fluids 13(7), 2097–2104 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  64. Weideman, J.A.C., Reddy, S.C.: A MATLAB differentiation matrix suite. ACM Trans. Math. Softw. 26(4), 465–519 (2000)

    Article  MathSciNet  Google Scholar 

  65. Wu, X.: Receptivity of boundary layers with distributed roughness to vortical and acoustic disturbances: a second-order asymptotic theory and comparison with experiments. J. Fluid Mech. 431, 91–133 (2001)

    Article  MATH  Google Scholar 

  66. 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  Google Scholar 

  67. Yeo, K.S.: The stability of boundary-layer flow over single-and multi-layer viscoelastic walls. J. Fluid Mech. 196, 359–408 (1988)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Khristianovich Institute of Theoretical and Applied Mechanics, Novosibirsk, Russia, 630090

    A. V. Boiko, A. V. Ivanov, Yu. S. Kachanov & D. A. Mischenko

  2. Tyumen State Oil and Gas University, Tyumen, Russia, 625000

    A. V. Boiko

  3. Institute of Numerical Mathematics, Moscow, Russia, 119333

    Yu. M. Nechepurenko

Authors
  1. A. V. Boiko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. V. Ivanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yu. S. Kachanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. D. A. Mischenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Yu. M. Nechepurenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Boiko.

Additional information

Communicated by Dr. M.R. Malik.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Boiko, A.V., Ivanov, A.V., Kachanov, Y.S. et al. Excitation of unsteady Görtler vortices by localized surface nonuniformities. Theor. Comput. Fluid Dyn. 31, 67–88 (2017). https://doi.org/10.1007/s00162-016-0404-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00162-016-0404-y

Keywords

Search

Navigation