Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-15T22:10:06.146Z Has data issue: false hasContentIssue false
  • English
  • Français

Discrete linear local eigenmodes in a separating laminar boundary layer

Published online by Cambridge University Press:  27 September 2012

Olaf Marxen,
Matthias Lang and
Ulrich Rist
Olaf Marxen*
Affiliation:
Institut für Aerodynamik und Gasdynamik, Universität Stuttgart, Pfaffenwaldring 21, D-70550 Stuttgart, Germany
Matthias Lang
Affiliation:
Institut für Aerodynamik und Gasdynamik, Universität Stuttgart, Pfaffenwaldring 21, D-70550 Stuttgart, Germany
Ulrich Rist
Affiliation:
Institut für Aerodynamik und Gasdynamik, Universität Stuttgart, Pfaffenwaldring 21, D-70550 Stuttgart, Germany
*
†Present address: Aeronautics and Aerospace Department, von Kármán Institute for Fluid Dynamics, Chaussée de Waterloo, 72, B-1640 Rhode-St-Genèse, Belgium. Email address for correspondence: olaf.marxen@vki.ac.be
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

The evolution of two- and three-dimensional small-amplitude disturbances in the laminar part of a laminar separation bubble is investigated in detail. We apply a combination of local linear stability theory, results from different experimental measurement campaigns and direct numerical simulations to identify two different discrete eigenmodes in the laminar part of the bubble. A stable eigenmode, the outer mode, governs unsteady oscillations in the upstream part of the bubble. However, this perturbation is quickly overtaken by an unstable eigenmode, the inner mode, which eventually leads to transition of the detached shear layer. Such a behaviour is observed due to an acceleration region with a favourable pressure gradient preceding the adverse-pressure-gradient region. The flow is stable in the acceleration region, in which the outer mode is only moderately damped, while the inner mode is strongly damped. At the onset of instability for the unstable eigenmode upstream of separation, both viscous Tollmien–Schlichting and inviscid Kelvin–Helmholtz instability mechanisms contribute to amplification, while deeper inside the bubble only the inviscid mechanism is active. If the explicit forcing is moved to a region downstream of the favourable pressure gradient, only the unstable eigenmode appears. The same behaviour is found for two-dimensional and weakly oblique waves.

JFM classification

Boundary Layers: Boundary layer separationBoundary Layers: Boundary layer stabilityInstability: Transition to turbulence

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 711 , 25 November 2012, pp. 1 - 26
Copyright
Copyright © Cambridge University Press 2012

Footnotes

‡

Present address: GE Global Research, Freisinger Landstrasse 50, 85748 Garching b. München, Germany.

References

1. Ashpis, D. E. & Reshotko, E. 1990 The vibrating ribbon problem revisited. J. Fluid Mech. 213, 531–547.CrossRefGoogle Scholar
2. Boiko, A., Dovgal, A., Hein, S. & Henning, A. 2011 Particle image velocimetry of a low-Reynolds-number separation bubble. Exp. Fluids 50, 13–21.CrossRefGoogle Scholar
3. Corke, T. C. & Gruber, S. 1996 Resonant growth of three-dimensional modes in Falkner–Skan boundary layers with adverse pressure gradients. J. Fluid Mech. 320, 211–233.CrossRefGoogle Scholar
4. Diwan, S. S. & Ramesh, O. N. 2009 On the origin of the inflectional instability of a laminar separation bubble. J. Fluid Mech. 629, 263–298.CrossRefGoogle Scholar
5. Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
6. Fedorov, A. & Tumin, A. 2011 High-speed boundary-layer instability: old terminology and a new framework. AIAA J. 49 (8), 1647–1657.CrossRefGoogle Scholar
7. Gaster, M. 1967 The Structure and Behaviour of Separation Bubbles, Aeronautical Research Council, Reports and Memoranda No. 3595, London.Google Scholar
8. Gruber, K., Bestek, H. & Fasel, H. 1987 Interaction between a Tollmien–Schlichting wave and a laminar separation bubble. AIAA Paper 1987–1256.CrossRefGoogle Scholar
9. Häggmark, C. P., Bakchinov, A. A. & Alfredsson, P. H. 2000 Experiments on a two-dimensional laminar separation bubble. Phil. Trans. R. Soc. Lond. A 358, 3193–3205.CrossRefGoogle Scholar
10. Hain, R., Kähler, C. J. & Radespiel, R. 2009 Dynamics of laminar separation bubbles at low-Reynolds-number aerofoils. J. Fluid Mech. 630, 129–153.CrossRefGoogle Scholar
11. Jones, L. E., Sandberg, R. D. & Sandham, N. D. 2008 Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence. J. Fluid Mech. 602, 175–207.CrossRefGoogle Scholar
12. Kachanov, Y. S. & Levchenko, V. Y. 1984 The resonant interaction of disturbances at laminar-turbulent transition in a boundary layer. J. Fluid Mech. 138, 209–247.CrossRefGoogle Scholar
13. Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 The three-dimensional nature of boundary-layer instability. J. Fluid Mech. 12, 1–34.CrossRefGoogle Scholar
14. Kloker, M. 1998 A robust high-resolution split-type compact FD scheme for spatial direct numerical simulation of boundary-layer transition. Appl. Sci. Res. 59, 353–377.CrossRefGoogle Scholar
15. Kotapati, R. B., Mittal, R., Marxen, O., Ham, F., You, D. & Cattafesta III, L. N. 2010 Nonlinear dynamics and synthetic-jet-based control of a canonical separated flow. J. Fluid Mech. 654, 65–97.CrossRefGoogle Scholar
16. Lang, M., Rist, U. & Wagner, S. 2004 Investigations on controlled transition development in a laminar separation bubble by means of LDA and PIV. Exp. Fluids 36, 43–52.Google Scholar
17. Lele, S. K. 1992 Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16–42.CrossRefGoogle Scholar
18. Levin, O., Chernoray, V. G., Löfdahl, L. & Henningson, D. S. 2005 A study of the Blasius wall jet. J. Fluid Mech. 539, 313–347.CrossRefGoogle Scholar
19. Mack, L. M. 1975 Linear stability theory and the problem of supersonic boundary-layer transition. AIAA J. 13 (3), 278–2285.CrossRefGoogle Scholar
20. Marxen, O. 2005Numerical studies of physical effects related to the controlled transition process in laminar separation bubbles. Dissertation, Universität Stuttgart.Google Scholar
21. Marxen, O. & Henningson, D. S. 2011 The effect of small-amplitude convective disturbances on the size and bursting of a laminar separation bubble. J. Fluid Mech. 671, 1–33.CrossRefGoogle Scholar
22. Marxen, O., Lang, M., Rist, U., Levin, O. & Henningson, D. S. 2009 Mechanisms for spatial steady three-dimensional disturbance growth in a non-parallel and separating boundary layer. J. Fluid Mech. 634, 165–189.CrossRefGoogle Scholar
23. Marxen, O., Lang, M., Rist, U. & Wagner, S. 2003 A combined experimental/numerical study of unsteady phenomena in a laminar separation bubble. Flow Turbul. Combust. 71, 133–146.CrossRefGoogle Scholar
24. Marxen, O. & Rist, U. 2010 Mean flow deformation in a laminar separation bubble: separation and stability characteristics. J. Fluid Mech. 660, 37–54.CrossRefGoogle Scholar
25. Marxen, O., Rist, U. & Henningson, D. S. 2006 Steady three-dimensional streaks and their optimal growth in a laminar separation bubble. In New Results in Numerical and Experimental Fluid Mechanics V (ed. Rath, H. J., Holze, C., Heinemann, H.-J., Henke, R. & Hönlinger, H. ), Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), vol. 92 , Springer, Contributions to the 14th STAB/DGLR Symposium, Nov. 16–18, 2004, Bremen, Germany.Google Scholar
26. Marxen, O., Rist, U. & Wagner, S. 2004 Effect of spanwise-modulated disturbances on transition in a separated boundary layer. AIAA J. 42 (5), 937–944.CrossRefGoogle Scholar
27. Maucher, U., Rist, U. & Wagner, S. 2000 Refined interaction method for direct numerical simulation of transition in separation bubbles. AIAA J. 38 (8), 1385–1393.CrossRefGoogle Scholar
28. Meyer, D., Rist, U. & Kloker, M. 2003 Investigation of the flow randomization process in a transitional boundary layer. In High Performance Computing in Science and Engineering’03 (ed. Krause, E. & Jäger, W. ), Transactions of the HLRS 2003 , pp. 239–253. Springer.Google Scholar
29. Postl, D., Balzer, W. & Fasel, H. F. 2011 Control of laminar separation using pulsed vortex generator jets: direct numerical simulations. J. Fluid Mech. 676, 81–109.CrossRefGoogle Scholar
30. Rist, U. & Augustin, K. 2006 Control of laminar separation bubbles using instability waves. AIAA J. 44 (10), 2217–2223.CrossRefGoogle Scholar
31. Rist, U. & Maucher, U. 2002 Investigations of time-growing instabilities in laminar separation bubbles. Eur. J. Mech. (B/Fluids) 21, 495–509.CrossRefGoogle Scholar
32. Rist, U., Maucher, U. & Wagner, S. 1996 Direct numerical simulation of some fundamental problems related to transition in laminar separation bubbles. In Computational Fluid Dynamics’96 (ed. Désidéri, J.-A., Hirsch, C., Le Tallec, P., Pandolfi, M. & Périaux, J. ), pp. 319–325. John Wiley & Sons.Google Scholar
33. Roberts, S. K. & Yaras, M. I. 2006 Large-eddy simulation of transition in a separation bubble. Trans. ASME: J. Fluids Engng 128, 232–238.Google Scholar
34. Sandham, N. D. 2008 Transitional separation bubbles and unsteady aspects of aerofoil stall. Aeronaut. J. 112 (1133), 395–404.CrossRefGoogle Scholar
35. Saric, W. S., Reed, H. L. & Kerschen, E. J. 2002 Boundary-layer receptivity to free stream disturbances. Annu. Rev. Fluid Mech. 34, 291–319.CrossRefGoogle Scholar
36. Schlichting, H. 1933 Zur Entstehung der Turbulenz bei der Plattenströmung. In Nachr. Ges. Wiss. Göttingen, Math.-Phys. Klasse, pp. 181–208, also Z. Angew. Math. Mech. 13 (3), 171–174.Google Scholar
37. Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows, 1st edn. Springer.CrossRefGoogle Scholar
38. Spalart, P. R. & Strelets, M. K. 2000 Mechanisms of transition and heat transfer in a separation bubble. J. Fluid Mech. 403, 329–349.CrossRefGoogle Scholar
39. Tollmien, W. 1929Über die Entstehung der Turbulenz. 1. Mitteilung, pp. 21–44. English translation: The production of turbulence. NACA TM 609 (1931).CrossRefGoogle Scholar
40. Tumin, A. 2003 Multimode decomposition of spatially growing perturbations in a two-dimensional boundary layer. Phys. Fluids 15 (9), 2525–2540.CrossRefGoogle Scholar
41. Tumin, A. & Aizatulin, L. 1997 Instability and receptivity of laminar wall jets. Theor. Comput. Fluid Dyn. 9, 33–45.CrossRefGoogle Scholar
42. Watmuff, J. H. 1999 Evolution of a wave packet into vortex loops in a laminar separation bubble. J. Fluid Mech. 397, 119–169.CrossRefGoogle Scholar
43. Yarusevych, S., Sullivan, P. E. & Kawall, J. G. 2009 On vortex shedding from an airofoil in low-Reynolds-number flows. J. Fluid Mech. 632, 245–271.CrossRefGoogle Scholar