Hostname: page-component-5c6d5d7d68-sv6ng Total loading time: 0 Render date: 2024-08-18T02:59:11.878Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of zero-pressure-gradient boundary layer distorted by unsteady Klebanoff streaks

Published online by Cambridge University Press:  31 May 2011

NICHOLAS J. VAUGHAN  and
TAMER A. ZAKI
NICHOLAS J. VAUGHAN
Affiliation:
Department of Mechanical Engineering, Imperial College, London SW7 2AZ, UK
TAMER A. ZAKI*
Affiliation:
Department of Mechanical Engineering, Imperial College, London SW7 2AZ, UK
*
†Email address for correspondence: t.zaki@imperial.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

The secondary instability of a zero-pressure-gradient boundary layer, distorted by unsteady Klebanoff streaks, is investigated. The base profiles for the analysis are computed using direct numerical simulation (DNS) of the boundary-layer response to forcing by individual free-stream modes, which are low frequency and dominated by streamwise vorticity. Therefore, the base profiles take into account the nonlinear development of the streaks and mean flow distortion, upstream of the location chosen for the stability analyses. The two most unstable modes were classified as an inner and an outer instability, with reference to the position of their respective critical layers inside the boundary layer. Their growth rates were reported for a range of frequencies and amplitudes of the base streaks. The inner mode has a connection to the Tollmien–Schlichting (T–S) wave in the limit of vanishing streak amplitude. It is stabilized by the mean flow distortion, but its growth rate is enhanced with increasing amplitude and frequency of the base streaks. The outer mode only exists in the presence of finite amplitude streaks. The analysis of the outer instability extends the results of Andersson et al. (J. Fluid Mech. vol. 428, 2001, p. 29) to unsteady base streaks. It is shown that base-flow unsteadiness promotes instability and, as a result, leads to a lower critical streak amplitude. The results of linear theory are complemented by DNS of the evolution of the inner and outer instabilities in a zero-pressure-gradient boundary layer. Both instabilities lead to breakdown to turbulence and, in the case of the inner mode, transition proceeds via the formation of wave packets with similar structure and wave speeds to those reported by Nagarajan, Lele & Ferziger (J. Fluid Mech., vol. 572, 2007, p. 471).

JFM classification

Boundary Layers: Boundary layer stability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 681 , 25 August 2011 , pp. 116 - 153
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Andersson, P., Brandt, L., Bottaro, A. & Henningson, D. S. 2001 On the breakdown of boundary layer streaks. J. Fluid Mech. 428, 29–60.CrossRefGoogle Scholar
Arnal, D. & Juillen, J. C. 1978 Contribution expérimentale a l'étude de la réceptivité d'une couche limite laminaire, a la turbulence de l'écoulement general. Tech. Rep., ONERA Rap. Tech. 1/5018 AYD.Google Scholar
Asai, M., Minagawa, M. & Nishioka, M. 2002 The instability and breakdown of a near-wall low-speed streak. J. Fluid Mech. 455, 289–314.CrossRefGoogle Scholar
Blennerhassett, P. & Bassom, A. 2006 The linear stability of high-frequency oscillatory flow in a channel. J. Fluid Mech. 556, 1–25.CrossRefGoogle Scholar
Boiko, A. V., Westin, K. J. A., Klingmann, B. G. B., Kozlov, V. V. & Alfredsson, P. H. 1994 Experiments in a boundary layer subjected to free stream turbulence. Part 2. The role of TS-waves in the transition process. J. Fluid Mech. 281, 219–245.CrossRefGoogle Scholar
Brandt, L. & Henningson, D. S. 2002 Transition of streamwise streaks in zero-pressure-gradient boundary layers. J. Fluid Mech. 472, 229–261.CrossRefGoogle Scholar
Brandt, L., Sclatter, P. & Henningson, D. S. 2004 Transition in boundary layers subject to free-stream turbulence. J. Fluid Mech. 517, 167–198.CrossRefGoogle Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637–1650.CrossRefGoogle Scholar
Cossu, C. & Brandt, L. 2002 Stabilization of Tollmien–Schlichting waves by finite amplitude optimal streaks in the Blasius boundary layer. Phys. Fluids 14 (8), 57–60.CrossRefGoogle Scholar
Cossu, C. & Brandt, L. 2004 On Tollmien–Schlichting-like waves in streaky boundary layers. Eur. J. Mech. B/Fluids 23 (6), 815–833.CrossRefGoogle Scholar
Fasel, H. F. 2002 Numerical investigation of the interaction of the Klebanoff-mode with a Tollmien–Schlichting wave. J. Fluid Mech. 450, 1–33.CrossRefGoogle Scholar
Goldstein, M. E. & Wundrow, D. W. 1995 Interaction of oblique instability waves with weak streamwise vortices. J. Fluid Mech. 284, 377–407.CrossRefGoogle Scholar
Goldstein, M. E. & Leib, S. J. 1993 Three-dimensional boundary-layer instability and separation induced by small-amplitude streamwise vorticity in the upstream flow. J. Fluid Mech. 246, 21–41.CrossRefGoogle Scholar
Goldstein, M. E. & Sescu, A. 2008 Boundary-layer transition at high free-stream disturbance levels: beyond Klebanoff modes. J. Fluid Mech. 613, 95–124.CrossRefGoogle Scholar
Grosch, C. & Salwen, H. 1968 The stability of steady and time-dependent plane Poiseuille flow. J. Fluid Mech. 34, 177–205.CrossRefGoogle Scholar
Herbert, T. 1988 Secondary instability of boundary layers. Annu. Rev. Fluid Mech. 20, 487–526.CrossRefGoogle Scholar
Hernon, D., Walsh, E. J. & McEligot, D. M. 2007 a Instantaneous fluctuation velocity and skewness distributions upstream of transition onset. Intl J. Heat and Fluid Flow 28 (6), 1272–1279.CrossRefGoogle Scholar
Hernon, D., Walsh, E. J. & McEligot, D. M. 2007 b Experimental investigation into the routes to bypass transition and the shear-sheltering phenomenon. J. Fluid Mech. 591, 461–479.CrossRefGoogle Scholar
Hultgren, L. S. & Gustavsson, L. H. 1981 Algebraic growth of disturbances in a laminar boundary layer. Phys. Fluids 24, 1000–1004.CrossRefGoogle Scholar
Hunt, J. C. R. 1977 A review of rapidly distorted turbulent flows and its applications. Fluid Dyn. Trans. 9, 121–152.Google Scholar
Jacobs, R. G. & Durbin, P. A. 2001 Simulations of bypass transition. J. Fluid Mech. 428, 185–212.CrossRefGoogle Scholar
Kendall, J. M. 1990 Boundary layer receptivity to free stream turbulence. AIAA Paper 90-1504.CrossRefGoogle Scholar
Kendall, J. M. 1991 Studies on laminar boundary-layer receptivity to freestream turbulence near a leading edge. In Boundary Layer Stability and Transition to Turbulence (ed. Reda, D. C., Reed, H. L. & Kobayashi, R.), vol. 114, pp. 23–30. Fluid Engineering Division, ASME.Google Scholar
Kerczek, V. C. & Davis, S. H. 1974 Linear stability theory of oscillatory Stokes layers. J. Fluid Mech. 62, 753–773.CrossRefGoogle Scholar
Klebanoff, P. S. 1971 Effect of freestream turbulence on the laminar boundary layer. Bull. Am. Phys. Soc. 16 (11), 1323.Google Scholar
Kleiser, L. & Zang, T. A. 1991 Numerical simulation of transition in wall-bounded shear flows. Annu. Rev. Fluid Mech. 23, 495–537.CrossRefGoogle Scholar
Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243–251.CrossRefGoogle Scholar
Lehoucq, R., Sorensen, D. & Yang, C. 1998 In ARPACK Users' Guide. SIAM Philadelphia.CrossRefGoogle Scholar
Leib, S., Wundrow, D. & Goldstein, M. 1999 Effect of free-stream turbulence and other vortical disturbances on a laminar boundary layer. J. Fluid Mech. 380, 169–203.CrossRefGoogle Scholar
Liu, Y., Zaki, T. A. & Durbin, P. A. 2008 a Boundary-layer transition by interaction of discrete and continuous modes. J. Fluid Mech. 604, 199–233.CrossRefGoogle Scholar
Liu, Y., Zaki, T. A. & Durbin, P. A. 2008 b Floquet analysis of secondary instability of boundary layers distorted by Klebanoff streaks and Tollmien–Schlichting waves. Phys. Fluids 20, 124102.CrossRefGoogle Scholar
Luo, J. & Wu, X. 2010 On the linear instability of a finite Stokes layer: instantaneous versus Floquet modes. Phys. Fluids 22, 054106.CrossRefGoogle Scholar
Mandal, A. C., Venkatakrishnan, L. & Dey, J. 2010 A study of boundary layer transition induced by freestream turbulence. J. Fluid Mech. 660, 114–146.CrossRefGoogle Scholar
Mans, J., Kadijk, E. C., Lange, H. C. & Steenhoven, A. A. 2005 Breakdown in a boundary layer exposed to free-stream turbulence. Exp. Fluids 39 (6), 1071–1083.CrossRefGoogle Scholar
Mans, J., de Lange, H. C. & van Steenhoven, A. A. 2007 Sinuous breakdown in a flat plate boundary layer exposed to free-stream turbulence. Phys. Fluids 19 (8), 088101.CrossRefGoogle Scholar
Matsubara, M. & Alfredsson, P. 2001 Disturbance growth in boundary layers subjected to free-stream turbulence. J. Fluid Mech. 430, 149–168.CrossRefGoogle Scholar
Morkovin, M. V. 1969 On the many faces of transition. In Viscous Drag Reduction, pp. 1–31. Plenum.Google Scholar
Morkovin, M. V., Reshotko, E. & Herbert, T. 1994 Transition in open flow systems: a reassessment. Bull. Am. Phys. Soc. 39, 1882.Google Scholar
Nagarajan, S., Lele, S. K. & Ferziger, J. H. 2007 Leading-edge effects in bypass transition. J. Fluid Mech. 572, 471–504.CrossRefGoogle Scholar
Nolan, K. P., Walsh, E. J. & McEligot, D. M. 2010 Quadrant analysis of a transitional boundary layer subject to free-stream turbulence. J. Fluid Mech. 658, 310–335.CrossRefGoogle Scholar
Orszag, S. A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50, 689–703.CrossRefGoogle Scholar
Phillips, O. M. 1969 Shear-flow turbulence. Annu. Rev. Fluid Mech. 1, 245–264.CrossRefGoogle Scholar
Reshotko, E. 1976 Boundary-layer stability and transition. Annu. Rev. Fluid Mech. 8 (1), 311–349.CrossRefGoogle Scholar
Rosenfeld, M., Kwak, D. & Vinokur, M. 1991 A fractional step solution method for the unsteady incompressible Navier–Stokes equations in generalized coordinate systems. J. Comput. Phys. 94 (1), 102–137.CrossRefGoogle Scholar
Saric, W. S. 1994 Görtler vortices. Annu. Rev. Fluid Mech. 26, 379–409.CrossRefGoogle Scholar
Saric, W. S., Reed, H. L. & Kerschen, E. J. 2002 Boundary-layer receptivity to freestream disturbances. Annu. Rev. Fluid Mech. 34, 291–319.CrossRefGoogle Scholar
Schrader, L.-U., Brandt, L., Mavriplis, C. & Henningson, D. S. 2010 Receptivity to free-stream vorticity of flow past a flat plate with elliptic leading edge. J. Fluid Mech. 653, 245–271.CrossRefGoogle Scholar
Swearingen, J. D. & Blackwelder, R. F. 1987 The growth and breakdown of streamwise vortices in the presence of a wall. J. Fluid Mech. 182, 255–290.CrossRefGoogle Scholar
Westin, K. J. A., Boiko, A. V., Klingmann, B. G. B., Kozlov, V. V. & Alfredsson, P. H. 1994 Experiments in a boundary layer subjected to free stream turbulence. Part 1. Boundary layer structure and receptivity. J. Fluid Mech. 281, 193–218.CrossRefGoogle Scholar
Wu, X. & Choudhari, M. 2003 Linear and nonlinear instabilities of a Blasius boundary layer perturbed by streamwise vortices. Part 2. Intermittent instability induced by long-wavelength Klebanoff modes. J. Fluid Mech. 483, 249–286.CrossRefGoogle Scholar
Zaki, T. A. & Durbin, P. A. 2005 Mode interaction and the bypass route to transition. J. Fluid Mech. 531, 85–111.CrossRefGoogle Scholar
Zaki, T. A. & Durbin, P. A. 2006 Continuous mode transition and the effects of pressure gradient. J. Fluid Mech. 563, 357–388.CrossRefGoogle Scholar
Zaki, T. A. & Saha, S. 2009 On shear sheltering and the structure of vortical modes in single and two-fluid boundary layers. J. Fluid Mech. 626, 111–148.CrossRefGoogle Scholar
Zaki, T. A., Wissink, J. G., Rodi, W. & Durbin, P. A. 2010 Direct numerical simulations of transition in a compressor cascade: the influence of free-stream turbulence. J. Fluid Mech. 665, 57–98.CrossRefGoogle Scholar