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Receptivity, instability and breakdown of Görtler flow

Published online by Cambridge University Press:  11 July 2011

LARS-UVE SCHRADER ,
LUCA BRANDT  and
TAMER A. ZAKI
LARS-UVE SCHRADER
Affiliation:
Linné Flow Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden
LUCA BRANDT
Affiliation:
Linné Flow Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden
TAMER A. ZAKI*
Affiliation:
Department of Mechanical Engineering, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

Receptivity, disturbance growth and breakdown to turbulence in Görtler flow are studied by spatial direct numerical simulation (DNS). The boundary layer is exposed to free-stream vortical modes and localized wall roughness. We propose a normalization of the roughness-induced receptivity coefficient by the square root of the Görtler number. This scaling removes the dependence of the receptivity coefficient on wall curvature. It is found that vortical modes are more efficient at generating Görtler vortices than localized roughness. The boundary layer is most receptive to zero- and low-frequency free-stream vortices, exciting steady and slowly travelling Görtler modes. The associated receptivity mechanism is linear and involves the generation of boundary-layer streaks, which soon evolve into unstable Görtler vortices. This connection between transient and exponential amplification is absent on flat plates and promotes transition to turbulence on curved walls. We demonstrate that the Görtler boundary layer is also receptive to high-frequency free-stream vorticity, which triggers steady Görtler rolls via a nonlinear receptivity mechanism. In addition to the receptivity study, we have carried out DNS of boundary-layer transition due to broadband free-stream turbulence with different intensities and frequency spectra. It is found that nonlinear receptivity dominates over the linear mechanism unless the free-stream fluctuations are concentrated in the low-frequency range. In the latter case, transition is accelerated due to the presence of travelling Görtler modes.

JFM classification

Boundary Layers: Boundary layer receptivity Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 682 , 10 September 2011 , pp. 362 - 396
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Asai, M., Minagawa, M. & Nishioka, M. 2002 The instability and breakdown of a near-wall low-speed streak. J. Fluid Mech. 455, 289–314.Google Scholar
Bassom, A. P. & Seddougui, S. O. 1995 Receptivity mechanisms for Görtler vortex modes. Theor. Comput. Fluid Dyn. 7, 317–339.CrossRefGoogle Scholar
Berlin, S. & Henningson, D. 1999 A nonlinear mechanism for receptivity of free-stream disturbances. Phys. Fluids 11 (12), 3749–3760.Google Scholar
Bertolotti, F. P. 1993 Vortex generation and wave-vortex interaction over a concave plate with roughness and suction. ICASE Rep., pp. 93–101.Google Scholar
Bippes, H. 1972 Experimentelle Untersuchung des laminar-turbulenten Umschlags an einer parallel angeströmten konkaven Wand. Heidel. Akad. Wiss., Math.-naturwiss. Kl., Sitzungsber. 3, 103–180.Google Scholar
Bippes, H. & Deyhle, H. 1992 Das Receptivity-Problem in Grenzschichten mit längswirbelartigen Störungen. Z. Flugwiss. Weltraumforsch. 16, 34–41.Google Scholar
Boiko, A. V., Ivanov, A. V., Kachanov, Y. S. & Mischenko, D. A. 2010 Steady and unsteady Görtler boundary-layer instability on concave wall. Eur. J. Mech. B/Fluids 29, 61–83.CrossRefGoogle Scholar
Bottaro, A. & Luchini, P. 1999 Görtler vortices: are they amenable to local eigenvalue analysis? Eur. J. Mech. B/Fluids 18 (1), 47–65.Google Scholar
Brandt, L., Henningson, D. S. & Ponziani, D. 2002 Weakly non-linear analysis of boundary layer receptivity to free-stream disturbances. Phys. Fluids 14, 1426–1441.Google Scholar
Brandt, L., Schlatter, P. & Henningson, D. S. 2004 Transition in boundary layers subject to free-stream turbulence. J. Fluid Mech. 517, 167–198.CrossRefGoogle Scholar
Cabal, A., Szumbarski, J. & Floryan, J. M. 2001 Numerical simulation of flows over corrugated walls. Comput. Fluids 30, 753–776.Google Scholar
Chevalier, M., Schlatter, P., Lundbladh, A. & Henningson, D. S. 2007 A Pseudo-spectral solver for incompressible boundary layer flows. Tech. Rep. TRITA-MEK 2007:07. Royal Institute of Technology (KTH), Department of Mechanics, Stockholm.Google Scholar
Cossu, C., Chomaz, J.-M., Huerre, P. & Costa, M. 2000 Maximum spatial growth of Görtler vortices. Flow Turbul. Combust. 65, 369–392.Google Scholar
Denier, J. P., Hall, P. & Seddougui, S. O. 1991 On the receptivity problem for Görtler vortices: vortex motions induced by wall roughness. Phil. Trans. R. Soc. Lond. A 335, 51–85.Google Scholar
Fischer, P., Kruse, J., Mullen, J., Tufo, H., Lottes, J. & Kerkemeier, S. 2008 NEK5000: Open Source Spectral Element CFD solver. Available at: https://nek5000.mcs.anl.gov/index.php/MainPage.Google Scholar
Floryan, J. M. 1991 On the Görtler instability of boundary layers. Prog. Aerosp. Sci. 28 (3), 235–271.CrossRefGoogle Scholar
Floryan, J. M. & Saric, W. S. 1982 Stability of Görtler vortices in boundary layers. AIAA J. 20 (3), 316–324.CrossRefGoogle Scholar
Fransson, J. H. M., Matsubara, M. & Alfredsson, P. H. 2005 Transition induced by free-stream turbulence. J. Fluid Mech. 527, 1–25.Google Scholar
Girgis, I. G. & Liu, J. T. C. 2006 Nonlinear mechanics of wavy instability of steady longitudinal vortices and its effect on skin friction rise in boundary layer flow. Phys. Fluids 18 (024102).CrossRefGoogle Scholar
Görtler, H. 1941 Instabilität laminarer Grenzschichten an konkaven Wänden gegenüber gewissen dreidimensionalen Störungen. Z. Angew. Math. Mech. 21, 250–252.Google Scholar
Grosch, C. E. & Salwen, H. 1978 The continuous spectrum of the Orr–Sommerfeld equation. Part 1. The spectrum and the eigenfunctions. J. Fluid Mech. 87, 33–54.CrossRefGoogle Scholar
Hall, P. 1982 Taylor–Görtler vortices in fully developed or boundary-layer flows: linear theory. J. Fluid Mech. 124, 475–494.Google Scholar
Hall, P. 1983 The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41–58.Google Scholar
Hall, P. 1990 Görtler vortices in growing boundary layers: the leading-edge receptivity problem, linear growth and the nonlinear breakdown stage. Mathematika 37 (74), 151–189.Google Scholar
Hall, P. & Horseman, N. J. 1991 The linear inviscid secondary instability of longitudinal vortex structures in boundary layers. J. Fluid Mech. 232, 357–375.Google Scholar
Ito, A. 1980 The generation and breakdown of longitudinal vortices along a concave wall. J. Japan. Soc. Aeronaut. Space Sci. 28, 327–333 (in Japanese).Google Scholar
Jacobs, R. G. & Durbin, P. A. 1998 Shear sheltering and continuous spectrum of the Orr–Sommerfeld equation. Phys. Fluids 10 (8), 2006–2011.CrossRefGoogle Scholar
Jacobs, R. G. & Durbin, P. A. 2001 Simulations of bypass transition. J. Fluid Mech. 428, 185–212.CrossRefGoogle Scholar
Kurian, T. & Fransson, J. H. M. 2009 Grid-generated turbulence revisited. Fluid Dyn. Res. 41, 021403.CrossRefGoogle Scholar
Lee, K. & Liu, J. T. C. 1992 On the growth of mushroom-like structures in nonlinear spatially developing Görtler vortex flow. Phys. Fluids A 4 (1), 95–103.CrossRefGoogle Scholar
Li, F. & Malik, M. R. 1995 Fundamental and subharmonic secondary instabilities of Görtler vortices. J. Fluid Mech. 297, 77–100.Google Scholar
Liu, W. & Domaradzki, J. A. 1993 Direct numerical simulation of transition to turbulence in Görtler flow. J. Fluid Mech. 246, 267–299.Google Scholar
Luchini, P. & Bottaro, A. 1998 Görtler vortices: a backward-in-time approach to the receptivity problem. J. Fluid Mech. 363, 1–23.CrossRefGoogle Scholar
Mitsudharmadi, H., Winoto, S. H. & Shah, D. A. 2004 Development of boundary-layer flow in the presence of forced wavelength Görtler vortices. Phys. Fluids 16, 3983–3996.Google Scholar
Mitsudharmadi, H., Winoto, S. H. & Shah, D. A. 2005 Splitting and merging of Görtler vortices. Phys. Fluids 17, 124102.CrossRefGoogle Scholar
Ng, L. L. & Crouch, J. D. 1999 Roughness-induced receptivity to crossflow vortices on a swept wing. Phys. Fluids 11 (2), 432–438.Google Scholar
Park, D. S. & Huerre, P. 1995 Primary and secondary instabilities of the asymptotic suction boundary layer on a curved plate. J. Fluid Mech. 283, 249–272.Google Scholar
Patera, A. T. 1984 A spectral element method for fluid dynamics: laminar flow in a channel expansion. J. Comput. Phys. 54, 468–488.Google Scholar
Peerhossaini, H. & Wesfreid, J. E. 1988 On the inner structure of streamwise Görtler rolls. Intl J. Heat Fluid Flow 9 (1), 12–18.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, 101–137.Google Scholar
Saric, W. S. 1994 Görtler vortices. Annu. Rev. Fluid Mech. 26, 379–409.CrossRefGoogle Scholar
Schrader, L.-U., Brandt, L. & Henningson, D. S. 2009 Receptivity mechanisms in three-dimensional boundary-layer flows. J. Fluid Mech. 618, 209–241.Google 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.Google Scholar
Schultz, M. P. & Volino, R. J. 2003 Effects of concave curvature on boundary layer transition under high freestream turbulence conditions. J. Fluids Engng 125, 18–27.Google Scholar
Swearingen, J. D. & Blackwelder, R. F. 1983 Parameters controlling the spacing of streamwise vortices on a concave wall. AIAA Paper 83-0380.Google 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
Tandiono, S., Winoto, H. & Shah, D. A. 2008 On the linear and nonlinear development of Görtler vortices. Phys. Fluids 20, 094103.Google Scholar
Tandiono, S., Winoto, H. & Shah, D. A. 2009 Wall shear stress in Görtler vortex boundary layer flow. Phys. Fluids 21, 084106.CrossRefGoogle Scholar
Wu, X. & Durbin, P. A. 2001 Evidence of longitudinal vortices evolved from distorted wakes in a turbine passage. J. Fluid Mech. 446, 199–228.CrossRefGoogle Scholar
Yu, X. & Liu, J. T. C. 1994 On the mechanism of sinuous and varicose modes in three-dimensional viscous secondary instability of nonlinear Görtler rolls. Phys. Fluids 6 (2), 736–750.Google Scholar
Zaki, T. A. & Durbin, P. A. 2005 Mode interaction and the bypass route to transition. J. Fluid Mech. 531, 85–111.Google 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, 111–147.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.Google Scholar