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Excitation of steady and unsteady Görtler vortices by free-stream vortical disturbances

Published online by Cambridge University Press:  08 July 2011

XUESONG WU ,
DIFEI ZHAO  and
JISHENG LUO
XUESONG WU*
Affiliation:
Department of Mechanics, Tianjin University, Tianjin 30072, China Department of Mathematics, Imperial College London, London SW7 2AZ, UK
DIFEI ZHAO
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
JISHENG LUO
Affiliation:
Department of Mechanics, Tianjin University, Tianjin 30072, China
*
†Email address for correspondence: x.wu@ic.ac.uk
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Abstract

Excitation of Görtler vortices in a boundary layer over a concave wall by free-stream vortical disturbances is studied theoretically and numerically. Attention is focused on disturbances with long streamwise wavelengths, to which the boundary layer is most receptive. The appropriate initial-boundary-value problem describing both the receptivity process and the development of the induced perturbation is formulated for the generic case where the Görtler number GΛ (based on the spanwise wavelength Λ of the disturbance) is of order one. The impact of free-stream disturbances on the boundary layer is accounted for by the far-field boundary condition and the initial condition near the leading edge, both of which turn out to be the same as those given by Leib, Wundrow & Goldstein (J. Fluid Mech., vol. 380, 1999, p. 169) for the flat-plate boundary layer. Numerical solutions show that for a sufficiently small GΛ, the induced perturbation exhibits essentially the same characteristics as streaks occurring in the flat-plate case: it undergoes considerable amplification and then decays. However, when GΛ exceeds a critical value, the induced perturbation exhibits (quasi-) exponential growth. The perturbation acquires the modal shape of Görtler vortices rather quickly, and its growth rate approaches that predicted by local instability theories farther downstream, indicating that Görtler vortices are excited. The amplitude of the Görtler vortices excited is found to decrease as the frequency increases, with steady vortices being dominant. Comprehensive quantitative comparisons with experiments show that the eigenvalue approach predicts the modal shape adequately, but only the initial-value approach can accurately predict the entire evolution of the amplitude. An asymptotic analysis is performed for GΛ ≫ 1 to map out distinct regimes through which a perturbation with a fixed spanwise wavelength evolves. The centrifugal force starts to influence the generation of the pressure when x* ~ ΛRΛG−2/3Λ, where RΛ denotes the Reynolds number based on Λ. The induced pressure leads to full coupling of the momentum equations when x* ~ ΛRΛGΛ−2/5. This is the crucial regime linking the pre-modal and modal phases of the perturbation because the governing equations admit growing asymptotic eigensolutions, which develop into fully fledged Görtler vortices of inviscid nature when x* ~ ΛRΛ. From this position onwards, local eigenvalue formulations are mathematically justified. Görtler vortices continue to amplify and enter the so-called most unstable regime when x* ~ ΛRΛGΛ, and ultimately approach the right-branch regime when x* ~ ΛRΛG2Λ.

JFM classification

Boundary Layers: Boundary layer receptivity Instability: Instability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 682 , 10 September 2011 , pp. 66 - 100
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Bassom, A. P. & Hall, P. 1994 The receptivity problem for O(1) wavelength Görtler vortices. Proc. R. Soc. Lond. A 446, 499–516.Google Scholar
Bippes, H. & Görtler, 1972 Dreidimensionale Störungen in der Grenzschicht an einer konkaven Wand. Acta Mech. 14, 251–267.CrossRefGoogle Scholar
Boiko, A. V., Ivanov, A. V., Kachanov, Y. S. & Mischenko, D. A. 2007 Quasi-steady and unsteady Görtler vortices on a concave wall: experiment and theory. In Advances in Turbulence XI: Proc. 11th Euromech Euro. Turbulence Conf., Porto (ed. Palma, J. M. L. M. & Silva Lopes, A.), pp. 173–175. Springer.CrossRefGoogle Scholar
Boiko, A. V., Kachanov, Y. S. & Mischenko, D. A. 2010 Steady and unsteady Görtler boundary-layer instability on concave wall. Euro. 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, 47–65.CrossRefGoogle Scholar
Crane, R. I. & Sabzvari, J. 1982 Laser-Doppler measurements of Görtler vortices in laminar and low-Reynolds number turbulent boundary layers. In Proc. Intl Symp. Applications of Laser Doppler Anemometry to Fluid Mechanics, Lisbon.Google Scholar
Day, H. P., Herbert, T. & Saric, W. S. 1990 Comparing local and marching analyses of Görtler instability. AIAA J. 28 (6), 1010–1015.CrossRefGoogle 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
Finnis, M. V. & Brown, A. 1997 The linear growth of Görtler vortices. J. Heat Fluid Flow 18, 389–399.CrossRefGoogle Scholar
Floryan, J. M. 1991 On the Görtler instability of boundary layers. Prog. Aerospace Sci. 28, 235–271.CrossRefGoogle Scholar
Floryan, J. M. & Saric, W. S. 1982 Stability of Görtler vortices in boundary layers. AIAA J. 20 (3), 316–323.CrossRefGoogle Scholar
Goldstein, M. E. 1983 The evolution of Tollmien–Schlichting waves near a leading edge. J. Fluid Mech. 127, 59–81.CrossRefGoogle Scholar
Görtler, H. 1940 Über eine dreidimensionale Instabilität laminarer Grenzschichten an konkaven Wänden. Des. D. Wiss. Göttingen, Nachr. 1 (2); translated as ‘On the three-dimensional instability of laminar boundary layers on concave walls’. NACA TM 1375, 1954.Google Scholar
Hall, P. 1982 Taylor–Görtler vortices in fully developed or boundary-layer flows: linear theory. J. Fluid Mech. 124, 475–494.CrossRefGoogle Scholar
Hall, P. 1983 The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41–58.CrossRefGoogle Scholar
Hall, P. 1988 The nonlinear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 193, 243–266.CrossRefGoogle Scholar
Hall, P. 1990 Görtler vortices in growing boundary layers: The leading edge receptivity problem, linear growth and the nonlinear breakdown stage. Mathematika 37, 151–189.CrossRefGoogle 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.CrossRefGoogle Scholar
Kim, J., Simon, T. W. & Russ, S. G. 1992 Free-stream turbulence and concave curvature effects on heated transitional boundary layers. ASME J. Heat Transfer 114, 338–347.CrossRefGoogle Scholar
Kottke, V. 1988 On the instability of laminar boundary layers along concave walls towards Görtler vortices. In Propagation in Nonequilibrium Systems (ed. Wesfreid, J. E. & Brand, H.), pp. 390–398. Springer.CrossRefGoogle Scholar
Kovasznay, L. S. G. 1953 Turbulence in supersonic flow. J. Aero. Sci. 20, 657–682.CrossRefGoogle Scholar
Leib, S. J., Wundrow, D. W. & Goldstein, M. E. 1999 Effect of free-stream turbulence and other vortical disturbances on a laminar boundary layer. J. Fluid Mech. 380, 169–203.CrossRefGoogle Scholar
Li, F. & Malik, M. R. 1995 Fundamental and subharmonic secondary instabilities of Görtler vortices. J. Fluid Mech. 297, 77–100.CrossRefGoogle Scholar
Saric, W. S. 1994 Görtler vortices. Annu. Rev. Fluid Mech. 26, 379–409.CrossRefGoogle Scholar
Schultz, M. P. & Volino, R. J. 2003 Effects of concave curvature on boundary layer transition under high free-stream turbulence conditions. ASME J. Turbomach. 125, 18–27.Google Scholar
Smith, A. M. O. 1955 On the growth of Taylor–Görtler vortices along highly concave walls. Q. J. Math. 13 (3), 233–262.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
Tani, I. 1962 Production of longitudinal vortices in the boundary layer along a concave wall. J. Geophys. Res. 67 (8), 3075–3080.CrossRefGoogle Scholar
Tani, I. & Aihara, Y. 1969 Görtler vortices and boundary-layer transition. Z. Angew. Math. Phys. 20, 609–618.CrossRefGoogle Scholar
Timoshin, S. N. 1990 Asymptotic analysis of a spatially unstable Görtler vortex spectrum. Izv. Akad. Nauk. SSSR Mekh. Zhid. i Gaza 25 (1), 25–33.Google Scholar
Volino, R. J. & Simon, T. W. 1995 Bypass transition in boundary layers including curvature and favourable pressure gradient effects. ASME J. Turbomach. 117, 166–174.CrossRefGoogle Scholar
Volino, R. J. & Simon, T. W. 2000 Spectral measurements in transitional boundary layers on a concave wall under high and low free-stream turbulence conditions. ASME J. Turbomach. 122, 450–457.CrossRefGoogle Scholar
Winoto, S. H. & Crane, R. I. 1980 Vortex structure in laminar boundary layers on a concave wall. Intl J. Heat Fluid Flow 2, 221–231.CrossRefGoogle Scholar
Wortman, F. X. 1969 Visualization of transition. J. Fluid Mech. 38, 473–480.CrossRefGoogle Scholar