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Interaction of oblique instability waves with weak streamwise vortices

Published online by Cambridge University Press:  26 April 2006

M. E. Goldstein  and
David W. Wundrow
M. E. Goldstein
Affiliation:
NASA, Lewis Research Center, Cleveland, OH 44135, USA
David W. Wundrow
Affiliation:
Nyma, Inc., Lewis Research Center Group, Cleveland, OH 44135, USA
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Abstract

This paper is concerned with the effect of a weak spanwise-variable mean-flow distortion on the growth of oblique instability waves in a Blasius boundary layer. The streamwise component of the distortion velocity initially grows linearly with increasing streamwise distance, reaches a maximum, and eventually decays through the action of viscosity. This decay occurs slowly and allows the distortion to destabilize the Blasius flow over a relatively large streamwise region. It is shown that even relatively weak distortions can cause certain oblique Rayleigh instability waves to grow much faster than the usual two-dimensional Tollmien–Schlichting waves that would be the dominant instability modes in the absence of the distortion. The oblique instability waves can then become large enough to interact nonlinearly within a common critical layer. It is shown that the common amplitude of the interacting oblique waves is governed by the amplitude evolution equation derived in Goldstein & Choi (1989). The implications of these results for Klebanoff-type transition are discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 284 , 10 February 1995 , pp. 377 - 407
Copyright
© 1995 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions. US National Bureau of Standards.
Bennett, J. & Hall, P. 1988 On the secondary instability of Taylor–Görtler vortices to Tollmien—Schlichting waves in fully developed flows. J. Fluid Mech. 186, 445–469.Google Scholar
Bennett, J., Hall, P. & Smith, F. T. 1991 The strong nonlinear interaction of Tollmien—Schlichting waves and Taylor–Görtler vortices in curved channel flow. J. Fluid Mech. 223, 475–495.Google Scholar
Benney, D. J. & Bergeron, R. F. 1969 A new class of nonlinear waves in parallel flows. Stud. Appl. Maths 48, 181–204.Google Scholar
Bodonyi, R. J. & Smith, F. T. 1981 The upper branch stability of the Blasius boundary layer including non-parallel flow effects. Proc. R. Soc. Lond. A 375, 65–92.Google Scholar
Choudhari, M., Hall, P. & Streett, C. 1992 On the spatial evolution of long-wavelength Görtler vortices governed by a viscous—inviscid interaction. Part 1: the linear case. ICASE Rep. 92–31.
Cowley, S. J. 1987 High frequency Rayleigh instability of Stokes layers. In Stability of Time Dependent and Spatially Varying Flows (ed. D. L. Dwoyer & M. Y. Hussaini), pp. 261–275. Springer.
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18, 487–488.Google Scholar
Goldstein, M. E. 1976 Aeroacoustics. McGraw-Hill.
Goldstein, M. E. 1994 Nonlinear interactions between oblique instability waves on nearly parallel shear flows. Phys. Fluids 6, (part 2), 724–735.Google Scholar
Goldstein, M. E. & Choi, S. W. 1989 Nonlinear evolution of interacting oblique waves on two-dimensional shear layers. J. Fluid Mech. 207, 97–120. Also Corrigendum, J. Fluid Mech. 216, 659–663.Google Scholar
Goldstein, M. E. & Durbin, P. A. 1986 Nonlinear critical layers eliminate the upper branch of spatially growing Tollmien—Schlichting waves. Phys. Fluids 29, 2344–2345.Google Scholar
Goldstein, M. E. & Lee, S. S. 1992 Fully coupled resonant-triad interaction in an adverse-pressure-gradient boundary layer. J. Fluid Mech. 245, 523–551.Google 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.Google Scholar
Goldstein, M. E., Leib, S. J. & Cowley, S. J. 1992 Distortion of a flat-plate boundary layer by free-stream vorticity normal to the plate. J. Fluid Mech. 237, 231–260.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
Hall, P. & Seddougui, S. 1989 On the onset of three-dimensionality and time dependence in Görtler vortices. J. Fluid Mech. 204, 405–420.Google Scholar
Hall, P. & Smith, F. T. 1988 The nonlinear interaction of Tollmien—Schlichting waves and Taylor—Görtler vortices in curved channel flows. Proc. R. Soc. Lond. A 417, 255–282.Google Scholar
Hall, P. & Smith, F. T. 1991 On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641–666.Google Scholar
Hama, F. R. & Nutant, J. 1963 Detailed flow observation in the transition process in a thick boundary layer. In Proc. Heat Transfer and Fluid Mech. Inst., pp. 77–93. Stanford University Press.
Henningson, D. S. 1987 stability of parallel inviscid shear flow with mean spanwise variation. FFA TN 1987-57. The aeronautical research institute of sweden, aerodynamics department.
Herbert, T. & Lin, N. 1993 Studies of boundary-layer receptivity with the parabolized stability equations. AIAA Paper 93-3053.
Horseman, N. J. 1991 Some centrifugal instabilities in viscous flows. PhD thesis, Exeter University.
Hultgren, L. S. & Gustavsson, L. H. 1981 Algebraic growth of disturbances in a laminar boundary layer. Phys. Fluids 24, 1000–1004.Google Scholar
Kachanov, Yu. S. 1987 On the resonant nature of the breakdown of a laminar boundary layer. J. Fluid Mech. 184, 43–74.Google Scholar
Kachanov, Yu. S., Kozlov, V. V., Levchenko, V. Ya. & Ramazanov, M. P. 1985 On nature of K-breakdown of a laminar boundary layer. In Laminar—Turbulent Transition (ed. V. V. Kozlov). pp. 61–74. Springer.
Kachanov, Yu. S. & Levchenko, V. Ya. 1984 The resonant interaction of disturbances at laminar—turbulent transition in a boundary layer. J. Fluid Mech. 138, 209–247.Google Scholar
Klebanoff, P. S. & Tidstrom, K. D. 1959 Evolution of amplified waves leading to transition in a boundary layer with zero pressure gradient. NASA TN, D-195.
Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 The three-dimensional nature of boundary layer instability. J. Fluid Mech. 12, 1–34.Google Scholar
Kovasznay, L. S. G., Komoda, H. & Vasudeva, B. R. 1962 Detailed flow field in transition. In Proc. Heat Transfer and Fluid Mech. Inst., pp. 1–26. Stanford University Press.
Landahl, M. T. 1990 On sublayer streaks. J. Fluid Mech. 212, 593–614.Google Scholar
Lee, S. S. 1994 Critical-layer analysis of fully coupled resonant-triad interaction in a boundary layer. Submitted to J. Fluid Mech.Google Scholar
Leib, S. J. & Lee, S. S. 1994 Nonlinear evolution of a pair of oblique instability waves in a supersonic boundary layer. To appear in J. Fluid Mech.Google Scholar
Mankbadi, R. R., Wu, X. & Lee, S. S. 1993 A critical-layer analysis of the resonant triad in boundary-layer transition: nonlinear interactions. J. Fluid Mech. 256, 85–106.Google Scholar
Messiter, A. F. 1970 Boundary-layer flow near the trailing edge of a flat plate. SIAM J. Appl. Maths 18, 241–257.Google Scholar
Nayfeh, A. H. 1981 Effect of streamwise vortices on Tollmien—Schlichting waves. J. Fluid Mech. 107, 441–453.Google Scholar
Nayfeh, A. H. & Al-Maaitah, A. 1988 Influence of streamwise vortices on Tollmien—Schlichting waves. Phys. Fluids. 31, 3543–3549.Google Scholar
Nishioka, M., Asai, M. & Iida, S. 1979 In Laminar—Turbulent Transition. IUTAM Mtg. Stuttgart.
Prandtl, L. 1935 Aerodynamic Theory 3, p. 3. Springer.
Rozhko, S. B. & Ruban, A. I. 1987 Longitudinal—transverse interaction in a three-dimensional boundary layer. Fluid Dyn. 22(3), 362–371.Google Scholar
Rudman, S. & Rubin, S. G. 1968 Hypersonic viscous flow over slender bodies with sharp leading edges. AIAA J. 6, 1883–1890.Google Scholar
Smith, F. T. & Burggraf, O. R. 1985 On the development of large-sized short-scale disturbances in boundary layers. Proc. R. Soc. Lond. A 399, 25–55.Google Scholar
Smith, F. T. & Walton, A. G. 1989 Nonlinear-interaction of near-planar TS waves and longitudinal vortices in boundary-layer transition. Mathematika 36, 262–289.Google Scholar
Stewartson, K. 1969 On the flow near the trailing edge of a flat plate II. Mathematika 16, 106–121.Google Scholar
Stuart, J. T. 1965 The production of intense shear layers by vortex stretching and convection. NPL Aero. Res. Rep. 1147. Also NATO AGARD Rep. 514.Google Scholar
Wu, X., Lee, S. S. & Cowley, S. J. 1993 On the weakly nonlinear three-dimensional instability of shear layers to pairs of oblique waves: the Stokes layer as a paradigm. J. Fluid Mech. 253, 681–721.Google Scholar
Wundrow, D. W. & Goldstein, M. E. 1994 Nonlinear instability of a uni-directional transversely sheared mean flow. NASA TM 106779.
Wundrow, D. W., Hultgren, L. S. & Goldstein, M. E. 1994 Interaction of oblique instability waves with a nonlinear plane wave. J. Fluid Mech. 264, 343–372.Google Scholar