Skip to main content Accessibility help
×
Home

Interaction of oblique instability waves with weak streamwise vortices

  • M. E. Goldstein (a1) and David W. Wundrow (a2)

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.

Copyright

References

Hide All
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, 445469.
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, 475495.
Benney, D. J. & Bergeron, R. F. 1969 A new class of nonlinear waves in parallel flows. Stud. Appl. Maths 48, 181204.
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, 6592.
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. 261275. Springer.
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18, 487488.
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), 724735.
Goldstein, M. E. & Choi, S. W. 1989 Nonlinear evolution of interacting oblique waves on two-dimensional shear layers. J. Fluid Mech. 207, 97120. Also Corrigendum, J. Fluid Mech. 216, 659–663.
Goldstein, M. E. & Durbin, P. A. 1986 Nonlinear critical layers eliminate the upper branch of spatially growing Tollmien—Schlichting waves. Phys. Fluids 29, 23442345.
Goldstein, M. E. & Lee, S. S. 1992 Fully coupled resonant-triad interaction in an adverse-pressure-gradient boundary layer. J. Fluid Mech. 245, 523551.
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, 2141.
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, 231260.
Hall, P. & Horseman, N. J. 1991 The linear inviscid secondary instability of longitudinal vortex structures in boundary layers. J. Fluid Mech. 232, 357375.
Hall, P. & Seddougui, S. 1989 On the onset of three-dimensionality and time dependence in Görtler vortices. J. Fluid Mech. 204, 405420.
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, 255282.
Hall, P. & Smith, F. T. 1991 On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641666.
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. 7793. 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, 10001004.
Kachanov, Yu. S. 1987 On the resonant nature of the breakdown of a laminar boundary layer. J. Fluid Mech. 184, 4374.
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. 6174. 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, 209247.
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, 134.
Kovasznay, L. S. G., Komoda, H. & Vasudeva, B. R. 1962 Detailed flow field in transition. In Proc. Heat Transfer and Fluid Mech. Inst., pp. 126. Stanford University Press.
Landahl, M. T. 1990 On sublayer streaks. J. Fluid Mech. 212, 593614.
Lee, S. S. 1994 Critical-layer analysis of fully coupled resonant-triad interaction in a boundary layer. Submitted to J. Fluid Mech.
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.
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, 85106.
Messiter, A. F. 1970 Boundary-layer flow near the trailing edge of a flat plate. SIAM J. Appl. Maths 18, 241257.
Nayfeh, A. H. 1981 Effect of streamwise vortices on Tollmien—Schlichting waves. J. Fluid Mech. 107, 441453.
Nayfeh, A. H. & Al-Maaitah, A. 1988 Influence of streamwise vortices on Tollmien—Schlichting waves. Phys. Fluids. 31, 35433549.
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), 362371.
Rudman, S. & Rubin, S. G. 1968 Hypersonic viscous flow over slender bodies with sharp leading edges. AIAA J. 6, 18831890.
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, 2555.
Smith, F. T. & Walton, A. G. 1989 Nonlinear-interaction of near-planar TS waves and longitudinal vortices in boundary-layer transition. Mathematika 36, 262289.
Stewartson, K. 1969 On the flow near the trailing edge of a flat plate II. Mathematika 16, 106121.
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.
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, 681721.
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, 343372.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Related content

Powered by UNSILO

Interaction of oblique instability waves with weak streamwise vortices

  • M. E. Goldstein (a1) and David W. Wundrow (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.