Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T19:37:53.707Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear evolution of Rayleigh waves in an initial value context: non-symmetric input and cross-flow

Published online by Cambridge University Press:  26 February 2010

T. Allen ,
S. N. Brown  and
F. T. Smith
T. Allen
Affiliation:
Room 247, Ocean Applications, The Met. Office, London Road, Bracknell, RG12 2SZ
S. N. Brown
Affiliation:
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT
F. T. Smith
Affiliation:
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT
Get access

Abstract

In recent papers the present authors considered the effects of small cross-flow on the evolution of two unequal oblique waves. In these studies the relative size of the crossflow meant that a diffusion (or buffer) layer was required around the critical layer to smooth out the algebraic growth in the mean-flow distortion generated by the nonlinear critical-layer interactions. The present analysis increases the cross-flow to an order of magnitude such that the buffer and critical layers coalesce. In this instance the nonlinear critical layer contains viscous as well as nonequilibrium effects. The resulting amplitude equations are solved for perturbations initiated at a fixed station in the flow.

Type
Research Article
Information
Mathematika , Volume 45 , Issue 2 , December 1998 , pp. 217 - 243
Copyright
Copyright © University College London 1998

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

Allen, T., Brown, S. N. and Smith, F. T.. On vortex/wave interactions. Part 2: Originating from axisymmetric flow with swirl. J. Fluid Mech., 325 (1996), 145–161.CrossRefGoogle Scholar
Bassom, A. P. and Gajjar, J. S. B.. Non-stationary cross-flow vortices in three-dimensional boundary-layer flows. Proc. R. Soc. Land. A, 417 (1988), 179–212.Google Scholar
Benney, D. J. and Chow, C.. A mean flow first harmonic theory for hydrodynamic instabilities. Stud. Appl. Maths., 80 (1989), 37–73.CrossRefGoogle Scholar
Brown, P. G., Brown, S. N., Smith, F. T. and Timoshin, S. N.. On the starting process of strongly nonlinear vortex/Rayleigh-wave interactions. Mathematika, 40 (1993), 7–29.CrossRefGoogle Scholar
Brown, S. N. and Smith, F. T.. On vortex/wave interactions. Part 1: Non-symmetric input and cross-flow in boundary layers. J. Fluid Mech., 307 (1996), 101–133.Google Scholar
Davis, D. A. R. and Smith, F. T.. Influence of cross-flow on nonlinear Tollmien Schlichting/vortex interaction. Proc. R. Soc. Lond. A, 446 (1994), 319–340.Google Scholar
Gajjar, J. S. B.. On the nonlinear evolution of a stationary cross-flow vortex in a fully threedimensional boundary layer flow. NASA Contractor Report, 198405 (1995).CrossRefGoogle Scholar
Gajjar, J. S. B.. Nonlinear stability of non-stationary cross-flow vortices in compressible boundary layers. Stud. Appl. Math., 96 (1996), 53–84.CrossRefGoogle Scholar
Goldstein, M. E. and Choi, S. -W.. Nonlinear evolution of interacting oblique waves on two-dimensional shear layers. J. Fluid Mech., 207 (1989), 97–120.Google Scholar
Goldstein, M. E. and Leib, S. J.. Nonlinear evolution of oblique waves on compressible shear layers. J. Fluid Mech., 207 (1989), 73–96.CrossRefGoogle Scholar
Hall, P. and Smith, F. T.. The nonlinear interaction of Tollmien Schlichting waves and Taylor-Gortler vortices in curved channel flow. Proc. R. Soc. Lond. A, 417 (1988), 255–282.Google Scholar
Hall, P. and Smith, F. T.. Nonlinear Tollmien-Schlichting/vortex interaction in boundary layers. Eur. J. Mech. B/Fluids, 8 (1989), 179–205.Google Scholar
Hall, P. and Smith, F. T.. Proc. ICASE workshop on instability and transition, vol. II, Hussaini, M. Y. & Voigt, R. G., eds., (Springer, 1990), 5 39.Google Scholar
Hall, P. and Smith, F. T.. On strongly nonlinear vortex/wave interactions in boundary layer transition. J. Fluid Mech., 227 (1991), 641–666.CrossRefGoogle Scholar
Hickernell, F. J.. Time-dependent critical layers in shear flows on the beta plane. J. Fluid Mech., 142 (1984), 431–449.CrossRefGoogle Scholar
Savin, D. J.. Linear and nonlinear aspects of interacting boundary-layer transition. Ph.D. Thesis (University of London, 1996).Google Scholar
Smith, F. T.. On transition over surface roughnesses. A.I.A.A. paper no. 96-1992, presented at New Orleans meeting, June 17–20, 1996.CrossRefGoogle Scholar
Smith, F. T., Brown, S. N. and Brown, P. G.. Initiation of three-dimensional non-linear transition paths from an inflectional profile. Eur. J. Mech. B/Fluids, 12 (1993), 447–473.Google Scholar
Wu, X.. The non-linear evolution of high-frequency resonant-triad waves in an oscillatory Stokes’ layer at high Reynolds number. J. Fluid Mech., 245 (1992), 553–597.CrossRefGoogle Scholar
Wu, X., Private Communication. (1996).Google Scholar
Wu, X., Lee, S. S. and Cowley, S. J.. 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 (1993), 681–721.CrossRefGoogle Scholar