Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-18T21:58:23.900Z Has data issue: false hasContentIssue false
  • English
  • Français

Three-dimensional boundary-layer instability and separation induced by small-amplitude streamwise vorticity in the upstream flow

Published online by Cambridge University Press:  26 April 2006

M. E. Goldstein  and
S. J. Leib
M. E. Goldstein
Affiliation:
NASA Lewis Research Center, Cleveland, OH 44135, USA
S. J. Leib
Affiliation:
Sverdrup Technology, Inc., Lewis Research Center Group, Cleveland, OH 44135, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the effects of a small-amplitude, steady, streamwise vorticity field on the flow over an infinitely thin flat plate in an otherwise uniform stream. We show how the initially linear perturbation, ultimately leads to a small-amplitude but nonlinear cross-flow far downstream from the leading edge. This motion is imposed on the boundary-layer flow and eventually causes the boundary layer to separate. The streamwise velocity profiles within the boundary layer become inflexional in localized spanwise regions just upstream of the separation point. The flow in these regions is therefore susceptible to rapidly growing inviscid instabilities.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 246 , January 1993 , pp. 21 - 41
Copyright
© 1993 Cambridge University Press

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

Benney, D. J. 1964 Finite amplitude effects in an unstable boundary layer. Phys. Fluids 7. 319– 326.Google Scholar
Benney, D. J. & Lin, C. C. 1960 On the secondary motion induced by oscillations in shear flow. Phys. Fluids 3, 656– 657.Google Scholar
Blackwelder, R. F. 1983 Analogies between transitional and turbulent boundary layers. Phys. Fluids 26, 2807– 2815.Google Scholar
Brown, S. N. 1965 Singularities associated with separating boundary layers. Phil. Trans. Roy. Soc. A 257, 409– 444.Google Scholar
Cebeci, T., Khattab, A. A. & Stewartson, K. 1981 Three-dimensional laminar boundary layers and the ok of accessibility. J. Fluid Mech. 107, 57– 87.Google Scholar
Cebeci, T. & Su, W. 1988 Separation of three-dimensional laminar boundary layers on a prolate spheroid. J. Fluid. Mech. 191, 47– 77.Google Scholar
Crow, S. C. 1966 The spanwise perturbation of two dimensional boundary layers. J. Fluid Mech. 24, 153– 164.Google Scholar
Ersoy, S. & Walker, J. D. A. 1985 Viscous flow induced by counter-rotating vortices. Phys. Fluids 28, 2687– 2698.Google Scholar
Ersoy, S. & Walker, J. D. A. 1986 Flow induced at a wall by a vortex pair. AIAA J. 24, 1597– 1605.Google Scholar
Goldstein, M. E. 1978 Unsteady vortical and entropic distortions of potential flows round arbitrary obstacles. J. Fluid Mech. 89, 433– 468.Google Scholar
Goldstein, M. E. 1983 The evolution of Tollmien-Schlichting waves near a leading edge. J. Fluid Mech. 127, 59– 81.Google Scholar
Goldstein, M. E. 1985 Scattering of acoustic waves into Tollmien-Schlichting waves by small streamwise variations in surface geometry. J. Fluid Mech. 154, 509– 529.Google Scholar
Goldstein, M. E., Leib, S. J. & Cowley, S. J. 1987 Generation of Tollmien-Schlichting waves on interactive marginally separated flows. J. Fluid Mech. 181. 485– 517.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 (referred to herein as I).Google Scholar
Goldstein, M. E., Sockol, P. M. & Sanz, J. 1983 The evolution of Tollmien-Schlichting waves near a leading edge. Part 2. Numerical determination of amplitudes. J. Fluid Mech. 129, 443– 453.Google Scholar
Hall, P. 1988 The nonlinear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 193, 243– 266.Google Scholar
Hall, P. 1990 Görtler vortices in growing boundary layers: the leading edge receptivity problem, linear growth and the nonlinear breakdown stage. Mathematica 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.Google Scholar
Hunt, J. C. R. & Carruthers, D. J. 1990 Rapid distortion theory and the ‘problems’ of turbulence. J. Fluid Mech. 212, 497– 532.Google Scholar
Johnston, L. J. 1990 An upwind scheme for the three dimensional boundary layer equations. Intl J. Numer. Meth. Fluids 11, 1043– 1073.Google Scholar
Lighthill, M. J. 1963 In Laminar Boundary Layers (ed. L. Rosenhead), p. 74. Dover.
Michalke, A. 1965 On spatially growing disturbances in an inviscid shear layer. J. Fluid Mech. 23, 521– 544.Google Scholar
Noble, B. 1958 Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations. Pergamon.
Prandtl, L. 1935 Aerodynamic Theory. Springer.
Raetz, G. S. 1957 A method of calculating three dimensional boundary layers of compressible flows. Northrop Corp. Rep. NAI 58– 73.Google Scholar
Roache, P. J. 1976 Computational Fluid Dynamics. Hermosa.
Rozhko, S. B. & Ruban, A. I. 1987 Longitudinal-transverse interaction in a three dimensional boundary layer. Fluid Dyn. 23, 362– 371.Google Scholar
Stewartson, K., Cebeci, T. & Chang, K. C. 1980 A boundary layer collision in a curved duct. Q. J. Mech. Appl. Maths 33, 59– 75.Google Scholar
Stewartson, K. & Simpson, C. J. 1982 On a singularity initiating a boundary layer collision. Q. J. Mech. Appl. Maths 35, 1– 16.Google Scholar
Stuart, J. T. 1988 Nonlinear Euler partial differential equations: singularities in their solution. In Proc. Symp. Honor of C.C. Lin (ed. D. J. Benney, Chi Yuan, F. H. Shu), pp. 81– 95. World Scientific.