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The onset of instability in unsteady boundary-layer separation

Published online by Cambridge University Press:  26 April 2006

K. W. Cassel ,
F. T. Smith  and
J. D. A. Walker
K. W. Cassel
Affiliation:
Department of Mechanical Engineering and Mechanics, Lehigh University, 19 Memorial Drive West, Bethlehem, PA 18015, USA
F. T. Smith
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
J. D. A. Walker
Affiliation:
Department of Mechanical Engineering and Mechanics, Lehigh University, 19 Memorial Drive West, Bethlehem, PA 18015, USA
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Abstract

The process of unsteady two-dimensional boundary-layer separation at high Reynolds number is considered. Solutions of the unsteady non-interactive boundary-layer equations are known to develop a generic separation singularity in regions where the pressure gradient is prescribed and adverse. As the boundary layer starts to separate from the surface, however, the external pressure distribution is altered through viscous—inviscid interaction just prior to the formation of the separation singularity; hitherto this has been referred to as the first interactive stage. A numerical solution of this stage is obtained here in Lagrangian coordinates. The solution is shown to exhibit a high-frequency inviscid instability resulting in an immediate finite-time breakdown of this stage. The presence of the instability is confirmed through a linear stability analysis. The implications for the theoretical description of unsteady boundary-layer separation are discussed, and it is suggested that the onset of interaction may occur much sooner than previously thought.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 315 , 25 May 1996 , pp. 223 - 256
Copyright
© 1996 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions. US Government Printing Office, Washington, DC.
Acarlar, M. S. & Smith, C. R. 1987a A study of hairpin vortices in a laminar boundary layer. Part 1. Hairpin vortices generated by hemisphere protuberances. J. Fluid Mech. 175, 1–41.Google Scholar
Acarlar, M. S. & Smith, C. R. 1987b A study of hairpin vortices in a laminar boundary layer. Part 2. Hairpin vortices generated by fluid injection. J. Fluid Mech. 175, 43–83.Google Scholar
Bhaskaran, R., Smith, F. T. & Timoshin, S. N. 1995 The appearance of boundary-layer instabilities as a velocity minimum develops. Proc. R. Soc. Lond. A 451, 331–339.Google Scholar
Bouard, R. & Coutanceau, M. 1980 The early stage of development of the wake behind an impulsively-started cylinder for 40 < Re < 104. J. Fluid Mech. 101, 583–607.Google Scholar
Brown, S. N., Cheng, H. K. & Smith, F. T. 1988 Nonlinear instabilities and break-up of separated flow. J. Fluid Mech. 193, 191–216.Google Scholar
Cassel, K. W. 1993 The effect of interaction on boundary-layer separation and breakdown. PhD dissertation, Lehigh University.
Cebeci, T. 1982 Unsteady separation. In Numerical and Physical Aspects of Aerodynamic Flows (ed. T. Cebeci), pp. 265–277. Springer.
Chuang, F. S. & Conlisk, A. T. 1989 Effect of interaction on the boundary layer induced by a convected rectilinear vortex. J. Fluid Mech. 200, 337–365.Google Scholar
Collins, W. M. & Dennis, S. C. R. 1973 Flow past an impulsively-started circular cylinder. J. Fluid Mech. 60, 105–127.Google Scholar
Cowley, S. J., Hocking, L. M. & Tutty, O. R. 1985 The stability of solutions of the classical unsteady boundary-layer equations. Phys. Fluids 28, 441–443.Google Scholar
Cowley, S. J., Dommelen, L. L. Van & Lam, S. T. 1990 On the use of Lagrangian variables in descriptions of unsteady boundary-layer separation. Phil. Trans. R. Soc. Lond. A 333, 343–378.Google Scholar
Doligalski, T. L., Smith, C. R. & Walker, J. D. A. 1994 Vortex interactions with walls. Ann. Rev. Fluid Mech. 26, 573–616.Google Scholar
Doligalksi, T. L. & Walker, J. D. A. 1984 The boundary layer induced by a convected two-dimensional vortex. J. Fluid Mech. 139, 1–28.Google Scholar
Elliott, J. W., Cowley, S. J. & Smith, F. T. 1983 Breakdown of boundary layers: (i) on moving surfaces; (ii) in semi-similar unsteady flow; (iii) in fully unsteady flow. Geophys. Astrophys. Fluid Dyn. 25, 77–138.Google Scholar
Head, M. R. & Bandyopadhyay, P. 1981 New aspects of turbulent boundary-layer structure. J. Fluid Mech. 107, 297–338.Google Scholar
Henkes, R. A. W. M. & Veldman, A. E. P. 1987 On the breakdown of the steady and unsteady interacting boundary-layer description. J. Fluid Mech. 179, 513–529.Google Scholar
Hoyle, J. M., Smith, F. T. & Walker, J. D. A. 1991 On sublayer eruption and vortex formation. Comput. Phys. Commun. 65, 151–157.Google Scholar
Krasny, R. 1986 A study of singularity formation in a vortex sheet by the point-vortex approximation. J. Fluid Mech. 167, 65–93.Google Scholar
Lam, S. T. 1988 On high-Reynolds-number laminar flows through a curved pipe, and past a rotating cylinder. PhD dissertation, University of London.
McCroskey, W. J. 1982 Unsteady airfoils. Ann. Rev. Fluid Mech. 14, 285–311.Google Scholar
Peridier, V. J., Smith, F. T. & Walker, J. D. A. 1991a Vortex-induced boundary-layer separation. Part 1. The unsteady limit problem Re → ∞. J. Fluid Mech. 232, 99–131.Google Scholar
Peridier, V. J., Smith, F. T. & Walker, J. D. A. 1991b Vortex-induced boundary-layer separation. Part 2. Unsteady interacting boundary-layer theory. J. Fluid Mech. 232, 133–165.Google Scholar
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. 1989 Numerical Recipes, The Art of Scientific Computing. Cambridge University Press.
Riley, N. & Vasantha, R. 1989 Unsteady high-Reynolds-number flows. J. Fluid Mech. 205, 243–262.Google Scholar
Ryzhov, O. S. & Smith, F. T. 1984 Short-length instabilities, breakdown and initial value problems in dynamic stall. Mathematika 31, 163–177.Google Scholar
Sears, W. R. & Telionis, D. P. 1975 Boundary-layer separation in unsteady flow. SIAM J. Appl. Maths 28, 215–235.Google Scholar
Shelley, M. J. 1992 A study of singularity formation in vortex-sheet motion by a spectrally accurate vortex method. J. Fluid Mech. 244, 493–526.Google Scholar
Smith, C. R., Walker, J. D. A., Haidari, A. H. & Sobrun, U. 1991 On the dynamics of near-wall turbulence. Phil. Trans. R. Soc. Lond. A 336, 131–175.Google Scholar
Smith, F. T. 1982 On the high Reynolds number theory of laminar flows. IMA J. Appl. Maths 28, 207–281.Google Scholar
Smith, F. T. 1988 Finite-time break-up can occur in any unsteady interacting boundary layer. Mathematika 35, 256–273.Google Scholar
Smith, F. T. & Bodonyi, R. J. 1985 On short-scale inviscid instabilities in flow past surface-mounted obstacles and other non-parallel motions. Aero. Q. June/July.
Smith, F. T. & Elliott, J. W. 1985 On the abrupt turbulent reattachment downstream of leadingedge laminar separation. Proc. Roy. Soc. Lond. A 401, 1–27.Google Scholar
Stewartson, K. 1964 The Theory of Laminar Boundary Layers in Compressible Fluids. Oxford University Press.
Tutty, O. R. & Cowley, S. J. 1986 On the stability and the numerical solution of the unsteady interactive boundary-layer separation. J. Fluid Mech. 168, 431–456.Google Scholar
Dommelen, L. L. Van 1981 Unsteady boundary-layer separation. PhD dissertation, Cornell University.
Dommelen, L. L. Van 1991 Lagrangian description of unsteady separation. Lect. Appl. Maths 28, 701–718.Google Scholar
Dommelen, L. L. Van & Shen, S. F. 1980 The spontaneous generation of the singularity in a separating laminar boundary layer. J. Comput. Phys. 38, 125–140.Google Scholar
Dommelen, L. L. Van & Shen, S. F. 1982 The genesis of separation. In Numerical and Physical Aspects of Aerodynamic Flows (ed. T. Cebeci), pp. 293–311. Springer.
Walker, J. D. A. 1978 The boundary layer due to a rectilinear vortex. Proc. R. Soc. Lond. A 359, 167–188.Google Scholar
Walker, J. D. A. 1990 Models based on the dynamical features of the wall layer. Appl. Mech. Rev. 43, S232–S239.Google Scholar