Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-01-09T22:18:54.447Z Has data issue: false hasContentIssue false
  • English
  • Français

Concerning marginal singularities in the boundary-layer flow on a downstream-moving surface

Published online by Cambridge University Press:  26 April 2006

S. N. Timoshin
S. N. Timoshin
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UKandCentral Aerohydrodynamical Institute, Zhukovsky, Russia, 140160
Get access Rights & Permissions [Opens in a new window]

Abstract

The formation of separation singularities in solutions of the classical boundary-layer equations is studied numerically and analytically for the case of a two-dimensional incompressible steady flow near a solid surface moving in the direction of the main stream. Unlike the previously studied regime of the incipient separation located at the maximum point in the external pressure distribution, the breakdown in this work occurs under an adverse pressure forcing and involves a regular flow field upstream of the Moore-Rott-Sears point with an algebraic non-analyticity downstream. Small deviations from the precisely regular approach to the singular point are shown to result in an exponential amplification of linear disturbances; in the subsequent nonlinear stage the solution terminates in a finite-distance blow-up singularity or, alternatively, continues in a regular fashion across the singular station. The case of asymptotically small slip velocities is considered and a connection with marginal separation on a fixed wall is discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 308 , 10 February 1996 , pp. 171 - 194
Copyright
© 1996 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

Brown, S. N. & Stewartson, K. 1983 On an integral equation of marginal separation. SIAM J. Appl. Maths 43, 1119–1126.Google Scholar
Chang, P. K. 1976 Control of Flow Separation. McGraw-Hill.
Cowley, S. J., Hocking, L. M. & Tutty, O. R. 1985 The stability of solutions of the classical unsteady boundary-layer equation. 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
Elliott, J. W., Smith, F. T. & Cowley, S. J. 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
Goldstein, S. 1948 On laminar boundary-layer flow near a position of separation. Q. J. Mech. Appl. Maths 1, 43–69.Google Scholar
Lam, S. T. 1988 On high-Reynolds-number laminar flows through a curved pipe, and past a rotating cylinder. PhD thesis., University of London.
Negoda, V. V. & Sychev, V. V. 1987 Concerning boundary layer on a rapidly rotating cylinder. Izv. Akad. Nauk. SSSR, Mekh. Zhidk. i Gaza 5, 36–45.Google Scholar
Nikolayev, K. V. 1982 Origin of the boundary-layer separation on a rotating cylinder in the flow of an incompressible fluid. Trudy TsAGI 13 (6), 32–39.Google Scholar
Riley, N. 1975 Unsteady laminar boundary layers. SIAM rev. 17, 274–297.Google Scholar
Ruban, A. I. 1981 Singular solution of the boundary-layer equations which can be extended continuously through the point of zero surface friction. Izv. Akad. Nauk. SSSR, Mekh. Zhidk. i Gaza 6, 42–52.Google Scholar
Ruban, A. I. 1982a Asymptotic theory of short separation regions on the leading edge of a slender airfoil. Izv. Akad. Nauk. SSSR, Mekh. Zhidk. i Gaza 1, 42–51.Google Scholar
Ruban, A. I. 1982b On stability of preseparating boundary layer on the leading edge of a slender airfoil. Izv. Akad. Nauk. SSSR, Mekh. Zhidk. i Gaza 6, 55–63.Google Scholar
Ruban, A. I. 1990 Marginal separation theory. In Separated Flows and Jets, Proc. IUTAM Symp., pp. 47–54. Springer.
Smith, F. T. 1982a On the high Reynolds number theory of laminar flows. IMA J. Appl. Maths 28, 207–281.Google Scholar
Smith, F. T. 1982b Concerning dynamic stall. Aeronaut. Q. 33, 331–352.Google Scholar
Smith, F. T. 1984 Concerning upstream influence in separating boundary layers and downstream influence in channel flow. Q. J. Mech. Appl. Maths 37, 389–399.Google Scholar
Smith, F. T. 1986 Steady and unsteady boundary-layer separation. Ann. Rev. Fluid Mech. 18, 197–220.
Smith, F. T., Brighton, P. W. M., Jackson, P. S. & Hunt, J. C. R. 1981 On boundary layer flow past two-dimensional obstacles. J. Fluid Mech. 113, 123–152.Google Scholar
Smith, F. T. & Elliott, J. W. 1985 On the abrupt turbulent reattachment downstream of leadingedge laminar separation. Proc. R. Soc. Lond. A 401, 1–27.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. 1958 On Goldstein's theory of laminar separation. Q. J. Mech. Appl. Maths 11, 399–410.Google Scholar
Stewartson, K. 1974 Multistructured boundary layers on flat plates and related bodies. Adv. Appl Mech. 14, 145–239.Google Scholar
Stewartson, K., Smith, F. T. & Kaups, K. 1982 Marginal separation. Stud. Appl. Maths 67, 45–61.Google Scholar
Stuart, J. T. 1963 In Laminar Boundary Layers (ed. L. Rosenhead), chap. IX. Oxford University Press.
Sychev, V. V. 1980 On certain singularities in solutions of the equations of boundary layer on a moving surface. Prikl. Matem. Mekh. 44, 831–838.Google Scholar
Sychev, V. V. 1987 Singular solution of the boundary-layer equations on a moving surface. Izv. akad. nauk. sssr, mekh. zhidk. i gaza 2, 43–52.Google Scholar
Sychev, V. V., Ruban, A. I., Sychev, V. V. & Korolev, G. L. 1987 Asymptotic Theory of Separated Flows. Nauka, Moscow.
Telionis, D. P. 1981 Unsteady Viscous Flows. Springer.
Telionis, D. P. & Werle, M. J. 1973 Boundary-layer separation from downstream moving boundaries. Trans. ASME J. Appl. Mech. 40, 369–374.Google Scholar
Timoshin, S. N. 1988 Removal of marginal separation in a pulsatile flow. Prikl. Matem. Mekh. 52, 77–81.Google Scholar
Timoshin, S. N. 1995 On short-wavelength instabilities in three-dimensional classical boundary layers. Eur. J. Mech. B/Fluids 14, 409–437.Google Scholar
Timoshin, S. N. & Smith, F. T. 1995 Singularities encountered in three-dimensional boundary layers under an adverse or favourable pressure gradient. Phil. Trans. R. Soc. Lond. A 352, 45–87.Google Scholar
Walton, A. G. & Smith, F. T. 1992 Properties of strongly nonlinear vortex/Tollmien-Schlichting-wave interactions. J. Fluid Mech. 224, 649–676.
Williams, J. C. 1977 Incompressible boundary-layer separation. In Ann. Rev. Fluid Mech. 9, 113–144.Google Scholar
Zametaev, V. B. 1987 Singular solution of the boundary-layer equations on a slender cone. Izv. akad. nauk. sssr, mekh. zhidk. i gaza 2, 65–72.Google Scholar
Zametaev, V. B. 1989 Formation of singularities in a three-dimensional boundary layer. Izv. Akad. Nauk. SSSR, Mekh. Zhidk. i Gaza 2, 58–64.Google Scholar