Hostname: page-component-5c6d5d7d68-pkt8n Total loading time: 0 Render date: 2024-08-22T23:21:02.462Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear instability of the wake behind a flat plate placed parallel to a uniform stream

Published online by Cambridge University Press:  26 April 2006

D. T. Papageorgiou  and
F. T. Smith
D. T. Papageorgiou
Affiliation:
Department of Chemical Engineering, City College of New York, Convent Avenue and 140th Street, New York, NY 10031, USA
F. T. Smith
Affiliation:
University College London, Department of Mathematics, Gower Street, London WC1 E6BT, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The growth of linear disturbances in the high-Reynolds-number laminar wake of a flat plate aligned with a uniform stream is investigated. The theory is developed rationally by use of appropriate wake profiles which originate at the trailing edge as double Blasius distributions and thereafter satisfy the equations of motion, in contrast to previous theoretical work where model profiles are used. We also emphasize the structures and scales of the instability in order to provide a rational basis for the development of nonlinear analyses as opposed to existing ad hoc ones. Disturbances, in the near wake, respond according to the Rayleigh equation which is considered analytically for short-, long- and neutral-wave solutions. For more general stability characteristics eigensolutions must be obtained numerically. We calculate these at successive wake stations for ‘improved’ basic flow profiles which are obtained as solutions of the wake boundary-layer equations. Our numerical results indicate fairly good agreement with the asymptotic theory and some experimental data (see §7).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 208 , November 1989 , pp. 67 - 89
Copyright
© 1989 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

Betchov, R. & Criminale, W. O. 1967 Stability of Parallel Flows. Academic.
Cebeci, T., Thiele, F., Williams, P. G. & Stewartson, K. 1979 On the calculation of symmetric wakes. I. Two-dimensional flows. Numer. Heat Transfer 2, 35–60.Google Scholar
Curle, N. 1957 On hydrodynamic stability in unlimited fields of viscous flow. Proc. R. Soc. Lond. A 238, 489–501.Google Scholar
Drazin, P. G. & Howard, L. N. 1962 The stability to long waves of parallel inviscid flow. J. Fluid Mech. 14, 257–283.Google Scholar
Drazin, P. G. & Howard, L. N. 1966 Hydrodynamic stability of parallel flow of inviscid fluid. Adv. Appl. Mech. 9, 1–89.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Gaster, M. 1968 Growth of disturbances in both space and time. Phys. Fluids 11, 723–727.Google Scholar
Goldstein, S. 1930 Concerning some solutions of the boundary layer equations in hydrodynamics. Proc. Camb. Phil. Soc. 26, 1–30.Google Scholar
Hollingdale, S. H. 1940 Stability and configuration of the wakes produced by solid bodies moving through fluids. Phil. Mag. (7) 29, 209–257.Google Scholar
Keller, H. B. & Cebeci, T. 1971 2nd Intl Conf. Num. Meth. Fluid Dyn., Berkeley, p. 92. Springer.
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
McKoen, C. H. 1957 Aero. Res. Coun., Land., Rep. 303.
Mattingly, G. E. & Criminale, W. O. 1972 The stability of an incompressible wake. J. Fluid Mech. 51, 233–272.Google Scholar
Messiter, A. F. 1970 Boundary layer flow near the trailing edge of a flat plate. SIAM J. Appl. Maths 18, 241–257.Google Scholar
Miksad, R. W., Jones, F. L., Kim, Y. C. & Khandra, L. 1982 Experiments on the role of amplitude and phase modulation during transition to turbulence. J. Fluid Mech. 123, 1–29.Google Scholar
Miksad, R. W., Jones, F. L. & Powers, E. J. 1983 Measurements of nonlinear interactions during natural transition of a symmetric wake. Phys. Fluid 26, 1402–1409.Google Scholar
Papageorgiou, D. T. & Smith, F. T. 1988 Theoretical investigation of the nonlinear instability of the wake behind a flat plate placed parallel to a uniform stream. Proc. R. Soc. Lond. A 419, 1–28.Google Scholar
Sato, H. & Kuriki, K. 1961 The mechanism of transition in the wake of a thin flat plate placed parallel to a uniform flow. J. Fluid Mech. 11, 321–352.Google Scholar
Smith, F. T. 1974 Boundary layer flow near a discontinuity in wall conditions. J. Inst. Maths Applies 13, 127–145.Google Scholar
Stewartson, K. 1969 On the flow near the trailing edge of a flat plate-II. Mathematika 16, 106–121.Google Scholar
Taneda, S. 1965 The stability of two-dimensional laminar wakes at low Reynolds numbers. J. Phys. Soc. Japan 18, 288.Google Scholar
Tollmien, W. 1935 General stability criterion of laminar velocity distributions (translated from the German original). Tech. Memo. Nail Adv. Comm. Aero. Wash. 792.Google Scholar