Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-18T21:43:57.002Z Has data issue: false hasContentIssue false
  • English
  • Français

Generation of Tollmien–Schlichting waves on interactive marginally separated flows

Published online by Cambridge University Press:  21 April 2006

M. E. Goldstein ,
S. J. Leib  and
S. J. Cowley
M. E. Goldstein
Affiliation:
Lewis Research Center, Cleveland, OH 44135, USA
S. J. Leib
Affiliation:
Lewis Research Center, Cleveland, OH 44135, USA
S. J. Cowley
Affiliation:
Imperial College of Science and Technology, Department of Mathematics, London SW7 2BZ, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is concerned with the interaction of very long-wavelength free-stream disturbances with the small but abrupt changes in the mean flow that occur near the minimum-skin-friction point in an interactive marginally separated boundary layer. We choose the source frequency so that the eigensolutions with that frequency have an ‘interactive’ structure in the region of marginal separation. The eigensolution wavelength scale must then differ from the lengthscale of the marginal separation and a composite expansion technique has to be used to obtain the solution.

The initial instability wave amplitude turns out to be exponentially small, but eventually dominates the original disturbance owing to its exponential growth. It then begins to decay but ultimately turns into a standard spatially growing Tollmien-Schlichting wave much further downstream.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 181 , August 1987 , pp. 485 - 517
Copyright
© 1987 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

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions. National Bureau of Standards.
Ackerberg, R. C. & Phillips, J. H. 1972 The unsteady laminar boundary layer on a semi-infinite flat plate due to small fluctuations in the magnitude of the free-stream velocity. J. Fluid Mech. 51, 137–157.CrossRefGoogle Scholar
Arbey, H. & Bataille, J. 1983 Noise generated by airfoil profiles placed in a uniform laminar flow. J. Fluid Mech. 134, 33–47.Google Scholar
Bers, A. 1975 Linear waves and instabilities. In Plasma Physics (ed. C. DeWitt & J. Peyraud), pp. 113–216. Gordon & Breach.
Briggs, R. J. 1964 Electron Stream Interaction With Plasmas. Massachusetts Institute of Technology Press.
Carrier, G. F., Krook, M. & Pearson, C. E. 1966 Functions of a Complex Variable. McGraw-Hill.
Crighton, D. G. & Leppington, F. G. 1974 Radiation properties of the semi-infinite vortex sheet: the initial-value problem. J. Fluid Mech. 64, 393–414.Google Scholar
Dingle, R. B. 1973 Asymptotic Expansions: Their Derivation and Interpretation. Academic.
Elliott, J. W. & Smith, F. T. 1987 Dynamic stall due to unsteady marginal separation. J. Fluid Mech. 179, 489–512.Google Scholar
Fink, M. R. 1975 Prediction of airfoil tone frequencies. J. Aircraft 12, 118–120.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., 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
Goldstein, S. 1948 On laminar boundary layer flow near a position of separation. Q. J. Mech. Appl. Maths 1, 43–69.Google Scholar
Kachanov, Y. S., Kozlov, V. V. & Levchenko, V. Y. 1978 Occurrence of Tollmien—Schlichting waves in the boundary layer under the effect of external perturbations. Fluid Dyn. 13, 704–711 (Engl. Transl.).Google Scholar
Lam, S. H. & Rott, N. 1960 Theory of linearized time-dependent boundary layers. AFOSR TN 60–1100.
Lighthill, M. J. 1954 The response of laminar skin friction and heat transfer to fluctuations in the stream velocity. Proc. R. Soc. Lond. A224, 1–23.Google Scholar
Lyne, W. H. 1970 Steady streaming associated with some unsteady viscous flows. Ph.D. thesis, Dept. of Mathematics, Imperial College of Science and Technology, London.
Morkovin, M. V. 1969 Critical evaluation of transition from laminar to turbulent shear layer with emphasis on hypersonically traveling bodies. AFFDL-TR-68–149.
Moore, D. W. 1979 The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc. R. Soc. Lond. A365, 105–119.Google Scholar
Murdock, J. W. 1980 The generation of a Tollmien—Schlichting wave by a sound wave. Proc. R. Soc. Lond. A372, 517–534.Google Scholar
Nayfeh, A. H. 1973 Perturbation Methods. Wiley.
Olver, F. W. J. 1959 Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders. J. Res. Natn. Bur. Stand. B63, 131–169.Google Scholar
Paterson, R. W., Vogt, P. G., Fink, M. R. & Munch, C. L. 1972 Vortex noise of isolated airfoils. AIAA Paper 72–656.
Polyakov, N. F. 1973a Method of study of flow characteristics in low turbulence wind tunnel and transition phenomena in an incompressible boundary layer. Candidate's dissertation, Institute of Theoretical and Applied Mechanics, Siberian Section USSR Academy of Sciences, Novosibirsk.
Polyakov, N. F. 1973b Aerophysical research, issue 2. Sb. nauch. trudov [Collected Scientific Works] p. 88. Institute of Theoretical and Applied Mechanics, Siberian Section USSR Academy of Sciences, Novosibirsk.
Reshotko, E. 1976 Boundary-layer stability and transition. In Ann. Rev. Fluid Mech. 8, 311–349.Google Scholar
Reid, W. H. 1965 The stability of parallel flows. In Basic Developments in Fluid Mechanics, Vol. 1 (ed. M. Holt), pp. 249–307. Academic.
Ruban, A. I. 1982a Asymptotic theory of short separation regions on the leading edge of a slender airfoil. Fluid Dyn. 17, 33–41 (Engl. Transl.).Google Scholar
Ruban, A. I. 1982b Stability of preseparation boundary layer on the leading edge of a thin airfoil. Fluid Dyn. 17, 860–867 (Engl. Transl.).Google Scholar
Ruban, A. I. 1985 On the generation of Tollmien—Schlichting waves by sound. Fluid Dyn. 19, 709–716 (Engl. Transl.).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
Smith, F. T. 1979 On the non-parallel flow stability of the Blasius boundary layer. Proc. R. Soc. Lond. A336, 91–109.Google Scholar
Smith, F. T. 1982 Concerning dynamic stall. Aero. Q. 33, 331–352.Google Scholar
Smith, F. T. 1986 Steady and unsteady boundary-layer separation. In Ann. Rev. Fluid Mech. 18, 197–220.Google Scholar
Smith, F. T. & Elliot, J. W. 1985 On the abrupt turbulent reattachment downstream of a leading edge separation. Proc. R. Soc. Lond. A 399, 25–55.Google Scholar
Stewartson, K. 1981 Marginally stable inviscid flows with critical layers. IMA J. Appl. Maths 27, 133–175.Google Scholar
Stewartson, K., Smith, F. T. & Kaups, K. 1982 Marginal separation. Stud. Appl. Maths 67, 45–61.Google Scholar
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics. Parabolic.
Vorob'Yev, N. F., Polyakov, N. F., Shashkina, G. N. & Shcherbakov, V. A.1976 Physical gasdynamics. Sb. nauch. trudov [Collected Scientific Works], p. 101. Institute of Theoretical and Applied Mechanics, Siberian Section USSR Academy of Sciences, Novosibirsk.