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The generation of Tollmien-Schlichting waves by free-stream turbulence

Published online by Cambridge University Press:  26 April 2006


P. W. Duck
Affiliation:
Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
A. I. Ruban
Affiliation:
Central Aerohydrodynamic Institute (TsAGI), Zhukovsky-3, Moscow reg., 140160, Russia
C. N. Zhikharev
Affiliation:
Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015, USA

Abstract

The phenomenon of Tollmien-Schlichting wave generation in a boundary layer by free-stream turbulence is analysed theoretically by means of asymptotic solution of the Navier-Stokes equations at large Reynolds numbers (Re → ∞). For simplicity the basic flow is taken to be the Blasius boundary layer over a flat plate. Free-stream turbulence is taken to be uniform and thus may be represented by a superposition of vorticity waves. Interaction of these waves with the flat plate is investigated first. It is shown that apart from the conventional viscous boundary layer of thickness O(Re−1/2), a ‘vorticity deformation layer’ of thickness O(Re−1/4) forms along the flat-plate surface. Equations to describe the vorticity deformation process are derived, based on multiscale asymptotic techniques, and solved numerically. As a result it is shown that a strong singularity (in the form of a shock-like distribution in the wall vorticity) forms in the flow at some distance downstream of the leading edge, on the surface of the flat plate. This is likely to provoke abrupt transition in the boundary layer. With decreasing amplitude of free-stream turbulence perturbations, the singular point moves far away from the leading edge of the flat plate, and any roughness on the surface may cause Tollmien-Schlichting wave generation in the boundary layer. The theory describing the generation process is constructed on the basis of the ‘triple-deck’ concept of the boundary-layer interaction with the external inviscid flow. As a result, an explicit formula for the amplitude of Tollmien-Schlichting waves is obtained.


Type
Research Article
Copyright
© 1996 Cambridge University Press

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References

Bodonyi, R. J., Welch, W. J. C., Duck, P. W. & Tadjfar, M. 1989 A numerical study of the interaction between unsteady free-stream disturbances and localized variations in surface geometry. J. Fluid Mech. 209, 285.Google Scholar
Brown, S. N. & Stewartson, K. 1973a On the propagation of disturbances in a laminar boundary layer I. Proc. Camb. Phil. Soc. 73, 493.Google Scholar
Brown, S. N. & Stewartson, K. 1973b On the propagation of disturbances in a laminar boundary layer II. Proc. Camb. Phil. Soc. 73, 503.Google Scholar
Dryden, H. L. 1956 Recent investigations of the problem of transition. Z. Flugwiss. 4, 89.Google Scholar
Duck, P. W. 1985 Laminar flow over unsteady humps: the formation of waves. J. Fluid Mech. 160, 465.Google Scholar
Duck, P. W. 1988 The effect of small surface perturbations on the pulsatile boundary layer on a semi-infinite flat plate. J. Fluid Mech. 197, 259.Google Scholar
Duck, P. W. 1990 Triple-deck flow over unsteady surface disturbances: the three-dimensional development of Tollmien-Schlichting waves. Computers Fluids 18, 1.Google Scholar
Fedorov, A. V. 1982 Generation of instability waves in a boundary-layer flow of compressible gas exposed to an acoustic field. Numer. Meth. Continuum Mech. (Novosibirsk) 13, 106.Google Scholar
Goldstein, M. E. 1985 Scattering of acoustic waves into Tollmien-Schlichting waves by small streamwise variation in surface geometry. J. Fluid Mech. 154, 509.Google Scholar
Goldstein, M. E. 1983 The evolution of Tollmien-Schlichting waves near a leading edge. J. Fluid Mech. 127, 59.Google Scholar
Goldstein, M. E. & Leib, S. J. 1993a Three-dimensional boundary-layer instability and separation induced by small amplitude streamwise vorticity in the upstream flow. J. Fluid Mech. 246, 21.Google Scholar
Goldstein, M. E. & Leib, S. J. 1993b A note on the distortion of a flat plate boundary layer by free-stream vorticity normal to the plate. J. Fluid Mech. 248, 531.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.Google Scholar
Guliaev, A. N., Kozlov, V. E., Kuznetsov, V. R., Mineev, B. I. & Secundov, A. N. 1989 Interaction of a laminar boundary layer with external disturbances. Izv. Akad. Nauk SSSR Mekh. Zhid. Gaza 6, 55.Google Scholar
Hunt, J. C. R. & Graham, J. M. R. 1978 Free-stream turbulence near plane boundaries. J. Fluid Mech. 84, 209.Google Scholar
Kerschen, E. J. 1991 Linear and Nonlinear Receptivity to Vertical Free-Stream Disturbances. ASME FED, vol. 114, p. 43.
Lin, C. C. 1946 On the stability of two-dimensional parallel flows. Part 3. Stability in a viscous fluid. Q. Appl. Maths 3, 277.Google Scholar
Morkovin, M. V. 1969 Critical evaluation of transition for laminar to turbulent shear layers with emphasis on hypersonically traveling bodies. Air Force Flight Dynamics Laboratory Rep. AFFDL-TR-68149.
Ruban, A. I. 1984 On Tollmien-Schlichting Wave Generation by Sound. Izv. Akad. Nauk SSSR Mekh. Zhid. Gaza 5, 44.Google Scholar
Ryzhov, O. S. & Timpeev, O. A. 1995 Interaction of a potential vortex with a local roughness on a smooth surface. J. Fluid Mech. 287, 2133.Google Scholar
Schubauer, G. B. & Skramsted, H. K. 1948 Laminar Boundary-Layer Oscillations and Transition on a Flat Plate. NACA Rep. 909.
Smith, F. T. 1979 On the non-parallel flow stability of the Blasius boundary layer. Proc. R. Soc. Lond. A 366, 91.Google Scholar
Smith, F. T., Doorly, D. J. & Rothmayer A. P. 1990 On displacement-thickness, wall-layer and mid-flow scales in turbulent boundary layers, and slugs of vorticity in channel and pipe flows. Proc. R. Soc. Lond. A 428, 255.Google Scholar
Stewartson, K. 1970 On laminar boundary layers near corners. Q. J. Mech. Appl. Maths 23, 137.Google Scholar
Sychev, V. V., Ruban, A. I., Sychev, Vic.V. & Korolev, G. L. 1987 Asymptotic Theory of Separated Flows. Moscow: Nauka.
Tanveer, S. & Speziale, C. G. 1992 Singularities of the Euler equation and hydrodynamic stability. ICASE Rep. 9254.Google Scholar
Terent'ev, E. D. 1981 Linear problem for vibration in subsonic boundary layer. Prikl. Mat. Mech. 45, 1049.Google Scholar
Terent'ev, E. D. 1984 Linear problem for vibration performing harmonic oscillations with supercritical frequency in subsonic boundary layer. Prikl. Mat. Mech. 48, 264.Google Scholar
Zhuk, V. I. & Ryzhov, O. S. 1980 Free interaction and stability of boundary layer in incompressible fluid flow. Proc. Akad. Nauk SSSR 253, 1326.Google Scholar

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