Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T06:53:23.140Z Has data issue: false hasContentIssue false
  • English
  • Français

Boundary-layer receptivity for a parabolic leading edge

Published online by Cambridge University Press:  26 April 2006

P. W. Hammerton  and
E. J. Kerschen
P. W. Hammerton
Affiliation:
Department of Aerospace & Mechanical Engineering, University of Arizona, Tucson, AZ 85721, USA Present address: School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK.
E. J. Kerschen
Affiliation:
Department of Aerospace & Mechanical Engineering, University of Arizona, Tucson, AZ 85721, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The effect of the nose radius of a body on boundary-layer receptivity is analysed for the case of a symmetric mean flow past a body with a parabolic leading edge. Asymptotic methods based on large Reynolds number are used, supplemented by numerical results. The Mach number is assumed small, and acoustic free-stream disturbances are considered. The case of free-stream acoustic waves, propagating obliquely to the symmetric mean flow is considered. The body nose radius, rn, enters the theory through a Strouhal number, S = ωrn/U, where ω is the frequency of the acoustic wave and U is the mean flow speed. The finite nose radius dramatically reduces the receptivity level compared to that for a flat plate, the amplitude of the instability waves in the boundary layer being decreased by an order of magnitude when S = 0.3. Oblique acoustic waves produce much higher receptivity levels than acoustic waves propagating parallel to the body chord.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 310 , 10 March 1996 , pp. 243 - 267
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

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover
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.Google Scholar
Brown, S. N. & Stewartson, K. 1973 On the propagation of disturbances in a laminar boundary layer. Proc. Camb. Phil. Soc. 73, 493–514.Google Scholar
Garrick, I. E. 1957 Nonsteady wing characteristics. In Aerodynamic Components of Aircraft at High Speeds (ed. A. F. Donovan & H. R. Lawrence), pp. 658–794. Princeton University Press.
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. & Hultgren, L. S. 1989 Boundary-layer receptivity to long-wave free-stream disturbances. Ann. Rev. Fluid Mech. 21, 137–166.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
Hammerton, P. W., & Kerschen, E. J. 1996a Boundary-layer receptivity for a parabolic leading edge: the small Strouhal number limit. To be submitted to J. Fluid Mech.Google Scholar
Hammerton, P. W., & Kerschen, E. J. 1996b Leading-edge receptivity for bodies at angle-of-attack. To be submitted to J. Fluid Mech.Google Scholar
Heinrich, R. A. E. 1989 Flat-plate leading-edge receptivity to various free-stream disturbance structures PhD Thesis, University of Arizona.
Heinrich, R. A. E. & Kerschen, E. J. 1989 Leading-edge boundary layer receptivity to various free-stream disturbance structures. Z. Angew. Math. Mech. 69, T596–598.Google Scholar
Keller, H. B. & Cebeci, T. 1970 Accurate numerical methods for boundary layer flows- I. Two-dimensional laminar flows. In Proc. Second Intl Conf. on Numerical Methods in Fluid Mechanics (ed. M. Holt), pp. 92–100. Springer.
Kerschen, E. J. 1990 Boundary layer receptivity theory. Appl. Mech. Rev. 43, 152–157.Google Scholar
Lam, S. H. & Rott, N. 1960 Theory of linearized time-dependent boundary layers. Cornell Univ. Grad. School of Aero. Engng Rep. AFOSR TN-60–1100.
Lam, S. H. & Rott, N. 1993 Eigen-functions of linearized unsteady boundary layer equations. Trans. ASME I: J. Fluids Engng 115, 597–602.Google Scholar
Libby, P. A. & Fox, H. 1963 Some perturbation solutions in laminar boundary layer theory. Part 1. The momentum equation. J. Fluid Mech. 17, 433–449.Google Scholar
Lighthill, M. J. 1954 The response of laminar skin friction and heat transfer to fluctuations in the free-stream velocity. Proc. R. Soc. Lond. A 224, 1–23.Google Scholar
Lin, N., Reed, H. L. & Saric, W. S. 1992 Effect of leading-edge geometry on boundary-layer receptivity to freestream sound. In Instability, Transition & Turbulence (ed. M. Y. Hussaini, A. Kumar & C. L. Streett). Springer.
Morkovin, M. V. 1969 Critical evaluation of transition from laminar to turbulent shear layers with emphasis on hypersonically traveling bodies. Air Force Flight Dyn Lab., Wright-Patterson AFB, Ohio. AFFDL-TR-68–149.
Murdoch, J. W. 1981 Tollmien-Schlichting waves generated by unsteady flow over parabolic cylinders. AIAA 81-0199.
Noble, B. 1958 Methods Based on the Wiener-Hopf Technique. Pergamon
Saric, W. S., Wei, W. & Rasmussen, B. 1994 Effect of leading edge on sound receptivity. In Laminar-Turbulent Transition (ed. R. Kobayashi), pp. 413–420. Springer.
Sedov, L. I. 1965 Two-Dimensional Problems in Hydrodynamics and Aerodynamics. Interscience Publishers
Van Dyke, M. D. 1964a Higher approximations in boundary layer theory. Part 3. Parabola in uniform stream. J. Fluid Mech. 19, 145–159.Google Scholar
Van Dyke, M. D. 1964b Perturbation Methods in Fluid Mechanics. Academic.