Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-24T15:29:25.710Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear and nonlinear receptivity of the boundary layer in transonic flows

Published online by Cambridge University Press:  30 November 2015

A. I. Ruban ,
T. Bernots  and
M. A. Kravtsova
A. I. Ruban*
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2BZ, UK
T. Bernots
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2BZ, UK
M. A. Kravtsova
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2BZ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we analyse the process of the generation of Tollmien–Schlichting waves in a laminar boundary layer on an aircraft wing in the transonic flow regime. We assume that the boundary layer is exposed to a weak acoustic noise. As it penetrates the boundary layer, the Stokes layer forms on the wing surface. We further assume that the boundary layer encounters a local roughness on the wing surface in the form of a gap, step or hump. The interaction of the unsteady perturbations in the Stokes layer with steady perturbations produced by the wall roughness is shown to lead to the formation of the Tollmien–Schlichting wave behind the roughness. The ability of the flow in the boundary layer to convert ‘external perturbations’ into instability modes is termed the receptivity of the boundary layer. In this paper we first develop the linear receptivity theory. Assuming the Reynolds number to be large, we use the transonic version of the viscous–inviscid interaction theory that is known to describe the stability of the boundary layer on the lower branch of the neutral curve. The linear receptivity theory holds when the acoustic noise level is weak, and the roughness height is small. In this case we were able to deduce an analytic formula for the amplitude of the generated Tollmien–Schlichting wave. In the second part of the paper we lift the restriction on the roughness height, which allows us to study the flows with local separation regions. A new ‘direct’ numerical method has been developed for this purpose. We performed the calculations for different values of the Kármán–Guderley parameter, and found that the flow separation leads to a significant enhancement of the receptivity process.

JFM classification

Boundary Layers: Boundary layer receptivity Boundary Layers: Boundary layer separation Aerodynamics: High-speed flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 786 , 10 January 2016 , pp. 154 - 189
Check for updates
Copyright
© 2015 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: With Formulas, Graphs and Mathematical Tables. Dover.Google Scholar
Bernots, T.2014 Receptivity of the boundary layer in transonic flow past an aircraft wing. PhD thesis, Imperial College London.Google Scholar
Bogdanov, A. N., Diesperov, V. N., Zhuk, V. I. & Chernyshev, A. V. 2010 Triple-deck theory in transonic flows and boundary layer stability. Comput. Math. Phys. 50 (1), 2095–2108.Google Scholar
Bowles, R. I. & Smith, F. T. 1993 On boundary-layer transition in transonic flow. J. Engng Maths 27, 309–342.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
Kravtsova, M. A., Zametaev, V. B. & Ruban, A. I. 2005 An effective numerical method for solving viscous–inviscid interaction problems. Phil. Trans. R. Soc. Lond. A 363 (1830), 1157–1167.Google Scholar
Lin, C. C. 1946 On the stability of two-dimensional parallel flows. Part 3. Stability in a viscous fluid. Q. Appl. Maths 3, 277–301.Google Scholar
Ruban, A. I. 1984 On the generation of Tollmien–Schlichting waves by sound. Fluid Dyn. 19, 709–717.Google Scholar
Ruban, A. I., Bernots, T. & Pryce, D. 2013 Receptivity of the boundary layer to vibrations of the wing surface. J. Fluid Mech. 723, 480–528.Google Scholar
Ryzhov, O. S. 2012 Triple-deck instability of supersonic boundary layers. AIAA J. 50, 1733–1741.Google Scholar
Schubauer, G. B. & Skramstad, H. K.1948 Laminar-boundary-layer oscillations and transition on a flat plate. NACA Tech. Rep. TR 909.Google Scholar
Smith, F. T. 1979a Nonlinear stability of boundary layers for disturbances of various sizes. Proc. R. Soc. Lond. A 368, 573–589.Google Scholar
Smith, F. T. 1979b On the nonparallel flow stability of the Blasius boundary layer. Proc. R. Soc. Lond. A 366, 91–109.Google Scholar
Terent’ev, E. D. 1981 The linear problem of a vibrator in a subsonic boundary layer. Z. Angew. Math. Mech. 45, 791–795.Google Scholar
Timoshin, S. N. 1990 Asymptotic form of the lower branch of the neutral curve in transonic boundary layer. Uch. Zap. TsAGI 21 (6), 50–57.Google Scholar
Tullio, N. & Ruban, A. I. 2015 A numerical evaluation of the asymptotic theory of receptivity for subsonic compressible boundary layers. J. Fluid Mech. 771, 520–546.CrossRefGoogle Scholar
Tumin, A. 2006 Biorthogonal eigenfunction system in the triple-deck limit. Stud. Appl. Maths 117, 165–190.CrossRefGoogle Scholar