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Receptivity of the boundary layer to vibrations of the wing surface

Published online by Cambridge University Press:  16 April 2013

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
D. Pryce
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2BZ, UK
*
Email address for correspondence: [email protected]

Abstract

In this paper we study the generation of Tollmien–Schlichting waves in the boundary layer due to elastic vibrations of the wing surface. The subsonic flow regime is considered with the Mach number outside the boundary layer $M= O(1)$. The flow is investigated based on the asymptotic analysis of the Navier–Stokes equations at large values of the Reynolds number, $\mathit{Re}= {\rho }_{\infty } {V}_{\infty } L/ {\mu }_{\infty } $. Here $L$ denotes the wing section chord; and ${V}_{\infty } $, ${\rho }_{\infty } $ and ${\mu }_{\infty } $ are the free stream velocity, air density and dynamic viscosity, respectively. We assume that in the spectrum of the wing vibrations there is a harmonic that comes in to resonance with the Tollmien–Schlichting wave on the lower branch of the stability curve; this happens when the frequency of the harmonic is a quantity of the order of $({V}_{\infty } / L){\mathit{Re}}^{1/ 4} $. The wavelength, $\ell $, of the elastic vibrations of the wing is assumed to be $\ell \sim L{\mathit{Re}}^{- 1/ 8} $, which has been found to represent a ‘distinguished limit’ in the theory. Still, the results of the analysis are applicable for $\ell \gg L{\mathit{Re}}^{- 1/ 8} $ and $\ell \ll L{\mathit{Re}}^{- 1/ 8} $; the former includes an important case when $\ell = O(L)$. We found that the vibrations of the wing surface produce pressure perturbations in the flow outside the boundary layer, which can be calculated with the help of the ‘piston theory’, which remains valid provided that the Mach number, $M$, is large as compared to ${\mathit{Re}}^{- 1/ 4} $. As the pressure perturbations penetrate into the boundary layer, a Stokes layer forms on the wing surface; its thickness is estimated as a quantity of the order of ${\mathit{Re}}^{- 5/ 8} $. When $\ell = O({\mathit{Re}}^{- 1/ 8} )$ or $\ell \gg {\mathit{Re}}^{- 1/ 8} $, the solution in the Stokes layer appears to be influenced significantly by the compressibility of the flow. The Stokes layer on its own is incapable of producing the Tollmien–Schlichting waves. The reason is that the characteristic wavelength of the perturbation field in the Stokes layer is much larger than that of the Tollmien–Schlichting wave. However, the situation changes when the Stokes layer encounters a wall roughness, which are plentiful in real aerodynamic flows. If the longitudinal extent of the roughness is a quantity of the order of ${\mathit{Re}}^{- 3/ 8} $, then efficient generation of the Tollmien–Schlichting waves becomes possible. In this paper we restrict our attention to the case when the Stokes layer interacts with an isolated roughness. The flow near the roughness is described by the triple-deck theory. The solution of the triple-deck problem can be found in an analytic form. Our main concern is with the flow behaviour downstream of the roughness and, in particular, with the amplitude of the generated Tollmien–Schlichting waves.

Type
Papers
Copyright
©2013 Cambridge University Press 

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References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions, 3rd edn. Dover.Google Scholar
Choudhari, M. 1994 Roughness-induced generation of crossflow vortices in three-dimensional boundary layers. Theor. Comput. Fluid Dyn. 6, 130.Google Scholar
Denier, J. P., Hall, P. & Seddougui, S. O. 1991 On the receptivity problem for Görtler vortices: vortex motion induced by wall roughness. Phil. Trans. R. Soc. A 335, 5185.Google Scholar
Duck, P. W., Ruban, A. I. & Zhikharev, C. N. 1996 The generation of Tollmien–Schlichting waves by free stream turbulence. J. Fluid Mech. 312, 341371.CrossRefGoogle Scholar
Goldstein, M. E. 1985 Scattering of acoustic waves into Tollmien–Schlichting waves by small streamwise variation in surface geometry. J. Fluid Mech. 154, 509529.Google Scholar
Hall, P. & Smith, F. T. 1984 On the effects of non-parallelism, three-dimensionality and mode interaction in nonlinear boundary-layer stability. Stud. Appl. Maths 70, 91120.Google Scholar
Kachanov, Yu. S., Kozlov, V. V. & Levchenko, V. Ya. 1982 The Appearance of Turbulence in the Boundary Layer. Nauka, Novosibirsk.Google Scholar
Kerimbekov, R. M. & Ruban, A. I. 2005 Receptivity of boundary layers to distributed wall vibrations. Phil. Trans. R. Soc. A 363, 11451155.CrossRefGoogle ScholarPubMed
Liepmann, H. W. & Roshko, A. 1967 Elements of Gasdynamics. John Wiley & Sons.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, 277301.Google Scholar
Messiter, A. F. 1970 Boundary-layer flow near the trailing edge of a flat plate. SIAM J. Appl. Maths 18 (1), 241257.CrossRefGoogle Scholar
Morkovin, M. V. 1969 Critical evaluation of transition from laminar to turbulent shear layers with emphasis on hypersonically travelling bodies. Tech. Rep. AFFDL-TR 68-149. US Air Force Flight Dynamics Laboratory, Wright Patterson Air Force Base, Ohio.Google Scholar
Neiland, V. Ya. 1969 Theory of laminar boundary layer separation in supersonic flow. Izv. Akad. Nauk SSSR Mech. Zhidk. Gaza (4), 5357.Google Scholar
Ng, L. L. & Crouch, J. D. 1999 Roughness-induced receptivity to crossflow vortices on a swept wing. Phys. Fluids 11 (2), 432438.Google Scholar
Ruban, A. I. 1983 Nonlinear equation for the amplitude of the Tollmien–Schlichting wave in the boundary layer. Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza (6), 6067.Google Scholar
Ruban, A. I. 1984 On Tollmien–Schlichting wave generation by sound. Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza (5), 4452.Google Scholar
Schneider, W. 1974 Upstream propagation of unsteady disturbances in supersonic boundary layers. J. Fluid Mech. 63, 465485.Google Scholar
Schrader, L.-U., Brandt, L. & Henningson, D. S. 2009 Receptivity mechanisms in three-dimensional boundary-layer flows. J. Fluid Mech. 618, 209241.CrossRefGoogle Scholar
Schubauer, G. B. & Skramsted, H. K. 1948 Laminar boundary-layer oscillations and transition on a flat plate. NACA Tech. Rep. 909.Google Scholar
Smith, F. T. 1979a Nonlinear stability of boundary layers for disturbances of various sizes. Proc. R. Soc. Lond. A 368, 573589.Google Scholar
Smith, F. T. 1979b On the nonparallel flow stability of the Blasius boundary layer. Proc. R. Soc. Lond. A 366, 91109.Google Scholar
Stewartson, K. 1969 On the flow near the trailing edge of a flat plate. Mathematika 16 (1), 106121.Google Scholar
Stewartson, K. & Williams, P. G. 1969 Self-induced separation. Proc. R. Soc. Lond. A 312, 181206.Google Scholar
Sychev, V. V., Ruban, A. I., Sychev, Vic. V. & Korolev, G. L. 1998 Asymptotic Theory of Separated Flows. Cambridge University Press.CrossRefGoogle Scholar
Terent’ev, E. D. 1981 Linear problem for a vibrator in subsonic boundary layer. Prikl. Mat. Mekh. 45, 10491055.Google Scholar
Wu, X. 2001 Receptivity of boundary layers with distributed roughness to vortical and acoustic disturbances; a second-order asymptotic theory and comparison with experiments. J. Fluid Mech. 431, 91133.CrossRefGoogle Scholar
Wu, X., Zhao, D. & Luo, J. 2011 Excitation of steady and unsteady Görtler vortices by free stream vortical disturbances. J. Fluid Mech. 682, 66100.Google Scholar
Zhigulev, V. N. & Tumin, A. M. 1987 The Appearance of Turbulence. Nauka, Novosibirsk.Google Scholar