Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-04T19:37:51.820Z Has data issue: false hasContentIssue false

A numerical evaluation of the asymptotic theory of receptivity for subsonic compressible boundary layers

Published online by Cambridge University Press:  21 April 2015

Nicola De Tullio*
Affiliation:
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
Anatoly I. Ruban
Affiliation:
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
*
Email address for correspondence: [email protected]

Abstract

The capabilities of the triple-deck theory of receptivity for subsonic compressible boundary layers have been thoroughly investigated through comparisons with numerical simulations of the compressible Navier–Stokes equations. The analysis focused on the two Tollmien–Schlichting wave linear receptivity problems arising due to the interaction between a low-amplitude acoustic wave and a small isolated roughness element, and the low-amplitude time-periodic vibrations of a ribbon placed on the wall of a flat plate. A parametric study was carried out to look at the effects of roughness element and vibrating ribbon longitudinal dimensions, Reynolds number, Mach number and Tollmien–Schlichting wave frequency. The flat plate is considered isothermal, with a temperature equal to the laminar adiabatic-wall temperature. Numerical simulations of the full and the linearised compressible Navier–Stokes equations have been carried out using high-order finite differences to obtain, respectively, the steady basic flows and the unsteady disturbance fields for the different flow configurations analysed. The results show that the asymptotic theory and the Navier–Stokes simulations are in good agreement. The initial Tollmien–Schlichting wave amplitudes and, in particular, the trends indicated by the theory across the whole parameter space are in excellent agreement with the numerical results. An important finding of the present study is that the behaviour of the theoretical solutions obtained for $\mathit{Re}\rightarrow \infty$ holds at finite Reynolds numbers and the only conditions needed for the theoretical predictions to be accurate are that the receptivity process be linear and the free-stream Mach number be subsonic.

Type
Papers
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

Balakumar, P. & Malik, M. R. 1992 Discrete modes and continuous spectra in supersonic boundary layers. J. Fluid Mech. 239, 631656.CrossRefGoogle Scholar
Borodulin, V. I., Ivanov, A. V., Kachanov, Y. S. & Roschektaev, A. P. 2013 Receptivity coefficients at excitation of cross-flow waves by free-stream vortices in the presence of surface roughness. J. Fluid Mech. 716, 487527.Google Scholar
Carpenter, M. H., Nordstrom, J. & Gottlieb, D. 1999 A stable and conservative interface treatment of arbitrary spatial accuracy. J. Comput. Phys. 148, 341365.Google Scholar
Choudhari, M. & Street, C. L. 1992 A finite Reynolds-number approach for the prediction of boundary-layer receptivity in localized regions. Phys. Fluids A 4 (11), 24952514.Google Scholar
Crouch, J. D. 1992 Localized receptivity of boundary layers. Phys. Fluids A 4 (7), 14081414.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. Lond. A 335, 5185.Google Scholar
De Tullio, N.2013 Receptivity and transition to turbulence of supersonic boundary layers with surface roughness. PhD thesis, School of Engineering Sciences, University of Southampton.Google Scholar
De Tullio, N., Paredes, P., Sandham, N. D. & Theofilis, V. 2013 Laminar–turbulent transition induced by a discrete roughness element in a supersonic boundary layer. J. Fluid Mech. 735, 613646.CrossRefGoogle Scholar
De Tullio, N. & Sandham, N. D. 2010 Direct numerical simulation of breakdown to turbulence in a mach 6 boundary layer over a porous surface. Phys. Fluids 22, 094105.CrossRefGoogle Scholar
Dietz, A. J. 1999 Local boundary-layer receptivity to a convected free-stream disturbance. J. Fluid Mech. 378, 291317.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
Fucciarelli, D., Reed, H. & Lyttle, I. 2000 Direct numerical simulation of leading-edge receptivity to sound. AIAA J. 38 (7), 11591165.CrossRefGoogle Scholar
Goldstein, M. E. 1985 Scattering of acoustic waves into Tollmien–Schlichting waves by small streamwise variations in surface geometry. J. Fluid Mech. 154, 509529.Google Scholar
Goldstein, M. & Hultgren, L. S. 1987 A note on the generation of Tollmien–Schlichting waves by sudden surface-curvature change. J. Fluid Mech. 181, 519525.CrossRefGoogle Scholar
Jones, R. D., Sandberg, R. D. & Sandham, N. D. 2010 Stability and receptivity characteristics of a laminar separation bubble on an aerofoil. J. Fluid Mech. 648, 257296.Google Scholar
Kachanov, Y. S. 1994 Physical mechanisms of laminar-boundary-layer transition. Annu. Rev. Fluid Mech. 26, 411482.Google Scholar
Kachanov, K., Kozlov, V. V. & Levchenko, V. Y. 1979 Origin of Tollmien–Schlichting waves in boundary layer under the influence of external disturbances. Fluid Dyn. 13, 704711.CrossRefGoogle Scholar
Kerimbekov, R. M. & Ruban, A. I. 2005 Receptivity of boundary layers to distributed wall vibrations. Phil. Trans. R. Soc. Lond. A 363, 11451155.Google Scholar
Kozlov, V. V. & Ryzhov, O. S. 1990 Receptivity of boundary layers: asymptotic theory and experiment. Phil. Trans. R. Soc. Lond. A 429, 341373.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.CrossRefGoogle Scholar
Reshotko, E. 1976 Boundary layer stability and transition. Annu. Rev. Fluid Mech. 8, 311349.Google Scholar
Reynolds, O. 1883 An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Phil. Trans. R. Soc. Lond. 174, 935982.Google Scholar
Ruban, A. I. 1984 On the generation of Tollmien–Schlichting waves by sound. Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza 5, 4452; translation in Fluid Dyn. 19 (5), 709–717, 1984.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, 480528.CrossRefGoogle Scholar
Sandham, N. D., Li, Q. & Yee, H. C. 2002 Entropy splitting for high-order numerical simulation of compressible turbulence. J. Comput. Phys. 178, 307322.Google Scholar
Saric, W. S., Hoos, J. A. & Radeztsky, R. H. 1991 Boundary-layer receptivity of sound with roughness. In Boundary Layer Stability and Transition to Turbulence (ed. Reda, D. C., Reed, H. L. & Kobayashi, R.), FED, vol. 114, pp. 1722. ASME.Google Scholar
Saric, W. S., Reed, H. L. & Kerschen, E. J. 2002 Boundary-layer receptivity to freestream disturbances. Annu. Rev. Fluid Mech. 34, 291319.Google Scholar
Saric, W. S. & White, E. B.1998 Influence of high-amplitude noise on boundary-layer transition to turbulence. AIAA Paper 98-2645.Google Scholar
Schubauer, G. B. & Skramstad, 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
Tempelmann, D., Schrader, L. U., Hanifi, A., Brandt, L. & Henningson, D. S. 2012 Swept wing boundary-layer receptivity to localized surface roughness. J. Fluid Mech. 711, 516544.Google Scholar
Terent’ev, E. D. 1981 Linear problem of a vibrator in subsonic boundary layer. Prikl. Mat. Mekh. 45, 10491055; translation in Appl. Math. Mech. 45 (6), 791–795, 1981.Google Scholar
Thomson, K. W. 1987 Time dependent boundary conditions for hyperbolic systems. J. Comput. Phys. 68, 124.Google Scholar
Thomson, K. W. 1990 Time dependent boundary conditions for hyperbolic systems, II. J. Comput. Phys. 89, 439461.Google Scholar
Wanderley, J. B. & Corke, T. C. 2001 Boundary layer receptivity to free-stream sound on elliptic leading edges of flat plates. J. Fluid Mech. 429, 121.Google Scholar
White, F. M. 2005 Viscous Fluid Flow, 3rd edn. McGraw-Hill.Google Scholar
Wray, A. A.1990 Minimal storage time advancement schemes for spectral methods. Rep. M.S. 202 A-1. NASA Ames Research Centre.Google Scholar
Wu, X. 2001 Receptivity of boundary layers with distributed roughness to vortical and acoustic disturbances; a second-order theory and comparisons with experiments. J. Fluid Mech. 431, 91133.Google 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. & Fedorov, A. V. 1987 Boundary-layer receptivity to acoustic disturbances. J. Appl. Mech. Tech. Phys. 28 (1), 2834.Google Scholar