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Receptivity of supersonic boundary layers over smooth and wavy surfaces to impinging slow acoustic waves

Published online by Cambridge University Press:  14 June 2019

Carlos G. Hernández  and
Xuesong Wu Open the ORCID record for Xuesong Wu [Opens in a new window]
Carlos G. Hernández
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK
Xuesong Wu*
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK
*
†Email address for correspondence: x.wu@ic.ac.uk
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Abstract

In this paper, we investigate the receptivity of a supersonic boundary layer to impinging acoustic waves. Unlike previous studies of acoustic receptivity, where the sound waves have phase speeds comparable with or larger than the free-stream velocity $U_{\infty }$, the acoustic waves here have much slower ($O(R^{-1/8}U_{\infty })$) phase velocity, and their characteristic wavelength and frequency are of $O(R^{-3/8}L)$ and $O(R^{1/4}U_{\infty }/L)$ respectively, compatible with the triple-deck structure, where $L$ is the distance to the leading edge and $R$ the Reynolds number based on $L$ and $U_{\infty }$. A significant feature of a sound wave on the triple-deck scale is that an $O(\unicode[STIX]{x1D700}_{s})$ perturbation in the free stream generates much stronger ($O(\unicode[STIX]{x1D700}_{s}R^{1/8})$) velocity fluctuations in the boundary layer. Two receptivity mechanisms are considered. The first is new, involving the interaction of two such acoustic waves and operating in a boundary layer over a smooth wall. The second involves the interaction between an acoustic wave and the steady perturbation induced by a wavy wall. The sound–sound, or sound–roughness, interactions generate a forcing in resonance with a neutral Tollmien–Schlichting (T–S) wave. The latter is thus excited near the lower branch of the neutral curve, and subsequently undergoes exponential amplification. The excitation through sound–sound interaction may offer a possible explanation for the appearance of instability modes downstream of their neutral locations as was observed in a supersonic boundary layer over a smooth wall. The triple-deck formalism is adopted to describe impingement and reflection of the acoustic waves, and ensuing receptivity, allowing the coupling coefficient to be calculated. The two receptivity processes with the acoustic waves on the triple-deck scale are much more effective compared with those involving usual sound waves, with the coupling coefficient being greater by a factor of $O(R^{1/4})$ and $O(R^{1/8})$ in the sound–sound and sound–roughness interactions, respectively. A parametric study for both the reflection and coupling coefficients is conducted for representative T–S waves, to assess the influence of the streamwise and spanwise wavenumbers, and the phase speed (or frequency) of the acoustic wave.

JFM classification

Boundary Layers: Boundary layer receptivity Boundary Layers: Boundary layer stability Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 872 , 10 August 2019 , pp. 849 - 888
Copyright
© 2019 Cambridge University Press 

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References

Balakumar, P.2003 Transition in a supersonic boundary layer due to roughness and acoustic disturbances. AIAA Paper 2003-3589.Google Scholar
Balakumar, P.2005 Transition in a supersonic boundary layer due to acoustic disturbances. AIAA Paper 2005-0096.Google Scholar
Blackaby, N. D., Cowley, S. J. & Hall, P. 1993 On the instability of hypersonic flow past a flat plate. J. Fluid Mech. 247, 369–416.Google Scholar
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, 285–308.Google Scholar
Choudhari, M. 1993 Boundary-layer receptivity due to distributed surface imperfections of a deterministic or random nature. Theor. Comput. Fluid Dyn. 4 (3), 101–117.Google Scholar
Choudhari, M. & Streett, C. L. 1992 A finite Reynolds-number approach for the prediction of boundary-layer receptivity in localized regions. Phys. Fluids A 4 (11), 2495–2514.Google Scholar
Crouch, J. D. 1992 Localized receptivity of boundary layers. Phys. Fluids 4, 1408–1414.Google Scholar
Crouch, J. D. 1994 Distributed excitation of Tollmien–Schlichting waves by vortical free-stream disturbances. Phys. Fluids 6, 217–223.Google Scholar
De 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.Google Scholar
Dietz, A. J. 1999 Local boundary-layer receptivity to a convected free-stream disturbance. J. Fluid Mech. 378, 291–317.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, 341–371.Google Scholar
Fedorov, A. V.2003a Receptivity of hypersonic boundary layer to acoustic disturbances scattered by surface roughness. AIAA Paper 2003-3731.Google Scholar
Fedorov, A. V. 2003b Receptivity of a high-speed boundary layer to acoustic disturbances. J. Fluid Mech. 491, 101–129.Google Scholar
Fedorov, A. V. 2011 Transition and stability of high-speed boundary layers. Annu. Rev. Fluid Mech. 43 (1), 79–95.Google Scholar
Fedorov, A. V. 2013 Receptivity of a supersonic boundary layer to solid particulates. J. Fluid Mech. 737, 105–131.Google Scholar
Fedorov, A. V. & Khokhlov, A. P. 1991 Excitation of unstable modes in a supersonic boundary layer by acoustic waves. Fluid Dyn. 26 (4), 531–537.Google Scholar
Fedorov, A. V. & Khokhlov, A. P. 2001 Prehistory of instability in a hypersonic boundary layer. Theor. Comput. Fluid Dyn. 14 (6), 359–375.Google Scholar
Gao, J., Luo, J. & Wu, X. 2015 Receptivity of hypersonic boundary layer due to fast-slow acoustics interaction. Acta Mechanica Sin. 31 (6), 899–909.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
Goldstein, M. E. & Hultgren, L. S. 1989 Boundary-layer receptivity to long-wave free-stream disturbances. Annu. Rev. Fluid Mech. 21 (1), 137–166.Google Scholar
Goldstein, M. E., Leib, S. J. & Cowley, S. J. 1987 Generation of Tollmien–Schlichting waves on interactive marginally separated flows. J. Fluid Mech. 181, 485–517.Google Scholar
Goldstein, M. E. & Ricco, P. 2018 Non-localized boundary layer instabilities resulting from leading edge receptivity at moderate supersonic Mach numbers. J. Fluid Mech. 838, 435–477.Google Scholar
Hall, P. & Smith, F. T. 1984 On the effects of nonparallelism, three-dimensionality, and mode interaction in nonlinear boundary-layer stability. Stud. Appl. Maths 70 (2), 91–120.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, 704–711.Google Scholar
Kendall, J. M.1975 Wind tunnel experiments relating to supersonic and hypersonic boundary-layer transition. AIAA Paper 74-133.Google Scholar
Kendall, J. M.1987 Experimental study of laminar boundary layer receptivity to a traveling pressure field. AIAA Paper 87-1257.Google Scholar
King, R. A. & Breuer, K. S. 2002 Oblique transition in a laminar Blasius boundary layer. J. Fluid Mech. 453, 177–200.Google Scholar
Kovasznay, L. S. G. 1953 Turbulence in supersonic flow. J. Aero. Sci. 20, 657–682.Google Scholar
Laufer, J. 1961 Aerodynamic noise in supersonic wind tunnels. J. Aero. Sci. 28, 685–692.Google Scholar
Lin, C. C. 1945 On the stability of two-dimensional parallel flows. Part II. Stability in an inviscid fluid. Q. Appl. Maths 3 (4), 117–142.Google Scholar
Ma, Y. & Zhong, X. 2003 Receptivity of a supersonic boundary layer over a flat plate. Part 2. Receptivity to free-stream sound. J. Fluid Mech. 488, 79–121.Google Scholar
Ma, Y. & Zhong, X. 2005 Receptivity of a supersonic boundary layer over a flat plate. Part 3. Effects of different types of free-stream disturbances. J. Fluid Mech. 532, 63–109.Google Scholar
Mack, L. M. 1984 Boundary-layer linear stability theory. In AGARD Rep. 709, Paper 3. Special Course on Stability and Transition of Laminar Flow. Jet Propulsion Laboratory.Google Scholar
Maslov, A. A., Shiplyuk, A. N., Sidorenko, A. A. & Arnal, D. 2001 Leading-edge receptivity of a hypersonic boundary layer on a flat plate. J. Fluid Mech. 426, 73–94.Google Scholar
Mengaldo, G., Kravtsova, M., Ruban, A. I. & Sherwin, S. J. 2015 Triple-deck and direct numerical simulation analyses of high-speed subsonic flows past a roughness element. J. Fluid Mech. 774, 311–323.Google 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, pp. 68–149.Google Scholar
Morkovin, M. V., Reshotko, E. & Herbert, T. 1994 Transition in open flow systems – a reassessment. Bull. Am. Phys. Soc. 39, 1882.Google Scholar
Nayfeh, A. H. & Ashour, O. N. 1994 Acoustic receptivity of a boundary layer to Tollmien–Schlichting waves resulting from a finite-height hump at finite Reynolds numbers. Phys. Fluids 6 (11), 3705–3716.Google Scholar
Qin, F. & Wu, X. 2016 Response and receptivity of the hypersonic boundary layer past a wedge to free-stream acoustic, vortical and entropy disturbances. J. Fluid Mech. 797, 874–915.Google Scholar
Reshotko, E. 1976 Boundary layer stability and transition. Annu. Rev. Fluid Mech. 8, 311–349.Google Scholar
Ricco, P. & Wu, X. 2007 Response of a compressible laminar boundary layer to free-stream vortical disturbances. J. Fluid Mech. 587, 97–138.Google Scholar
Ruban, A. I. 1983 Nonlinear equation for the amplitude of Tollmien–Schlichting waves in the boundary layer. Izv. Akad. Nauk. SSSR Mekh. Zhidk. Gaza 6, 60–67.Google Scholar
Ruban, A. I. 1984 On Tollmien–Schlichting wave generation by sound. Izv. Akad. Nauk. SSSR Mekh. Zhidk. Gaza 5, 44. (English translation: Fluid Dyn. 19, 1985, 709–716).Google Scholar
Ruban, A. I., Bernots, T. & Kravtsova, M. A. 2016 Linear and nonlinear receptivity of the boundary layer in transonic flows. J. Fluid Mech. 786, 154–189.Google Scholar
Saric, W. S., Reed, H. L. & Kerschen, E. J. 2002 Boundary-layer receptivity to freestream disturbances. Annu. Rev. Fluid Mech. 34 (1), 291–319.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
Schneider, S. P. 2001 Effects of high-speed tunnel noise on laminar-turbulent transition. J. Spacecr. Rockets 38 (3), 323–333.Google Scholar
Semionov, N. V. & Kosinov, A. D. 2008 An experimental study of receptivity of supersonic boundary layer on a blunted plate. Intl J. Mech. 2 (3), 87–95.Google Scholar
Shapiro, P. J.1977 The influence of sound upon laminar boundary layer instability. Tech. Rep. MIT Acoustics and Vibration Laboratory.Google Scholar
Smith, F. T. 1979 On the non-parallel flow stability of the Blasius boundary layer. Proc. R. Soc. Lond. A 366 (1724), 91–109.Google Scholar
Smith, F. T. 1989 On the first-mode instability in subsonic, supersonic or hypersonic boundary layers. J. Fluid Mech. 198, 127–153.Google Scholar
Smith, F. T. & Brown, S. N. 1990 The inviscid instability of a Blasius boundary layer at large values of the mach number. J. Fluid Mech. 219, 499–518.Google Scholar
Smith, F. T., Sykes, R. I. & Brighton, P. W. M. 1977 A two-dimensional boundary layer encountering a three-dimensional hump. J. Fluid Mech. 83 (1), 163–176.Google Scholar
Stewartson, K. 1964 The Theory of Laminar Boundary Layers in Compressible Fluids. Oxford Mathematical Monographs.Google Scholar
Sykes, R. I. 1978 Stratification effects in boundary layer flow over hills. Proc. R. Soc. Lond. A 361 (1705), 225–243.Google Scholar
Terent’ev, E. D. 1981 The linear problem of a vibrator in a subsonic boundary layer. Prikl. Math. Mech. 45, 1049–1055.Google Scholar
Wu, X. 1999 Generation of Tollmien–Schlichting waves by convecting gusts interacting with sound. J. Fluid Mech. 397, 285–316.Google Scholar
Wu, X. 2001a On local boundary-layer receptivity to vortical disturbances in the free stream. J. Fluid Mech. 449, 373–393.Google Scholar
Wu, X. 2001b 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, 91–133.Google Scholar
Wu, X. & Hogg, L. W. 2006 Acoustic radiation of Tollmien–Schlichting waves as they undergo rapid distortion. J. Fluid Mech. 550, 307–347.Google Scholar
Wu, X. & Zhang, J. 2008 Instability of a stratified boundary layer and its coupling with internal gravity waves. Part 2. Coupling with internal gravity waves via topography. J. Fluid Mech. 595, 409–433.Google Scholar
Würz, W., Herr, S., Wörner, A., Rist, U., Wagner, S. & Kachanov, Y. S. 2003 Three-dimensional acoustic-roughness receptivity of a boundary layer on an airfoil: experiment and direct numerical simulations. J. Fluid Mech. 478, 135–163.Google Scholar
Zhigulev, V. N. & Fedorov, A. V. 1987 Boundary-layer receptivity to acoustic disturbances. J. Appl. Mech. Tech. Phys. 28 (1), 28–34.Google Scholar
Zhong, X. & Wang, X. 2012 Direct numerical simulation on the receptivity, instability, and transition of hypersonic boundary layers. Annu. Rev. Fluid Mech. 44 (1), 527–561.Google Scholar